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
13 The Sorites, Linguistic Preconceptions, and the Dual Picture of Vagueness 228 ´ MARIO GOMEZ-TORRENTE
21 Scope Confusions and Unsatisfiable Disjuncts: Two Problems for Supervaluationism
NICHOLAS J. J. SMITH
List of Contributors ˚ Jonas Akerman Jc Beall University of Connecticut Richard Dietz Cian Dorr
Hartry Field New York University Graeme Forbes University of Colorado, Boulder Manuel Garc´ıa-Carpintero
Mario G´omez-Torrente Institute of Philosophical Research, UNAM Patrick Greenough
Leon Horsten University of Bristol and University of Leuven Dominic Hyde University of Queensland Andrea Iacona
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
Agust´ın Rayo Massachusetts Institute of Technology and Arch´e, and University of St Andrews Mark Richard
List of Contributors Nathan Salmon University of California, Santa Barbara Stephen Schiffer
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
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
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).
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).
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
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
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
of reasoning in vague languages, and that we need to reinterpret probability in a way characteristic of vagueness. 3
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
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
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
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
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
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’
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,
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.
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.
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
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.
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’).
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
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
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).
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.
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
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 ’.
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.
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.
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.
1.1.1 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).
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.
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.
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 ).
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.
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.]
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
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.
1.2.1 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 .
‘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
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.
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
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,
Good approx. / High degree
Bad approx. / Low degree C1 / M1
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.
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
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.
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.
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
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.
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.
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
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.
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).)
(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
∀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.
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.
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).
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.
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.
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
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
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.
[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).
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).
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.
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
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.
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
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.
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.
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.
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
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
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).
The Possibility of Partial Definition
59
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).
60
Scott Soames
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).
The Possibility of Partial Definition
61
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).
62
Scott Soames
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.
63
64
Matti Eklund
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).
Vagueness and Second-Level Indeterminacy
65
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).
66
Matti Eklund
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
Vagueness and Second-Level Indeterminacy
67
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
68
Matti Eklund
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.
Vagueness and Second-Level Indeterminacy
69
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.
70
Matti Eklund
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.
Vagueness and Second-Level Indeterminacy
71
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).
72
Matti Eklund
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).
Vagueness and Second-Level Indeterminacy
73
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.
74
Matti Eklund
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.
Vagueness and Second-Level Indeterminacy
75
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.
76
Matti Eklund
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.
77
78
Brian Weatherson
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
Vagueness as Indeterminacy
79
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
80
Brian Weatherson
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.
Vagueness as Indeterminacy
81
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
82
Brian Weatherson
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
Vagueness as Indeterminacy
83
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.
84
Brian Weatherson
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.
Vagueness as Indeterminacy
85
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
86
Brian Weatherson
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.
Vagueness as Indeterminacy
87
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
88
Brian Weatherson
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.
Vagueness as Indeterminacy
89
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.
90
Brian Weatherson
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).
91
92
Dorothy Edgington
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.
Sorensen on Vagueness and Contradiction
93
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.
94
Dorothy Edgington
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).
Sorensen on Vagueness and Contradiction
95
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.)
96
Dorothy Edgington
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.
Sorensen on Vagueness and Contradiction
97
( 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: . . . [N]o two possibilities can differ about what’s true unless they also differ in what things there are, or in how they are. In saying just this much, we do not join Armstrong in demanding truthmakers for negative existentials [and other problem cases]. Yet I think we do justice to the insight behind the demand: truths are about things, they don’t float in the void. (Lewis 1992, in 1999, 206)⁹
Lewis’s formulation is significantly weaker than the demand typically made by truthmaker theorists: it does not require an entity as a truthmaker for every truth; truth supervenes not on just what things there are, but on how they are. The no–no cases, ⁷ As a reader pointed out to me, matters would be more complex with an accompanying bottle of mouthwash bearing the legend ‘Ten per cent extra free’! But, I think Sorensen would say, this is not different in principle from his example. ⁸ It is debatable whether Sorensen’s is the right line to take on the no–no paradoxes. Some will say that they are too similar to the liar paradox to be accommodated so readily. I leave that aside. Their interest here lies in the light they are supposed to shed on the sorites. ⁹ Lewis goes on to say that, on pain of trivialization, his formulation should be interpreted thus: truth is supervenient on what things there are and which perfectly natural properties and relations they instantiate.
98
Dorothy Edgington
Sorensen claims, violate even this. These truths do just ‘float in the void’. That is, there are possible worlds which agree in all other respects but disagree about the distribution of truth values to the sentences on the list. Similarly, according to Sorensen, for the sorites: keep the world constant in all respects other than the threshold for ‘noonish’ and anything consequent thereon; there remains a range of possibilities open for the threshold. That is, there is a range of possible worlds which agree on everything other than the threshold, but disagree about where the threshold is. ‘[T]here are distinct, qualitatively identical worlds. In particular, there are worlds that have the same truthmakers but vary as to which propositions are true in those worlds’, he says (2005b, 727). If this is right, there is no necessary truth of the form ‘n is the threshold for noonish’; therefore, no statement of the form ‘n seconds after noon is noonish and n + 1 seconds after noon is not’ is necessarily true. In my 2005 paper, I assumed that analytic truths were supposed to be necessary truths. If so, no statement of this form is analytic. This conflicts with the message of the earlier part of the book, discussed in §2. Sorensen responds by denying my assumption (2005b, 716–20). ‘Unmade truths’, as he now calls them, are analytic yet contingent: Any proposition that does not owe its truth value to the world is analytic. So unmade propositions (ones that lack truth-makers or false-makers) are analytic. Yet they are also contingent. There are possible worlds that have the same truth-makers yet differ in that the unmade proposition is true in one and false in the others. Therefore, all unmade propositions are contingently analytic. (2005b, 716)
He gives examples from other parts of philosophy of what might count as contingentbut-analytic: Kaplan’s (1989) ‘You are the addressee of my utterance’ (alternatively ‘I am here’); Kripke-style examples such as ‘Neptune is responsible for the perturbations of Uranus’. If we accept Kripke’s contention that there are contingent truths which are knowable a priori, we might accept that (some of ) these deserve to be called contingent analytic.¹⁰ But these are recognizable as true a priori, while the borderline cases are not recognizable as true at all. The quoted passage above gives a very negative characterization of analyticity: ‘does not owe its truth to the world’. It is also in danger of proving too much. For surely, statements like ‘Jack is bald’ or ‘It was noonish when she arrrived’ (by contrast with ‘A person with n hairs is bald’ and ‘n seconds after noon is noonish’) are not analytic by any stretch of the imagination, yet they may be borderline, and hence lack a truthmaker. Anyway, what is more interesting, to my mind, is that the threshold statements are contingent: true in some possible worlds, false in others. And I can see no reason why they should not be more variable still. Why should there not be possible worlds in which borderline cases switch truth values at regular intervals, so that the threshold changes at regular intervals? Why should not the English word ‘red’ and the French ¹⁰ This does not accord with Kripke’s own usage (1980, 39).
Sorensen on Vagueness and Contradiction
99
word ‘rouge’ have different absolutely unknowable thresholds at the actual world, the difference never showing up in practice?
5.4 Given Sorensen’s picture, the following terminology is immensely natural: call something determinately true if it has a truthmaker, and determinately false if its negation has a truthmaker (or it has a falsemaker); or in Lewis’s terminology, it is determinately true/false if and only if it is true/false at all possible worlds which do not differ in what there is and how things are. If it is on an epistemic island, it is neither determinately true nor determinately false. It is determinately true that one toothbrush was bought and one was not, but neither determinately true nor determinately false that this toothbrush was bought. It is determinately true that the two no–no sentences have opposite truth values, but it is not determinate which is true. It is determinately true that there is an n such that n seconds after noon is noonish and n + 1 seconds after noon isn’t; but it is not determinate what that n is, i.e. there is no n such that it is determinate that n is the last noonish second. Now I am struck by the close correspondence between Sorensen’s position and some varieties of supervaluationism. Some supervaluationists claim that a statement involving vague terms is true if and only if it is true on all legitimate sharpenings of the vague terms. On this view, a borderline case (call it B) is neither true nor false; but the law of excluded middle still holds: B ∨ ¬B is true for all sharpenings of the vague term. Thus they must reject the equivalence principles for truth and falsehood: ‘B’ is true iff B; ‘B’ is false iff ¬B. For given these equivalences, one can derive the principle of bivalence ‘ ‘‘B’’ is true or ‘‘B’’ is false’, from the law of excluded middle, ‘B ∨ ¬B’. This kind of supervaluationism is mentioned occasionally by Sorensen. It is not the view I have in mind. Others (e.g. McGee and McLaughlin 1995; Field 1994) keep the equivalence principles for truth and falsehood. They do not say that a borderline case, B, is neither true nor false. The law of excluded middle and the principle of bivalence go hand in hand. They say that a statement is determinately true/false if and only if it is true/false on all sharpenings.¹¹ B is either true or false, but it is not determinate which: it is not determinately true, and not determinately false. Our concepts, and the way the world is, do not determine it either way. Put differently, these authors distinguish a thin notion of truth defined by the equivalence principle, from a thick notion: what is determined by our concepts and the way the world is. And Sorensen does the same. This strikes me as the better form of supervaluationism, for it keeps the equivalence principles, and, relatedly, does justice to the idea that a borderline case is not definitely not true, and not definitely not false. Just as a disjunction can be determinately true while it is not determinate which disjunct is true, so an existential quantification can be determinately true while it ¹¹ McGee and McLaughlin used ‘definitely’, but added a note: ‘perhaps ‘‘determinately’’ would be a better word, since ‘‘definitely’’ has so many uses, but we realized this too late to make the change’ (1994, 210).
100
Dorothy Edgington
is not determinate which of its instances are true. So it is determinate—it follows straightforwardly from premises we are committed to—that there is an n such that n is the last noonish second. On the innocuous, thin reading of ‘true’, it is determinate that there is an n such that it is true that n is the last noonish second. But it’s not the case that there is an n such that it is determinate that it is the last noonish second. Or, on the thick reading of ‘true’, as determined by our concepts and the way the world is, it’s not the case that there is an n such that it is true that it is the last noonish second. McGee and McLaughlin see this disambiguation as defusing the sorites paradox, but not in epistemicist terms. There is no determinate fact of the matter about which the last noonish second is. This theory also differs from older versions of supervaluation by appealing not to precise languages—precisifications of, say, English—but instead to precise interpretations of, or models for English, such that the definite truths are those which are true in all of a set of suitable models: Each possible world is associated with a set of models, each of which conforms to the facts that obtain in that world. Whereas, in a fully precise language, there is one model that describes each world, in a vague language there will be many models that conform to the facts of the world in the sense that every sentence that is definitely true in the world is true in the model. (1995, 228)
This picture is somewhat similar to Sorensen’s. There is this difference: for Sorensen each model is associated with a possible world; and one of these worlds is actual, we know not which. But one wonders how much work is done by these additional elements—or whether the explanatory work is done with the elements his view shares with McGee and McLaughlin’s approach. In my earlier piece on Sorensen (2005), I doubted whether Sorensen’s view, resting on truth-maker gaps, really deserved to be called epistemicist. He justifiably replied that an epistemicist is entitled to give a metaphysical explanation of our ignorance (2005b, 726–8). I accept that. Nevertheless there is something unusual about the dialectic situation. A motive for epistemicism is dissatisfaction with revisionary, contentious metaphysical or semantic views. Sorensen accepts the revisionary, ‘gappy’ metaphysical/semantic view as a ground for his epistemicism, and one wonders whether the metaphysical/semantic view is not enough on its own, without the extra epistemicist component. In Section 5.6 I shall criticize the metaphysical/semantic view which I find in common between McGee and McLaughlin and Sorensen.
5.5 I want now to discuss Sorensen’s chapter 2, ‘Intellectual embarrassment without vagueness’. Doing so enables me to bring into focus what I take to be an important feature of sorites arguments, ignored by Sorensen. The scene is a long row of 100 trees, increasing in height, from very short to very tall. A subject, X, is on trial. First X is asked to say which of the trees are short. He
Sorensen on Vagueness and Contradiction
101
protests that ‘short’ is vague. ‘Very well’, says Madame Inquisitor, ‘let us substitute ‘‘less than 50 feet tall’’ for ‘‘short’’ ’. [I add: there is no tree around 50 feet tall with leaves wafting in the breeze so that it could be a borderline case of less than 50 feet tall, i.e. every tree is either clearly, or clearly not, less than 50 feet tall.] She produces a list of statements: 1. Tree 1 is less than 50 feet tall. . . 100. Tree 100 is less than 50 feet tall. She asks him to mark which are true and which are false. He has normal eyesight and no access to special equipment. X believes 1 and disbelieves 100. Neighbouring statements in the sequence do not significantly differ in credibility. So it would seem that neighbouring statements should be both believed or both disbelieved. These principles would lead to his believing all the statements, and to his disbelieving all the statements. This is absurd. So there must be some n such that X believes n and does not believe n + 1. There must be somewhere he stops believing, and somewhere (else) he starts disbelieving—though the positions are unlikely to be stable, and he is left feeling extremely uncomfortable about the switches in attitude. The message here seems to be that many of the features of a sorites are present even when we are not dealing with a vague term like ‘short’, although in this case we have a clear conception of how, with more equipment and effort, it could be settled which judgements are correct. Contrary to the title of the chapter, it seems to me that this is not intellectual embarrassment without vagueness. ‘Believe’ is vague, and we have been given a sorites argument which exploits the vagueness of belief. In a set-up like this, there is no determinate point at which one stops believing, or stops disbelieving: belief just gradually fades away, or dawns. (And not just in a set-up like this: the same can be said when evidence gradually builds up in favour of, or against, a proposition.) We get a better representation of X’s epistemic attitudes if we allow him a spectrum of degrees of confidence in the propositions on the list, varying from 100% or virtually 100% certain, to 0% certain, and changing only very gradually. This does not eliminate vagueness or arbitrariness, but we have a more accurate, less jarring representation, as insignificant differences are represented by insignificant differences: the closer a tree is to clearly under 50 feet, the higher his degree of confidence that a tree is under 50 feet. We could represent X’s epistemic attitudes pictorially, with the statements 1–100 on the horizontal axis and X’s degree of confidence, from 100% to 0%, on the vertical axis (see Figure 5.1). We could ask him at what odds he would bet that a given tree is under 50 feet, and he could answer in a way that does not induce intellectual embarrassment. There will be some arbitrariness, but the feeling that X is being irrational in suddenly making the big jump from belief to neutrality, or neutrality to disbelief, has disappeared: small differences in the apparent height of the trees induce small differences in his degree
102
Dorothy Edgington 100%
0%
0
20
40
60
80
100
Figure 5.1
of confidence. Of course, the all-or-nothing conception of belief wins on simplicity, serves us well in many situations, and deserves to survive for those reasons; but where it is inadequate, because it involves big leaps while only tiny steps are being taken, we do better to think instead in terms of differing degrees of closeness to certainty. The moral of this particular sorites involving belief applies to any sorites, in my view. To state the obvious, the constitution of any sorites argument is that you start with a clear case of F , end with a clear case of ¬F , and get there via a chain of tiny differences. And so we get a better representation of the phenomena by setting aside the vague true/false classification, and replacing it by one which, though still vague, classifies cases as of varying degrees of distance from clear cases. Each ‘tolerance conditional’ comes out as very close to clearly true. The argument is valid. But all the small departures from clear truth mount up, and enable us to get from a clearly true premise to a clearly false conclusion. If the many tolerance conditionals are conjoined into the statement (∀n)(Fn → Fn ), where n is the successor of n, there are enough small departures for that generalization to be clearly false, its negation (∃n)(Fn & ¬Fn ) clearly true. Thus I agree with Sorensen, and with McGee and McLaughlin, that the induction step is false. But I reach this conclusion by a different route. In my view the induction step is false because of the small departure from clear truth of a large number of its instances. When speaking of the long no–no paradox, epistemic islands and truthmaker gaps, Sorensen said ‘Thus the probability that any given member is true is 0.99. Normally, probabilities are dynamic, changing in the light of new evidence. The probabilities of propositions belonging to an epistemic island are static. They are impervious to news’ (175). He could have distinguished two kinds of more ordinary probabilities, epistemic and objective: a person’s degree of closeness to certainty in a proposition; and the
Sorensen on Vagueness and Contradiction
103
objective chance of a given outcome. Both are dynamic. The former can change as new information is acquired; the latter can change with the occurrence of intermediate events, until finally settling down to 1 or 0. To avoid confusion it has become customary to give these two kinds of probability different names—in Lewis’s terminology, credence and chance. The ‘probabilities’ associated with an epistemic island are indeed static, and so are neither credence nor chance. They arise because of indeterminacy in the way our concepts relate to the world. They represent the degree of closeness to a clear case. Elsewhere I have called them ‘verities’ (see Edgington 1996). In my view they give us an improved way of understanding vagueness and of assessing reasoning in vague languages. 5.6 My preference for theorizing about reasoning in vague languages in terms of degree of closeness to clear truth is not necessarily incompatible with either epistemicism or a supervaluationist theory like that of McGee and McLaughlin. For an epistemicist my ‘verities’ may be assimilated to degrees of closeness to certainty of a person with no relevant ignorance other than the location of the boundary. For the supervaluationist they may be thought of as the proportion (or weighted proportion) of permissible models on which a sentence comes out true. I did distance myself from both these interpretations. Against an epistemic reading, I argued that ordinary uncertainty plays a very different role in our mental lives from the judgement that something is a borderline case, or is a certain distance from a clear case (1996, 312–4). The epistemicist may reply that this is not ordinary uncertainty, and one would expect it to play a different role. About supervaluationism, I said that thinking in terms of proportions of sharpenings may be a useful heuristic device, but I did not take it to be part of a deep explanation of the nature of vagueness (ibid., 315–16). Neither point is conclusive. I now want to distance myself a little more from some aspects of the picture I find in common between Sorensen, and McGee and McLaughlin—the truthmaker gaps, the lack of determination by our use of words and how things are—while still accepting the structure of this solution to the sorites. I have been influenced by David Barnett’s (2009) ‘Is vagueness sui generis?’ A question worth asking is: why think that the phenomenon of vagueness can be given a reductive account, in terms of anything else—be it a particular kind of ignorance, or a particular kind of lack of determination by our words and the world (or both)? We have learned the hard way, in philosophy, that reductive analyses of interesting concepts such as knowledge, or causation, or possibility, are not likely to succeed: why should we be able to find an informative way of saying, in other words, what it is to know something, or what it is for x to cause y? This is not to say, of course, that there cannot be interesting theories linking these notions to other concepts. Although there are notorious puzzles about reasoning in vague languages, the phenomenon of vagueness is very easy to grasp: what could be simpler than the fact that red just shades off into orange, rich into not-rich, noonish into later-than-noonish, and so on? It is a phenomenon we meet every day of our lives.
104
Dorothy Edgington
This was the spirit in which I wrote: The word ‘definitely’ occurs a lot in this section. It is meant as a theoretically innocent operator, understandable by anyone who concedes (and who can deny?) that vagueness exists, whatever the right account of it. Vagueness is about borderline cases. a is a borderline case of F iff a is not definitely F and a is not definitely not F . A proposition is a borderline case of truth, iff it is not definitely true, and not definitely false. Hence ‘definitely’ denies that what it operates on is a borderline case. . . . ‘[C]learly’, ‘definitely’ and ‘determinately’ I use interchangeably. (1996, 309)
I had not paid sufficient attention to the fact that McGee and McLaughlin do not use these operators in such a non-committal way: they put forward the substantive thesis that the indefinite (indeterminate) is what is not settled by our use of words and how things are—as equivalent to Sorensen’s lack of a truthmaker or a falsemaker. For instance, they say ‘The total history of the world, past present and future, does not determine an answer to the question ‘‘Who was the wealthiest poor person in the world on June 15, 1994?’’ ’ This, I shall argue, may be a mistake. I follow Barnett in stipulating that I shall use ‘clearly’ as the neutral, non-reductive term: a is clearly F iff (a) a is F and (b) it is not vague whether a is F . Clearly p iff (a) p and (b) it is not vague whether p. The following three paragraphs owe much to his paper. One can see McGee and McLaughlin’s kind of indeterminacy, and Sorensen’s truthmaker gaps, as the last in a progression of ‘gappy’ responses to problems about vagueness. Let x be a borderline case of ‘red’. First response: deny the Law of Excluded Middle (LEM): it is neither red nor not red. Unfortunately, this is a contradiction. Suppose one can get round this by inventing different senses of ‘not’. It is still a bad idea, for the vagueness of ‘red’ has been lost. It is no longer vague whether x is red: it isn’t; and it isn’t not red either. More tellingly, suppose that LEM is clearly false for borderline cases, and ‘x is red’ is clearly borderline. Then x is clearly not red.¹² This is plain wrong, of a borderline case. Further, this story is at odds with our ambivalence about x’s colour, our inclination to see it as ‘sort of ’ red and ‘sort of ’ not red. Second response: keep LEM but deny Bivalence, as the classical supervaluationists do. x is red or x is not red, but it is neither true nor false that x is red. As we have seen, this has the disadvantage of denying the equivalence principles, e.g. it is true that x is red iff x is red. It follows that either x is red but it’s not true that x is red, or x is not red but it’s not false that x is red. Again, suppose Bivalence is clearly false for borderline cases, and x is a clear borderline case. Then it is clearly not true that x is red, and clearly not false that x is red. This seems wrong. ‘x is red’ should be a borderline case of truth, not a clear case of non-truth. Again, this is in tension with our ambivalence about the question whether ‘x is red’ is true, our temptation to see it as ‘sort of ’ true. Third response (McGee and McLaughlin; Sorensen): keep LEM and Bivalence, but reject the stronger principle: a vague sentence is not determinately true and not ¹² This rests on the unconentious principle: if A entails B, then ‘Clearly A’ entails ‘Clearly B’.
Sorensen on Vagueness and Contradiction
105
determinately false, where this is explained as our words and the world not fixing whether it is true, or whether it is false¹³—in Sorensen’s terms, the absence of a truthmaker or falsemaker. This is also puzzling. How can it be true that x is red, if our words and the world don’t make it true, or false that x is red when our words and the world don’t make it false? Would it not be preferable to keep LEM, Bivalence and the stronger principle in line, and hence the equivalences: x is red iff it is true that x is red, iff it is fixed by our words and how things are that x is red? On this view, if it is vague whether x is red, it is vague whether it is true that x is red, i.e. it is vague whether the world makes x red, or makes x not red. It is not that we have the absence of a truthmaker. It is simply unclear (vague) whether we have a truthmaker or a falsemaker—whether the world makes it true, or makes it false. The world does not, as it were, do nothing about the status of ‘x is red’: the world, as it were, renders it ‘true-ish’ and ‘false-ish’. ( This is a difference between vagueness and, e.g. the no–no paradox.) Friends of gappiness sometimes evoke artificial partially-defined predicates like Kit Fine’s (1975) ‘nice∗ ’, a predicate of numbers, such that (a) if n > 15, n is nice∗ ; and (b) if n < 15, n is not nice∗ . But there is a difference between this and a case of vagueness. We have no intuitions at all about whether 15 is nice∗ : we have no inclination to think it is, no inclination to think it isn’t. ( Thus the case of nice∗ is more like the no–no paradox than the case of vagueness.) With vague predicates, we are ambivalent, we feel the pull in both directions.¹⁴ We can keep the advantages of the shape of McGee and McLaughlin’s account without the costs. Remember that ‘clearly’ is our neutral, non-reductive word. A disjunction can be clearly true while neither disjunct is. Take something which is clearly green, but it is not clear whether it is dark green, or green-but-not-dark-green. One more example, from Barnett, enables us to make a further point: As the unfortunate result of an accident, Ning is now on an operating table having one of her legs reattached to her body. At the present stage of the procedure, it is vague whether the leg qualifies as attached to the body. The surgeon asks an attending student ‘What is the patient’s current weight?’ Well prepared, the student knows that Ning’s body weighs precisely 100 pounds without the leg, and the leg weighs precisely 20 pounds. She offers the natural response ‘Clearly, the patient weighs something. The only two candidates are 100 and 120 pounds. Hence, clearly she weighs 100 or 120 pounds, even though it is vague which’. (Barnett, 2009)
Now, Ning clearly weighs something. She clearly does not weigh nothing. Yet there is no weight such that she clearly weighs it. So this is a nice example of an existential statement being clearly true, though none of its instances are clearly true.¹⁵ Similarly, it is vague where the cut-off for (e.g.) being rich is (not: there is no cut-off; if there were no cut-off, it would not be vague where the cut off is). ¹³ Maybe it is better to say ‘our concepts and the way the world is’. I use ‘words and the world’ as short for a more adequate phrasing. ¹⁴ Philosophers who have highlighted ambivalence as a feature of our attitude to vagueness are Wright (2001) and Stephen Schiffer (2003, ch. 5). ¹⁵ It is harder to come up with convincing relatively natural examples of this phenomenon in the case of existentially quantified statements than it is for disjunctions.
106
Dorothy Edgington
So the sorites is defused, as before. We reject the tolerance principle and the induction step, but accept something in their vicinity: for every vague predicate F , small enough differences in F make no clear difference to whether F applies. There is no n such that n seconds after noon is clearly noonish and n + 1 seconds after noon is clearly not noonish. There remains, of course, much to be said. To take vagueness as primitive is not to be reduced to silence on metaphysical, epistemological, semantic and logical issues concerning vagueness. I persist in thinking that one gets a more satisfying account of reasoning in vague languages by using the notion of degree of closeness to clear cases. But I will not pursue that matter here. Re f e re n c e s Barnett, David (2009), ‘Is vagueness sui generis?’ The Australasian Journal of Philosophy 87, 5–34. Edgington, Dorothy (1996), ‘Vagueness by degrees’ in R. Keefe and P. Smith, eds., Vagueness: A Reader, Cambridge MA, MIT Press, 294–316. (2005), ‘The mystery of the missing boundary’, Philosophy and Phenomenological Research 71, 3, 704–11. Field, Hartry (1994), ‘Disquotational truth and factually defective discourse’, Philosophical Review, 103, 405–52. Reprinted in his Truth and the Absence of Fact, Oxford, Oxford University Press, 2001, 222–58. Fine, Kit (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. Horgan, Terry (1997), ‘Deep ignorance, brute supervenience and the problem of the many’, Philosophical Issues, 8, 229–36. Kaplan, David (1989), ‘Demonstratives’ in Themes from Kaplan, ed. Joseph Almog, John Perry, and Howard Wettstein. Oxford, Oxford University Press. Keefe, Rosanna (2000), Theories of Vagueness, Cambridge, Cambridge University Press. Kripke, Saul (1980), Naming and Necessity, Oxford, Basil Blackwell. Lewis, David (1992), ‘Armstrong on combinatorial possibility’, Australasian Journal of Philosophy, 70, 211–24. Reprinted in his Papers in Epistemology and Metaphysics, Cambridge, Cambridge University Press, 1999, 196–214. Page reference to this volume. McGee, Vann and McLaughlin, Brian (1995), ‘Distinctions without a difference’, The Southern Journal of Philosophy 33, 203–51. Schiffer, Stephen (2003), The Things We Mean, Oxford, Oxford University Press. Sorensen, Roy (2001), Vagueness and Contradiction, Oxford, Oxford University Press. (2005a), ‘Pr´ecis of Vagueness and Contradiction’, Philosophy and Phenomenological Research, 71, 678–85. (2005b), ‘A reply to critics’, Philosophy and Phenomenological Research 71, 712–28. Williamson, Timothy (1994), Vagueness, London, Routledge. Wright, Crispin (1975), ‘On the coherence of vague predicates’, Synthese 30, 325–65. (2001), ‘On being in a quandary: Relativism, vagueness, logical revisionism’, Mind, 110, 45–98.
II Vagueness in Reality
This page intentionally left blank
6 Vague Properties Stephen Schiffer
6.1
VAG U E P RO PE RT I E S A N D T H E P RO B L E M O F VAG U E N E S S
The philosophical problem of vagueness is to say what vagueness is in a way that helps to resolve the sorites paradox. Saying what vagueness is requires saying what kinds of things can be vague and in what the vagueness of each kind consists. Philosophers dispute whether things of this, that, or the other kind can be vague, but no one disputes that there are vague linguistic expressions. Among vague expressions, predicates hold a special place in the problem of vagueness, for it’s their vagueness that is sorites-generating. That puts the vagueness of predicates at the hub of the problem of vagueness, and there can be little doubt that we’ll be a short step from home if we can account for it. Any account of vagueness will of course require commitment to theses that are themselves foci of philosophical debate, but one can’t expect to get anywhere without taking on some as working hypotheses and then striving to say something that will be plausible if those hypotheses are plausible. One of the working hypotheses of this chapter is that propositional attitudes and propositional speech acts are relations to propositions of some stripe or other, in the generic sense in which a proposition is an abstract, mind- and language-independent entity that has a truth condition, and has that truth condition both essentially and absolutely (i.e. without relativization to anything).¹ The existence of propositions requires the existence of properties, in the generic sense in which a property is an abstract, mind- and language-independent entity that has an instantiation condition, and has that instantiation condition both essentially and absolutely. For present purposes it will be harmless to pretend that the propositions we believe and assert are Russellian propositions—structured entities whose basic constituents are the objects and properties our beliefs and speech acts are about. When a propositionalist speaks in loosey-goosey mode, she is apt to say that a sentence token is true just in case the proposition expressed in its utterance is true. But that way of speaking is tendentious when vague sentences are at issue; for as regards ¹ See e.g. Schiffer (2003), ch. 1.
109
110
Stephen Schiffer
vagueness there is the following important division among those who recognize the need for propositions, and therefore properties, as semantic values: (a) Some philosophers hold that there are no vague propositions, and therefore no vague properties; for them, the vagueness of a sentence token consists in its indeterminately (or ‘partially,’ or ‘sort of ’) expressing each of myriad absolutely precise propositions, and the vagueness of a predicate token consists in its indeterminately (or ‘partially,’ or ‘sort of ’) expressing each of myriad absolutely precise properties. (b) Other philosophers hold that there are vague properties, and therefore vague propositions; for them the vagueness of a predicate token consists in its expressing a vague property, and likewise, mutatis mutandis, for the vagueness of a sentence token.² I have argued against (a) and for (b) elsewhere,³ but rather than revisit those arguments here, I’ll adopt (b) as a second working hypothesis of this chapter. So, regardless of what you may think about ‘Harry’, the proposition expressed by a token of ‘Harry is bald’ is bound to be vague just by the fact that tokens of ‘bald’ always express vague properties. The theory of vagueness turns on the nature of vague properties. This chapter is a renewed effort to make some headway towards explaining in what the vagueness of a property consists. I shall be content if I can give a plausible sketch of the form that such an explanation should take. A property is vague just in case it either admits of borderline cases or, like the property of being both bald and not bald, is in some to-be-specified way a function of properties one or more of which admit of borderline cases. This suggests that we shall essentially have accounted for vague properties if we can say what it is for something to be a borderline instance of a property. In the end, however, that may not be the best way to state the philosophical question vague properties pose. I allude in part to the way the problem of borderline cases quickly explodes into difficult questions about so-called higher-order vagueness. Funny properties like being both bald and not bald aside, what is distinctive of vague properties is that each vague property φ has a shadow vague property, the property of being a borderline instance of φ (‘β-φ’). Since the shadow properties are themselves vague, we are quickly challenged to know what to make of the infinite series φ β-φ β(β-φ) β(β(β-φ)) β(β(β(β-φ))) .. . I begin by taking seriously the question of what it is for a thing to be a borderline instance of a property. If asking that question leads us to appreciate its naivet´e, then we will have been moving in at least one right direction. ² By this criterion, the epistemic theorist counts as holding that there are vague properties, even though his vague properties have absolutely precise extensions and wouldn’t be recognized as vague by theorists of vagueness who don’t take vagueness to be a kind of ignorance. ³ Schiffer (2003), ch. 5.
Vague Properties 6.2
111
QUA N D A RY A N D Q - C O N S T R A I N T
Crispin Wright, Hartry Field, and I have independently argued that there is a psychological state that is characteristic of taking a thing to be a borderline case, independently of whether or not one believes—or even has the conceptual resources to believe (one may not have the concept of a borderline case)—that the thing is a borderline case.⁴ While each of us has in mind the same psychological state, we have offered different, albeit overlapping, accounts of how that state should be characterized. I have critically discussed these differences elsewhere,⁵ and won’t be going into them here—except briefly to indicate how the characterization I’m about to sketch corrects the one offered in The Things We Mean. Wright calls the characteristic psychological state quandary; but then he goes on to define ‘quandary’ in terms of features some of which the state in question doesn’t have, while the remaining features don’t capture the state’s specific connection to vagueness.⁶ Still, I like the label ‘quandary’, so I propose to use ‘Quandary’ as a proper name of what I take to be the psychological state that is characteristic of taking a thing to be a borderline case, but whose exact nature remains to be specified. One way to take a man to be borderline bald would be to believe that he is borderline bald. But even a person who lacks the conceptual resources to have that belief may take a man to be borderline bald. In such a case the psychological state that constitutes taking the man to be borderline bald is Quandary as regards whether or not the man is bald. Now Harry is standing before Jane, a rational nonphilosopher, in circumstances that are optimal for her to determine whether or not he is bald. Her being in Quandary as regards whether or not Harry is bald reveals that she takes him to be borderline bald. If she has the concept of a borderline case we should expect her also to believe that he is borderline bald, in which case we should also expect her belief to be explained by her Quandary; but let’s not try to say everything at once. Jane’s Quandary as regards whether or not Harry is bald is not, and is not thought by her to be, a state of ignorance or uncertainty. It is a state of ambivalence in which she is stymied to make any judgment at all as to whether or not Harry is bald, and this because of the way she finds his resemblance to men she would say are bald to be in competition with his resemblance to men she would say are not bald. It is no surprise that she would be stymied in those circumstances. In the normal case she would know, or at least be poised to know, that if her perceptual faculties and circumstances are as she confidently takes them to be, nothing can come to light that she would take to be evidence that Harry is, or that he isn’t, bald; but she would not take that to show how well hidden the fact of the matter about his baldness was.⁷ ⁴ See e.g. Wright (2003b) and (forthcoming); Field (2001b) and (forthcoming); and Schiffer (2003), ch. 5, (forthcoming-a), and (forthcoming-b). ⁵ Schiffer op. cit. ⁶ Wright (2003b). ⁷ Not only does Jane’s being in her particular state of Quandary qualify her as taking Harry to be borderline bald; it also shows where she implicitly locates him in the penumbra of the (contextually relevant) property of being bald. We may infer this location (which will of course itself be very vague) from how she implicitly weighs the competing resemblances. If she thinks Harry’s resemblance to men she would judge to be bald is counterbalanced by his resemblance to men she would judge not
112
Stephen Schiffer
Moreover, when Jane is in Quandary as regards whether or not Harry is bald, the way in which she is stymied to make any judgment about the truth status of the proposition that Harry is bald does not represent her as having classical partial beliefs about his being bald—partial beliefs about his being bald that are normatively governed (under suitable idealization) by the axioms of classical probability theory. For suppose Jane is stymied because she takes Harry’s resemblance to men she would judge to be bald to be counterbalanced by his resemblance to men she would judge not to be bald. That would not mean that she believes to degree 0.5 both that Harry is bald and that he isn’t bald, as she would believe to degree 0.5 both that the fair coin about to be flipped will land heads and that it will land tails. When Jane believes to degree 0.5 both that the coin will land heads and that it will land tails, she might report her partial beliefs by saying she thinks there is a 50–50 chance that the coin will land heads. She will have no inclination at all to say she thinks there is a 50–50 chance that Harry is bald. Believing to degree 0.5 both that the coin will land heads and that it will land tails is the opposite of being stymied to make any judgment as to whether or not the coin will land heads, for in this case one is making an entirely unstymied judgment: one is judging to degree 0.5 both that the coin will land heads and that it will land tails. Partially believing and disbelieving p to degrees that sum to 1 requires believing that p is either true or else false, that it’s either a fact that p or a fact that not p. Jane doesn’t take herself to be in a situation in which it’s either a wellhidden fact that Harry is bald or else a well-hidden fact that he isn’t bald. Moreover, one who believes to degree 0.5 both that the coin will land heads and that it will land tails has no inclination to judge either that the coin will land heads or that it will land tails. But when Jane is in Quandary as regards whether or not Harry is bald she might well experience some inclination to judge that Harry is bald and some inclination to judge that he isn’t bald. The distance between Jane’s Quandary and classical partial belief is also brought home by supposing her circumstances also to be optimal for her to determine whether or not Harry is tall and whether or not he is thin, and that, by some great coincidence, she not only takes Harry to be borderline tall and borderline thin, but also takes him to occupy the midpoint of the penumbras of the three vague properties baldness, tallness, and thinness, properties she also takes to be probabilistically independent. Now, what would you expect Jane’s attitude to be as regards the proposition that Harry is bald, tall, and thin and the proposition that Harry isn’t bald, tall, and thin?⁸ If being in Quandary as regards whether or not p entailed partially believing and disbelieving p to positive degrees that sum to 1, then Jane should believe the conjunction to degree 0.125 and believe its negation to degree 0.875. Believing a proposition tout court is arguably just believing it to some contextually relevant high degree, and in to be bald, then that counts as her locating Harry in the middle of the penumbra; if she thinks his resemblance to men she would judge to be bald is greater than his resemblance to men she would judge not to be bald, then that counts as her locating him closer to the definitely bald men than to the definitely not bald men, and vice versa if the weights of the resemblances are reversed. ⁸ Cf. MacFarlane’s discussion (this volume) of Jim, who is borderline bald, borderline tall, and borderline smart.
Vague Properties
113
most contexts believing a proposition to degree 0.875 would count as believing it. But Jane wouldn’t say that she believes that Harry isn’t bald, tall, and thin, and I wager that she would be as stymied to judge whether or not he is bald, tall, and thin as she is to judge whether or not he is bald, whether or not he is tall, or whether or not he is thin. Quandary motivates a constraint on theories of vague properties. The constraint is that a theory of vague properties must be consistent with what I’ll call Q-Constraint: If one is in Quandary as regards whether or not x is F in circumstances that one knows are optimal for determining whether or not x is F, then there is nothing incorrect about one’s being in that state.
6.3
T RU T H - S TAT U S T H E O R I E S V S . Q - C O N S T R A I N T
If φ is vague, then it’s shadowed by another vague property, the property of being a borderline instance of φ. For any vague properties φ, ψ, how must those properties be related in order for ψ to be the property of being a borderline instance of φ? If ψ is the property of being a borderline instance of φ, then knowing that x is an instance of ψ should tell us something important about the proposition that x is an instance of φ. Many philosophers would say it tells us about the truth-status of the proposition that x is an instance of φ. A truth-status theory of vague properties, as I shall use that label, is a theory of vague properties that, for any thing x and property φ, assigns a truth-status to the proposition that x is an instance of φ if, according to the theory, x is a borderline instance of φ. Truth-status theories divide into two mutually exclusive and jointly exhaustive kinds: those that accept the principle of bivalence and thus hold that even borderline propositions must be true or false (familiar epistemic theories are the prime example); and those that deny bivalence and assign to borderline propositions a truth status other than truth or falsity (these are what Wright has called third-possibility theories, since such theories take a proposition’s being borderline to consist at least partly in its having ‘some kind of status other than truth and falsity—a lack of truth value, perhaps, or the possession of some other truth value.’)⁹ No truth-status theory is consistent with Q-Constraint. For suppose the correct theory of vague properties were a bivalent truth-status theory. Then when Harry comes across to Jane as borderline bald, the bivalent theorist must say that, even if Jane is subjectively justified in not believing that it’s either a fact that Harry is bald or else a fact that he isn’t bald, believing that there is such a fact of the matter is the objectively correct state for her to be in, and being in that state entails that she ought objectively to believe and disbelieve that Harry is bald to positive degrees that sum to 1. But Jane is in Quandary as regards whether or not Harry is bald, and her being in that state precludes her having any classical partial belief about whether or not he is bald. So the bivalent truth-status theorist must say that Quandary as regards whether ⁹ Wright (2003b), 475.
114
Stephen Schiffer
or not Harry is bald is the objectively incorrect state for Jane to be in, and that therefore Q-Constraint is false. But to what error theory will the bivalent truth-status theorist appeal in order to explain why Jane doesn’t believe that it’s either a fact that Harry is bald or else a fact that he isn’t bald? Epistemic theorists have been known to suggest that the lingering influence of verificationism explains why philosophers don’t accept the bivalent truth-status theory of vague properties. But that doesn’t explain why nonphilosophers don’t attribute a thing’s coming across to them as a borderline case as due to how well hidden the truth of the matter is. Verificationism requires indoctrination; ordinary people seem not to have any trouble believing that there are, or might be, facts they can’t even in principle know (at least while they live). Besides, in order for an error theory to be plausible it needs to tell a plausible story about how to get the erring person to see the light. What will the bivalent truth-status theorist say, that Jane will have to take his course on vagueness? I submit that no bivalent truth-status theorist has yet said anything to make it plausible that Q-Constraint isn’t plausible. No nonbivalent truth-status theory is consistent with Q-Constraint either. Jane’s Quandary as regards whether or not Harry is bald is a state of ambivalence in which she is stymied to make any pronouncement as to whether or not Harry is bald but in which she might well experience conflicting inclinations, one to judge that he is bald, the other to judge that he isn’t bald. Jane’s ambivalent state of mind strikes me as in every way a perfectly correct state for her to be in, given the way our use of vague terms engenders borderline cases by creating penumbral regions wherein our use of a term hovers between two impulses thanks to our being in a region of cases whose resemblance to cases where we would apply the term compete with its resemblance to cases where we would apply its negation. But if a nonbivalent truth-status theory were correct, Jane, once again, would not be in the epistemic state she objectively ought to be in. For if such a theory were correct, then it would be neither true that borderline Harry was bald nor true that he was not bald, and Jane would have no business being in a state of ambivalence (or uncertainty): objectively, she ought to be in the entirely unambivalent and unstymied state of being certain that it’s neither a fact that Harry is bald nor a fact that he isn’t bald. This problem for the nonbivalent truth-status theorist is well expressed by Crispin Wright: [I]t is quite unsatisfactory in general to represent indeterminacy as any kind of determinate truth-status—any kind of middle situation, contrasting with both the poles (truth and falsity)—since one cannot thereby do justice to the absolutely basic datum that in general borderline cases come across as hard cases: as cases where we are baffled to choose between conflicting verdicts about which polar verdict applies, rather than as cases which we recognize as enjoying a status inconsistent with both.¹⁰
So the nonbivalent truth-status theorist who wants to stay that way needs an error theory to explain why Jane is in her objectively incorrect Quandary state. Error theories in this area are difficult to motivate. What will the theorist say—that Jane, who is not given to philosophical reflection, somehow unwittingly came in some way ¹⁰ Wright (2003b), 475–6.
Vague Properties
115
subpersonally to accept a false theory of vagueness? If so, is there a remedy for her other than intensive tutoring in whatever nonbivalent truth-status theory of vague properties is correct? A theorist who has gotten herself to this point will now be looking for a nontruth-status theory of vague properties which satisfies Q-Constraint. As a non-truthstatus theorist, she is assuming something significant about the relation that must hold between properties φ and ψ if ψ is to be the property of being a borderline instance of φ —namely, that if the proposition that x has ψ is true, then there is nothing to determine any truth status for the proposition that x has φ, not even that it has no truth-status (since having no truth-status would itself be a truthstatus). It would make life easier for us if that were all there were to one property’s being the property of being a borderline instance of another property. But we know that can’t be all there is, since being borderline is only one way for a proposition to be indeterminate, and we should suppose (at least initially) that if no truth-status can be assigned to borderline propositions, then none can be assigned to any indeterminate proposition.¹¹ Given the way Quandary has proved to be a stumbling block for truth-status theories, it’s reasonable at this stage of inquiry to suppose that what we need to be looking for is not merely a non-truth-status theory which satisfies Q-Constraint, but also a theory that somehow explains a property’s being vague in terms of Quandary, the psychological state characteristic of one who takes a thing to be a borderline case. Perhaps this should be a theory whose guiding inchoate thought is that ‘borderline cases are cases where a competent judge may be put in quandary despite conditions obtaining which are optimal for judgement in clear cases.’¹² Or perhaps the striking connection between vagueness and Quandary derives not from vagueness being definable in terms of Quandary but merely from the tendency of borderline cases to induce Quandary.
6.4
P L ATO ’ S H E AV E N A N D T WO M O D E L S O F T H E N A M E – N A M E D R E L AT I O N
According to a prevalent misconception of Plato’s Heaven, one can mentally survey the properties residing there, fix one’s mental gaze on any one of them, and stipulate that a certain expression will signify it, just as one might stipulate a name for a dog or a person. Part of this conception is right. The properties our predicates express are in Plato’s Heaven, which in less eschatological terms means that these abstract entities have an existence that is independent of speakers or thinkers. Where the metaphor of Plato’s Heaven goes wrong is not in seeking an analogy between the way general terms ¹¹ My use of ‘determinate’ is pre-theoretic and presupposes no particular account of it. For example, I don’t presuppose that indeterminate propositions are neither true nor false, and as I use the term, a proposition is indeterminate if it’s borderline. ¹² Wright (2003b, 487).
116
Stephen Schiffer
may come to signify properties and the way names may come to signify things. Where it goes wrong is in supposing that there is only one model of the name–named relation. There are in fact two models. One of them would sustain the false conception of Plato’s Heaven if it were applicable to properties, which it isn’t; the other is applicable to properties, and when appreciated it makes plain the mistake that lies behind the false conception. I’ll try to explain. A raison d’ˆetre of names of any kind is to give people an easy way to refer to their bearers in a wide variety of contexts. What makes n a name of x is the way n enables speakers to refer to x. Roughly speaking, when a speaker uses a name n to refer to one of its bearers x, she expects her hearer to recognize that she is using n to refer to x by virtue, in large part, of its being mutual knowledge between them that there is a practice of using n to refer to x. It’s because n has been used to refer to x that people who share that knowledge are able to use n to refer to x when speaking with one another. So one thing the use of a name may achieve is simply the upshot of its being a name, viz. the provision of a useful way to refer to whatever it names. In many cases—for example, all names of persons and dogs—that is pretty much the only way communication can be aided by the use of the name. When this is true of a name–named pair, it exemplifies what I’ll call the ‘Fidelle’–Fidelle model. Not every name–named pair exemplifies this model; only those that satisfy this condition: it’s metaphysically possible for someone to identify the thing named in a way that distinguishes it from other things well enough to provide reference-enabling knowledge about it prior to anyone’s ever having referred to it. The pair of you and your name exemplifies the ‘Fidelle’–Fidelle model. Every name has the raison d’ˆetre already mentioned, but some also have a second raison d’ˆetre, a second job they do in addition to the job that goes with the first raison d’ˆetre; and these names couldn’t name the things they name if they didn’t do this second job. The pairs of these names and their bearers don’t exemplify the ‘Fidelle’–Fidelle model, but they do exemplify what I’ll call the ‘Midtown’–Midtown model. Nothing can exemplify both models. Fidelle is in her doghouse and the number of dogs in the doghouse is 1. This doesn’t mean that Fidelle’s temporal and physical boundaries aren’t vague. They are vague: there is no precise moment at which she determinately came into existence, there is no precise moment at which she will determinately cease to exist, and there is no location in space where she is determinately located. But thanks to the fact that no two dogs can even partially occupy all of a given space at the same time, we are able to individuate and have knowledge about Fidelle in an entirely name independent way. Like Fidelle, the area of Manhattan named ‘Midtown’ has vague boundaries. Times Square is definitely in Midtown, and if from Times Square you walk in any direction you will eventually definitely not be in Midtown; but no matter in which direction you walk, you will never come to an invisible line on one side of which you are definitely in Midtown and on the other side of which you are definitely not in Midtown. Nevertheless, <‘Midtown’, Midtown> doesn’t exemplify the ‘Fidelle’–Fidelle model. We can have some intuitive evidence that it doesn’t before we know why it doesn’t. Suppose you’re being held captive in the desert and your captors tell you they
Vague Properties
117
will kill you unless you draw a square in the sand and then identify a unique area α such that anything within the square would be a borderline case of a thing that was in α. If your ingenuity is no better than mine, this is the time to start pleading for your life. Now suppose Midtown had never enjoyed a name and ask yourself how you might stipulate a name for it. Were the question about Fidelle, you could answer in a flash. The notion of a sortal concept helps to locate the difference between Fidelle and Midtown which explains why only the former can belong to a name–named pair that exemplifies the ‘Fidelle’–Fidelle model. A sortal concept is a category concept that enables the enumeration of things as falling under it. Dog is a sortal concept: my possession of it often enables me to know by counting how many dogs there are within a certain area. If it were ever indeterminate whether one of the creatures in the area was a dog, then my counting would be stymied. Thing, by contrast, is not a sortal concept; it makes no coherent sense to speak of the number of ‘things’ in an area. I believe it’s because I can individuate Fidelle under the sortal dog that I am able to acquire individuating knowledge of her that would make it possible for me to stipulate a name for her, or even to be the first person ever to refer to the dog. I believe it’s because I don’t have a sortal for Midtown (area certainly isn’t a sortal) that I can’t first acquire individuating knowledge of that vague area and then be the first person ever to refer to it, let alone give it its first name. But then how does it come about that we are able to use the name ‘Midtown’ to refer to the vaguely defined area Midtown? Because—to a first approximation—it’s the very practice of using ‘Midtown’ which itself fixes the individuation of Midtown in a way that enables there to be any reference to it. Whence the second raison d’ˆetre some names enjoy and the criterion for exemplifying the ‘Midtown’–Midtown model: the very practice of using the name is itself what enables us to pick out its vaguely defined referent well enough for there to be any reference to it. In the name–named pairs that exemplify the ‘Midtown’–Midtown model, the thing named is in a sense a shadow of its name, and its individuating contours mirror the contours of the name-using practice which makes its name its name. When, and only when, this happens does a name–named pair exemplify the ‘Midtown’–Midtown model. Here is a caricature —no more than that—of how this might work. Growing up in Wasilla, Alaska, Harvey had barely heard of Manhattan, let alone of the section of it called ‘Midtown’. Upon graduation from Alaska State University, he landed a job in Manhattan and moved there. He gradually became acclimated, and in the process acquired the use of ‘Midtown’ as the name of an area of Manhattan. As part of that acquisition process, he learned from observing referential uses of the name that certain locations are unquestionably in Midtown and that other locations are unquestionably not in Midtown. Since Harvey also learned that it’s possible to go from any one place in Midtown to any other without leaving Midtown, he was able to know by deduction that many places are unquestionably in, or unquestionably not in, Midtown. He also learned that in between the locations that unquestionably count as being in, or not in, Midtown, there were many for which there was no governing precedent. In certain contexts, such a location might be said to be in Midtown, in other contexts said not to be in Midtown, and in still other contexts there was
118
Stephen Schiffer
evidently nothing one could say one way or the other. Today Harvey’s mastery of ‘Midtown’ is as good as anyone’s, and his use of the name has these features: (i) There are locations λ+ such that, for any x, Harvey’s being certain that x is in λ+ will result in his also being in a state I’ll call Positive Confidence as regards whether or not ‘Midtown’ applies to x, where to be in that state is to judge that ‘x is in Midtown’ is true and to be disposed not to count anything as a reason to doubt its truth as long as one is certain that x is in λ+ . Witnesses to λ+ include the locations of the Empire State Building, The Chrysler Building, Grand Central Station, Times Square, and Rockefeller Center. (ii) There are locations λ− such that, for any x, Harvey’s being certain that x is in λ− will result in his also being in a state I’ll call Negative Confidence as regards whether or not ‘Midtown’ applies to x, where to be in that state is to judge that ‘x is not in Midtown’ is true and to be disposed not to doubt its truth as long as one is certain that x is in λ− . There are many more witnesses to λ− than there are to λ+ . These include, for example, every location in the universe outside of Manhattan. (iii) There are locations λ+/− such that, for any x, Harvey’s being certain that x is in λ+/− will result in his being stymied to accept any verdict on the truth-status of ‘x is in Midtown’ and to be disposed not to count anything as a reason to accept any verdict on the sentence’s truth-status as long as he is certain that x is in λ+/− , where his being stymied is explained by the extent to which the pull of λ+/− ’s resemblance to the λ+ locations is offset by the opposite pull of its resemblance to the λ− locations. While it is very easy to find context-invariant indubitable witnesses both to λ+ and to λ− , for a reason to be discussed in connection with vague properties, it is considerably more difficult, if not impossible, to find context-invariant indubitable witnesses to λ+/− . The meaning that a word has for a person is always at least partly determined by its wide conceptual role in her talk and thought (henceforth ‘wide’ will be unvoiced in ‘conceptual role’).¹³ For words that have a deferential component, conceptual role won’t completely determine meaning. But I see no reason to suppose that conceptual role doesn’t completely determine meaning for those words of which one knows that one’s mastery equals anyone else’s. While conceptual role always plays a role in determining meaning, it mustn’t be assumed that it plays the same role in every case, not even for terms of the same syntactic kind. For example, it may be reasonable to suppose that what makes ‘Fidelle’ your neural name of Fidelle is the way Fidelle enters into causal explanations of the fact that certain sentences containing ‘Fidelle’ are tokened in your belief box. The causal conditions that would make ‘Fidelle’ your name for Fidelle won’t serve to individuate the dog; that individuation is presupposed by the causal account of how the dog acquired the name it has for you. The vaguely defined area Midtown couldn’t replace Fidelle in a similar causal story if its life depended on it. What we need for ‘Midtown’–Midtown isn’t a causal story, but an account that shows how conditions that make ‘Midtown’ a name of Midtown also individuate it. My sketch of the conceptual role that ‘Midtown’ has for Harvey is intended to suggest just such an account. More specifically, it is intended to suggest ¹³ ‘Wide’ conceptual role, unlike ‘narrow’ conceptual role, takes into account causal relations to things outside the head. See Harman (1988).
Vague Properties
119
something to the effect that, thanks to its having features (i)–(iii), respectively, the conceptual role ‘Midtown’ has for him secures that in his idiolect the name refers to a vaguely defined area αm that is partially characterized by the conditions’ securing that every λ+ is included in αm , every λ− is not included in αm , and every λ+/− is a borderline case of a location that is in αm . Even as a caricature of the conceptual role ‘Midtown’ has for Harvey, the foregoing would have to be amended and qualified before it could serve as a point of departure for a proper portrait. But the sketch isn’t intended to be a prelude to a portrait of vague areas; it’s intended to be a prelude to an attempt to extract lessons applicable to vague properties from the ‘Midtown’–Midtown model of the name–named relation, and I think I have said enough for that purpose. With that application in mind, I would like to draw attention to an asymmetry on display in the sketch of Harvey’s use of ‘Midtown’ from which I will presently attempt to extract a generalization applicable to vague properties. The asymmetry on display is between, on the one hand, what would make an application of ‘Midtown’ determinately correct or determinately incorrect, and, on the other hand, what would make an application borderline correct/incorrect. When an application of ‘Midtown’ is correct, it is owing to a feature of the word’s use that determines it to be correct, and likewise, mutatis mutandis, for when an application is incorrect. But being a borderline application isn’t a status like being a correct or incorrect application, one that enjoys a feature specifiable in isolation from those that determine correct or incorrect applications. Rather, being borderline is owing to the absence of either of the other two conditions. Borderline cases are those for which no status is determined. Moreover, borderline applications aren’t merely ones that fall between two stools; they are ones that bounce around between them because of the resemblance-based attraction each stool exerts. But enough of this attempt at rigor and precision; let’s return to this chapter’s eponymous subject. 6.5
VAG U E P RO PE RT I E S : A S K E TC H W I T H D I S TO RT I O N S
Semantically simple general terms are like names. When I say to you ‘Harry is bald’ in order to tell you that Harry is bald, I intend our mutually knowing that ‘Harry’ is used to refer to Harry to be the primary thing that enables you to know that what I’m saying is about Harry. Similarly, I also intend our mutually knowing that ‘bald’ is used to express the property of being bald to be the primary thing that enables you to know that what I’m saying about Harry is that he is bald. Is ‘bald’ like ‘Harry’ or like ‘Midtown’? ‘Midtown’, clearly. The theory of vague properties should mirror the theory of vague areas. Start with the ‘Midtown’–Midtown model of the name–named relation, swap ‘bald’ for ‘Midtown’ and baldness for Midtown, tinker a little more with the model, and you have the ‘bald’-baldness model of the relation between predicates and the properties they express. That is why the prevalent misconception of Plato’s Heaven is a misconception: you can’t stumble upon a previously unnoticed vague property and pick it out well enough to introduce a word to express it. You can’t have epistemic access to hitherto unnoticed properties and then attach words to
120
Stephen Schiffer
them. You need the words in order to gain epistemic access to them in the first place. But how can this be? Because the conceptual role by virtue of which a term expresses a vague property also fixes the conditions that enable us to pick out that property well enough to express it. Just as the contours of a vague area mirror those of one’s use of one’s name for it, so, too, the contours of a vague property mirror those of one’s use of one’s predicate for expressing it. This doesn’t mean that vague properties don’t exist with the individuating conditions they actually have in possible worlds in which no one has words to express them, or, for that matter, in possible worlds in which there are neither thinkers nor speakers. It does, however, mean that the only way to gain epistemic access to them in any possible world is to have expressions that signify them in that world. Vague properties—and that probably includes every property we can express—are unlike dogs and volcanoes in that respect.¹⁴ Properties, to use David Armstrong’s metaphor, are shadows of predicates.¹⁵ It’s tempting to see the following sketch as the one whose distortions we should bother to correct. Jane’s use of F is non-deferential, so that it’s her use of F that determines the meaning F has for her. Some feature of Jane’s use of F determines a condition C + which entails that whatever property F expresses will be such that anything that satisfies C + will instantiate that property. In other words, a thing’s satisfying C + will secure that F is true of it, but not because F already expresses a property that x instantiates, but rather because C + determines F to express a property that anything satisfying C + will instantiate. And the further idea here is not that there is some property φ such that we will discover that F expresses φ by virtue of there being a certain sort of isomorphism between the conditions that individuate φ and certain features of F ’s conceptual role, as though we might just as well have learned that F failed to express any property because it transpired that no property was correlated in the right way with F ’s conceptual role. The idea is rather that the fact that F ’s conceptual role has certain features simply makes it the case that F expresses a property whose individuative contours mirror those of F ’s conceptual role. I take condition C + to be the analogue in Jane’s use of F of the Positive Confidence condition in my caricature of Harvey’s use of ‘Midtown’, and like that condition I think it’s plausible that C + will be some sort of (relatively) indefeasible acceptance condition; but my present purposes don’t require me to have that much of my neck exposed. It’s obvious that very many properties enjoy numerous sufficient conditions for being instantiated. For example, being 7 feet tall is a sufficient condition for being a tall man, and for every real number n greater than 7, being n feet tall is a fortiori also a sufficient condition for being tall. Since we can’t identify sharp cutoffs anywhere, we know that no vague property has a condition we can identify as its unique minimally sufficient condition for its instantiation. It’s therefore important that I not be construed as saying that there is some condition C + such that it specifically is singled out by some feature of Jane’s use of F . I am instead saying that a feature of Jane’s use of F determines there to be a condition C + which entails that whatever property F ¹⁴ See the discussion of pleonastic properties in Schiffer (2003), ch. 2. ¹⁵ Armstrong (1989).
Vague Properties
121
expresses will be such that anything that satisfies C + will instantiate that property. In the same way, some feature of Jane’s use of F determines that there is some condition C − which entails that whatever property F expresses will be such that a thing does not instantiate it if the thing satisfies C − (thus making satisfaction of the condition ¬C − a necessary condition for a thing’s having the property F expresses). In the way that F ’s conceptual role determines there to be conditions C + and C − , does it also, in the very same way, determine there to be a condition C b , specifiable without reference to C + or C − , which entails that whatever property F expresses will be such that anything that satisfies C b will be a borderline instance of that property? It is of the essence of the account of vague properties I’m in the midst of sketching that the answer is a resounding no. It’s the absence of such a condition that accounts for how a vague property’s penumbra is determined. F ’s conceptual role determines a thing to be a borderline instance of whatever property F expresses in part by failing to enable Jane to apply F or not F to that thing. More specifically, x will be a borderline instance of whatever property F expresses when, owing to the way x’s resemblance to the things to which Jane would confidently apply F competes with its resemblance to the things to which Jane would confidently apply not F , the conceptual role of F fails to determine any truth-status for x is F . It’s that failure which explains why Jane is stymied to make any judgment about the truth-status of the proposition expressed by x is F when she is in, and knows herself to be in, circumstances that are optimal for determining whether or not that proposition is true, and why in those circumstances she is apt to be in Quandary as regards whether or not F applies to a thing. A spinoff of the way F ’s conceptual role determines the vague property F expresses is that it simultaneously determines that vague property’s shadow, the vague property of being a borderline instance of that property. Earlier I asked how properties ψ and φ must be related in order for ψ to be the property of being a borderline instance of φ, and I suggested that part of the answer was that if ψ is the property of being a borderline instance of φ, then the truth of the proposition that x has ψ entails that there is nothing to determine any truth status for the proposition that x has φ. That suggestion might have seemed puzzling, given that vague properties have borderline instances even in possible worlds devoid of speakers or thinkers. If we can go by the distorted sketch of vague properties on offer, then the resolution of the puzzle lies in the way it’s the very use of predicates that individuates the properties they express in a way that enables users of the predicates to express those properties in the first place. Are we then to understand that a thing’s being a borderline instance of a property is to be defined in terms of there being a conceptual role that would determine a predicate to express the property? Not unless we are also prepared to define a thing’s being an instance of a property in terms of the same conceptual role. We shouldn’t confuse the conceptual-role features that determine a property’s individuating conditions with the conditions they determine. You may have noticed that the sketch-implied account of being a borderline case doesn’t seem directly to invoke Quandary. What happened to the thought that may have seemed the upshot of the early part of this chapter, that ‘borderline cases are cases where a competent judge may be put in quandary despite conditions obtaining
122
Stephen Schiffer
which are optimal for judgement in clear cases’?¹⁶ Quandary has indeed been moved from center stage. The account of vague properties I’m sketching gives pride of place to the idea that what makes x a borderline instance of φ is that the predicate-using practice that individuates φ fails to determine a truth-status for the proposition that x is an instance of φ, instead of to the idea that the use-practice determines a positive condition defined in terms of Quandary whose satisfaction accounts for x’s being a borderline instance of φ. When the conceptual role of the predicate that expresses φ for one fails to determine a truth-status for the proposition that x is an instance of φ, and one is confronted with x in the best possible circumstances for judging whether or not x is an instance of φ, one is apt to feel stymied to make any judgment and thus to be in Quandary as regards whether or not x is an instance of φ. But that is an accident of the way we’re built and not essential to taking a thing to be a borderline case: one might with total aplomb realize that there is no judgment one can make and not give it a second’s thought. One might even mistake one’s situation vis-`a-vis the borderline case—perhaps under the influence of a philosophical theory of vagueness—and, having convinced oneself that, say, the proposition that Harry is bald has just got to be true or false, misattribute one’s inability to determine whether or not Harry is bald to how well hidden the truth about his scalp is. As if to add insult to injury, the sketch, as it so far stands, implies that Quandary isn’t a genuine propositional-attitude state like believing or any other propositionalattitude state named by a gerund. It’s rather the inability to be in such a propositionalattitude state. The Great Computer Designer who created us had a good reason to give us belief states, desire states, etc. She gave us those states so that we might survive in the environments in which She intended to place us. But She didn’t plan to give us states of being stymied; our occasionally going into one of those states was the unintended consequence of the unavoidable limitations on our information-processing abilities. She thought of those unintended consequences of Her handiwork—those Quandary states—as unavoidable conceptual noise. If that is right, then, contrary to what I suggested in The Things We Mean, Quandary isn’t a kind of nonclassical partial belief. I was encouraged to think that it was in part because I thought that a nontruth-status account of vague properties required a propositional-attitude condition of its own for a thing’s being a borderline instance of a property, a condition that was on a par with the Positive Confidence condition I thought was needed to determine a sufficient condition for having the property in question and with the Negative Confidence condition I thought was needed to determine a sufficient condition for not having that property. That misstep caused me to take a second—viz. my attempt to define being a borderline instance of a property in a way that made being borderline look like a response-dependent property, the response vagueness-related partial belief, which is what I then called Quandary, in a sort of ideal epistemic agent, in something like the way one might try to define being funny in terms of a disposition to make certain idealizations of ourselves amused. The view now being adumbrated suggests an account of the conditions that will account for a thing’s being a borderline case, but is under no burden to ‘analyze’ or ‘define’ the property of being a borderline instance ¹⁶ Wright (2003b, 487).
Vague Properties
123
of a property. Yet both the account being sketched here and the one offered in Things remain the same in an extremely important respect: both are non-truth-status theories that respect Q-Constraint, and because they are non-truth-status theories, they offer the same response to the sorites in that both imply that the sorites premises of standard sorites inferences have no determinate truth-status, which in turn implies that it’s indeterminate whether those inferences are classically valid.¹⁷
6.6
VAG U E P RO PE RT I E S : TOWA R D S D I S TO RT I O N R E M OVA L
Most of the distortions consist in things left out of the sketch, so removing them requires adding to the sketch. I’m relieved to see that I’m too close to my word limit to do much by way of correcting for the sketch’s missing details, but I will try to inventory some of what needs to be done. 1. I need to qualify what I said about the inability to find a hitherto unknown property in Plato’s Heaven without the aid of a predicate to express it. I think that even without a predicate for expressing it, it’s conceivable that we might come across an entirely new basic color, one that couldn’t be obtained by mixing familiar colors. We could refer to this by a demonstrative—e.g. ‘that color’—and then introduce a name for it. What this possibility may seem to threaten is the idea that properties are individuated by the conceptual roles of expressions that express them. But I don’t think it does threaten that. What it shows is that the conceptual roles that individuate members of a kind of property—colors or natural kind properties like doghood, for example—fix a template for the individuation of any new members of the kind. This wouldn’t show that language wasn’t needed to discover any property, only that in some cases the language needed to discover a property is at one remove from a term that actually expresses that property. 2. The sketch pretty much ignores the way the penumbras of vague terms can vary from one context of utterance to the other, thus implying that for no vague predicate type (as opposed to token) is there ever such a thing as the property it expresses. I don’t think this context variability will prove difficult to accommodate. The competing resemblances that make for borderline cases are always resemblances with respect to certain weighted features. The features and their weights are interest-sensitive, and the relevant interests of the speaker are highly context dependent. I see the contextvariability of penumbras as a corollary of the way resemblance works to determine a given person’s application of a term at a given time. Here, however, is a slightly bolder speculation about how things work, one that borrows selectively from views in cognitive psychology. The cases in which Jane would predicate F of a thing x are ¹⁷ I recognize that what I just said is hardly self-contained, but I can’t here give it the elaboration it needs. For an elaboration that might suffice, see Schiffer (2003), ch. 5, (forthcoming-a), and (forthcoming-b). It’s important to note that my saying that it’s indeterminate whether sorites inferences are classically valid is entirely consistent with my thereby recognizing the need for a nonclassical logic to accommodate vague language.
124
Stephen Schiffer
of two kinds: those in which the predication is based on x’s satisfying a condition that Jane’s use of F implicitly stores as a sufficient condition for a correct application of F , and those in which the predication is based on x’s resemblance to exemplars of things of the first kind (the same goes for not F ). Resemblance-based predications are highly plastic. As noted, resemblance is always resemblance in respects deemed relevant, and the importance of the various ‘respects’ are weighted. Presumably we are innately disposed to conceptualize the world in some ways and not others, and therefore to notice and attach importance to some kinds of resemblances and not others, and of course these factors vary from one kind of property to another. But within those more-or-less fixed constraints for any given kind of property there is considerable room for variation. Is this thing enough like those definite F s to itself be an F ? We have already noticed that how one answers may to some extent depend on how one weighs respects of resemblance, and to some extent this will be a function of one’s interests at the time, or the purpose and nature of the conversation in which the question arises. Then of course even along a certain weighted dimension, resemblance isn’t all or nothing: even with respect to some fixed parameter things resemble each other to some degree or other, and the degree of relevant resemblance is another thing that is easily influenced by pragmatic contextual considerations. These reflections help to explain an interesting feature of vague terms noted in the discussion of Harvey’s use of ‘Midtown’, but which I’ll now make with the focus on predicates. For virtually every vague predicate it’s very easy to find context-independent examples to which the predicate determinately and unquestionably applies. Yul Brynner counts as bald in any context of utterance, and Michelle Obama counts as not bald in any context of utterance. But it’s extremely difficult, if not impossible, to find contextindependent examples of things that are determinately borderline cases of things to which the predicate applies. I credit this to the fact that when in a given context a thing comes across to us as a borderline case of a thing to which a certain predicate applies, there are usually possible contexts in which one would deem the predicate either to apply or not to apply. But what of the idea, made much of in the unrefined sketch, that the contours of a property mirror the contours of the conceptual role of the predicate that expresses it? That may need to be complicated slightly. When vagueness is not at issue, a philosopher who thinks that use—i.e. wide conceptual role—determines meaning will speak as though at any given time the predicate F has a single meaning-determining conceptual role for Jane that it contributes to the meaning-determining conceptual role of every sentence of Jane’s mentalese in which F occurs. That way of speaking would be strictly OK only if we could say that at any given time there is a single vague property that is the meaning F has for Jane at that time. But we can’t say that. Since we will want to allow that, even without a change in the hair situation on Harry’s scalp, there may be both true and false utterances of ‘Harry is bald’ by Jane, one possible way to go with a change to accommodate context-shift would be to construe Jane’s mentalese (whose sentences might be thought of as representations of the logic forms of public language surface forms) as somehow realizing the following description. F has a conceptual role in Jane’s mentalese but, strictly speaking, no sentence
Vague Properties
125
containing F can be tokened in her belief box, and F can’t, strictly speaking, be said to express any vague property. Rather, what we find is that F interacts with Jane’s interests of the moment to create new predicates, F , F , F , and so on. These ‘new’ predicates are functionally individuated by conceptual roles created when F ’s conceptual role connects with a momentary interest of Jane’s, and it’s that conceptual role that individuates the vague property expressed by a public language manifestation of F with an unvoiced superscript. Clearly, the issue raised by the context-variability of vague predicates needs considerably more work, but I hope I’ve said enough to allow me to think that the complications will merely require complicating, as opposed to junking, the sketch now on the easel. 3. Up to this point I’ve been discussing vague properties as though the topic could be confined to what is going on in a given person’s idiolect, and this leaves out something important. The missing detail concerns a way in which a thing may be borderline F even though each speaker’s use of ‘F ’ counts the thing either as F or as not F . For example, Mike’s scalp appears as hairless as a billiard ball, but it’s common knowledge that he shaves his scalp and that no one would apply ‘bald’ to him if he stopped shaving it and let his hair grow out. Carl and Carla are native speakers of English neither of whom uses ‘bald’ incorrectly. They are together in circumstances that are optimal for judging the hair situation on Mike’s scalp, when the question ‘Is Mike bald?’ arises. Carl, true to his use of ‘bald’, answers ‘Yes’; Carla, true to her use of ‘bald’, answers ‘No’. How should we describe what is going on? Should we say that, owing to their different uses of ‘bald’, they aren’t really disagreeing and each person’s answer is true? Or should we say that they are both addressing the same question—viz. whether or not Harry is bald—and disagreeing about the answer? Our criteria for individuating what is said in an utterance, and thus the criteria we use for deeming two people to have said the same thing, are course-grained, context-dependent, and suffer large areas of indeterminacy, so much so that I doubt that the question just posed has a determinate answer. Its not having a determinate answer is, however, compatible with the further claim that, once apprised of the relevant facts about how English speakers divide on what they are inclined to say about shaved heads, it ought to be clear to them both that as ‘bald’ is used among English speakers, it is indeterminate whether it applies to someone who wouldn’t be bald if he didn’t shave his scalp. This sort of inter-personal vagueness raises interesting questions and adds to our understanding of the context variability of vague predicates, but it seems to me that its complications are easily accommodated by the sketch even as it now stands. For once we fix well enough which vague property is in question, we have no reason to think that the cases of inter-personal vagueness in question requires any revision of what has so far been suggested about being a borderline instance of a property. 4. Nothing has been said about the vague properties expressed by semantically complex predicates, in the simplest case the properties expressed by predicates of the form ‘is an F G’ in straightforward predications of the form ‘a is an F G’. In some of these predications ‘a is an F G’ means the same as ‘a is F and a G’, as e.g. ‘Betsey is a pregnant cow’ means the same as ‘Betsey is pregnant and a cow’. The sketch needs no alteration to account for these. More challenging are predications which mean the
126
Stephen Schiffer
same as ‘a is F for a G’.¹⁸ Here it doesn’t seem implausible to treat F as a functor that expresses a vague function that maps properties onto properties. If that is so, then these cases also don’t threaten the sketch. 5. I have already said that Quandary has been demoted from the feature role it played in The Things We Mean: it has been demoted from a genuine propositional attitude, billed as a kind of nonclassical partial belief, to mere ‘conceptual noise’; and it has been demoted from something in terms of which being a borderline case needs to be defined to something like a causal spinoff of that which determines the individuation of vague properties. Neither demotion was claimed on the basis of direct objections to anything in Things, but as consequences of a better, positively motivated account of vague properties. Yet even in the book I indicated two places where the account of vagueness-related partial belief (VPB)—aka Quandary—might well be in trouble. One place was the flimsy warrant I knew I had for saying that VPB was a form of partial belief. Suppose momentarily that I was justified in taking VPB to be a genuine propositional attitude that comes in degrees—call it V ing—so that one can say, for example, that the degree to which one V s p is greater than the degree to which one V s ¬p. That of course wouldn’t show that there was any rationale for saying V ing was a form of partial believing. I appreciated that a propositional attitude ‘that can’t be used to explain behaviour in the way beliefs standardly do has no business being called a belief.’¹⁹ My initial reason for saying that V ing was a kind of belief was that it explained what appeared to me to be qualified assertions: ‘Is Harry bald?’ ‘Well, it’s really hard to know what to say; he sort of is, but then again he sort of isn’t.’ But I never doubted that I had to do better than that if I was to show that V ing was a kind of belief. I made a not very convincing effort to do better, but I also said that the issue was of no great moment for me, because even if I couldn’t explain vagueness in terms of a kind of partial belief, I nevertheless explained it ‘in terms of a new kind of propositional attitude, one that comes in degrees and that precludes [classical] partial beliefs.’²⁰ I should have realized, however, that even to show that Quandary was a genuine propositional attitude on a par with the familiar ones, it was important to find a unique role for it in the production of behavior. And this is what I now doubt can be done. Hartry Field and John MacFarlane agree with me that Quandary precludes classical partial belief, and they agree with my book’s effort of trying to explain Quandary in terms of a novel kind of partial belief.²¹ But neither of them has shown that the ‘belief ’ states that define Quandary have a role to play in the production of behavior in a way that is even remotely like the roles belief, desire, and intention have. The second place where the book was tentative was in its attempt to find neat laws that normatively govern logically complex VPBs and the interaction of VPBs and classical partial beliefs. I made some suggestions but was unable to find laws that ¹⁸ Semantically simple predicates like ‘tall’ in ‘Phil is tall’, which can’t be uttered without an intended reference class (‘tall for an X ’), should be considered parts of complex predicates like ‘tall girl’ for present purposes. ¹⁹ Schiffer (2003), 231. ²⁰ op. cit. 232. ²¹ Field (2001b) and MacFarlane (this volume).
Vague Properties
127
completely did the job, and John MacFarlane gave a convincing counterexample to a key suggestion that I did make.²² But if Quandary is simply conceptual noise, we shouldn’t expect there to be such neat laws. None of this is to go back on the claim that Quandary precludes classical partial belief, nor to deny the obvious fact that Quandary shows that the classical conception of partial belief can’t be right, since we will need a nonclassical notion of partial belief to explain the appropriate degree of belief to have in, say, the disjunction of p and q when one believes and disbelieves p to positive degrees but q is borderline true/false. That merely shows that we need to go back to the drawing board on partial belief, not that there should be laws that normatively govern partial beliefs when only they are involved and laws that normatively govern conceptual noise (an oxymoron if ever there was one) when only it is involved, and then laws that normatively govern the intermingling of these two disparate kinds of states. 6. The sketch may seem to have puzzling implications for higher-order vagueness. It has it that x is borderline F just in case the way x is positioned between its resemblance to the F s and the non-F s secures that no truth-status is determined for the proposition that x is F . That evidently entails that x is borderline borderline F just in case the way x is positioned between its resemblance to the borderline F s and either its resemblance to the F s or to the non-F s secures that no truth-status is determined for the proposition that x is borderline F . Earlier I remarked that while for vague properties it was quite easy to find examples of things that unquestionably instantiated them or didn’t instantiate them, it wasn’t at all clear that one could find examples of things that were unquestionably borderline instances of a vague property. The conundrum intensifies when we look into the mind of a person who might take a thing to be, say, borderline red but doesn’t explicitly have the concept of a borderline case, and so can’t judge that the thing is borderline red. What would a mental state of this person have to be like in order for us to be stymied to say whether he took the thing to be red or took it to be borderline red? At the same time, we have the concept of a thing’s being borderline red, and there can be no doubt that the property of being borderline red is every bit as vague as the property of being red. Under what conditions would we judge that a thing was borderline borderline red? Suppose you are examining a fabric sample in circumstances that you know are as good as they can get for you to determine whether or not the sample is red according to your concept of red (so that your judgment won’t depend on how you think speakers of English use ‘red’). Is it that your judging the sample to be borderline borderline red would manifest your being stymied to judge whether or not your current mental state is belief or Quandary? I find it hard to imagine what that would be like. I think the resolution of any puzzle lurking here lies in the way context can affect the penumbras of vague terms. For consider this thought experiment. You see Barbara’s new car while speaking to a friend on your mobile phone and report ‘Barbara’s new car is red’. OK, now focus on the property your utterance of ‘red’ expressed, and imagine having to say whether or not some object not contemplated when ²² MacFarlane (2006).
128
Stephen Schiffer
you reported the color of Barbara’s car was an instance of that color or a borderline instance of it. You might easily find yourself stymied to give an answer, and this because there was nothing in the circumstances of your utterance to determine any answer to that unanticipated question. Had the question been asked when you spoke, it would have created a new context and your next use of ‘red’ might well have expressed a property with a slightly different penumbra. I submit that it’s worth considering whether insight into higher-order vagueness may be gained by focusing on the way contexts of utterance affect the penumbral boundaries of the properties expressed by the predicates uttered in those contexts. 7. Finally, the view of vague properties being sketched challenges a number of widely accepted views about meaning and language understanding. There is space for only the briefest glimpse of what is likely to be the tip of an iceberg. To get the glimpse, suppose Bob reports, ‘Jack said that Paula’s wealthy brother lives in Midtown’. A familiar view would hold that Bob’s report is true only if, for some sentence σ , meaning μ, and proposition p, (a) σ means μ; (b) Jack uttered σ and in doing so meant p; (c) p ‘fits’ μ;²³ and (d) Jack’s utterance of the that-clause ‘that Paula’s wealthy brother lives in Midtown’ refers to p. Along with that the view also holds that, if a speaker said p in uttering σ , then in order to understand her utterance one must know that she said p in uttering σ . It would seem that none of this can be true if the view of vagueness I’ve sketched is true. For suppose the sentence Jack uttered was ‘Paula’s wealthy brother lives in Midtown’. Then Jack’s utterance of ‘wealthy’ expressed a certain vague property ω and his utterance of ‘Midtown’ referred to a certain vague area α, which means that Bob’s that-clause refers to the proposition Jack said only if Bob’s utterance of ‘wealthy’ also expressed ω and his utterance of ‘Midtown’ also referred to α. But given the way the penumbra’s of vague terms may vary from one context of utterance to the next, it’s very unlikely that the vague property expressed by Bob’s utterance of ‘wealthy’ will be identical to the vague property expressed by Jack’s utterance of ‘wealthy’, and it’s very unlikely that the vague area to which Bob’s utterance of ‘Midtown’ referred will be identical to the vague area to which Jack’s utterance of ‘Midtown’ referred. For the same reason, it is extremely unlikely that anyone will know exactly which property Jack’s utterance of ‘wealthy’ expressed or to which vague area his utterance of ‘Midtown’ referred, and thus extremely unlikely that anyone would understand Jack’s utterance. It’s also difficult to see how there could be such things as sentence meanings. For what would correspond to ‘wealthy’ in the meaning of the sentence type ‘Paula’s wealthy brother lives in Midtown’? Surely not one of the myriad vague properties that might be expressed in a literal utterance of ‘wealthy’, and parity of reasoning would show that it can’t be anything all those properties have in common, since that, too, would have to be a vague property, and we would be back with the same problem. Whether it’s the familiar view of meaning and understanding or the view of vague properties I’ve sketched which should be abandoned depends on the plausibility of ²³ i.e. σ ’s meaning μ requires one who speaks literally in uttering σ to mean a proposition of a certain type and p is of that type.
Vague Properties
129
the best theory of meaning and language understanding to be conjoined with the view of vague properties. It’s indisputable that the vagueness of predicates is at the center of the philosophical problem of vagueness. This chapter recognizes the indisputable, but adds to it the hypothesis that the semantics of vague predicates must appeal to the properties utterances of them express, and that those properties are themselves vague. That shifts the focus of inquiry to the nature of vague properties, and since (for all that matters) a property is vague just in case it admits of borderline cases, it’s recognized that a theory of vague properties will perforce also be a theory of borderline instances of vague properties. Quandary is a psychological state that is characteristic of taking a thing to be a borderline case. Q-Constraint is the claim that there need be nothing incorrect about being in Quandary. A truth-status theory of vague properties is a theory that, for some particular truth-status (e.g. being true or false, having a degree of truth, having no truth value), entails that the proposition that x is an instance of φ has that truth status if x is a borderline instance of φ. A central claim of this chapter is that no truth-status theory is true because none is consistent with Q-Constraint. The remainder of the chapter sketches a non-truth-status theory of vague properties that satisfies Q-Constraint. The essence of the sketch is that vague properties are individuated in a way that enables us to have words to express them by the use-practices that determine which vague properties words express. And the key idea here is that those individuating conditions determine no truth-status for the proposition that x is an instance of φ when x hovers between a sufficient condition for being an instance of φ and a sufficient condition for not being an instance of φ thanks to the way x resembles satisfiers of both conditions, this hovering being what is apt to cause one to be stymied in the way of Quandary to judge either that x is, or isn’t, an instance of φ. The sketch is no more than a sketch because of its lack of detail about the use-practices that individuate vague properties, and because of the several challenging questions it leaves waiting for answers. Re f e re n c e s Armstrong, D. (1989), Universals: An Opinionated Introduction, Westview Press. Field, H. (2001a), Truth and the Absence of Fact, Oxford University Press. (2001b), ‘Indeterminacy, degree of belief, and excluded middle’, in Field (2001a). (forthcoming), ‘Vagueness, partial belief, and logic’ in Ostertag (forthcoming). Harman, G. (1988), ‘Wide functionalism’ in Schiffer and Steele (1988). MacFarlane, J. (2006), ‘The things we (sorta, kinda) believe’, Philosophy and Phenomenological Research 73, 218–24. (2009), ‘Fuzzy epistemicism’, this volume. Miller, A., ed. (forthcoming), Essays for Crispin Wright: Logic, Language and Mathematics, Oxford University Press. Ostertag, G. (forthcoming), Meanings and Other Things: Essays on Stephen Schiffer, MIT Press. Schiffer, S. (2003), The Things We Mean, Oxford University Press. (forthcoming-a). ‘Vagueness and indeterminacy: Responses to Edgington, Field, and Wright’ in Ostertag (forthcoming).
130
Stephen Schiffer
Schiffer, S. (forthcoming-b), ‘Quandary and intuitionism: Crispin Wright on vagueness’ in Miller (forthcoming). Schiffer, S. and Steele, S., eds. (1988), Cognition and Representation, Westview Press. Wright, C. (2003a), Saving the Differences: Essays on Themes from Truth and Objectivity, Harvard University Press. (2003b), ‘On being in a quandary: Relativism, vagueness, logical revisionism’ in Wright (2003a), 443–509. (forthcoming), ‘On the Characterisation of Borderline Cases’ in Ostertag (forthcoming).
7 Vagaries about Vagueness Nathan Salmon
7.1 There are at least two broad approaches for dealing with certain sorts of vague expressions, those that admit the possibility of borderline cases, including terms like ‘heap’, ‘bald’, and ‘adult’.¹ One approach, perhaps currently dominant, provides a metaphysically tidy account of the familiar phenomenon of there being no fact of the matter. This approach regards the world of facts as complete and fully determinate, and imputes vagueness and indeterminacy to a misfit between language and the objects and attributes (properties or relations) that make up the facts. Where there is ‘no fact of the matter’ concerning a particular borderline case, this is not due to a paucity of facts. The facts are all there; rather it is entirely because our language fails in its aspiration to capture the facts. Vagueness is a semantic misfiring of a certain sort. This approach locates vagueness not in the world of objects and their attributes but solely in our means of representing the world. On this vagueness-in-language approach, having a property and standing in a relation is always a matter of definite fact, in a particular sense that is difficult to express correctly. For present purposes, I shall formulate the central thesis as follows: for any property P and for any (sortally I presented some of these ideas to the Santa Barbarian discussion group in 2005, organized by C. Anthony Anderson. I am grateful to fellow Barbarians Anderson, Anthony Brueckner, Luke Manning, Michael Rescorla, Chris Tennberg, and David Vander Laan, and to my audiences at the University of California, San Diego, the University of Glasgow, the University of St. Andrews, and the University of Oslo in 2006 for their comments and queries. I thank also Teresa Robertson for discussion and suggestions. ¹ There is a third approach. According to epistemicism, vagueness is entirely a matter of ignorance. Cf. Sorensen (1986); and Williamson (1994). Here I ignore epistemicism. I think it excessively implausible that removing a single grain from a heap of sand can make for a non-heap, and that plucking a single strand of hair from the head of someone who is not bald can make him bald. Still, I suspect there is considerably more merit to epistemicism than is generally realized. I provide one reason for thinking so in Salmon (1986), at 343–44 of Salmon (1981). Another thing to be said for epistemicism (also discussed in Salmon (1986a)) is that even if the doctrine is false and there is no sharp cut-off between those configurations of grains of sand that are heaps and those that are not, and instead there is a range of borderline cases, there may still be a sharp cut-off, although we do not know where it is, between those configurations that are determinately heaps and the borderline cases, and again between the borderline cases and those configurations that are determinately not heaps.
131
132
Nathan Salmon
appropriate) object o, it is either true or it is false that o is a P.² It is either true or false, for example (in fact, it is false), that the real number π is an even natural number. Likewise, for any n-ary relation (n > 1) R, and any (sortally appropriate) n-tuple of objects, it is either true or it is false that those objects are R (i.e. are Rrelated one to another). It is either true or false, for example (in fact, it is true), that π is less than 3.1416. Consequently—and this captures the core idea—every (sortally correct) atomic singular proposition is either true or false by virtue of a corresponding fact. For present purposes, we may reformulate this thesis by saying that for any n-ary attribute (n > 0) and any (appropriate) n-tuple of objects, those objects either definitely bear that attribute or they definitely bear the attribute’s complement.³ In particular, where there is a property (unary attribute) P and an object, there is always a corresponding fact of the matter—either that the object is a P or that it is not (and consequently is a non-P). In short, there is no predication without correspondence.⁴ The world is just the way it is and there is no vagueness about it itself. Although every singular proposition is supposed to be truth valued, the vaguenessin-language approach recognizes that sentences involving borderline cases are semantic-truth-value-challenged. This is allegedly because it is not settled which among a range of singular propositions—some true, some false—the sentence expresses. On the vagueness-in-language approach, slippage occurs in the semantic connection between language and the world. Vague terms are semantically under-defined. The noun ‘heap’, for example, is one whose semantic content is supposed to be a property of certain material structures, but the word is only partially defined according to a rule that certain specific sorts of configurations (e.g. of grains of sand) qualify as ‘heaps’ and certain smaller configurations qualify as ‘non-heaps,’ with nothing being settled concerning configurations in between. These undecided structures are the borderline cases. (It is semantically decided in advance—hence analytic—that any heap-like structure of grains of sand larger than some heap is also a heap, and any heap-like structure smaller than some non-heap is also a non-heap.) Which property of physical structures is it on the vagueness-in-language approach that the word ‘heap’ expresses (or designates)? On this approach, there are very many ² An object o is sortally appropriate to a property P if o is the right sort of thing to be a candidate for having P even if it does not have P. ³ It is to be understood here that for something of the appropriate sort to bear the complement of a property P is simply for it not to be a P (e.g. not to be an even natural number). For present purposes one might reformulate the thesis instead (albeit somewhat misleadingly) by saying that for any n-ary attribute (n > 0) and any (sortally appropriate) n-tuple of objects, those objects either definitely bear that attribute or they definitely lack it—where lacking an attribute is equivalent to bearing its complement. The core idea is that every sortally correct singular proposition is either true or false. ⁴ I am using the term ‘predication’ in a metaphysical rather than linguistic sense, for a pairing of an n-tuple of (appropriate) objects with an n-ary attribute (rather than a verbal subject-predicate ascription or its utterance). Strictly speaking, the vagueness-in-language approach confines the motto of no predication without correspondence to those objects and attributes associated with vague terms. It is consistent with this approach that objects fall short of having attributes, or else having their complements, for reasons other than vagueness—for example, sortal incorrectness (category mistakes). Henceforth I ignore potential sources of there being no fact of the matter other than borderline cases due to vagueness and semantic truth valuelessness due to failure of designation.
Vagaries about Vagueness
133
properties P of configurations of grains of sand that coincide with the partial specification of a heap, in that any structure that qualifies as a ‘heap’ according to the partial specification definitely has P and any structure that qualifies as a non-heap definitely has the complementary property, non-P. But not vice versa; borderline heaps definitely have some of these properties P while definitely having the complements of others. It is semantically settled that ‘heap’ is a term for one or another of these many properties P, but it is not settled which. The word does duty as a term for one property from the range of candidates, but it is arbitrary which one. David Lewis, champion of a version of the vagueness-in-language approach, called this alleged phenomenon ‘semantic indeterminacy’ and ‘semantic indecision’.⁵ On the vagueness-in-language approach, it is not settled which property is expressed or which object is designated by a vague term precisely because in nearly every case it does not matter. The borderline cases are (supposedly) almost never problematic. If a situation requires resolution of the indeterminacy—as in the case of ‘adult’ and regulations governing certain activities—we can ‘sharpen’ or ‘precisify’ the under-defined term by artificial stipulation. Alternatively, we might use, or introduce, a more fully defined term (‘person 18 or older’). When we sharpen a previously vague word, we decide an issue previously left undecided. Until such time, we can happily live with the semantic under-determination. The task of fixing a particular content for the term is unfinished, but no one in his right mind cares. It ain’t broke, so why fix it? Or as Mark Twain said on behalf of procrastinators everywhere, ‘Never put off until tomorrow what can be done the day after tomorrow.’ On the vagueness-in-language approach, the world is fact-rich while the language is a work in progress. This is vagueness for fuzzy speakers in an exact world. The alternative approach locates some indeterminacy in the bearing of attributes, and in this sense, in the world itself and not merely in our means for representing it. Indeterminacy of this sort is sometimes also called ‘ontic vagueness’. On this vagueness-in-the-world approach, there can be a specific property P (e.g. being a heap) and a specific object o of the right sort for P (e.g. a physical structure consisting entirely of sand), with no problem of settling just which property and object these are, such that the corresponding singular proposition is still neither true nor false, just because o is a borderline case of P and consequently P is undefined for o. The world is seen as factually incomplete, with a host of definite questions left undecided because of a shortage of facts. There are attributes and (appropriate) objects for which a corresponding fact is missing; there is predication without correspondence. Where there is ‘no fact of the matter’ concerning a particular borderline case, this is because, although there is a matter there is no corresponding fact. On the vagueness-in-theworld approach, a word like ‘heap’ is only partially defined, but not in the sense that it is not semantically settled which property it expresses.⁶ It is fixed semantically that the word expresses a particular property—that of being a heap—to the exclusion ⁵ Lewis (1988). See also Lewis (1993), at 169–70 of Lewis (1999); and Thomason (1982). ⁶ The terminology of ‘vagueness-in-language’ and ‘vagueness-in-the-world’ is not meant to suggest that the two approaches so designated are the only approaches that locate vagueness in language or in the world, respectively.
134
Nathan Salmon
of all other properties. But for some configurations of grains of sand there is no fact that the structure in question has the property in question, but neither is there a fact that the structure has the complementary property of being a non-heap (i.e. that it is not a heap). The borderline cases are such that it is indeterminate (neither true nor false, there is no objective fact of the matter) whether they are heaps. Borderline heaps—real and definite objects—neither definitely have the property of being a heap nor definitely have its complement—real and definite properties both. The borderline cases are indeterminate with respect to the property. If the question is raised whether a given borderline heap is a heap—yes or no—it is settled which matter is the issue at hand, but there is no fact to settle the matter itself. On this approach, when we sharpen a previously vague word, we do not decide an issue previously left undecided. Rather, we alter the meaning by assigning a new and different semantic content in place of the old. On the vagueness-in-the-world approach, language is a finished product while the world is factually impoverished. This is vagueness for exact speakers in a fuzzy world.⁷ On either approach, certain atomic sentences—those predicating a vague term of a borderline case—suffer a lack of truth value, but the approaches differ dramatically in their diagnoses of the source (and possibly also the nature) of the gap: one approach blames the world of objects and their attributes, the other the means of representation. If the world is fuzzy, the language can hardly be blamed for the resulting semantic truth valuelessness. On the contrary, it is to the language’s credit. On the other side of the coin, the principal appeal of the vagueness-in-language approach is precisely the hope it holds out for accommodating indeterminacy due to vagueness without countenancing objects and attributes with no corresponding fact—the utopian goal of no predication without correspondence. Both approaches must face the question of how the truth values of molecular sentences are fixed on the basis of the truth values of the components, when they are and when a component lacks truth value. Whereas either approach may invoke the method of supervaluation,⁸ the vagueness-in-language approach seemingly all but demands it. According to that approach, if the structure of sand before us definitely has all of the candidate properties compatible with the word ‘heap’, we will interact with it the same way regardless of how the semantic under-determination might be resolved, correctly calling the configuration a ‘heap’ on each such admissible sharpened variant of the word. If the configuration before us definitely has the complements of all of the candidate properties, again we will interact with it the same way, correctly denying it is a ‘heap’ regardless of the sharpening.⁹ It is nearly always noted ⁷ I proffer an account following this approach in Salmon (1981), 298–300. ⁸ van Fraassen (1966); Fine (1975). ⁹ Using supervaluation in the metalanguage, the vagueness-in-language theorist will sincerely utter, or assent to, such pronouncements as ‘There is a definite property—that of being a heap—expressed by the English word ‘‘heap’’ ’—even though such constructions in fact express precisely what the theorist rejects as not corresponding to a fact. Any such pronouncement by the vagueness-in-language theorist is thus extremely misleading (and will be revealed as such when the theorist spells out the view more thoroughly by adding ‘It is under-determined which property from among a multiplicity of properties ‘‘heap’’ semantically expresses’). The language of the present essay
Vagaries about Vagueness
135
in this connection that one highly significant advantage of supervaluation over, for example, the Kleene strong three-valued truth-tables is that the former accommodates all the classical truths of logic: If I point to a nearby borderline heap and utter ‘This is a heap’, my sentence suffers a lack of truth value. On the Kleene truth tables, the excluded-middle disjunction ‘Either this is a heap or else it is not’ suffers the same truth value gap. On the method of supervaluation, the disjunction is true—or else it is true in a new, more refined sense, ‘super-true’: no matter which of the candidate contents the word is taken to express, the disjunction comes out true (in the traditional sense). 7.2 John Hawthorne has argued that the phenomenon of vagueness provides for a short refutation of the theory of direct reference for proper names.¹⁰ One example from Lewis may serve as illustration.¹¹ There are two municipalities in New Jersey both called ‘Princeton’: Princeton Borough and the surrounding Princeton Township (not counting nearby Princeton Junction, on the rail line between New York and Philadelphia). Residents of either municipality specify their city in their address simply as ‘Princeton, New Jersey’. According to Lewis, the appellation ‘Princeton, New Jersey’ is semantically under-defined in the same way as ‘heap’. It is semantically determined that the appellation designates a part of New Jersey, but it is not settled which part. On one admissible resolution of the indeterminacy—on one sharpening or ‘precisification’—the appellation designates the borough, on another the fusion of borough and township (not a municipality).¹² The sentence ‘Princeton, New Jersey is a borough’ is true on some admissible sharpenings, false on others; it is neither super-true nor super-false. But ‘Princeton, New Jersey is in New Jersey’ is every bit as analytic as ‘Either Princeton, New Jersey is a borough or it is not’, since in all admissible models it is true on all admissible sharpenings. Hawthorne argues as follows. Consider someone, Godfrey (Hawthorne chooses God), who knows the fact expressed by ‘Princeton Borough is a borough’. The directreference theory embraces the following schema, where α is any designating proper name: DR: Godfrey knows that α is thus-and-so, if and only if α is something Godfrey knows is thus-and-so. is a metalanguage for vague object languages. It is my metalanguage, and I herewith stipulate that its sentences are not to be supervaluated. Any vagueness-in-language sympathizer who obstinately persists in supervaluating engages in deliberate misinterpretation (which is not nice). Supervaluation will be criticized below. ¹⁰ Hawthorne (2005), 8–9. Hawthorne’s objection to direct reference is presented in passing. ¹¹ Lewis (1988), 128. ¹² The resulting ‘semantic indecision’ with regard to the appellation, ‘Princeton, New Jersey’, arises even if (perhaps contrary to followers of the vagueness-in-language approach) there is no additional semantic indecision with regard to ‘Princeton Borough’ or ‘Princeton Township’ or ‘the fusion of Princeton Borough and Princeton Township’.
136
Nathan Salmon
The left-to-right conditional is relatively uncontroversial but not the right-to-left. Some philosophers (evidently including Hawthorne) hold that Godfrey can know, and hence believe, of Ortcutt (de re) that he is a spy without even believing, let alone knowing, that Ortcutt is a spy. Let us assume DR for a reductio ad absurdum. We are given that the following sentence is true: (1) Godfrey knows that Princeton Borough is a borough. From DR and (1) we infer (2) Princeton Borough is something Godfrey knows is a borough. (Notice that this application of DR invokes the uncontroversial left-to-right conditional.) It will be generally agreed that the sentence S: Princeton, New Jersey is a borough. is not simply true (i.e. not super-true), since on some admissible sharpenings it is false. What is not true is not known. Hence, (3) Godfrey does not know that Princeton, New Jersey is a borough. Applying DR to (3) yields (4) Princeton, New Jersey is not something Godfrey knows is a borough. It follows from (2) and (4) by a suitably restricted form of Leibniz’s Law that (5) Princeton, New Jersey = Princeton Borough. But (5) is false on one admissible sharpening, and hence not super-true. Every transition in the derivation seems unobjectionable except one: the move from (3) to (4). This transition invokes precisely the controversial half of DR.¹³ Hawthorne concludes, ‘it thus seems that considerations of vagueness, as applied to proper names, provide compelling grounds to give up’ DR (9). Evidently, in particular, we must give up the controversial right-to-left conditional—not because we have uncovered a false instance (although some believe there are such), but rather because we have uncovered an instance that, through relatively uncontroversial considerations of vagueness, is seen to be un-super-true (and un-super-false): If Princeton, New Jersey is something Godfrey knows is a borough, then Godfrey knows that Princeton, New Jersey is a borough. Hawthorne is sufficiently wedded to the vagueness-in-language approach that he does not pause to consider whether the alternative vagueness-in-the-world approach calls this objection into question. Before considering whether it does, let us consider (3) on the vagueness-in-language approach more thoroughly. ¹³ Hawthorne uses a significantly more complicated example, also from Lewis, in which the term corresponding to our use of ‘borough’ is also relevantly vague. Although Hawthorne’s derivation is more prolix, I believe the derivation presented here captures his intent. (Hawthorne could have made his argument against DR confining his observations simply to the inference from (3) and DR to (4).)
Vagaries about Vagueness
137
As Hawthorne realizes, he cannot legitimately build it into the example by hypothesis that Godfrey does not know that Princeton, New Jersey is a borough (for example, because Godfrey does not even believe it). If the appellation ‘Princeton, New Jersey’ is semantically under-defined, this fact will manifest itself in our evaluation of (3). Hawthorne believes (3) must be true since its complement clause, S, is vague in a way that prevents it from being true on all admissible sharpenings, and epistemology tells us that (3) is true if S is not. But this manner of argumentation flies in the face of the very method of valuation (i.e. of assigning truth values to complex sentences) that the vagueness-in-language approach employs. With supervaluation what matters are not the truth values of the components per se, but the truth value of the whole on all admissible truth valued sharpenings of the components. And on at least one such sharpening specifically mentioned by Lewis, (3) is false given that (1) is true—or at least, so the direct-reference theorist can be expected to argue. Moreover, the supervaluationist may consistently deem both S and (3) neither super-true nor super-false while accommodating the epistemological truism, If Godfrey knows that Princeton, New Jersey is a borough, then Princeton, New Jersey is a borough, since it is true on all admissible sharpenings. Hawthorne considers this response (or something evidently nearly identical to it) and dismisses it as excessively weak. He argues for (3) precisely from the vagueness, and consequent non-super-truth, of S. Regarding the claim that (3) instead suffers a lack of truth value due to vagueness if S does, he says, ‘That diagnosis seems less plausible, refusing as it does to acknowledge that vagueness brings a lack of knowledge in its wake’ (9). 7.3 It must be granted that one cannot know what is untrue; a fortiori one cannot have knowledge of a proposition that is neither true nor false. For example, given that France is not presently a monarchy, one cannot have knowledge that the present king of France is bald, since it is untrue. But which untrue proposition is it that, according to (3), Godfrey does not know? We are here taking seriously Lewis’s idea that ‘Princeton, New Jersey’ is under-defined. If it is, then S inherits the appellation’s semantic under-determinacy. And if S is semantically under-determined, then (3) is as well. On Lewis’s view, it is ‘not settled’ which proposition S semantically expresses as its semantic content from among a set of candidate contents, some of them true, some false. This makes for three possibilities concerning S’s semantic content on Lewis’s account: (i) S is ambiguous among the candidate contents; (ii) S does not express any of the candidate contents (and so does not express any proposition at all); and (iii) there is no fact of the matter concerning whether S expresses any particular candidate content. I find (i) the most plausible of these as an account of S. But on (i) S is semantically over-determined, not under-determined. On this possibility, S has both truth values, not neither. (It has only one truth value per reading.) Plausible though it may
138
Nathan Salmon
be as an account of S, (i) is evidently incompatible with the vagueness-in-language approach.¹⁴ Possibility (ii) seems to capture the original idea that the task of assigning a semantic content to ‘Princeton, New Jersey’ was left unfinished. For if the process was left unfinished, then the task was not performed and nothing was assigned as content. If nothing was assigned as content, then nothing is the content. But recall that the schema DR is restricted to proper names α that designate. All bets are off if α does not designate.¹⁵ Lewis’s talk of ‘semantic indeterminacy’ and ‘semantic indecision’ strongly suggests that he had in mind (iii) rather than (ii). Possibility (iii) will be discussed further in §V. Here it is sufficient to note that a similar problem arises on both of (ii) and (iii). In either case—whether S expresses none of the candidate contents or there is no fact of the matter—we lack the motivation sketched above for (3). For there is then no fact that the proposition that Princeton, New Jersey is a borough is untrue. There is only a fact that the sentence S suffers a lack of semantic truth value. The gap is due not to the fact that the proposition expressed is untrue—there being no such fact—but to a more radical kind of failure: either no proposition or no fact of the matter which. On the vagueness-in-language approach this provides a much more plausible motivation for the thesis that (3) is neither super-true nor super-false—at least given that Godfrey knows that Princeton Borough is a borough. It is instructive to consider the transformation of Hawthorne’s argument under the replacement of ‘Godfrey knows’ with ‘it is true’, where the latter is interpreted as a predicate for metaphysical as opposed to semantic truth, i.e. truth of a proposition as opposed to truth of a sentence. The analogue of DR is the following: DR : It is true that α is thus-and-so, if and only if α is something that truly is thusand-so, where as before, α is any designating proper name. This principle seems beyond reasonable doubt. It justifies the transition to (2 ), ‘Princeton Borough is something that truly is a borough’. On the other hand, if Lewis’s vagueness-in-language account of ‘Princeton, New Jersey’ is correct, then nothing can justify (3 ) It is not true that Princeton, New Jersey is a borough. The fact that S is false under some admissible sharpenings directly yields only (3 ) The English sentence S is not super-true. An additional premise is required to move from this to (3 )—perhaps, ‘The English sentence S semantically expresses the proposition that Princeton, New Jersey is a borough’. But on Lewis’s account of ‘Princeton, New Jersey’, no such premise could be true. Instead, (3 ) should be counted neither super-true nor super-false. Similarly for (3). ¹⁴ I find the claim that ‘Princeton, New Jersey’ simply and unambiguously designates the fusion of borough and township not in the least implausible. This is incompatible with Lewis’s account of the appellation. ¹⁵ Cf. Salmon (1998), 305–10.
Vagaries about Vagueness
139
Closer scrutiny reveals that on the vagueness-in-language approach even the sentence ‘Princeton, New Jersey is in New Jersey’ suffers a lack of truth value of some sort. For on that approach, either the sentence expresses no proposition at all or there is no fact about which proposition among several it expresses—possibilities (ii) and (iii) above. On the first alternative, the sentence cannot have a truth value. At first blush the second alternative seems to accord better with the idea that the sentence in question is true after all, since each of the candidate contents is true. Still, on this alternative, there is no fact concerning which proposition the sentence expresses. Now the semantic truth value of a sentence is defined to coincide with the metaphysical truth value of the proposition expressed. To say that a sentence is semantically true is to say that the proposition expressed is metaphysically true, and to say that a sentence is semantically false is to say that the proposition expressed is metaphysically false, i.e. that the denial of the proposition is metaphysically true. If the sentence suffers semantic indeterminacy concerning its semantic content, it is difficult to see that there can be a fact nevertheless concerning its semantic truth value, unless that fact is that there is no truth value. Lewis downplayed this feature of his theory as if it were not a difficulty.¹⁶ He wrote: ‘Super-truth, with respect to a language interpreted in an imperfectly decisive way, replaces truth simpliciter as the goal of a cooperative speaker attempting to impart information’ (ibid., 172). Granted, by uttering ‘That is a heap’ or ‘There is a prestigious university in Princeton, New Jersey’ one imparts a great deal of information. It must be noted in response, however, that imparting information is not the same thing as literally expressing the information by uttering a sentence that semantically encodes it.¹⁷ On the vagueness-in-language approach, however useful these sentences may be as instruments for pragmatically imparting information, at best it remains indeterminate what information the sentences semantically encode. Insofar as some sentences are semantically under-determined, the phenomenon of truth under all admissible semantically determinate interpretations is not the same thing as truth. Socalled super-truth falls considerably short of genuine truth; it is merely the absence of the potential to be reinterpreted, in a certain manner, as expressing a false proposition. As such, it is fake truth, a pretense more aptly termed ‘pseudo-truth’. To take an analogous albeit more extreme case, suppose a new adjective ‘glubulatory’ is about to be defined when an unforeseen circumstance aborts the attempt before any stipulative decision can be made (or even begun) except for this: that the new term shall correctly apply to Woody Allen. The yet-to-be interpreted sentence ‘Woody Allen is glubulatory’ will in the fullness of time, if our intentions are brought to fruition, come to express some truth or other, though it is not yet settled which. ¹⁶ Hawthorne appears unaware of the difficulty. He uses a concept of definiteness for propositions, characterized (ibid., 3–4) in such a way that, allegedly, semantic ascent may be avoided by saying of the propositions expressed (‘meant’) by (particular utterances of) ‘Princeton, New Jersey is in New Jersey’ and ‘Princeton, New Jersey is a borough’ that they are, respectively, definite and indefinite—not the sentences themselves but their semantic contents (whatever that means). Which propositions are those? (Although Lewis sometimes also spoke this way, he did not explicitly regard it as a way of avoiding semantic ascent by speaking about propositions instead of expressions.) ¹⁷ Cf. Salmon (1986b), 58–60 and passim.
140
Nathan Salmon
It can hardly be said that the sentence is already true; it does not yet even mean anything. Very well, let us introduce a new honorific term, ‘super-duper-deluxe-true’, to apply to any sentence that either expresses a truth or is as yet uninterpreted but, if all goes according to plan, will someday come to express something that is true. The sentence ‘Woody Allen is glubulatory’ is thus super-duper-deluxe-true. This sounds like something very special, but it is in fact of no semantic significance whatsoever, since the sentence is meaningless. Things are no better if there is not even a fact of the matter concerning what a given sentence means. So it is with so-called supertruth. It makes no difference, for example, whether the conclusion of an argument is true under all admissible sharpenings under which the premises are true, if a premise is semantically under-determined to begin with. Such an argument offers no genuine support for its conclusion. A sentence that is semantically under-determined but comes out true under this reinterpretation or under that, or even under all admissible semantically determinate interpretations (i.e. is pseudo-true), is—when all is said and done—a sentence that is semantically under-determined. Calling the sentence truein-a-different-sense conceals, but does nothing to change, the fact that there is either no truth value or no fact concerning truth value. Obfuscation is a blunt philosophical scalpel. In fact, the vagueness-in-language theorists’ reliance on supervaluation suggests that they may have neither possibility (ii) nor possibility (iii) in mind after all but (i).¹⁸ Maybe the vagueness-in-language approach is vague, and there is no fact of the matter concerning which possibility the vagueness-in-language theorist has in mind. (Derridians, who thrive on vagueness, will probably conclude that the vagueness-inlanguage theorist has all three possibilities in mind and he has none of them in mind.) Problems for the vagueness-in-language approach do not end there. The excludedmiddle disjunction ‘Either Princeton, New Jersey is a borough or it is not’ should suffer a lack of truth value of the same sort. For on the vagueness-in-language approach, this disjunction too either expresses no proposition at all or else is semantically indeterminate with regard to content. On the first alternative, it clearly has no truth value. On the second, either there is no fact concerning truth value or there is such a fact, which is that there is no truth value.
7.4 There is a closely related flaw in the general method of supervaluation—whether it is applied to compound sentences with components straightforwardly neither true nor false, to ambiguous sentences, to sentences with content-less components, or to semantically under-determined sentences.¹⁹ What is nearly universally cited as a significant advantage of supervaluation is in fact a serious disadvantage. For linguistic intuition dictates that whenever an English disjunction is acknowledged as definitely true (for example, ‘Either I worked late last night and that is why I came home late, ¹⁸ Cf. Lewis (1993), 180.
¹⁹ Cf. Tye (1989).
Vagaries about Vagueness
141
or I got into an accident on the way home’), then it makes perfectly good sense to ask, ‘Very well, which is it?’. If the disjunction is definitely genuinely true, the correct answer will be one of: ‘the former’, ‘the latter’, or ‘both’. (Likewise, whenever an English conjunction is acknowledged as definitely false, then it makes perfectly good sense to ask, ‘Very well, which is not the case?’, with the same options regarding the correct answer.) Though the correct answer need not always be known, it need always exist if the disjunction is genuinely true. Contrapositively, if none of the three is a correct answer—not ‘the former’, not ‘the latter’, and not ‘both’—then whatever else the disjunction may be (e.g. pseudo-true), it is not determinately, genuinely, straightforwardly true. This is so as much for an excluded-middle disjunction as for any other (making allowance for the possibility of scope ambiguity). In particular, if Russell’s example ‘The present king of France is bald’ is (contrary to Russell) not false, then the disjunction, ‘Either the present king of France is bald or he isn’t’, is not true—precisely as Frege and Strawson held—contrary to valuation by supervaluation.²⁰ Ordinary linguistic intuition strongly supports the conclusion that excluded-middle disjunctions with non-bivalent disjuncts are not evidence supporting supervaluation, but in fact counterexamples to it. The claim that a disjunction is true though neither disjunct is, is sufficiently counterintuitive that charity virtually demands reinterpretation. (Likewise the claim that a conjunction is false though neither conjunct is.) I submit that the supervaluationist might typically mean something quite different by disjunction than what disjunction means—something with both weaker truth conditions and weaker falsehood conditions. Specifically, the supervaluationist, in uttering or assenting to a disjunction φ or ψ, might typically mean or interpret it as an abbreviation for a more complex construction, such as: Either φ or ψ, or else it is logically true that if the propositions that φ and that ψ are both truth valued, then either φ or ψ; and furthermore it is not logically true that if the propositions that φ and that ψ are both truth valued, then neither φ nor ψ.
References to logic might be replaced with references to some more specific theory or body of information (e.g. psychology). Where φ and ψ are classical sub-contraries (i.e. where Either φ or ψ is a classical analytic truth) this interpretation is true, even if neither φ nor ψ is true and one or both is neither true nor false. (Let φ be ‘The present king of France is bald’ and let ψ be the negation.) And where φ and ψ are both classical analytic falsehoods this is false, even if not both φ and ψ are false and one or both is neither true nor false. (Let φ be any inconsistency and let ψ be a non-bivalent contradiction, perhaps ‘The present king of France is bald and the present king of France is not bald’.) The other standard, dyadic, classical-logical ²⁰ Russell agreed that his excluded-middle disjunction is not true on any of the primaryoccurrence readings. Contrast this with the following dialogue: Bert: ‘Either the present king of France is bald, or else not.’ Pete: ‘Very well, which is it?’ Bert: ‘Plainly the latter.’
142
Nathan Salmon
connectives—‘if’, ‘and’, and ‘if and only if’—are then definable for the supervaluationist in the standard way in terms of ‘or’ and ‘not’, with the result that for the supervaluationist none of the classical dyadic connectives is truth-functional.²¹ Other charitable interpretations are possible. There are formidable complications in providing a charitable interpretation of the vagueness-in-language theorist’s use of disjunctions—indeed of the vagueness-in-language theorist’s use of any sentence involving a vague expression.²² The vagueness-in-language theorist would do better simply to mean by ‘or’ Kleene strong disjunction, and to come clean that compound sentences with semantically under-determined components are themselves semantically under-determined. Pseudo-truth and pseudo-falsity, by whatever name, are beside the point.
7.5 Let us consider more carefully possibility (iii): that on the vagueness-in-language approach there is no fact of the matter whether, for example, ‘Princeton, New Jersey’ designates the fusion of Princeton Borough and Princeton Township. It can be shown that on this construal, the vagueness-in-language approach, if thoroughgoing, leads to a curious infinite regress, one that renders the approach devoid of any genuine account of vagueness.²³ Let us ask the vagueness-in-language theorist: What exactly does semantic underdetermination come to? Specifically, what does it mean exactly to say that it is semantically indeterminate—that it is not ‘settled’—which object a given designator designates? If the meta-linguistic claim that ‘Princeton, New Jersey’ designates the ²¹ In uttering or assenting to a conjunction φ and ψ, the supervaluationist might typically mean or interpret it as an abbreviation for: Either both φ and ψ or it is logically true that if the propositions that φ and that ψ are both truth valued, then both φ and ψ; and furthermore, it is not logically true that if the propositions that φ and that ψ are both truth valued, then not both φ and ψ. Where φ and ψ are both classical analytic truths this is true, even if not both of φ and ψ are true and one or both is neither true nor false. And where φ and ψ are classical contraries (i.e. where Not both φ and ψ is a classical analytic truth) this is false, even if neither φ nor ψ is false and one or both is neither true nor false. Similar results obtain for the conditional and the biconditional. (On this proposal, the supervaluationist means by ‘not’ classical, three-valued, truth-functional, choice negation.) ²² The vagueness-in-language theorist, in uttering a sentence φ that invokes one or more vague expressions, might be charitably interpreted as meaning thereby the conjunction of all of (what the theorist regards as) the coordinated sharpenings of φ —where a sentential sharpening is coordinated if it sharpens related sub-sentential vague expressions compatibly with one another. Notice that this conjunction is false if any coordinated sharpening of φ is, whereas in uttering the negation ∼φ the vagueness-in-language theorist is interpreted as meaning thereby something that is true if and only if each coordinated sharpening of φ is false. ²³ I discovered after writing this chapter a closely related critique of the vagueness-in-language approach by Trenton Merricks (2001). I believe Merricks’ critique may suffer from his failing to distinguish between possibilities (ii) and (iii) and to construe the latter possibility in conformity with the characteristic feature of the vagueness-in-language approach: that there is no shortage of facts whereas the language fails to capture them. As will be obvious to Merricks’ readers, however, I am very much in sympathy with the general spirit of Merricks’ insightful discussion.
Vagaries about Vagueness
143
municipality-fusion suffers from semantic under-determination, and if vagueness-inlanguage is thoroughgoing, this semantic under-determination cannot be attributed to some deficiency in the designation relation itself (designation-in-English, for example). To do so would be to locate vagueness in objects and their attributes—in the municipality-fusion and the property of being designated in English by the appellation ‘Princeton, New Jersey’. This would be to renege on a campaign promise: No predication without correspondence. According to the thoroughgoing vagueness-inlanguage point of view, for any relation R between expressions and objects there is always a fact whether a given expression bears R to a given object or instead bears the complement non-R. To each appropriate pairing of attributes with objects corresponds a fact. It may seem, ironically, as if this tenet in itself in fact precludes possibility (iii), since for every semantic relation and for every pairing of an expression with a potential semantic value, there is always a corresponding fact either that the expression bears the relation to the potential semantic value or that it bears the complement.²⁴ Every purely semantic question then has a correct answer. But to conclude that this requires a fact whether ‘Princeton, New Jersey’ designates the municipality-fusion is evidently to misunderstand how vagueness is supposed to arise according to a thoroughgoing vagueness-in-language approach, assuming possibility (iii). On that approach, there is no fact concerning whether ‘Princeton, New Jersey’ designates the municipality-fusion not because of a missing fact—all the semantic facts are in place—but because of semantic indecision with regard to some expression, or expressions, in predicates like ‘designates the fusion of Princeton Borough and Princeton Township’. In this case the indecision is evidently meta-semantic, located in semantic terms like ‘designate’, ‘refer’, ‘denote’, ‘stands for’, ‘extension’, etc. (See note 12.) Some designation-like relations obtain between ‘Princeton, New Jersey’ and the fusion while the complements of other designation-like relations do so, and it is not settled which candidate designation-like relation ‘designate’ metadesignates, i.e. it is not settled exactly which semantic relation the word designates in the metalanguage. Now if there is indeterminacy whether ‘designate’ meta-designates a particular designation-like relation R, this also cannot be attributed to a deficiency in the meta-designation relation, else again we lapse into vagueness-in-the-world, a relation and appropriate objects with no corresponding fact. Instead the indeterminacy must be located in the word ‘meta-designate’ (or in the phrase ‘designate in the metalanguage’, etc.). But if there is indeterminacy whether ‘meta-designate’ meta-metadesignates a particular meta-designation-like relation, this cannot be attributed to a deficiency in the meta-meta-designation relation and must instead be located in the word ‘meta-meta-designate’, and so on. Hence, if any object-language expression suffers from semantic under-determination, the entire hierarchy of semantic expressions—‘designate’, ‘meta-designate’, ‘meta-meta-designate’, etc.—is infected with indeterminacy all the way up. This infinite hierarchy of vagueness may not seem especially problematic, until one recalls our original question: What exactly does semantic under-determination ²⁴ Merricks so concludes (op. cit., 150–1).
144
Nathan Salmon
amount to? We are told that there is no fact of the matter whether ‘Princeton, New Jersey’ designates the fusion of Princeton Borough and Princeton Township but that this is due to a deficiency of language (the appellation ‘Princeton, New Jersey’), not of the world. When we ask what the semantic under-determination of ‘Princeton, New Jersey’ amounts to then, we are told it is due to semantic underdetermination with regard to the semantic word ‘designate’. The attempt to explain semantic under-determination does not reduce the phenomenon to a ‘previous case’; instead it ‘reduces’ it to a new case: semantic under-determination of ‘designate’. When we ask what the semantic under-determination of ‘designate’ amounts to then, we are told it is not some deficiency in the putative designation relation, but rather semantic under-determination with regard to ‘meta-designates’. When we ask what this last amounts to, we are told it amounts to semantic under-determination with regard to ‘meta-meta-designates’, and so on. At no stage in our deduction is any account ever offered, only a ‘reduction’ to a new case. At each stage, the explanation passes the buck to the next stage up. This is analogous to each commander in a chain of command delegating responsibility to a subordinate. Only here there are no buck privates; the buck never stops, the responsibility is never met. In short, assuming possibility (iii) the vagueness-in-language approach, if thoroughgoing, is no approach at all. 7.6 One sort of thing that would count as an explanation of semantic underdetermination is this: Contrary to a thoroughgoing vagueness-in-language ‘account,’ there is a unique designation relation (for English), but for some pairs of expressionand-object—in particular, with regard to the pair, ‘Princeton, New Jersey’ and the municipality-fusion—there is no fact of the matter whether the first bears the designation relation to the second or instead its complement, non-designation. There is the municipality-fusion and there is the property of being designated by ‘Princeton, New Jersey’, but there is no fact whether the first has the second or instead has its complement. There is no fact whether the fusion is designated by the appellation. This reduces vagueness-in-language to a special case of vagueness-in-the-world, the world of objects and linguistic attributes. The issue of whether the appellation designates the fusion is a linguistic matter of which there is no fact. There is predication without correspondence after all. This is what the vagueness-in-language approach eventually comes to, and, I suspect, this is the very position that Lewis and company adopt. Although vagueness ultimately traces to a paucity of facts, the vaguenessin-language theorist will take solace in that the missing facts are always linguistic in nature. (See again note 4.) There is vagueness in the world, it is reluctantly conceded, but only because language is a part of the world (even if philosophers are wont to pretend otherwise). The metaphysically tidy account of indeterminacy as being due to a misfit between language and the world rather than to a worldly shortage of facts is seriously compromised. It is the fusion of Princeton Borough and Princeton Township—an extra-linguistic object—that now falls short of definitely having a particular property
Vagaries about Vagueness
145
or instead its complement. This is predication without correspondence. It would be mysterious, in fact it would be downright weird, if the municipality-fusion falls through the cracks with regard to certain linguistic attributes but not with regard to any non-linguistic attributes whatsoever—attributes of shape, pulchritude, climate, and so on. If there is an object, an appellation, and a designation relation but no corresponding fact whether the appellation stands in the designation relation or its complement to the object, one should expect that there can just as easily be a non-linguistic property—being a heap, being bald, being an adult, being pretty, etc.—and no corresponding fact whether a given object definitely has that property or instead its complement. Once vagueness is imputed to the world of objects and their attributes at some level, the masquerade party is over. Vagueness might as well be acknowledged at Level One—the level of extra-linguistic objects and their non-linguistic attributes.²⁵ In fact, once one countenances the prospect of objects and linguistic attributes but no corresponding fact, other things being equal it is best simply to face the reality of such non-linguistic properties as those of being bald and being pretty (and not merely pretend to do so by going through the motions—see note 9). If there is such a property as that of being bald and Godfrey is a borderline case, then there is no fact about whether he is bald but there is a fact about which proposition is expressed by ‘Godfrey is bald’, and there is also a fact about which truth value it has: neither. We know what Hegelians, who love a synthesis, will probably conclude about Godfrey. Those of us less enamored of a synthesis should seriously consider abandoning vagueness-inlanguage in favor of vagueness-in-the-world. The most serious problem with the vagueness-in-language approach—and the most powerful consideration favoring the vagueness-in-the-world approach—is that the former approach imputes to any sentence involving a vague expression a failure with regard to semantic content—either no proposition expressed or, at best (even though more radical), no fact of the matter concerning which proposition is expressed. The great bulk of any human language suitable for everyday discourse (as opposed, for example, to the language of pure mathematics) is shot through and through with vague expressions. It is remarkably difficult in fact to provide clearcut examples of English general terms (common nouns, adjectives, or verbs) that do not admit at least the possibility of a borderline case. It is a consequence of the vagueness-in-language approach that the vast majority of sentences that make up natural language—including those sentences that the approach deems ‘super-true’ or ‘super-false’—suffer the relevant sort of failure of content (either no content or, more likely, no fact of the matter). This consequence of the approach is quite extreme. The continued popularity of the approach despite its strikingly radical commitments is a remarkable phenomenon in itself, one that undoubtedly reveals more about the discipline of philosophy than about the phenomenon of vagueness. I argued that if vagueness is imputed at Level Two, it should be imputed at Level One. The converse is not so. Once vagueness is imputed at Level One, there is no longer any rationale for imputing vagueness at Level Two, or at least not the same ²⁵ Cf., Merricks, ibid., 156.
146
Nathan Salmon
rational as before. Borderline cases of heaps are fully accommodated by recognizing the property of being a heap, and recognizing that borderline heaps neither definitely have this very property nor definitely have its complement. To recognize this much does not require us to turn our backs on the many heap-like properties postulated by the vagueness-in-language approach. We may let a thousand flowers bloom. We simply acknowledge the existence of a pivotal, user-friendlier property—being a heap—and the resulting prospect of occasional singular propositions without truth value, occasional predication without corresponding facts. This misfit is not between language and propositions (or between language and the objects and attributes that make up the facts); it is between the propositions and the facts. There simply are not enough facts to adjudicate between every proposition and its anti-twin. The vagueness-in-the-world approach imputes indeterminacy to the bearing of attributes by objects, not to the objects that are indeterminate with respect to those attributes. In a borderline case of a property there is the property and a definite object but no corresponding fact. Objects are not vague or indeterminate. Of course, some objects—ordinary physical objects, for example—have indeterminate boundaries. It might also be indeterminate whether a given object exists, e.g. as it is fading away into nonexistence. Both of these phenomena are cases of indeterminacy among an object’s properties, not indeterminacy of the object itself. An object itself is just the thing that it is, and as Bishop Butler astutely observed, not another thing. An object is not the bundle of its qualities; it is that which has the qualities. The object is not it-withsuch-and-such-properties. It is the very object itself, without even the clothes on its back. (Some objects have better appearances than others.) An object’s properties are not the object itself, or even a part of the object. The object has whatever properties it has, lacks whatever properties it lacks, and is indeterminate with respect to whatever properties it is indeterminate with respect to. Names for objects either definitely designate or they definitely do not. A name for an object whose boundary is fuzzy designates an object with a fuzzy boundary. A name for an object that does not exist (e.g. ‘Socrates’) designates a nonexistent object (Socrates).²⁶ A name for an object whose present existence is questionable presently designates an object of questionable ontological status, but the object in question is just what it is, nothing more and nothing less. In short, objects themselves are not vague and neither are their names. What is indeterminate is whether certain objects bear certain attributes or their complements.²⁷ What, then, of Hawthorne’s argument against direct reference? The phrase, ‘the New Jersey municipality called ‘‘Princeton’’ ’, is an improper definite description, ²⁶ Cf. Salmon (1998), 286–7. ²⁷ One attribute that is not subject to indeterminacy of this sort is strict, numerical identity—the equivalence relation that any object bears to nothing other than itself. I provide proof in Salmon (1981), at 241–5 and 338–43. For a very similar but also significantly different argument, see Evans (1978). Both Evans’ argument and mine have come under severe criticism, the most developed of which is Parsons (2000). I provide a comparison and adjudication between Evans’s argument and my own, as well as a response to Parsons and other critics, in Salmon (2002).
Vagaries about Vagueness
147
since there are two such municipalities. But it would be draconian to declare the appellation, ‘Princeton, New Jersey’, therefore non-designative. In any case, if the appellation is non-designative, DR is inapplicable. Perhaps the appellation is ambiguous. Certainly a speaker can use it to designate the borough, or the township, or their fusion. If the appellation is ambiguous, then so are sentences that employ it, like S. The readings of the appellation on which (3) is true in Hawthorne’s example are readings on which (4) and (5) are true as well. Perhaps ‘Princeton, New Jersey’ officially designates the fusion, end of story. In this case (5) is true, end of story. (This is the option I find most plausible of the three; see note 14.) Whether the appellation is non-designating, ambiguous, or univocal, Hawthorne’s reductio of direct reference ad absurdum fails—although which is the false step depends on a verdict concerning a purely semantic issue.²⁸ Re f e re n c e s Fine, Kit (1975), ‘Vagueness, truth, and logic’, Synthese 30, 265–300. Evans, Gareth (1978), ‘Can there be vague objects?’ Analysis 38, 208. Hawthorne, John (2005), ‘Vagueness and the mind of God’, Philosophical Studies 122, 1–25. Lewis, David (1988), ‘Vague identity: Evans misunderstood’, Analysis 48, 128–30. (1993), ‘Many, but almost one’, K. Campbell, J. Bacon, and L. Reinhardt, eds, Ontology, Causality, and Mind: Essays on the Philosophy of D. M. Armstrong, Cambridge, Cambridge University Press; reprinted in Lewis (1999), 164–82. (1999), Papers in Metaphysics and Epistemology, Cambridge, Cambridge University Press. Merricks, Trenton (2001), ‘Varieties of vagueness’, Philosophy and Phenomenological Research, 62(1), 145–57. Parsons, Terence (2000), Indeterminate Identity: Metaphysics and Semantics, Oxford, Oxford University Press. Salmon, Nathan (1981), Reference and Essence, Amherst, NY, Prometheus Books. (1986a), ‘Modal paradox: Parts and counterparts, points and counterpoints’ in Peter French, Theodore Uehling, Jr., and Howard Wettstein, eds., Midwest Studies in Philosophy XI: Studies in Essentialism, Minneapolis, University of Minnesota Press, 75–120; reprinted in Salmon (1981), 273–344, at 343–4. (1986b), Frege’s Puzzle, Atascadero, CA, Ridgeview, 1986. (1998), ‘Nonexistence’, Noˆus 32(3), 227–319; reprinted in Salmon (2005), 50–90. (2002), ‘Identity facts’ in C. Hill, ed., Philosophical Topics 30(1) (Spring), 237–67; reprinted in Salmon (2005), 165–91. (2005), Metaphysics, Mathematics, and Meaning, Oxford, Oxford University Press. Sorensen, Roy (1986), Blind Spots, Oxford, Oxford University Press. Thomason, Richmond (1982), ‘Identity and vagueness’, Philosophical Studies 42, 329–32. ²⁸ The vagueness-in-the-world approach offers a simple, straightforward, and I believe obviously correct diagnosis of sorites arguments: They are perfectly valid but unsound. The inductive premise, i.e. the major premise For every n: If F (n), then F (n ± 1) (e.g. ‘The result of removing a single grain from a heap of sand is still a heap’), is not false. In fact, the vast majority of its instances are true. But not all are. Specifically, each of the conditionals whose antecedent or consequent is about a borderline case is neither true nor false. The inductive claim itself is also therefore neither true nor false. For more detail, see Salmon (1986a), especially 298–9, 330 of Salmon (1981).
148
Nathan Salmon
Tye, Michael (1989), ‘Supervaluationism and the law of excluded middle’, Analysis 49(3), 141–3. Williamson, Timothy (1994), Vagueness, London, Routledge. van Fraassen, Bas C. (1966), ‘Singular terms: Truth value gaps, and free logic’, Journal of Philosophy 63, 481–95.
8 Vagueness, Metaphysics, and Objectivity Stewart Shapiro
One much-discussed issue concerns the metaphysical nature, or the source or cause, of vagueness. Is vagueness a purely linguistic matter, concerned (merely) with how the world gets represented, via language, or is there a sense in which the world itself is vague? Is the vagueness of, say, baldness or the boundaries of a given mountain, a fact about the world, as it is, in itself, independent of human schemes of representation? Or is the vagueness of baldness and mountains merely a function of how we represent the world? The purpose of this chapter is to argue against the intelligibility of the question of metaphysical vagueness, at least if it is put in such a straightforward and perhaps naive manner. Reflection on the origins and nature of vagueness in language suggests that the phenomenon is due to both the way the world is and the way that we, the users of language, and our faculties of representation are, and that there is no sense to separating those factors and attributing vagueness to one of them exclusively. This depends on some issues concerning metaphysics or, better, meta-metaphysics. So let us begin with those. 8.1
W H I T H E R M E TA PH Y S I C S ?
There are a number of competing philosophical traditions that have it that, in one way or another, there is no way to sharply separate the ‘human’ and the ‘world’ contributions to our theorizing. The idea goes back at least to the saying attributed to Protagoras: ‘man is the measure of all things’. Extreme versions of idealism, not to mention some postmodern perspectives, give the world itself a human character, holding that the universe is somehow shaped by our judgements, observations, etc. A more modest outlook is Kant’s doctrine that the ding an sich is inaccessible to human inquiry. Against idealism and what would later be postmodern views, Kant agrees that there is a ding an sich, but he insists that we humans can only approach the world through our categories, concepts, and intuitions; and it is not possible to get beyond those, to the world as it is, independently of said categories, concepts, and intuitions. In short, there is no way to get at the world independently of how we get at the world. Among contemporary philosophers, a widely held view, championed by W. V. O. Quine, (some temporal stages of) Hilary Putnam, Donald Davidson, 149
150
Stewart Shapiro
and John Burgess (Burgess and Rosen (1997)), is that there is, to use a common phrase, no God’s eye view to be had, no perspective from which we can compare our best theories of the world to the world itself, to figure out which are the ‘human’ parts of the theories and which are the world parts, getting at things as they are, independently of how we get at them. As Burgess and Rosen (1997, 240) put it, ‘our theories of life and matter and number are to a significant degree shaped by our character, and in particular by our history and our society and our culture’. To be sure, this is not an idealism or a postmodern perspective. Burgess and Rosen do not declare that the world itself is somehow shaped by ‘our character’. It is our theories of the world that are so shaped. The view in question is that there is no way to sharply delineate the components, to get at the world, as it is, independent of human conventions, forms of representation, and the like. Let us call the underlying perspective, or perspectives, here the Kant–Quine thesis. The Kant–Quine thesis allows that there are, or may be, some features of a world view that are on or closer to the ‘human’ side and some features that are on or are closer to the ‘world’ side of the mix. The fact that the Hebrew words for ‘man’, ‘red’, and ‘ground’ have the same root does not say anything at all about people, colors, and the earth. That connection is on the human/linguistic side. And the fact that the earth spins on its axis has nothing to do with human judgements, linguistic conventions, etc. The Kant–Quine thesis is that there is, in general, no way to sharply separate the human and the world aspects to our theorizing. It is not a repudiation of the very notion of objectivity. Let us take a brief look at an ‘exchange’ between Gary Ebbs (1997) and Mark Wilson (2006). Ebbs articulates some features of some views opposed by the Kant–Quine thesis: The idea behind metaphysical realism is that we can conceive of the entities and substances and species of the ‘external’ world independently of any of the empirical beliefs and theories we hold or might hold in the future. To accept this picture, we must conceive of the relationships between our words and the ‘external’ world from an ‘external’ perspective. We must imagine that we can completely distinguish between what we believe and think about the things to which we refer, on the one hand, and the pure truth about these things, on the other. In this imagined ‘external reality’, things, species, and substances are individuated by their own natures or constituting principles. This picture generates questions about what these principles of individuation are, and thus drives philosophers to theorize about the metaphysical structure of the things, species, and substances in the ‘external’ world. (Ebbs 1997, 203)
After quoting this passage, Mark Wilson (2006, 79), notes that when ‘metaphysical realism’ is put in such ‘sweeping terms, [it] certainly sounds . . . foolish’. However, he asks us to contemplate just what Ebbs is saying. What, exactly, is the content of what I am calling the Kant–Quine thesis (or theses), and what is the content of the opposing ‘metaphysical realism’? Wilson invites us to consider a more mundane instance of the ‘metaphysical’ view, presenting a parody of Ebbs’s passage: The idea behind metaphysical realism is that we can conceive of rabbits and their liking for carrots independently of any of the empirical beliefs and theories we hold or might hold about
Vagueness, Metaphysics, and Objectivity
151
such mammals and their vegetative preferences in the future. To accept this picture, we must conceive of the relationships between our words and rabbits from an ‘external’ perspective. We must imagine that we can completely distinguish between what we believe and think about rabbits and their favorite foods, on the one hand, and the pure truth about these issues, on the other. In this imagined ‘external reality’, rabbits and their affection for carrots are individuated by their own natures or constituting properties [quite independently of our thoughts] [sic] This picture generates questions about what these principles of individuation are, and thus drives philosophers to theorize about the rabbits and food preferences of the ‘external world’. (Wilson 2006, 79)
Wilson says that he ‘utterly fail[s] to see what is odd about this position, except that the task of ‘‘theorizing’’ about rabbits and their favorite foods seems more the prerogative of animal husbandry than philosophy.’ By focusing on this particular case, instead of the global claims of ‘metaphysical realism’, Wilson claims to be expressing little beyond the banal observation that rabbits (at least in the wild) pretty much go about their own business, independently of how we happen to think about them. I think we should be loath to blithely abandon our commonsensical assumption that we can sort out such issues of conceptual contribution to our ‘rabbit’ talk quite crisply (although doing so in other kinds of circumstance may require a good deal of strenuous scientific investigation). (Wilson 2006, 80)
All of this sounds quite sensible, with the possible exception of Wilson’s use of the word ‘quite crisply’. The Kant–Quine thesis does not, or at least need not, undermine the notion of objectivity. Of course rabbits (in the wild) have their preferences independently of how we theorize about them, although, in this case, one might complain of the anthropomorphism involved in words ‘liking’ and ‘preference’. To repeat the above example, surely the earth spins on its axis independently of how we talk about it and theorize about it. No danger of anthropomorphism there; just commonsense objectivity—whatever that comes to. As indicated by the parenthetical remark at the end of the just-quoted passage, it is also a truism that hard insights won by science shed light on which features of our experience and our justified beliefs are due to objective facts about the world, and which features are tied to our resources—to the particular ways that we manage to get at the world. We know a lot, for example, about what is responsible for our perceptions and judgements concerning color and sound, although a whole host of interesting philosophical issues are broached along the way, issues which would take us too far afield. None of this undermines the Kant–Quine thesis, which, once again, is that there is in general no way to separate the human and the world side to our theorizing. Wilson’s comment that the ‘sweeping terms’ of Ebbs characterization of ‘metaphysical realism’ sound ‘foolish’ seem to me to represent an insight. To be sure, that fact (if it is a fact) that a statement of a certain view sounds foolish does not constitute an argument against the view, nor one for the opposing Kant–Quine thesis. Another item that would be required to further shed light on the general, ‘sweeping’, metaphysical issue would be an articulation of the notion of objectivity which,
152
Stewart Shapiro
again, is an entrenched part of common sense, and of scientific practice. What, exactly, is it for a given statement, belief, or theory, to be true or false independently of human judgements, conceptual/representational schemes, forms of life, or whatever? Rather than try to discharge this obligation here, which would again take us pretty far afield—well beyond any competence I might claim—let me turn to the issue of metaphysical or ontic vagueness. In typical holistic fashion, I suggest that this will shed some light on the general ‘sweeping’ issues surrounding the Kant–Quine thesis, and will serve to articulate and even argue for at least part of that thesis. 8.2
W H I T H E R O N TO LO G I C A L VAG U E N E S S ?
Putnam is credited with a practice of writing words in small capitals when one means to, as Terence Horgan (1994, 99) put it, ‘be talking about denizens of the mind-independent, discourse-independent, world’. According to this convention, words written in small capitals are metaphysically loaded, and thus not subject to irrealist reinterpretation or explanation. A Berkeley-style idealist, for example, would agree with the vulgar that there are baseballs, but he would insist that there are no baseballs. The use of small caps is an attempt—successful or otherwise—to refer to the ultimate constituents, the underlying fabric, of reality, as it is, independent of any attempt to describe or apprehend it. Of course, this literary device presupposes that we can somehow talk about, or at least refer to, the ultimate constituents of the world as they are, independently of language, conventions, form of life, etc.—despite the obvious fact that we have to use language to do this talking. I do not know the extent to which the claim underlying a small caps pronouncement is to go beyond the commonsense objectivity noted in the previous section. I also do not know if the small caps convention is supposed to presuppose the falsity of the Kant–Quine thesis, nor do I assume that the attempt to refer to the world in this way is ever successful. Nevertheless, I propose to go along with the Putnam convention, for a while, in order to state and discuss the underlying issues concerning metaphysical vagueness. The issue of metaphysical vagueness is often formulated in terms of indeterminate identity statements and (putatively) vague objects. Consider a supposedly indeterminate statement in the form a = b. There are (at least) two ways to understand it. One is that there are these objects, a and b, and it is indeterminate whether those very objects are identical or distinct. On this metaphysical reading, the singular term ‘a’ and the singular term ‘b’ each fulfils its task of denoting one particular object. The identity statement concerns the object or objects so denoted. The identity statement is indeterminate if these (two?) objects are a borderline case of the identity relation. The other interpretation of the indeterminate identity statement is more directly linguistic, or semantic. On this second reading, it is indeterminate which object ‘a’ denotes and/or it is indeterminate which object ‘b’ denotes. The terms might denote the same object, and they might denote (determinately) different objects. As Mark Sainsbury [1994, §5] puts it, in this case there is ‘vagueness concerning
Vagueness, Metaphysics, and Objectivity
153
what, if anything, our words denote’. On our second reading, the blame for the indeterminacy of the statement a = b is placed on the reference relation, and not on the world. This is (mere) semantic, representational, or linguistic vagueness. I’ll use those terms interchangeably in what follows. Sometimes a similar distinction is made for vague predicates and/or properties. Let Charles be a borderline case of ‘thin’. On most account of vagueness, the sentence ‘Charles is thin’ is indeterminate. How are we to understand this? The metaphysical reading is that the English predicate ‘thin’ expresses (or stands for, or refers to) a single property, the property of thinness, and this very property is vague, having borderline cases, Charles being one such. An opposing, semantic thesis is that there are no vague properties. If the universe contains properties at all, they have sharp boundaries. Accordingly, since the predicate ‘thin’ is vague, it does not pick out a single property. Perhaps the word indeterminately picks out a number of properties, as some kinds of supervaluationists might put it, or perhaps the word fails to pick out any properties at all, and the sentence ‘Charles is thin’ has some more indirect truth conditions (`ala Horgan 1994). In any case, this is mere linguistic vagueness. Timothy Williamson [1994, 249] once noted that the idea of metaphysical vagueness has attracted some and repelled others. The idea attracts, because it promises to allow a rather direct relation between our vague ordinary words and the facts we use those words to describe, for example between an utterance of ‘blood is red’ and the fact that the substance blood has the property of being red. The idea repels, because it promises to forbid a complete description of all the facts in precise scientific words. Opposed metaphysical proclivities underlie the ensuing debate.
The attraction, or at least one attraction, of metaphysical vagueness is that it makes possible a straightforward model-theoretic semantics for our ordinary vague natural languages. Grammatically, words like ‘Mount Everest’ and ‘The Outback’ are proper names, and in our logic texts, singular terms are models of proper names. The role of a singular term is to pick out an object in the domain of each interpretation of the language. So, it would be nice if terms like ‘Mount Everest’ and ‘The Outback’ picked out an individual object in the world, namely Mount Everest and The Outback, respectively. Similarly, words like ‘red’ and ‘bald’ are common nouns or adjectives, and these are modeled by predicates. The role of a predicate is to pick out a property, or at least an extension in each interpretation. So it would be good if ‘red’ and ‘bald’ did so; thus the properties of redness and baldness. In other words, metaphysical vagueness allows for a straightforward account of word–world relations, or should I say word–world relations (or word–world relations)? The account is just the familiar picture we have inherited from model theory, although I should point out that model theory was developed with mathematical languages as its target, and not the messy natural languages under discussion here. Of course, stepping on a bump in a rug often causes a bump (or the same bump) to arise somewhere else. On the metaphysical picture, and putting epistemicism aside, the messiness of vagueness is located in how semantic values get determined. We end up with indeterminacy in the world: objects that are
154
Stewart Shapiro
indeterminately identical and possibly also properties that have indeterminate instances. This last, of course, is the very reason that metaphysical vagueness ‘repels’, as Williamson puts it. The view precludes ‘a complete description of all the facts in precise scientific words’. Of course, the advocate of metaphysical vagueness would regard this as a feature, rather than a bug. It is all but analytic that metaphysical vagueness precludes a description of all the facts in precise terms. If Charles is borderline bald, then (continuing to put epistemicism aside) there is a fact that the property bald applies indeterminately to Charles, and, by definition, no precise terms can express that fact. Williamson’s term ‘scientific’ is an honorific. Some people may hold that science in fact does, or should, or ideally will, employ only precise terms, and others may add that anything expressible at all can be expressed in scientific terms (of which more later). Such folks may indeed be repelled by the specter of metaphysical vagueness.
8.3
S O RT I N G I T O U T — O R T RY I N G TO : H OW W E D E S C R I B E A N D W H AT W E D E S C R I B E .
Putnam (1981, 49) once defined ‘metaphysical realism’ to be the view that ‘the world consists of some fixed reality of mind-independent objects. There is exactly one true and complete description of ‘‘the way the world is’’.’ If one accepts the first clause in this definition, the issue of metaphysical vs. linguistic vagueness is fairly clear. It comes down to whether there are any vague specimens among the fixed reality of mind-independent objects and, presumably, mind-independent properties. If there are, then there is metaphysical vagueness, if not, then not. The second clause in Putnam’s definition of metaphysical realism is of-a-piece with what ‘repels’ some thinkers against the possibility of metaphysical vagueness. According to Putnam’s second clause, the issue of metaphysical vagueness comes down to the nature of the supposed ‘one true and complete description of ‘‘the way the world is’’.’ The idea is that the world is vague if, and only if, the complete description of it is vague. If the complete description contains a vague predicate or a vague singular term, then one would think that the predicate stands for something, a vague property, or the singular term refers to something, a vague object. If, on the other hand, the one true and complete description contains no vague vocabulary, then there is no metaphysical vagueness. In that case, it would seem, we live in a crisp world. The second clause of Putnam’s definition of metaphysical realism goes to the heart of present concern, the Kant–Quine thesis. Putnam’s clause invokes the presumably semantic notion of description. What does the possibility (or impossibility) of various kinds of descriptions have to do with the metaphysical nature, the underlying fabric, of reality? According to the metaphysician, presumably, the world is one thing; what we can and cannot say about it is something else. Metaphysics aside, it is commonsense that the world is one thing, and what we can and cannot say about it is something else.
Vagueness, Metaphysics, and Objectivity
155
At the time, Putnam’s agenda was to reject what he calls ‘metaphysical realism’, in favor of some version of the Kant–Quine thesis. Suppose that the second clause in the characterization of metaphysical realism is false. That is, suppose that it is not the case that there is one true and complete description of ‘the way the world is’. Is this because of the way the world is, independently of how we describe it? Or is the negation of Putnam’s clause merely a statement about what descriptions we are capable of? In other words, to what extent is the lack of certain descriptions a function of us and our descriptive resources, and to what extent is it a function of the way the world is? Unless this can be answered, one can ponder the extent to which Putnam has characterized a metaphysical view, at least with his second clause, despite the name given to the view. Most theorists, with the exception of some epistemicists (see Williamson (1994, ch. 9)), agree that there is such a thing as linguistic, semantic vagueness. The issue concerns whether there is, in addition, a phenomenon of metaphysical vagueness. I think it fair to say that the most popular, but not universal, view is that there is no metaphysical vagueness (but see Rosen and Smith (2004) and Akiba (2004) for opposing views). When it comes to vague objects, David Lewis (1986, 212) is typical: The only intelligible account of vagueness locates it in our thought and language. The reason it’s vague where the outback begins is not that there’s this thing, the outback, with imprecise borders; rather, there are many things, with different borders, and nobody’s been fool enough to try to enforce a choice of one of them as the official referent of the word ‘outback’.
Lewis’s underlying metaphysical thesis is that the world itself consists only of crisp, non-vague objects and crisp, non-vague properties. Vagueness comes in only when we use imprecise language to describe and negotiate this world. Gareth Evans (1978) once gave a short, but influential argument against the indeterminacy of identity. The details need not detain us here. Lewis pointed out that linguistic vagueness is not the target of the Evans argument. Neither Lewis or Evans has an objection to vague statements of identity, indeterminate sentences in the aforementioned form ‘a = b’ Rather, Evans argued against indeterminate identities. The article begins thus: It is sometimes said that the world might itself be vague. Rather than vagueness being a deficiency in our mode of describing the world, it would then be a necessary feature of any true description of it.
Defenders of vague objects, and thus vague identities try to find ways to resist the ensuing argument. Sainsbury (1994) agrees with Lewis, in arguing that there is no coherent and even prima facie plausible thesis of metaphysical vagueness: ‘when we make enough concessive assumptions to have an intelligible thesis of [metaphysical] vagueness, we have a thesis which can be shown to be false by a few short lines of proof ’, namely the aforementioned Evans (1978) argument. B. Jack Copeland (1994) is even less concessive, arguing that Sainsbury has not ‘managed to find a way of giving coherent expression to the thesis of [metaphysical] vagueness’—even a view that is easily refuted.
156
Stewart Shapiro
It is worth seeing how an attempt at defending, or at least articulating, metaphysical vagueness might go. Copeland writes that there is . . . a perfectly intelligible thesis about representations that a supporter of [metaphysical] vagueness might regard as capturing part of what it is they want to say. The fundamental thought of the friend of [metaphysical] vagueness seems to be that it is not (or might not be) up to us whether to be vague or not. That is, a believer in [metaphysical] vagueness denies what I will call the crispness postulate: Beyond a certain point of sophistication the physical sciences will use only non-vague, or crisp representations of the world. To maintain that good science cannot be rid of vague representations is to say that the world constrains us to be vague even where we would most like to be precise. (Copeland 1994, § 4)
The crispness postulate is that when science has become sufficiently advanced, it will invoke only precise, non-vague predicates and singular terms. To avoid making spurious predictions, let us say that the thesis is that, in principle, science can get by, without loss, with only precise predicates and terms. Is the crispness postulate true? Thinkers as diverse as the logical positivists, Frege, and Quine have insisted that the ideal, proper, regimented, language of science—the Begriffschrift—will or should contain only precise terms. Vagueness in natural language, and in contemporary science, is regarded as a defect, to be eliminated as logic and science progress. To be sure, this is more of an attitude than a developed and defended thesis. It may be due to the advent and overwhelming success of mathematical logic. Once again, however, recall that this logic was developed to describe and explain features of the languages of mathematics, languages which, presumably, are not vague. Classical model theory works well for mathematics, and we understand that semantics very well, thanks to a few generations of work by the giants of that field. It is natural to want to apply the tool of classical model theory to all languages, and, in particular, the languages of future or finished or ideal science, and to think of our textbook formal languages as the model for regimented versions of natural language. And model theory is a branch of classical set theory, which recognizes only sets with sharp boundaries. Of course, this is not an argument in favor of the crispness postulate, but it may help explain its attractiveness—or why a view that opposes it may be found repellant. The same may go for classical logic and, perhaps, principles like Leibniz’s law of the indiscernibility of identicals, which are often put to service in metaphysical discourse, such as the aforementioned Evans argument. The intuitions behind these things may come from thinking about mathematical languages, or languages idealized to resemble mathematical languages. Or it may be wishful thinking on the part of some. Suppose that we accept the crispness postulate, for whatever reason, and supplement it with a naturalistic assumption that future, or ideal, science gives us a true and complete account of reality. We are supposing that there is no aspect of the universe that lies beyond the purview of science, and that science can get by with only precise terms. Then, it would seem, there is no metaphysical vagueness. Any vagueness that we suffer from is chalked up to the imperfect and, hopefully, temporary languages
Vagueness, Metaphysics, and Objectivity
157
that we currently employ. We will work to overcome this flaw, and, the assumptions go, we will, or at least can, succeed in overcoming the flaw. Nevertheless, the crispness postulate, alone, does not warrant this conclusion. A defender of metaphysical vagueness can maintain Sainsbury’s crispness postulate by rejecting what I just called the ‘naturalistic assumption’ that future, or ideal, science gives us a complete account of the universe. She might maintain that there are aspects of reality that this future or ideal science does not deal with, namely its vagueness. The issue might come down to what counts as ‘science’, or what counts as ‘ideal’, and I will not pursue this option any further. In any case, I do not know how someone might argue for the crispness postulate. In the history of thought, there is, of course, a trend toward sharpening terms as we go. Some of this is motivated by scientific considerations. Better and more general theories and explanations often require sharper, more precise vocabulary. In part, this is what allows them to be tested so thoroughly. Prima facie, if we want to invoke the rigor of mathematics, we need to deal with sharp terms—for that is what mathematics employs, at least for now. In line with the Kant–Quine thesis, however, the trend toward sharpening could be motivated by human considerations: we are more comfortable with sharper terms; we know what to do with them, how to rigorously formulate the theories, etc. In any case, it can hardly be denied that the tendency towards sharpening our terms has been successful. There is pressure from social and legal directions as well. We agree that children should not be able to buy cigarettes and alcohol. To come up with a workable policy of enforcing this, we need to get sharper about what counts as a child. We agree that state residents should pay less tuition, and, to implement this, we need to get sharper about what counts as a state resident. But the fact that we can, and have, sharpened our language in myriad ways does not entail that we can ever completely sharpen it, eliminating all vagueness—even if we wanted to. One might support the crispness postulate by arguing that (1) the world itself is crisp, so there is no metaphysical vagueness, and (2) if we try hard enough, we will find the right way to describe this world, as it is. Clearly, this would beg the question, both against the friend of metaphysical vagueness and against advocates of the Kant–Quine thesis. It seems to me that nowadays, the prevailing opinion is that the crispness postulate is false: vagueness is here to stay, like it or not. This could simply be a result of frustration at attempts to completely eliminate vagueness, a sort of pessimistic induction. Or it might be the result of profound reflection. Although he does not use the term, Dominic Hyde (1998) provides some detailed, compelling arguments against the crispness postulate, or at least against some straightforward ways one might articulate and defend the postulate. An advocate of the crispness postulate might think that talk of predicates like heat and color, and objects like tables, persons, and apples (not to mention electrons) will be reduced to sharp, scientific vocabulary, or else such talk will be eliminated in favor of whatever the postulated future science postulates. Or else our advocate may think that vague languages and theories supervene on precise ones. Hyde argues that on plausible views concerning elimination, reduction, and supervenience, crispness is either ruled out a priori, is highly implausible, or else the
158
Stewart Shapiro
holistic ‘costs’ of crispness are inordinately high. The details of the arguments need not detain us here. What is interesting for present purposes is that Hyde and his opponents cast the issue as one concerning what sort of descriptions are possible. In terms similar to the second clause of Putnam’s definition of metaphysical realism, Hyde and his opponents query the extent to which precise language is, or can be, ‘descriptively complete’. To take one example, Hyde quotes Christopher Peacocke (1981, 132–33): Suppose we have a language L containing vague expressions. Then the suggestion that the world itself is not vague is the suggestion that there will be some conceivable language L1 which contains no vague expressions and which has the following property: it is a priori that if two situations agree in all respects describable using the language L1 , then they agree in all respects describable using the language L. This is a form of supervenience.
(first emphasis mine). Hyde glosses this: ‘the world is precise if and only if all possible natural-language descriptions supervene, in the above sense, on precise descriptions’. He agrees that this thesis, ‘coupled with the view that all vagueness is ‘‘superficial’’ would justify a purely semantic account of vagueness. Though natural language is vague, the world is not’. Hyde then goes on to argue against these premises. Again, why can we draw conclusions about the nature of the world from the impossibility of certain kinds of descriptions or certain kinds of languages? How do we get from a premise concerning the inadequacy of a completely precise scientific theory to a conclusion about how things are, independently of our abilities to describe and theorize about them? To spell this out a bit, let us now assume that the crispness postulate is false, at least for the sake of argument. So, as Copeland puts it, it is not up to us whether to be vague. Like it or not, we have to use vague language to conduct our scientific, philosophical, and social business. The falsehood of crispness postulate, by itself, does not locate the source of the vagueness, and it does not settle the issue of metaphysical vagueness. Crispin Wright (1976), (1987) speculates that vagueness is due to the human need for predicates that can be applied on the basis of casual observation. To communicate effectively, we often have to quickly describe the color, size, shape, smell, feel, and taste of various objects and organisms. It would defeat much of the purpose of communication if we had to use a digital meter every time someone needed to know the color of a certain poisonous fruit or dangerous animal, or the size of a certain hill. And human powers of observation, even among the young and healthy, are limited. Eyes, ears, noses, taste buds, and hands can discriminate only so much. This suggests a principle of tolerance for certain crucial predicates—small differences do not matter and large differences do. And, of course, tolerant predicates are prone to sorites arguments. Even sophisticated science, no matter how far into the future we dream, requires some predicates that can be applied on the basis of observation, casual or not. After all, even idealized science is not an a priori affair, rationalism aside. Moreover, the human powers of observation, even careful non-casual observation, remain limited. If vagueness is to be eliminated, we will need to design and use only equipment that
Vagueness, Metaphysics, and Objectivity
159
stays well within our abilities to discriminate. Everything will have to be digital, and use only relatively small numbers. The falsehood of the crispness postulate may be due to a fact that equipment like this is not sufficient to cut nature at its joints, and tell us everything there is to tell about our universe. Of course, I do not intend to argue for this speculative principle. For present purposes, it is sufficient that a story along these lines is conceptually possible. If it is, then it very well might be that every sufficiently rich, true description of the world that we are capable of making is infested with vagueness. If so, then it is indeed not up to us whether to be vague. We have to be. Yet the ‘responsibility’ for the falsehood of the crispness postulate lies with us, and our limited powers of observation and detection, and not with the world—except to the extent that human observers and talkers are part of the world. Following Copeland, the negation of the crispness postulate does not entail anything in the neighborhood of metaphysical vagueness, at least not by itself. Of course, someone who denies the crispness postulate in favor of metaphysical vagueness might not be limiting the ‘representations’ to those that we finite beings are capable of making. Our metaphysician might be saying that no being, human or otherwise, can completely describe the world without invoking vague language. Even those who have unlimited powers of observation and discrimination—even God—have to use vague language. If this is what our metaphysician means, then the denial of crispness is just a statement of metaphysical vagueness, and does not lend any content or clarification to it. Of course, if it is the world that is vague, then no being can describe it completely without using vague language. So, with Copeland, I do not see how we have given content to the thesis of metaphysical vagueness. Along what might be similar grounds, Sainsbury (1994, §7) concludes that the very question of whether our language is vague because the world is vague, or whether the world is vague because language is vague . . . suggests that we have a choice of picture. According to one, our world, before we find it, is an undifferentiated sludge. In finding it, we divide it up. If we divide with vague tools, we see the world as containing vague objects . . . but the explanation will lie with us and our tools and not with the world. According to the alternate picture, the world is a certain way before we find it. Our job is to fashion concepts to mirror it. Because it contains vague objects, we find vague objects, and fashion vague concept[s] to match . . . I doubt whether either picture is intelligible, and I therefore doubt whether this route will lead to an intelligible thesis of [metaphysical] vagueness. We cannot think of our world except through our concepts, . . .
I presume that Sainsbury does not mean to challenge the common wisdom that the world ‘is a certain way before we find it’, idealism aside. Before we and any users of language ‘found’ the universe, it contained stars and other animals. Again, I do not wish to undermine the ordinary notion of objectivity. Clearly, some of the concepts that we apply to the world are vague, and, arguably, these concepts have to be vague, given the sorts of beings we are. The point, I take it, is that this is about all that there is to say. Sainsbury’s conclusion is, in essence, the Kant–Quine thesis. Its application here is the rejection of the very issue of metaphysical vagueness, at least if it is put in present terms.
160
Stewart Shapiro
I perhaps should not be completely definitive here. What is needed for either a defense or a rejection of metaphysical vagueness is a detailed account of the relationship between language and the world—a full blown theory of word—world relations. The theory should also have a plausible epistemology, showing just how we learn to use and deploy natural and scientific languages. And, of course, it should have a semantics, showing just what makes each true sentence true (assuming that the grand theory invokes a notion of truth for this purpose). And we would need to know something about the meta-language, the language in which this grand theory is formulated. Is that language vague or precise? And the theory should apply to this very meta-language as well, showing how it latches onto the (relevant parts of) the world. Conceivably, an acceptable grand theory like this might make it plausible that there is metaphysical vagueness—vagueness in the world itself, having nothing to do with language and representation. The theory would show how this happens. Conversely, it is also conceivable that a grand theory might make it plausible that there is no metaphysical vagueness—that all vagueness is due to how we represent a non-vague world. Again, the theory would show how this happens. In either of these cases, we might also have an argument against the Kant–Quine thesis, as the human and worldly aspects of representing and theorizing might be sharply delimited in the grand theory. But all of this is more or less idle speculation, at least for the time being. Trenton Merricks (2001) argues that linguistic vagueness presupposes either metaphysical vagueness or epistemicism, in the semantics itself. Let me conclude with a brief look at that most interesting argument, as it dovetails with present considerations, although probably not in ways that Merricks would approve. As above, suppose that a person Charles is borderline thin. Merricks invites us to consider a sentence like (1) ‘Thin’ describes Charles On its surface, sentence (1) is of the form Rab. It says that the description relation holds between a given word and a given person. Suppose, first, that sentence (1) determinately expresses a single proposition. Suppose, second, that this proposition is not determinately true and that this proposition is not determinately false. In present terms, the two assumptions are that it is indeterminate whether the word ‘thin’ stands in the relation of being described by to Charles. In other words, the two suppositions entail that there is metaphysical vagueness concerning the semantic relation itself. A fortiori, the suppositions entail that either epistemicism is correct or there is metaphysical vagueness. Now suppose that sentence (1) determinately expresses a single proposition, but that this proposition is either determinately true or determinately false. Given that Charles is borderline thin, it would follow that the truth value of (1) is unknown, and so the semantic relation of being described by gets an epistemicist treatment. Thus, epistemicism, and so either epistemicism or metaphysical vagueness. So it seems that Merrick’s opponent must claim that sentence (1) does not express a single proposition. She might contend that sentence (1) expresses ‘many propositions, some true, some false’, as some supervaluationists and some contextualists have it, or she could contend that (1) expresses no proposition, but would express different
Vagueness, Metaphysics, and Objectivity
161
propositions if sharpened in various ways, as other supervaluationists or contextualists contend. Merricks goes through several ways of understanding such claims, showing that in each case (or each case that is remotely plausible), there is either metaphysical vagueness concerning the semantic relations between words and world, or else some sort of epistemicism is correct for semantics. At the start of the article, Merricks claims that his argument ‘is so straightforward that it should suggest itself every time the linguistic theorist says there is no settled fact of the matter about whether a vague word describes a certain object’ (147). He expresses surprise that no one has formulated the argument before, ‘even if just for the purpose of being shot down’. As above, I agree with Merricks that there is a clear sense in which vagueness is not entirely linguistic. However, I submit that the proper conclusion is not that we are saddled with either metaphysical vagueness or epistemicism concerning the semantic relations, but rather that the linguistic and worldly components to vagueness are thoroughly intertwined and cannot be disentangled (at least pending a grand theory of the sort sketched just above). If there is fault to be found in the Merricks argument, I suggest that it is in the underlying metaphysics of propositions, which seems to run counter to the Kant–Quine thesis. The coherence of the issue of metaphysical vs. merely linguistic vagueness business presupposes something in the neighborhood of or analogous to the analytic-synthetic distinction. Quineans and Davidsonians agree with the truism that the truth or falsehood of any sentence is due to both what the words in the sentence mean and to the way the world is. They argue, however, that there is no clean way to separate those factors from each other. The present thesis is that this goes for vagueness. Let S be the sentence ‘Charles is thin’, recalling that Charles is borderline thin—whatever this comes to. The status of S as indeterminate is due in part to the meaning of the words ‘Charles’, ‘is’, and ‘thin’, and the semantic status of S is due in part to the way the non-linguistic world is, to how much body-fat Charles has and his waist size, for example. But, on the philosophical perspective advocated here, there is no clean way to separate out those factors. As I see things, vagueness is a linguistic phenomenon, tied to the kinds of languages that humans have evolved, and are continuing to evolve. But it is just as true to say that vagueness is due to the world we find ourselves in and try to make sense of. It is also due, just as much, to the kinds of beings we are. The world, and perhaps the world, is a certain way, and we are destined to represent it, or it, using vague language. But there is no indicating whether one of those drives the other. Re f e re n c e s Akiba, Ken (2004), ‘Vagueness in the world’, Noˆus 38, 407–29. Burgess, J. and G. Rosen (1997), A Subject with no Object: Strategies for Nominalistic Interpretation of Mathematics, Oxford, Oxford University Press. Copeland, B. Jack (1994), ‘On vague objects, fuzzy logic and fractal boundaries’, The Southern Journal of Philosophy 33, supplement, 83–96.
162
Stewart Shapiro
Ebbs, Gary (1997), Rule-Following and Realism, Cambridge, Massachusetts, Harvard University Press. Evans, Gareth (1978), ‘Can there be vague objects?’, Analysis 38, 208; reprinted in Keefe and Smith (1997), 317. Horgan, T. (1994), ‘Transvaluationism: A Dionysian approach to vagueness’, The Southern Journal of Philosophy 33, supplement, 97–126. Hyde, Dominic (1998), ‘Vagueness, ontology and supervenience’, Monist 81, 297–312. Keefe, R. and P. Smith (1997), Vagueness: A Reader, Cambridge, Massachusetts, MIT Press. Lewis, D. (1986), On the Plurality of Worlds, Oxford, Basil Blackwell Ltd. Merricks, Trenton (2001), ‘Varieties of vagueness’, Philosophical and Phenomenological Research 62, 145–57. Peacocke, Christopher (1981), ‘Are vague predicates incoherent?’, Synthese 46, 121–41. Putnam, H. (1981), Reason, Truth and History, Cambridge, Cambridge University Press. Rosen, Gideon and Nicholas J. J. Smith (2004), ‘Worldly indeterminacy: a rough guide’, Australasian Journal of Philosophy 82, 185–98. Sainsbury, R. M. (1994), ‘Why the world cannot be vague’, The Southern Journal of Philosophy 33, supplement, 63–81. Williamson, T. (1994), Vagueness, London and New York, Routledge Publishing Company. Wilson, Mark (2006), Wandering Significance, Oxford, Oxford University Press. Wright, C. (1976), ‘Language mastery and the sorites paradox’ in G. Evans and J. McDowell, ed., Truth and Meaning: Essays in Semantics, Oxford, Oxford University Press, 223–47; reprinted in Keefe and Smith (1997), 151–73. (1987), ‘Further reflections on the sorites paradox’, Philosophical Topics 15, 227–90; reprinted in Keefe and Smith (1997), 204–50.
III Tolerance and Paradox
This page intentionally left blank
9 Agnosticism and Vagueness Sven Rosenkranz
One of the lessons to be learnt from Michael Dummett’s writings on the topic is that there are intimate connections between issues in metaphysics and issues in the philosophy of logic and language. That metaphysical views such as realism entail global claims about the nature of truth should not be all too surprising; and that such global claims have some impact on the semantics for statements to which truth applies should be natural for anyone who conceives of semantics within the truthconditional paradigm. Dummett’s central insight, whose origins he traces back to Wittgenstein, is that such global claims about the semantics for statements, and hence any metaphysical views entailing them, are answerable to an account of the use of these statements. Whether a given semantics serves as an adequate basis for a theory of meaning for our language crucially depends on whether the use of that language bears it out. So philosophers of language cannot turn a blind eye on this constraint when applying semantics to natural language. Accordingly, whether some statements have truth-conditions whose obtaining it is beyond our powers to recognize, as the realist would have it, or whether all statements are determinate in truth value, however high or low the demands they make on precision, as the classicist contends, the right to making such claims must be earned by a plausible account, however rough in outline, of how the use of statements can invest them with any such semantic properties. Although most philosophers nowadays accept, in one way or other, that meaning supervenes on use, it has become fashionable to discount any such obligation. One of the main motives for this tendency undoubtedly is that Dummett’s case for antirealism, made on these grounds, has remained unconvincing. However, even if Dummett’s anti-realist argument should be resisted, its failure at most demonstrates that realism has not been shown not to comply with the constraint, and not that realism Some of the main ideas of this chapter were first aired in September 2003 at the first Arch´e workshop on vagueness. In April 2007 I had another go at the topic when giving a talk at the Arch´e vagueness seminar. I am grateful to all discussants present on either occasion for helpful comments, in particular to Patrick Greenough, Sebastiano Moruzzi, Stewart Shapiro, Crispin Wright, and Elia Zardini. I am also indebted to Stephen Schiffer and Timothy Williamson who gave me critical feedback on part of an earlier draft, and to two anonymous referees for their valuable criticism. Continued work on this chapter was made possible by a generous Heisenberg Research Fellowship from the Deutsche Forschungsgemeinschaft.
165
166
Sven Rosenkranz
complies with it. For all that follows, agnosticism may be the right attitude to take towards the realist’s view. Classicism is the thesis that natural language is subject to the rules of classical logic and its standard bivalent semantics. Realism, as here understood, is the thesis that there are truths we cannot know.¹ Standard epistemicism implies both classicism and realism in this sense. It results from the combination of two claims, that if vague terms draw sharp boundaries, it will be unknowable where these boundaries lie, and that vague terms do indeed draw such sharp boundaries. The first of these claims is meant to be underwritten by an account of ignorance specific to vagueness, the second is thought to be guaranteed by classicism in conjunction with certain truths about clear cases. Taking the latter for granted, standard epistemicism can thus be seen to proceed from classicism to realism. Classicism plays a double role in standard epistemicist thinking: classical rules of inference serve as a means to establish the truth of the general claim that some cut-off exists, and bivalence then ensures that this claim is true in virtue of the truth of a particular instance. Standard epistemicism thus treats classical rules of inference as a means to discover the existence of unknowable facts.² The Dummettian constraint thus equally applies to standard epistemicism. If meaning supervenes on use, then in order for vague terms to have sharply bounded intensions, their use must be a certain way; and no purely semantic considerations can assure us that it is that way. In other words, even if meaning supervenes on use ‘in an unsurveyably chaotic way’ (Williamson 1992, 157), we cannot simply take that as a license to make demands on the semantics for statements as we please or think fruitful. If the relation between meaning and use really is unsurveyably chaotic, this should rather make us wary of any wholesale claims about meaning that go beyond the uncontentious. It is for this reason that the strategy to hold on to the epistemicist view for as long as it has not been refuted is highly suspect. Again, agnosticism about the view may be all we are justified in endorsing. In what follows, I defend such an agnosticism about the existence of sharp boundaries. The argument for that view has two parts. First, it is argued that by presupposing classicism, standard epistemicism is hostage to commitments in the theory of reference which we do not know how to redeem (Sections 9.4–9.5). Secondly, it is argued that if this is so, agnosticism about the existence of sharp boundaries is justified and classicism must be abandoned (Section 9.2–9.3). I furthermore show that this agnostic view underwrites a non-classical solution to the Sorites and can account for borderline cases in terms of our ignorance of facts about referencedetermination (Section 9.6). Although this agnostic view favours an intuitionistic restriction on classical reasoning, it contrasts with other views of this kind, such as Crispin Wright’s, in foregoing any commitment to anti-realism (cf. Wright 2001 and 2003).³ ¹ Anti-realism will be understood to be the antithesis that all truths are knowable. Elsewhere I have argued that there are views foregoing commitment to unknowable truths that nonetheless deserve to be called ‘realist’ (see Rosenkranz 2003a). ² Roy Sorensen would replace ‘facts’ by ‘truths’ (Sorensen 2001, 165–84). ³ For a criticism of Wright’s views, see Rosenkranz 2003b, 2005 and forthcoming.
Agnosticism and Vagueness 9.1
167
T H E N O - S H A R P - B O U N D A RY PA R A D OX
Imagine a series of men aj , of equal stature and size, each with a different number j of evenly distributed hairs of equal thickness and length on their scalp, where 0 ≤ j ≤ 90, 000. Let them be lined up and ordered in such a way that for any two adjacent members of the series, the one on the left has exactly one hair less on his scalp than the one on the right. Let F be the predicate ‘bald’ and let m be 90,000. Assume that if anyone satisfies ‘bald’ so does a0 , and if anyone satisfies its contradictory so does am . It can be shown, using only conjunction introduction, existential generalization and reductio, that the following piece of reasoning is logically valid (where the variable ‘n’ ranges over the natural numbers): Fa0 ∼ Fam ∼ ∃n(Fan & ∼ Fan+1 ) ⊥ Here the third premise denies that there are any two adjacent members in the series, differing by just one hair, such that the one on the left is bald, while the one on the right is not bald. In other words, it denies that somewhere in the series there is a sharp cut-off between the bald and the not-bald, and hence that ‘bald’ has a sharply bounded intension. The proof that shows this premise to be inconsistent with the conjunction of the other two is a version of the Sorites paradox, the socalled ‘no-sharp-boundary paradox’. It deserves the label ‘paradox’ in so far as each of the premises, taken individually, seems to have something to be said in its favour, while the logical rules employed in the proof seem beyond reproach. There are several ways to respond to this logical finding. Some—the nihilists —take it to prove that, appearances notwithstanding, vague predicates like ‘bald’ are internally incoherent (Dummett 1978) or just empty (Unger 1979). Neither nihilist position seems the least attractive, as our everyday vocabulary is shot through with vague terms, and there are aspects of how reality strikes us which we could not talk about at all if we were denied the use of such terms (cf. Williamson 1994, 2). Others who therefore regard nihilism as unacceptable deny the third premise. They accordingly hold on to the premises of the following argument and accept its conclusion: Fa0 ∼ Fam ∼∼ ∃n(Fan & ∼ Fan+1 ) In order to restore consistency and to guarantee the coherence of vague predicates this would already do. This is not to say that ‘∼∼ ∃n(Fan & ∼Fan+1 )’ need not independently be motivated. After all, what it denies—i.e. ‘∼ ∃n(Fan & ∼Fan+1 )’—seems initially plausible. The plausibility of ‘∼ ∃n(Fan & ∼Fan+1 )’ may ultimately derive from the thought that, by denying it, we would be forced to say that ‘bald’ has a
168
Sven Rosenkranz
sharply bounded intension, which is precisely what we do not seem to be in a position to say. This thought will be compelling if classical logic is assumed to be valid. For, in classical logic—unlike in intuitionistic logic—double negation elimination holds so that ‘∼∼A’ logically entails A. By contrast, if intuitionistic logic is adopted, ‘∼∼ ∃n(Fan & ∼Fan+1 )’ will not collapse into ‘∃n(Fan & ∼Fan+1 )’; and then it will no longer be clear what remains of the initial plausibility of ‘∼ ∃n(Fan & ∼Fan+1 )’. That ‘∃n(Fan & ∼Fan+1 )’ must not be endorsed may turn out to be all our intuitions suggest. Commitment to classical logic, then, commits the foe of nihilism to the existence of sharp boundaries: Fa0 ∼ Fam ∃n(Fan & ∼ Fan+1 ) Although nihilists can of course be classicists, I will call this solution to the no-sharpboundary-paradox ‘the classical solution’. The classical solution is perplexing not only because that which we know if we know the premises is said to require the truth of the conclusion, something we positively seem not to have any reason to believe—which, if correct, would upset our claim to know the premises. The classical solution is also perplexing because, at the same time, knowing the premises is said to afford a route to knowledge of the conclusion—so that it seems that our first worry can immediately be allayed. That the validity of classical logic has this latter implication is a corollary of externalism about logical knowledge. According to this species of externalism, if a given inference is logically valid and logically competent subjects are disposed to draw that inference, then they are in a position to know its conclusion if they are in a position to know its premises. Here logical competence is conceived in terms of reliability in reasoning and the capacity to break up deductive inferences into simple inferential transitions corresponding to the successive application of basic logical rules.⁴ Thus, if the inference from ‘Fa0 ’ and ‘∼Fam ’ to ‘∃n(Fan & ∼Fan+1 )’ is logically valid—as the classical solution assumes—then the following principle of closure will hold for logically competent subjects who are disposed to draw that inference: K(F0 ), K(∼ Fam ) K(∃n(Fan & ∼ Fan+1 )), where ‘Kφ’ is short for ‘One is in a position to know φ’ (cf. Williamson 2000: 95, 117–18). A lot has lately been written on the distinction between closure, on the one hand, and the possibility of aquiring knowledge by inference, on the other. Some have argued that closure might hold without exception even if, in some contexts, logically valid inferences fail to proscribe routes to knowledge of their conclusion. To mark the ⁴ It is notoriously difficult to analyse the notion of reliability in modal terms when necessary connections are concerned. But this tells more against modal analyses than against the idea that there is such a thing as reliability in logical reasoning.
Agnosticism and Vagueness
169
difference, these writers refer to such failures as ‘failures of warrant transmission’ (see Wright 2000, Davies 2000). However, they base their case exclusively on examples in which knowledge of—or warranted belief in—the conclusion is a prerequisite for properly regarding the evidence adduced in favour of the premises as providing sufficient warrant for them. This is very plausibly not the case here: in order to properly regard recognition that a man has no hair on his scalp as a sufficient warrant for calling this man bald, and to properly regard recognition that another man has 90,000 hairs on his scalp as a sufficient warrant for calling that man not-bald, one need not presuppose in any compromising way that there is a sharp boundary between the bald and the not-bald. So both our aforementioned worries remain. These worries should not merely be taken to express that, confronted with paradox, our confidence is shaken so that we are no longer disposed to employ double negation elimination. For, that would so far be compatible with the idea that others, whose confidence is not shaken, need not listen. Rather, our worries are of such a kind that, unless they are somehow allayed, they cast doubt on the legitimacy of investing any such confidence. Some—the supervaluationists —try to allay these worries by preserving the letter of the classical solution, while suggesting that what that solution commits us to is less dramatic than it might initially seem. To this end, they introduce the notion of admissible sharpenings of vague terms and define truth as applied to statements containing such terms as truth under all their admissible sharpenings. What makes a sharpening of ‘bald’ admissible is partly determined by the constraint that ‘A man with no hairs on his scalp is bald’ and ‘A man with 90,000 hairs on his scalp is not bald’ come out true under all admissible sharpenings. The classical solution accordingly shows that ‘∃n(Fan & ∼Fan+1 )’ will be true under all admissible sharpenings. But—and this is supposed to be the comforting thought—different admissible sharpenings will determine different cut-offs so that there is no single statement of the form ‘Anyone with no more than n hairs on his scalp is bald but no one with n + 1 hairs on his scalp is’ which is true under all admissible sharpenings. Accordingly, there is no fact which the classical solution holds out the promise of securing indirect knowledge of. All we may learn about on its basis is a constraint we must comply with were we to precisify F: as long as the admissibility of any such precisification, or sharpening, requires the truth of both ‘Fa0 ’ and ‘∼Fam ’, the truth of some statement of the form ‘Fan & ∼Fan+1 ’ is also required. Supervaluationism faces problems of its own, most notably because the distinction between admissible and inadmissible sharpenings is itself vague (Williamson 1994) and because our responses to clear cases of being borderline do not cohere with the assignment of any third truth-status (Wright 2003). But it should in any case be clear that supervaluationism is not at the service of an epistemicist account of vagueness according to which there are unknowable facts determining sharp boundaries: epistemicists make use of classical logic with the intent to show that in situations in which we arguably cannot overcome our ignorance of a statement’s truth value, there is still an independent fact making that statement either true or false. In other words, they assume that borderline statements have bivalent truth-conditions, and
170
Sven Rosenkranz
this is precisely what supervaluationists deny. For the sake of argument, I will here set supervaluationism aside and conceive of the classical solution in its standard—i.e. bivalent—interpretation. 9.2
F RO M PA R A D OX TO R EV I S I O N
Given the perplexing nature of the classical solution, it seems fair to say that, unless we have some independent reassurance that the inference from ‘Fa0 ’ and ‘∼Fam ’ to ‘∃n(Fan & ∼Fan+1 )’ is truth-preserving—independent, that is, from the fact that classical logic is well-entrenched—we are entitled to claim that even those otherwise logically competent subjects who are disposed to draw the disputed inference are not in a position to acquire, by the disputed inference, knowledge of the conclusion solely on the basis of knowing the premises. In sections 9.3 to 9.5, I argue that we do in fact lack any such independent reassurance. In the remainder of this section, I spell out what we are entitled to conclude if this contention should prove correct. As already indicated, if we knew (1) we would be entitled to claim (2): (1) We lack any independent reassurance that the inference from ‘Fa0 ’ and ‘∼Fam ’ to ‘∃n(Fan & ∼Fan+1 )’ is truth-preserving. (2) Even those otherwise logically competent subjects disposed to draw the disputed inference are not in a position to arrive, by the disputed inference, at knowledge of the conclusion solely on the basis of knowing the premises. If worries about transmission-failure can properly be set aside, (2) will be tantamount to (3) The principle of closure ‘K(Fa0 ), K(∼Fam ) K(∃n(Fan & ∼Fan+1 ))’ is invalid even for those otherwise logically competent subjects disposed to draw the inference from ‘Fa0 ’ and ‘∼Fam ’ to ‘∃n(Fan & ∼Fan+1 )’. But then, modulo externalism about logical knowledge, we are entitled to infer (4) The inference from ‘Fa0 ’ and ‘∼Fam ’ to ‘∃n(Fan & ∼Fan+1 )’ is not logically valid. Although entitlement to (4) does not entail the truth of (4), friends of classical logic are likely to object to this result, and hence to at least one of the premises on which it depends. As long as no alternative to externalism is forthcoming—and any such alternative faces serious problems⁵—rejection of the externalist premise does not seem very promising. Since our entitlement to (4) is said to be conditional on our knowledge of (1), it will not be to the point to deny the latter premise at this stage. Any misgivings about the line of reasoning set out so far should rather be directed against ⁵ If logical competence was deemed insufficient for acquisition of knowledge by logical inference, then how could one ever come to acquire such knowledge? If the idea is that one must be in possession of additional warrants for thinking the inference truth-preserving, then since any such warrants will be inferentially acquired, either a regress threatens or competent employment of a logical rule at some point suffices for acquisition of inferential knowledge. But if it sometimes does, it should always do.
Agnosticism and Vagueness
171
(*) If we know (1), we are thereby entitled to claim (3). For, modulo externalism, (*) yields (**) If we know (1), we are thereby entitled to claim (4). Most friends of classical logic conceive of adherence to classical logic as the default position deviation from which must be argued for (see e.g Williamson 1994: 186; 1997: 215, 217–18; Sorensen 2001: 8, 11). If (**) holds, this order is reversed. So it would seem that assuming (**) begs the question. Actually, it is the other way round: it would be the classicist who begs the question. Classical logic could acquire the status of an entrenched body of doctrine partly because the phenomenon of vagueness was systematically and deliberately ignored (see e.g. Frege [1903]: § 56). Vagueness does not merely affect certain confined areas of discourse outside of which classical logic could unconditionally win the day: vagueness is ubiquitous and permeats all our thought and talk (Williamson 1994, 2). So whatever credentials classical logic has ‘until proven guilty’, these credentials cannot be relied upon once the question is raised whether it can handle vagueness well. There is accordingly no reason to deny that (*) and (**) impose a legitimate constraint that adequately reflects the worries that befall us when confronted with the classical solution. But then, knowledge of (1) will imply that we are not in a position to know that ‘∃n(Fan & ∼Fan+1 )’ is true. It is agreed on all sides that, if F is vague, we are not in a position to locate any sharp cut-off between those things which satisfy F and those which satisfy ‘∼F’ (cf. Greenough 2003). If this is so, however, then the only presently available way of knowing that such a cut-off exists would have to be by indirect proof. Yet, there is presently no other conceivable way of knowing ‘∃n(Fan & ∼Fan+1 )’ by indirect proof save by an inference of the disputed kind; and as just argued, if we know (1), we are entitled to conclude that we are not in a position to acquire knowledge in this way. So, if we know (1), we are entitled to conclude (5) We are not in a position to know whether ‘∃n(Fan & ∼Fan+1 )’ is true. 9.3
T H E L I M I TS O F E L I M I N AT I O N
The main burden of the foregoing argument evidently rests on a vindication of (1). This vindication, I suggest, can and should proceed as follows. The further information warranting the claim that the disputed inference is truth-preserving will have to be evidence for a semantic theory for vague terms of which that claim is a consequence. However, as I shall argue in sections 9.4 and 9.5, the following holds: (6) Any semantic theory for vague terms that subscribes to the principle of bivalence is hostage to assumptions about reference we do not know how to redeem. Now (6), if true, would already seem bad enough. But it would be premature to conclude from the fact that semantic theories of the requisite kind assume what we do not know how to redeem, that no evidence for such theories can be given at all. Before we are tempted into drawing this conclusion, we must be clear what kind of evidence can
172
Sven Rosenkranz
reasonably be asked for. And here it seems the following type of answer is the only one that stands a chance of being correct: a semantic theory for vague terms is justified to the extent that it is an integral part of a theory of vagueness which can claim to yield the best overall explanation of the sources of vagueness and our dealings with it. If such is the case with a given semantic theory, the fact that it rests on assumptions we do not know how to redeem will be no hurdle to giving whole-hearted assent to it. The logic of ‘best explanation’ does not, of course, permit that there are two rival theories of vagueness that equally satisfy this description. As long as there are two competing theories that are equally good and each better than any third, we cannot but concede that, pending further argument, neither is known to be correct. That there can be two mutually incompatible theories between which extant evidence does not decide is a thought familiar from the philosophy of science. It is also widely agreed that if we were to be faced with such a stalemate in the empirical case, it would nonetheless be rational for us to choose one of the rival theories rather than none, and to do so on the basis of such invidious ‘virtues’ as simplicity and architectonic beauty. But this is so because, first, there are practical concerns whose pursuit might otherwise be crippled and, second, the external constraints on empirical theories are understood well enough so as to make adopting a coherent theory which is false the best strategy to find out that it is. Nothing of the sort can plausibly be claimed in the context of philosophical theorizing about vagueness. Yet, even if it could somehow be made plausible that it is rational to endorse a theory of vagueness, although one must concede not to be in a position to know it, this will leave the agnostic unimpressed. He is home and dry once that concession is being made, and need not argue any further. So unless the epistemicist wants to give in to the agnostic straightaway, she should set out to show that her favoured theory of vagueness, of which bivalent semantics is an integral part, fares better than any other in respect of explanatory power. In the empirical realm, such explanatory advantages will typically be sufficient to warrant endorsement of a theory and hence endorsement of the claim that any genuinely incompatible theory is false. Thus, acquisition of warrant for an empirical theory by inference to the best explanation does not require the antecedent refutation of rivals. This is so because, in the empirical realm, producing the best overall explanation of a certain range of phenomena and refuting rival explanations are quite distinct achievements, in so far as the explaining facts and the phenomena to be explained are themselves distinct. In philosophy, however, explanations are more akin to coherent descriptions of the phenomena themselves; and arguing for the adequacy of some such description is always arguing overtly against any other description incompatible with it. So, already in the course of showing, of a given philosophical theory, that it best explains a given set of phenomena, one must show its rivals to be false. To devise a theory of vagueness is to diagnose what the phenomenon of vagueness is; and the explanatory power of such a theory will in part consist in its power to yield a coherent description of all the aspects of the phenomenon. Its coherence will have to be not only internal, but also external: the description it offers must be made to cohere with what we think we are in a position to say in response to other, yet related philosophical problems. It is for this reason that being hostage to assumptions we do not
Agnosticism and Vagueness
173
know how to redeem can seriously compromise a philosophical theory’s explanatory power. But, as said, its explanatory power need not thereby be compromised to such an extent that it no longer counts as warranted. If all alternatives to it prove either internally incoherent or hostage to even farther-reaching assumptions we likewise do not know how to redeem, it may still qualify. If this cannot be shown, however, we will not be entitled to endorsing it. Against the backdrop of these reflections, we may accordingly lay down: (7) (6) will not be a decisive objection against the idea that some theory of the disputed kind is warranted, only if it can be shown that all alternatives are either known to be false or hostage to even farther-reaching assumptions which we do not know how to redeem. So the only way how such a theory can still be warranted is by indirect means, viz. by elimination of alternatives (cf. Williamson 1994). One way to undermine this strategy of warranting theories of the disputed kind is to show that some of the considered alternatives are coherent after all and are not hostage to any farther-reaching assumptions we do not know how to redeem. But let all considered alternatives be shown to fail in these respects. Is this then the end of the matter? Are we then forced to say that the classical solution has been vindicated? No. For mind the following: (8) In order for the eliminative strategy to be successful, the list of considered alternatives must exhaust all alternatives. So, another way to undermine the eliminative strategy is to devise a hitherto unconsidered alternative that is neither known to be false nor hostage to any farther-reaching assumptions we do not know how to redeem. Note that it is no constraint on the success of this negative strategy that the alternative be shown to be better than theories of the disputed kind: as already argued, theories of vagueness that subserve classical logic and its bivalent semantics do not enjoy any default status. Thus, the mere availability of an undiscounted alternative is sufficient to undermine the suggested justification of theories of the disputed kind. If the alternative should prove to be of the agnostic type which the argument in section 9.2 showed to be justified on condition that no reason for the truth-preserving nature of classical inferences could be given, then we can conclude that the alternative not only has not been discounted, but is further justified. Accordingly, all I need to argue for in order to discount any semantic theory subserving the classical solution, and to supplant it by a better alternative, is the following: (9) An agnostic theory can be devised which is neither known to be false nor hostage to fortune and yet is a genuine competitor to theories of the disputed kind in that it denies the logical validity of relevant classical rules. Should my case be successful, there will then no longer be sufficient evidence for theories that subserve the classical solution. Accordingly, there is then no further information to be had which would warrant the claim that the disputed inference is truth-preserving—and (1) is vindicated.
174
Sven Rosenkranz
In order for the agnostic theory to qualify as a theory of vagueness, it must satisfy all the desiderata such a theory has to satisfy: it must identify the sources of vagueness and explain our dealings with it. This will include giving a solution to the no-sharpboundary paradox as well as an account of borderline cases. As was already suggested in section 9.1, rejection of classical logic may be all we need in order to dispel the air of paradox. For, then, it is no longer clear that Fa0 ∼ Fam ∼∼ ∃n(Fan & ∼ Fan+1 ) presents a paradox: once we have shown that endorsing the conclusion is compatible with refusal to endorse ‘∃n(Fan & ∼Fan+1 )’, the pull to deny the latter loses its intuitive force. So, there is reason to be optimistic that agnosticism about classical logic yields a solution to the no-sharp-boundary paradox. The argument for (6), set out in the next two sections, will provide the backdrop against which a non-classical account of borderline cases is devised in section 9.6.
9.4
C L A S S I C A L LO G I C A N D T H E R E F E R E N C E O F VAG U E TERMS
If classical logic is a means of discovery, as standard epistemicists assume it to be, then what it allows us to infer from a given set of premises should be determined as true or false independently of the fact that it can be so inferred. For, employment of the rules of classical logic is then meant to provide access to what is already there to be discovered. Accordingly, the rules themselves must not be taken to determine, but rather to respect, whatever factors determine a statement as true or false. Whether a given statement is true or false depends both on how things are and on what its ingredient terms refer to. If the statement is either true or false, then these are the only factors that determine it to be the one rather than the other. It follows that what a given term refers to is not determined by logic. Logic, we may say, has no reference-determining powers.⁶ If it is the task of the theory of reference to identify whatever it is in virtue of which terms refer, then logic accordingly is answerable ⁶ Even the assignment of truth-functions to logical constants must respect the independently determined reference of the non-logical vocabulary, as can be illustrated by the following example: if an atomic statement B, which is not itself a logical truth, can be shown to follow from A and , and to follow from ‘∼ A’ and , then the classical rule of dilemma allows us to infer B from and . This inference is acceptable only if ‘B v ∼ B’ is available as a premises on which to perform a modus tollendo ponens step. If it were argued that ‘B v ∼ B’ holds because the reference of ‘v’ and ‘∼’ is determined in such a way that ‘B v ∼ B’ comes out true quite independently from the way reference is assigned to the terms in B, then this would amount to saying that, given and , B will be true quite independently from the way reference is assigned to the terms in B —which is absurd (cf. Dummett 1978: 166–85, 1991: 11).
Agnosticism and Vagueness
175
to the theory of reference. This idea can be expressed in the form of a general constraint: (R) Logic must not require the intensions of predicates and their contradictories to comprise more than what is assigned to these intensions by the referencedetermining mechanisms identified by the theory of reference. So, in particular, if the reference-determining mechanisms are said to be such that they determine that, given how things are with both a and b, a is to belong to the intension of F and b is to belong to the intension of ‘∼F’, then the conjunction of ‘Fa’ and ‘∼Fb’ should not have any logical consequences whose truth requires the intensions of F and ‘∼F’ to comprise more than what is assigned to these intensions by the reference-determining mechanisms in question. As we have seen, classical logic validates the inference Fa0 ∼ Fam ∃n(Fan & ∼ Fan+1 ) where F is again the predicate ‘bald’, aj is a (possible) man with j hairs on his scalp, and m is 90,000. If bivalence holds, the conclusion of this inference, if true, must be true in virtue of the truth of one of its instances. That is, in so far as the premises are true, there must be a pair of statements of the form ‘Fan ’ and ‘∼Fan+1 ’ which are both true. Accordingly, there must be reference-determining mechanisms, identified by the theory of reference, and two (possible) men an and an+1 such that these mechanisms effect the assignment of an to the intension of ‘bald’ and the assignment of an+1 to the intension of ‘not bald’, if they assign a0 to the intension of ‘bald’ and am to the intension of ‘not bald’. (I here assume that there is an abundance of possible men with any number of hairs up to 90,000.) If we are to be reassured that classical logic is valid then, given (R), we must be given reasons for thinking there to be such reference-determining mechanisms. The question accordingly is: are there any such reasons? Note that even if we accept, as we should, that the truth of predications of ‘bald’ and ‘not bald’ supervenes on truths about numbers of hairs (their distribution, thickness, length etc.), this alone will not yield any reason for believing that if ‘Fa0 ’ and ‘∼Fam ’ are both determined to be true, whatever determines them to be true also determines the truth of ‘Fan & ∼Fan+1 ’, for some number n. For, even if the reference-fixing mechanisms can be shown to vindicate the following supervenience thesis (S), they are not thereby shown to determine as true any statement of the form ‘Fan & ∼Fan+1 ’: (S) If a man x with exactly n hairs of equal thickness and length, evenly distributed on his scalp, is bald, then any man y with exactly n hairs of the same thickness and length, evenly distributed on y’s scalp of the same size and shape as x’s, will also be bald; and if a man x with exactly n hairs of equal thickness and length,
176
Sven Rosenkranz evenly distributed on his scalp, is not bald, then any man y with exactly n hairs of the same thickness and length, evenly distributed on y’s scalp of the same size and shape as x’s, will also be not bald.
So how can epistemicism’s theoretical commitment be redeemed? Stephen Schiffer raises just this question when he contends that, first, the epistemicist ‘has no direct reason stemming from views specifically on reference or meaning for thinking that there are such exquisitely fine-tuned reference-determining factors [which determine sharp boundaries]. Second, that there are such factors seems extremely implausible given what we do understand about how reference is determined’ (Schiffer 1999, 492). In his response to Schiffer, Timothy Williamson takes on the challenge to explain how, according to the epistemicist, the reference of vague predicates can be determined in such a way as to bear out bivalence: Consider a context in which, on an epistemicist view, ‘bald’ refers to a property P. Everything either has P or lacks it (and not both). The reference-determining factors (including the context) determine that (a) ‘bald’ refers rather than failing to refer and (b) ‘bald’ refers to P rather than to some other property. How can they determine so much? I start with (b). [. . .] To determine which property ‘bald’ refers to, the reference-determining factors must determine of each thing x, time t and possible world w whether x at t in w is to have the property, in other words, whether the ordered triple x, t, w is to belong to the intension of ‘bald’. Nothing more is needed. [. . .] As a matter of classical logic, either the reference-determining factors do enough to determine x, t, w to belong to the intension of ‘bald’ or they do not. If they do not, then by that very fact they determine x, t, w not to belong to the intension of ‘bald’, for it cannot belong without being determined to do so by the reference-determining factors. [. . .] Thus if there are not enough facts about use and the environment to determine x, t, w to belong to the intension of ‘bald’, then that very shortfall is enough to determine it not to belong, and is itself determined by the facts about use and the environment. Reference can go by default. [. . .] Bivalence is structurally guaranteed, independently of the nature of the reference-determining factors. [. . .] It remains to mention point (a): the reference-determining factors also determine that ‘bald’ refers rather than failing to refer. Not every use determines a referent. To refer in a given context, an expression must be used with some degree of success in the relevant environment. But lack of success does not consist in the failure of use and the environment to determine some aspect of reference, for they leave nothing undetermined about reference. Rather, in such situations they determine that the expression as used in that context does not apply to any case, and the circumstances do not permit the assignment of a null referent (such as a property nothing could have), as opposed to no referent at all. (Williamson 1999: 509–11)
Let me make three preliminary comments. First, even if there is a way of talking about languages according to which predicates of these languages have their intensions essentially and omnitemporally, the sense in which we are here concerned with reference-determination and assignment of intensions is a rather different one. On the understanding relevant to the present discussion, reference-determination and assignment of intensions to predicates take place in time. Accordingly, any statement to the effect that a predicate F is determined to have an intension, to refer to a property or to apply to a given item needs to be appropriately temporally qualified. Secondly, although there is a sense in which identity of intensions is necessary but not
Agnosticism and Vagueness
177
sufficient for identity of properties, let us for the time being assume, with Williamson, that properties can be identified with intensions. Accordingly, until further notice we will use the phrases ‘F refers to a property’ and ‘F has an intension’ interchangeably (cf. Williamson 1996: 333). Thirdly, since we are not here concerned with ambiguity, any F with which we deal will be understood to uniquely refer to a property (have a unique intension), if it refers (has any intension) at all. Given classical logic, it is indeed ‘structurally guaranteed’ that the following holds: (10) For every x, t, w , either x, t, w is determined to belong to the intension of F at some time or it never is.⁷ Here the negation in the second disjunct must take wide scope: satisfaction of that disjunct must be consistent with there being nothing which is determined to be the intension of F. Otherwise, (10) would not be an instance of the law of excluded middle. Conditionalizing on there being something that is determined to be F’s intension, we get: (11) If something is determined to be the intension of F, then for every x, t, w , either x, t, w is determined to belong to the intension of F at some time or x, t, w never is determined to belong to the intension of F. Satisfaction of the second disjunct of the consequent is now meant to be inconsistent with there being nothing that is determined to be the intension of F. If (10) is ‘structurally guaranteed’ to the extent that it is a truth of classical logic, so is (11). Following Williamson, let us also assume that ‘reference can go by default’ in the following sense: (12) If something is determined to be the intension of F, then for every x, t, w , if x, t, w is never determined to belong to that intension, x, t, w is thereby determined to belong to F’s anti-intension. Conditionalizing on the antecedent is necessary as F may fail to be assigned any intension at all, in which case failure ever to be determined to belong to the intension of F does not imply being determined to belong to F’s anti-intension: if F has no intension, it has no anti-intension either. (11) and (12) jointly entail (13) If something is determined to be the intension of F, then for every x, t, w , either x, t, w is determined to belong to that intension at some time or x, t, w is determined to belong to F’s anti-intension. Satisfaction of the antecedent condition does not imply that there is a unique time at which what is determined to be the intension of F is so determined. For all that has so far been said, it may be thus determined sub specie aeternitatis by the fact that its members are at different times determined to satisfy F and everything else is never determined to satisfy F. ⁷ To say that at time t but at no earlier time, F is determined to have an intension, to apply to a given item or to refer to property P is not meant to suggest that such determination ever is instantaneous, but only that it has not been completed before t.
178
Sven Rosenkranz
Call an intension ‘classical’ if everything either belongs to it or, if not, belongs to its complement (i.e. the corresponding anti-intension). The classicist will of course maintain that every intension is classical in this sense. (13) will be acceptable to the intuitionist only if ‘intension’ is read in this way. Suppose then (13) is accepted to the extent that ‘intension’ is understood in terms of ‘classical intension’. The crucial question then becomes: what would be sufficient in order for F to be determined to have a (classical) intension? Here Williamson suggests the first two of the following three principles that jointly entail the third: (14) At t F is not determined to have any intension iff at t F is determined to have no intension. (15) At t F is determined to have no intension iff at t F is determined not to apply to any x, t , w and at t the circumstances do not permit assignment of the empty set as its intension. (16) If at t F is not determined to have any intension, then at t F is determined not to apply to any x, t , w . By contraposition, double negation elimination and the truism that if at some time F is determined to apply to some triple, then at that time F is not determined not to apply to any triple, (16) entails (17) If there is a time t and an x, t , w such that F is at t determined to apply to x, t , w , then F is at t determined to have an intension. That (17) is non-trivial even if classical logic is assumed can be brought out as follows. First, recall that if an intension is classical, then everything either belongs to it or, if not, to its complement (i.e. the corresponding anti-intension). But now: (18) For every x, t , w , if x, t , w belongs to the intension which F is at t determined to have, x, t , w is at t determined to belong to F’s intension. (19) For every x, t , w , if x, t , w does not belong to the intension which F is at t determined to have, x, t , w is at t determined to belong to F’s anti-intension. From (17) to (19) we thus get (20) If there is a time t and an x, t , w such that F is at t determined to apply to x, t , w , then for every x , t , w , either x , t , w is at t determined to belong to F’s intension or x , t , w is at t determined to belong to F’s antiintension. (20) amounts to a rejection of the idea, mentioned earlier, that the intension of F is complete only sub specie aeternitatis. Note also that, even if classical logic is assumed valid, (20) is not ‘structurally guaranteed’, and thence neither is (17): they are neither truths of classical logic nor derivable from classical logic in conjunction with the default principle (12). (20) and (17) are substantive claims about reference.
Agnosticism and Vagueness 9.5
179
A L L AT O N C E
Accordingly, we have to ask: what must the reference-determining factors be like in order for (20) to be borne out? It seems that the only viable answer to this question is this: if at t F is determined to apply to x, t , w , then it is so determined by two factors, F’s being determined at t to refer to a property and x, t , w ’s having that property. For, if at t F is determined to refer to a property, then to the extent to which everything either has or lacks that property,⁸ everything is at t either determined to belong to the intension of F or to its anti-intension.⁹ But now, if we stick to the identification of properties with intensions, there must be a natural, use-cum-environment-dependent reference-relation R that at some time picks out the intension of F as a whole. And how can this be so unless the members of this intension have something in common over and above their satisfying F? In other words, it would then seem that there has to be a characterization of F’s intension, or of the property it refers to, that says what members of that intension have in common and that differs from all of the following: ‘the intension of F’, ‘the property of satisfying F as we actually use it’, ‘the property of being determined to belong to the intension of F ’ and the like. For, none of the following claims yields a possible explanation of how x, t, w is determined to belong to that intension if it does: x, t, w belongs to the intension of F because F refers to the intension of F and x, t, w belongs to that intension. x, t, w belongs to the intension of F because F refers to the property of satisfying F as we actually use it and x, t, w has that property, i.e. satisfies F as we actually use it. x, t, w belongs to the intension of F because F refers to the property of being determined to belong to the intension of F and x, t, w has that property. What could a characterization of the required kind be like? Here we may for example say that what members of F’s intension have in common is that they all have fewer than n hairs, for some particular number n (where F is ‘bald’). Then the unresolved question is how there could be a natural, use-cum-environment-dependent relation R that picks out all of these members at once. At least, (17) now becomes highly questionable: evidently, the mere fact that at some time a given item is determined to belong to the intension of F will not be sufficient to ensure that F is determined to uniquely refer to the property of having fewer than n hairs, because someone with fewer than n hairs has fewer than n + 1 hairs. (Recall that F is here assumed to have a unique intension if it has any intension at all.) If (17) is questionable, then so is (20). Even if all and only bald people turn out to have fewer than n hairs, it is doubtful that possessing that property could ever explain why use and environment determine ⁸ Note that this is quite a concession to the classicist, a concession intuitionists typically are unwilling to make. ⁹ Provided that baldness does not supervene on any tensed properties like being the only G so far, if a possible-object-at-a-time has the relevant property at some time, it has it at all times.
180
Sven Rosenkranz
them to belong to the intension of ‘bald’. It seems that some story must be told about why reference to such a property might possibly matter to us. Absent any intention to uniquely refer to some such property, this story is most likely to invoke considerations about the psychological, biological or sociological function served by reference to a unique property of this kind. The assumption that there is some such function is certainly needed in order to render plausible that ‘bald’ is determined to refer to the property of having fewer than n hairs, for some n; and there is so far no reason to believe in its truth. If this is Schiffer’s challenge, then Williamson has not met it. Here Williamson is likely to object that explanation is an epistemic notion and that it is part and parcel of the epistemicist view that, owing to our epistemic limitations, we are unable to explain to ourselves in virtue of what all and only the bald men are bald. However, the point is not so much that we be given the means to identify what members of the intension of ‘bald’ have in common, and in virtue of sharing which they can all be explained to be members of that intension, but that we be assured that there are at all such commonalities among the members of each of the various candidate intensions, corresponding to each n within a certain range, which could pull off the feat. Talk about such commonalities may involve abandoning the conception of properties as intensions. But to abandon this conception is not yet to take a step towards anti-realism. It is the mere conceivability of the referential mechanism that is at issue, not the identifiability of the right one; and while conceivability may be a poor guide to possibility, commitment to possibilities does not sit well with commitment to their inconceivability. Alternatively, we may say that what the members of F’s intension have in common is that they are conceived by us to form a class, where to say this involves more than just saying that they all satisfy F as we actually use it. Then it may turn out not to be too problematical to suggest that there is a natural, use-cum-environmentdependent relation R that picks out that intension as a whole. Thus, for example, if all and only the Fs bear a nomic relation to tokenings of a mental concept C, then use and environment may determine F to express C. Suppose, then, that this is indeed the suggestion: what all and only the Fs have in common is that they bear a nomic relation to tokenings of some mental concept C (Fodor 1998, 137–47). Note that, so far, we do not need to presuppose anything about the exact nature of the nomic relation or the relevant tokenings. In particular, the relevant tokenings need not be those that occur in judgement and the relevant nomic relation need not be one that renders such judgements true (an idea Williamson would surely abhor; see his 1999: 512). What matters here is just that there has to be a possible answer to this question which relates to the actual constitution of a cognizer’s mind (Fodor 1998, (137n). Here is Fodor on doorknobs: The basic idea is that what makes something a doorknob is just: being the kind of thing from experience with which our kind of mind readily acquires the concept doorknob. And, conversely, what makes something the concept doorknob is just: expressing the property that our kinds of minds lock to from experience with good examples of instantiated doorknobhood. [. . .] [C]ontent is constituted by some sort of nomic, mind-world relation. Correspondingly, having a concept (concept possession) is constituted by being in some sort of nomic, mindworld relation. [. . .] But how could there be laws about doorknobs? The answer [. . .] is that
Agnosticism and Vagueness
181
there is really only one law about doorknobs (qua doorknobs); viz. that we lock to them in consequence of certain sorts of experience. And this law isn’t really about doorknobs because, of course, it’s really about us. [. . .] Doorknobs aren’t a natural kind, but we are. (Fodor 1998, 137–8, 146–7)
Accordingly, on Fodor’s view, what doorknobs have in common is just that we do or would selectively respond to them in thought by tokening the concept doorknob. This is here meant to imply that there is some nomic relation between those things that are doorknobs and our tokenings of the concept under which they fall. (The nomic relation in question will correspond to a ceteris-paribus-law which leaves room for the causal role of certain contextual factors; cf. Fodor 1990, 152.) However, it is unclear whether, given the way our minds actually work, there can be any such Fodorian property for ‘bald’ to refer to that vindicates both bivalence and the a priori supervenience thesis (S). For all we know, it may be constitutive of the structure of our minds that two borderline cases a and b with the same supervenience base differ in such a way that a is nomically related to tokenings of the concept bald, while b is not, and yet the factors that account for this difference cannot reasonably be taken to suggest a revision of (S) that would include them in the supervenience base for being bald. In other words, it may happen that, at least with respect to borderline cases, there is no supervenience base for being nomically related to tokenings of bald that coincides with the supervenience base for being bald. An analogy may help bring out what is here being suggested as a genuine possibility. If a person z is choleric, certain things are disposed to outrage z no matter what the circumstances are, while certain other things are not disposed to outrage z no matter what the circumstances are. But a choleric’s reactions are not always in this sense predictable. That is, there are certain things that are disposed to outrage z in some circumstances and not so disposed in other circumstances. Whether z will be outraged by these things, when confronted with them, will also depend on the circumstances and not just on whatever properties these things have irrespectively of the circumstances. What it is for a subject to possess the concept bald may consist in a complex pattern of dispositions of just this character. The problem is that, at least with respect to borderline cases, causally relevant features of the circumstances under which that concept is tokened may well be extraneous to what we take baldness to supervene upon. For all our evolutionary success, this success certainly does not require that our minds be such as would be needed in order for our use of bald to comply with both (S) and classical semantics. Now this would amount to a falsification of (S) only if ‘bald’ had to be considered to be assigned an intension that can be characterized in ways that make it intelligible that there is a natural, use-cum-environment-dependent reference-relation R that uniquely picks out that intension all at once. If this claim is foregone, we have no longer any reason to believe in (20). But then we have no longer any reason to think that, if it has application at all, ‘bald’ must be determined to have a classical intension. Consequently, we will not have any license to detach the consequent of (13). What would be the alternative? For all we know, satisfaction of ‘bald’ is determined by our accepted criteria of application about what we ought to say and what
182
Sven Rosenkranz
we ought not to say, of which (S) would then be one.¹⁰ Notably, these accepted criteria of application would not classify all borderline cases as either satisfying ‘bald’ or as satisfying ‘not bald’. Reference would not ‘go by default’. And what would hold good for ‘bald’ would also hold good for any other vague predicate for which the corresponding instance of (20) could not be redeemed. For all we know, this is the correct account of how vague predicates and their negations are determined to apply. But then, for all we know, vague predicates are determined to refer to classical intensions. In the next and final section, I argue that this kind of ignorance already provides the basis for an alternative theory of vagueness that is neither committed to bivalence nor to any kind of anti-realism. Combined with an account of how our ignorance in this matter affects our use of vague predicates, this ignorance will shed light on the nature of borderline cases and the reference of vague terms and pave the way to a non-classical solution to the no-sharp-boundary paradox.¹¹ Since we do not know whether classical logic complies with constraint (R), and since the alternative theory of vagueness to be suggested exploits this very ignorance, we can then conclude that (1) is justified. But if (1) is justified, then by the argument set out in section 9.2, so will be agnosticism about sharp boundaries.
9.6
A N AG N O S T I C T H E O RY O F VAG U E N E S S
As was argued in sections 9.4 and 9.5, we are not in a position to know whether for each vague predicate F, it is determined that F refers to a classical intension so that there is a time at which, for each item, this item has either been determined to belong to the intension of F or been determined to belong to the intension of ‘∼F’. Let (∃P) be the claim that there is a classical intension, or property, that ‘bald’ uniquely refers to. If we knew that (∃P) was false, then we would know that our ordinary criteria of application for ‘bald’ and its negation would be all the reference-determining factors that there were. We would then have to regiment our ordinary criteria of application for ‘bald’ and its negation in such a way as to make them intersubjectively authoritative. A regimentation of this kind would determine the reference of ‘bald’ in such a way that the supervenience thesis (S) would hold. For it is part of our understanding of ‘bald’ and ‘not bald’ that their correct use complies with (S); and since our understanding of ‘bald’ and ‘not bald’ informs our ordinary criteria of application for them, and in the envisaged circumstances these criteria are the only reference-determining factors, (S) will be respected. But given that there is persistent interpersonal disagreement and intrapersonal variation over time with respect to verdicts on borderline ¹⁰ Although (S) is not phrased metalinguistically, this does not prevent it from serving the role of a criterion of application, just as the fact that ‘Nothing red all over is green’ is about colours, and not about colour-terms, does not prevent it from being treated as a constraint on our use of colour-terms. ¹¹ For another example of how agnosticism about the semantics of a discourse may motivate rejection of classicism for that discourse, see Rosenkranz 2006. That agnosticism, suitably stabilized, can qualify as a genuine stance is argued in Rosenkranz 2007.
Agnosticism and Vagueness
183
cases, such a regimentation would most probably involve leaving some borderline cases of ‘bald’ undetermined. Just as there is a distinction between something not being what one ought to say and something being what one ought not to say, there would be a distinction between applications of ‘bald’ that are not licensed by these criteria and applications of its negation that are. Pace Williamson, reference would not then go by default. To be sure, the question would then arise as to whether there was anything that determined a sharp boundary between those applications which are licensed by our criteria of application and those which are not. If there was nothing that determined such a sharp boundary, our regimentation could not succeed; and if we knew that there was no such thing, our attempt to effect this regimentation would be idle. But the lesson is not that we know there to be something that determines a sharp boundary between those applications of ‘bald’ and ‘not bald’ that are and those that are not licensed by our criteria, nor that we know (∃P) to be true after all. For we do not know any of this. The lesson rather is that we do not know that there is nothing of the relevant kind. That we are neither in a position to know that (∃P) is true nor in a position to know that (∃P) is false, has an effect on the criteria of application for ‘bald’ and ‘not bald’ that we take to be in force. We are insecure as to whether or not a subject’s, positive or negative, verdict on a borderline case is the result of her better attunement to instantiations of some property that makes (∃P) true. We cannot then discount as incorrect predications of ‘bald’ or ‘not bald’ in borderline cases. For all we know they are true, in that the objects, of which ‘bald’ or ‘not bald’ are being predicated, have or lack a property which ‘bald’ is determined to refer to. Our ordinary criteria of application will accordingly get permissive. Given our ignorance, we will not then accept any use of ‘bald’ and ‘not bald’ as correct which is inconsistent with ‘∃x(x is bald & x+1 is not bald)’ and fashion our criteria of application accordingly. Knowledge requires mandatory warrants. So, unless better criteria come into play there is no knowledge to be had in borderline cases (Rosenkranz 2005). The better criteria would precisely be of such a kind as would be licensed by the identification of a classical intension, or property, that satisfies (∃P). As long as we do not know whether (∃P) is true, we are a fortiori not in a position to identify any such property. Given that we do not know whether (∃P) is true, we are ignorant which standards of correctness our use of ‘bald’ and ‘not bald’ is subject to: is it merely subject to the standards set by our ordinary criteria of application about what we ought and what we ought not to say, or also to those set by classical truth? This ignorance has an effect on our ordinary criteria of application: they are fashioned in such a way that they (i) never rule in what classical truth would rule out and (ii) never rule out what classical truth would rule in and (iii) rule out every verdict violating these two constraints. Thus, not only do our ordinary criteria of application get permissive with respect to positive or negative verdicts on borderline cases—they neither rule them in nor rule them out—the standards of correctness set by them forbid any verdict to the effect that claiming ‘bald’ to have a sharp cut-off is incorrect. If the standards set by these criteria are the only relevant standards of correctness, then this will collapse into its being incorrect to say that it is incorrect to say that there is a sharp cut-off—so that the double negation of ‘∃x(x is bald & x+1 is not bald)’ will hold. If these standards
184
Sven Rosenkranz
are not the only relevant standards of correctness, but our verdicts are further subject to classical truth, then ‘∃x(x is bald & x+1 is not bald)’ will be correct and hence also its double negation. So, ‘∼∼ ∃x(x is bald & x+1 is not bald)’ will come out correct, no matter whether our verdicts are merely subject to the standards of correctness set by our ordinary criteria of application or furthermore to those set by classical truth. Accordingly, despite our ignorance on the matter, we can and should endorse ‘∼∼ ∃x(x is bald & x+1 is not bald)’ anyway. But ‘∃x(x is bald & x+1 is not bald)’ itself will not yet be determined as correct, if all the standards of correctness we have to go on should turn out to be those set by our ordinary criteria of application. It will only be so determined if our thought and talk is further answerable to classical truth. So, given our ignorance, our criteria of application will not rule in any verdict to the effect that ‘∃x(x is bald & x+1 is not bald)’ is correct, in so far as they will not assign each borderline case either to the intension of ‘bald’ or to the intension of ‘not bald’. For to repeat, if assignment to an intension is a matter of what our criteria of application dictate, then not to assign an item to the intension of ‘bald’ and to assign it to the intension of ‘not bald’ are two distinct achievements—just as to issue the obligation not to say something and not to issue the obligation to say it are distinct types of decree. And in borderline cases, our ordinary criteria of application neither dictate that ‘bald’ be applied nor dictate that ‘not bald’ be applied. Accordingly, it should not come as a surprise that the classical solution to the no-sharp-boundary paradox turns out logically invalid. Where is the line to be drawn between what our ordinary criteria mandate and what they merely permit, and between what they still permit and what they already forbid? We are not in a position to say; and as long as this is so, our criteria will themselves be unstable. This instability will be a constitutive feature of them. What we do, if need arises, is to negotiate what it is permissible to say on any given occasion. But there are no rules or constraints that prescribe the outcome these negotiations ought to have (save (S) and the familiar penumbral connections). It might be suggested that the agnostic theory outlined here is at least as absurd as epistemicism, in that it mistakenly, and quite implausibly, ascribes to ordinary speakers a tendency to reflect on the epistemic possibility that there are properties referred to by terms like ‘bald’ that would bear classicism out. Of course, ordinary speakers do no such thing. But then, contrary to what is here being suggested, the theory just sketched does not assume that they do. What one does not know need not be something one thinks—let alone, knows—one does not know. Nor, thence, need what one does because one does not know X be something one knows one does because one does not know X . All that is required, by this theory anyway, is that what ordinary speakers do when they use vague terms like ‘bald’ be something they would not do if they knew, or believed to know, that (∃P) was true or that it was false. Explanations of behaviour in terms of ignorance have the nice feature that they do not have to assume anything in particular about what goes on in an agent’s mind. It is the theoretician who ascribes the ignorance, and not the ordinary speaker to whom it is ascribed. The theoretician does so after reflecting on theories like epistemicism; and while accepting epistemicism may be deemed absurd, reflecting on epistemicism ought not to be. In order to ascribe ignorance about whether or not
Agnosticism and Vagueness
185
p—this is another nice feature—one does not have to commit oneself to the truth or falsity of p. So, at least in this respect, the theoretician need not claim to be in any better epistemic position. According to this agnostic theory, then, the reference or satisfaction conditions of vague terms are either determined in the way epistemicists assume or by means of our ordinary criteria of application, but we are not in a position to know which is the correct account. Thus, it is not hostage to the assumptions about reference which epistemicists are bound to make and which we do not know how to redeem. Borderline cases are said to be cases in which we do not know what to say and do not know whether there is anything correct to be said. So, quite naturally, this theory lends itself to a non-classical solution to the no-sharp-boundary paradox. It shares with standard intuitionism the refusal to assign the law of excluded middle and the rule of double negation elimination the status of logical principles. However, unlike standard intuitionism, of which Wright’s view can be seen to be an example, this theory is not committed to any form of anti-realism (cf. Wright 2001 and 2003). The theory’s central tenet is that we do not know whether there is, for each vague predicate F, some property which F uniquely refers to and reference to which secures that there is a time at which it is determined, for every item, that it either satisfies F or satisfies ‘∼F’. In order to disprove it, one would accordingly have to show, for every vague predicate F—and a fortiori for every vague predicate of any metalanguage—, that there is a classical intension it is determined to have. But we do not know how to go about proving any such thing. We lack both the methods and the background knowledge necessary for the task. Re f e re n c e s Davies, Martin (2000), ‘Externalism and armchair knowledge’ in Boghossian, P./Peacocke, C., eds., New Essays on the A Priori, Oxford, Oxford University Press, 384–414. Dummett, Michael (1978), Truth and Other Enigmas. London, Duckworth. (1991) The Logical Basis of Metaphysics, London, Duckworth. Fodor, Jerry A. (1990), A Theory of Content and Other Essays, Cambridge/MA, MIT Press. (1998), Concepts, Oxford, Clarendon. Frege, Gottlob (1903), Grundgesetze der Arithmetik, vol. 2, Jena, Hermann Pohle. Greenough, Patrick (2003), ‘Vagueness: A minimal theory’, Mind 112, 235–81. Rosenkranz, Sven (2003a), ‘Realism and understanding’, Erkenntnis 58, 353–78. (2003b), ‘Wright on vagueness and agnosticism’, Mind 112, 449–63. (2005), ‘Knowledge in borderline cases’, Analysis 65, 49–55. (2006), ‘Metaethics, agnosticism, and logic’, Dialectica 60, 47–61. (2007), ‘Agnosticism as a third stance’, Mind 116, 55–104. (forthcoming), ‘Liberalism, entitlement, and verdict exclusion’ in Kallestrup, J. and Pritchard, D., eds. (forthcoming 2009), special issue of Synthese on the philosophy of Crispin Wright (published online first). Schiffer, Stephen (1999), ‘The epistemic theory of vagueness’, Philosophical Perspectives 13, 481–503. Sorensen, Roy (2001), Vagueness and Contradiction, Oxford, Oxford University Press Unger, Peter (1979), ‘There are no ordinary things’, Synthese 41, 117–54.
186
Sven Rosenkranz
Williamson, Timothy (1992), ‘Vagueness and ignorance’. Proceedings of the Aristotelian Society Supplement 66, 145–62. (1994), Vagueness, London: Routledge. (1996), ‘What makes it a heap?’, Erkenntnis 44, 327–39. (1997), ‘Imagination, stipulation and vagueness’, Philosophical Issues 8, 215–28. (1999), ‘Schiffer on the epistemic theory of vagueness’, Philosophical Perspectives 13, 505–17. (2000), Knowledge and its Limits, Oxford, Oxford University Press. Wright, Crispin (2000), ‘Cogency and question-begging: Some reflections on McKinsey’s paradox and Putnam’s proof’, Philosophical Issues 10, 140–63. (2001), ‘On being in a quandary: Relativism, vagueness, logical revisionism’, Mind 110, 45–98. (2003), ‘Vagueness: A fifth column approach’, in Beall, Jc ed., Liars and Heaps: New Essays on Paradox, Oxford, Oxford University Press, 84–105.
10 Vague Intensions: A Modest Marriage Proposal Jc Beall
The hard nut of vagueness arises from two strong appearances: •
Full Tolerance. There are no cutoffs. If x and y stand in some tolerance relation with respect to (vague) F , then they both satisfy F or neither does. • Utility. Vague predicates cut genuine distinctions in the world; they are useful in truly describing the world. There are no accounts of vague expressions that preserve both appearances, at least in the given simple forms.¹ What makes vagueness hard is the great strength of those appearances coupled with their apparent joint inconsistency. Utility demands cutoffs; Tolerance prohibits cutoffs. The given tension is crystalized in the sorites puzzle.² Tolerance forces us to say that everything satisfies F if anything does; but that would make F useless, thereby rubbing against—indeed, rubbing out—Utility. As Graham Priest (2003) puts it, ‘[t]he sorites phenomenon . . . arises simply because we are forced to recognize the existence of cutoff points where both common sense and philosophical intuition scream that there are none.’ Let a sharp predicate be a predicate that admits ‘cutoffs’ (e.g. some shift of semantic value along its range of application, or a sorites series, or etc.).³ Priest’s point, then, is that the sorites forces us to say that our ‘vague’ predicates are sharp where both common sense and philosophical intuition scream that they aren’t. Standard responses to the sorites reject Full Tolerance (so understood) and attempt to explain why we find cutoffs so counter-intuitive. In this chapter, I pursue a different For comments and discussion, I thank Dominic Hyde, E. J. Lowe, Graham Priest, Greg Restall, Stewart Shapiro, Achille Varzi, Brian Weatherson, Robbie Williams, Crispin Wright, and participants in the Arch´e vagueness workshop. Thanks, too, to colleagues at the University of Connecticut, especially Don Baxter, Tim Elder, John Troyer, and Sam Wheeler. Thanks, finally, to the work of Richard Dietz and Sebastiano Moruzzi who not only organized very interesting workshops but also edited this volume. ¹ I am also assuming normal extensional connectives. ² I skip a rehearsal of the sorites. See Moruzzi and Dietz’s Introduction to this volume. ³ Throughout this discussion, I use ‘sharp’ as above. (Notice that ‘sharp’, so understood, needn’t be classical.)
187
188
Jc Beall
course. I shall assume that our language contains vague predicates, and that such predicates (or expressions, in general) are essentially fully tolerant. The question concerns what our language is like if vague predicates are essentially (fully) tolerant, and in what way, if any, such predicates might be useful. In effect, the project is a modest one: to give a plausible reconstruction of what our language is like if vague predicates are fully tolerant. One desideratum is to have a framework in which other standard approaches to ‘vagueness’ (e.g. contextualism, supervaluationism, subvaluationism, epistemicism) find a place—at least in broad outline. Of course, since standard proposals reject Full Tolerance (though many admit different, more limited notions of tolerance), such proposals will not be seen as accounts of vague expressions, since the guiding assumption is that vague expressions are essentially fully tolerant. Still, such accounts are accounts of some vagueness-related phenomenon, and a desideratum of the project is to have such accounts naturally and a place in the overall, broad picture. My proposal, like the project, is modest. In short, vague predicates are essentially fully tolerant, and so without cutoffs, and so, for that reason, Utility fails for vague predicates. On the other hand, there is a genuine sense in which vague predicates (their intensions) are useful: they provide sharp lookalikes—sharp, homonymous relatives—of the vague. In a sense, nihilists (very generally construed) about vague predicates are right; however, non-nihilist, more standard accounts are also right, but not about vague predicates, which are essentially fully tolerant and, so, without cutoffs. The chapter remains at a very abstract level, and is intended to be brief.⁴ The next few sections sketch the proposed framework: Section 10.1 states, in abstract terms, what vague intensions are like; Section 10.2 specifies satisfaction conditions for such predicates; Section 10.3 addresses the issue of Utility, and how it fits into the proposed framework; and Section 10.4 discusses the place of standard accounts of ‘vagueness’ in the sketched framework. Section 10.5 briefly summarizes the (admittedly abstract) picture, and Section 10.6 provides replies to a few objections. Section 10.7 offers a few closing remarks.
10.1
VAG U E I N T E N S I O N S QUA F U L LY TO L E R A N T
The intensions of sharp predicates yield unique extensions; they are, or may be modeled by, functions—for simplicity, functions that take a predicate to a unique extension.⁵ What of vague predicates, which, by assumption, are essentially fully tolerant? My suggestion is that, unlike the intensions of sharp predicates, vague intensions yield ⁴ This chapter is a very slightly revised version of a brief but ‘big picture’ talk at St Andrews (2004). The aim of the talk was modest, and this chapter, as above, retains the modest aim. ⁵ The thought may be generalized in the usual way, so that the given intensions are functions from worlds to sets of objects, or to ‘truth values’ or etc. I simplify by just talking about intensions as taking predicates to extensions. (If the sharp predicate is non-classical, then the given intension will similarly assign a unique antiextension.)
Vague Intensions
189
many extensions; they are (or may be modeled by) non-functional relations, relating their arguments to many extensions. Concentrating on atomic sentences, the idea, in a nutshell, is as follows. The vague intension of F is a relation ρ that relates F to (many) extensions.⁶ In particular, ρ ‘determines’ some ‘core extension’ of F but also relates F to other extensions that are ordered by inclusion. ( Think of some initial ‘core set’ E0 , which comprises a bunch of objects in a soritical series. Then E1 is a superset of E0 comprising ‘the next’ element, and E2 a superset of E1 comprising the next, and so on.) Intuitively, ρ strives to be fully tolerant, strives to be all-inclusive. A picture: •
Let RF be a tolerance relation with respect to predicate F , and let ρ relate F to Ej , that is, ρ(F , Ej ). If x1 ∈ Ej and x1 , x2 ∈ RF , then x2 ∈ Ej+1 and ρ(F , Ej+1 ), where Ej+1 = Ej ∪ {x2 }.
The ‘all-inclusive’ nature of a vague intension ρ, so understood, ensures that for any element x of a soritical series, x finds itself in some extension to which ρ relates a vague predicate. It is precisely that feature of vague intensions—their abundant tolerance—that distinguishes them from sharp predicates. What distinguishes vague predicates from sharp predicates is that the former have (non-functional) relations as intensions; they relate a predicate to more than one extension, where the given extensions are ordered by inclusion, guided by ‘tolerance’, as above. The suggestion is not that vague intensions have a parameter for contexts or the like; those, at least on standard treatments, are simply more functional relations (viz., functions with a parameter for context or the like). The proposal, rather, is that vague intensions are non-functional, and that the various extensions, to which such intensions relate a predicate, are ordered via inclusion—driven by the vague intension’s quest for tolerance. It should be clear that, as hoped, sharp intensions may be seen as a special case of our (overly generous) vague intensions. In short, sharp intensions are simply ‘cut off ’ vague intensions; they are relations from predicates to a unique extension. In short, sharp intensions are functional relations; they are special, cutoff cousins of the broader—and, again, wildly generous—vague intensions. (I will return to this.)
10.2
S AT I S FAC T I O N A N D TO L E R A N C E
The big question, of course, concerns satisfaction conditions for vague predicates, which conditions, in turn, dictate ‘truth conditions’ (at least for atomics). Here, there are two salient options.⁷ ⁶ I write ρ(F , Ei ) to indicate that ρ relates F to Ei , where Ei ⊆ On , with O a non-empty domain of objects. As above, one can just as easily—and perhaps more plausibly—say that ρ relates worlds (or whatever) to the various extensions, but for simplicity I will just say that ρ relates a given predicate to various extensions. ⁷ It’s interesting to think about variations on these options, but towards keeping this chapter short, I skip such discussion and go only with the two salient options.
190 • •
Jc Beall
Fregean: a predicate F is satisfied by x iff x is in the unique extension of F . Liberal: a predicate F is satisfied by x iff x is in an extension of F .
If we follow the Fregean condition, we will say that, (apparently) contrary to appearance,⁸ no genuinely vague sentence—that is, a sentence using genuinely vague, fully tolerant predicates—is true. If we go with the Liberal intuition, we will wind up saying that all vague sentences are true. I will go with Frege. No vague sentence is true, since no vague intension yields a unique extension, contrary to the demands of the Fregean truth conditions.⁹ What is important to notice is that, on either route, we get Full Tolerance: we have no sharp boundaries in either case. On the Fregean approach, we hardly get cutoffs when no vague predicates are satisfied—all premises (and conclusion) of a sorites argument will have the same semantic value (viz., untrue). On the liberal approach, we get no cutoffs for the same reason, although now all premises and conclusion are true. What is also worth noting is that, again, the above satisfaction conditions cover the general case (viz., vague predicates) and the special, cut off case (viz., sharp predicates). The Fregean approach, which I’m assuming, requires a unique extension. As such, sharp predicates, whose intensions are functions, are candidates for truth (as it were), while vague predicates, having non-functional intensions that yield more than one extension, are not candidates for truth. On the Liberal approach, vague predicates are candidates for truth, since the only requirement is some extension or other, in which case sharp predicates, which always have a unique extension—and, so, some extension or other—are likewise candidates for truth. So, again, the framework is one in which standard, sharp predicates are merely a narrow, special case of the broader (and overly generous) vague predicates. Before turning to the issue of Utility, it is worth briefly addressing the issue of negation. On either the Fregean or Liberal approach, one might wonder whether inconsistency arises. The answer, of course, depends on the logical behavior of negation. As I am assuming the Fregean approach to satisfaction conditions, there are two basic options.¹⁰ One is to treat negation as exhaustive, in the sense that A or ¬A holds for all A (vague or otherwise). If disjunction is normal, then the exhaustive approach yields that A ∨ ¬A is valid.¹¹ Defining falsity to be truth of negation (i.e. A is false ⁸ In the next section, I suggest that the appearance is really an appearance of something else, in which case the apparent conflict is only apparent. ⁹ Braun and Sider (2007) have recently defended a ‘Fregean view’, according to which no vague sentence is true or false. They aim to answer Utility via complicated pragmatic principles and a version of supervaluationism. I will take a different route, although ultimately the Braun–Sider view, like (I hope) other accounts of ‘vague’ expressions, will naturally find a place within the overall framework. ¹⁰ The Liberal approach, which, for brevity, I am setting aside, will likely require a paraconsistent negation, at least if symmetry of sorites arguments is to be respected. (The symmetry in question is that every—or, at least, many—positive sorites (e.g., starting with a − 1 is F and running to a − n is F ) corresponds to a symmetric negative sorites (i.e., starting with a − n is not F and running to a − 1 is not F ). This is not the only option, but it is likely the most natural. ¹¹ NB: sometimes, the term ‘exclusive’ is used for the target negation behavior, with the idea that such a negation ‘excludes gaps’. I think that the term ‘exhaustive’ is more appropriate, since such a
Vague Intensions
191
just if ¬A is true), this approach would have it that, for any vague A (which, given Fregean satisfaction conditions, is not true), A is false since ¬A is true. The other approach is along Strong Kleene (SK) lines, wherein negation toggles truth and falsity but is fixed at any other ‘semantic value’. In the current context, the SK approach would have it that the ‘semantic value’ of A is the same as A if A is vague—namely, ‘untrue’. For present purposes, I think that either approach is viable given the ‘Fregean’ assumption on satisfaction. On the other hand, given that most sorites arguments have both a ‘positive’ and ‘negative’ version, it seems that symmetry pushes for the SK approach. So, that is the approach that I’ll assume—though, as mentioned, the chief aim of this chapter doesn’t force one approach over the other.¹² In addition to SK negation, I will assume a basic SK framework, in which our extensional connectives are ‘normal’.¹³ With an SK approach to negation (and extensional connectives, generally), coupled with the proposal above, we have it that no vague sentence is true or false. The obvious question, then, concerns the appearance of Utility (see Section 10.1).
10.3
UTILITY
As above, we have Tolerance; however, we lose Utility. How can vague predicates be useful if they’re never true (or, derivatively, false) of anything? The short answer is that they can’t be useful, at least not useful for purposes of expressing true claims!¹⁴ Admittedly, one might try to salvage Utility by invoking complicated pragmatic mechanisms; however, I will explore another route.¹⁵ Instead of trying to preserve Utility via pragmatic mechanisms, I suggest a different, twofold route towards making sense of the appearance of Utility. •
First, find some other way in which vague intensions are useful (towards the aim of expressing true claims, and so on); • Second, trace the strong appearance of Utility to something other than vague predicates, something related to vague predicates but not vague predicates. negation, assuming normal disjunction (which is standardly assumed), serves to cut two exhaustive categories—the true and untrue. ¹² The SK approach will have it that Fa is false exactly if a is in the unique anti-extension of F . (I leave use-mention to context here.) ¹³ For an elementary discussion of some of the standard, non-classical options, see Beall and van Fraassen 2003. ¹⁴ The matter is less clear on the Liberal approach; it depends on what one says about negation, and also the general logic. As above, I am assuming the ‘Fregean’ approach, and so will ignore Liberal issues. ¹⁵ I should note that putting the full burden of Utility on pragmatics, which, I take it, is the approach of Braun and Sider (2007), seems to me misguided, since pragmatic ‘goals of assertion’ will likely reinvigorate the initial tension between Tolerance and Utility, where these are now reformulated in terms of the given ‘goals of assertion’. But I set this aside, in an effort to keep this chapter short.
192
Jc Beall
Given that vague predicates are essentially (fully) tolerant, they’re not useful in the Utility sense; they don’t cut distinctions in the world, and are generally not capable of truly describing the world (at least given the ‘Fregean’ approach to satisfaction, which I’m assuming). The above, twofold route towards accommodating the appearance of Utility calls for finding some other job for the vague predicates—in particular, their intensions—to do towards achieving a useful language. In turn, the above route calls for explaining the strong appearance of Utility, with respect to vague predicates, as an appearance of something else—some other class of predicates that, unlike our (essentially fully tolerant) vague predicates, do satisfy Utility. Following the above course, my suggestion is that we understand the service of vague predicates—in particular, their intensions—as affording many riches that serve to give us genuinely useful predicates. In particular, what makes vague intensions useful is not that they cut distinctions in the world or are otherwise employable in true descriptions; rather, their intensions give us—what may well, in some cases, appear to be arbitrarily drawn—useful, sharp predicates, predicates that do have cutoffs in virtue of (as it were) forcing a well-behaved (functional) intension. In other words, while vague predicates aren’t useful for truly describing the world, since they aren’t true (or false) of anything, they are useful: their intension yields a plenitude of extensions that, in turn, yield sharp predicates, ones that are intended for cutting distinctions, ones that have a unique extension and, so, are capable of being true along Fregean lines. How is the appearance of Utility preserved in all of this? While our vague predicates are essentially tolerant and thereby buck Utility, the appearance of utility is nonetheless preserved via look-alikes, sharp predicates that are homonyms of the vague. Such homonyms, I suggest, are what we typically—but erroneously—call ‘vague predicates’. Such sharp look-alikes are what we normally use in our daily, true descriptions of the world, and it is them—not their vague progenitors—that reflect Utility. How do our vague intensions give birth to the given sharp offspring? One—among many—ways in which we might get sharp offspring from the vague intensions is as follows. (See §5 for other accounts.) Let F be a vague predicate the intension of which yields various extensions Ei , as in §2. A sharp descendent of F , which I’ll write ‘F ’ (even though, in our real language, the two are homonymous), naturally emerges. • x is in the extension of F exactly if x is in F, where F comprises each Ei such that ρ(F , Ei ). So, for example, while the vague intension of ‘is tall’ yields too many extensions to make the (vague) predicate useful in cutting distinctions, one quickly gains a sharp and useful relative of ‘is tall’—for clarity, call it ‘is tall ’—by generalized intersection: something is tall exactly if it is in every extension of ‘is tall’.¹⁶ ¹⁶ The similarity with supervaluational approaches to ‘vague predicates’, I hope, is obvious. Note, however, that there’s no hint of invoking other languages or other valuations; instead, the idea is that our Tolerant vague predicates yield all the requisite resources for sharp, useful ones—all within the same language. Moreover, and more importantly, there’s no need to define ‘supertruth’ as the
Vague Intensions
193
The suggestion, then, is that while vague predicates afford no cutoffs, they none the less afford sharp ‘look-alikes’ that, in virtue of being sharp, immediately yield cutoffs. We think that vague predicates are not only Tolerant but Useful (i.e. satisfy Utility); however, we are conflating the Tolerant vague predicates with their Useful—but non-Tolerant—sharp relatives. This raises the question of what to make of ‘rival’ accounts of ‘vague’ predicates (or expressions, generally), and also raises the prospect for a marriage of nihilists (very broadly construed) and non-nihilist, standard accounts. 10.4
M A R R I AG E : N I H I L I S TS A N D N O N - N I L H I L I S TS
The modest proposal is that if, as herein assumed, vague predicates are Fully Tolerant, they fail to give us Utility; however, they nonetheless play a role in providing Useful—and, hence, non-Tolerant—predicates. The appearance of Utility for vague predicates is merely an appearance; it arises from conflating the sharp offspring of the vague with their fully tolerant, vague progenitors. The proposal, then, affords a sense in which nihilists about vagueness are right, but non-nihilists—in effect, standard theorists about ‘vagueness’—are also right. In effect, nihilists maintain that vague predicates are essentially (fully) tolerant, and hence are utterly useless, or at least don’t satisfy Utility—cutting distinctions in the world via cutoffs.¹⁷ On the current proposal, nihilists (at least broadly conceived) are right about vagueness. On the other hand, non-nihilists may also be right; it’s just that they’re right about something else, some vagueness-related but non-vague fragment of our language. In particular, non-nihilists—inasmuch as their accounts yield cutoffs and, hence, are non-Tolerant—may be seen as giving an account of the sharp offspring of our vague predicates, not an account of vagueness itself. What are standard theorists of ‘vagueness’ doing? If, as suggested, our vague predicates—in particular, their intensions—yield useful, sharp offspring, there’s a question about how such ‘offspring’ behave. What is the right theory of them? It is this question, in the end, towards which standard, non-nihilist accounts of ‘vagueness’ are aimed. Such accounts tell us how the sharp offspring of the vague work, how their logic or semantics goes. To see as much, consider that each of the standard three (non-nihilist) approaches, qua account of ‘sharp offspring’, finds a natural place in the proposed framework, at least from a sufficiently abstract—and, admittedly, simplified—point of view. How do the sharp offspring come about? How do they work? Here are standard options, very broadly construed.¹⁸ aim of assertion. Truth itself remains the aim, and we never properly assert vague claims—since they’re one and all untrue. It’s the sharp offspring of vague predicates that we use in proper assertion. ( Thanks to Achille Varzi for discussion on this point.) ¹⁷ For versions of nihilism, not all of which would fully subscribe to this characterization, see Wheeler 1979 and Unger 1979. (On my rather crude characterization, Braun and Sider (2007) would count as nihilists too, at least on my understanding of their position.) ¹⁸ Other accounts, like Weatherson’s (2005), can also find a place, I think.
194
Jc Beall
E1. Supervaluational. Vague intensions provide the material (viz., a bunch of extensions) for a sharp predicate that is defined via generalized intersection: where F is our vague predicate, we have F the extension of which is the generalized intersection of all of F ’s extensions.¹⁹ E2. Contextualism. Vague intensions provide the material (viz., a bunch of extensions) for a sharp predicate that is sensitive to context: the (unique) extension of F , relative to a context c, is exactly one of F ’s many extensions, namely, the one ‘selected’ in c (where the details of such ‘selection’ depends on the exact, contextualist account). (See Fara 2000; Raffman 1996; Shapiro 2006) E3. Epistemicism. Vague intensions provide the material (viz., a bunch of extensions) for a sharp predicate the (unique) extension of which is unknown to us, and perhaps in principle unknowable to us. (It might be that how the sharp predicate arises out of the vague predicate’s material is similarly unknown, perhaps unknowable.) (See Sorensen 2001; Williamson 1994) While the foregoing is only crudely sketched (and obviously far from exhaustive!), it is clear that such standard stories may be seen—at least abstractly—as giving accounts of the ‘sharp offspring’ that emerge from our overly generous, fully tolerant vague intensions. The usual versions of such stories, of course, all aim to be giving accounts of vague intensions (vague predicates); however, each such account encounters the screams of ‘philosophical intuition’ as each gives up Full Tolerance by admitting inevitable cutoffs—inevitable, because such accounts purport to preserve Utility for their target predicates. What I suggest, however, is that they needn’t and shouldn’t give up the full tolerance of vague predicates. The current (admittedly modest) suggestion is that we can have our Full Tolerance for genuinely vague predicates; what we must recognize is that such ‘standard accounts’ are really just accounts of the sharp homonyms of vague predicates—the sharp and Useful predicates that fall out of fully tolerant vague ones. In some sense, then, the current proposal affords a marriage of nihilists (very broadly conceived) and their non-nihilist ‘rivals’. The former are right about genuinely vague intensions, and the latter provide candidate theories of the sharp offspring of the vague. Our vague intensions give us everything we need to get our useful, sharp predicates; and the non-nihilist, standard theories tell us how such sharp predicates work. I have said nothing about which account of our useful (sharp) predicates, (E1)–(E3), is correct, and I have little to offer on that score. Indeed, I am not sure that exactly one of the given standard accounts is right; it might be that more than one is right, that we have various different kinds of sharp predicates that have emerged from the vague. I leave the matter for future debate. ¹⁹ As mentioned in §4, this is along the lines of a supervaluationist account (Fine 1975, Keefe 2000, Varzi 1999), though it needn’t exactly be supervaluationism, for reasons given above (in an earlier footnote). Importantly, there needn’t be any appeal to supertruth—certainly no conflating ‘supertruth’ and truth. NB: the so-called subvaluationist approach also finds a place within the framework. Here, the subvaluationist denes F as per above, but now switches to the Liberal truth conditions for sharp predicates! (For subvaluationism, see Hyde 1997, and for discussion Beall and Colyvan 2001.)
Vague Intensions
195
What I think is worthwhile about the proposed framework is that, while modest, it gives us a way to have Tolerance and have useful predicates. There is no one phenomenon that yields both Full Tolerance and Utility—a point on which all parties, as far as I know, agree. The ongoing complaint against standard ‘accounts of vagueness’ is that they ignore the essential feature of Full Tolerance—no cutoffs. My modest suggestion is that, irrespective of our account of (the derivative) sharp predicates, we need not give up Full Tolerance. Vague intensions are not by themselves useful for describing the world, but they are useful in affording predicates for the job. 10.5
S U M M A RY O F P RO P O S A L
The hard nut of vagueness is the grating tension between two strong appearances: namely, Full Tolerance and Utility. The former precludes cutoffs; the latter demands them. The standard course is to give up on Full Tolerance in favor of Utility. Nihilists, at least very broadly conceived, buck the standard course. They take full tolerance to be essential to vague predicates. If R F is a tolerance relation for vague F such that xR F y, then both or neither of x and y satisfy F . The result, of course, is the loss of Utility. No one phenomenon satisfies both Full Tolerance and Utility, at least as the terms are used here. The current proposal suggests that the appearance of Utility emerges from vagueness-related phenomena, namely, the sharp offspring of our fully tolerant vague intensions. The hard nut of vagueness is resolved by accepting that, for vague predicates, there are no cutoffs, just as ‘common sense and philosophical intuition’ demand. Nihilism, then, is partly right, but it is rejected as the full story. We also have Utility; it’s just that Utility is achieved via the sharp predicates that emerge from our overly tolerant vague predicates. It is those predicates, the sharp offspring of the vague, on which much standard work on ‘vagueness’ has focused; it’s just that such work needs to be re-conceived. In short, the screams of common sense and philosophical intuition are misplaced when directed at standard accounts, since such accounts are accounts of sharp predicates (or expressions, generally). Nobody should scream at cutoffs if the predicates are sharp. The proposal, of course, remains very abstract, with few details given. The aim has been only to gesture at a framework that, as far as I know, has been neglected, a framework that might resolve the long-standing ‘screams’ by providing, in some sense, a marriage of two very different approaches: nihilist and non-nihilist, standard accounts of ‘vagueness’. Yet, even at the abstract level, objections arise. I close by answering a few objections. 10.6
OBJECTIONS AND REPLIES
Objection. ‘Penumbral connections’ are central to vague predicates (Fine 1975). ‘If x is red, then x is not pink’ is a conceptual truth about redness and pinkness. But, on the proposed account, such connections are entirely lost.
196
Jc Beall
Reply. The short reply is that such connections are not lost; they never governed genuinely vague predicates. What the given principles govern are the sharp relatives of vague predicates—the predicates that we typically use, and use in accord with the given principles. How, and to what extent, such principles hold is a pressing matter for accounts of the sharp offspring of the vague, and a matter to which most standard account of ‘vagueness’ (i.e. accounts of our sharp offspring) have much to say. What’s important, for present purposes, is that penumbral connections are connections among our Useful predicates, which, as said, are not the genuinely vague predicates but, rather, the sharp derivatives on which standard (non-nihilistic) accounts focus.²⁰ Objection. You’ve preserved Full Tolerance, and you’ve done so in the only possible fashion: having no cutoffs. But while you don’t have typical cutoffs—some shift in semantic value—you still have a sort of cutoff: namely, the ‘core extension’ to which a vague intension relates a (vague) predicate. Why should the intension ‘yield’ that core extension, as opposed to some other—e.g. the next one? Reply. I have no worked out answer to the question, but I also see no reason to think that an answer is required. Full Tolerance, as above, requires us to recognize something strange about how the language works. My suggestion is that the strangeness emerges via relational intensions—intensions that, as it were, ‘determine’ an initial extension but ‘keep going’. Admittedly, there is something apparently arbitrary in the way that such (relational) intensions keep going; but there is also something arbitrary at any point at which they would stop (were they to stop, contrary to fact). What is nice about the current proposal is that it preserves the insight that any of our sharp, ‘offspring’ predicates are, in some sense, arbitrarily drawn. What is also nice is that vague intensions, as here characterized, provide the resources for sharpness, however seemingly arbitrary they may be. Objection. There is a related problem with the ‘core extensions’. Perhaps there is no need to speculate why this, rather than that, is the core extension that ρ relates to a vague predicate F . Even so, your ‘core extension’ still draws a boundary where, intuitively, vague intensions ought not draw boundaries. (See, e.g. Sainsbury 1997.) Reply. The push for boundary-less intensions arises, I think, from what appear to be inevitable shifts of semantic value—there’s an x and a y such that xR F y, but exactly one of x and y satisfies F . Such shifts are not part of the proposed account of vague predicates. A virtue of the present account is that, while there are indeed ‘boundaries’, at least in as much as there are extensions, no single such ‘boundary’ is privileged with respect to what makes a vague claim true. In that respect, there is a sense in ²⁰ The matter is actually more involved and largely turns on what sort of conditional is involved. I’ve been assuming a Strong Kleene setting for the overall language. As such, the only conditional with which to express ‘penumbral connections’ is the material conditional (the hook). Since vague predicates, according to the proposal, are one and all ‘gappy’, such a conditional wouldn’t do the trick; however, as above, the reply is that penumbral connections hold only for the sharp predicates. Whether a material conditional will work for such sharp predicates depends, in the end, on the account of sharpness (e.g. if they’re classical, then the hook might do the job).
Vague Intensions
197
which vague intensions are ‘boundary-less’, since any given extension is as good as any other, at least as far as the (ever-tolerant) vague intensions are concerned. (Such shifts, of course, are essential to the sharp offspring of the vague, which are our useful predicates.) Objection. Your proposed marriage preserves Full Tolerance for vague predicates, predicates that are thereby Useless (fail to satisfy Utility). Utility, in turn, is achieved by the sharp offspring of the vague, predicates that enjoy cutoffs and, hence, buck Full Tolerance. Such sharp offspring, according to the proposal, are the predicates that we typically use—since, again, the vague predicates themselves are entirely useless (except for the material that they provide for sharp predicates). The problem, though, is that the so-called offspring of the vague—the predicates towards which standard (non-nihilist) accounts are directed—are useful in virtue of tolerance, as Wright (1975) pointed out. Wherein does this fit? Reply. What Wright is talking about, of course, are our normal, sharp offspring of the vague. If Wright is right, then any account of those predicates will have to yield the sort of tolerance—far short of Full Tolerance—that is said to be required of such predicates. This is not an easy task, but it’s a task that, if Wright is right, must be met. That said, the requisite tolerance of our sharp predicates is not ruled out by the (modest) marriage. What is ruled out is that such predicates, given that they’re Useful, are fully tolerant, in the target sense (see Section 10.1). What’s interesting is that, on the current account, there’s a straightforward way in which the sharp offspring, regardless of the exact account (e.g. E1, E2 or so on from Section 10.5), immediately achieve a sort of tolerance—perhaps just the sort that Wright requires (though only details would tell). Consider the initial example, from Section 10.4, of how we might get some of our sharp predicates, namely, the supervaluational approach. On this account (which, admittedly, remains entirely abstract), we have a vague intension ρ for (vague) F . ρ, being a vague intension, delivers a host of extensions for F . We get a sharp descendent of F , namely, F , via general ized intersection: x is in the extension of F exactly if x is in F, where F comprises each Ei such that ρ(F , Ei ). What’s interesting is that F enjoys a natural sort of tolerance (though certainly short of Full Tolerance): for any x, if x satisfies F , and xR F y (i.e. stands in the vague predicate’s tolerance relation), then y is in some close extension of F . Given that, according to the going account, F and F are homonyms, it’s not surprising that we might think of F as tolerant—since it is, in at least the given respect. Objection. Standard (non-nihilist) accounts, you say, can be wedded to a nihilistic account of vagueness (according to which, e.g. vague predicates are neither true nor false of anything), and then be seen as accounts of how the sharp (cutoff-carrying) predicates work. The trouble with this suggestion is that many such standard accounts typically strive to retain classical logic. A marriage of nihilism (understood as above) and some such standard account would force a non-classical logic, at least on the proposed ‘Fregean’ approach to satisfaction (that results in vague predicates being neither true nor false of anything), but perhaps also on the Liberal approach (on which all
198
Jc Beall
vague claims are true).²¹ So, the proposed marriage is doomed from the start, or at least not something into which the standard, classically driven accounts would happily enter. Reply. It is true that, at least on the Fregean approach to satisfaction, the proposed marriage requires a non-classical logic. What to make of this depends on the arguments for classical logical theory, and I leave that topic for another venue. What should be noted is that, at least on the proposed Strong Kleene account, classical logic remains an extension (a proper extension) of the proposed logic. (In other words, if some argument is valid in SK, then it’s classically valid.) This sort of situation allows for an entirely classical fragment (proper fragment) of the language, wherein the logic may be (in effect) entirely classical. The proposal, of course, is that any such classical fragment would at best involve the sharp offspring of our vague predicates; it wouldn’t involve our vague predicates. Given that we normally use the sharp predicates—since the ‘real vague’ predicates are entirely useless (except for the material that they provide for our useful sharp predicates)—such a situation shouldn’t be overly troubling to those who think that, for the most part, classical logic is ‘right’. Objection. It’s plain that, on the proposal, the sorites puzzle is at best unsound if it uses genuinely vague predicates, since such predicates are neither true nor false of anything (at least given the proposed Fregean condition for satisfaction). But what of sorites that employ the normal, sharp offspring of the vague predicates? What is the reply? Reply. The reply will depend on the preferred account(s) of sharp offspring. As said, the proposed framework, at least in the abstract, is largely compatible with standard accounts; it’s just that they’re not, in the end, accounts of vague predicates, but rather accounts of vagueness-related predicates (viz., the so-called sharp offspring). As such, a classical, epistemicist account of the sorites might be the response, or perhaps a standard supervaluational account. Some such account must be given, I agree, but the issue turns on which account of our normal, sharp offspring is ultimately accepted. In this chapter, I offer no arguments one way or the other on that matter. On the other hand, the proposal does have something to say about typical ‘screams’ against cutoffs: they are misplaced if directed against accounts of the sorites that involve sharp offspring. Being useful predicates, the sharp offspring obviously have cutoffs, and as such their respective sorites arguments will be at least unsound. But screams are misplaced. The only fully tolerant predicates are the vague ones—and they’re thereby useless, except for providing the material for sharp offspring.
10.7
C LO S I N G R E M A R K S
Vagueness has gained a lot of attention in recent years. The sorites puzzle challenges us to arrest its slide without transgressing Full Tolerance—without requiring cutoffs. ²¹ As in Section 10.3, whether the Liberal approach requires a non-classical logic depends on the account of negation. If symmetry of sorites arguments is respected, then one will probably require a non-classical (paraconsistent) logic.
Vague Intensions
199
But that challenge is met only if vague predicates are Useless. The standard line is that vague predicates are obviously useful, and so we must find a way to explain away the strong appearance of Full Tolerance—accept cutoffs, but perhaps explain why they’re hard to accept. In this chapter I have suggested an alternative course: we may accept Full Tolerance (the absence of cutoffs) as a central feature of genuinely vague predicates, but explain the apparent Utility of such predicates as arising from something else—namely, sharp homonyms of the vague predicates. Genuinely vague predicates fail to be Useful in the usual sense, but they still provide a service; they provide the ingredients for cutoff intensions of (homonymous) sharp predicates. It is such sharp (homonymous) predicates to which typical (non-nihilist) accounts are directed. The proposal, in the end, is very simple, but it is also potentially liberating. When we recognize that the pull of Full Tolerance and Utility is rooted in conflation—the conflation of vague predicates and their sharp (look-alike) offspring—the tension might well be diminished. There is still work to do on what sort of ‘tolerance’, if any, the sharp offspring may enjoy; but we know, in advance, that such tolerance will be short of Full Tolerance, which is enjoyed only by non-Useful, vague progenitors of our normal, sharp look-alikes. As such, we can expect cutoffs from accounts of the sharp look-alikes. At the very least, the site of such cutoffs is no place to scream. Re f e re n c e s Beall, Jc and Colyvan, Mark (2001), ‘From heaps of gluts to Hyde-ing the sorites’, Mind 110: 401–8. Beall, Jc and van Fraassen, Bas C. (2003), Possibilities and Paradox: An Introduction to NonClassical Logic, Oxford University Press, Oxford. Braun, David and Sider, Ted (2007), ‘Vague, so untrue’, Noˆus 41: 133–56. Fara, Delia Graff (2000), Shifting sands: an interest-relative theory of vagueness’, Philosophical Topics 28: 45–81. Fine, Kit (1975), ‘Vagueness, truth and logic’, Synthese 30: 265–300. Hyde, Dominic (1997), From heaps and gaps to heaps of gluts’, Mind 106: 641–60. Keefe, Rosanna (2000), Theories of Vagueness, Cambridge University Press, Cambridge. Priest, Graham (2003), ‘A site for sorites’ in Jc Beall, ed., Liars and Heaps: New Essays on Paradox Oxford University Press, Oxford, 9–23. Raffman, Diana (1996), ‘Vagueness and context sensitivity’, Philosophical Studies 81: 175–92. Sainsbury, Mark (1990), ‘Concepts without boundaries’ in Rosanna Keefe and Peter Smith, ed., Vagueness: A Reader, MIT, Cambridge, MA, 1997. This was a lecture at King’s College, London. Shapiro, Stewart (2006), Vagueness in Context, Oxford University Press, Oxford. Sorensen, Roy (2000), Vagueness and Contradiction, Clarendon, Oxford. Unger, Peter (1979), ‘There are no ordinary things’, Synthese 4: 117–54. Varzi, Achille (1999), An Essay in Universal Semantics, vol. 1 of Topoi Library, Kluwer Academic Publishers, Boston, 1999. Weatherson, Brian (2005), ‘True, truer, and truest’, Philosophical Studies 123: 47–70. Wheeler, Samuel C. III (1979), ‘On that which is not’, Synthese 41: 155–94. Williamson, Timothy (1994), ‘Vagueness, Routledge. Wright, Crispin (1975), ‘On the coherence of vague predicates’, Synthese 30: 325–65.
11 This Magic Moment: Horwich on the Boundaries of Vague Terms Hartry Field
Consider the following argument: (1) Bertrand Russell was old at age 3 × 1018 nanoseconds (that’s about 95 years) (2) He wasn’t old at age 0 nanoseconds (3) So there is a number N such that he was old at N nanoseconds and not old at k nanoseconds for any k < N . Presumably he was old for k > N as well as for k = N . Given this, (3) says that there’s a sharp boundary between when he was old and when he wasn’t. (You could of course make it sharper than a nanosecond, by further division of the unit.)¹ Many people find the conclusion extremely counterintuitive; but premises (1) and (2) seem incontrovertible, and (3) follows from them using the least number principle. Could it be that there’s a problem with the unrestricted application of the least number principle to vague or otherwise indeterminate concepts? Paul Horwich considers this in ‘The sharpness of vague terms’ (chapter 4 of Horwich 2005), but says (I) that such a position is an act of desperation, and (II) that when (3) is properly understood it shouldn’t seem particularly counterintuitive. I will consider these claims in reverse order. 11.1
ARE SHARP BOUNDARIES COUNTERINTUITIVE? – I
One worry about (3) is that our inability to know which number is the critical number N (i.e. which moment is the magic moment) doesn’t seem at all like other cases of This chapter was presented at the Conference on Paul Horwich’s Minimalist Theory of Meaning and Truth, at the University of Pecs, on May 16 2006. Thanks to Paul Horwich for helpful discussions of these issues. ¹ Obviously, ‘old’ is context-dependent, so the argument is really that ‘old’ as used on a given occasion has a sharp boundary. I’ll follow Horwich in ignoring the context-dependence of vague terms in what follows.
200
On the Boundaries of Vague Terms
201
inevitable ignorance (e.g. the impossibility of knowing certain details of what’s going on inside a specific black hole). Can we do justice to the evident difference? Horwich thinks we can do justice to the difference. To this end he offers the following account of our inability to know the location of the boundary: (a) Learning to employ ‘old’ involves something like acquiring a pattern of conditional degrees of inclination to accept ‘S is old’ or ‘S is not old’ on the basis of beliefs or assumptions about ‘underlying parameters’, e.g. the time since S’s birth. (b) (i) This learned pattern of conditional degrees of inclination doesn’t include any stable inclination to apply either ‘S is old’ or ‘S is not old’ when the assumed value of the ‘time since birth’ parameter is in a mid-range. (We might have unstable inclinations.) (ii) The learned pattern may even include a prohibition against having a stable inclination to accept either ‘S is old’ or ‘S is not old’ in these cases. ( This is of course compatible with our stably accepting the disjunction ‘S is either old or not old’, and Horwich thinks we should stably accept that.) (c) Were we to substantially change the inclinations, say by starting to stably call people known to be in the mid-range ‘old’, this would constitute a change of meaning in the term. (d) Knowledge requires stable belief, not based on error about the values of the underlying parameters; so one couldn’t know of someone in the midrange that he is old, on the current meaning of ‘old’. Let’s grant that this explains why we can’t know the the location of the borderline for our term ‘old’. (I have doubts about the way that the notion of change of meaning is being deployed, but this isn’t the place to pursue them.) But one of the things that seems to separate our ignorance of the borderline from ignorance of the details inside a black hole is that in the latter case there seems to be no conceptual incoherence in supposing an omniscient god who does know the details inside the black hole; whereas not even an omniscient god could know ‘the location of the boundary between the old and the non-old’. More generally, the following seems to be part of our conception of vagueness: (∗ ) Not only can’t we know any claim that purports to give ‘the location of the boundary’, no superior intelligence (god, Martian, whatever) could know such a claim either, even if that superior being thought in a system of representation very unlike ours. At first blush it would seem that Horwich would have to deny the italicized claim, and reject (∗ ). For if there is a unique critical number N (i.e. a unique N such that Russell was old at N nanoseconds and not old a nanosecond before), then there’s a truth about what that number is, and so a fact about what that number is, and an omniscient being by definition knows all the facts. Some might try to dispute this with a highly ‘inflationary’ notion of truth or fact, but certainly Horwich can’t. So it initially surprised me to find Horwich asserting (∗ ); asserting, indeed, that there is a conceptual
202
Hartry Field
incoherence in supposing that a god or a Martian knows the boundary of ‘old’. How can he consistently say that? The answer, I think, is that Horwich is implicitly telling us that the concept of an omniscient being is conceptually incoherent: since it is conceptually incoherent to suppose that anyone knows the location of the boundary, and since an omniscient being would have to know the location, omniscience is conceptually incoherent. Postulating an omniscient being is like postulating a square circle. (A surprising contribution to the philosophy of religion!) So far so good. But if there really is a critical number N , how are we to argue for (∗ )? (a)–(d) above explain (let’s assume) why we can’t know the value of N , but why couldn’t someone else who thought or spoke only in a different language know it? Horwich’s argument that they couldn’t appeals to a version of deflationism, according to which the only way for someone (e.g. god or Martian) to argue that one of our terms α is true of an object k is to first argue that our term α means the same as some term β of his own language, and then argue that his own term β applies to k. So if his term β really does mean the same as our term ‘old’, it is governed by the same rule for belief, and so he is prohibited from stably applying either β or its negation to the mid-range case. And so he can have no stable view about whether our term α applies to that case. This is an ingenious argument, but I don’t think it’s believable. Suppose I hear mathematicians employing a word ‘borniforous’ that’s completely unfamiliar to me, and hear them say that borniforous things are mathematical objects of some sort. An anolog of Horwich’s argument says that until I’m in a position to come up with a synonymous expression that I understand, I can’t legitimately believe that their word doesn’t apply to snails. That seems absurd. A deflationist should say not that we translate ‘borniforous’ into a pre-existing expression of our language but that we incorporate it into our language, with whatever conceptual role we can give it. In this case the conceptual role we give it will have to be more impoverished than what the mathematicians give it, but this doesn’t keep us from regarding it as equivalent in extension to theirs (especially if we insist, as Horwich must, that it has a definite extension). We’ll then hold that ‘borniforous’ as the mathematicians use it is true of precisely the borniforous things. Similarly, the god can incorporate our ‘old’ into her language, and use it to state the extension of our ‘old’. If ‘old’ has precise boundaries, her omniscience requires her having true stable beliefs about them, but now there’s no reason why this should be impossible. 11.2
ARE SHARP BOUNDARIES COUNTERINTUITIVE? – II
I turn to a quite different worry. Suppose we grant that Horwich has explained why no one, no matter how superior to us, can have knowledge of or stable belief in the location of the alleged sharp boundary between old and non-old. Still, that isn’t enough to explain away reasonable doubts about there being such a boundary. For instance, imagine someone, Roger, who thinks his life will go better if the number he chooses for his bank account password ends in the same digit as the last digit of
On the Boundaries of Vague Terms
203
the critical number N than it would if he had chosen a different last digit. (He realizes he’ll never know if he made the right choice.) Compare Roger to Sam, who thinks his life will go better if the number he chooses for his bank account password ends in the seventeenth significant digit of the Centigrade temperature at the currently hottest point in the interior of the sun. Sam’s belief is thoroughly irrational, but Roger’s intuitively seems even worse: it seems based on a kind of conceptual confusion about vagueness. It’s hard to see how one could explain this on Horwich’s account; certainly showing the impossibility of knowledge of the location of the boundary, or stable belief about it, doesn’t address this. There are many other examples to the same effect. Imagine a person who knows his own age very precisely (and knows his own level of physical decrepitude, and any other physical facts that might be related to whether he counts as old), but who wonders whether he’s passed the alleged critical point that marks the unknown boundary between the old and non-old, and hopes that he hasn’t. It isn’t enough to say that his wondering about this is idle, and that his hope is irrational since it can’t affect anything of interest to him; that’s true of someone who wonder’s or hopes about the seventeenth significant digit of the Centigrade temperature at the currently hottest point in the interior of the sun, but as above, the critical number case seems different in being conceptually confused. (Indeed, it isn’t clear that irrationality without conceptual confusion can explain the critical age example. Many people idly wonder or hope about things of no independent interest: consider a dying man who wonders whether the Red Sox will win the pennant, and hopes that they will, though he knows he won’t be around to savor it. For a believer in a sharp boundary, why should wondering and hoping about whether he’s passed the boundary be any more irrational than that?) A third example involves moral attitudes. To make the example somewhat realistic, it’s better to switch from ‘the nanosecond where Russell became old’ to ‘the nanosecond where a given life begins’. Obviously the argument carries over to this case; that is, classical logic supplies an argument that there is a precise moment at which life begins. Given this, it is hard to see why someone shouldn’t attach moral significance to that moment, and be deeply troubled by the thought that the magic moment might be earlier than he’d assumed (say at conception rather than at birth), making some action that he’s already performed at a point in between deeply immoral. But it seems to me that this attitude, while sensible according to a supernaturalist metaphysics according to which at some unknown point our bodies are (instantaneously) infused by ‘vital substance’, is not a sensible attitude for anyone who regards the question of when life begins as vague. And it seems hard to explain why this should be, if there is a true answer to the question ‘At which moment does life begin?’ Horwich sometimes says that the answers to questions like whether the critical number N is odd or even are ‘indeterminate’, but this is no help: for him this just means that the answers aren’t stably believable, so it in no way reduces the worries raised in this section. ( This is in contrast to the role of indeterminacy in views which restrict excluded middle: there, believing of a specific claim that it is indeterminate commits one to rejecting the corresponding instance of excluded middle.)
204
Hartry Field
I’m sure there’s nothing in what I’ve said that a defender of sharp boundaries would be unable to swallow. I do think, though, that there’s something quite counterintuitive in the postulation of sharp boundaries; an explanation of why knowledge of or stable belief in their location is conceptually impossible does not make the counterintuitiveness go away.
11.3
AC T O F D E S PE R AT I O N ?
What is required to avoid the conclusion that there is a sharp line separating when Russell was old from when he wasn’t? We certainly don’t need to completely jettison the classical least number principle (CLNP) ∃nF (n) implies ∃N [F (N ) ∧ (∀k < N )¬F (k)]. We need only to weaken it slightly, in a way that allows it to have significant application even if excluded middled isn’t assumed to hold generally: (GLNP) ∃n[F (n) ∧ (∀k < n)(F (k) ∨ ¬F (k))] implies ∃N [F (N ) ∧ (∀k < N ) ¬F (k)]. (‘G’ is for ‘generalized’.) When F obeys excluded middle, (GLNP) reduces to (CLNP). Presumably excluded middle holds throughout mathematics, and indeed whenever vagueness and related phenomena are not at issue. This suffices to explain why in dealing with precise language (or language that can be taken to be precise for the practical purposes at hand), (CLNP) can be assumed. But when vagueness is at issue, we can avoid the argument for sharp boundaries by restricting excluded middle where vague concepts are concerned, and recognizing that if we do so then the appropriate form of least number principle is (GLNP). To put it in a slightly sloppy but picturesque manner, the idea is that for numbers n in a certain range, the claim that Russell was old at n nanoseconds is ‘fuzzy’: it’s inappropriate to assume that at that age he was either old or not old. ( The range where this is fuzzy is itself fuzzy; this is crucial to the plausibility of the approach, and I will discuss it in a moment.) Given this, it’s inappropriate to assume that there is a first n at which he was old. For to say of any given n that it is the first is to say that he was old at n nanoseconds of age and not old at n − 1, and this will be fuzzy at best: it will be fuzzy if n or n − 1 falls into the fuzzy region, false if they both fall outside it. (If it’s fuzzy whether one of them falls into the fuzzy region, then it will be fuzzy whether the claim that n is the first one at which Russell was old is fuzzy: we will have higher order fuzziness. But in any case, the claim that n is the first won’t be clearly true.) Supervaluationists argue that even if all claims of form ‘n is the first natural number such that Russell was old at n nanoseconds of age’ are at best fuzzy, still the existential generalization is clearly true. It seems to me that such a view does not avoid positing sharp boundaries, though I will not pause to argue that. My view is that if one wants to avoid positing sharp boundaries, one should take an existential generalization of claims that are fuzzy at best to be fuzzy. If so, it is inappropriate to believe ‘There is a first natural number n such that Russell was old at n nanoseconds
On the Boundaries of Vague Terms
205
of age’. ( That doesn’t mean one ought to believe that there is no first nanosecond at which he was old. Negations of fuzzy claims are fuzzy too, hence inappropriate to believe.) Views of this sort obviously raise a huge number of questions, and it is not my purpose here to discuss any of them in detail. Indeed, I won’t discuss most of them at all, but I do want to mention one worry very briefly. The worry is that an approach like this can avoid a sharp line between the old and the not-old only by introducing other sharp lines, e.g. between one might call the determinately old and the not determinately old. The thought is that even if the law of excluded middle doesn’t apply to the predicate ‘old’, it must apply to the predicate ‘determinately old’; in which case there must be a first nanosecond at which Russell is determinately old. If so, little progress would have been made. My answer to this is that on any reasonable way of introducing the notion of determinately old, excluded middle cannot be assumed to hold of it. Consider a few representative attempts: (A) Russell is determinately old at those nanoseconds for which (i) he’s old and (ii) it is true that he’s either old or not old; (B) Russell is determinately old at those nanoseconds for which (i) he’s old and (ii) it is appropriate to assume that he’s either old or not old; (C) Russell is determinately old at those nanoseconds for which it is neither the case that he is not old nor that it’s ‘fuzzy’ whether he’s old. But on (A), there’s no distinction between ‘determinately old’ and ‘old’. As Horwich rightly insists, ‘True()’ is equivalent to ‘p’; so A(ii) is equivalent to ‘he’s either old or not old’, which is strictly weaker than A(i); so when conjoined with A(i) one just gets A(i), i.e. ‘he’s old’. With (B), adding (ii) does produce a genuine strengthening. But given that ‘appropriate’ is obviously vague, there’s still no reason to think that ‘determinately old’ has sharp boundaries. The situation with (C) is similar to that of (B): it’s unclear how exactly to explain ‘fuzzy’, but it seems like however one explains it, it’s bound to itself be vague. Still, there’s a substantial worry: that we could produce a sharp border by iterating a ‘non-fuzziness’ operator into the transfinite. That is, why doesn’t the sequence old; old and not fuzzy whether old; old, not fuzzy whether old, and not fuzzy whether fuzzy whether old; and so forth
collapse to a bivalent predicate by level ω or by some higher transfinite level γ ? If this were to happen—and it does happen in many standard proposals for nonclassical logics for vagueness, e.g. the Lukasiewicz continuum-valued logic—then that would be a disaster. For then there would be a number N such that Russell was determinatelyγ old at nanosecond N but not determinatelyγ old a nanosecond before; we’d have a sharp boundary for ‘determinatelyγ old’, so why not just take this as the sharp boundary for ‘old’?
206
Hartry Field
If the determinately operator collapses to bivalence in this way, nothing would be gained by going non-classical. And it is a delicate matter to get a non-classical logic of vagueness in which such a collapse is avoided. Still, it can be done: there are reasonable logics of vagueness in which such a collapse never occurs.² Admittedly, a non-classical logic appropriate to vagueness is somewhat complicated. Given that almost every term is somewhat vague, wouldn’t the non-classical approach make proper reasoning about ordinary subjects difficult? I think this worry is exaggerated. It might be useful to compare the case to geometric reasoning. We all know that space is not quite Euclidean, and indeed fails to be Euclidean in a quite complicated way; nonetheless, we are safe in using Euclidean reasoning except in special contexts, because the error involved in doing so is so slight. That is the policy I recommend for logic: reason classically, except for those situations where there is reason to think that the errors induced by such reasoning are significant. Situations where we derive boundaries for vague terms look like just the sort of situation to worry about! 11.4
B ROA D E N I N G T H E R A N G E O F C O N S I D E R AT I O N S
How do we decide between a classical logic approach to vagueness, which must postulate sharp borders, and a non-classical approach that avoids this but complicates the logic? It’s a matter of weighing costs and benefits. I haven’t tried to argue that the weight of the benefits is on the non-classical side: that would be a big task. Rather, I’ve just tried to argue that the non-classical approach is not without motivation (especially if it avoids the danger of collapsing determinately operators). I’d like to conclude by mentioning an additional item on the nonclassical side of the ledger: Berry’s paradox. Say that a 1-place formula F (x) of English is uniquely true of an object c if it is true of c and of nothing else. Let an S-formula be a formula of English with less than a thousand symbols. Then 1. There are only finitely many S-formulas; since there are infinitely many natural numbers, there must be natural numbers that no S-formula is uniquely true of. So, by the least number principle, 2. There is a smallest natural number M such that no S-formula is uniquely true of it. But ‘x is the smallest natural number such that no formula of English with less than a thousand symbols is uniquely true of it’ is an S-formula. So 3. ‘x is the smallest natural number such that no formula of English with less than a thousand symbols is uniquely true of it’ isn’t uniquely true of M ; that is, it isn’t uniquely true of the smallest natural number such that no formula of English with fewer than 1000 symbols is uniquely true of it. In other words, ² The logic advocated in Field 2008 is one such.
On the Boundaries of Vague Terms
207
3∗ . ‘x is the smallest natural number such that no formula of English with fewer than 1000 symbols is uniquely true of it’ either (i) isn’t true of the smallest natural number such that no formula of English with fewer than 1000 symbols is uniquely true of it, or (ii) is true of things other than the smallest natural number such that no formula of English with fewer than 1000 symbols is uniquely true of it. Either option is thoroughly counterintuitive, and a gross violation of the schema ( T ) ‘F (x)’ is true of c if and only if F (c). Horwich often emphasizes the centrality and importance of the truth-of schema, and for good reason. But we see that the unrestricted least number principle forces a violation of that schema. This is a substantial consideration in favor of restricting the least number principle in the context of vagueness, and hence in favor of restricting the law of excluded middle which underlies it. Horwich takes the opposite stance, of restricting not excluded middle but the truth-of schema. But that has a high cost. Let’s look at the point of the notions of truth and truth-of. Sticking to truth for simplicity: suppose I forget the details of what a doomsayer said yesterday, but remember the gist well enough to conclude: If everything he said yesterday is true, then we’re in trouble. On the assumption that what he said was p1 , . . . , pn , this had better be equivalent to If p1 and . . . and pn , then we’re in trouble. This requires the intersubstitutivity of True(
) with p in extensional contexts. Given the very minimal law p ↔ p, this yields the truth schema True(
)↔ p. ( The situation with ‘true of ’ is similar.) Restricting intersubstitutivity restricts the ability to generalize in a reasonable way, leading to extreme pathologies in theories of truth that reject intersubstitutivity or the truth schema. But we can keep the intersubstituvity principle and the truth schema unrestricted if we weaken excluded middle (and more or less equivalently, the least number principle); similarly for truth-of. We can do this in a way that allows for fully classical reasoning when no ‘ungrounded’ uses of ‘true’ are present (and when vagueness isn’t at issue): for instance, we can accept classical reasoning within mathematics without restriction. And the logics that keep the truth schema (and the intersubstitutivity of True(
) with p) seem to be fully suited to deal with vague and indeterminate concepts in the way sketched earlier. I think this is no accident: there’s a strong intuitive connection between the Sorites example (Russell is old) and the Berry paradox example. Moreover, the obstacles that must be overcome in getting a logic that adequately handles vagueness and the semantic paradoxes are pretty much the same in
208
Hartry Field
both cases: for instance, in both cases we must make sure that no determinacy operator collapses to bivalence when iterated. ( This is why the Lukasiewicz logic fails both as a logic for vagueness and as a logic for the semantic paradoxes.) That there is a connection here is of no surprise. Vague concepts and ‘true’ seem species of indeterminate concepts. ‘True’ initially seems determinate, because it seems that the truth schema True(
) ↔ p settles its extension. But once we reflect on ‘ungrounded’ sentences (such as Truthteller sentences, which assert their own truth; and Liar sentences, which assert their own untruth), we see that this is an illusion. This connection makes it natural to use the same logic for such ‘ungrounded’ applications of ‘true’ as for vague predicates. That there is a link between the semantic paradoxes and the paradoxes of vagueness is perhaps further suggested by another paradox of the same ilk, which seems to have ties to both. (I think I first heard of it many years ago in Martin Gardner’s Scientific American column.) Some natural numbers aren’t very interesting. So there must be a smallest one that isn’t very interesting. The smallest one that isn’t very interesting! What an interesting number! Contradiction. (In case anyone is tempted to regard this as a proof that every natural number is very interesting, it’s worth remarking that an analogous proof using the classically correct least ordinal principle yields that every ordinal number is very interesting. Since for any cardinal number c, there are more than c ordinal numbers, it seems quite surprising that interest can extend so far!) Another kind of paradox that suggests a connection is what Sorenson calls a ‘nono’ paradox: Person A asserts that what person B is saying is not true, at the same time that person B says that what person A says isn’t true. Classically, either what A says is true and what B says isn’t, or vice versa; and yet A and B seem symmetrically placed. (We might even imagine that A and B are Doppelgangers in a completely symmetric universe; in which case we have a failure of truth to supervene on non-semantic facts.) Intuitively this is a kind of underdetermination reminiscent of vagueness, and the paradox arises only from the supposition of excluded middle. To summarize, I think there is considerable pressure in the vagueness case to slightly weaken the logic so as to avoid postulating counterintuitive boundaries, and even more pressure in the semantic paradox case to weaken the logic in the same way to enable us to keep the truth and truth-of schemas. These two pressures to weaken the logic are, I think, mutually reinforcing, and succumbing to this joint pressure is not the desperate measure that Horwich suggests it is. Re f e re n c e s Field, Hartry (2008), Saving Truth from Paradox, Oxford University Press, Oxford. Horwich, Paul (2005), Reflections on Meaning, Oxford University Press, Oxford.
12 Perceptual Indiscriminability and the Concept of a Color Shade Leon Horsten
12.1
I N D I S C R I M I N A B I L I T Y A N D C O LO R S H A D E S
We shall be concerned with visual indiscriminability, and more specifically on perceptual indiscriminability of colors. But, as usual, the discussion is intended to carry some more generality. It intends to contain some lessons for the philosophical theory of perceptual qualia in general. According to the received view, perceptual indiscriminability is a nontransitive relation. Until about a decade ago, there was a nigh consensus in the philosophical literature that the received view is correct. Today, this consensus has dissolved. This has prompted me to make an attempt at reevaluating the question of the transitivity of perceptual indiscriminability. In the first sections of this chapter, recent challenges to the thesis of the nontransitivity of perceptual indiscriminability will be critically examined. Such challenges can take different forms. Some of them concentrate on philosophical arguments for the nontransitivity thesis; some concentrate on alleged empirical evidence for the claim that indiscriminability is nontransitive. Fara’s challenge belongs to the first kind. She has criticized Wright’s influential philosophical argument that was intended to establish that indiscriminability is nontransitive. We shall investigate to what extent Wright’s argumentation can be upheld in the face of Fara’s reply. Raffman has emphasized that the question of the transitivity of the relation of perceptual indiscriminability is ultimately empirical in nature. And she believes that the alleged empirical evidence in favor of the nontransitivity thesis is far from I am indebted to the participants in the Arch´e Vagueness Conference (St. Andrews, 2007) for insightful comments and helpful suggestions. My thanks go especially to Paul Egr´e (who commented on this chapter), Diana Raffman, Patrick Greenough, Crispin Wright, Nathan Salmon, and Anthony Everett. Section 12.3 and section 12.6 are based on earlier joint work with Rafael De Clercq, to whom I am also grateful for discussions specifically relating to the problems with which the present chapter is concerned. It should not be assumed that Rafael De Clercq agrees with the overall position that is articulated and defended in this chapter.
209
210
Leon Horsten
conclusive. She argues that a contextualist interpretation of the data can be given which leaves ample room for the thesis that perceptual indiscriminability is a transitive relation after all. We shall see that whereas her arguments are not conclusive, there is much to be learned from her considerations. If perceptual indiscriminability is transitive, then at least a criterion of identity for color shades is readily obtained by Frege’s familiar method of abstraction. But if perceptual indiscriminability is a nontransitive relation after all, then we cannot rely on the method of abstraction to yield an identity criterion for color shades—at least not in any straightforward manner. The question then becomes acute whether the concept of a color shade is coherent in the first place, and, if so, which form it could take. It shall be argued that in this situation, it must be conceded that the concept of a color shade is to some extent theoretical in nature. But since the concept of a color shade is invoked in the first place to make sense of our color discrimination judgements, it must respect these as much as possible. We shall therefore inquire how a concept of a color shade can be obtained which is, in the face of nontransitivity of indiscriminability, maximally faithful to our color indiscriminability judgements. But faithfulness to indiscriminability judgements is not the only requirement. The resulting concept of a color shade should also be well in agreement with the way in which the concept of a color shade is used in natural language. So we must also ask ourselves whether this requirement, too, is satisfied. In what follows, I have in mind the notion of indiscriminability in color for a given agent which is kept fixed throughout the discussion. This agent is assumed to have capacities for discriminating between colors of objects that are or fairly average for a human being. Two objects are said to be indiscriminable in color by the agent if she cannot discriminate between them with respect to color on the basis of a direct color comparison. Thus the relation of discriminability with which we are operating coincides with what Goodman called the matching relation. Indiscriminability with respect to color does not exclude that there are other ways in which the agent can distinguish the colors of objects. Suppose, for instance, that there is a nontransitive triad of objects x, y, and z. On the basis of a direct color comparison, x cannot be discriminated from y, y cannot be discriminated from z, but x can be discriminated from z. Then our agent can reason that since x is discriminable from z whereas y is not, x and y must have a slightly different color. Still, in our sense of the word, x and y remain indiscriminable. In other words, the concept of indiscriminability as it will function in our discussion is assumed to be phenomenal. Indiscriminability will also be taken to be a judgemental relation. For two objects to be indiscriminable in color for our agent, she has to be able to judge them to be indiscriminable when she visually compares them. Likewise, for these objects to be discriminable for her, she has to be able to judge them to be discriminable on the basis of a visual comparison.
Perceptual Indiscriminability and the Concept of a Color Shade 12.2
211
W R I G H T ’ S N O N T R A N S I T I V I T Y A RG U M E N T
Perceptual indiscriminability is nontransitive if it is possible to have three items x, y, z such that x is perceptually indiscriminable from y, y is perceptually indiscriminable from z, and x is perceptually discriminable from z. This entails that precisely one of the following two theses must hold: The Transitivity Thesis Perceptual indiscriminability is a transitive relation. The Nontransitivity Thesis Perceptual indiscriminability is a nontransitive relation. Until fairly recently, it was common philosophical practice to accept the nontransitivity thesis (and thus to reject the transitivity thesis) on the strength that we can imagine a process of gradual change in which a series of unnoticeably small changes finally add up to a noticeable change (in respect of a given quality). Several authors have sought to show that perceptual indiscriminability is nontransitive in this way. Here we discuss an influential philosophical argument by Crispin Wright (Wright 1975, 345–7). Wright presents his proof as a reductio showing that the nontransitivity thesis follows from the possibility of phenomenal continua. The argument can be paraphrased as follows. Suppose that indiscriminability is transitive. Then consider a process of change in respect of some observable property (think of it as a determinable such as color, position or pitch). The process is composed of stages between which there is no seemingly abrupt transition, and is non-recurrent in that for two distinct stages x and y, with x preceding y, there is no later stage z such that z is more like x (in respect of the observable property) than y is. Take any two stages Di and Dj such that Dj is discriminable from Di and yet close enough to it to guarantee that all stages lying in between are either indiscriminable from Di or indiscriminable from Dj . In other words, the intermediate stages will appear to have the same determinate of the determinable as either of the two surrounding stages (e.g. the same shade of color). They cannot be indiscriminable from both Di and Dj since being-indiscriminable-from is supposed to be a transitive relation. As a result, the region between Di and Dj will divide into two adjacent sub-regions, one consisting of stages indiscriminable from Di , the other consisting of stages indiscriminable from Dj . Since indiscriminability is supposed to be transitive and since Di is discriminable from Dj any stage belonging to the first sub-region will likewise be discriminable from any stage belonging the second sub-region. However, if this is true, then, contrary to what we have been assuming, a seemingly abrupt change must occur between Di and Dj . In recent years, the nontransitivity thesis has been called into doubt. These challenges tend not to consist of arguments that directly support the thesis of the transitivity of indiscriminability. Rather, they consist in the first place of attempts to undermine the alleged evidence for the nontransitivity thesis. In the next section, I shall scrutinize Fara’s critique of Wright’s philosophical argument. In section 12.4, we shall turn to Raffman’s attempt to deflect the charge that we have conclusive empirical support for the nontransitivity thesis.
212
Leon Horsten 12.3
U N N OT I C E D A P PA R E N T C H A N G E S
According to Fara, Wright’s proof relies on two assumptions: the possibility of phenomenal continua and the finiteness of human discriminatory powers (Fara 2001, 931). The first assumption is needed to deny the existence of a seemingly abrupt transition from one stage to another. The second assumption allows for perceptually indiscriminable stages in the process. According to Fara these two assumptions ‘are, taken individually, not implausible [but] they are in so much tension with each other that it is utterly unreasonable to accept them jointly when neither has anything remotely like adequate support’ (Fara 2001, 931). Closer inspection of Wright’s argument reveals that the first assumption, concerning the possibility of phenomenal continua, is not necessary for his argument. In fact, it is not hard to see that Wright’s argument, reduced to its essentials, is exceedingly simple. Aside from a plausible physical assumption, the finiteness of our powers of discrimination is all that is needed for the argument. Let there be given an observable physical quantity Q. Suppose that the value of this quantity can be expressed as a real number. (Thus, the quantity Q can be regarded as a determinable with specific values as determinates). And adopt the physical continuity assumption that the value of Q varies according to some smooth continuous function (in the mathematical sense of the word!) through time. Let ri refer to the value of quantity Q at time i. Now we assume finite discriminability in the sense that (i) there are ra , rb such that the subject can discriminate between them and (ii) there is a d ∈ R such that if ri − rj < d , then a given person is unable to perceptually discriminate between Q at i and Q at j. Now consider a finite chain ra = r0 , r1 , . . . , rn = rb , such that for each ri in the chain, ri+1 − ri < d . The foregoing assumptions entail that such a chain exists. Moreover, finite discriminability entails that a subject perceiving the chain will not notice ‘an abrupt change’, which means that the change in Q will be perceived as continuous in Wright’s (phenomenal) sense. Elementary mathematical considerations show immediately that this chain must contain a violation of transitivity of indiscriminability. After all, since each element in the chain is indiscriminable from the next with respect to Q, transitivity would imply that the first element is indiscriminable from the last. However, by assumption ra is discriminable from rb with respect to Q. So in the final analysis, all rests on the assumption of finite discriminability.¹ Fara does not find this assumption evident. In her discussion of the phenomenon of ‘slow motion’, she writes (Fara 2001, 928): . . . we have two competing explanations of what is going on when the hour-hand of a clock looks to have moved over some long [time] interval, but also seems to have looked still during every sufficiently short sub-interval. The first explanation is that when we judge the hour-hand to look still, say for every twenty-second period, it does in fact look to be in the same position at the end of each period as at the start. The alternative explanation is that when we judge the hour-hand to look still, although there is at least one twenty-second period for which it does ¹ One may wonder at this point why Fara believes that the phenomenal continuity assumption is in tension with the finite discriminability assumption. A diagnosis is offered in De Clercq and Horsten (2004, section II).
Perceptual Indiscriminability and the Concept of a Color Shade
213
not look in the same position at the end as at the start, we do not notice this. Noticing the change in an apparent position requires not only that there be an apparent change, but also that we believe there to be one. [emphasis in the original]
In other words, according to one explanation of what happens when the hour-hand of a clock changes unnoticeably, there is no apparent change because there does not appear to be a change: at least at a conscious level, things look exactly the same before and after the change. This explanation seems plausible enough. However, Fara’s sympathy lies with the other explanation: the apparent position of the hour-hand of a clock—the position it appears to have—changes constantly, i.e. even within time intervals that are so short that we are unable to tell (‘notice’) whether there has been a change. Neuro-psychological research has born out that we routinely respond to visual stimuli of which we have no conscious awareness.² In the situation that Fara describes, too, it may be that we are behaviorally able to respond in some way to the change of the position of the hour-hand in a twenty seconds period even though we are not consciously aware of a position change. So, in this sense, it might be said that the position change may not completely escape our attention even if it escapes our conscious attention. Elsewhere in her paper, Fara argues that accepting the nontransitivity of ‘looking the same as’ does insufficient justice to the phenomenal character of looks (Fara 2001, 932). After all, if ‘looking the same as’ is transitive, then looks can simply be taken to be equivalence classes of the relation; and if ‘looking the same as’ is nontransitive, then one must either maintain that there are things which look the same (in some respect) but nevertheless do not have the same look, or that there are things which look different but have the same look, or both. However, if this objection is justified, then it might be argued that Fara stands guilty of a similar charge. Fara’s concept of indiscriminability is in the end a subjudgemental and subconscious relation; it can fail to obtain in a comparison test without the agent being conscious that it fails to obtain. By separating the notion of apparent change from the notion of noticed change, Fara may then be said to deprive the notion of apparent change of its phenomenal nature. At best, the notion of apparent change becomes one that is determined by our (partially unconscious) response-behavior rather than by the contents of our explicit consciousness. Nothing can prevent Fara from abstracting a notion of color shade from such a behavioral relation. But this notion of color shade will fail to qualify as fully phenomenal. 12.4
A P PE A R I N G I N C O N T E X T
Raffman has articulated a contextualist position on the basis of which she criticizes Wright’s argument for the nontransitivity of indiscriminability (Raffman 2000). She ² See Weiskrantz (1986).
214
Leon Horsten
is concerned with the relation of perceptual indiscriminability of objects, in respect of a perceptual property (Raffman 2000, 155). This should be interpreted loosely. It is intended to include situations where someone is asked to discriminate visual images or parts of visual images with respect to color. Raffman’s arguments are not intended to establish but only to make room for the hypothesis that perceptual indiscriminability is a transitive relation (Raffman 2000, 154–5). She says of her her argumentation that much of it is ‘speculative [...] and in need of empirical test’ (Raffman 2000, 155). The contextualist hypothesis can roughly be formulated in the following way: The Contextualist Hypothesis When objects x and y are mentally compared with respect to perceptual property P with the aim of reaching a discrimination judgement, contextual factors can and typically do influence the agent’s perception of x and of y. This hypothesis seems eminently plausible, and is supported by a large body of empirical data, which go under the rubric of ‘contrast effects’. The contextualist hypothesis is used by Raffman to undermine arguments for the nontransitivity of perceptual indiscriminability such as Wright’s as logical fallacies. Suppose first x and y are compared by someone with respect to color; then y and z are compared; and then x and z are compared. And suppose that the person’s discrimination judgements are no, no and yes, respectively, thereby forming a prima facie case for nontransitivity of the relation of perceptual indiscriminability. Then the contextualist will point out that, to a first approximation, in the first comparison, x constitutes the context of y, and in the second comparison, z constitutes the context of y. If the contextual hypothesis is correct, this causes the person’s perception of y in the first comparison to differ from the person’s perception of y in the second comparison. So one is not allowed to ‘carry over the middle term’ in the argument for nontransitivity (Raffman 2000, 161).³ In a contextualist framework, the transitivity thesis takes the following form. If objects are compared in view of a perceptual property P in such a way that there is no contextual disturbance of perception, indiscriminability is a transitive relation. According to Raffman, this hypothesis is not excluded by what is presently known. And we have seen that Raffman expresses the hope that empirical evidence might be brought to bear on it. It is clear that even in experiments set up to determine whether one participant can discriminate between the color of stimuli, her answers will display a statistical distribution. Hardin emphasized the importance of this phenomenon for the philosophical discussion about the (non-)transitivity of indiscriminability.⁴ It will happen that even for a single pair of color patches, she will sometimes answer that she can discriminate them, whereas at other times she will judge them to be indiscriminable in color. So it must be conceded at the outset that our assumption that an agent always makes ³ Schroer (2002) develops a similar, but less fully articulated, line of reasoning. ⁴ See Hardin (1988).
Perceptual Indiscriminability and the Concept of a Color Shade
215
the same discriminability judgement makes us guilty of grossly oversimplifying the situation. We shall have to come back to this: in section 12.6, the implications of retracting this assumption will be (all too briefly) discussed. Raffman considers possible counter-evidence to the contextualist version of the transitivity thesis. She considers circles, presented against a uniform background color, and divided into 3, 4, 5 or more equally large sectors, where adjacent sectors receive very similar colors. In particular, she considers such a circle divided into 3 sectors (Raffman 2000, 163ff). Participants are asked to compare the three sectors two by two with respect to color. The advantage of such a set-up is that the ‘context’ of y, for example, when compared to x is the same as when compared to z, for in each act of comparison the whole circle remains firmly in view of the participant.⁵ Raffman contends that for circles divided in three sectors, solid empirical evidence pointing in the direction of nontransitivity has not been forthcoming. It appears to be impossible to construct a circle divided into three sectors x, y and z of slightly different colors in such a way that the agent systematically judges x and y as well as y and z indiscriminable, but also systematically judges x and z discriminable. Here the statistical nature of the distribution of discriminability answers may play a substantial role. The statistical distribution is presumably just not sufficiently sharply peaked to come to a reliable conclusion in such a triad situation. This point is not without importance. It entails that reports such as the following just cannot be accepted without further ado: Suppose I focus on just patches 1–3 and claim sincerely to attend simultaneously to the colorappearances of all three patches. ([. . .] I can attest from my own case—as you can probably attest from yours—that we can [focus on three patches at a time].) At a given instant, I claim, patches 1 and 2 looks the same to me, as do patches 2 and 3, but patch 1 and 3 look different. (Mills 2002, 395)
At this point, it is important that a background condition for success is firmly kept in mind. This condition was highlighted by J. A. Burgess (Burgess 1990, 209), but it has not received much attention in the literature. For the experiment to succeed, x, y and z have to be perceived as uniformly colored. This forces us to design the experiment carefully. A first question is whether the three sectors are separated from each others by lines of a uniform color which clearly differ from the colors of x, y and z. And a second question is whether the three sectors really touch each other or, alternatively, are ‘pulled apart’ some distance against a uniform background. If the answer to both these questions is no, then it can legitimately be questioned whether the background condition is satisfied. When one tries to construct a pie consisting of three contiguous sectors that forms a counterexample to the transitivity hypothesis, the participants in the experiment have some tendency to see it as a phenomenal continuum. To the extent that that is the case, the participants will be hesitant to affirm that each of the three sectors are uniformly colored. This problem can be mitigated by pulling the sectors apart against a uniform background color. Schroer notes that the sectors should ⁵ Raffman admits that she is ‘helping [her]self to the notion of a visual context more or less unexamined’ (Raffman 2000, 159), but let us grant her that.
216
Leon Horsten
not be pulled apart far (Schroer 2002, 265–7). If the sectors occupy parts of the visual field of the participant that are not very close to each other, then the participant cannot compare them directly. So on account of physiological limitations she will have to shift her focal visual attention from one sector to the other. And this entails that in her act of comparison, she has to rely on her memory, which is much less precise than occurrent appearances are. Let us assume in the sequel that the experiments are set up in such a way that the participants perceive the monochromatic colors as uniformly colored. Then one may ask: how could empirical evidence bear out that the nontransitivity hypothesis is false? In any actual mental comparison of two objects x and y with respect to P, representations of x and y are necessarily present in a special way. When a person compares x and y, she is focused on x and y in a manner that she is not focused on z. Suppose, perhaps contrary to the facts, that it were possible to construct a three-sector circle in such a way that it would reliably generate prima facie evidence for the nontransitivity thesis. Then the contextualist is free to conjecture that in experiments of this sort, concentrating on x and y (in a comparison of x and y) versus concentrating on y and z (in a comparison of y and z) makes a difference in the perception of y. Thus the prima facie evidence for nontransitivity would on closer inspection disqualify as genuine evidence for nontransitivity of the indiscriminability relation. Just this move is executed by Raffman when she considers variants of this experiment with circles divided in at least five sections (Raffman 2000, section III). A knee-jerk reaction to the difficulty of constructing a three-sector circle which generates prima facie evidence for the nontransitivity hypothesis is to divide the circle into more sectors—as many sectors as it takes! One might, for instance, divide the circle into five sectors such that sector 1 is very close in color to sector 2, which is in turn very close in color to sector 3, and so on, until one arrives at sector 5, which is not very close in color to its neighboring sector 1. Raffman thinks that on closer inspection, even prima facie evidence of this kind does not refute the transitivity hypothesis (Raffman 2000, 169): . . . it may be that in any series whose adjacent members are indiscriminable (would be judged the same in a same / difference comparison), [. . .] at any given time at least one member looks different in its two hypothetical comparisons with adjacent items, even in the case the entire series is viewed simultaneously.
But there is no reason why a contextualist could not take this line even concerning circles divided into three sectors. In fact, this kind of move is always open to the contextualist. It even provides a scheme for reacting to diachronic comparison experiments. Suppose we modify the experiment so that at a given time, the whole visual field is occupied by exactly one monochromatic paint chip, and the agent is asked whether she can discriminate the color presently in her field of view from the color she saw two seconds ago. Even though the color sample is now not presented against a visual background, it is presented against a ‘visual memory context’. Presumably visual memory contexts can influence the content of a color perception in a way similar to the way in which visual contexts influence colors perception of objects.
Perceptual Indiscriminability and the Concept of a Color Shade
217
Raffman’s attempt to give hard empirical content to the contextualist defense against the prima facie five-sector counterevidence against the transitivity hypothesis is unsuccessful. She suggests the following diachronic comparison experiment (Raffman 2000, 167): Suppose you are given a new task, involving just the pairs #3 / #4 and #4 / #5: you are to attend first to #3 and #4, then to #4 and #5, this time with the aim of judging whether #4 looks the same or different in the two pairings—in the two attendings, one might say. In other words, rather than making two comparisons, of #3 to #4 and #4 to #5 as in the serial scenario, your new task is to perform a single, cross-contextual comparison. As I will put it, your task is to compare #4 viewed in an act of focused attention to #3 and #4, with #4 viewed in an act of focused attention to #4 and #5. First you attend to the pair #3 / #4, then you shift your attention to #4 / #5, keeping #4 in view continuously, and you make a same / difference judgement of #4. Isn’t it possible that #4 should look different?
It is not clear that Raffman’s diachronic comparison experiment is well-conceived in the first place. The memory-image of #4 may not be sharp enough for an average participant to compare with the occurrent experience of #4 (Wright 1975, 336), (Schroer 2002, 267). But even if memory-images are sufficiently crisp for the task, there is no reason why the contextualist should feel cramped by consistent responses that in the two comparison acts #4 looks the same, if such responses were forthcoming. She is free to conjecture that after the second comparison task, the memory of #4 in the first comparison task is influenced by the subsequent focus on #5. This is after all, for all we know, possible. And if it is the case, then ‘no difference’-responses to the diachronic task have little import for the transitivity hypothesis. At this point it becomes evident that conclusively refuting the contextualist defense of the transitivity hypothesis on the basis of empirical evidence, is impossible. This is just a consequence of the fact that was noted earlier, namely that in an explicit color comparison between two items x and y, these items are present in a special manner; they are present in a way that is unlike the way in which features of the context are present in consciousness. This always leave the contextualist free to deny that it is permissible to ‘carry over the middle term’. One might say that the situation with the transitivity hypothesis is no different from that with scientific hypotheses in general. One may hold as fast to them as one likes. But there is a point at which rescue missions begin to look somewhat ad hoc. It seems that if empirical circumstances would force Raffman to ascribe contextual effects to focusing acts, i.e. when empirical evidence would force her to question whether in the five-sector circle experiment #4 looks the same when compared to #3 as when compared to #5, this point has been reached. For all that has been said, the transitivity hypothesis may still be correct. It may be correct even in the face of the sort of hypothetical prima facie empirical counterevidence that we have discussed. Still, as Raffman has emphasized, it remains at least in part an empirical matter. For it may well be that the empirical data look much more favorably upon the contextualist theory than in the pessimistic scenarios that we have been considering.
218 12.5
Leon Horsten C O LO R S H A D E S : PH E N O M E N A L O R T H E O R E T I C A L ?
Raffman was right to insist that it is at least to some extent an empirical question whether perceptual indiscriminability is a transitive notion. But we have seen that it is not as easy to test the transitivity hypothesis as it appears at first sight. If we want to empirically determine color shades, how should we proceed? What are the possible outcomes of such an investigation? And what is their significance? As mentioned before, color shades are supposed to make sense of our sense experience. But the discussion of Fara’s position entails that we ought to be more precise. We must decide whether we want color shades to make sense only of what we are consciously aware of, or whether it should also make sense of what we are unconsciously or behaviorally aware of. This is purely a matter of decision. But it appears best in line with the historical development of this enterprise if we restrict ourselves to conscious sense experience, which the agent is able to make explicit in discriminability judgements. From a contextualist point of view, the situation then looks as follows. Contextual factors typically influence indiscriminability judgements. So when indiscriminability judgements are combined in arguments, the context must be kept as uniform as possible. Raffman advises us to keep the context constant (as much as possible) throughout the indiscriminability experiment by making sure that all the stimuli remain clearly within the visual field of the participant throughout the successive comparison tasks. For concreteness, let us suppose that the participants are presented with sectors of a circle which are slightly differently colored. To ensure that the sectors are perceived as uniformly colored, they are pulled apart a bit. And the sectors are presented against a uniformly colored background. To conclude, let us suppose that we are working with a large but finite number of color stimuli (‘paint chips’). In this way, we obtain a total indiscriminability graph in a fairly straightforward way. But on closer inspection even this much is far from clear, for the following reason. Surely there will be a maximum on the number of sectors that can be presented in one experiment (Raffman 2000, 170). And it seems likely that this number will be less than the number of color stimuli that are of a slightly different color from which we want to abstract the color shades. Hence not all distinct stimuli can be present in one run of the experiment. Instead, many runs will have to be done with many sextuples (say) of colored sectors. Even if the transitivity hypothesis can explain each run of the experiment individually, the question remains how the results of the different runs of the experiment should be patched together. It will not be sufficient to assume without further ado that if one stimulus is present as a sector of the circle in different runs of the experiment, the appearance in one run of it is the same as the appearance of it in another run. After all, the same stimulus was offered in different contexts, and the contextualist hypothesis will predict that contextual differences typically change the appearance of the stimulus. In sum, it is not at all clear how the results of the sequence of experiments can be taken to give rise to one global graph on the class of stimuli. It cannot be excluded
Perceptual Indiscriminability and the Concept of a Color Shade
219
that the simultaneous focal presence of one color may influence the appearance of the other—this is unavoidable. And this would mean that the contextualist may still cry foul: it may be that it is impermissible to combine the individual discrimination judgements into a discriminability graph. But from an operational point of view, this appears the best we can do. We shall not pursue this problem further. Instead we shall for the sake of argument assume that it can be dealt with so that in the end, an indiscriminability graph is obtained. So let us suppose that an experiment is set up and carried out in this way. Then there are in general two possible outcomes. We shall examine them in turn. First, it may turn out that the empirical data can be explained very well by the transitivity hypothesis in the framework of a version of contextualism which does not appear ad hoc. Then, it seems, color shades can be abstracted in the familiar way from explicit indiscriminability judgements that are corrected as much as possible for contextual effects. Abstraction does not yet tell us what the nature of color shades consist in. But it does give us a criterion of identity for color shades: The Innocent Criterion The color shade of paint chip x is numerically the same as the color shade of paint chip y if and only if x and y are perceptually indiscriminable with respect to color. In these circumstances, and given the supposition that the transitivity hypothesis can be upheld in the light of the data, there will be sharp cutoff points between color shades. A marked advantage of the innocent criterion is that it ensures that color shades harmonize perfectly with the indiscriminability relation. Color shades are admitted into our ontology in the first place in order to make sense of our visual experiences. So if we can best make sense of the discrimination experiment on the basis of a global transitive graph on the stimuli, then we are in the fortunate situation that the abstracted collection of color shades will be completely faithful to our (corrected) discrimination judgements. One may wonder whether a theoretical element has nevertheless entered into the contextualist defense of the transitivity hypothesis. After all, on this account there are no ‘pure’ indiscriminability judgements; it is still the case that all we have are objects that are indiscriminable in a context. But this appears unproblematic. It just seems hard to deny that context is a parameter that plays a role in the discriminability of objects with respect to color—even though until recently philosophers have hardly recognized it. A second possible outcome is that empirical evidence makes it hard even for a contextualist to maintain the transitivity hypothesis without resorting to seemingly ad hoc maneuvers. Then it is impossible to abstract color shades from indiscriminability judgements in the familiar way. Either some patches that cannot be distinguished in color will be judged to have different color shades, or some patches that are discriminable in color will be judged to have the same color shade, or both.
220
Leon Horsten
This is what Fara finds objectionable (Fara 2001, 909). In his earlier writings, Wright also deemed this unacceptable (Wright 1975, 352).⁶ Color shades (‘looks’) were supposed to make sense of our sense experience first and foremost. If the concept of a color shade does not fully respect our indiscriminability judgements, then it is not a phenomenal concept. Wright went so far as to say that in such a situation we have a transcendental concept (Wright 1975, 357). Yet what are our options, in the situation under consideration? We have access to color shades only via color comparisons. In this sense, indiscriminability judgements are prior to the color shades themselves. First, one can deny that color shades (or sense data in general) exist. This is not an uncommon stance. Armstrong takes this position (Armstrong 1961). Fara does not explore this option; she merely mentions in passing the possibility to give up on ‘looks’ altogether (Fara 2001, 916, fn 13). Raffman does not embrace this position either, although she is careful not to rely anywhere in her argumentation on the assumption that color shades exist (Raffman 2000, 160–1). Perhaps from a contextualist point of view denying the existence of shades should appear as an attractive option. For would it not be natural to say, from a contextualist perspective, that color shades (or ‘looks’) are essentially relative to a context? This would mean that it makes sense to say that in a given visual context in which two objects x and y are present, the color shade of x is the same as the color shade of y. But it does not make sense to ask whether the color shade of an object z presented in one visual context is the same as the color shade of u presented in another visual context. (As far as I am aware, the prospects of this position have not yet been fully explored in the literature.) A second option consists in conceding that the concept of a color shade is to some extent a theoretical concept. This line was taken by J. A. Burgess (Burgess 1990, 218–19):⁷ . . . theories that respect the phenomenology of perception are just that: theories. This means that they might not only be required to postulate properties that are (in some sense) not presented in experience in order to do justice to some facts of experience [. . .]; it also means that they might need to discard some apparent data as illusory.
The concept of a color shade is not clearly a philosophers’ concept, but it is also not clearly a pre-theoretical concept that is as pure as driven snow. It appears to be a low level theoretical concept. This should not be taken to contradict the fact that it is intimately tied to experience. Indeed, it can and should be upheld that its aim remains first and foremost to make sense of our indiscriminability judgements. Whether it is fruitful to develop a theoretical concept of color shade depends on what use it can be put to. In ordinary language, color shades play an important communicative and judgemental role. We routinely make judgements about color shades, for instance when we say ‘this shade is the same as that shade’ while pointing at parts of the surfaces of two objects. If we develop a theoretical concept of color shade that respects indiscriminability judgements as much as possible, we may hope to validate ⁶ He has since then sought to qualify his position on this issue. See below, section 12.7. ⁷ Linsky defends a similar position (Linsky 1984).
Perceptual Indiscriminability and the Concept of a Color Shade
221
many of the judgements of this kind. In communicating our visual experiences to others, we use the concept of a color shade. We can construct a theoretical model of what such color shades could be. In the quotation above, Burgess appears to take a realist view of color shades. Such a view is certainly tenable, but it is not forced upon us. One can also take color shades to be useful theoretical fictions. Which of these two stances is more appropriate is, as in all forms that the realism debate takes, difficult to adjudicate. 12.6
A P P ROX I M AT I N G B Y E QU I VA L E N C E R E L AT I O N S
Let us proceed on the assumption that we want to recognize the existence of color shades in our ontology. More specifically, let us assume that we want to find a criterion of identity for color shades in the face of apparent nontransitivity of the indiscriminability relation. Then we are faced with two requirements that are pulling in opposite directions. On the one hand, we want color shades to respect the indiscriminability relation as much as possible. On the other hand, we need an equivalence relation to base our abstraction on. In this predicament, we should try to strike a good compromise. Being phenomenal or observational is a matter of degree in this context, and should not be taken to be a matter of principle. We should look for an equivalence relation that is somehow as close as possible to the nontransitive indiscriminability relation. From such an equivalence relation, a criterion of identity for color shades can be obtained. The advocates of color shades in the light of nontransitivity have been aware of this task. But some of them have underestimated the subtlety of the problem. A first proposal was made by Nelson Goodman (Goodman 1966, ch. IX). He thinks that the innocent criterion should be replaced by (roughly) the following criterion:⁸ Goodman’s Criterion The color shade of paint chip x is numerically identical with the color shade of paint chip y if and only if for every paint chip z, x is perceptually indiscriminable with respect to color from z if and only if y is perceptually indiscriminable with respect to color from z. Even if indiscriminability is nontransitive, the right-hand-side of Goodman’s identity criterion is an equivalence relation. If Goodman’s criterion is correct then discriminability of paint chips x and y is a sufficient, but not a necessary condition for the paint chips to be of a different color shade. So the effect of Goodman’s criterion is to loosen the tie of color shades with indiscriminability in exchange for having an equivalence relation. Goodman’s criterion will result in what Wright calls Goodman Shades. Wright has observed that under fairly general circumstances, Goodman Shades will be very finely grained (Wright 1975, 354). Indeed, in many circumstances, each pair of paint chips ⁸ Burgess (1990) also defends a version of Goodman’s criterion. Linsky (1984) shows sympathy for it, but he has reservations and does not quite endorse it. See Linsky (1984, 379).
222
Leon Horsten
will be judged by the criterion to be of different color shades. This restricts the usefulness of Goodman’s criterion. For surely the color shades that we attempt to communicate and quantify over in ordinary language are not so fine grained. Williamson has proposed methods for finding equivalence relations which are much closer to the nontransitive indiscriminability relation (Williamson 1986, 1990). We are given an indiscriminability relation G which is reflexive and symmetrical but nontransitive. Our task is to approximate G by an equivalence relation. We can approximate G from above by taking the transitive closure of G. Or we can approximate G from below by considering maximal equivalence relations G − ⊂ G. Either way, we obtain equivalence relations that are in a qualitative sense close to G: we call these qualitatively best equivalence-approximations. In the present context, the approximation from above does not make sense. For if the indiscriminability graph is connected (as in most cases it will be) then the transitive closure is the total graph on the underlying collection of paint chips. And that would mean that we have only one color shade. In other words, compared to Goodman’s criterion we would have landed in the other extreme. Qualitatively best equivalence-approximations from below seem more promising. They are of the same kind as Goodman Shades: they ensure that discriminability is a sufficient but not in general a necessary condition for having different color shades. The difference is that qualitatively best equivalence-approximations are more coarsely grained than Goodman Shades. Consequently, qualitatively best equivalence-approximations from below are in general more faithful to the indiscriminability relation than Goodman Shades. One drawback of qualitatively best equivalence-approximations from below is that they are in general not unique. For all nontransitive indiscriminability graphs, there exist more than one best equivalence-approximations from below. This problem can be mitigated by choosing the most coarse grained best equivalence-approximation from below. But even imposing this as an extra requirement does not always ensure that there is a unique best approximation. This entails that a conventional element is inherent in the proposal of choosing a qualitatively best equivalenceapproximation as the basis for an identity criterion for color shades. Sometimes all the best equivalence-approximations are equivalent to each other up to a simple transformation. In such cases, the conventional element does not play a deep role in the theory of color shades. But there are indiscriminability graphs for which the equivalence-approximations are not equivalent up to a simple scaling factor. In such situations, the conventional component plays a deeper role. Williamson’s methods can be somewhat improved upon by taking a quantitative point of view (De Clercq and Horsten 2005).⁹ Our task again is to approximate the indiscriminability graph as closely as possible by a transitive graph. Taking a quantitative view of the matter, we consider those transitive graphs that result from G by cutting and/or pasting a minimum number of edges from/in G. Equivalence ⁹ The method that we are about to discuss can be fairly straightforwardly extended to infinite domains. See again De Clercq and Horsten (2005).
Perceptual Indiscriminability and the Concept of a Color Shade
223
approximations in this sense are quantitative approximations. It is easy to show that often the quantitatively closest approximation to G results from cutting and pasting edges. In other words, often a closest equivalence approximation is one that partially overlaps with G. It is also clear from this that quantitatively best approximations that only cut edges are qualitatively best approximations from below (in Williamson’s sense). Nontransitive indiscriminability graphs are more likely to have a unique quantitatively best equivalence-approximation from below than to have a unique qualitatively best equivalence-approximation from below. But uniqueness can still not be guaranteed in all circumstances. So the conventional element will play a smaller role, but it cannot be guaranteed that it plays no role whatsoever. But in general, quantitatively best equivalence-approximation will be more faithful to the indiscriminability relation than qualitatively best equivalence-approximations. So quantitatively best equivalence-approximations can be said to be more ‘phenomenal’ than Williamson’s qualitative approximations. This may be an appropriate place to illustrate the differences between the different strategies for abstracting color shades from nontransitive relations. As a basis for this illustration, we use the indiscriminability graph G = V , E = {e1 , e2 , e3 , e4 }, {e1 e2 , e2 e3 , e3 e4 } . It scarcely needs to be mentioned that this is a highly simplified indiscriminability graph. In an actual experiment, we shall want a much larger domain of paint chips. And it is very unlikely that the associated indiscriminability graph will be serial in the way that G is. If we follow Goodman’s criterion, then there is a one-to-one correspondence between the paint chips and the shades, for the transitive graph E g to which it gives rise is the totally unconnected graph. Since G is connected, the qualitatively best equivalence-approximation from above E + is the total graph. There are two qualitatively best equivalence-approximations E1− , E2− from below, of which the edges are, respectively: e2 e3 ; e1 e2 , e3 e4 . We see that according to E1− there are three color shades ({e1 }, {e2 , e3 }, {e4 }) whereas according to E2− there are only two ({e1 , e2 }, {e3 , e4 }). So if we want the most course grained qualitative approximation from below, we must choose E2− . Also, we see that from a quantitative point of view there is only one best approximation: E2− . The latter makes only one ‘mistake’ against G, whereas E1− makes two mistakes and E g and E + contain even more discrepancies with G. One advantage of the innocent criterion and of Goodman’s criterion is that they specify a rule for deciding, at least in principle, whether two paint chips are of the same color shade. On finite domains, best equivalence-approximations (both in the qualitative and in the quantitative sense) are of course decidable in principle. But
224
Leon Horsten
quantitatively best equivalence-relations even on a decidable infinite nontransitive graph need not be decidable at all. And even in the finite situations to which we confine our discussion, best equivalence-approximations do not give us a rule in any natural sense of the word for deciding equality of color shade for paint chips. Also, and related to this, most discriminability graphs are such that their best equivalenceapproximations cannot be succinctly expressed in natural language. For some indiscriminability graphs, the quantitatively best equivalence-approximation both cuts and pastes edges.¹⁰ In other words, sometimes the quantitatively best equivalence-approximation is an equivalence relation that partially overlaps the original graph. When closeness to the agent’s discriminability judgements is our primary consideration, it seems that one should in such a situation opt for such an overlapping approximation. But this would entail that we count some patches that are discriminable in color as having the same shade of color. Surely many participants in the debate will find this objectionable. But is it really? Are we not sometimes willing to count certain objects as having the same shade even if we can discriminate them in color? Don’t we sometimes say ‘yes, I see a slight difference between them, but they are the same shade’? So if fitting common language usage is an important consideration, then it is not clear that our concept of color shade is not permitted to function in this way. Admittedly such considerations fall short of being decisive. Perhaps in such cases what is meant is that the relevant objects are of roughly the same shade. So we shall in the sequel refrain from challenging the thesis that it is part of the meaning of the concept of a color shade that discriminability entails difference in color shade. As mentioned before, indiscriminability with respect to color is in practice not an all or nothing affair. Participants are typically more sure of some of their indiscriminability judgements than of others. This can be captured by assigning weights to the edges in the discriminability graph (De Clercq and Horsten 2005, 388). At the very least we should allow participants to opt out of some indiscriminability judgements. This would be captured by setting the set of weights equal to {0, 0.5, 1}, where an edge with weight 0.5 corresponds to a pair of paint chips for which the participant is unsure whether they are indiscriminable with respect to color. Assigning weights to edges of course limits the scope of the non-uniqueness problem even further. But it will still not dissolve the problem completely. So what should we do when we have a weighted indiscriminability graph for which there exist two or more quantitatively best equivalence-approximations? As the reader will expect, familiar maneuvers present themselves. For instance, it is possible to maintain that two paint chips are only of the same (a different) color shade if the corresponding edge belongs to all (no) quantitatively best equivalence-approximation. Where some equivalence-approximations disagree, we could say that there is no matter of fact whether the relevant paint chips are of the same color shade. Suppose that we have a global nontransitive indiscriminability graph of which there are more than one quantitatively best equivalence-approximations. Then how do we individuate color shades? ¹⁰ A simple example is given in De Clercq and Horsten (2005, 377).
Perceptual Indiscriminability and the Concept of a Color Shade
225
One option is to take the stance that color shades are low-level theoretical constructs seriously and simply abstract them from one particular quantitatively best equivalence-approximation—perhaps from one that is particularly easy to describe. Another option would be to argue that the ‘belonging to’ relation between color stimuli and color shades is to some extent vague. A supervaluation idea could be applied in the following way. Take any two color stimuli a and b from the global indiscriminability graph. Say that a and b (determinately) belong to the same color shade if a and b are indiscriminable according to all quantitatively best equivalence-approximations. a and b (determinately) do not belong to the same color shade if a and b are discriminable according to all quantitatively best equivalenceapproximations. And in the remaining case, it is indeterminate whether a and b belong to the same color shade. If one wants to remain as close as possible to the way in which the concept of a color concept functions in natural language, then the latter might look like the best option. But this position posits vagueness in the world, and arguments have been formulated which purport to show that the only vagueness that can exist is linguistic vagueness (Evans 1978). But in the present situation the vagueness involved—if there is any—cannot be easily shifted to the meaning of expressions. The reason is that natural languages do not contain simple names for all color shades. Perhaps natural languages do not even for every color shade contain a definite description that singles it out.¹¹ So if one is persuaded by Evans’ arguments against worldly vagueness, then it is not easy to see in which way an appeal to vagueness can solve the non-uniqueness problem. 12.7
C O LO R S H A D E S A N D N AT U R A L L A N G UAG E SEMANTICS
From the foregoing we may conclude that the question whether indiscriminability relation is a transitive relation, is at least to some extent an empirical one. But to some extent, it is also a theoretical one: the answer depends to some extent on theoretical interpretations of the evidence. Of course, it is often that way in empirical science. If in the final analysis indiscriminability turns out to be transitive, then a concept of a color shade can be defined that is completely faithful to the indiscriminability judgements. If indiscriminability is not a transitive relation, then it is futile to look for a (non-contextual) concept of color shade that completely respects the relation of indiscriminability. But in these circumstances it is still open to us to define a concept of color shade that is as faithful as possible to our indiscriminability judgements. This would result in a concept of color shades that falls short of being completely phenomenal, even though it will be as phenomenal as possible. As adumbrated before, for Wright an important theoretical function of (in his view phenomenal) concepts such as that of a color shade consists in assisting us in ¹¹ Cf. the next section.
226
Leon Horsten
constructing a semantics for natural language. A concept of color shades abstracted from a best equivalence-approximation to our indiscriminability judgements could fit this bill. Hardin doubts that color shades modeled on the basis of the indiscriminability relation are of much direct relevance for natural language semantics. The reason is that natural languages have names for only a few of them (Hardin 1988, 226–7). But considerations such as these should not be taken as decisive objections against Wright’s project. Aside from color predicates, English contains the concept of a color shade. And this concept is not so easily dispensable. Many statements containing the term ‘color shade’ can be paraphrased by statements that contain the concept of indiscriminability instead. Thus, instead of saying These two walls are painted in the same shade of orange, we can equivalently say: These two orange walls are indiscriminable with respect to color. But the paraphrases of many statements containing the expression ‘color shade’ sound awkward. And for some statements containing the expression ‘color shade’ it is difficult to see how they can be paraphrased in terms of the concept of indiscriminability at all. An example of such a statement may be: There are more shades of colors than we are able to produce samples of in our laboratory.
So it is not excluded that a concept of a color shade developed along the lines that have been outlined in this article may be of value for natural language semantics. Such a concept of color shades might count certain paint chips that are indiscriminable with respect to color as being of a different color shade. In this sense, the concept of a color shade coheres imperfectly with indiscriminability judgements. But this is only problematic if indiscriminability somehow gives the rule by which identity judgements of color shades are made. Already in his (Wright 1975), Wright thought that that this is precisely the assumption that should be given up anyway. Our semantic competence does not consist exhaustively in using expressions according to implicit rules. In his more recent publications, Wright has connected this with a softening of his stance on the observability of color predicates (Wright 1987, 246): ‘How could the appropriate kind of sensitivity operate selectively among indiscriminabilia?’ [. . .] it should now seem as if this question has rather disappeared. ‘Looks red’ ought certainly to qualify as observational on this count. But the kind of sensitivity to appearance which someone who understands ‘looks red’ [. . .] must have does operate selectively among items which, in respect to apparent color, cannot be told apart. [. . .] The suggestion that there is some kind of tension between such selectivity of response and its being the sort of response appropriate in the case of an observable predicate, depends on the thought that it cannot then be purely in response to appearance—either it is unprincipled or it is a principled response to more than the appearances. But ‘unprincipled’ here just means: not guided by rules correlating responses with appearances. So we should embrace the first alternative: such responses are indeed unprincipled, and no less appropriate or less purely ‘to’ appearances on that account.
These remarks are in perfect agreement with the theory that proposes identity criteria for color shades in terms of best equivalence-approximations to the indiscriminability relation. An identity criterion for color shades expressed in terms of equivalenceapproximations points away from the picture of indiscriminability furnishing the
Perceptual Indiscriminability and the Concept of a Color Shade
227
ingredients of a compact rule that language users are implicitly guided by when forming judgements of identity and difference of color shades. Indeed, we do not even have succinct expressions in natural language for most of these equivalenceapproximations. But in case indiscriminability is nontransitive, best equivalenceapproximations give us identity criteria for entities that cohere best, albeit not perfectly, with our indiscriminability judgements. Re f e re n c e s Armstrong, D. (1961), A Materialist Theory of Mind, Routledge and Kegan Paul. Burgess, J. A. (1990), ‘Phenomenal qualities and the nontransitivity of matching,’ Australasian Journal of Philosophy 68, 206–20. De Clercq, R. and Horsten, L. (2004), ‘Perceptual indiscriminability: In defense of Wright’s proof,’ Philosophical Quarterly 54, 439–44. (2005), ‘Closer,’ Synthese 146(2005), 371–93. Evans, G. (1978), ‘Can there be vague objects?’ Analysis 38, 208. Fara, D. G. (2001), ‘Phenomenal continua and the sorites,’ Mind 110, 905–35. Goodman, N. (1966), The Structure of Appearance, 2nd edition, Bobbs-Merill, 1966. Hardin, C. L. (1988), ‘Phenomenal colors and sorites,’ Noˆus 22, 213–34. Linsky, B. (1984), ‘Phenomenal qualities and the identity of indiscernibles,’ Synthese 59, 363–80. Mills, E. (2002), ‘Fallibility and the phenomenal sorites,’ Noˆus 36, 384–407. Raffman, D. (2000), ‘Is perceptual indiscriminability nontransitive?’ Philosophical Topics 28, Vagueness, ed. Christopher Hill, 153–75. Schroer, R. (2002), ‘Matching sensible qualities: A skeleton in the closet for representationalism,’ Philosophical Studies 107, 259–73. Weiskrantz, L. (1986), Blindsight: A Case Study and its Implications, Oxford University Press. Williamson, T. (1986), ‘Criteria of identity and the axiom of choice,’ Journal of Philosophy 83, 380–94. (1990), Identity and Discrimination, Basil Blackwell. Wright, C. (1975), ‘On the coherence of vague predicates,’ Synthese 30, 325–65. (1987), ‘Further reflections on the sorites paradox,’ reprinted in R. Keefe and P. Smith (eds.), Vagueness: A Reader, MIT Press, 1997, 204–50.
13 The Sorites, Linguistic Preconceptions, and the Dual Picture of Vagueness Mario G´omez-Torrente
Following roughly a formulation of Delia Graff Fara, we may say that we have an instance of the sorites paradox when, in a particular occasion of use, we are confronted¹ with a group of sentences of the following form,² each of which seems highly compelling in that occasion of use: (A) (∃x1 ) . . . (∃xn )([Kx1 & Kx2 & . . . & Kxn &]Rax1 & Rx1 x2 & . . . & Rxn−1 xn & Rxn z); (B) [Ka &] Fa; (C) (∀x)(∀y)([Kx & Ky ⊃](Fx & Rxy ⊃ Fy)); (D) [Kz &] ∼ Fz (cf. Fara 2000, 49f ). The brackets indicate that the bracketed parts will appear in some instances of the sorites paradox (those involving a comparison class; see below) but will not appear in others. Here ‘F’ is to be replaced with the sorites susceptible predicate, ‘a’ with a name of a case of application of the predicate which is intuitively clear in the occasion of use, ‘z’ with a name of a case of negative application which is intuitively clear in the occasion of use, and ‘R’ with a name of some binary relation. The occasion of use is understood as containing some factors that supply at least a universe of discourse (and range for the quantifiers) which includes the things named by ‘a’ and ‘z’. In the case of at least some sorites susceptible predicates (such as ‘is small’ or ‘is expensive’), interpretation seems to require the occasion of use to supply a comparison class, which may coincide with or be properly included in the universe of discourse. We may view the comparison class as the interpretation of ‘K’ in the bracketed parts (even though sometimes there may not be a predicate naming the I am indebted to many people for their comments. Richard Dietz, Manuel Garc´ıa-Carpintero, Sven Rosenkranz, Timothy Williamson, and audiences at the UNAM, the University of Barcelona, the University of Saint Andrews, the University of Lisbon, and the Institut Jean Nicod in Paris deserve special gratitude. Research was supported by DGAPA (IN 401909–3) and by the Spanish MICINN (FF 12008–04263). ¹ The sentences need not be physically uttered for the paradox to arise, but at least a mental utterance of some sort will occur if the paradox is to be considered by a thinker at all. ² There are other versions of the sorites; the basic considerations of this chapter will apply to them without substantive changes.
228
The Dual Picture of Vagueness
229
comparison class, which may be left implicit). In these cases interpretation also seems to require the occasion of use to provide further standards of some sort for the application of ‘F’ and its relationship with ‘R’, which are partly responsible for the fact that (B), (C) and (D) are compelling in the occasion of use. And there may be other things that full interpretation requires the occasion of use to provide. The paradox is that every set of fully interpreted utterances of sentences of the form of (A), (B), (C) and (D) (whether we include the bracketed parts or not) is inconsistent according to classical semantics and logic, and yet many such sets are highly compelling in their corresponding occasions of use.³ For example, ‘F’ may be replaced with ‘is small’, ‘R’ with ‘has a population of 1 inhabitant less than’, ‘a’ with ‘Smalltown’, the name of a town with just 100 inhabitants, and ‘z’ with ‘Nonsmalltown’, the name of a town with 49,900 inhabitants; the relevant comparison class (the interpretation of ‘K’) may be taken to be the set of towns in the world that at present have 50,000 inhabitants or less, and the universe of discourse may be any set that includes that set of towns. Then (Bsmall ), (Csmall ) and (Dsmall ) are all highly compelling under normal standards,⁴ and we may also suppose for the sake of the example that we know (Asmall ) to be true as a matter of fact: (Asmall ) (∃x1 ) . . . (∃x49,799 )(Kx1 & . . . & Kx49,799 & Smalltown has a population of 1 inhabitant less than x1 & x1 has a population of 1 inhabitant less than x2 & . . . & x49,798 has a population of 1 inhabitant less than x49,799 & x49,799 has a population of 1 inhabitant less than Nonsmalltown); (Bsmall ) K(Smalltown) & Smalltown is small; (Csmall ) (∀x)(∀y)(Kx & Ky ⊃ (x is small & x has a population of 1 inhabitant less than y ⊃ y is small)); (Dsmall ) K(Nonsmalltown) & ∼ Nonsmalltown is small. Assuming that the truth of (Asmall ) is not in dispute, a solution of the paradox must convince us that one or more of (Bsmall ), (Csmall ), and (Dsmall ) is not true, or that classical logic or semantics do not apply. The same holds, of course, of every highly compelling set of fully interpreted utterances of sentences of the form of (A), (B), (C), and (D). Not all predicates that have been held to be sorites susceptible seem sorites susceptible for the same reasons. Many (though not all) sorites susceptible degree adjectival predicates (such as ‘is small’ or ‘is expensive’) give rise to very compelling (A)–(D) sets, as in the example above. Many predicates whose predicative element is a scalar noun (such as ‘is a heap’ or ‘is a youngster’)⁵ do not seem substantively different in ³ The use of (semi-)formalized language is not essential to the formulation and existence of the paradox; it just helps make somewhat clearer its formulation and its relevance for classical semantics and logic. ⁴ But not under all standards. For example, it is imaginable that we can work with a standard under which even Smalltown counts as not small, and only some very small towns of under fifty inhabitants count as small. (See related remarks in Fara 2000, 65; see also section 13.2 below.) ⁵ I understand a scalar noun as one that has an analytically associated dimension of comparison, usually also analytically connected with a degree adjective—a heap, on the acceptation that I take to be relevant to the sorites discussion, is a big pile of suitable things lying one on another.
230
Mario G´omez-Torrente
this respect. However, other predicates, and in particular some whose predicative element is a non-scalar noun, e.g. a noun for a typical natural kind (such as ‘is a dog’) do not seem to give rise to (A)–(D) sets that are compelling for the same reasons. And the situation with other predicates may not be subsumable under any of these two usually discussed types. In this chapter I will sketch a picture of the workings of sorites susceptible predicates in English and similar natural languages—including an outline of a solution of the sorites paradox—that is especially applicable to many adjectives and scalar nouns. The picture is applicable also to other predicative words and phrases, but lack of space will prevent extended discussion of this topic here (see footnote 28 and surrounding text). The picture is a dual picture, because it is based on a division of occasions of use of a sorites susceptible predicate into regular and irregular, according as to whether the predicate has a reference (extension) in the occasion of use or not. It is also based on two distinct sub-pictures of how language, and in particular the mechanisms for the fixing of reference (and, more generally, of intension), work in regular occasions of use and fail to work in irregular occasions of use. On the picture, the meaning of a typical word is pretty meager, though it comes together with a number of firmly accepted sentences containing the word, its associated ‘preconceptions’, which are not part of its meaning but are somehow designed to help fix its extension (and intension) in particular occasions of use. Typically, some of these preconceptions intuitively state that a certain predicate has some paradigmatic cases of application and negative application, while other preconceptions intuitively postulate generic principles for the expansion of the extension of the predicate beyond its paradigmatic range. These ideas are explained in section 13.2. According to the dual picture, in occasions of use where the preconceptions and the facts of the matter about a typical degree adjective (or scalar noun) give rise to an instance of the sorites paradox, the occasion of use is of the irregular kind: the adjective (or scalar noun) lacks an extension (and an intension) and as a result the utterances of sorites-relevant sentences containing it don’t have truth conditions. (But sorites-paradoxicality is only one source of irregularity.) In particular, for example, (Bsmall ), (Csmall ), and (Dsmall ), as uttered in the occasion of use described above, are neither true nor false. This thesis about sorites-paradoxical occasions of use is related to treatments that postulate reference or truth value gaps to deal with other paradoxes and inconsistencies, but it is radically different from standard theories of vagueness that postulate truth value gaps exclusively for alleged so-called ‘borderline cases’ of sorites susceptible predicates. The latter theories postulate that a sorites susceptible predicate has some kind of non-classical extension, and that the logical expressions operate on that extension through some suitably ad hoc non-classical semantics or logic. On the dual picture, a typical sorites susceptible degree adjective (or scalar noun)⁶ has no extension at all in those occasions of use. But the picture as developed here will also postulate (in section 13.4) that this is psychologically obscured by the fact that the default mechanism designed to fix a classical extension for these words often succeeds in ⁶ Henceforth I will omit the parenthetical addition of ‘and/or scalar noun(s)’ to ‘typical degree adjective(s)’ in many cases in which it should be tacitly understood.
The Dual Picture of Vagueness
231
doing so, even in closely connected regular (thus non-paradoxical) occasions of use. Section 13.4 also compares the ‘extension gap’ thesis about sorites-paradoxical occasions of use with an analogous and independently plausible thesis about extension (and intension) gaps arising from certain linguistic conflicts among paradigm and generic preconceptions for other adjectives and other nouns. This comparison makes plausible the extension gap thesis for typical sorites susceptible degree adjectives in sorites-paradoxical contexts. It will also help me defend some further theses about the psychological reasons why an appearance is created, even after exposure to the sorites paradox, that utterances of sentences of the forms (B), (C), and (D) have truth conditions even in paradoxical occasions of use, and in particular of why people exposed to the sorites paradox tend to give a preference to their intuitions about the truth value of the (B) and (D) sentences over their intuitions about the truth value of the (C) sentence. These theses will appeal to plausible conjectures about the psychology of any reference-fixing mechanism that relies heavily on paradigms. It is plausible to assume that if a certain mechanism of reference-fixing exists (and persists), it must be successful in at least a vast number of occasions. The dual picture postulates that in a vast number of occasions of use of a typical sorites susceptible degree adjective, the default reference-fixing mechanism of preconceptions works successfully and turns the occasions of use into regular ones in which even (B), (C), and (D) sentences have truth conditions. This happens when the paradigm and generic preconceptions and the facts of the matter about the adjective determine an extension and an anti-extension which are mutually exclusive and jointly exhaustive with respect to the universe of discourse in the occasion of use; in these cases no compelling (A) sentence is in sight. The hypothesis that these occasions of use are very numerous is a purely empirical one, and cannot be fully decided on the basis of a priori linguistic reflection on the semantics of sorites susceptible predicates. Some considerations that favor it are offered in section 13.3, where the picture’s semantic treatment of regular occasions of use is sketched as well. It may also be tempting to assume, additionally, that successful communication with grammatically declarative sentences must nearly always use utterances with truth conditions. But I don’t take this to be a compulsory thesis. What I do take as a very reasonable thesis is that communication with declarative utterances generally occurs under the tacit assumption that these utterances have truth conditions. Plausibly, it also involves some understanding, however implicit and inchoate, of how those truth conditions should be determined if they in fact exist, as well as of how the referents of particular classes of words should be determined if they in fact exist. I take it that semantic theory often appeals to this implicit understanding as evidence in the construction of theories. So I take it as a reasonable burden on semantic theory to describe reference-fixing mechanisms for classes of words that plausibly underlie our tacit understanding of the referential properties of those words. But this is compatible with the possibility that there may be frequent instances of successful communication by means of utterances without truth conditions. In the presence of an adequate theory of the mechanisms of reference-fixing, successful communication by means of such utterances is explained by the fact that speakers can elicit from hearers all sorts of desired responses under the tacit common understanding of how the truth conditions
232
Mario G´omez-Torrente
of those utterances would be determined (if they had in fact existed). To take a simple example, a mother can say to her child ‘Santa Claus will bring you presents tomorrow’ and get the child to expect happily the presents from an unknown man that at some point someone has named ‘Santa Claus’. This particular instance of communication is not prevented by the lack of a referent for ‘Santa Claus’ or by the presumable lack of truth conditions for the mother’s sentence. The search for a convincing, or even promising, idea for a solution of the sorites paradox(es) has proved to be very elusive, perhaps surprisingly so. Standard attempts are predictably affected by some sophisticated problem or other, but more importantly, they are also generally unpersuasive even when taken as rough pictures of the workings of the sorites susceptible predicates and their interaction with the logical expressions. I will start (in section 13.1) with a brief survey of what I take to be some fundamental (as opposed to sophisticated) problems afflicting the most familiar standard theories. The survey is based on a tripartite classification of standard solutions of the sorites as ‘paradigmatist’, ‘genericist’, or ‘strongly nihilist’, which I think is illuminating in its own right. The dual picture is a neutral picture, in that it is intermediate between all these standard (and extreme) positions. Later it will be useful to have made explicit the fundamental problems of standard theories, for much of the support I wish to marshal for the neutral dual picture will consist in noting that it does not suffer from those problems. The picture is also, I hope, not unpersuasive at first sight. It is of course not free from potential objections, only some of which can I try to describe and defuse here. Nevertheless, I hope this initial exposition can convince some that it is a step in the right direction. I also hope that it can be expanded and refined in future work. 13.1
A C L A S S I F I C AT I O N O F FA M I L I A R T H E O R I E S , A N D T H E I R P RO B L E M S
The theories of the sorites paradox that we may call optimistic claim that, despite appearances, (C) utterances, regardless of the occasion of use, must be false, and do not postulate a semantics or logic for the logical expressions distinct from the classical. For example, they propose that, for any one of the sorites series t1 , t2 , . . ., t49,798 , t49,799 making true (the matrix of ) (Asmall ) that can be drawn from the comparison class above, the negation ‘it is not the case that for all ti and ti+1 , if ti is small then ti+1 is small’ is true; or, what is equivalent given classical semantics and logic, the existential quantification ‘there is a town with i inhabitants that is small and a town with i + 1 inhabitants that is not small’ is true. Optimism is often (but need not be) accompanied by epistemicism, the additional claim that we cannot know which number i is. (See e.g. Cargile 1969, Williamson 1994.) If epistemicism is true, it provides a certain kind of explanation of the natural repugnance we feel for optimism: we cannot accept the existential quantification according to which there are such towns because we cannot find out which number i is. There are a number of sophisticated problems with epistemicist theories (see e.g. G´omez-Torrente 2002), but the basic problem would seem to be that, no matter how sophisticated the defenses of optimism get, it
The Dual Picture of Vagueness
233
is very hard to believe that some fact about the actual semantics of ‘small’ makes those existential quantifications true, at least in occasions of use such as the one above.⁷ And indeed, we have no plausible semantic model of how they could be true.⁸ One appealing feature of optimism, however, is that it does not postulate a non-classical semantics or logic for the standard logical expressions. Supervaluationists (Fine 1975 is the prime source) also say that those existential quantifications must be true. But unlike optimists, they claim that the existential quantifier does not have its classical semantics, at least when it interacts with sorites susceptible predicates. They claim that ‘(∃x)(∃y)(Kx & Ky & x is small & x has 1 inhabitant less than y & ∼y is small)’ may be true without there being some specific towns that make true ‘(x is small & x has 1 inhabitant less than y & ∼y is small)’. This is supposed to calm our worries about how the existential quantification could be true. But it sounds very implausible: assuming, as seems reasonable, that classical semantics is the semantics that we intuitively ascribe to the existential quantifier, why should we believe that it adopts an ad hoc semantics in certain cases? Despite occasional timid claims to the contrary, I see no evidence that we intuitively occasionally ascribe this semantics to the existential quantifier. Other theories of the sorites are based on even more radical departures from classical semantics. For example, typical degree theorists (such as Machina 1976) claim that the existential quantifier works in such a way that the existential quantification in question comes to have some alternative degree-theoretic truth value—it is (approximately) ‘half-true’. For what we might call primitivists, sorites predicates also determine a sui generis semantics for the logical expressions, but they claim that we need not know (or even that we may be unable to know) what this semantics is, at most that it is different from the classical. ( This is the way I read Sainsbury 1990.) As with the supervaluationist, the fundamental problem for the degree theorist and the primitivist is that postulating a (possibly unknowable) non-classical semantics for the logical expressions merely because they interact with sorites susceptible predicates is counterintuitive and ad hoc. Like optimists, I find the thesis that the logical expressions are in all essential respects governed by classical semantics and logic more than compelling. We may call all of these views paradigmatist, since they stick to the truth of the (B) and (D) sentences in each particular occasion of use, affirming the application and ⁷ Some optimists have theories compatible with anti-epistemicism that appeal to a special kind of contextualism to explain why (C) sentences seem true to us. Fara (2000) claims that some factor that context contributes to interpretation makes it the case that the sharp cut-off point between the extension and the anti-extension of ‘small’ in the context is never ‘where we look’, which explains why we believe of any pair of towns ti and ti+1 that it is not the case that ti is small and ti+1 is not small. But even if this were true, it would give us no reason to believe that the cut-off point is in some place where we don’t look (see e.g. Heck 2003, 120). Soames (1999), ch. 7, uses a similar strategy, although he is not strictly an optimist, since he postulates that there is no sharp cut-off point between the small and the non-small towns. But he postulates an equally implausible semantically determined sharp cut-off point between the small towns and towns which are supposed to be neither in the extension of ‘small’ in the context nor in its anti-extension (the ‘borderline’ towns). ⁸ As emphasized e.g. by Schiffer (1999) and Wright (2003).
234
Mario G´omez-Torrente
negative application of the sorites predicate in contextually paradigmatic cases, and they claim that the (C) sentence is somehow not true. An altogether different kind of option is to claim that somehow the (C) sentence is made true and either the (B) or the (D) sentence, or both, are made false by the relevant semantic mechanisms. For this reason we may call these views genericist. In the case above, and assuming classical semantics and logic, there are three suboptions: (a) to claim that ‘Smalltown is small’ is true and ‘∼Nonsmalltown is small’ is false—and presumably that all towns are in fact small; (b) to claim that ‘Smalltown is small’ is false and ‘∼Nonsmalltown is small’ is true—and presumably that no town is in fact small;⁹ (c) to claim that ‘Smalltown is small’ and ‘∼Nonsmalltown is small’ are both false. (c) is absurd and can safely be discarded. One general problem with (a) and (b) is that they give a preference to the intuitions about (C) utterances over the intuitions about (B) and (D) utterances. However, to me they seem more or less equally strong intuitions, and if I were forced to choose, I would say that the intuitions about the truth value of (B) and (D) utterances feel somewhat stronger after exposure to the sorites paradox. Also, the idea that, if sorites susceptible predicates do have extensions, then these are non-trivial extensions that affect semantically real distinctions between objects seems very plausible.¹⁰ But for the genericist, the extensions of sorites predicates never manage to do that. Williamson (1994, 165 ff ) has placed the (b) suboption together with other views on which the (logically atomic) sorites susceptible predicate also lacks application, although in this case because the predicate is nonsensical or in some milder way semantically defective, but in any case lacks an extension altogether. All these views on which the sorites susceptible predicate lacks application he calls nihilist. The views on which the sorites predicate is in some way semantically defective we might call (in order to distinguish them from the (b) suboption above) strongly nihilist (Dummett 1975 is an (imperfect) example; Eklund 2002 is closer to the idea). Strong nihilists are certainly not paradigmatists, but they are not genericists either, since the sorites predicate is just as defective in the (C) sentence as it is in the (B) and (D) sentences. Something that seems to me to be a problem for strong nihilists (or at least for representative strong nihilists) is that they seem to think that all of the (B), (C) and (D) sentences are compelling because the semantic rules for the sorites susceptible predicate in some way dictate that they must be accepted as true, if one is to abide by the meaning or broadly speaking the semantics of the predicate. (More exactly, what they claim is that, e.g. given the meaning of ‘small’ and uncontroversial facts about the population of Smalltown, (Bsmall ) must be accepted.) But I doubt that (Bsmall ), (Csmall ), and (Dsmall ) seem compelling because of that. If they did, the paradigmatist and genericist positions alike would seem to us to be excluded ⁹ Unger (1979) embraces this view for logically atomic copular predicates whose predicative element is a count noun applying to what he calls ‘ordinary things’: predicates like ‘is a table’, ‘is a house’, etc. have an empty extension. Elsewhere he held the same view for natural kind predicates, but he changed his view about these in Unger (1984). I am unsure of what he would say of predicates like ‘is small’. ¹⁰ As I will note in footnote 23, however, options like (a) or (b) are clearly the right options in some special non-paradoxical occasions of use of sorites predicates.
The Dual Picture of Vagueness
235
on purely semantic grounds; we would have the feeling that abandoning either of (Bsmall ), (Csmall ), or (Dsmall ) would inevitably amount to changing the meaning or the semantics of ‘small’. But I don’t think we have that intuition. I think that when in theoretical discussion we are exposed to optimism or to a (b)-type genericism we just see them as weird, not analytically excluded speculations that might conceivably uncover the concealed semantics of the sorites susceptible predicates. (We don’t see them the way we would see a theory which claimed that being unmarried is not an analytically necessary condition of bachelors.) In any case, a problem for all varieties of strong nihilism is that it is incompatible with the assumption (made at the beginning of this chapter) that the default reference-fixing mechanism for typical sorites susceptible degree adjectives must work successfully in a vast number of occasions. In fact, for the strong nihilist, just like for the genericist, we can never effect semantically real distinctions between objects by means of sorites susceptible degree adjectives. A final problematic consequence of strong nihilism is that it’s self-referentially ‘instable’: if it were true, it could not be stated with truth, given that, in all probability, it would have to use sorites susceptible predicates in its own statement. My purpose in what follows is to sketch a basis for a picture of vagueness that is not affected by the fundamental problems of the theories that we have reviewed in this section. How the picture avoids each of the problems will be pointed out along the way.
13.2
LINGUISTIC PRECONCEPTIONS
The picture, or at least my defense of it, relies on a broadly Kripkean view of language as evolving through the appearance and modification or abandonment of what we might call preconceptions. (Kripke has used the word ‘prejudices’ for this or a closely related concept.¹¹ I prefer ‘preconception’ both because it indicates that there may be differences between the two concepts and because it seems less negatively charged than ‘prejudice’.) In general, these preconceptions are sentences which are very resolutely assented to by minimally sophisticated normal people at relatively pretheoretical levels of use, sentences that are very difficult or even nearly impossible to abandon without exposure to relatively extensive reflection or empirical research. Preconceptions need not be a priori, necessary, analytic or even in any sense dictated to be of obligatory acceptance by the semantics of an expression; they just have to be very hard to give up. Nevertheless, among preconceptions some (perhaps all) are what we might call ‘linguistic’: they have a bearing on the extensions (and the intensions) that we assign to words, and in particular on the extensions that we assign to predicates. But not even linguistic preconceptions are invariably or even usually analytic or dictated to be of obligatory acceptance by the semantics of an expression. The Kripkean picture I have in mind postulates that we may view most predicates (and most words) as having a pretty meager meaning that usually does not suffice ¹¹ For exposition of the Kripkean notion of a prejudice see G´omez-Torrente (forthcoming).
236
Mario G´omez-Torrente
to fix their extension (or their intension),¹² but also as ‘introduced’ in some way together with a number of linguistic preconceptions involving them, preconceptions that are somehow designed to help fix their extensions, possibly with respect to particular occasions of use. Now a further idea, less definitely Kripkean, but certainly suggested by Kripke’s presentations, is that an ‘initial’ set of preconceptions {(P), (P), X(P), . . .} about the extension of a predicate ‘P’ is ‘introduced’ together with a tacit conditional instruction. We may also view this as a linguistic preconception (perhaps one that in some sense is a priori or dictated to be of obligatory acceptance by the semantics of the predicate), having a form similar to ‘If there is exactly one set Q such that (Q), (Q), X(Q), . . ., then the extension of ‘‘P’’ is that set’. These instructions help fix the extension of ‘P’ when their antecedents are satisfied. But when their antecedents are not satisfied, i.e. when the preconceptions in the initial set are not jointly uniquely satisfied, it may often be unclear that an extension is determined for the predicate. There are presumably other general, but less definite or less compelling linguistic preconceptions that may help in some cases in which those antecedents are not satisfied. For example, there may be general preconceptions exhorting us to try to assign extensions to predicates by abandoning those of the preconceptions in the initial set that intervene less in the use of the predicate, or by abandoning those whose abandonment provides for the simplest way of obtaining an extension (if any), etc. But even after (implicit) attempts to apply these further preconceptions, conflicts of unsatisfiability or other problems may often remain unresolved. The mentioned conditional instructions are similar to Kripkean conditionals by means of which in some way an explicit or implicit attempt is made to fix the reference of some proper names and general terms for natural kinds, substances or phenomena. One example is ‘If there is exactly one planet causing the perturbations in the orbit of Uranus, then ‘‘Neptune’’ refers to that planet’, which successfully fixes the extension (reference) of ‘Neptune’. (As well as its intension, which on account of its rigidity is simply the function assigning Neptune to each possible world.)¹³ Here the sentence ‘There is exactly one planet causing the perturbations in the orbit of Uranus’ cannot be called a preconception in the strict sense above, as it is certainly not resolutely accepted by normal people at relatively pretheoretical levels of use, though it is clearly not analytic or a priori, and it has a role in fixing the reference of ‘Neptune’. But stricter examples are provided by general terms for natural kinds, substances and phenomena. In the case of ‘dog’, it is natural to assume the existence of some such conditional instruction as ‘If there is exactly one set of which (most of ) a, b, c,. . . are members and such that the things that are in it are exactly the instances of a certain natural kind, then the extension of ‘‘is a dog’’ is that set’ (where ‘a’, ‘b’, ‘c’, . . . are names of things which are taken as paradigms of dogs). In this case ‘(Most of ) a, b, c,. . . are dogs’ and ‘The things that are dogs are exactly the instances of a certain natural kind’ are plausibly viewed as initial linguistic preconceptions about ‘dog’. ¹² Henceforth I will omit the parenthetical addition of ‘intension(s)’, ‘its(their) intension(s)’, etc. in many cases in which it should be tacitly understood. ¹³ Assuming that ‘Neptune’ is ‘obstinately rigid’ in the sense of Nathan Salmon (1982).
The Dual Picture of Vagueness
237
They have an apparently successful (implicit) role in fixing both the extension and the intension of ‘is a dog’.¹⁴ (On account of the rigidity of ‘dog’, the latter is simply the constant function assigning the set of possible dogs to each possible world.¹⁵) One distinction between two important kinds of preconceptions stands out, and it is already illustrated in the case of ‘dog’. It’s the distinction between paradigm preconceptions and generic preconceptions. Paradigm preconceptions are relatively simple preconceptions whose intuitive content is either that a certain predicate applies to a certain specific object or objects (positive paradigm preconceptions) or that it negatively applies to a certain specific object or objects (negative paradigm preconceptions). (One example of (positive) paradigm preconception would be ‘(Most of) a, b, c,. . . are dogs’.) Generic preconceptions, on the other hand, are simply preconceptions which are not paradigm preconceptions. Often they intuitively state necessary, sufficient or other sorts of general conditions for the application or negative application of a predicate. They are designed to guide us in the expansion of our use of the predicate beyond its paradigmatic range of use. (One example would be ‘The things that are dogs are exactly the instances of a certain natural kind’.) One specific proposal of the picture in this chapter is that typical sorites susceptible degree adjectives (and scalar nouns) provide yet another example of predicative words associated with a set of preconceptions that includes paradigm and generic preconceptions designed to help fix the extension of those words.¹⁶ (B) and (D) sentences in occasions of use where they are compelling are examples of paradigm preconceptions; (C) sentences in occasions of use where they are compelling are examples of generic preconceptions. But there is one important difference one must emphasize with respect to ‘is a dog’. The intuitive truth value of paradigm and generic sentences for typical degree adjectives varies extremely with the occasion of use, unlike in the case of ‘dog’ and related words. It is natural to postulate the existence of abstract linguistic preconceptions associated with typical sorites susceptible degree adjectives that are not intended to help fix an absolute extension but one relative to an occasion of use, or in other words, regulatory principles for the adoption of concrete paradigm and generic preconceptions relative to particular occasions of use.¹⁷ An abstract preconception regulating the acceptability of specific paradigm preconceptions relative to an occasion of use for ‘small’ presumably takes a form similar to this: In an occasion of use, that already provides a universe of discourse U and a comparison class K included in U , one may take members r1 , r2 , etc. of K as cases of small things and/or ¹⁴ Successful at least over usual universes of discourse, one of which presumably constitutes the intended domain of quantification in these preconceptions. See the text surrounding footnote 28 for more on this qualification. ¹⁵ Assuming that a natural kind predicate is rigid because its designation in all possible worlds is the same set of possible objects. (Cf. the notion of ‘obstinate essentiality’ in G´omez-Torrente 2006.) ¹⁶ The importance of paradigms for our understanding of sorites susceptible predicates is emphasized by Sainsbury (1990). ¹⁷ This is not to imply that preconceptions associated with ‘dog’ and related nouns do, when successful, fix an absolute extension for them, even if in a way they may be so intended. See again the text surrounding footnote 28.
238
Mario G´omez-Torrente
members s1 , s2 , etc. of K as negative cases of small things, just as long as the relevant sizes of r1 , r2 , etc. are smaller than the relevant sizes of s1 , s2 , etc.
This would account for the fact that the intuitive truth value of paradigm sentences for ‘is small’ varies extremely with the standards in operation in the occasion of use. In one occasion of use (e.g. a conversation between wealthy people looking for an apartment) 100 square meters is a positive paradigm of a small size for an apartment; in another (a conversation between people with very modest incomes) 100 square meters is a negative paradigm of a small size. On the other hand, even the intuitive truth value of a concrete generic preconception for ‘is small’ in the occasion of use involving the wealthy (or even the not so wealthy) people, such as ‘(∀x)(∀y)(Kx & Ky ⊃ (x is small & x has 1 square meter less than y ⊃ y is small))’, can vary with the occasion of use. In a conversation between people in Hong Kong looking to buy a micro-apartment where they can fit all their furniture, 1 square meter may make all the difference between smallness and non-smallness. So an abstract preconception regulating the acceptability of concrete generic preconceptions for ‘is small’ presumably takes a form similar to this: In an occasion of use, that already provides a universe of discourse U and a comparison class K included in U , and that may provide members r1 , r2 , etc. of K as cases of small things, and/or members s1 , s2 , etc. of K as negative cases of small things, one may take ‘(∀x)(∀y)(Kx & Ky ⊃ x is small & x has a size inferior by 1 u to y ⊃ y is small))’, where u is a relevant size unit, as a generic principle holding in the occasion of use, provided just that the difference between the ri with the greatest size and the si with the smallest size is greater than 1 u.
Note that ‘(∀x)(∀y)(Kx & Ky ⊃ (x is small & x has a size inferior by 1 u to y ⊃ y is small))’ is logically equivalent to ‘(∀x)(∀y)(Kx & Ky ⊃ (∼ y is small & x has a size inferior by 1 u to y ⊃ ∼x is small))’, which will thus be a concrete generic preconception in play in those occasions of use where its equivalent is in play. It is also natural to postulate that ‘small’ has associated with it an abstract preconception regulating the acceptability of concrete conditional instructions for the fixing of extension/anti-extension pairs relative to particular occasions of use. It would be something like this: In an occasion of use, that already provides a universe of discourse U and a comparison class K included in U , that provides members r1 , r2 , etc. of K as cases of small things, and/or members s1 , s2 , etc. of K as negative cases of small things, and that provides some general principle(s) ‘(∀x)(∀y)(Kx & Ky ⊃ (x is small & x has a size inferior by 1 u1 to y ⊃ y is small))’, ‘(∀x)(∀y)(Kx & Ky ⊃ (x is small & x has a size inferior by 1 u2 to y ⊃ y is small))’, etc., a principle of this form is acceptable: If there is a unique pair <E, A> of subsets of U which are mutually exclusive and jointly exhaustive over U , and are such that • r1 , r2 , etc. are in E, • everything in K that has a size inferior to something in E is in E, • everything in K that has a size superior by 1 u1 to something in E is in E, • everything in K that has a size superior by 1 u2 to something in E is in E, etc. • s1 , s2 , etc. are in A, • everything in K that has a size superior to something in A is in A,
The Dual Picture of Vagueness
239
• everything in K that has a size inferior by 1 u1 to something in A is in A, • everything in K that has a size inferior by 1 u2 to something in A is in A, etc. • U -K is included in A,¹⁸
then <E, A> is the extension/anti-extension pair of ‘is small’ relative to the occasion of use.¹⁹
If something like this principle underlies the fixing of a reference for ‘small’ relative to an occasion of use, then in occasions of use where a unique pair <E, A> satisfies the antecedent of the concrete conditional instruction in operation, a reference for ‘small’ (in the sense of an extension/anti-extension pair) gets fixed;²⁰ in other occasions of use, a reference may not get fixed. Williamson (1999) has argued that, if some mechanism fixes the extension of a predicate F, then the same mechanism, by default, fixes an anti-extension for F: the set of things that (are in the universe of discourse and) are not in the extension of F; as Williamson puts it, fixing the extension and the anti-extension of F are not ‘independent achievements’ (509). This is a reasonable idea, and is not contradicted by the just postulated mechanism for the fixing of a reference for ‘small’ relative to an occasion of use. Note that no proposal is made that there is a set of preconceptions giving intuitively jointly necessary and sufficient conditions for membership in the extension of ‘small’ (unlike what happened in the case of ‘dog’); and no proposal is made that there is an independent set of preconceptions giving intuitively jointly necessary and sufficient conditions for membership in its anti-extension. The extension/anti-extension pair is fixed (when it is) in a ‘coordinated’ fashion, i.e. when the positive and negative paradigm preconceptions and the generic preconceptions are jointly satisfied by a pair of classes. If someone wished to use Williamson’s remark as the basis for an objection to the mechanism postulated here, he might try to argue that there is some independent reason to think that the extension of a predicate must in successful cases be fixed by a ‘non-coordinated’ mechanism, and that its anti-extension must only then be fixed by default. However, I see no reason why this should be so in general, and the apparent possibility of the mechanism just postulated in the text goes against this radical thesis. Furthermore, there are special reasons to think that the thesis is false for actual typical degree adjectives. As emphasized by Sainsbury (1990), a typical degree adjective generally comes together with an antonym (e.g. ‘big’ in the case of ‘small’), and the antonym is analytically connected with a sufficient condition for membership in the anti-extension of the original adjective (e.g. ‘If something is big, it’s not small’). Assuming only that positive paradigm preconceptions play a role in fixing the extension of each lexically different degree adjective by giving sufficient conditions for ¹⁸ I assume (somewhat artificially) that everything that is not in the comparison class is not small in the sense relevant to the occasion of use, and thus that it is in the anti-extension of ‘small’. ¹⁹ There are surely preconceptions about ‘small’ (and as we will see, also about ‘dog’) other than the paradigm and generic preconceptions postulated in the text, and thus the hypothesized conditional instructions are simplifications. ²⁰ As noted above, reference-fixing conditional instructions may often be analytic or a priori. In particular, the preceding abstract principle and its concrete instances may well be analytic or a priori.
240
Mario G´omez-Torrente
membership in it, it follows that the positive paradigm preconceptions corresponding to two antonyms F and G must be ‘coordinated’ in some way if the predicates are to possess suitable extension/anti-extension pairs, for there should be no overlap of the set of positive paradigms of F (which should be in the extension of F) with the set of positive paradigms of G (which should be in the anti-extension of F). It can already be seen from what has been said so far that the present picture implies that the semantic rules for ‘is small’ do not per se (or even in conjunction with uncontroversial facts about the populations of Smalltown and Nonsmalltown) dictate that (Bsmall ), (Csmall ), or (Dsmall ) must be accepted as true. (Bsmall ), (Csmall ), and (Dsmall ) are merely preconceptions, analogous to the non-analytic paradigm and generic principles that have a role in (implicitly) fixing the reference of natural kind predicates. This is consistent with the intuition, mentioned in section 13.1, that (Bsmall ), (Csmall ), and (Dsmall ) are not really analytic or dictated as of obligatory acceptance by the semantics of ‘is small’.
13.3
THE FIXING OF REFERENCE IN REGULAR OCCASIONS OF USE
The classical logic and semantics of the logical expressions, in particular of the quantifiers, requires essentially one thing of the interpretation of a predicate: that the interpretation fix an extension and an anti-extension for the predicate which are mutually exclusive and jointly exhaustive over the previously given universe of discourse. This will not be sufficient by itself for the predicate to be endowed with an intension, but in some cases the fixing of an extension may determine the fixing of an intension if some additional factors are in play. In many occasions of use, the (concrete) paradigm and generic preconceptions and conditional instructions for a sorites susceptible degree adjective or scalar noun provide a classical extension/anti-extension pair for them, and perhaps they are also enough to fix an intension. Let’s consider the following example. A couple of modest income are looking to buy an apartment, and they are having a conversation in which they will try to decide which one to buy. There aren’t that many options. Their choice is reduced to four apartments, with sizes of 65, 70, 100, and 105 square meters; call them ‘A65’, ‘A70’, ‘A100’, and ‘A105’. We may take this set K of apartments as the relevant comparison class for ‘small’ in the conversation. Given their standards in the situation, A65 counts as small for them, and A105 as not small. Also given their standards, they take it that 5 square meters don’t make a difference as to whether an apartment is small or not. We may also postulate that the following concrete conditional instruction (licensed in this occasion of use by the last abstract preconception of section 13.2) is in operation: If there is a unique pair <E, A> of subsets of the universe of discourse U which are mutually exclusive and jointly exhaustive over U , and are such that • A65 is in E, • everything in K that has 5 square meters more than something in E is in E, • A105 is in A,
The Dual Picture of Vagueness
241
• everything in K that has 5 square meters less than something in A is in A, • U -K is included in A,
then <E, A> is the extension/anti-extension pair of ‘is small’ relative to the occasion of use.
Under all these assumptions, the conditional instruction and the paradigm and generic preconceptions in play fix a classical extension/anti-extension pair for the predicate ‘is small’: its extension in the occasion of use is {A65, A70}, and its antiextension is the set containing A100 and A105 plus any other thing that is not in K . Extension and anti-extension are mutually exclusive and jointly exhaustive over the universe of discourse relevant in the conversation. Perhaps also a (classical) intension for ‘is small’ is fixed in the mentioned occasion of use with the help of the mentioned principles and others that might be plausibly postulated. Given that a classical extension/anti-extension pair has been fixed, this pair in turn may induce in the obvious way a pair of scales of associated numbers (measuring sizes in square meters, in this case), which we may take simply as a pair of sets; in the example, the pair of scales would be <{65, 70}, {100, 105}>. Then an intension for ‘is small’ in the mentioned occasion of use might be computed with the help of this principle: the extension of ‘is small’ over the previously given universe of discourse U in a world w contains an element a of K just in case a’s size in w is less than or equal to one of the sizes in the first scale, {65, 70}; and the anti-extension of ‘is small’ over U in a world w contains an element a of K just in case a’s size in w is greater than or equal to one of the sizes in the second scale, {100,105} (and it also contains everything that is not in K ). The resulting intension puts A100 in the extension of ‘is small’ in worlds in which, say, its builder changed the architect’s plans and gave it a size of just 68 square meters.²¹ An occasion of use of a degree adjective or scalar noun may presuppose a comparison class that is even more reduced than the comparison class in the apartment example, and that does not create any obstacle to the fixing of a classical extension/anti-extension pair. If we are talking about a figure such as the following
in most occasions of use we will be able to say felicitously and truly things like ‘The small circle is to the left’, ‘There is a small circle to the left’, ‘The circle to the left is small’, ‘The non-small circle is to the right’, etc.²² In occasions of use like these, ²¹ This mechanism is of course very sketchy and leaves questions unanswered. In worlds w where one of the apartments of the given 4-element set has a size in the open interval (70,100), the mechanism doesn’t assign a classical extension/anti-extension pair to ‘is small’ in w. This seems tolerable and not incompatible with usual possible worlds semantics, which contemplates intensions which are partial functions. The description in the text is not meant as a complete one, but only as indicative of the direction a more complete description might take. ²² These two-element comparison classes (and figures similar to the one in the text) are considered in Klein (1980). See also Kennedy (2007).
242
Mario G´omez-Torrente
it is clear that the extension of ‘small’ relative to the universe of discourse will have been taken to be the set consisting of the circle to the left, and its anti-extension will have been taken to contain the circle to the right. In these cases, the paradigm preconceptions in play suffice to fix the extension/anti-extension pair, given that no appropriate generic preconception will enter into conflict with them. In general, mutatis mutandis the same can be said of cases in which, as in the apartment case or the two circles case, the comparison class is clearly divided into two mutually exclusive and jointly exhaustive subsets consisting, respectively, of positive paradigms and/or individuals that can be ‘reached’ from the positive paradigms by generic preconceptions, and of negative paradigms and/or individuals that can be ‘reached’ from the negative paradigms by generic preconceptions. A great number of uses of typical degree adjectives do not seem to presuppose large, sorites-prone, or even not clearly divided comparison classes. Many of them, on the contrary, seem to be what we might call contrastive uses: they seem to presuppose precisely a comparison class consisting of two clearly separated sets of objects, not infrequently sets of just one element, that need to be forcefully contrasted for the conversational purposes of the situation. Consider the italicized degree adjectives in the following passages, all taken from the first page of a widely used reference work: abbess. (. . .) In the Middle Ages wide powers were claimed by some abbesses, but the Council of Trent put an end to most special prerogatives. Abbot, George (1562–1633), Archbishop of Canterbury from 1611. (. . .) he won James I’s favour by his mission to Scotland (1608) (. . .). As archbishop he was severe on Roman Catholics and partial to Calvinists at home and abroad. (. . .) The strong line which he took over the Essex nullity suit (1613) won him respect and a temporary popularity. In 1621 he accidentally shot a gamekeeper and his position was considered to have become irregular; James decided in his favour and he resumed his duties. He crowned Charles I but had little influence in his reign. (Livingstone (2006), 1)
It is clearly forced to view the use of ‘wide’ as applied to the powers claimed by some abbesses as presupposing a large comparison class consisting of (classes of) powers that people have claimed, or even of powers that the heads of monasteries have claimed; certainly, no knowledge of such comparison class is required of the reader for the understanding of the sentence. Its use simply seems to presuppose a contrast with the powers of the other abbesses who did not claim those same powers. This is made somewhat clearer by the next sentence, in which these powers are called ‘special’, in a use that does not even seem to admit of comparatives. The use of ‘severe’ as applied to Abbot on the Catholics could hardly presuppose an extended class of men or of acts of severity; presumably Abbot’s actions against the Catholics do not rank especially high in the universal classification of acts of severity. The author apparently means to contrast Abbot’s attitude toward the Catholics with his attitude toward the Calvinists, as made clear by the next clause. The use of ‘partial’ presumably should be understood along the same lines. The use of ‘strong’ as applied to Abbot’s line over the Essex nullity suit is nearly impossible to understand as presupposing a comparison class of several ‘lines’ ordered by strength; the author just means to contrast Abbot’s adverse attitude with the favorable attitude of the other side. The use of ‘irregular’ (if
The Dual Picture of Vagueness
243
it’s a use that admits comparatives) seems again not based on a comparison with several positions men (or archbishops) could have, but simply with the position Abbot enjoyed before the shooting incident as an archbishop who had not been involved in any strange circumstances. Finally, the use of ‘little’ as applied to his influence in Charles I’s reign is not meant to locate that influence at a low point in a ranking of ‘influences’ of people or Canterbury archbishops, but simply to contrast that influence with the influence he enjoyed under James I, from whose favor he had benefited. In contrastive uses like these (and the examples could be multiplied at will), there is no obstacle to the fixing of a classical extension/anti-extension pair over the contextual universe of discourse by means of the mechanism of preconceptions sketched above, given that the relevant comparison class is smallish (or even a two-element one) and clearly divided.²³ The ease with which these numerous and useful uses are accommodated without abandoning the presuppositions of classical semantics suggests that, even though other uses are problematic, the linguistic practice involving the employment of sorites susceptible adjectives is sustainable in the face of paradox. If speakers using typical degree adjectives were constantly faced with uses which did seem to create problems for classical semantics or logic, that linguistic practice would probably be hard to sustain. The above examples of uses of adjectives, together with the preconceptions picture of how they obtain classical extensions (and thus of how the utterances in which they appear obtain classical truth conditions), vindicates the plausible idea that an often successful mechanism for the fixing of reference underlies our use of degree adjectives. On the present picture, then, justice is done to the convincing idea that we manage to effect semantically real distinctions between objects with the help of typical sorites susceptible degree adjectives, and even that we do so in a vast number of occasions. Furthermore, the picture does this without postulating an ad hoc non-classical semantics for the logical expressions when they interact with those adjectives. On the picture the intuition is preserved that when we deal with typical sorites susceptible adjectives, we use the classical semantics for the logical ²³ As advanced in footnote 10, in yet other non-problematic cases the adjective will have either an empty or a universal extension over the comparison class; i.e. either an (a) or a (b) genericist suboption (in the sense of section 13.1 above) will be the right option in some special occasions of use. Suppose the comparison class and the universe of discourse are the same, the set of natural numbers; suppose that we take the first ten numbers as (positive) paradigms of small numbers, but we abstain from taking any number as a negative paradigm of smallness; and suppose that we accept the principle ‘(∀x)(∀y)(x is small & x + 1 = y ⊃ y is small)’. The mechanism of preconceptions postulated above then generates as the extension of ‘small’ the whole set of natural numbers: every number is small under the exacting standards in the situation. (Not unreasonable standards, if we reflect that every number is only greater than finitely many numbers but smaller than infinitely many.) This is a (b) case; analogous (a) cases are also easily imagined. And similar cases can be created with many adjectives for other universes of discourse if we suppose the comparison class to be greatly unrestricted, e.g. when it contains many merely possible objects. It seems that there could have been towns of all finite numbers of inhabitants. Suppose then that our universe of discourse contains all such possible towns, and that the comparison class is the set of all possible towns. In a perfectly acceptable occasion of use with these features the extension of ‘is small’ will be universal with respect to the comparison class: every town will count as small. By analogous arguments, one could argue that in some occasions of use in which the comparison class is greatly unrestricted every man counts as bald, no man counts as tall, every man counts as poor, etc. (Perhaps occasions of use of this kind have motivated the proposal of genericist theories; but I’m unaware that they have.)
244
Mario G´omez-Torrente
expressions. We are simply working under the assumption, however tacit or inchoate, that our linguistic preconceptions fix a classical extension/anti-extension pair for the adjectives, and thus no theoretical hypothesis that the logical expressions operate in a non-classical way on a non-classical extension is called for.
1 3 . 4 T H E FA I LU R E O F R E F E R E N C E I N I R R E G U L A R , E S PE C I A L LY PA R A D OX I C A L , O C C A S I O N S O F U S E In the case of names and general terms for natural kinds, substances and phenomena, the Kripkean mechanism of preconceptions described at the beginning of section 13.2 plausibly fails to generate extensions (and intensions) in some cases. These include cases in which the preconceptions enter directly into contradiction with relevant truths that are not preconceived (and they may include non-conflictive cases in which they are nevertheless insufficiently specific to generate a unique extension). Similarly, in the case of typical sorites susceptible degree adjectives and scalar nouns the mechanism of preconceptions described later in section 13.2 fails to generate extensions in cases in which the preconceptions are in conflict with a truth of the form of (A), with the content that a sorites series can be drawn from the comparison class; and the preconceptions also fail to generate extensions in some cases in which no true sentence of the form of (A) is in sight, but they are nevertheless insufficiently specific to divide the comparison class (and hence the universe of discourse) uniquely into two mutually exclusive and jointly exhaustive subclasses. I will begin this section explaining and illustrating these failures. I take it to be fairly uncontroversial that in some cases the descriptive identifications and conditional instructions by means of which an attempt is made to fix a reference for certain names fail to do so. A well-known example is ‘If there is exactly one planet causing the perturbations in the orbit of Mercury, then ‘‘Vulcan’’ refers to that planet’, which fails to fix a reference for ‘Vulcan’. It seems also most reasonable to think that, even though in a vast number (or even a majority) of cases of terms for natural kinds, substances and phenomena, the initial linguistic preconceptions and conditional instructions about a predicate successfully fix an extension (and an intension) for it, in at least some cases they fail to do so, just as in the proper name case.²⁴ Consider ‘If there is exactly one set of which most of a, b, c,. . . are members and such that the things that are in it are exactly the instances of a certain disease, then the extension of ‘‘is an instance of madness’’ is that set’. Here ‘Most of a, b, c,. . . are instances of madness’ and ‘The things that are instances of madness are exactly the instances of a certain disease’ can be viewed as initial linguistic preconceptions about ‘is an instance of madness’. If the former mentions paradigms of all traditional kinds, the two preconceptions are not jointly satisfied, for it has turned out that there are many kinds of equally frequent traditional paradigms, which are instances of diseases ²⁴ It is natural to conjecture, as we did with degree adjectives, that the vast number of cases in which the mechanism of preconceptions fixes a reference for terms for natural kinds makes the linguistic practice involving them sustainable even in the presence of problematic cases.
The Dual Picture of Vagueness
245
or other phenomena that don’t have anything to do one with the other—epilepsy, tetanus, dementia praecox, delirium tremens, all kinds of so called neuroses and psychoses, etc. Now, of course normal people at a relatively pretheoretical level of use do not sense any problem preventing sentences containing ‘madness’ or ‘mad’ to have truth values. And it seems to me that, after exposure to the theoretically varied nature of the diseases or other phenomena that prompt attributions of madness, most people tend to reject the generic preconception ‘The things that are instances of madness are exactly the instances of a certain disease’ and stick to the preconception to the effect that at least a majority of paradigm cases must fall under the extension of ‘madness’ and ‘mad’ (Charles Manson, Dr Samuel Johnson,²⁵ my extremely agoraphobic neighbor, and so on). However, on what seems to me to be the most reasonable view after reflection, there is actually no fact of the matter as to whether, e.g. Manson, Dr Johnson, my neighbor, etc. are instances of madness, or there is no such thing as an instance of madness among them; thus no extension is fixed for ‘is an instance of madness’, as there isn’t even a fact of the matter whether its extension should contain them or not. Of course we may use ‘madness’ with a definite extension if either we stick to the preconception that most people in our paradigms list are instances of madness but accept that madness is not a (common) disease; or if we stick to the preconception that the instances of madness are precisely the instances of a certain disease but accept that it’s not a disease exemplified by most people in the list, in which case it’s false that most of them are instances of madness and ‘is an instance of madness’ has a reduced extension, possibly empty, consisting of the instances of a single disease, possibly an imaginary one. But regardless of any initial inclination we may have, reflection suggests that these options ultimately require arbitrary decisions not justified by preexisting usage. In particular, if we decide to stick to most of our paradigm preconceptions, it is unclear that we can appeal to any principle determining exactly which majoritarian subset of these preconceptions we should stick to. And even if there is such a principle (e.g. if for some reason ‘the’ principle is to stick to all of our paradigm preconceptions), once we abandon the idea that most instances of madness must be instances of a common disease, it becomes unclear how to evaluate new cases for membership in the extension of ‘madness’, and thus how to obtain a determinate extension merely from the paradigms. Is ‘the’ general principle to include in the extension all the new cases which exhibit the same descriptive symptoms (assuming we can specify these) as the initial paradigms? Is ‘the’ principle to include all the new cases which fall under one of the diseases or other phenomena exemplified by the initial paradigms? Is it something else? Of course, we may revert to sticking to the generic preconception to the effect that the instances of madness ought to be precisely the instances of a certain disease, and then probably to considering ‘madness’ as naming some sort of imaginary disease having no real instances. But this seems no more compulsory than any of the paradigmatist options. The most reasonable stand seems to be to acknowledge that ‘madness’ is in some way defective, and that the reason is that its preexisting semantics ²⁵ A well-known case of a personality with extremely obsessive-compulsive habits.
246
Mario G´omez-Torrente
together with its preexisting associated preconceptions fail to fix an extension for it in the presence of a conflict they were not designed to cope with. A crucial proposal of this chapter is that, provided we accept that typical sorites susceptible degree adjectives and scalar nouns are governed by something like the mechanism of preconceptions of section 13.2, they fail to have a reference (a classical extension/anti-extension pair) in occasions of use in which the preconceptions are in conflict with a truth of the form of (A), stating that a sorites series can be drawn from the comparison class. Consequently, utterances of usual sentences containing typical sorites susceptible degree adjectives and scalar nouns in such occasions of use will lack truth conditions;²⁶ these are paradoxical occasions of use. The sorites reasoning makes explicit the existence of a conflict between, e.g. the truth (Asmall ), on the one hand, and the paradigm preconceptions (Bsmall ) and (Dsmall ), and the generic preconception (Csmall ) on the other, as uttered or considered in a paradoxical occasion of use such as the one described at the beginning of this chapter.²⁷ We can ask again: does preexisting usage determine that some of these sentences are true while at the same time the others are false? Is it the generic preconception that is false, or is it some of the paradigm preconceptions? (Or is it the case that we have been under some illusion that classical semantics and logic govern our use of ‘small’?) Needless to say, the conflict is unresolved, as reflected in the existence of genericist and strongly nihilist theories, even though, as noted earlier, there is some initial intuitive pressure for paradigmatism. The most reasonable view seems again to be that ‘small’, as used in paradoxical occasions of use, is defective, as its semantics and associated preconceptions are not enough to get an extension for it in the presence of the unexpected sorites conflict. Despite the by now predictable initial intuitive pressure for paradigmatism, the sorites case is one in which it is particularly clear that paradigmatism is not the right option. It is not only that, as in the case of ‘madness’, no paradigmatist option for obtaining a full extension from the paradigms seems singled out by preexisting usage, but also that all standard paradigmatist options seem clearly false after some reflection (as noted in section 13.1). Abandoning (Csmall ) means either postulating and ad hoc semantics or logic for the logical expressions, or else accepting the negation of (Csmall ) as classically understood, and hence the truth of the corresponding optimistic, and so hard to believe, existential quantification. One virtue of the present picture is that it explains in a simple way why this existential quantification is so hard to believe. In all probability there is no further preconception that provides for the determination of the sharp cut-off point that is needed in this case; hence abandoning (Csmall ) does not provide any way of assigning an extension to ‘is small’. The presumable scarcity of preconceptions, and especially the inexistence of a preconception providing for the determination of needed sharp cut-off points, explains our natural ²⁶ Perhaps some utterances of sentences containing sorites susceptible predicates in some irregular occasions of use have truth conditions, e.g. some where the predicates are in the scope of locutions of propositional attitude. ²⁷ In speaking of (Bsmall )–(Dsmall ) as sentences in conflict, or as elements of reasoning, etc., I am of course not implying that they do after all have a truth value. They have those properties roughly in the same sense that schemata can be inconsistent or are usable in schematic reasonings.
The Dual Picture of Vagueness
247
repugnance for optimism, and even implies that it is false as a matter of fact, at least if we further accept that sharp cut-off points could only be determined by some feature of either the meaning or the non-analytic linguistic preconceptions about a predicate. The present proposal thus satisfies in a very strong sense the desideratum that a picture of the sorites phenomenon should not be optimistic. More generally, the picture has the welcome implication that paradigmatist options as a group are not really semantically superior to genericism, despite initial appearances. Many other kinds of predicates besides degree adjectival predicates have been claimed to be sorites susceptible, including natural kind predicates. For example, the following has been claimed to be a compelling (A)–(D) set, where the quantifiers range over a greatly unrestricted universe of discourse containing billions of particle aggregates, n is some huge number, ‘Rigo’ is the name of a dog, and ‘Molly’ the name of a single molecule of some sort: (Adog ) (∃x1 ) . . . (∃xn )(x1 results from Rigo by the removal of a single molecule & x2 results from x1 by the removal of a single molecule & . . . & xn results from xn−1 by the removal of a single molecule & Molly results from xn by the removal of a single molecule); (Bdog ) Rigo is a dog; (Cdog ) (∀x)(∀y)(x is a dog & y results from x by the removal of a single molecule ⊃ y isa dog); (Ddog ) ∼ Molly is a dog. If these are real paradoxes, the present picture suggests that the solution for them may lie in acknowledging that, while the more natural preconceptions associated with a natural kind predicate (mentioned in section 13.2) are in some sense designed to fix an absolute extension for it, they may only manage to fix one over the tame universes of discourse which are presumably quantified over in those preconceptions. If (Bdog ), (Cdog ), and (Ddog ), for example, are further preconceptions associated with ‘dog’, then, in occasions of use involving a universe of discourse containing billions of suitably weird aggregates of particles, the preconceptions associated with ‘dog’ will not be jointly satisfied, ‘dog’ will not get an extension and (Bdog ), (Cdog ), and (Ddog ) will all lack a truth value. Nevertheless, the natural preconceptions for ‘dog’ mentioned in section 13.2 surely fix an extension for it if they quantify, as they presumably do, over more usual universes of discourse, that contain only normal objects and don’t contain billions of weird aggregates of particles. Even if sorites conflicts create obstacles to the fixing of an extension in weird occasions of use involving greatly unrestricted universes of discourse, a vast majority of occasions of use involving ‘dog’, and other non-scalar nouns for natural kinds, artifacts, etc. will be non-problematic occasions of use (similarly for non-degree adjectives like ‘canine’). Space limitations prevent detailed examination of these predicates here.²⁸ ²⁸ Another case that I can only mention cursorily is that of ‘appearance’ predicates like ‘looks red (to John)’. Such a predicate is often thought to give rise to a sorites paradox when the relation in the relevant (A) and (C) sentences is ‘looks the same in color (to John)’. Again the present picture suggests the possibility that ‘looks red (to John)’ fails to get an extension over universes of discourse
248
Mario G´omez-Torrente
There are also reference failures in some cases in which no true (A) sentence is in view, now due simply to insufficient specificity of the paradigm and generic preconceptions in play. These will be non-paradoxical but still irregular occasions of use. Think of this variant of the apartment example. A different couple (with a less modest income) must choose an apartment from a set K containing 65, 70, 100, 130, and 135 square meters apartments (A65, A70, A100, A130, and A135). A65 and A135 count clearly as small and not small for them, respectively, and 5 square meters don’t make a difference for them as to whether an apartment is small or not. The relevant preconceptions then imply that A70 is small, and that A130 is not small, but fail to imply that A100 is either in the extension of ‘small’ or in its anti-extension. In a case like this, even though there is no paradox, it is most reasonable to think that the predicate ‘is small’ fails to have a uniquely determined reference, for there seems to be no preconception determining that A100 should be either in its extension or in its antiextension.²⁹ In both irregular paradoxical and irregular non-paradoxical occasions of use, an impression is created that some objects far away from the positive and negative paradigms along the relevant dimension of comparison are ‘borderline cases’, objects that fall outside the extension and anti-extension of the adjective in question. The explanation of this impression according to the present picture is that these objects, besides being neither positive nor negative cases of application of the adjective, are not even preconceived as paradigms, and are psychologically far away from them along the relevant dimension of comparison.³⁰ The objects in question are not ‘borderline cases’ in the sense that they fall outside the extension and anti-extension of the adjective while the paradigms and objects easily reached from them by the generic that contain suitable sorites series of color patches, while it gets an extension in tamer universes. However, it is also quite possible that this case is in fact like other related cases which sometimes are thought to be sorites paradoxical but are not really so. We could train a pigeon to peck at big heaps of seed and to refrain from pecking at small heaps. Substitute ‘is pecked at (by the pigeon)’ for ‘F’ in (B)–(D) and ‘is indiscriminable for pecking purposes (by the pigeon)’ for ‘R’ in (A) and (C), and think of n as some suitably large number. Here the relevant (A) sentence seems simply false (while ‘is pecked at (by the pigeon)’ does get an extension). For any sorites series h1 , h2 , . . ., hn there will be some number i of seeds for which the pigeon will eventually fail to peck at hi+1 , after having pecked at hi , and thus it will after all discriminate in some way between the sizes of hi and hi+1 . In the same way, the camel’s back will break with a number j+1 of straws even though it did not break with j straws. (These ‘cut-off’ numbers will vary from circumstance to circumstance, but this doesn’t show that the predicates involved have any interesting semantic peculiarity; it only shows that the constitutions of the pigeon and the camel suffer minute changes from circumstance to circumstance.) ²⁹ There may also be cases in which the preconceptions about a term for a natural kind are insufficiently specific to generate a unique extension, even if they don’t enter into conflict with any truths. ³⁰ A speaker who considers successively the items in a sorites series will presumably reach a point where, e.g. though he is (already baffled but) ready to count as small a certain town ti , he is baffled and not ready to count ti+1 as small, and he may perhaps in some cases be (baffled and) ready to count ti+1 as not small. Contextualist theorists (see footnote 7) may explain these baffled shifts as arising from subtle context changes. The present theory postulates that the judgments in question lack a truth value, and that presumably the shifts (and the bafflement) of the speaker are to be explained by the truth value gaps rather than by any concealed context change.
The Dual Picture of Vagueness
249
preconceptions fall inside. There is of course no extension/anti-extension pair in the problematic cases, and so no ‘borderline cases’ in the mentioned semantic sense.³¹ Nor are there ‘borderline cases’ in an epistemic sense, i.e. objects that fall either in the extension or in the anti-extension but are not known to fall in any of the two places. There are ‘borderline cases’ in a purely psychological sense. Normal speakers at relatively pretheoretical levels of use tend strongly to believe that utterances containing degree adjectives, of both paradigm and generic sentences, have truth values even in irregular occasions of use; the thought of an extension gap and accompanying truth value gaps is very hard to elicit from them. This fact was of course to be expected in speakers not exposed to the sorites paradox and to sufficient theoretical reflection on it, and constitutes no problem at all for the dual picture of this chapter. This case is no different from the case of failed natural kind terms, in which the plausible lack of reference and of truth conditions is nevertheless accompanied by a resolute acceptance by normal speakers of both the relevant generic and paradigm preconceptions. It is important, however, to stress the fact that, under the assumptions made earlier, this resolute acceptance would be hard to explain if we did not have at hand an often successful reference-fixing mechanism for natural kind terms that plausibly underlay normal people’s tacit understanding of how these terms come to have a reference (when they do). Analogously, in the case of degree adjectives this is precisely what is provided by the description of the mechanism of preconceptions in section 13.2. A further psychological factor that may contribute to the resoluteness with which normal speakers accept the relevant preconceptions is what we may call the closeness phenomenon. When we are confronted with a compelling (B)–(C)–(D) set in a particular paradoxical occasion of use, and in fact even when we are confronted with a compelling (B)–(C)–(D) set in an irregular but non-paradoxical occasion of use, there are potential very close regular occasions of use in which the same paradigm and generic sentences work as preconceptions, but generate a classical extension/anti-extension pair. The existence of these close occasions of use may even ³¹ The notion of a ‘borderline case’ has been closely associated with attempts to characterize vagueness. It is by now generally accepted that the existence of ‘borderline cases’ in a semantic sense could not characterize what it is for a predicate to be vague, for, assuming that ‘borderline cases’ in the semantic sense were possible, one could define predicates with precise cut-off points separating the positive and negative cases from the ‘borderline’, thus predicates with ‘borderline cases’ but not sorites susceptible. An alternative, and by now apparently popular proposal is to say that a vague predicate is a ‘boundaryless’ predicate (in the sense of Sainsbury 1990), i.e. one for which there is simply no semantically determined precise border separating the positive and the negative cases from the ‘borderline’. But think of the occasion of use in the second apartment example, and imagine that we introduce an adjective ‘small*’ stipulating that the sentences resulting from replacing ‘small’ with ‘small*’ in the paradigm and generic preconceptions of the original example are to be taken as of required acceptance by the semantics of ‘small*’ (and no other principle governs its semantics). Then, in a clear sense, no semantically determined precise border exists separating the positive and negative cases of apartment smallness* from the borderline (we don’t stipulate jointly necessary and sufficient conditions for membership in either the extension or the anti-extension of ‘is small*’). And yet there is no sorites susceptibility, because no sorites series is in sight in the occasion of use. Assuming that sorites susceptibility is a necessary condition on vagueness, it follows that ‘boundarylessness’ does not characterize it.
250
Mario G´omez-Torrente
divert to some extent our attention from paradox or irregularity in general. Let’s go back to the (Asmall )–(Dsmall ) set of our initial example and its described original occasion of use. A very close occasion of use is one in which the comparison class consists exclusively of Smalltown and/or a few tiny towns ‘reachable’ from it with the help of the generic preconception(s) in play in the original occasion of use, plus Nonsmalltown and/or a few big towns ‘reachable’ from it with the help of that(those) same generic preconception(s). In the new occasion of use the comparison class provides no sorites series, and the paradigm and generic preconceptions in play (the same as in the original occasion of use) suffice to divide the comparison class (and hence the universe of discourse) into two mutually exclusive and jointly exhaustive subclasses. Similarly, in the second apartment example, which was irregular but not paradoxical, a very close occasion of use is one in which the comparison class consists exclusively of the 65 square meters apartment and/or the 70 meters apartment, plus the 135 meters apartment and/or the 130 meters apartment. Here again there is no sorites series, but neither is there any psychological ‘borderline case’, and the paradigm and generic preconceptions in play suffice to divide the comparison class (and hence the universe of discourse) into two mutually exclusive and jointly exhaustive subclasses. In general, the strong tendency to believe that paradigm and generic sentences, and other sentences, are true even in irregular occasions of use, may to some extent be reinforced by the existence of closely similar and simpler occasions of use of the basic contrastive, regular kind described in section 13.3.³² In my view, one main strength of the picture of this chapter is the understanding it can provide of the fact that, even after reflection on paradox, people tend to give a preference to their intuitions on the truth value of paradigm preconceptions over their intuitions on the truth value of generic preconceptions—the phenomenon of the preference for paradigm intuitions, for short. Most theories of the sorites paradox have taken the preference for paradigm intuitions at semantic face value, and thus have assumed that paradigm preconceptions must be true. However, as argued in section 13.1 and recalled a few paragraphs ago, reflection suggests that standard paradigmatist theories of the sorites paradox are all false. I take this as a datum, and I think that what is needed is a theory that, while implying the falsity of paradigmatism, ³² In (2009) Peter Pagin has proposed a much more ambitious contextualist thesis, according to which in most contexts of use of a sorites susceptible predicate, some contextual factor restricts the domain of quantification so that a classical extension/anti-extension pair over the restricted universe is delivered for the predicate, and in such a way that sentences involving the predicate retain their intuitive truth value. However, it seems implausible that any factor determining contextually the domain of quantification works in such fine-tuned coordination with the mechanisms for predicate reference-fixing. For one thing, it is unlikely that any contextual factor determines two unique sharp cut-off points, between the extension of the predicate and the intermediate excluded cases and between these and the anti-extension. Pagin seems to agree with this, and to propose his theory not as a theory of the determination of reference, but as a theory about the determination of a class of extension/anti-extension pairs that might all equally well play the role of referents for a sorites predicate in paradoxical occasions of use. However, I doubt that even such a class can be determined in paradoxical cases: in the (Asmall )–(Dsmall ) example, it is implausible that there is a biggest town that will not appear in the extension of any of the extension/anti-extension pairs that might all equally well play the role of referents of ‘small’. (For discussion of Pagin’s theory I am indebted to Sven Rosenkranz.)
The Dual Picture of Vagueness
251
can at the same time explain its appeal. The picture in this chapter does precisely that, when it is supplemented with a number of allied plausible conjectures about the psychology of paradigm beliefs. The preference for paradigm preconceptions has several plausible sources in the reliance on paradigms for reference-fixing and its associated psychology. Probably paradigm preconceptions are psychologically more basic than generic preconceptions in many respects. For example, it is well known that the inclination to classify under a common predicate certain paradigms or prototypes develops earlier in children than any implicit generic idea as to how one should expand the extension of the predicate starting from the paradigms. This inclination is also probably of more adaptive or practical value at pretheoretical levels of use than the development of any generic idea; one can classify and contrast particular objects by means of paradigm preconceptions (even if these turn out to contain semantically defective predicates), thus getting a means to influence and react to one’s hearer’s responses to particular objects, but one cannot do that merely with generic preconceptions. At a more theoretical level of use, and especially after exposure to paradox or conflict, other factors may contribute psychologically to the preference for paradigm preconceptions. It is clear that stipulatively rejecting the paradigm preconceptions would involve a more radical departure from established usage than stipulatively rejecting the generic preconceptions. We might, for example, fix a generic principle determining the extension of ‘madness’ by stipulation, but we could not stipulate paradigm preconceptions about ‘madness’ to be false without suppressing our ability to effect distinctions with the help of ‘madness’; similarly, we might fix cut-off points for ‘small’ by stipulation in irregular occasions of use, but we could not stipulate paradigm sentences about ‘small’ to be false without suppressing our ability to effect distinctions with the help of ‘small’. There is also the presumable fact that occasionally, after the emergence of paradox or conflict, and without the help of explicit stipulations, linguistic practice settles on some generic principle that is compatible with the original paradigm preconceptions. The psychologically evident possibility of stipulations or implicit choices of generic principles compatible with the original paradigm beliefs may well cause to some extent the preference for paradigm preconceptions. But of course, this possibility does not imply that preexisting usage does, or even can, single out non-arbitrarily any generic principle that fixes the extension of a problematic predicate. These psychological factors are presumably in operation whenever a referencefixing mechanism relies heavily on paradigm preconceptions. Other plausible psychological factors contributing to the preference for paradigm preconceptions are specific to the use of degree adjectives. Paradigm preconceptions about these adjectives are often less variable than generic ones, even with respect to the same comparison class. Presumably the 65 meters apartment will count as small under any standard with respect to all (or most) comparison classes in which it is the smallest apartment; but a generic preconception with the intuitive content that five meters don’t make a difference as to smallness will vary widely in perceived truth value even with respect to a fixed comparison class where the 65 meters apartment is the smallest apartment. Other paradigm preconceptions are stable even across all occasions of use of a
252
Mario G´omez-Torrente
predicate; for example, a man with no hairs counts as intuitively bald in all occasions of use. It seems to me that together, all these probable psychological factors provide considerable support for the thesis that the preference for paradigm preconceptions does not have a semantic root. If we take it as a datum that paradigmatist semantic theories of the sorites paradox are not determined to be correct by preexisting usage, the existence of these psychological explanations goes quite a bit of the way toward eliminating the paradigmatist inclination suggested by the preference for paradigm preconceptions. Finally, I should stress that, unlike strong nihilist theories, the dual picture is not necessarily self-referentially instable. Its proponent says ‘All sorites susceptible predicates, in paradoxical occasions of use, lack an extension, and all sorites susceptible predicates, in occasions of use in which the mechanism of preconceptions works, have an extension’. In order for his utterance to be true, the occasion of use in which he makes it must be one in which the predicates appearing in that sentence have an extension. Do they have an extension in the relevant occasion of use? Assuming that predicates in general have their extensions determined (when they do) by mechanisms of preconceptions related to the ones postulated for degree adjectival predicates and natural kind predicates, the problem is basically the problem of what is the typical universe of discourse presupposed in that occasion of use or similar ones, and of whether the preconceptions associated with the predicates appearing in the sentence fix classical extension/anti-extension pairs for them over that universe. This in turn reduces to the question whether the proponent of the dual picture needs to quantify over things which, in the relevant occasion of use, are neither clear predicates nor clear non-predicates, or neither clear occasions of use nor clear non-occasions of use, etc. I conjecture that he doesn’t need so to quantify. Metaphysical theories often quantify over large universes of discourse. But it’s unclear that an appropriate theory of the basic linguistic phenomena surrounding the sorites must be a metaphysical theory. It may be a linguistic theory that doesn’t need to quantify over universes of discourse containing inordinately large numbers of things. For example, it might be claimed that some types of sounds emitted by humans in some counterfactual, imaginable or even real cases are neither clear cases of predicates nor clear non-cases, perhaps because they are neither clear cases of words nor clear cases of non-words. But the proponent of the dual picture doesn’t need to consider the properties of those sounds, just as a syntactician doesn’t typically theorize about sounds or expressions that are not clear words. The dual picture is intended only for things of the type we find in dictionaries, in English and similar languages. In the universe of things it quantifies over, its proponent can assume a clear division between words and non-words, and presumably between predicates and non-predicates. Related remarks hold for the picture’s use of ‘occasion of use’ and other predicates that appear in its formulation. Unfortunately, space limitations again prevent further discussion of this issue in this preliminary presentation.
The Dual Picture of Vagueness
253
Re f e re n c e s Cargile, J. (1969), ‘The sorites paradox’, British Journal for the Philosophy of Science 20, 193–202. Dummett, M. (1975), ‘Wang’s paradox’, Synthese 30, 301–24. Eklund, M. (2002), ‘Inconsistent languages’, Philosophy and Phenomenological Research 64, 251–75. Fara, D. G. (2000), ‘Shifting sands: An interest-relative theory of vagueness’, Philosophical Topics 28, 45–81. (Originally published under the name Delia Graff.) Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. G´omez-Torrente, M. (2002), ‘Vagueness and margin for error principles’, Philosophy and Phenomenological Research 64, 107–25. (2006), ‘Rigidity and essentiality’, Mind 115, 227–59. (forthcoming), ‘Kripke on color words and the primary/secondary quality distinction’ in A. Berger (ed.), Saul Kripke, Cambridge University Press, Cambridge, forthcoming. Heck, R. G. (2003), ‘Semantic accounts of vagueness’ in Jc Beall (ed.), Liars and Heaps, Oxford University Press, New York, 106–27. Kennedy, C. (2007), ‘Vagueness and grammar: The semantics of relative and absolute gradable predicates’, Linguistics and Philosophy 30, 1–45. Klein, E. (1980), ‘A semantics for positive and comparative adjectives’, Linguistics and Philosophy 4, 1–45. Livingstone, E. A. (2006), The Concise Oxford Dictionary of the Christian Church, 2nd rev. edn., Oxford University Press, Oxford. Machina, K. (1976), ‘Truth, belief and vagueness’, Journal of Philosophical Logic 5, 47–78. Pagin, P. (2009), ‘Vagueness and Central Gaps,’ in this volume. Sainsbury, M. (1990), ‘Concepts without boundaries’, reprinted in R. Keefe and P. Smith (eds.), Vagueness. A Reader, MIT Press, Cambridge, MA, 1996, 251–64. Salmon, N. (1982), Reference and Essence, Blackwell, Oxford. Schiffer, S. (1999), ‘The epistemic theory of vagueness’, Philosophical Perspectives 13, 481–503. Soames, S. (1999), Understanding Truth, Oxford University Press, New York. Unger, P. (1979), ‘There are no ordinary things’, Synthese 41, 117–54. (1984), Philosophical Relativity, University of Minnesota Press, Minneapolis. Williamson, T. (1994), Vagueness, Routledge, London. (1999), ‘Schiffer on the epistemic theory of vagueness’, Philosophical Perspectives 13, 505–17. Wright, C. (2003), ‘Vagueness: A fifth column approach’ in Jc Beall (ed.), Liars and Heaps, Oxford University Press, New York, 84–105.
14 Vagueness and Central Gaps Peter Pagin
Ordinary intuitions that vague predicates are tolerant, or cannot have sharp boundaries, can be formalized in first-order logic in at least two non-equivalent ways, a stronger and a weaker. The stronger turns out to be false in domains that have a significant central gap for the predicate in question, i.e. where a sufficiently large middle segment of the ordering relation (such as taller for ‘tall’) is uninstantiated. The weaker principle is true in such domains, but does not in those domains induce the sorites conclusion. This fact can be used for interpreting ordinary uses of vague expressions by means of a new kind of contextual quantifier domain restriction. A central segment is cut from the domain, if consistent with speaker intentions. As long as this is possible, tolerance, bivalence and consistency can all be retained. This chapter focuses on the basic semantic properties in a model-theoretic setting. The natural language application is sketched and the nature of the approach briefly discussed.
14.1
TO L E R A N C E P R I N C I P L E S
A sorites argument in the inductive format is normally taken to have the following form: (1) 1 F (k1 ) 2 ∀i(F (ki ) → F (ki+1 )) 3 F (kn ) This chapter was presented as a paper at the 6th Arch´e vagueness workshop in St Andrews, in March 2006, at the Logic and Language conference in Birmingham, in April 2006, and at the Stockholm Logic and Language seminar in September 2006. I am grateful to the audiences on those occasions, in particular to Herman Cappelen, Manuel Garc´ıa-Carpintero, Richard Dietz, Patrick Greenough, Patrick Grim, Jeff King, Andrew McGonigal, Augustin Rayo, Tim Williamson, and Crispin Wright. I owe especially much to Sven Rosenkranz who was my commentator in Birmingham, and who also provided me with further helpful comments on a later version. I also owe much to Kathrin Gl¨uer for discussions of these ideas over several years. The work on this paper was funded by a research grant from The Swedish Research Council.
254
Vagueness and Central Gaps
255
The terms ‘k1 , . . . kn ’ are taken to denote objects a1 , . . . an in a sequence A along some ordering relation, such as taller than. We have an instance of the sorites paradox if the premises are apparently true while the conclusion is apparently false. We get a typical example by choosing ‘is tall for a man’ for ‘F ’, and a1 , . . . an as a sequence of men so that a1 is 200 cm tall, and each ai+1 is 1 mm shorter than his predecessor, while an is 150 cm tall. In such a case both premises do seem true, while the conclusion also does seem false. The argument is inductive, and the second premise is the inductive premise. The apparent acceptability of the inductive premise derives from a basic intuition about typically vague terms: that they do not have sharp boundaries. In the case imagined, that intuition would typically be expressed by means of (2) 1 mm cannot make the difference between being tall and not being tall. Put in a more inductive format, we would say, in this case: (3) If a man of n + 1 mm is tall, then a man of n mm is tall. When put in the format of (3), the formulation is apt for expressing a version of the basic intuition: that the predicate in question is insensitive to small differences. The property of being insensitive to small differences is what Crispin Wright has called ‘tolerance’ (Wright 1976, 156). Wright says What is involved in treating these examples as genuinely paradoxical is a certain tolerance in the concepts which they respectively involve, a notion of a degree of change too small to make any difference, as it were. . . . Then F is tolerant with respect to φ if there is also some positive degree of change in respect of φ insufficient ever to affect the justice with which F applies to a particular case.
For reasons that are briefly given later (section 14.5), I prefer to speak of linguistic expressions as being vague or tolerant, rather than concepts, but that is for now less significant.¹ Wright’s introduction of the notion of tolerance, over and above the notion of lacking sharp boundaries, was important, not just because a term may in fact lack sharp boundaries for one reason or other,² but because the tolerance of a vague term intuitively explains why it lacks sharp boundaries. It is conceivable that a term lack sharp boundaries in all worlds where it has the same meaning, even though there is a nomological or metaphysical explanation of why this is so that has nothing to do with vagueness.³ For present purposes I shall refer to principles like (3) as tolerance principles, even though, for the reasons just stated, they can be true because of other factors than ¹ I also find they claim, that the justice with which the term applies isn’t affected, unnecessarily strong. It would be enough for purposes of an account of vagueness, it seems to me, if the degree of justification drops by approximating a limit value that is high enough. ² This could be, for instance, because by nomic regularity, animal species cannot be very similar. As long as we treat vagueness extensionally, it can also be because the predicate has an empty extension. ³ We cannot explain in general how the distinction between being true in virtue of meaning properties and being true in virtue of properties of what is denoted is to be applied (there does not seem to be an analytic/synthetic dichotomy in this sense), but we may be able to do somewhat better when it comes to tolerance.
256
Peter Pagin
insensitivity. A tolerance principle is an inductive principle that involves a tolerant predicate—‘tall’ in the case of (3)—and a tolerance level —1 mm in the case of (3). A tolerance level, as I use the term, then, does not depend on any particular sequence of objects, but makes explicit what the difference is to be in a sequence of objects that is appropriate for an inductive premise like the one in 1. Since tolerance principles are themselves inductive principles we can as well state a sorites argument with their help, without relying on any particular sequence. In order to so, we must make the logical form precise. It quickly turns out, however, that the informal statement (3) admits of two different formalizations. The form of antecedent and consequent, (4) A man of x mm is tall as it occurs in (3), has a generic interpretation, which, I assume, is equivalent to (5) ∀y(Man(y) & H (y) ≥ x → Tall(y)) with ‘H ’ for ‘the height of . . . in mm’.⁴ Accordingly, given that (3) itself is of conditional form, the natural predicate logic rendering would be (T1) ∀y(Man(y) & H (y) ≥ n + 1 → Tall(y)) → ∀y(Man(y) & H (y) ≥ n → Tall(y)) On the other hand, we would also take (3) to have a reading like this: (6) For any two men, if the one is (not more than) n + 1 mm tall and the other is (at least) n mm tall, then if the former is tall, so is the latter. And, the natural formalization of (6) is (T2) ∀x∀y((Man(x) & Man(y) & H (x) ≤ n + 1 & H (y) ≥ n) → (Tall(x) → Tall(y))) But although we seem to get (T1) as well as (T2) out of the intuitive formulation (3), (T1) and (T2) are not equivalent. The difference is made more perspicuous by noting that (T2) is equivalent with (T2 ) ∃y(Man(y) & H (y) ≤ n + 1 & Tall(y)) → ∀y(Man(y) & H (y) ≥ n → Tall(y)) I shall call tolerance principles of the form (T1) strong tolerance principles, and those of the form (T2)/(T2 ) weak tolerance principles.⁵ Weak and strong tolerance principles work differently. With respect to some cases they induce the same result, but not in all. Consider the original setup, with the sequence A = a1 , . . . , an , and Let G(n) =def ∀y(H (y) ≥ n → Tall(y)) ⁴ The generic reading of (4) involves an element of nomicity that is not captured by the quantification in (5), but I shall not concern myself with that aspect. ⁵ I shall henceforth take quantification to be restricted to a domain of men, and so the ‘Man’ conjunct will be dropped.
Vagueness and Central Gaps
257
We then get with (T1) a modified sorites argument: (7)
1 2 3 4 5 6
G(2000) H (kn ) = 1500 & ¬ Tall(kn ) ∀i(G(i + 1) → G(i)) G(1500) Tall(kn ) ⊥
assump. assump. (T1) 1, 3 2, 4 2, 5
Note that the absurdity conclusion requires the assumption (2), for by itself, (4), i.e. ∀y(H (y) = 1500 → Tall(y)) is true if there are no men of 150 cm height in the universe of discourse. Similarly, 1 is true if there are no men of height 200 cm in the universe of discourse. We could add a dependence on actual instantiation and drop the assumption 1. We then replace it with assumption about k1 , together with a number uniformity principle, stating that anyone is tall whose height is at least equal to that of someone who is tall:⁶ (U) ∀x, y(H (y) ≥ H (x) → (Tall(x) → Tall(y))) We then have the simple derivation (8)
1 H (k1 ) = 2000 & Tall(k1 ) 2 G(2000)
assump. 1, (U)
which can replace the first assumption in (7). In that case the contradiction depends on the (U), (T1), and the facts about the heights and tallness attributes of k1 and kn . In either case, at least one singular fact is needed. If instead we use the weaker tolerance principle (T2), the derivation will be rather different. First, another abbreviation: Let E(n) =def ∃x(H (x) ≤ n & Tall(x)) With respect to the same domain of men, we will then have the sorites argument: (9)
1 2 3 4 5 6 7
H (k1 ) = 2000 & Tall(k1 ) H (kn ) = 1500 & ¬Tall(kn ) ∀i(E(i + 1) → G(i)) E(2000) G(1999) H (k2 ) = 1999 → Tall(k2 ) H (k2 ) = 1999
assump. assump. (T2) 1 (T2), 4 5 assump.
⁶ This is of course a simplification, since a man of 200 cm height and 250 cm width would not count as tall, but I shall disregard shape. Note that (U) implies both the claim that men of the same height have the same tallness attribute, and the so-called penumbral principle that anyone taller than someone who is tall, is tall. The penumbral principle follows on the assumption that x is taller than y iff H (x) > H (y).
258
Peter Pagin 8 9 10 11
Tall(k2 ) H (k2 ) = 1999 & Tall(k2 ) E(1999) G(1998) .. .
12 13 14 15 16
G(1500) H (kn ) ≥ 1500 → Tall(kn ) Tall(kn ) ¬Tall(kn ) ⊥
6, 7 7, 8 9 10, (T2)
12 2, 13 2 14, 15,
For deriving the contradiction with (T2), we need as premises both the heights and tallness attributes of a positive and a negative specimen (i.e. premises 1 and 2 of sorites argument (9)), but also each of the height instantiations (including 7). That is, the sorites argument with (T2) requires that there be an unbroken sequence of individuals from a positive instance to a negative instance, where adjacent members differ by at most the tolerance level (in the example, 1 mm). Initially, this may seem to mean nothing more than that a simple and elegant sorites argument can be made complex and cumbersome. However, the difference between (T1) and (T2) gains importance in relation to a domain where the chain in question has a significant central gap. Informally, a significant central gap is a gap in the chain which is bigger than any admissible tolerance level, and such that there are only positive instances on the one side of the gap and only negative instances on the other side. With respect to our domain of men of different heights, we would have such a gap if, say, all members of heights between 190 cm and 160 cm were removed. Thus, suppose that from the vocabulary, the terms k102 , . . . , k400 , denoting men shorter than 190 cm and taller than 160 cm, are removed. What happens with the sorites arguments? As regards (T1), the argument (7) goes through as before, since (T1) only requires one negative instance (and in the expanded version, one positive and one negative instance). As regards (T2), however, the argument (9) clearly does not go through: in order to derive the intermediate conclusion that all men of a height of 1898 mm are tall, we need as a premise that there is a man of at most 1899 mm who is tall, but since this time there are no men in the domain of heights between 1900 mm and 1898 mm, the derivation is broken at this step. The instantiation picks up again at the term k401 , denoting a man of 160 cm height, but that man is already clearly short, and so the premise Tall(k401 ) cannot be justifiably added. Therefore, the derivation cannot be resumed at this step, and because of that the conclusion Tall(kn ) cannot be reached. That is, because the tolerance principle that would be needed to bridge the gap postulates an unacceptably high tolerance level—30 cm—the conclusion cannot be reached, and the sorites argument fails.
Vagueness and Central Gaps
259
That the (T2) argument fails is not of great significance if the (T1) argument, which does not rely on an unbroken chain of instantiations goes through anyway. However, it doesn’t. Interpreted in a domain with a significant central gap, the (T1) principle is straightforwardly false. With respect to our example, consider the instance of (T1) at the lower end of the gap: (10) G(1601) → G(1600) i.e. (10) ∀y(H (y) ≥ 1601 → Tall(y)) → ∀y(H (y) ≥ 1600 → Tall(y)) To make things explicit, we add as an extra premise (11) ¬ Tall(k401 ) where the term ‘k401 ’ denotes the man of 1600 mm height. Now, the antecedent of (10) is true: anyone in the domain that is 1601 mm or higher is 1900 mm or higher, and so tall. The consequence, however, is false: there is an individual of 1600 mm, the referent of ‘k401 ’, and that individual is not tall, by the truth of (11). Hence, the instance (10) of (T1) is false, and so (T1) itself is false. Therefore, in a gappy domain, the falsity of (T1) does not entail the existence of a sharp boundary. By contrast, (T2) is true at the lower end of the gap, for (12) ∃y(H (y) ≤ 1601 & Tall(y)) → ∀y(H (y) ≥ 1600 → Tall(y)) is true: the antecedent is false, since any domain member shorter than or equal to 1601 mm is shorter than or equal to 1600 mm, and therefore not tall. But since the antecedent is false, we can’t detach the consequent, and don’t have a contradiction. Similarly, (T2) is true at the upper end of the gap. (13) ∃y(H (y) ≤ 1900 & Tall(y)) → ∀y(H (y) ≥ 1899 → Tall(y)) The antecedent is true, but so is the consequent, for any domain member with a height of at least 1899 mm has a height of at least 1900 mm, and so is tall. So with respect to a domain with a significant central gap, the strong tolerance principle (T1) is false, but the falsity does not entail that there is a sharp boundary. The weak tolerance (T2) is true, but does not lead to inconsistency. Only the existence of a sharp boundary makes (T2) false (see next section). Again, these facts might seem to be of marginal interest, since the difficult problems concerning vagueness relate to non-gappy domains. I shall, however, make use of these facts for a semantics of natural language sentences where the existence of significant central gaps is not presupposed. In the next section, the (T1) and (T2) principles, and their semantic relation, will be characterized model-theoretically. It will turn out that under natural conditions, (T2) is a consequence of (T1). The non-technical reasoning, and the application to vagueness, will be resumed in section 14.3.
260
Peter Pagin 14.2
M O D E L S O F TO L E R A N C E A N D G A P S
To characterize the (T1) and (T2) principles formally, we define a class of models: Definition 2.1. A V–model M is a classical model D, I, T , H for a first-order language L, where D is a domain of individuals and I an interpretation function, T a function that for each predicate letter F in L as argument assigns a real number TF ≥ 0, and H a function that for each predicate letter F in L assigns a total function HF from D to R, such that (+) for any a, b ∈ D (HF (a) ≥ HF (b) → (b ∈ I(F ) → a ∈ I(F ))) By a classical model I mean a model where the truth definition gives the standard classical clauses for the first-order logical constants. These clauses will be assumed below. The function TF will be used below for assigning a tolerance level to the predicate F in the model. Definition 2.2. A V-model M has a central F-gap iff there are no objects a, b ∈ DM such that a ∈ I(F ), b ∈ I(F ) and 0 ≤ HF (a) − HF (b) ≤ TF Remark 2.3. It would be natural to identify a central gap with a pair i, j of real numbers such that i is the least upper bound and j the greatest lower bound of the gap. But although this can be done, there is no simple and uniform definition of the pair. For if we allow dense and continuous domains of objects, there need not be any smallest member of the set (for instance, for every long period of time in the domain, there may be a shorter period of time in the domain that is still long, even though not every period is), nor any largest number that is not the measure of a member (e.g. for every real number u that is not the length of a long period in the domain, there is greater number v that is also not the length of a long period in the domain, even though there are long periods). We are only guaranteed by the Least Upper Bound Axiom of real analysis that one of the two must exist. But then to identify a gap with a pair of numbers leads to a considerable increase in the complexity of definitions and proofs. Secondly, as the definition is stated, a domain has a central F -gap even in case all objects are F :s or no object is. Again, it simplifies the definition and the reasoning with it not to require both positive and negative instances in the domain. But if we don’t, there need not even be a definite pair i, j to identify with the gap. However, for the natural language semantics to be presented in section 14.4, it is more convenient to identify gaps with definite pairs of numbers. End of remark. Now we can characterize the two tolerance principles model theoretically: Definition 2.4. A (T1)-model M for a predicate F of L is a V-model D, I, T , H for L such that TF > 0, and (ti) For all k (if (for all a ∈ DM (if HF (a) ≥ k + TF , then a ∈ I(F )), then (for all a ∈ DM (if HF (a) ≥ k, then a ∈ I(F )))
Vagueness and Central Gaps
261
Definition 2.5. A (T2)-model M for a predicate F of L is a V-model D, I, T , H for L such that TF > 0, and (tii) For all k, (if (there is a ∈ DM (HF (a) ≤ k + TF , and a ∈ I(F )), then for all a ∈ DM (if HF (a) ≥ k, then a ∈ I(F ))) We shall now prove some elementary properties of the (T1) and (T2) models. Definition 2.6. A V-model M is (a) (b) (c) (d) (e)
HF -unbounded iff for all k ∈ R there is a ∈ DM such that HF (a) ≥ k empty iff DM = ∅ F -full iff DM ∩ I(F ) = DM F -empty iff DM ∩ I(F ) = ∅ F -free iff M has a central F -gap
Fact 2.7. For all V-models M, M is a (T1)-model for a predicate F of L iff M is empty, or M is non-empty and F -full, or M is non-empty and F -empty and HF unbounded. Proof. Left to right. Assume for reductio that M is a) non-empty, and b) not F -full, and c) not both F -empty and HF -unbounded. Because of a) and b) there is a non-F , i.e. an object b− ∈ DM − I(F ). Because of a) and c), either—case A—there is an F , i.e. an object b+ ∈ DM ∩ I(F ), or—case B—M is HF -bounded, i.e. there a k ∈ R such that for all a ∈ DM , HF (a) < k. Consider first case A. By clause (+) of the definition of a V-model, it holds that (i) for all a ∈ DM (if HF (a) ≥ HF (b+ ), then a ∈ I(F )) Then by repeated applications of (ti), from (i) we finally get (ii) for all a ∈ DM (if HF (a) ≥ HF (b− ), then a ∈ I(F )) Since HF (b− ) ≥ HF (b− ), by (ii) we can conclude that b− ∈ I(F ), contradicting the assumption. Hence, case A cannot hold. Consider then case B. Let kj > HF (a) for all a ∈ DM . Then (iii) for all a ∈ DM (if HF (a) ≥ kj , then a ∈ I(F )) Then we can again apply (ti) from (iii) repeatedly, until we have derived (ii), again concluding that b− ∈ I(F ), contradicting the assumption. Hence, case B cannot hold either. Right to left. Three cases: A) M is empty, B) M is non-empty and F -full, C) M is non-empty and F -empty and HF -unbounded. Consider case A. Since by assumption DM = ∅ it holds for any k ∈ R that (iv) for all a ∈ DM (if HF (a) ≥ k, then a ∈ I(F ))
262
Peter Pagin
Therefore, it also holds for any k ∈ R that (v) if for all a ∈ DM (if HF (a) ≥ kj + TF , then a ∈ I(F )), then for all a ∈ DM (if HF (a) ≥ kj , then a ∈ I(F )) and then (ti) holds as well. Case B. Since by assumption it holds for all a ∈ DM that a ∈ I(F ), (iv) will again hold for any k ∈ R, and so we have the same conclusion as in case A. Case C. Assume for reductio that (ti) is false, and hence there is a kj such that (vi) for all a ∈ DM (if HF (a) ≥ kj + TF , then a ∈ I(F )) is true, while (vii) for all a ∈ DM (if HF (a) ≥ kj , then a ∈ I(F )) is false. Since by the first C-case assumption it holds for any a ∈ DM that a ∈ I(F ), (vi) is true only if it holds that (viii) for all a ∈ DM (HF (a) < kj + TF ) But since by the second C-case assumption M is HF -unbounded, this does not hold. Hence (ti) is true. In all three cases, (ti) true, and therefore M is a (T1) model. Fact 2.8. For all V-models M, M is a (T2)-model for a for a predicate F of L iff M is empty, or M is non-empty and F -full, or M is non-empty and F -empty, or M is non-empty and F -free. Proof. Left to right. Assume for reductio that a) M is non-empty, b) not F -full, c) not F -empty, and d) not F -free. Because of a) and b), there is a non-F , i.e. an object b− ∈ DM − I(F ). Because of c), there is an F , i.e. an object b+ ∈ DM ∩ I(F ). Since M is not F -free, there are two objects a, b ∈ DM such that a ∈ I(F ), b ∈ I(F ) and 0 ≤ HF (a) − HF (b) ≤ TF . Hence (i) HF (a) ≤ HF (b) + TF Because a ∈ I(F ), we have (ii) there is c ∈ DM (HF (c) ≤ HF (b) + TF , and c ∈ I(F )) Applying (tii) we can conclude from (ii) (iii) for all c ∈ DM (if HF (c) ≥ HF (b), then c ∈ I(F ))) Instantiating, we have the conclusion that b ∈ I(F ), contrary to assumption. Therefore, M cannot be simultaneously non-empty, not F -full, not F -empty, not F -free. Right to left. We have four cases: A) M is empty, B) M is non-empty and F -full, C) M is non-empty and F -empty, and D) M is non-empty and F -free. Case A. By assumption, M is empty, and so (iv) there is a ∈ DM (HF (a) ≤ k + TF , and a ∈ I(F ))
Vagueness and Central Gaps
263
is false for all k ∈ R, and hence the conditional (v) if (there is a ∈ DM (HF (a) ≤ k + TF , and a ∈ I(F )), then (for all a ∈ DM (if HF (a) ≥ k, then a ∈ I(F )) is true for all k ∈ R, and so (tii) follows. Case B. By assumption, M is non-empty and F -full. Then, all a ∈ DM are also in I(F ), and so (vi) for all a ∈ DM (if HF (a) ≥ k, then a ∈ I(F )) is true for all k ∈ R. Hence, (tii) is again true for all k ∈ R. Case C. By assumption, M is non-empty and F -empty. As in case A, (iv) is false for all k ∈ R, and the rest follows. Case D. By assumption, M is non-empty and F -free. If M is F -empty or F -full, (tii) follows, so assume M is neither. In order for (v) to be false for a particular k ∈ R, there must a pair b+ , b− ∈ M such that b+ ∈ I(F ), b− ∈ I(F ), and (vii) HF (b+ ) ≤ HF (b− ) + TF Then (v) is false for k = HF (b− ). However, if there is such a pair b+ , b− ∈ M, then either HF (b− ) > HF (b+ ), or 0 ≤ HF (b+ ) − HF (b− ) ≤ TF . The first disjunct is ruled out by condition (+) of the definition of V-models. If the second disjunct is true, then by definition 2.2, there is no central F -gap in M. Since by assumption there is a central F -gap, the second disjunct is false as well. Then, (v) cannot be false for any k, and hence (tii) holds. In all four cases, (tii) holds, and therefore M is a (T2) model. Fact 2.9. (T 1) |V (T 2) but (T 2) |V (T 1) Proof. It follows from Facts 2.7 and 2.8 that the class of (T1)-models is a proper subclass of the class of (T2)-models. (T1) will be false in (T2)-models M that are nonempty, not F -full, not F -empty and F -free, as well as in (T2)-models M that are non-empty, not F -full, F -empty but not HF -unbounded.
14.3
M E T H O D O LO G I C A L I N T E R LU D E
The standard alternatives for coping with the sorites paradox are (a) (b) (c) (d)
reject the validity of the argument question or reject the (strict) truth of the inductive premise reject the truth of the minor premise or the falsity of the conclusion accept the whole reasoning and conclude that the vague predicate is incoherent.⁷ ⁷ Cf. Keefe and Smith 1996b, 10.
264
Peter Pagin
I think it is fair to say alternative b) has been the most popular one in the literature of recent decades. Epistemicists such as Timothy Williamson (1994) take the inductive premise to be straightforwardly false, although it is not knowable where the boundary is. Supervaluationists, such as Kit Fine (1975) or Rosanna Keefe (2000), take the inductive premise to be (super-)false, since false in every classical evaluation. Degree theorists, such as Kenton F. Machina (1976), take the inductive premise to be almost completely false, even though each of its instances is almost completely true. Contextualists, like Diana Raffman (1994, 1996), Scott Soames (1999), Delia Graff Fara (2000), and Stewart Shapiro (2003, 2006) take the inductive premise to be false in each context, although the boundary shifts between contexts. I shall propose that we accept the inductive premise, i.e. in the form of the surviving tolerance principle, exemplified by (T2), or rather, a certain revised version of it. Without going deeply into polemics with the dominant trend, I take it to be intuitively part of the semantics of vague predicates to be insensitive to small differences. Our inability to locate any sharp boundary of vague predicates, and even more the intuitive rejection by ordinary speakers of the very idea of a sharp boundary, suggests that it is part of the meaning of vague expression as used by ordinary speakers not to have them. Although it is well known that all the theories of vagueness that involve rejecting the inductive premise come at a high cost, rejection in any of the proposed forms may still seem to involve smaller costs than does acceptance. Since drastic revisions of logic are required to treat the standard sorites argument as invalid, and since it is implausible to deny the existence of tall men, or of nontall men, the only remaining option seems to be that the use of vague vocabulary is incoherent, or even inconsistent. This position has been advocated e.g. by Michael Dummett (1975). Dummett says Wang’s paradox merely reflects this inconsistency. What is in error is not the principles of reasoning involved, nor, as on our earlier diagnosis, the induction step. The induction step is correct, according to the rules of use governing vague predicates such as ‘small’: but these rules are themselves inconsistent, and hence the paradox. Our earlier model for the logic of vague expressions thus becomes useless: there can be no coherent such logic. (1975, 265)
A little before that, Dummett provides a gloss on ‘consistent’: ‘Consistent’ here means that it would be impossible to force someone, by appeal to rules that he acknowledged as correct, to contradict himself over whether the predicate applied to a given object. (1975, 264)
That the use of vague vocabulary is inconsistent may be seen as rendered plausible from considerations of so-called forced march sorites, which ‘is designed to force us, one step at a time, into a separate verdict on each successive pair of adjacent items in a sorites sequence’ (Horgan 1994, 173). Our inclination to respect initial intuitions about clear cases while not accepting any sharp boundary, naturally leads us into trouble.
Vagueness and Central Gaps
265
On the other hand, there is a frequent use of vague vocabulary that by common sense standards fulfill its communicative function pretty well. In fact, most applications of vague predicates appear unproblematic. This observation has been used in an objection against Dummett by Crispin Wright: [ . . . ] what is actually responsible, on this view, for the large degree of coherence and communicative success which our use of color vocabulary enjoys? Indeed, what is the justification for continuing to think of the use of such expressions as governed by rule? Knowledge of appropriate rules was supposed to constitute linguistic competence. But it cannot do so if competent usage essentially has a coherence which, in Dummett’s view, the rules lack. Dummett’s response needs supplementing with an explanation of our communicative success with such vocabulary in which the idea of knowledge of inconsistent rules has an ineliminable part to play. For either such knowledge is still to be a basic ingredient in competence or we should drop the idea. (Wright 1987, 212)
Directly following this, Wright makes a related point: That brings us to [ . . . ] a decisive objection to Dummett’s response. I do not see how we can rest content with the idea that certain implicitly known semantic rules are incoherent when nobody’s reaction, on being presented with the purported demonstration of the inconsistency, i.e. the paradox—even if they can find no fault with it—is to lose confidence in the unique propriety of the response—e.g. ‘That’s orange’—which the demonstration seems to confound. (Wright 1987, 213)
I agree with Wright. We should try to account for the apparent communicative success of the use of vague vocabulary. Most of natural language lexical items are vague, and to dismiss the use as governed by inconsistent rules is bad theory. On the other hand, it seems to me absurd to try to rescue every single sequence of applications of vague predicates. Speakers do contradict themselves. It cannot be the goal of semantic theory to represent natural language as a foolproof means of making good sense (after all, people do paint themselves into corners and cut off the branches they are sitting on). These considerations suggest that we try to find a semantic account of ordinary applications of vague predicates that does not reject all tolerance principles as unacceptable. For a forced-march sorites is not a sequence of ordinary applications. I shall therefore propose a combination of two strategies. For ordinary contexts of use, I propose that we opt for strategy a): reject the validity of the argument. For certain extreme contexts, on the other hand, I propose alternative d): the use of the vague predicate is incoherent. In the next section, I shall sketch such an account, based on the observations of the first section. It is a contextual account, but unlike the mainstream of current contextual accounts, the most important feature will not be the shift between contexts, but an extra element in the context itself.
266
Peter Pagin 14.4
CENTRAL GAP SEMANTICS
I think that the following correctly describes ordinary speaker psychology of ordinary applications of vague predicates: A predicate, like ‘tall’, is applied to a clear case of tallness or non-tallness, and although the speaker does not think that a sufficiently small difference, like 1 mm, can decide between being the one or the other, the difficult intermediate cases are simply ‘forgotten’, or ‘ignored’. It is enough that the case considered is clear, and that there are clear contrastive cases. There is no need to consider intermediate cases. We could say that intermediate cases are dismissed. It is this dismissal of intermediate cases that can be modeled by means of interpretation that introduces significant central gaps in the domain of discourse in those cases where it is needed. It is widely agreed that for interpreting (normal) utterances of sentences like (14) Everyone went to bed at midnight a contextually induced restriction on the quantifier domain is needed. The present proposal is that we extend the tool of quantifier domain restriction to give a context semantics for vague expressions that respects tolerance, bivalence as well as consistency for normal use. I shall propose that for a vague predicate F , like ‘. . . is tall’, a speaker that accepts it as tolerant with respect to a particular dimension, uses it with a tacit assumption that there is a restriction on the domain of discourse, such that the domain in question has a significant central F -gap. In this application, it is convenient to think of a central gap for a predicate F in a context c, relative to a dimension of variation, as a pair of real numbers (i, j) with respect to a measurement scale and a measure function HF . With an initially given domain D of individuals, the gap determines a proper subset of D, the set Fc = {a ∈ D : j ≤ HF (a) ≤ i} of individuals in D whose F measures are in the gap. The gap then forces a cut in the domain, consisting in subtracting Fc from D. That is, the restricted domain is D − Fc .⁸ The type of semantics that I suggest is a context semantics.⁹ It involves the assumption that in each context c, for each vague predicate F that is used in c, a tolerance level and a central gap is determined for F . It assumes the existence of a general gap function G that maps contexts of utterance on central gaps. Since the pair of number selected is arbitrary within limits, G must be a choice function with restrictions on the values it can give. It must be required that in each context a full semantics is given for the full fragment of a language that is used in the context, but not for linguistic material outside that fragment. This means in particular that any referring singular term that is used in c must have referent in the quantifier domain of c. For maintaining bivalence, the ⁸ Some predicates are associated with more than one dimension of variation, e.g. predicates formed from simple predicate by means of connectives. In those cases we will need n-tuples of simple real number pairs, one for each dimension of variation. ⁹ The account is more fully worked out in my Vagueness and Domain Restriction, to appear in a volume on vagueness and language use edited by Paul Egr´e and Nathan Klinedinst.
Vagueness and Central Gaps
267
relevant measure of that term for the predicate F cannot fall in the central gap for F at c. Hence, the central gap is required not to include the measure of that referent. So the location of the gap depends on the terms used in the context. This heads the list of restrictions, ordered according to importance. (GAP)
(i) The central gap must be selected so that the full fragment of language used in the context, including pragmatically determined contextual updates, is taken account of. (ii) The size of the central gap must be at least equal to the tolerance level. (iii) The location of the central gap must be selected so as to ensure consistency. (iv) The location of the central gap must be selected so that what the speaker says comes out as true, to the extent this is possible and reasonable.
(GAPi) cannot be compromised, but this has the consequence that the other three may not be jointly satisfiable, given collateral facts. When they are not, it is not always clear e.g. that consistency of the speaker should take precedence of the truth of individual judgments. But this is a matter of further investigation. That the gap must be at least equal to the tolerance level is necessary to preserve consistency in normal situations. If we make the further decision to let the size of the central gap for a predicate in a context be equal to the tolerance level for that predicate in the context, then the central gap can be regarded as determined by two well-known contextual standards: a standard of comparison and standard of precision.¹⁰ We can simply identify the standard of precision with the tolerance level. If we further let the standard of comparison correspond to the center of the gap, then the gap is determined as a function of the standard of comparison and the standard of precision: where i is the value of the standard of comparison and k is the standard of precision, the gap is simply (i + k/2, i − k/2). There are then two basic ideas for the semantics: The first is that for each predicate F for which a gap is introduced, the extension of F consists of the individuals a in D − Fc with measures above the gap (HF (a) > i) and the anti-extension of individuals b with measure below that gap (HF (b) < j). The second idea is that quantifiers are domain-restricted by means of the cut. We can then verify that a tolerance principle, stated with binary quantifiers, such as (15) Some x(man(x) & height(x) ≤ n + k mm, tall(x)) All x(man(x) & height(x) ≥ n mm), tall(x)
→
is true in any context c where the tolerance level for ‘tall’ is k mm or greater. We assume that the relevant measure function maps men on their heights in mm. Let’s assume here that ‘man’ is non-vague. Then the antecedent of (15) is true just in case some individual a in the restricted domain is a man and has a height above the upper edge of the ‘tall’ gap in c. Then any individual b in the c-restricted domain that is a man has a height at least that of the height of a minus k mm itself has a height above ¹⁰ Cf Lewis 1979, 244–6.
268
Peter Pagin
the upper edge of the gap, for there is no individual in the restricted domain that has a height in the gap, and any individual c in the domain with a height below the gap is more than k mm shorter than a. So the the consequent of (15) is true.¹¹
14.5
MEANING AND CONTENT
You might think that no adult human being, at least these days, who is 150 cm in height, could reasonably be counted as tall. That is, it would be part of the linguistic meaning of ‘tall’ that no interpretation function with an associated upper gap boundary below 150 cm is admissible. A consequence of this view is that if there is no sharp boundary between admissible and non-admissible gaps, there will be no sharp boundary between admissible and non-admissible gap functions, and thus it would be unclear how a speaker could be interpreted. This is not, I think, a severe problem. We cannot anyway reasonably hope that our entire meta-language vocabulary is precise, and there is no good prior reason, from the present perspective, to believe that the domain of admissible gap functions, in some particular context of utterance, is sharply delimited. A leading idea of the present approach is that it is enough for adequate interpretation that there is a least one clearly admissible gap function (see below). A more unwelcome result, however, is that two speakers A and B who disagree about how short an adult human can be and still count as tall, by such a view would be using ‘tall’ with different linguistic meanings. That is, they would not disagree substantially about the lower boundary of tallness, but would be speaking different languages, with phonetically and orthographically identical but semantically distinct predicates ‘. . . is tall’.¹² There would then be neither agreement nor disagreement between them on matters of tallness. I find that implausible. Rather, with one proviso, if two speakers agree on the number uniformity principle (U) (stating so-called penumbral connections) of the predicate in question, then they share the concept. If they agree on the uniformity principle, then they agree on the dimension of comparison, and they agree on the direction relevant for the predicate (e.g. if taller, then more disposed to be counted as ¹¹ The present account has some similarity to the account in Manor 2006, although the two were developed independently. Manor provides a non-standard semantics where the usual inductive premise fails if there is a suitable gap in the sequence of measures. According to Manor, the existence of a unique gap in the contextual domain provides a natural demarcation of the extension of a vague predicate in that context. To effect such a demarcation, the gap must be unique and well placed, and on Manor’s account vague terms are used with the presupposition that there is such a gap. The present account differs from Manor’s inter alia in that gaps are provided as part of interpretation, rather than declaring a speech act as failed when the presupposition isn’t met. ¹² For an epistemicist, like Williamson (1994, 205–12), there is a sharp boundary of tallness determined for the language of the speech community, and from this perspective at least one of the two speakers would be mistaken about what the lowest admissible boundary is. Moreover, on this view, both speaker would be mistaken in thinking that there is a range of admissible alternatives. As many other non-epistemicists, I find such a determination of sharp boundaries from non-uniform use implausible.
Vagueness and Central Gaps
269
tall). The proviso is that for something to have a property as depending on its position in some order, then it must be possible that something has a lower (or, depending on the predicate, higher) position in that order. So, an object x is not tall unless it is possible that some other object y is shorter, and is not non-bald unless it is possible that some other object y has less hair. This means that for agreeing on the meaning of the predicate, two speakers must also agree on such absolute limits. Accordingly, I take it e.g. to be part of the meaning of ‘bald’ that a person with no hairs on his scalp is bald, but for no number greater than zero is it part of the meaning of ‘bald’ that a person with that number of hairs is bald. Accordingly, I take two speakers who assign the same meaning in this respect to a tolerant predicate to agree on the general concept, such as the concept of tallness, or baldness or heapness. However, such a concept is not individuated directly by application conditions. That is, the concept of being tall, for instance, is not individuated by a set of conditions C such that if an object x satisfies the conditions in C, x is tall and if it does not, it is not. The sorites paradox itself is a reason against this view, provided the premise of tolerance is accepted. For then, if there is such a concept of tallness, it is tolerant, and hence a tolerance principle is valid for that concept, which together with facts about the distribution of heights among adults of the world, leads to a sorites-type contradiction. We cannot avoid such a contradiction by changing the semantics of the concept, for concepts have their semantics built-in. Only for linguistic expressions, or other sign-like entities, can we devise alternative semantic theories. The conceptual semantic paradox can be avoided only if such a concept of tallness does not exist. Rather, the concept of tallness, on the present view, is to be seen as a function from standards of application to extensions. And, on the present approach, admissible standards of application involve a contextual significant central gap. A central gap together with a measure function and the condition of belonging to the extension just in case one’s measure is at least as great as the upper boundary, does fix the extension (and correspondingly for the lower boundary and the anti-extension). For the issue of utterance and belief content, we must switch to an intensional framework. Within a possible-worlds framework, it is natural to regard the concept associated with a tolerant predicate as a function from possible worlds and standards of application to extensions. If we take the standard of application as the first argument, the value of the function is an ordinary intension, i.e. a function from possible worlds to extensions. For instance, an upper gap boundary for tallness of 180 cm determines an intension that for any possible world w fixes an extension consisting of the set of (adult male) humans in w that have a height of at least 180 cm. It is not reasonable, however, to attribute such a precise belief content to the normal speaker. More plausibly, for each speaker and context there is a range of standards that the speaker is prepared to count as admissible. And plausibly, that range is not itself sharply delimited. That is, we have an unsharply bounded set of propositions that are admissible as intensions of the sentence in the context of utterance. This provides one dimension along which the expression ‘believes that’ is itself tolerant, and the present suggestion is, accordingly, that central gap domain restriction is to be applied to sentences containing it.
270
Peter Pagin
It is in line with this suggestion to take two speakers A and B to agree on a particular statement in a context c, not just conceptually but also doxastically, just in case there is a standard of application that is admissible to both A and B. This entails that for each tolerant predicate F involved in c, there is a pair (i, j) which is admissible for both GFA and for GFB . As a consequence, doxastic agreement in a context is not transitive: there may be a standard that is admissible to both A and B, and a standard that is admissible to both B and C but still no standard that is admissible to both A and C. But note that if two values are both admissible, a switch between them will not affect the truth value distribution over sentences. Hence, if A and B, and B and C, respectively, agree doxastically, they agree on truth values, and hence so do A and C, for the given linguistic fragment and the given domain. The non-agreement between A and C is conceptual, not a doxastic dis-agreement.
14.6
P RO B L E M S
The basic strategy of the current proposal is to account for ordinary non-paradoxical use of vague expressions in one way, and allow inconsistency in the extreme cases where we have to do with sorites sequences. Accordingly, it is a necessary condition for the viability of the proposal that the account does give intuitively correct results for ordinary utterances, and in particular that ordinary utterances that do seem intuitively true or intuitively false, do not come out as incoherent on the proposed semantics. It is, however, not obvious that this is the case. Potential sources of trouble are the combining of tolerant predicates, and the use of quantifiers. Here, there is not space for a thorough investigation of the matter. I can only discuss a few examples. One thing that can happen is that objects are added consecutively to the domain in a way that eventually eliminates the possibility of a central gap. If we say (16) Julia is tall. So is Georgina, and so is Elsa, and so is Amanda . . . we may in the end populate the domain of quantification so that a sorites sequence results, given reasonable tolerance levels for ‘tall’. I don’t think this is a problem, however. Rather, this is one way in which the use of vague predicates can run into trouble.¹³ This is perfectly in line with the present proposal. The combination of tolerant predicates in a single context of utterance can give rise to the opposite effect. Since we need a gap and hence a domain cut for each vague predicate, the result might be that so much is cumulatively taken out of the domain of discourse that the topic is distorted. This consequence is avoided in case we are allowed context shifts that need not be conservative, in the sense that new cuts can be made and old cuts undone. This is an issue that requires further investigation. But there is a problem even with a small cut. Suppose a basketball coach says (17) Every player in my team is tall ¹³ The idea that verb phrase ellipsis provides problems for other contextualist accounts of vagueness is due to Jason Stanley (2003).
Vagueness and Central Gaps
271
and it so happens that, given what counts as tall in the context, and given the contextual tolerance level, there is a sorites sequence in the team between the tall and the non-tall. Then some domain cut is needed, let’s say with the effect that two players must go out of the contextually restricted domain. Have we thereby not misrepresented the content of the utterance of the coach? After all, the coach seems to be saying that every player of his team is tall, not that every member in a set consisting of all except two of the players in his team are tall. The content ascribed to the utterance is not identical to the intuitive content, a content which would be incoherently ascribed. Rather, the content ascribed only approximates the intuitive content. The general idea of imposing contextual domain restrictions on utterance interpretation is that of getting the utterance content right. Isn’t it used here with the opposite effect? Let’s note, first, that the intuitive truth value assignment is not affected. Intuitively, the utterance is false, since the team contains at least one non-tall player (or otherwise there would be no sorites sequence). But with a central gap, the cut in the domain is such as to leave anti-extension members in the domain. Hence, even with respect to the restricted domain the utterance comes out as false. Had all members been tall by the given standard, then no cut had been needed in the first place. So, either way, the intuitive truth value is preserved.¹⁴ Second: We have assumed that the utterance of the coach in itself makes sense. What doesn’t make sense is to interpret it with the contextual tolerance level and the initially determined domain, i.e. the entire team. To bring out what is intuitively right, or in this case intuitively wrong, with the utterance, we need to deviate from the intuitive content ascription. This could be done in other ways, e.g. by assigning greater precision, i.e. a lower tolerance or even a zero tolerance level, going beyond any discrimination that the speaker himself would be prepared to make. This too would, on the current assumption, amount to distorting the intuitive content. Applying it across the board would have the effect of making the tolerance principle (15) come out false, despite being held true by the speaker, and thus force an error theory about the speaker, and about natural speakers in general to the extent that such tolerance principles are generally affirmed. In order to assign content in a consistent way, some approximation has to be made. The type of approximation suggested here generally saves the intuitively assigned truth values and avoids the need of adopting an error theory about tolerance itself. Re f e re n c e s Dummett, M. (1975), ‘Wang’s paradox’, Synthese 30: 301–24. Reprinted in Dummett 1978. Page reference to the reprint. (1978), Truth and Other Enigmas, Harvard University Press, Cambridge, MA. Fara, D. G. (2000), ‘Shifting sands: an interest-relative theory of vagueness’, Philosophical Topics 28: 45–81. ¹⁴ We can indeed get more complicated cases with other quantifiers or determiners, like ‘most’, but there is not enough space here to discuss these cases. Some of them are more problematic, but then again problematic also on all standard accounts of vagueness.
272
Peter Pagin
Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30: 265–300. Horgan, T. (1994), ‘Robust vagueness and the forced-march sorites paradox’ in J. E. Tombelin (ed.), Logic and Language, vol. 8 of Philosophical Perspectives, Ridgeview, Atascadero, CA. Keefe, R. (2000), Theories of Vagueness, Cambridge University Press, Cambridge. Keefe, R. and Smith, P. (eds.) (1996a), Vagueness: A Reader, MIT Press, Cambridge, MA. 1996b, ‘Introduction: theories of vagueness’, in R. Keefe and P. Smith, eds., Vagueness: A Reader, 1–57, MIT Press. Lewis, D. (1979), ‘Scorekeeping in a language game’, Journal of Philosophical Logic 8: 339–59. Reprinted in Lewis 1983. Page reference to the reprint. 1983, Philosophical Papers. Volume I , Oxford University Press, Oxford. Machina, K. F. (1976), ‘Truth, belief, and vagueness’, Journal of Philosophical Logic 5: 47–78. Manor, R. (2006), ‘Solving the heap’, Synthese 153: 171–86. Raffman, D. (1994), ‘Vagueness without paradox’, Philosophical Review 103: 41–74. (1996), ‘Vagueness and context-relativity’, Philosophical Studies 81: 175–92. Shapiro, S. (2003), ‘Vagueness and conversation’ in Jc Beall, ed., Liars and Heaps, 39–72, Oxford University Press, Oxford. (2006), Vagueness in Context, Oxford University Press, Oxford. Soames, S. (1999), Understanding Truth, Oxford University Press, Oxford. Stanley, J. (2003), ‘Context, interest relativity and the sorites’, Analysis 63: 269–80. Williamson, T. (1994), Vagueness, Routledge, London. Wright, C. (1976), ‘Language-mastery and the sorites paradox’ in G. Evans and J. McDowell, eds., Truth and Meaning, 223–47, Clarendon Press, Oxford. (1987), ‘Further reflections on the Sorites paradox’, Philosophical Topics 15: 227–90. Reprinted in Keefe and Smith 1996a (with omission of section 5), 204–50. Page references to the reprint.
IV Vagueness in Context
This page intentionally left blank
15 Hold the Context Fixed—Vagueness Still Remains ˚ Jonas Akerman and Patrick Greenough
Contextualism about vagueness (hereafter ‘Contextualism’) is the view that vagueness consists in a particular species of context-sensitivity and that properly accommodating this fact into our semantic theory will yield a plausible solution to the sorites paradox.¹, ² But Contextualism, as many commentators have noted, faces the following immediate objection: if we hold the context fixed, vagueness still remains, therefore vagueness is not a species of context-sensitivity. Call this ‘the simple objection’.³ Absent a convincing reply to the simple objection, Contextualism is in very bad shape. Oddly enough, defenders of Contextualism have said very little in reply. Proponents of the objection have tended to assume that this is because no reply is in the offing—the simple objection is taken to be unassailable. In this paper, we sketch two replies to the simple objection which result in two very different kinds Parts of this were jointly presented at Seventh Arch´e Vagueness Workshop in November 2006 and at the Arch´e Audit in June 2007. Thanks to the following folk for very useful feedback (on either or both of those occasions): Elizabeth Barnes, Maria Cerezo, Richard Dietz, Dan L´opez de Sa, Aidan McGlynn, Sebastiano Moruzzi, Peter Pagin, Graham Priest, Diana Raffman, Sven Rosenkranz, Mark Sainsbury, Stewart Shapiro, Paula Sweeney, Jordi Valor, and Crispin Wright. Thanks also to Sven Rosenkranz and Elia Zardini for particularly valuable comments on the penultimate draft. This paper was completed while one of the authors (Greenough) was a Postdoctoral Fellow in the Epistemic Warrant Project at ANU (2007–8). Thanks go to the many philosophers at ANU for their hospitality—philosophical and otherwise. ¹ According to a generic version of Contextualism, the vagueness of the predicate type ‘is tall (for a Ugandan Pygmy)’, for example, consists, in part, in the fact that relative to different contexts of utterance (where these contexts of utterance differ only in respect of certain designated parameters), the extension of this predicate can differ (even though the heights of all people in Uganda remain fixed). For Fara (2000) (originally published under Delia Graff), the designated contextual parameters are the interests and purposes of the speaker (and their conversational participants). For Raffman (1994, 1996) the designated parameters concern the psychological states and dispositions of the speaker. For Lewis (1979), Soames (1999, 216–17), Shapiro (2003, 2006), the designated parameters concern ˚ the operative standards of precision. See Akerman and Greenough (2009) for a critical discussion of the various ways in which vagueness may consist in a particular species of context-sensitivity. ² There are two broad kinds of contextualist solutions to the sorites paradox (see Section 15.2). ³ The objections raised against Contextualism in Stanley (2003) and in Keefe (2007) are strictly ˚ independent of the simple objection discussed here. See Akerman and Greenough (2009) for a critical discussion of some of Keefe’s objections.
275
˚ Jonas Akerman and Patrick Greenough
276
of Contextualism: Epistemicist Contextualism and Radical Contextualism. With these two theories in hand, the simple objection loses much, if not most, of its force. 15.1
C O N T E X T UA L I S M A N D W E A K TO L E R A N C E
All extant forms of Contextualism are committed to something like the following principle of weak tolerance: (WT) It is not the case that: there is a context of utterance C and there is an x such that x and x are considered together as a pair by a single subject in C and ‘is F ’ (as used in C) is true of x and ‘is F ’ (as used in C) is false of x , (where x is adjacent to x in the sorites series running from F to not-F ).⁴ Roughly, WT says that, when considered pairwise, adjacent members of the series are never category different.⁵ WT is a principle of weak tolerance since it permits that (a) there can be a context C and a context C such that ‘is F ’ (as used in C) is true of x and ‘is F ’ (as used in C ) is false of x , and that (b) there can be a sharp boundary within C if x and x are not considered together as a pair in C. One of the characteristic symptoms of vagueness is that vague predicates draw no known boundary across their associated dimension of comparison.⁶ WT can explain how this symptom of vagueness arises: as we inspect each pair of adjacent items in the sorites series, WT ensures that the members of each adjacent pair cannot be category different. Given the factivity of knowledge, it follows that there is no context of utterance C such that there are two adjacent items x and x , which are considered together in C, such that a subject knows that ‘x is F and x is not-F ’ is true. Roughly, no (context in which there is a) boundary between saliently similar objects in the series entails no (context in which there is a) known boundary between those objects. (We shall encounter two further symptoms of vagueness in section 15.3.) But do vague predicates draw sharp boundaries or not? WT is compatible with either view. On this score, there is an important (and generally overlooked) distinction between what may be termed Boundary-Shifting Contextualism (BSC) and Extension-Shifting Contextualism (ESC).⁷ ⁴ It’s a further question whether WT holds in all contexts (see Shapiro 2003, 44, fn. 1, for some relevant remarks). ⁵ WT is cognate to both Raffman’s principle IP∗ which, with respect to ‘is red’, says that ‘for any n, if patch #n is red then patch #(n + 1) is red, relative to a pairwise presentational context’ (1994, 68) and Fara’s salient-similarity constraint which says that ‘if two things are saliently similar, then it cannot be that one is in the extension of the predicate, or in its anti-extension, while the other is not’ (Fara 2000, 57). Cf. Soames (1999, 214–16) and Shapiro (2003, 42–3). ⁶ In Greenough (2003), this symptom is called ‘epistemic tolerance’. ⁷ See Greenough (2005, 178–9) for more on this distinction. Raffman (1994) and Shapiro (2003, 2006) both defend forms of ESC, while Fara (2000) defends a form of BSC. Soames (1999, 216–17) appears to defend a form of BSC whereby there is a shifting boundary between the extension/anti-extension of a predicate and the undefined cases in the borderline area—cases for which there is a truth value gaps of sorts. Thus, while all forms of BSC are committed to sharp (variant) cut-offs, not all forms are committed to classical logic. Stanley (2003), Heck (2003), Priest (2003), and Keefe (2007) simply assume that Contextualism is exhausted by BSC.
Hold the Context Fixed—Vagueness Still Remains
277
1 5 . 2 B O U N D A RY- S H I F T I N G C O N T E X T UA L I S M A N D E X T E N S I O N - S H I F T I N G C O N T E X T UA L I S M BSC says that in every context there is a cut-off. That is, across a sorites series for ‘is F ’, for every context of utterance C, there is an x such that ‘is F ’ (as used in C) is true of x and ‘is not-F ’ (as used in C) is true of x .⁸ Thus, BSC is a form of epistemicism in that vague predicates draw sharp, bivalent, boundaries. Unlike the epistemicism of Sorensen (1988) and Williamson (1994), however, it is constitutive of vagueness that the boundary can shift as a function of changes in the context of utterance (see fn.1). Thus, the following principle is invalid: there is an x such that, for every context of utterance C, ‘is F ’ (as used in C) is true of x and ‘is not-F ’ (as used in C) is true of x . This latter principle amounts to the claim that there is a cut-off such that it obtains in every context. Furthermore, as we should expect, BSC plus WT entails that the cut-off drawn by a vague predicate is not only unknown but unknowable—at least via the method of inspecting adjacent items. What does BSC say about the standard sorites paradox? With respect to a typical sorites series for the predicate ‘is red’, it is given that the first colour patch in the series is red and the last colour patch is not red. The major premise of the standard version of the paradox says that, for all colour patches x in the series, if patch x is red then patch x is red. Given mathematical induction, it follows that all patches in the series are red. But that contradicts the fact that the last member is not red. In order to resolve the paradox, BSC—just like standard epistemicism—holds the major premise to be outright false. But if the major premise is false why did we find it so plausible (and so believe it) in the first place? Importantly enough, BSC and standard forms of epistemicism differ with respect to this key question. Standard epistemicism can offer something like the following ‘confusion’ diagnosis: in confronting the paradox we systematically confuse the (true and plausible) claim that there is no known boundary across a sorites series with the (false) claim that there is no sharp boundary. Such a confusion confers plausibility onto the stronger claim—explaining why we come to believe the stronger claim.⁹ BSC is able to offer a related, but distinct, ‘confusion’ diagnosis: in confronting the paradox we systematically confuse the (true and plausible) weak principle of tolerance WT (and kindred principles) with the following (false) strong principle of tolerance (and kindred principles): (ST) It is not the case that: there is a context of utterance C and there is an x such that ‘is F ’ (as used in C) is true of x and ‘is F ’ (as used in C) is false of x , (where x is adjacent to x in the sorites series running from F to not-F ).¹⁰ ⁸ Such a formulation assumes that we can never set standards so low or so high such that either everything or nothing counts as an F . ⁹ In fact this diagnosis is available to any theory which takes the major premise of the sorites to be false—such as a supervaluational or intuitionistic conception of vagueness (see Greenough 2003, 272–4 for further discussion). ¹⁰ ST entails that for all contexts of utterance C and for all x, if ‘is F ’ (as used in C) is true of x then ‘is F ’ (as used in C) is true of x . In other words, the predicate ‘is F ’ is tolerant in all contexts.
278
˚ Jonas Akerman and Patrick Greenough
Very roughly, we confuse the (true and plausible) claim that there is never a boundary between any two adjacent items considered together as a pair with the stronger (and false) claim that there is a never a boundary between adjacent items. Again, such a confusion confers plausibility onto the stronger claim—explaining why we come to believe the stronger claim.¹¹ (We shall return to these diagnoses in section 15.4.) ESC represents a radically different form of contextualism. Given ESC, in no context of utterance is there a cut-off.¹² For ESC there can only be ‘quasiboundaries’—boundaries which hold, as it were, across, but not within contexts.¹³ With respect to the standard sorites, the paradox is not to be resolved by taking the major premise to be unequivocal and false as in the case of BSC. Rather, the sorites is taken to exhibit a fallacy of equivocation.¹⁴ There is a true reading of the major premise: for all colour patches x in the series, if patch x is red then patch x is red relative to a pairwise presentational context whereby x and x are presented together as a pair to a competent judge. And there is a false reading: for all colour patches x in the series, if patch x is red relative to a singular presentational context then patch x is red relative to a singular presentational context, whereby the context in which x is presented to a competent judge may differ from the context in which x is presented to a judge.¹⁵ It follows that, in the present context, for all x, if x is F then x is F . That is, the major premise of the standard sorites follows from ST. Given classical logic, and the fact that the first member of the series is F and the last member of the series if not-F , then the major premise is outright false and so ST is outright false. ¹¹ Why does such a confusion take place? The thought is that subjects are typically (pretheoretically) unaware of the effect that context has in the determination of the extension of a predicate. ¹² The alert reader will have noticed that this is just to assert ST. But ST classically entails the major premise of the standard sorites. As it turns out, ESC can retain ST without fear of paradox because the classical consequence relation is restricted within contexts given ESC—in particular, the classical least number principle is not valid (see main text below). For the special case of the sorites paradox under which one uses the negation of ST to derive a contradiction, the solution given by ESC is as follows: the major premise ST is not equivocal at all but simply true, however the paradox does not arise because classical logic fails. ¹³ With respect to the forced march sorites, Raffman (1994, 46–7, passim) and Shapiro (2003, 51–3) allege that a (competent) subject will always ‘jump’ in the forced march—thus delivering a differential verdict with respect to adjacent items in the series. But this jump does not mark a boundary (within a context) but rather a shift in context. ¹⁴ See Raffman (1994, 68–9). Shapiro defends a form of ESC but, oddly, takes the major premise to be false (see Shapiro 2003, 53). In Greenough (2005, 178) it is argued that Shapiro should posit a fallacy of equivocation. ¹⁵ As it turns out, BSC can offer an alternative (and incompatible) explanation of the seductiveness of the major premise by also positing a fallacy of equivocation: the major premise equivocates between a strong (and false and implausible) reading (via ST and cognate principles) and a weak (and true and plausible) reading (via WT and cognate principles). We have resisted this way of presenting matters because extant defenders of BSC (e.g. Fara) represent themselves as taking the major premise to be false and so this premise is not, for Fara at least, equivocal. A further point of note is that it is not possible for ESC to co-opt the solution to the sorites posited by BSC under which ST and the major premise are taken to be false. The reason for this is that ESC takes ST to be true—see fn. 12. So, there is an asymmetry between BSC and ESC: both can offer a diagnosis under which the major premise equivocates between a true reading and false reading, while only BSC can offer a diagnosis under which the major premise is both false and yet taken to be true/plausible because a
Hold the Context Fixed—Vagueness Still Remains
279
Which of these two species of Contextualism is the better view? Here is a quick argument in favour of BSC over ESC: According to ESC, in no context of utterance is there a cut-off. It follows that within a context of utterance, whereby the first member of the series is F and the last member is not-F , the classical least number principle is invalid—otherwise we could derive that there is a cut-off between the F ’s and not-F ’s in that very context. Thus, classical logic fails given ESC. Given that BSC preserves classical logic, and ESC does not, then BSC is the more plausible view.¹⁶ The reason is simple: the contextualist has no need to both posit context-sensitivity and give up on classical logic in order to resolve the sorites paradox. This argument provides a pretty strong reason to prefer BSC over ESC. So, in what follows we shall only defend BSC against the simple objection.¹⁷ (From now on, by ‘Contextualism’, we shall mean BSC.) 15.3
THE SIMPLE OBJECTION
Some prominent exemplars of the simple objection are as follows: Vagueness remains even when the context is fixed. (Williamson 1994, 215) we should distinguish vagueness from paradigm context-dependence (i.e. having a different extension in different contexts) even though a term may have both features (e.g. ‘tall’). Fix on a context which can be made as definite as you like (in particular choose a specific comparison class): ‘tall’ will remain vague, with borderline cases, and fuzzy boundaries, and the sorites paradox will retain its force. This indicates that we are unlikely to understand vagueness or solve the [sorites] paradox by concentrating on context-dependence. (Keefe and Smith 1997, 6, see also Keefe 2000, page 10) the first blush response that almost everyone seems to have [towards Contextualism] is: OK, fix the context; the extension of ‘red’ in that context is still vague [. . .] The sorites reasoning is just as appealing when one nails the extension down as it is when one allows it to vary. (Heck 2003, 120)¹⁸
If we follow Keefe’s particular example and assume that the context-sensitivity which is constitutive of vagueness is exhausted by the sensitivity to a comparison class then the objection is persuasive. However, no extant or sensible form of Contextualism invokes that kind of context-sensitivity to make sense of vagueness.¹⁹ Even so, the subject when first confronting the paradox confuses it with a true and plausible principle of weak tolerance. ¹⁶ See Greenough (2005, 178–9). ¹⁷ We do not mean to imply that ESC is any worse off than BSC when it comes to the simple objection. However, ESC must offer a rather different range of responses to the simple objection than the range of responses that are available to BSC. ¹⁸ A form of the simple objection also appears in an unpublished paper ‘A problem for contextualism about vagueness’ by Max K¨olbel, 2007. ¹⁹ See fn. 1. In her (2007, 276), Keefe recognizes that vagueness-related context-sensitivity is independent of sensitivity to shifts in comparison class.
280
˚ Jonas Akerman and Patrick Greenough
objection has a more general form: suppose we hold all aspects of the context of utterance fixed (e.g. speaker, world, time, place, orientation, conversational partners, contextually salient comparison class, the operative standards of precision, the psychological states of the conversationalists, and so on) then the extension of ‘is red’ in that context will still exhibit all the symptoms of vagueness and will thus count as vague. Since, by hypothesis, the predicate ‘is red’ cannot vary its extension within the fixed context in hand, and since this predicate remains vague, then vagueness is not a species of context-sensitivity.²⁰ We’ve encountered one (epistemic) symptom of vagueness already: vague predicates draw no known boundary across their respective dimension of comparison. Two other symptoms are important. The second symptom is also epistemic: vague predicates give rise to borderline cases, cases such that we do not know whether or not the predicate applies.²¹ The third symptom is quasi-psychological in nature: vague predicates are sorites-susceptible—they are such that (pre-theoretically) we are seduced into accepting the major premise of the sorites paradox.²² For the purposes of this paper we will assume that these symptoms are individually necessary and jointly sufficient for the presence of vagueness.²³ WT as we have already seen can be used to explain why there is no known boundary across the series: when adjacent items in a sorites series are considered together as a pair, those items are never category different and so there is no known boundary between them. This means that when we employ the (very natural) method of inspecting adjacent members of the series in order to discover the whereabouts of the boundary we cannot locate the boundary since WT ensures that the boundary can never be where we are looking. Furthermore, the contextual factors which (in part) go to determine the extension cannot be held fixed through a complete inspection of the series using this method since successively considering adjacent items as pairs inevitably entails a change in those very factors.²⁴ Thus, WT ensures that there are ²⁰ Raffman’s distinction between internal (‘psychological’) contexts and external contexts (which concern the relevant comparison class, operative standards, and so forth) is of no help in resolving this more general form of the simple objection (for the distinction see Raffman 1994, 64–6; cf. Shapiro 2003, 60–1). ²¹ While extant forms of ESC (as given by e.g. Raffman 1994, Shapiro 2003, 2006) allow that first symptom of vagueness is a genuine symptom, these theories nonetheless permit a subject to know whether or not a predicate applies across the borderline area—and so the second symptom of vagueness is not a genuine symptom. This feature of these views issues from the fact that, in borderline cases, whether or not a predicate applies is taken to be a response-dependent matter such that what a (competent) subject judges to be the case determines what is the case (where such a judgment also puts the speaker in a position to know what is the case). Strictly speaking, such a response-dependent conception is not an essential feature of ESC. ²² We use the expression ‘quasi-psychological’ because in giving an explanation as to what gives rise to this third symptom of vagueness we not only need to give some psychological explanation as to why we come to believe that vague predicates are strongly tolerant, but we also need to establish why the claim that vague predicates are tolerant is so pre-theoretically intuitive. The two parts of this explanation are, of course, connected. ²³ Arguably, they are also individually sufficient, though substantiating that fact lies outside the scope of this paper. In Greenough (2003, 265–72) two proofs are given which show that the first two symptoms are equivalent given some pretty plausible background assumptions. ²⁴ It follows from WT that a subject cannot simultaneously bring all pairs in the series to salience.
Hold the Context Fixed—Vagueness Still Remains
281
certain conditions under which we cannot hold the context fixed. Under those conditions, the simple objection cannot arise. Even so, this only helps defuse a certain version of the simple objection. Even if the relevant contextual factors cannot be held fixed in the required way, it seems that we can introduce a new predicate via stipulation which is intuitively just as vague as the original one but is apparently not sensitive to differences in the context. Heck has a version of this objection as follows: Suppose I say, [in context C0 ]: Some of the patches are red; call them the reddies. I might ask which is the last of the reddies. [. . .] The question is why we cannot locate the last of the reddies. Maybe the extension of the word ‘red’ as we would then be using it would indeed shift, but the point does not seem relevant. There is no such shift in the extension of ‘the reddies’. (Heck 2003, 118–19)²⁵
Heck’s stipulation licenses the following double biconditional: (S) ‘is a reddie’ is true of x if and only if ‘is red in context C0 ’ is true of x if and only if ‘is red’ (as used in C0 ) is true of x. Heck assumes that the predicate ‘is a reddie’ cannot shift in extension (as a function of which pairs in the series we happen to be considering). Given (S), this assumption entails that the predicate-context pair ‘is red’ (as used in C0 ), and the predicate ‘is red in context C0 ’ likewise cannot shift in extension.²⁶ The general form of the puzzle then becomes: absent such shiftiness, what explains (a) why we don’t know the cutoff drawn by these predicates, (b) why these predicates give rise to borderline cases, and (c) why these predicates are sorites-susceptible? However, if this is the nub of the simple objection, then a further issue emerges: it’s not immediately obvious that the predicate ‘is red in context C0 ’ is genuinely soritessusceptible.²⁷ Here the immediate thought is that this predicate is a theoretical predicate of sorts—and we simply lack the requisite intuitions in natural language to say with conviction that this predicate exhibits the symptom of sorites-susceptibility. But ²⁵ Williamson also has a version of this objection (see Mills 2004, 640). ²⁶ Elia Zardini has suggested to us that if ‘is red’ is context-sensitive then the predicate ‘is red in context C0 ’ is a ‘monstrous’ predicate (in the sense of ‘monstrous’ given by Kaplan 1989, 510–11). If that is right then the open sentence ‘It is true in context C0 that x is red’ is also monstrous. However, that would only seem to be so under the assumption that the context-sensitivity of ‘is red’ is indexical context-sensitivity. Indexical context-sensitivity demands that the operator ‘It is true in context C0 that’ cannot operate upon character—because indexicals (in English at least) are such as to always take wide scope. In other words, if ‘is red’ is an indexical then this predicate, as used in a context in which the sentence ‘It is true in context C0 that x is red’ is uttered, determines an extension (relative to a circumstance of evaluation) given some value which is supplied from the context of utterance rather than from C0 itself. However, on a non-indexical model of context-sensitivity that need not be so. On such a model, the operator ‘It is true in context C0 that’ is akin to the modal ˚ operator ‘It is true at world W0 that’. See Akerman and Greenough (2009) for several arguments in favour of non-indexical over indexical contextualism. ²⁷ Cf. Stanley (2003, 279, fn. 13) who assumes without scruple, following Williamson, that a predicate such as ‘is tall at time t’ is sorites-susceptible. Presumably Stanley and Williamson would say the same concerning the predicate ‘is red in context C0 ’.
˚ Jonas Akerman and Patrick Greenough
282
if there is some doubt that ‘is red in context C0 ’ is genuinely sorites-susceptible and we then reflect on (S), then that doubt may spread to the predicate ‘is a reddie’ and, in turn, to the predicate-context pair ‘is red’ (as used at C0 ). Given that sorites-susceptibility is a necessary condition of the presence of vagueness then the simple objection lapses since vagueness is no longer present once we hold the context fixed. Perhaps all this shows is that the notion of sorites-susceptibility is too elusive to rely on as a reliable indicator of vagueness. After all, once one has been exposed to enough theory then it’s often hard to be drawn to think that vague predicates are strongly tolerant or think that the major premise of the standard sorites paradox simply must be true. In any case, it turns out that one can defuse the simple objection even if all the predicates in (S) are taken to be sorites-susceptible and so, for the purposes of argument, we shall assume that these predicates exhibit all three symptoms of vagueness. (To simplify matters, however, in much of what follows we shall focus on the predicate-context pair ‘is red’ as used in C0 .) What replies to the simple objection are in the offing? 15.4
R E P LY O N E : E PI S T E M I C I S T C O N T E X T UA L I S M
In brief, this reply runs as follows: Let it be granted that the predicate-context pair ‘is red’ (as used in C0 ) has a sharp and invariant extension. Let it also be granted that this predicate-context pair exhibits the first symptom of vagueness such that there is no known boundary between the extension of this predicate and its anti-extension. However, let the explanation for this ignorance be a purely epistemological explanation. One can flesh-out the required epistemological explanation by invoking something like a safety-based account of knowledge to explain our ignorance of the cut-off. On such an account, a belief that p is safe just in case there are no nearby worlds where I form the false belief that p on the same basis (see Williamson 1994, ch. 8, Williamson 2000 chs. 5, 7). The basic idea is that even if a subject formed a true belief, on a basis B, that the boundary for ‘is red’ (as used in C0 ) lies between a certain pair, this belief cannot constitute knowledge since the subject could easily have formed a false belief about the whereabouts of the cut-off on the same basis. Here the thought is that the extension of the predicate-context pair could easily have been different since the boundaries drawn by such predicates are unstable—even relative to a fixed context (see below). Such a story can also serve to explain why the second symptom of vagueness arises.²⁸ Suppose that a subject forms a true belief, on a basis B, that a certain item in the series belongs to the extension of the predicate-context pair ‘is red’ (as used in C0 ). Suppose also that this item lies near to the boundary drawn by the predicate-context pair. The subject’s belief fails to constitute knowledge because this belief could easily have been false. Again, the thought is that the extension of the predicate-context pair is unstable (relative to a fixed context) and so it could easily have been the case that the item failed to belong to the extension of the predicate (see below). ²⁸ And indeed the story can be used to explain why ‘is a reddie’ and ‘is red in context C0 ’ also exhibit the first two symptoms of vagueness.
Hold the Context Fixed—Vagueness Still Remains
283
A hybrid theory of vagueness is thus called for. A form of epistemicism is required to explain why we lack knowledge of the invariant cut-off for ‘is red’ (as used in C0 ), while a contextualist explanation, drawing on WT, would explain why we can’t know the cut-off for ‘is red’ relative to a fixed context where we are considering adjacent items together. Call this hybrid theory Epistemicist Contextualism. Is this reply ad hoc? Hybrid theories of vagueness are not uncommon. Ironically, Heck (2003, 124–5) himself sponsors a hybrid conception of vagueness under which first-order vagueness is taken to be semantic, but the boundary between the borderline area and the non-borderline regions is taken to be sharp (and unknowable). Heck says: ‘there is nothing ad hoc about the refusal to go epistemic at one point but not the other’ (ibid., 124). But then Heck can have no principled complaint with the reply in hand to the simple objection.²⁹ Even so, those who accept standard forms of epistemicism (e.g. Sorensen and Williamson) are likely to be unmoved by this reply on the grounds that considerations of simplicity and uniformity dictate that a non-hybrid theory of vagueness is called for.³⁰ This counter-reply can itself be resisted. The most well-worked out form of epistemicism—Williamson’s—is an impure form of epistemicism in that Williamson posits that the sharp boundaries drawn by vague predicates are themselves ‘unstable’ (1994, 231) such that ‘the extension of ‘‘thin’’ as used in a given context could very easily have been slightly different’ (ibid., 230). This (modal) instability in extension arises because the pattern of usage of ‘thin’ (even with respect to a fixed context) is itself unstable. Even though such usage may be invariant from context to context in the actual world, nonetheless, such usage could easily have been different. For Williamson, this instability in extension plays a key role in explaining why I cannot, for example, know the truth value of the sentence ‘Everyone with exact physical measurements x, y, z, is thin’. Suppose this sentence is true and I believe it to be so, why does my belief fail to constitute knowledge? If true, this sentence expresses a necessary truth (Williamson 1994, 204, 230). But since there are no worlds in which the proposition expressed by this sentence is false, then a fortiori there are no nearby worlds in which the proposition expressed by this sentence is false. Hence, my belief that the sentence is true is guaranteed to be safe. It thus seems a safety-based account of knowledge cannot explain the requisite kind of ignorance. However, if the sentence could easily have expressed a different, and indeed false, proposition (relative to a fixed context) then my belief that the sentence is true could easily have been false and so cannot constitute knowledge. It is for this reason that Williamson posits unstable cut-offs for vague predicates to fully explain the ignorance which may arise because of vagueness. A pure form of epistemicism, in contrast, posits only an epistemological explanation for our ignorance of cut-offs. Impure forms of epistemicism are hybrid theories because they posit a special vagueness-relevant semantic (or metaphysical) feature of vague predicates and invoke an epistemological story from there. For this reason, an ²⁹ Koons (1994, 447) sponsors a similar hybrid view. Goguen (1969) also seems to defend a hybrid of fuzzy logic and epistemicism, whereby the borderline area is also sharply-bounded. See also Simons (1992). ³⁰ See also Keefe and Smith for this objection (1997, 47).
284
˚ Jonas Akerman and Patrick Greenough
epistemicist form of Contextualism and Williamson’s impure epistemicism are simply on a par with respect to the desiderata of simplicity and uniformity.³¹ The preceding considerations show that Contextualism can not only allow, but even predicts, that the first two (epistemic) symptoms of vagueness arise even when the context is held fixed. But what about the third symptom of vagueness? Why are vague predicates sorites-susceptible? Recall from above that BSC offers the following ‘confusion’ diagnosis as to why we find the major premise of the standard sorites paradox so plausible: we confuse the following two principles (and their respective kin): (WT) It is not the case that: there is a context of utterance C and there is an x such that x and x are considered together as a pair in C and ‘is F ’ (as used in C) is true of x and ‘is F ’ (as used in C) is false of x , (where x is adjacent to x in the sorites series running from F to not-F ). (ST) It is not the case that: there is a context of utterance C and there is an x such that ‘is F ’ (as used in C) is true of x and ‘is F ’ (as used in C) is false of x , (where x is adjacent to x in the sorites series running from F to not-F ). The question then arises: does the diagnosis mooted by Contextualism above as to why we find the major premise of the sorites so compelling retain its force when the context is held fixed? According to the diagnosis in hand, ST derives its plausibility from being confused with WT. For the purposes of argument, let that part of the diagnosis stand. It is also the case that ST, as applied to ‘is red’ entails: for all x and for all contexts C, the predicate ‘is red’ (as used in C) is not true of x and false of x . In other words, take any context you like, the predicate ‘is red’ (as used in that context) does not draw a boundary. So, take the context C0 . It follows that ‘is red’ as used in C0 does not draw a boundary. So, if we are confused into accepting ST, then we are confused into accepting that the predicate-context pair ‘is red’ as used at C 0 draws no boundary. On that basis, we accept the major premise of the standard sorites as applied to the predicate-context pair ‘is F ’ (as used at C0 ). In other words, this predicate-context pair is sorites-susceptible even though it draws a sharp and invariant boundary across the dimension of comparison. Thus, not only can Contextualism allow that sorites-susceptibility remains even when the context has been held fixed, it predicts that such sorites-susceptibility will remain. The simple objection simply does not get a grip when it comes to the third symptom of vagueness. Even if one resists the details of the diagnosis just given, epistemicist forms of Contextualism have a fallback diagnosis. Recall that the standard epistemicist diagnosis as to why the major premise of the standard sorites is so plausible also posits a confusion. But this confusion is more humdrum: we confuse the (true and plausible) claim that vague predicates do not draw a known boundary with the (false) claim that they do not draw a (sharp) boundary.³² Given Epistemicist Contextualism, the predicates ‘is a reddie’, ‘is red in C0 ’, and the predicate-context pair ‘is red’ (as used at C0 ), all exhibit ³¹ In his most recent defence of epistemicism, Sorensen (2001) is also committed to a hybrid view of sorts since he posits a metaphysical explanation for the unknowability of the sharp cut-offs drawn by vague terms in terms of what he calls ‘truthmaker gaps’. ³² Where to lack a sharp boundary is to lack a boundary.
Hold the Context Fixed—Vagueness Still Remains
285
the first symptom of vagueness—they all draw no known boundary across the sorites series for ‘is red’. Given the ‘confusion’ diagnosis just posited, this first symptom is easily confused, when first thinking about the paradox, with the claim that they draw no (sharp) boundary. If we are confused into believing that these predicates draw no sharp boundary then we are confused into believing that the major premise of the sorites is valid. Hence, these predicates are sorites-susceptible. There are various ways in which one can finesse such a diagnosis.³³ However for our purposes it doesn’t matter whether such a diagnosis is compelling. What matters is that the simple objection posits no special objection to Contextualism since Contextualism can also draw on epistemicist resources to explain why the sorites-susceptibility of a predicate remains even when the context has been held fixed. The overall upshot, then, is that (an epistemicist form of) Contextualism can allow, and even predicts, that each of the three symptoms of vagueness arise when one holds the context fixed. The simple objection is no objection to Contextualism. The trouble with this reply is that it is committed to a form of epistemicism and so is unlikely to persuade everybody. Is there a viable alternative? 15.5
R E P LY T WO : R A D I C A L C O N T E X T UA L I S M
In brief, this reply runs as follows: Let it be granted that all three predicates in (S) give rise to our three symptoms of vagueness. So, ‘is a reddie’, ‘is red in context C0 ’, and the predicate-context pair ‘is red’ (as used in context C0 ) are all vague. But note that ‘is red’ (as used in C0 ) is true of x if and only if x satisfies ‘is red’ in context C0 . Given (S), this means that the vagueness of the object-language predicates ‘is a reddie’, ‘is red in context C0 ’, and the predicate-context pair ‘is red’ (as used in C0 ) will co-vary with the vagueness of the meta-linguistic predicate ‘x satisfies ‘‘is red’’ in context C0 ’. Meta-linguistic vagueness represents a kind of higher-order vagueness.³⁴ Thus, to ask whether the predicate-context pair ‘is red’ (as used at C0 ) is vague is a way of asking whether the predicate type ‘is red’ is higher-order vague (in the requisite sense of ‘higher-order vague’). The first-order vagueness of this predicate type consists in the fact that, relative to different contexts of utterance, this predicate type can differ in extension (relative to a given world). The second-order vagueness of this predicate type consists in the fact that a meta-linguistic predicate such as ‘x satisfies ‘‘is red’’ in context C0 ’ can itself differ in extension relative to different contexts of utterance. The vagueness of this metalinguistic predicate ‘x satisfies ‘‘is red’’ in context C0 ’ may have one of two sources. Either it is vague what the singular term ‘context C0 ’ refers to, or it is vague what the quotation name ‘ ‘‘is red’’ ’ refers to. We shall simply focus on the former source. In Heck’s statement of the simple objection we are ³³ One way to finesse the diagnosis would be to argue that this confusion itself arises from an internalist conception of meaning and understanding which licenses the transparency claim that for all n, if it is true that patch n in the series is red then one is in a position to know this (cf. Williamson 1994, 205–12). ³⁴ On this score, we agree with Keefe and Smith (1996, 15–16).
286
˚ Jonas Akerman and Patrick Greenough
supposed to be able to nail down a sharp and invariant extension for the reddies by saying, at a particular time T0 : ‘Some of the patches are red; call them the reddies.’ However, such a saying has temporal spread because all speech acts take place over time. It is thus unclear, and indeed vague, at what exact time the saying—the stipulation—has taken place. That is, it is vague just what time is picked by the name ‘T0 ’, and so vague just what context is picked out by the name ‘C0 ’. It follows that there are some objects such that it is vague whether or not these objects fall under the predicate ‘is a reddie’. On a contextualist model of vagueness, according to certain strict standards, the saying may be deemed to have taken place in a very narrow interval of time, while relative to more lax standards, the saying may be deemed to have taken place at a broader interval. Thus the predicate ‘x satisfies ‘‘is red’’ in context C0 ’ is itself subject to contextual variation. There is thus no difference in kind between the vagueness of the predicate type ‘is red’ and the vagueness of ‘is a reddie’. The status of the simple objection now ought to be clear: it amounts to the claim that Contextualism cannot allow for (a certain type of) higher-order vagueness. Epistemicist Contextualism can be seen as an attempt to offer a semantic model of first-order vagueness and an epistemic model of higher-order vagueness. Radical Contextualism can be seen as an attempt to offer a uniform characterization of all orders of vagueness. Is Radical Contextualism defensible? For our purposes it doesn’t matter. What matters is whether it is co-defensible with what the leading non-epistemic (non-contextualist) theories of vagueness say concerning higher-order vagueness. One way of making sense of higher-order vagueness given Radical Contextualism is to offer a type-theoretical model. According to such a suggestion, contexts of utterance should be typed to a level: level-1 contexts of utterance, level-2 contexts of utterance, and so on. Semantic closure is thus to be rejected and a hierarchy of increasingly expressive meta-languages is called for. So, for example, context C0 , is a level-1 context of utterance, whereas the context of utterance in which the predicate ‘x satisfies ‘‘is red’’ in context C0 ’ determines an extension is a level-2 context of utterance. This kind of radical model defuses the simple objection as follows: when it is said ‘hold all the features of the context fixed, vagueness still remains, therefore vagueness is not context-sensitivity’ this should simply be read as ‘hold all the features of the level-1 context fixed, vagueness still remains, therefore not all vagueness is level-1 contextsensitivity’. On this score, it is notable that perhaps the most sophisticated response to issue from the non-epistemic camp concerning the various puzzles of higher-order vagueness is given by Keefe (2000, ch. 8).³⁵ It turns out that Keefe can have no principled objection to the broad type-theoretic strategy just mooted with respect to the simple objection. That’s because she alleges that a Tarskian style hierarchy of increasingly expressive meta-languages is required if we are to address a central puzzle of higherorder vagueness given by Williamson. Williamson’s puzzle can be given as follows: Suppose we define a notion of absolute definiteness as follows: It is absolutely definite that A =df A and it is definite that A and it is definite that it is definite that A ³⁵ Every other non-epistemic theory of vagueness has notably failed to address all of the pressing puzzles concerning higher-order vagueness.
Hold the Context Fixed—Vagueness Still Remains
287
and . . . The notion of absolute definiteness intuitively ought to be vague. But it also follows from the definition, given some simple logic, that an S4 reduction schema for absolute definiteness is valid: If it is absolutely definite that A then it is absolutely definite that it is absolutely definite that A.³⁶ But if this schema holds then absolute definiteness cannot exhibit genuine higher-order vagueness.³⁷ If that is so, it is not genuinely vague at all. Contradiction.³⁸ Keefe concedes that absolute definiteness is vague but that it’s vagueness cannot be expressed within the meta-language. A richer metalanguage is needed. But within this richer meta-language we can define a new notion of absolute definiteness (‘absolute definiteness∗ ’), which itself is vague. Yet this new notion cannot be used to express the fact that it is vague without contradiction and so the problem re-occurs. A richer meta-meta-language is needed to express the vagueness of this new notion. To fully resolve the problem the hierarchy of meta-languages is non-terminating. To this she adds: If there is no general objection to the claim that the sequence of metalanguages for metalanguages is potentially infinite, then what is the difficulty with adding ‘and each of these languages is vague’? [. . .] There is no vicious infinite regress forced upon us. It is just that the vague is not reducible to the non-vague. (2000, 208)
Is Keefe’s model of higher-order vagueness defensible? Again, for our purposes it doesn’t matter.³⁹ What is clear is that it is broadly co-defensible with what Radical Contextualism is committed to with respect to higher-order vagueness in order to address the simple objection. The upshot is that what Radical Contextualism says in response to the simple objection yields a set of commitments which, broadly, are no more implausible than the commitments incurred by the most promising non-epistemic (noncontextualist) theories of vagueness with respect to higher-order vagueness. Likewise, what Epistemicist Contextualism says in response to the simple objection yields a ³⁶ We freely assume the closure of the D-operator here. That is not uncontroversial of course. ³⁷ If S4 (i.e. KT4) is the logic for absolute definiteness then there is only a finite number of modalities (in fact at most fourteen distinct modalities, see Chellas 1980, 149). Consequently, there cannot be borderline cases to borderline cases ad infinitum. ³⁸ See Williamson (1994, 160–1) for the puzzle and for the anticipation of Keefe’s reply. ³⁹ A further issue concerns the possibility of quantifying over all levels. If that is possible then a strengthened version of the simple objection can be formulated thus: hold all features of all contexts of whatever level fixed, the vagueness of the predicate relative to that (infinite) sequence of contexts/levels still remains, therefore vagueness is not context-sensitivity. Is there a reply? In the first place, note that Keefe’s model of higher-order vagueness also suffers from a strengthened version of Williamson’s puzzle of higher-order vagueness if we are allowed to quantify over all meta-languages. So, again, (Radical) Contextualism is no worse off than its most sophisticated competitors. Secondly, as Elia Zardini has pointed out to us, to make this strengthened version of the challenge stick we would need to make sense of the infinite embedding ‘. . . satisfies ‘‘satisfies ‘‘satisfies . . . ‘‘satisfies ‘‘red’’ in C1 ’’ in C2 ’’ in C3 ’’, . . .’. But it is far from clear that such a sentence can be understood. Its length can be arbitrarily large (well beyond omega), and already at omega (and then at any limit ordinal) the string is not going to be well founded (it has no starting point as can be seen from the initial dots).
288
˚ Jonas Akerman and Patrick Greenough
set of commitments which are, broadly, no more implausible than the commitments incurred by the leading epistemic theories with respect to vagueness and higher-order vagueness. Either way, the simple objection to Contextualism loses much, if not most, of its force. Re f e re n c e s ˚ Akerman, J. and Greenough, P. (2009), ‘Vagueness and non-indexical contextualism’ in New Waves in the Philosophy of Language, ed. Sarah Sawyer, Basingstoke, Palgrave Macmillan. Beall, Jc (2003), Liars and Heaps: New Essays on Paradox, New York, Oxford University Press. Chellas, B. F. (1980), Modal Logic, Cambridge, Cambridge University Press. Fara, D. G. (2000), ‘Shifting sands: an interest-relative theory of vagueness’, Philosophical Topics 28, 45–81. (Originally published under the name ‘‘Delia Graff’’). Goguen, J. A. (1969), ‘The logic of inexact concepts’, Synthese 19, 325–73. Greenough, P. (2003), ‘Vagueness: a minimal theory’, Mind 112, 235–81. (2005), ‘Contextualism about vagueness and higher-order vagueness’, Proceedings of the Aristotelian Society, Supplementary Volume, 79, 167–90. Heck, R. (2003), ‘Semantic accounts of vagueness’ in Liars and Heaps, ed. Jc Beall, 106–27, New York, Oxford University Press. Kamp, H. (1981), ‘The paradox of the heap’ in Aspects of Philosophical Logic, ed. U. M¨onnich, 225–77, Dordrecht, Reidel. Kaplan, D. (1989), ‘Demonstratives’ in Almog, Perry, and Wettstein (eds.), Themes from Kaplan, Oxford, Oxford University Press. Keefe, R. (2000), Theories of Vagueness, Cambridge, Cambridge University Press. (2007), ‘Vagueness without context change’, Mind 116, 275–92. Keefe, R., and Smith, P., eds. (1996), Vagueness: A Reader, Cambridge, MA, MIT Press. K¨olbel, M. (2007), ‘A problem for contextualism about vagueness’, Paper presented at the Joint Session of the Mind Association and Aristotelian Society, July 2007. Koons, R. (1994), ‘A new solution to the sorites problem’, Mind 103, 439–49. Lewis, D. (1979), ‘Scorekeeping in a language game’, Journal of Philosophical Logic 8, 339–59. Mills, E. (2004), ‘Williamson on vagueness and context-dependence’, Philosophy and Phenomenological Research, 68, 635–41. Priest, G. (2003), ‘A site for sorites’ in Liars and Heaps, ed. Jc Beall, 9–23, New York, Oxford University Press. Raffman, D. (1994), ‘Vagueness without paradox’, Philosophical Review, 103, 41–74. (1996), ‘Vagueness and context relativity’, Philosophical Studies 81, 175–92. Shapiro, S. (2003), ‘Vagueness and conversation’ in Liars and Heaps, ed. Jc Beall, 39–72, New York, Oxford University Press. (2006), Vagueness in Context, Oxford, Oxford University Press. Simons, P. (1992), ‘Vagueness and ignorance’, Aristotelian Society, suppl. 66, 163–77. Soames, S. (1999), Understanding Truth, Oxford, Oxford University Press. Sorensen, R. (1988), Blindspots, Oxford, Clarendon Press. (2001), Vagueness and Contradiction, Oxford, Oxford University Press. Stanley, J. (2003), ‘Context, interest-relativity, and the sorites’, Analysis 63, 269–80. Williamson, T. (1994), Vagueness, London, Routledge. (2000), Knowledge and Its Limits, New York, Oxford University Press.
16 Saying More (or Less) Than One Thing Andrea Iacona
In a paper called Definiteness and Knowability, Tim Williamson addresses the question whether one must accept that vagueness is an epistemic phenomenon if one adopts classical logic and a disquotational principle for truth. Some have suggested that one must not, hence that classical logic and the disquotational principle may be preserved without endorsing epistemicism.¹ Williamson’s paper, however, finds ‘no plausible way of substantiating that possibility’. Its moral is that ‘either classical logic fails, or the disquotational principle does, or vagueness is an epistemic phenomenon’.² The moral of this chapter, on the contrary, is that there is a plausible way of substantiating that possibility. The option it contemplates looks like a view that Williamson dismisses at the beginning of his paper, and that others regard as unworthy of serious consideration.
16.1 A couple of preliminary clarifications. The first concerns the expression ‘borderline case’. It is widely accepted as a matter of definition that a vague word is a word that admits of borderline cases. Typically, a vague predicate is a predicate that admits of borderline cases. Yet this leaves unsettled what ‘borderline case’ means exactly. According to one reading of the expression, a borderline case is a situation that involves actual problems of evaluation. We are ‘in’ a borderline case so understood when a speaker assertively utters a sentence and we don’t know whether the assertion is correct or incorrect. For example, it may happen that a speaker assertively utters ‘A is tall’ and we don’t know how to evaluate the assertion, because we don’t know whether it is correct to say that A falls into the extension of ‘tall’. According to another reading of the expression, a borderline case is an object that neither clearly belongs to the extension of a predicate neither clearly does not belong This chapter has a long history, as it went through a considerable number of changes and revisions before taking its present shape. In the course of this history, several people have helped me with it in various ways. Richard Dietz, Max K¨olbel and Tim Williamson are definitely among them. But there are many others. ¹ See Horwich 1998, 78–84 and Field 1994, 409–22. ² Williamson 1995, 171.
289
290
Andrea Iacona
to it. This reading hides an ambiguity. An object may be a borderline case for a predicate in the sense that it is unclear simpliciter whether it belongs to the extension of the predicate. That is, an object may neither paradigmatically have a property nor paradigmatically not have it. For example, if the height of A is 1.75, A is neither paradigmatically tall nor paradigmatically not tall. Thus A may be called a borderline case for ‘tall’. Alternatively, an object may be a borderline case for a predicate relative to a certain context in the sense that it is unclear in that context whether it belongs to the extension of the predicate. For example, A is a borderline case for ‘tall’ relative to a certain context if it is unclear in that context whether ‘tall’ applies to A. The distinction between borderline situations and borderline objects doesn’t really matter as far as borderline objects in the second sense are concerned. Since the characterization of such an object is parasitic on the notion of a borderline situation, it turns out to be trivial that borderline situations arise just in case borderline objects are involved. However, it is important to avoid confusion between borderline situations and borderline objects in the first sense, because there is no reason to think that they must always go together. It is plausible to say that—given the due qualifications—whenever a borderline situation arises, a borderline object in the first sense is involved. But the converse does not hold. Borderline objects in that sense do not necessarily make borderline situations. Suppose that A is 1.75. This does not entail that whenever ‘A is tall’ is assertively uttered, problems of evaluations arise. It is easy to imagine cases in which the sentence is assertively uttered, yet no unclarity affects the evaluation of the assertion. If A is teacher of a class of kids, and one of them assertively utters ‘A is tall’ in order to explain to another why A is able to write on a part of the board they can’t reach, the assertion may be taken to be correct. If instead ‘A is tall’ is assertively uttered by someone who is seriously considering A’s chances to join a basketball team, the assertion may be taken to be incorrect. Here only the first reading of ‘borderline case’ will be adopted. To avoid confusion, the expression will not be used to refer to objects. In the literature the two readings are often mixed, and the ambiguity of the second is often neglected. This is why sometimes philosophers talk as if whenever (and just because) a borderline object in the first sense is involved, a borderline case should arise. And given that borderline objects are as common as paradigmatic objects, this amounts to talking as if borderline cases were as common as unproblematic cases. But such talk is just a theoretical habit that takes us very far from ordinary linguistic practice. In reality, it almost never happens that a speaker assertively utters a sentence and we don’t know how to evaluate the assertion. We normally take for granted some way of understanding the sentence according to which the assertion turns out clearly correct or clearly incorrect. Borderline cases almost never arise. As a matter of fact, borderline objects in the first sense are correctly or incorrectly described this or that way depending on the occasion. The second clarification concerns the expression ‘what is said’. Roughly, what is said by uttering a sentence on a certain occasion depends on how the words occurring in the sentence are understood on that occasion, where understanding a word involves grasping its linguistic meaning and, possibly, specifying its reference in the context of utterance. This leaves room for at least two distinct notions. One is that,
Saying More (or Less) Than One Thing
291
when a sentence is assertively uttered by a speaker, there is something that the speaker has in mind. For example, a speaker may use the sentence ‘He is there’ to convey the information that A is in St. Andrews, while another speaker may use it to convey the information that A is in Mexico City. A natural way to express this difference is to say that the two speakers have different things in mind when they utter the sentence. So there is a reading of ‘what is said’—call it intentional reading—according to which what is said by uttering a sentence on a certain occasion is a matter of what understanding of the sentence can rightfully be ascribed to the speaker on that occasion. The other notion is that, when a sentence is assertively uttered by a speaker, there is something to which truth or falsity can be ascribed. The something in question is naturally understood as a specification of the reference of the words occurring in the sentence such that, according to it, either the sentence describes things as they are or it describes things as they are not. Suppose that A is in Mexico City. Then ‘He is there’ turns out true if ‘he’ refers to A and ‘there’ refers to Mexico City, while it turns out false if ‘he’ refers to A and ‘there’ refers to St. Andrews. So there is a reading of ‘what is said’—call it truth-conditional reading—according to which what is said by uttering a sentence on a certain occasion is a matter of what understanding makes the sentence evaluable as true or false on that occasion. It is a naive temptation to put together the two notions. This amounts to thinking that, when a sentence is assertively uttered by a speaker, there is one understanding of the sentence that can rightfully be ascribed to the speaker and involves a specification of the reference of the words occurring in it that makes it evaluable as true or false. Contrary to this temptation, here it will be assumed that the two readings of ‘what is said’ do not coincide. It may happen that a speaker assertively utters a sentence and has something in mind, yet no specification of the reference of the words in the sentence that makes it evaluable as true or false can rightfully be ascribed to the speaker.³
16.2 The thought entertained in this chapter is that borderline cases are cases in which there is no such thing as what is said in the truth-conditional sense. An example of borderline case may help illustrate. Suppose that the grandmother is in the kitchen and the cat is in the living room in such a position that half of its body lies on the mat and the other half lies on the floor. The grandmother makes a guess and assertively utters the sentence (1) The cat is on the mat Apparently, it is hard to tell whether the grandmother’s assertion is correct or incorrect, because it is hard to tell whether the word ‘on’ that occurs in (1) applies to the cat ³ Iacona (2006) deals with cases in which the two readings seem not to coincide.
292
Andrea Iacona
and the mat. This is not simply due to the fact that the cat is in the position described. It is easy to imagine situations in which (1) is uttered and the cat is in the very same position, yet no problem of evaluation arises. Imagine that the cat got muddy in the garden, and the mat is brand new and very expensive. The grandmother could be concerned about the mat, and wonder whether at least part of the body of the cat lies on it. Or imagine that it is winter and the floor is cold. The grandmother could be concerned about the health of the cat, and wonder whether its whole body lies on the mat. On the assumption that there are cases in which the two readings of ‘what is said’ do not coincide, this can be regarded as one of them. The grandmother has something in mind. That is, something can rightfully be ascribed to her as the actual understanding of (1). But that understanding does not amount to one specification of the reference of the words occurring in (1) that makes (1) evaluable as true or false. Whether (1) is true or false depends on whether or not the pair formed by the cat and the mat belongs to the extension of ‘on’. But the extension of ‘on’ can be specified both in such a way as to include the pair and in such a way as not to include it. As the two alternative situations imagined show, neither of the two possibilities is ruled out by the linguistic meaning of ‘on’. This is to say that there are ways of understanding ‘on’ according to which (1) is true and ways of understanding ‘on’ according to which (1) is false. The point is that the actual understanding of (1) does not decide between the former and the latter. What is said in the intentional sense does not uniquely determine something said in the truth-conditional sense. The case of the grandmother may be described as a case of underspecification, that is, as a case in which the actual understanding of a sentence is not sufficiently specific for the purpose of ascribing truth or falsity to the sentence. For there are at least two ways of understanding (1) that go beyond its actual understanding, in the sense of not being uniquely determined by it. A case of underspecification is not a normal case, as normally the actual understanding of a sentence is sufficiently specific. The account may be phrased in a more rigorous way by using familiar terminology. Let a valuation be an assignment of semantic properties to the sentences of a language that determines definite truth conditions for them. For example, in the case of (1) a valuation will assign an object to ‘the cat’, another object to ‘the mat’, and a set of pairs to ‘on’. Valuations amount to what supervaluationists call ‘precisifications’, that is, ways of making language precise.⁴ A valuation is admissible when it corresponds to a legitimate way of making language precise, that is, when it respects the constraints imposed by the linguistic meaning of its expressions. For example, a valuation that assigns a dog to ‘the cat’ will not be admissible. The actual understanding of a sentence—what is said in the intentional sense—may be described as a set of admissible valuations, namely, the set of all the admissible valuations that are not ruled out by that understanding.⁵ Truth in a valuation is defined in the standard way. In the case of a simple sentence, it depends on whether the predicate applies to the objects denoted by the terms. Thus (1) will be true in a valuation v just in case the pair formed by the objects that v ⁴ As in Fine (1975).
⁵ Admissibility is again as in Fine (1975).
Saying More (or Less) Than One Thing
293
assigns to ‘the cat’ and ‘the mat’ belongs to the set that v assigns to ‘on’. The truth of a complex sentence depends on that of its constituents, in accordance with the usual compositional rules. For example, if (1) is true in v and ‘Snow is white’ is true in v, then also ‘The cat is on the mat and snow is white’ will be true in v. Let us stipulate that two valuations overlap on a sentence s when they are alike as far as the truth or falsity of s is concerned. For example, two valuations overlap on (1) if they assign to ‘the cat’ and ‘the mat’ the same objects a and b, and they assign to ‘on’ two sets that differ only in that one includes a pair formed by a certain piece of paper and a certain table, while the other doesn’t include it. In both cases, the pair belongs to the extension of ‘on’. An interpretation of s is a set of admissible valuations that overlap on s. Informally speaking, an interpretation is a way of understanding a sentence that is sufficiently specific for the purpose of ascribing truth or falsity to the sentence on a given occasion. What is said in the truth-conditional sense is an interpretation. Truth in an interpretation is defined in terms of truth in a valuation. Let α be an interpretation of s. Then, s is true in α if and only if it is true in every valuation that belongs to α. Similarly, s is false in α if and only if it is false in every valuation that belongs to α. Let an interpretation of s be compatible with the actual understanding of s just in case it is a subset of it. The distinction between normal cases and cases of underspecification is then as follows. A normal case is a case in which there is a unique interpretation that is compatible with the actual understanding, namely, the actual understanding itself. A case of underspecification, instead, is a case in which there are at least two interpretations that are compatible with the actual understanding and yield opposite truth values. To see this, suppose that the actual understanding α of s is not an interpretation. Then there are at least two valuations v1 and v2 that belong to α but do not overlap on s. Let β be a set whose only member is v1 , and let γ be a set whose only member is v2 . β and γ are interpretations compatible with α that yield opposite truth values for s. Inversely, suppose that there are two interpretations β and γ that are compatible with α and yield opposite truth values for s. Then α is not an interpretation. For if we take a member v1 of β and a member v2 of γ , we get that v1 and v2 belong to α but do not overlap on s. The underlying thesis of the account is that truth and falsity apply to sentences relative to interpretations. On the truth-conditional reading of ‘what is said’, this means that the things we say are interpretations of the sentences we utter. The distinction between normal cases and cases of underspecification may thus be described in terms of attribution of things said to speakers. Let the condition of attribution be as follows for an interpretation α and a speaker S: α can be attributed to S when only α is compatible with the actual understanding manifested by S. There are cases in which something true is said, that is, cases in which an interpretation can be attributed to the speaker and the sentence uttered is true in that interpretation. Similarly, there are cases in which something false is said, that is, cases in which an interpretation can be attributed to the speaker and the sentence uttered is
294
Andrea Iacona
false in that interpretation. The case of the grandmother belongs to neither of these two categories, as it is a case in which there is no unique thing said. One way to make sense of this is to allow that more than one thing is said, in that more than one interpretation is compatible with the actual understanding. Another way is to hold that nothing is said, in that no interpretation can be attributed to the speaker. The latter seems preferable. First of all, it is quite implausible to hold that a speaker says both something true and something false at the same time. Secondly, although it is sensible to assume that any thing said in the truth-conditional sense is an interpretation, there seems to be no reason in addition to grant that any interpretation is something said in the truth-conditional sense. For example, any set containing one admissible valuation is by definition an interpretation of (1). But we wouldn’t describe any such set as something said by the grandmother. A case of underspecification differs from a normal case in that it involves a conflict of interpretations. The problem that characterizes it concerns the attribution of interpretations, it is not primarily a problem of truth and falsity. There is an obvious sense in which underspecification is compatible with the principle of bivalence, according to which truth and falsity are mutually exclusive and jointly exhaustive values. It is the sense in which bivalence holds in any interpretation. By definition, for any interpretation α of a sentence s, either all the valuations that belong to α make s true or they make s false. This entails two claims. One is that either s is true in α or s is false in α: truth and falsity are jointly exhaustive. The other is that it cannot be the case that s is true in α and s is false in α: truth and falsity are mutually exclusive. 16.3 So far it has been suggested that the case of the grandmother is a case of underspecification. Since the case of the grandmother is clearly a borderline case, this entails that at least some borderline cases are cases of underspecification. However, it does not entail that all borderline cases are cases of underspecification. The stronger claim may be justified as follows. Let C be a case in which S assertively utters s, and let α be the actual understanding of s manifested by S. Now suppose that α is an interpretation. Given that bivalence holds relative to interpretations, s is either true or false in α. If s is true in α, then an interpretation in which s is true can be attributed to S. This means that no problem of evaluation arises in C. Similarly, if s is false in α, then an interpretation in which s is false can be attributed to S. This means that no problem of evaluation arises in C. Therefore, if α is an interpretation then no problem of evaluation arises in C. Since C is a borderline case only if some problem of evaluation arises in C, we get that if α is an interpretation then C is not a borderline case. It follows that if C is a borderline case then α is not an interpretation. That is, underspecification obtains in all borderline cases. We saw that if α is not an interpretation, there are at least two interpretations compatible with α that yield opposite truth values for s. Therefore, the claim that underspecification obtains in all borderline cases entails that if C is a borderline case, there are at least two interpretations compatible with α that yield opposite truth values for
Saying More (or Less) Than One Thing
295
s. A fortiori, it entails that if C is a borderline case, there are admissible valuations that yield opposite truth values for s. This amounts to a presupposition that is at the core of semantic theories of vagueness, namely, that borderline cases are cases in which the linguistic meaning of the words occurring in the sentence does not settle the question of whether the assertion is correct or incorrect. For example, a case in which a speaker assertively utters ‘A is tall’ and we don’t know how to evaluate the assertion is a case in which the meaning of ‘tall’ rules out neither the possibility that the assertion is correct nor the possibility that it is incorrect. A general tenet of semantic theories of vagueness is that the kind of unclarity that is characteristic of borderline cases depends on the linguistic meaning of the words occurring in the sentences we utter. The argument above shows that this tenet holds, in that it is a necessary condition of borderline cases that the linguistic meaning of the words occurring in the sentence uttered does not determine whether the assertion is correct or incorrect. On the other hand, however, underdetermination in this sense is not a sufficient condition of borderline cases. The linguistic meaning of ‘on’ and the fact that the cat is in such a position that half of its body lies on the mat and the other half lies on the floor do not suffice to make the case borderline. The same goes for the linguistic meaning of ‘tall’ and the fact that the height of A is 1.75. In order to have a borderline case we need three ingredients: the linguistic meaning of the words occurring in the sentence, the state of affairs, and the actual understanding. The first two do not suffice. This point may be phased in terms of the customary distinction between speaker’s meaning and semantic meaning, that is, between what the speaker means by uttering certain words, and what those words mean. Since borderline cases are effects of the actual understanding of sentences, they are features of speaker’s meaning. Therefore, there is a sense in which vagueness concerns speaker’s meaning rather than semantic meaning. Semantic theories of vagueness usually do not recognize this sense. Yet there is also a sense in which vagueness concerns semantic meaning, that in which it amounts to underdetermination of linguistic meaning. The semantic meaning of a vague sentence may be identified with a set of valuations some of which are in conflict. In this sense it is correct to say that borderline cases are grounded on linguistic meaning. Asking which of these two senses comes first, or which is more fundamental, is like asking whether the chicken or the egg comes first. On the one hand, vagueness in speaker’s meaning depends on vagueness in semantic meaning, in that the latter is a condition of the possibility of the former. On the other, vagueness in semantic meaning depends on vagueness in speaker’s meaning, in that words have linguistic meaning as a result of the way speakers use them. 16.4 The account of borderline cases in terms of underspecification contradicts Williamson’s disjunctive moral. In the first place, borderline cases pose no threat to classical logic. For they involve no violation of bivalence. In the second place, the disquotational principle is preserved. Assuming that truth and falsity apply to
296
Andrea Iacona
sentences relative to interpretations, there are two ways of stating the principle. One is to phrase it as a schema that holds for sentences as they are actually understood and involves ascription of truth relative to interpretations. That is, for any interpretation α, the following biconditional is true relative to α: ( T1) ‘p’ is true if and only if p Here ‘p’ is a substitutional variable, and ‘true’ is a predicate whose extension varies with α. Borderline cases do not affect the principle, in that the only trouble they give concern the choice of α. The other way is to phrase the principle as a schema that holds for interpreted sentences. For any α, let ‘pα ’ stand for ‘p’ as it is interpreted according to α. The following biconditional is true simpliciter: ( T2) ‘pα ’ is true if and only if pα In this case, to see that borderline cases do not affect the principle it must be acknowledged that ( T2) follows from a more general schema: ( T3) If ‘p’ says that pα , then ‘p’ is true if and only if pα A borderline case is a case in which ‘p’ is uttered, but there is no α such that ‘p’ says that pα . This means that the antecedent of ( T3) is not satisfied, hence that ( T3) is vacuously true.⁶ In the third place, the account does not entail that vagueness is an epistemic phenomenon. According to epistemicism, the meaning of the words occurring in a sentence determines a truth value for the sentence, in borderline cases just as in normal cases. This is to say that in the case of the grandmother the extension of ‘on’ is such as to make (1) true or false. The idea is that words have sharp boundaries, but we don’t know exactly where these boundaries lie. That is, given any cat and any mat, either the pair formed by them belongs to the extension of ‘on’ or it doesn’t. The fact is that we don’t know exactly the borders of that extension, hence it may happen that we don’t know, of a certain cat and a certain mat, whether or not the pair formed by them belongs to it. The account in terms of underspecification entails nothing like that. Truth and falsity apply to (1) relative to interpretations, so only relative to this or that interpretation the extension of ‘on’ is such as to make (1) true or false. Since borderline cases are cases in which there is no such thing as ‘the’ correct interpretation, there is no such thing as the interpretation that makes (1) true or false. So there is nothing to be ignorant of. ⁶ ( T3) provides an argument against the hypothesis—whose plausibility is called into question in §2—that borderline cases are cases in which more than one thing is said. As it is shown in Andjelkovic and Williamson 2000, 225–6, ( T3) entails a principle of uniformity to the effect that everything said by a sentence on a certain occasion has the same truth value: if ‘p’ says that pα and ‘p’ says that pβ , then pα if and only if pβ . For suppose that there are two interpretations α and β such that ‘p’ says both that pα and that pβ . By instances of ( T3), this yields that ‘p’ is true if and only if pα and that ‘p’ is true if and only if pβ . It follows that pα if and only if pβ . The principle of uniformity rules out the hypothesis that there are cases in which ‘p’ is uttered, and there are two interpretations α and β such that ‘p’ says both that pα and that pβ but α and β yield opposite truth values.
Saying More (or Less) Than One Thing
297
This provides a response to an argument that is a mainstay in Williamson’s defence of epistemicism. The argument goes as follows. Suppose that ‘A is tall’ is uttered and we are in a borderline case. If one accepts a plausible unknowability principle for borderline cases, one gets that we don’t know whether A is tall. But if one assumes classical logic and the disquotational principle, one also gets that ‘A is tall’ is either true or false. This means that there is something to be known: ‘the speaker’s ignorance has an object’.⁷ The response is that it must not be taken for granted that classical logic and the disquotational principle entail that ‘A is tall’ is either true or false simpliciter. On the assumption that truth and falsity apply to sentences relative to interpretations, bivalence can be shown to hold relative to interpretations, in accordance with classical logic and the disquotational principle. Therefore, what one gets is that, in any interpretation, ‘A is tall’ is either true or false. This is consistent with the hypothesis that the speaker’s ignorance has no object in the case considered. For in that case there is no such thing as the correct interpretation of ‘A is tall’.⁸ 16.5 Showing that there is a coherent view that preserves classical logic and the disquotational principle without being epistemicist wouldn’t be enough if the view were implausible. For mere coherence does not justify acceptance. This is what Williamson seems to think of a view that looks pretty much like that outlined in the previous sections: At least one view does combine classical logic and ( T+) with the denial that vagueness is an epistemic phenomenon in a way that seems at any rate not formally inconsistent. This is the view that vague sentences do not say that anything is the case, in borderline cases if not elsewhere. Thus, if ‘b’ is a borderline case for ‘bald’, to say ‘b is bald’ or ‘b is not bald’ is to make an utterance without propositional content. One says nothing by uttering either sentence, so neither is true. Thus, the corresponding antecedents of ( T+), in ‘in c ‘‘b is bald’’ says that P’ and ‘in c ‘‘b is not bald’’ says that P’, fail whatever legitimate substitution is made for ‘P’, and ( T+) holds vacuously. For the same reason, ( T ) would have no relevant instance. Thus, there would be no relevant true sentence, and therefore nothing for speakers of the language to be ignorant of. [. . .] No attempt will be made here to argue against that extreme view. Many of the philosophers who wish to accept orthodoxy while denying that vagueness is an epistemic phenomenon are willing to instantiate ( T ) with vague sentences even in borderline cases. They allow that ‘b is bald’ is true if and only if b is bald; ‘b is bald’ says that b is bald, even if b is borderline for ‘bald’.⁹
Here ( T+) is a principle equivalent to ( T3), ( T ) is the disquotation schema, and ‘orthodoxy’ stands for the combination of classical logic with ( T ). The account outlined in the previous sections seems to fit the description. There is a sense in which ⁷ Williamson 1995, 174–5. ⁸ Obviously, here the assumption is that Williamson’s use of ‘borderline case’ does not differ from ours. ⁹ Williamson 1995, 173.
298
Andrea Iacona
one says nothing by uttering a sentence in a borderline case, namely, the truthconditional sense. In this sense it is correct to say that a principle equivalent to ( T+) holds vacuously, in that the antecedent of ( T3) is not satisfied. Why should the view be ‘extreme’? The alleged motivation is that an utterance of ‘b is bald’ in a borderline case would have no content. Consequently, the disquotation schema wouldn’t be instantiated, and we couldn’t say that ‘b is bald’ is true if and only if b is bald. But that isn’t so. It is plausible that an utterance of ‘b is bald’ in a borderline case has content. The obvious sense in which it has content is that some understanding of the sentence can rightfully be ascribed to the speaker. In other words, such a case differs from one in which, say, ‘He is there’ is written on the board to make a point of grammar. However, the view does not entail that ‘b is bald’ lacks content in that sense. For that sense is the intentional sense, and the view does not deny that something is said in the intentional sense.¹⁰ Similarly, it is plausible that ( T ) is instantiated with sentences uttered in borderline cases, as it is quite natural to say that ‘b is bald’ is true if and only if b is bald. But again, the view does not deny it. The disquotational principle can be phrased in terms of a schema—( T1)—that holds for sentences as they are actually understood, and these obviously include sentences uttered in borderline cases. So the view is not extreme, and perhaps some attempt should be made to argue against it. To appreciate its plausibility, it is important to distinguish it from another view that is indeed extreme, namely, that according to which only precise expressions have meaning, so whenever we utter a sentence containing vague expressions we say nothing at all.¹¹ What the account of borderline cases outlined requires is simply that that there are cases in which we say nothing sufficiently precise. This is compatible with there being cases—the normal ones—in which we say sufficiently precise things. For example, even if there are cases in which ‘on’ is understood in such a way as to say nothing sufficiently precise by uttering (1), most of the time ‘on’ is understood in ways that are sufficiently precise for the descriptive goals that guide our use of the sentences containing it. This is not to say that most of the time ‘on’ is understood in a completely precise way. The use of a predicate almost never involves complete specification of its extension. Rather, it involves a partial specification of it, in that the only part of extension that matters on each occasion concerns the objects that are salient on that occasion. In other words, the use of a predicate on a given occasion determines a set of valuations, the set of all the valuations that share the part specified. Thus if one uses (1) to describe a cat a and a mat b, one takes the extension of ‘on’ to include the pair . But there may be another pair , such that it is left indeterminate by that use whether belongs to the extension of ‘on’. Therefore, that use determines a set that includes at least two valuations: in one of them the extension ¹⁰ Garc´ıa-Carpintero 2007, draws a distinction that is analogous at least as far as Williamson’s appeal to intuitions is concerned. ¹¹ This is the view called ‘Nihilism’ in Williamson 1994, ch. 6. The same view is considered in McGee and McLaughlin 1995, 215.
Saying More (or Less) Than One Thing
299
of ‘on’ includes but not , in the other it includes both < a, b > and < c, d >. If we call completely specified a way of understanding a sentence that provides for all possible cases, then sufficient specification does not mean complete specification. A completely specified way of understanding a sentence amounts to a single valuation rather than to a set of valuations.
16.6 The foregoing sections show how borderline cases can be accounted for in accordance with the thesis that truth and falsity apply to sentences relative to interpretations. This section shows that the thesis provides a straightforward solution to the paradox of the sorites. In its original form, the sorites says that if the removal of one grain from a heap always leaves a heap, the successive removal of every grain still leaves a heap. The argument goes as follows: (2) 1000 grains make a heap (3) For every n, if n grains make a heap then n − 1 grains make a heap (4) 0 grains make a heap The inference from (2) and (3) to (4) seems legitimate, (2) and (3) seem acceptable, but (4) seems unacceptable. Therefore, it is natural to think that there is something wrong with the argument. To provide a solution to the paradox is to say what exactly is wrong. The thesis that truth and falsity apply to sentences relative to interpretations points to the following definitions of validity and soundness. An argument is valid if and only if, necessarily, every interpretation that makes its premises true also makes its conclusion true. The criterion involved is the classical one, that according to which validity is necessary truth preservation. But since the premises and conclusion of an argument can be true only relative to interpretations, truth can be necessarily preserved only relative to interpretations. Similarly, an argument is sound in an interpretation if and only if it is valid and all its premises are true in that interpretation. Again, the criterion involved is the classical one, that according to which a sound argument is a valid argument whose premises are true. But since the premises of an argument can be true only relative to interpretations, the argument can be sound only relative to interpretations. These two definitions tell us what is wrong with the sorites. Although the argument is valid, there is no interpretation that makes it sound. For there is no interpretation in which (3) is true. An interpretation of (3) is a way of understanding (3) relative to which truth or falsity can be ascribed to it. Since (3) is equivalent to a list of conditionals, the interpretation must allow ascription of truth or falsity to each of the conditionals in the list, hence it must allow ascription of truth or falsity to the antecedent and to the consequent of each of the conditionals in the list. This entails that, for each of the collections of grains featuring in the series that goes from 1000 to 0 grains,
300
Andrea Iacona
it has to be specified whether or not it is a heap. Therefore, an interpretation of (3) involves a way of understanding ‘heap’ according to which there is a cut-off point in the series, that is, there is a number n such that a collection of n grains belongs to the extension of ‘heap’ while a collection of n − 1 grains does not belong to it. The sorites draws its appeal from the fact that we normally use ‘heap’ without specifying its extension as required by (3). Thus, (2) seems acceptable in that ‘heap’ is normally understood in such a way that a collection of 1000 grains belongs to its extension. Similarly, (4) seems unacceptable in that ‘heap’ is normally understood in such a way that a collection of 0 grains does not belong to its extension. But such ways of understanding ‘heap’ do not involve complete specification of its extension. In particular, they do not involve a delimitation of that extension sensitive to differences of one grain. Normally, when we understand ‘heap’ in such a way that a collection of n grains belongs to its extension, we do not have in mind a specification which prescribes that a collection of n − 1 grains does not belong to it. Therefore, we are apt to exclude that a collection of n grains is a heap but a collection of n − 1 grains is not a heap. This is why (3) seems acceptable. In other words, what makes the existence of a cut-off point for ‘heap’ unwelcome is that we normally do not specify such a point. Whenever we use the word to describe a certain object, we take for granted that no cut-off point lurks in the vicinity of that object, namely, that relevantly similar objects may equally be described in the same way.¹²
16.7 The account of vagueness outlined in this chapter—call it the underspecification view —may be contrasted with two similar accounts that are well known. The first is standard supervaluationism. The underspecification view substantively differs from standard supervaluationism, as the latter does not contradict Williamson’s moral. The basic idea of supervaluationism is that the vagueness of natural language consists in its capacity in principle to be made precise in more than one way. Following this idea, the method of supervaluations is adopted in order to deal with sets of precisifications of the language. A supervaluation is an assignment of truth values based on a quantification over assignments of truth values relative to precisifications. A sentence is ‘supertrue’ if it is true on all precisifications, ‘superfalse’ if it is false on all precisifications, and neither otherwise. Supervaluationism identifies truth with supertruth. This is why it obeys Williamson’s moral. In the first place, supertruth does not conform to the disquotational principle. If a sentence is true on some precisifications and false on others, the biconditional obtained by plugging the sentence in the disquotation schema is not supertrue. In the second place, supertruth violates bivalence, hence ¹² Assuming that the use of ‘heap’ on a given occasion is guided by what is psychologically or conversationally salient on that occasion, one may say that in normal circumstances no cut-off point is located within the area of salience. This, however, is not quite the same thing as to say that a cut-off point exists but is located somewhere outside that area, as suggested in Fara 2000.
Saying More (or Less) Than One Thing
301
classical logic. Moreover, as it has been emphasized by its critics, supervaluationism fails to preserve some classical principles about consequence and rules of inference. This is due to the definition of validity as necessary preservation of supertruth, that is, the definition according to which an argument is valid just in case necessarily, if its premises are supertrue then its conclusion is also supertrue.¹³ The underspecification view shares with supervaluationism its basic idea. It is plausible to say that a vague language is a language that in principle can be made precise in more than one way. This is couched by the assumption that a sentence allows different admissible valuations. However, the underspecification view differs from supervaluationism in that it does not identify truth with supertruth. Truth in an interpretation conforms to the disquotational principle and does not violate bivalence.¹⁴ Moreover, the underspecification view has no problem with consequence and rules of inference, in that it does not involve a supervaluational definition of validity. The latter says that an argument is valid just in case, necessarily, if its premises are true on all precisifications then its conclusion is true on all precisifications, while the definition given in §6 says that an argument is valid just in case, necessarily, in all interpretations, if its premises are true then its conclusion is true. The difference between the two definitions may be seen as a difference in the scope of the quantification over precisifications or interpretations.¹⁵ The second account to be considered is the non-standard version of supervaluationism advocated by Van McGee and Brian McLaughlin, and then adopted by Cian Dorr. The version at issue is non-standard in that it does not identify truth with supertruth. McGee and McLaughlin claim that a distinction must be drawn between ‘truth’ and ‘definite truth’, where the former is defined in accordance with the disquotational principle and classical logic, while the latter is characterized by using a supervaluational model-theoretic apparatus. Given that this characterization of definite truth is not epistemic, their view is like the underspecification view—and unlike standard supervaluationism—in that it contradicts Williamson’s moral.¹⁶ But some significant differences remain. In the first place, it is not clear whether the actual understanding of a sentence plays some role in the determination of definite truth. McGee and McLaughlin say at a certain point that ‘the thoughts and practices of the speakers of the language, together with the non-linguistic facts, pick out a set of sentences as definitely true’.¹⁷ However, it is not clear whether this entails that what a speaker has in mind on a certain occasion may contribute to determine what ¹³ See Williamson 1994, 146–53. The problem concerns supervaluationism in its standard version. But other definitions may be adopted. See Keefe 2000 and Varzi 2007. ¹⁴ The simple fact that truth in an interpretation is defined in terms of a quantification over valuations does not mean that it is a form of supertruth, at least not in the sense that matters here. For the definition does not allow for a third status between truth and falsity. ¹⁵ Williamson 1994, 147–8, Keefe 2000, §3, and Varzi 2007 spell out this distinction. The two kinds of definitions are called ‘global’ and ‘local’ respectively. ¹⁶ McGee and McLaughlin 1995, Dorr 2003. Williamson 1997, however, questions that the characterization of definite truth in McGee and McLaughlin 1995 is not epistemic. ¹⁷ McGee and McLaughlin 1995, 227.
302
Andrea Iacona
is definitely true on that occasion. If it doesn’t, then in the case of the grandmother the view must be that the unclarity that affects the evaluation of (1) is independent of what the grandmother has in mind, although it may depend on the thoughts and behaviour of the linguistic community as a whole. This is to say that the unclarity at issue is due entirely to the fact that the linguistic meaning of ‘on’ and the position of the cat are as they are, contrary to what is assumed here.¹⁸ In the second place, even supposing that McGee and McLaughlin do take into account the actual understanding of a sentence, or that their view may be adjusted in such a way as to take it into account, a difference remains. When McGee and McLaughlin talk of truth and definite truth, they seem to have in mind what is said in the intentional sense, whereas here the bearer of truth and falsity is what is said in the truth-conditional sense. This is in part a matter of focus. The underspecification view could as well be phrased in terms of the intentional reading of ‘what is said’. Borderline cases could be described as cases in which a speaker asserts a unique thing that is not evaluable. This would not essentially differ from saying that borderline cases are cases in which the thing asserted is neither definitely true nor definitely false, given that ‘evaluable’ would mean ‘evaluable as true or false’. However, there is one crucial respect in which the difference is not simply a matter of focus. McGee and McLaughlin ascribe truth simpliciter to what is said in the intentional sense. This entails that the things we say in borderline cases are true or false simpliciter. By contrast, even if the underspecification view were phrased in terms of the intentional reading of ‘what is said’, such ascription would be ruled out. For its main thesis would be that the things we say are evaluable as true or false only relative to interpretations. Re f e re n c e s Andjelkovic M. and Williamson, T. (2000), ‘Truth, falsity, and borderline cases’, Philosophical Topics 28, 211–44. Dorr, C. (2003), ‘Vagueness without ignorance’, in Hawthorne, J. and Zimmerman, D., eds., Philosophical Perspectives 17, Blackwell, 83–114. Fara, D. G. (2000), ‘Shifting sands: an interest-relative theory of vagueness’, Philosophical Topics 28, 45–81. (Originally published under the name ‘‘Delia Graff ’’). Field, H. (1994), ‘Disquotational truth and factually defective discourse’, Philosophical Review 103, 405–52. Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. Garc´ıa-Carpintero, M. (2007), ‘Bivalence and what is said’, Dialectica 61, 167–90. Horwich, P. (1998), Truth, Oxford University Press, Oxford. Iacona, A. (2006), ‘True in a sense’, Grazer Philosophische Studien 72, 141–54. Keefe, R. (2000), ‘Supervaluationism and validity’, Philosophical Topics 28, 93–106. McGee, V. and McLaughlin, B. (1995), ‘Distinctions without a difference’, The Southern Journal of Philosophy 33 (Supplement), 203–51. Varzi, A. (2007), ‘Supervaluationism and its logics’, Mind 116, 633–75. ¹⁸ This is what they seem to think when they say things such as ‘Harry is bald’ is definitely true if Harry is ‘definitely bald’, 210. Similar considerations hold for Dorr 2003.
Saying More (or Less) Than One Thing
303
Williamson, T. (1994), Vagueness, Routledge, London. (1995), ‘Definiteness and knowability’, The Southern Journal of Philosophy 33 (Supplement), 171–91. (1997), ‘Imagination, stipulation and vagueness’ in Villanueva, E. (ed.), Philosophical Issues 8, Ridgeview, Atascadero (CA), 214–28.
17 Vagueness as Semantic Max K¨olbel
I shall argue that vagueness, understood as a semantic phenomenon, can be accommodated within standard semantics by assimilating it to contingency in standard modal semantics and suitably modifying the pragmatics. I claim that vagueness in natural language is not a defect and that accommodating it is therefore obligatory for semantic frameworks for natural languages. In section 17.2, I interpret the claim that vagueness is a semantic phenomenon as involving at least the claim that vague predicates do not determine an extension. I then outline three ways in which standard semantics can account for the failure of an expression to determine an extension, namely ambiguity, indexicality and relativity to circumstances of evaluation (e.g. contingency). I point out some problems with treating vagueness as a form of ambiguity or as a form of indexicality. Then I explain the view that vagueness is a form of relativity to circumstances of evaluation, and why such a view needs to provide an account of the normative significance of truth for assertion and belief. I show how this normative role is constrained by the two desiderata that we explain the seductiveness of sorites arguments and give an account of borderline cases. Finally I briefly consider higher-order vagueness and conclude by comparing the account given with other views of vagueness.
17.1
VAG U E N E S S I S N OT A D E F E C T
Many natural language predicates are vague in the sense that they seem subject to tolerance constraints and therefore generate sorites paradoxes. For example, the predicate ‘is rich’ is vague because it seems to be subject to the constraint that if someone is not rich then receiving a small amount of money such as one cent will not make that person rich. Thus a sorites paradox can be formulated as follows: (A) A person with possessions worth 0 Euros is not rich. This chapter was first presented at the Fifth Arch´e Vagueness Workshop, 18–19 November 2005, and subsequently at a few other occasions. I would like to thank the participants for their comments, especially Hartry Field, Manuel Garc´ıa-Carpintero, Mark Sainsbury and Achille Varzi.
304
Vagueness as Semantic
305
(B) If a person with possessions worth n Euros is not rich, then a person with possessions worth n + 0.01 Euros is not rich either.¹ (C) A person with possessions worth 100 million Euros is not rich. (C) seems to follow from (A) and (B), but while (A) and (B) seem clearly true, (C) seems clearly not true. This phenomenon is widespread. Countless natural language predicates are vague in this sense. The vagueness of these predicates does not seem to be an impediment to their usefulness in communication. Similarly, the concepts expressed by vague predicates do not seem to create any problems for our thought. The vagueness of natural language predicates and the concepts they express is therefore not some deficiency, shortfall or malfunction. Vagueness is perfectly normal. If vagueness is normal, then semantic frameworks for natural languages, ought to be able to accommodate it. If standard frameworks cannot accommodate vagueness, then they need to be abandoned in favour of new or modified frameworks that do accommodate vagueness. In the interest of continuity, it is therefore desirable to explore if and how standard semantics can make room for vagueness. As we shall see, there are several ways in which room can be made for vagueness within standard semantics, some better than others.
17.2
VAG U E N E S S A S E X T E N S I O N A L I N D E T E R M I N AC Y
There is a minority of philosophers, the epistemicists, who hold that vagueness is not a semantic problem, but rather reflects our inability to know the exact borderlines of the extensions of the predicates (and concepts) we use. On this view, premise (B) in the above sorites is simply false. There is a truth of the form (D) A person with possessions worth n Euros is not rich and a person with possessions worth n + 0.01 Euros is rich. But we cannot know that truth because of general principles concerning knowledge.² It is this fact that explains why (B), despite its falsity, is so attractive. According to epistemicism, then, vagueness is an epistemic, and not a semantic phenomenon. The meaning of vague as well as non-vague predicates determines for each object whether it is in that predicate’s extension or not. The majority, however, finds the epistemic view incredible, in large part because it remains mysterious how the precise extensions of vague predicates are determined. The majority instead believes that vagueness is a semantic phenomenon, i.e. that the meanings of vague predicates fail to determine exact extensions. I will not provide any reasons to favour semantic views over epistemicism. I shall merely assume that ¹ There are different ways of formalizing ‘a person’ in (B), see Pagin 2009. These differences will not matter for the current discussion. ² See Williamson 1994 and Sorensen 1988.
306
Max K¨olbel
the semantic view is correct. Starting from that assumption, I will make a case for a certain treatment of vagueness, understood as semantic, within standard semantic frameworks. Thus the conclusion of this chapter is a conditional one. Some of those who believe that vagueness is a semantic phenomenon may think that the failure of vague predicates to determine an extension is a kind of imperfection that it would be better to reform away. I have already argued above that vagueness should not be treated as an imperfection because it is a normal, widespread and unproblematic aspect of language use. Others may infer from the extensional indeterminacy of vague predicates that we must modify the semantic framework, e.g. by allowing three truth values. But before going down that path, we ought to examine the resources of the existing standard framework to accommodate the kind of extensional indeterminacy characteristic of vague predicates.
17.3
S TA N D A R D S E M A N T I C S F O R C O N T E X T- S E N S I T I V E L A N G UAG E S W I T H I N T E N S I O N A L O PE R ATO R S
The semantic framework I shall be using can be called ‘double-index semantics for context-sensitive languages’.³ According to this framework, the meanings of natural language sentences determine characters, and these are functions from contexts of use to contents (propositions). Contents in turn are (or determine) functions from circumstances of evaluation to truth values (see diagram below). In the language of intension and extension: the meaning of each expression determines, in a context of use, an intension, and an intension determines, in each circumstance of evaluation, an extension, as shown in the diagram. For example, the English sentence ‘I am hungry now.’ has a character that determines different contents in different contexts of utterance. Utterances of the sentence can express propositions about different people and different times, depending on who utters it when. Now, consider one of these propositional contents, about some person John and some time t. This content, i.e. the proposition that John is hungry at t, has a truth value. Which truth value it has will depend on how things are with John at t, or, in other words, it will depend on the circumstances of evaluation. If John goes long enough without eating before t and is otherwise normal, then the proposition is true. If John has a large breakfast just before t, then it is not true. Thus the proposition is true in some possible worlds and not in others. The extension of the concept is hungry at t varies from one possible world to another. To summarize: the determination of the truth value of an utterance is generally a matter of two stages: the meaning or character of the expression determines, ³ This type of framework is familiar from Kaplan (1977). Lewis (1980) argues against Kaplan’s two-stage approach and proposes a competing one-stage theory, which does not postulate contents expressed by sentences in contexts. I shall assume that Kaplan is right to introduce double-indexing, but that otherwise the difference between Lewis’s index theory and Kaplan’s two-stage theory is not directly relevant to present purposes. For discussion see Recanati 2007.
Vagueness as Semantic Sentence (type)
Meaning (character)
Intension (content, proposition)
Context of use
Circumstances of evaluation
307 Extension (truthvalue)
for each context of utterance, a content, and then that content determines, for each circumstance of evaluation, an extension. Such variation in extension with different circumstances is a feature of contents that is exploited in the standard semantics of modal operators. It is worth adding that natural language sentences, in addition to their characters, also exhibit force indicators. These force indicators indicate the communicative function of utterances of the sentence.⁴ Thus, the proposition that John is hungry at t could be expressed, for example, with assertoric force, or as a question. Even though semanticists are traditionally more concerned with the truth conditional content of utterances, their theories must ultimately connect up with the theory of speech acts. One particular link that will play a role below is the normative role the extension of an utterance has for assertoric speech acts. Usually it is thought that assertion in some sense aims at truth, so that asserting an untrue proposition constitutes some kind of mistake. A semantic theory for a language in some sense⁵ represents part of the competence of users of that language. One aspect of linguistic competence, however, is usually treated as pre-semantic: the ability to resolve ambiguities. The input from which a semantic theory can be used to derive intensions (and illocutionary forces) of utterances, is thought of as unambiguous syntactic forms. Which unambiguous syntactic form is expressed by the utterance of an ambiguous sentence (such as ‘The bill was huge.’) is something that will again be resolved by recourse to the linguistic and nonlinguistic context of the utterance. Despite some similarities, there is a difference between disambiguation and assignment of content to indexicals.⁶ Disambiguation is usually treated as pre-semantic, while assignment of contents to indexical elements is treated as part of semantics. This is why the diagram above does not represent the determination of character or meaning of an expression type as yet another semantic function, in addition to the functions from context to content and from circumstance to truth value. ⁴ See K¨olbel forthcoming for more detailed reflections on force indicators and assertoric force. ⁵ In what sense is a notoriously difficult question which I shall not broach here. ⁶ It is not easy to justify the relegation of disambiguation to the pre-semantic realm, or the strict separation of the two phenomena. There are great similarities between some of the phenomena that are standardly treated as cases of ambiguity and those that are standardly treated as cases of indexicality. Names, such as ‘John’ are often treated as ambiguous, and such treatment may explain why in some contexts one can say literally and coherently ‘John is home, but John isn’t.’. Some however, will treat this as evidence for the indexical character of personal proper names—compare ‘He is French and he isn’t.’ or ‘Now the lights are on and now they are not.’, when demonstrating different people or times at the moment of uttering the different occurrences of ‘he’ and of ‘now’.
308
Max K¨olbel 17.4
T H R E E S TA N D A R D S O U RC E S F O R E X T E N S I O N A L I N D E T E R M I N AC Y
As already noted, on the semantic view of vagueness, vague predicates fail to determine for each object whether the predicate applies to it. For short, vague predicates are extensionally indeterminate. Diagnoses of this sort are not at all alien to the semanticist. There are three ways in which a semantically non-deficient predicate (thought of as an expression type) can fail to determine an extension: the predicate may be ambiguous, indexical or its extension may vary with the circumstances of evaluation. Let me briefly review these three sources of extensional indeterminacy. The first source is ambiguity: ambiguous expressions fail to determine an extension. For example, the word ‘coach’ has several distinct and unrelated meanings in English. Each meaning determines a different extension. On one meaning, some people are, some people are not in the extension of ‘coach’. On the other meaning, no person is in the extension.⁷ When ambiguous expressions are used in communication, confusion is avoided because the context of use allows communicators correctly to disambiguate, i.e. to focus on one of the meanings of the ambiguous expression and to ignore the others. For example, if someone utters ‘The coach is waiting.’, then successful communication seems to require correct disambiguation. Correct disambiguation would here seem to involve at least that speaker and audience disambiguate in the same way. The second source is indexicality. A predicate may fail by itself to determine an extension because its character is a non-constant function. For example the predicate ‘is my uncle’ expresses different properties when used by different speakers. When used by you it expresses a (relational) property instantiated by your uncles (if any), and when expressed by me it expresses a property instantiated my uncles. The third source is sensitivity to circumstances of evaluation. A predicate may fail to determine an extension because the content it expresses is a non-constant function from circumstances of evaluation to extensions. The best known and least controversial type of example is that of predicates expressing contingent properties. The extension of the property of being a photographer varies according to what actually happens. Modotti is in the extension the property has in some circumstances of evaluation, including the actual circumstance. But had she met different people in her youth, she would not have become a photographer, and would consequently not have been in the extension of the property. Sensitivity to circumstances of evaluation is best known in the case of contingency, and often circumstances of evaluation are interpreted merely as possible worlds. However, the framework does in principle allow further parameters in the circumstances. For example, one circumstantial parameter that ⁷ Some prefer to use ‘word’ (‘expression’, ‘predicate’) in such a way that by definition each word has only one meaning. Thus instead of having one word with several meanings, we have several words that are phonetically and orthographically indistinguishable. On this terminology, it is not words, but, for example, phonetic types that are ambiguous. This is just a terminological variation.
Vagueness as Semantic
309
has been much discussed is the time parameter that temporalists (including Kaplan himself ) want to add.⁸ The standard framework thus allows for three sources of extensional indeterminacy, and if vagueness is a form of extensional indeterminacy, each of these is a potential source of vagueness. I shall discuss the three sources in turn.
17.5
VAG U E N E S S A S A M B I G U I T Y
Suppose we want to account for the phenomena of vagueness by treating vagueness as a special form of ambiguity. This is how the story goes: a vague predicate, such as ‘rich’ has many meanings, in fact countless meanings, each of which draws a different precise boundary between the rich and the non-rich. This explains why the predicate by itself does not determine an extension. Thus, the semantics is standard, but we have an especially complicated pre-semantics.⁹ There is an immediate worry. When successfully communicating with ambiguous expressions, communicators are generally required to disambiguate, and to do so correctly. This means, at least, that in successfully interpreting an ambiguous utterance, speaker and audience have in mind the same of the candidate meanings. But there does not seem to be an analogous requirement of disambiguation in the case of vague communication. It seems wrong to say that when I hear ‘Anita is rich.’ I need to select one of many precise meanings of ‘rich’, and then to use that meaning (and only that meaning) in interpretation. There are answers to this worry. The ambiguity theorist might argue that communication with ambiguous expressions does not always require disambiguation. Consider an uncontroversially ambiguous sentence: ‘The coach is waiting.’ True, understanding an utterance of this sentence will often require correct disambiguation. However, there may be occasions, when no disambiguation is required. Suppose the coach of a second division football club doubles as the team’s chauffeur. Everyone knows that after the match, the team’s bus is waiting iff the team’s trainer is waiting. For it’s the team’s trainer who conducts the bus and if the trainer is waiting after a match, he is always waiting in the bus, the engine running. Given this background knowledge, neither speaker nor audience may need to disambiguate ‘coach’ in an utterance of ‘The coach is waiting.’. Another example: in order successfully to argue about whether a chemical purifier factory ought to be built down the road, we may not need to disambiguate ‘chemical purifier factory’. The differences ⁸ For an overview of other forms of variation of extension with circumstances of evaluation, see K¨olbel 2008. ⁹ The actual position closest to this view is that defended by Linda Burns in her 1991. Her view in turn is inspired by some remarks in Lewis 1975. Kit Fine’s classic exposition of supervaluationism (Fine 1975) also has some affinity, as Fine calls vagueness ‘ambiguity on a grand and systematic scale’. However, supervaluationism is not usually read as a form of the ambiguity view presented in this section.
310
Max K¨olbel
between the various possible meanings may be irrelevant to our communicative purposes.¹⁰ No doubt, the ambiguity theorist of vagueness could devise a systematic account of what is involved in communicating ambiguously without disambiguation, and she could apply this account to the special case of vague predicates. But whatever that account is, it faces some further difficulties. On the standard account sketched above, it is assumed that any non-defective utterance expresses a unique content or proposition. So, what, on the view that vagueness is ambiguity, is the proposition asserted by an utterance of ‘Anita is rich.’? There are only two ways the ambiguity theorist can go. Either she retains the principle that non-defective utterances express a unique proposition, or she does not. Let’s consider the first case first. The predicate ‘rich’ has countless precise meanings. In a given utterance of ‘Anita is rich.’, each of these meanings corresponds to one non-vague proposition concerning Anita. Let’s call these propositions the ‘candidate propositions’. The ambiguity theorist’s account will devise a way that allows us to derive the proposition expressed by an utterance of ‘Anita is rich.’ in some way from the set of candidate propositions. For example, she might say that the context of use determines a certain range of relevant candidate propositions, and that the proposition expressed is a conjunction, or perhaps a disjunction, of the relevant candidate propositions. Whatever the merit of these proposals, it is clear that they are proposals that move away from the idea that ‘rich’ is ambiguous. For what the so-called ‘ambiguity theorist’ is now claiming is that the content expressed by utterances of ‘rich’ (not the meaning of ‘rich’) is determined systematically by the many meanings of ‘rich’ and the context of use. So, while there may be a viable theory in the neighbourhood, it is highly misleading to describe it as a theory according to which vagueness is a form of ambiguity. The resulting theory will belong to the group of views that treat vague predicates as varying in intension with the context of use, i.e. as being indexical at least in a wide sense. These views will be considered in the next section. Now consider the second case. Suppose the ambiguity theorist wants to give up the principle that each non-defective utterance of a declarative sentence expresses a unique proposition. She might say, for example, that in using a sentence like ‘Anita is rich.’, the utterer does not determinately assert any single proposition but indeterminately asserts a range of the candidate propositions. She might abandon the idea that assertion is a propositional act that relates a person to a single proposition and ¹⁰ This example is from Sainsbury 2001. Sainsbury uses it to support the view that certain unspecificities in compound expressions are due to very unspecific meanings, rather than to hidden indexical variables, as claimed by Stanley (2000), or to ambiguity in the pre-semantic sense, as defended by Travis (1985, 1996). Unlike Sainsbury, I am here taking it for granted that ‘chemical purifier factory’ is ambiguous. For whatever one may think about the unspecificity in the compounding operation (i.e. ‘purifier factory’ can be read as a factory that makes purifiers or as a factory that employs purifiers in making something), the phrase also exhibits a classic scope ambiguity (‘chemical’ can qualify ‘purifier’ or ‘factory’). Thus, even if Sainsbury’s view about the first unspecificity were correct, the example can still illustrate my thesis that disambiguation is not always necessary.
Vagueness as Semantic
311
instead think of assertion as an act relating the asserter to a range of propositions.¹¹ I cannot see any principled reason why such proposals could not be made to work. They would need to be complemented with an appropriate theory of assertion and indeterminate assertion, and similarly with an account of the belief states we express by vague utterances (e.g. an account of indeterminately believing a range of propositions, or belief as a relation towards a range of propositions). This may ultimately yield an explanation of the phenomena of vagueness. But any such proposal does require some major modifications to the standard framework. If, on the ambiguity view, utterances of vague sentences express multiple propositions then we are in fact dealing with a phenomenon quite different from ordinary ambiguity. I shall not here examine the ambiguity view further because the approach discussed in section 17.7 below seems to me to involve less of a departure from the standard framework.
17.6
VAG U E N E S S A S I N D E X I C A L I T Y
Let us now consider the view that vagueness is a special form of indexicality. Indexical expressions do not by themselves determine an extension because they determine a content (intension) only in a context of use (their character is not constant). Thus, the predicate ‘is my uncle’ does not have a specific content until it is used by someone in a suitable context of utterance. At that point, it expresses a specific content (namely the property of being that person’s uncle), and determines an extension in any possible world. If vague predicates are indexical, then the content expressed by them similarly varies with context. Which property is expressed by ‘is rich’ will vary from one context of utterance to another. But what are these variable contents, and how does the context of utterance determine which of these variable contents is expressed? Consider ‘rich’. The extension of ‘rich’ clearly varies with a comparison class. In a context where we are talking about the wealth of sub-Saharan refugees, the threshold for membership in the extension of ‘rich’ will be much lower than in a context where we are discussing the comparative wealth of European royalty. Richness for a refugee and richness for a royal are two different properties with different extensions. The same goes for many vague adjectives: ‘tall’, ‘small’, ‘poor’, ‘bald’, ‘young’, etc. However, this form of context-sensitivity, as obvious as it is, is not particularly useful in accounting for the phenomena of vagueness. For the tolerance constraints characteristic of vagueness, and responsible for sorites paradoxes, govern ‘rich for a sub-Saharan refugee’ just as much as they do ‘rich’. ¹¹ Soames 2003, as well as Cappelen and Lepore 2005 and Cappelen 2008 distinguish the proposition semantically expressed from the proposition(s) asserted by an utterance. I will not discuss this complication here for reasons of clarity of exposition. These theories face the challenge of specifying how the proposition semantically expressed constrains the proposition(s) asserted. If semantic properties of expressions are to be determined by their use (and use is in turn to be constrained by semantic properties) then this challenge cannot be ignored, as it is deliberately by Cappelen 2008 (see also Pagin and Pelletier 2007).
312
Max K¨olbel
Those who invoke context-relativity in trying to account for vagueness¹² usually appeal to forms of context-relativity that explain why we are unable to find the exact border between the members and the non-members of the extension of a vague predicate. The idea is that the extension of a vague predicate depends on the context in such a way that the border between the members and the non-members is never where we are currently looking. Thus, in a context where we are considering two people whose level of wealth is only marginally different, our very act of considering them ensures that the boundary between the rich and the non-rich (at that context) does not separate these two people.¹³ Thus, the borderline is never where we are looking, and each of the instances of the generalized conditional premise of the sorites ((B) above) will be true at every context. However, it does not follow that the generalized conditional premise is therefore also true at every context. With the right account of a context, it will not be true in any context.¹⁴ Whatever the details of such an account, it will crucially claim that when we make utterances concerning the F -ness of objects located at different parts of a sorites series for F -ness, then the contextrelativity of ‘F ’ will be such that the contexts of these utterances differ significantly, i.e. they differ in a way that triggers a change in ‘F ’ ’s content and therefore extension. Let’s introduce a neutral term for that feature of an utterance context that allegedly determines the extension of vague predicates. Let’s say that each vague predicate has a determinate extension only relative to a standard, and that one aspect of each relevant context of utterance is precisely such a standard. In other words, each context of utterance determines a precise standard of richness, poverty, youth, baldness etc. I shall leave open how exactly an ordinary context of utterance determines one such standard—my observations will be neutral as to the exact implementation of the indexical approach. The indexical approach faces several problems. Like epistemicism, it holds that vague predicates determine a precise extension at every context of use, but that we are ignorant of it.¹⁵ While the epistemicist explains our ignorance of the extension of ¹² Here I have in mind primarily Kamp 1981, Raffman 1994, 1996, Soames 1999, 2002, Fara 2000 and Shapiro 2005. I am not claiming that all these writers regard vague predicates as indexicals, just that they invoke context-dependence of some sort in resolving the paradox. In the last section I will say more about this question. For now I just want to consider the position (no matter whether actually held by anyone) that vague predicates are indexical, and that their indexicality is the source of the characteristic extensional indeterminacy. ¹³ Raffman’s 1994 constraint (IP*) is that two adjacent members of a sorites series that are being judged at a context must be both in the extension at that context or neither. Soames requires that if two objects are sufficiently and relevantly similar, and one of them is salient at a context, then either both or neither are in the predicate’s extension at that context. Fara 2000 speaks of a ‘similarity constraint’, which in her case requires that any two things that are relevantly and sufficiently similar and whose similarity is salient at a context, are either both in the extension at that context or neither is. Kamp’s (1981) treatment is different in that his context-dependence involves the semantics for the conditional, but this is beside the point here. ¹⁴ Kamp 1981 is most thorough on this point, by offering a formal theory of contexts that rules out a context in which an entire sorites series is salient as incoherent. Raffman’s account is reminiscent of Kamp’s but gives a psychological explanation of why usually, before we regard an entire sorites series, our inner context switches. ¹⁵ This means that it shares a problem with epistemicism: if the meaning, and a fortiori the extension, of an expression is ultimately determined by the way we use it, then it seems mysterious how vague predicates in context should have acquired these meanings (extensions).
Vagueness as Semantic
313
vague predicates with certain general limitations on knowledge, the indexical contextualist attempts to explain this with the way in which vague predicates vary in extension with context, which is governed by some similarity constraint. The moment we consider two sufficiently similar objects (or consider their similarity), we are forced to conclude that they are either both inside or both outside the extension (compare Fara 2000, 59). However, this explanation is not satisfactory. Consider another context-dependent predicate: the predicate ‘is an object I am currently not considering’. Clearly, whenever I consider whether some particular object o is in the extension of this predicate, my very act of considering o causes o to be excluded from the extension. This does not mean that the extension of the predicate is empty. The way to convince yourself of that is to think of some particular utterance of ‘There is an object I am currently not considering.’ in some context c. After c, you can retrospectively, consider which objects where in the extension of our predicate in c. Equipped with a sufficiently detailed and reliable introspective memory, you could then determine for any object whether it is in that extension. The same goes for vague predicates, if the indexical contextualist story is right. It should be possible to consider a particular utterance of ‘Bob is rich.’, made in context c1, and retrospectively to consider for any object whether the extension of ‘is rich’ in c1 includes that object. There is no danger that in so considering we change the context, because we are thinking about the extension of the predicate in c1, a context that can no longer be changed. However, it seems utterly mysterious how we should go about it. Holding the context fixed does not make the limits of the extension in any way less elusive.¹⁶ The second and third problem for the indexical approach is that vague predicates do not behave like typical indexicals in certain respects. It is important to be clear from the start that these two points can merely show that vague predicates are not typical indexicals. They leave open whether vague predicates are a special or unusual kind of indexical (where indexicality is understood to be the phenomenon of variation of content with context of use). The second problem concerns speech reports. In general, when reporting indexical speech one must adjust the words used in the report to any relevant changes in the context. For example, if reporting an utterance of ‘You are a fool.’, one can use the same words in the report as originally uttered only if the addressee of the context of the report is the same as the addressee of reported utterance. If Otto addresses Peter saying ‘You are a fool.’, then I can report his utterance with the words ‘Otto said you were a fool.’ only if in making the report I am also addressing Peter. Otherwise I would have to adjust and say something like ‘Otto said Peter was a fool.’ Thus reporting indexical speech follows the following general rule: ¹⁶ A related problem is that the indexical contextualist does not have an appropriate way to characterize borderline cases. For, if she is to remain within the standard classical framework, a vague predicate in a context of use determines a precise concept. So the indexical contextualist will have to say that borderline cases are objects that either are or are not in the extension of the vague concept, but that we simply do not know which. But why this should be so is mysterious. Perhaps the indexical contextualist would at this point show her true colours and adopt one of the epistemicists’ explanations.
314
Max K¨olbel
(SR) If a sentence s is indexical in such way that the content expressed by s varies with contextual feature f , and utterance u is an utterance by S of s in context c1 , and context c1 differs from context c2 in feature f in a way that would alter the content of s in c2 as compared to its content in c1 , then an utterance in context c2 of ‘S said that s’ is incorrect. Now, clearly, if vague predicates are indexical in the way described above (i.e. if their extension varies with the context of use in such a way that a similarity constraint is met), then they do not comply with this rule. Consider a speaker who is being ‘force-marched’ from left to right through a sorites series of 50 coloured patches which range from paradigmatic red at the left to paradigmatic orange to the right. Suppose the speaker at some point utters ‘Patch 25 is red.’. Later on, she is forcemarched through the same series from right to left. This time she utters ‘Patch 25 is not red.’. Remembering her earlier utterance, she might add ‘but a while ago I said that it is red’. I believe that this would be a correct report. However, if the indexical contextualist theory of vagueness is correct, then (SR) predicts that this is not a correct report. The conclusion is that if vague predicates are indexical in the suggested way, then they are reported in an exceptional way at relevantly changed contexts. The third problem is related. When we evaluate the correctness of indexical utterances retrospectively, we evaluate them with respect to the original context of utterance. Thus, if I utter ‘I am hungry.’ before lunch and then consider the correctness of my utterance after lunch, I will evaluate what I said before lunch as correct just if I believe that I was hungry then. Thus, we would expect that if the above-mentioned speaker re-evaluates her earlier verdict on patch 25 when considering it in the later context, she should without hesitation evaluate the earlier utterance as correct. However, it would seem decidedly odd, if she said ‘Patch 25 is not red. A while ago I said that it was red, and what I said is true.’. Again, the conclusion is that if vague predicates are indexical in the way proposed then these indexicals behave unexpectedly when utterances of them are evaluated at a relevantly changed context. These three problems do not conclusively refute the indexical approach to vagueness. However, I believe that the first represents a serious challenge, while the second and third show that at the very least we are dealing with indexicals of an exceptional variety. This should be sufficient motivation for exploring the third potential source of the extensional indeterminacy of vague predicates.
17.7
R E L AT I V I T Y TO C I RC U M S TA N C E S O F EVA LUAT I O N
The third potential source of the extensional indeterminacy of vague predicates is a variation of extensions with circumstances of evaluation. In order to illustrate this possibility, I shall first briefly discuss another, better known case in which such variability has been debated, namely the case of tensed sentences.
Vagueness as Semantic
315
17.7.1 Eternalism and temporalism Eternalism and temporalism are two alternative ways of construing the semantics of tensed sentences, such as ‘MK is hungry.’ or ‘The root canal treatment is over.’. Eternalism treats tensed sentences as expressing different propositions with eternal truth value at different times of use. Thus, the indeterminacy of truth value exhibited by tensed sentences is treated as indexical, and as being resolved by placing the sentence in a context of use. Temporalism on the other hand treats tensed sentences (qua tensed sentences) as non-indexical.¹⁷ Tensed sentences (qua tensed sentences) express the same proposition in all contexts of use. However, these propositions are so-called ‘tensed propositions’. Tensed propositions do not have absolute truth values. Like contingent propositions, they vary in truth value with circumstances of evaluation. The difference is that while contingent propositions are sensitive merely to a possible world parameter in the circumstances, tensed propositions are sensitive to a time parameter in the circumstances. Thus the sentence ‘MK is hungry.’ expresses the tensed proposition that MK is hungry, and this proposition changes its truth value regularly. Before lunch, on most days, it is true at the actual world, and after lunch, on most days it is false (when events take their normal course). One advantage of, and motivation for, temporalism is the fact that it can accommodate certain intuitions about propositions conceived of as the objects of belief, assertion, etc. For example, believing the tensed proposition expressed by the sentence ‘My root canal treatment is over.’, will typically cause relief. However, believing a corresponding eternal proposition (the proposition that my root canal is over 12 March 2006 at noon) does not, by itself warrant any relief, for one might believe that proposition truly even before the 12 March at noon, and relief would be out of place then.¹⁸ The disadvantage of temporalism may be that there are also different intuitions regarding the objects of assertion and belief, which it does not accommodate. For example, suppose I sincerely use the sentence ‘Clinton is US president.’ twice, once in 1996 and once in 2000. Clearly, what I have asserted first (the belief I expressed) is true, and what I have asserted on the second occasion false. So it would seem that the objects of assertion cannot be the same on both occasions.¹⁹ Put in this form, the argument can be resisted: for why should it follow from a difference in truth value that the propositions expressed are also different? If tensed propositions have different truth values at different times then we can continue to maintain that both utterances expressed the same tensed proposition. However, there does seem to be a robust intuition that in some sense the object of belief changes. The most sensible reaction would seem to be the ecumenical one of allowing both tensed and eternal propositions, and to say that I expressed the same tensed, but different eternal propositions on the two occasions.²⁰ ¹⁷ See, e.g. Kaplan 1977 and Prior 1967. ¹⁸ See Recanati 2007, Book I, for a detailed defence of temporalism. ¹⁹ Compare Richard 1981. ²⁰ Temporalism and eternalism are also different in their treatment of tenses and other temporal qualifications, such as ‘sometimes’. Thus, temporalists construe tenses and temporal qualifications as operators while eternalists treat them as quantifiers. While the quantifier treatment is clearly more
316
Max K¨olbel
Whatever we may think about the debate between temporalists and eternalists, I here merely want to draw attention to the way in which temporalists need to refine the way they think about the normative significance of propositional truth. A tensed proposition, such as the proposition that my root canal treatment is over, does not have an absolute truth value, but varies its truth value not just from possible world to possible world but also from time to time. This raises the question (also asked by Evans 1979): truth at which time is relevant for the correctness of an assertion, or for the correctness of a belief ? Suppose at t1 I assert (and believe) the tensed proposition p, that the root canal is over. At t2 (later than t1) we can ask: is the assertion (and the belief ) correct? The question is unclear. We could look at the truth value of p at t1, we could look at the truth value of p at t2, or we could look at the truth value of p at any other time or even range of times. In principle all these manners of evaluation could be interesting and legitimate. But it is obvious that only one manner of evaluation is relevant if we want to test our semantics against language use: the truth value of p at t1, the time at which the assertion was made (or at which the belief occurred). Temporalism in the semantics of natural language makes sense only on the background of certain assumptions of how truth at a circumstance is relevant for the evaluation of assertions (or beliefs) as correct. The obvious principle expressing this relevance is: ( TP) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at the time of c.
17.7.2
Vague propositions
Let us turn to the third potential source of extensional indeterminacy, according to which the extensions of vague predicates vary not with the context of use, but with the circumstances of evaluation. The proposal is to mimic the temporalist by adding another parameter to the circumstances of evaluation, and saying that the propositions expressed by sentences containing vague predicates vary in truth value with this parameter. They vary in this way because the vague predicates used to express these propositions express vague concepts which themselves vary their extensions with this parameter. What are the values of the circumstantial parameter with which the truth values (extensions) of vague propositions (predicates) vary? They are ways of making vague predicates precise consistently with clear cases and with certain a priori principles, i.e. functions that assign to the vague concepts expressed by vague predicates precise extensions. We could call these functions ‘reasonable standards of precisification’. But I will here rely on the terminology familiar from supervaluationism and call them ‘(admissible) sharpenings’. On this view, then, vague predicates express properties that are extensionally sensitive to a sharpening component in the circumstances of evaluation (just as contingent properties are extensionally sensitive to a possible world popular among semanticists, there does not seem to be any compelling reason for this preference (for discussion see King 2003, Recanati 2007).
Vagueness as Semantic
317
component in the circumstances of evaluation). Consequently sentences containing vague predicates will sometimes express vague propositions, i.e. propositions that are sensitive in their truth value to a sharpening parameter in the circumstances of evaluation.²¹ Just as in the case of tensed propositions, in order now to make sense of vague propositions as the objects of assertions or beliefs, we again need to spell out the normative significance of propositional truth. We need to know which sharpening or sharpenings are relevant for the evaluation as correct of an assertion or belief. In the case of tensed propositions, it was plausible to say that for each assertion (or belief ) there was just one time for evaluation, namely the time at which the assertion (the belief ) occurs. In the case of vague propositions, the situation will be more complicated. What we need is an appropriate completion of the following schematic principle: (VP) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at . . . I would like to broach this task by first distinguishing in the abstract two dimensions in which completions of (VP) can vary, and then argue for each of these dimensions what our completion should look like. In principle, the completion of (VP) could either (a) privilege a unique sharpening or (b) privilege a range of several sharpenings and it could either (1) privilege the same sharpening(s) in each situation of assertion/belief or (2) privilege a different sharpening (or different sharpenings) in different situations of assertion/belief. In the next two sections I shall explain why I advocate a b-2 completion of (VP). 17.8
L E A R N I N G F RO M C O N T E X T UA L I S TS : T H E S O R I T E S
I believe that contextualists about vagueness teach us how option (2) helps us avoid sorites paradoxes. Let’s consider a non-inductive version of the sorites of section 17.1. Consider a sorites series of people, P0 , P1 , P2 , . . . P1,000,000 , such that P0 has ¤0, P1 has ¤1, P2 , has ¤2 and so on, each Pi having exactly i Euros. ²¹ For simplicity, I am ignoring the indexicality of many vague predicates, such as their sensitivity to a contextually salient comparison class. Thus I am strictly speaking considering only a subclass of vague predicates, namely those whose character is constant—such as, perhaps, ‘is tall for a British male born between 1975 and 1980’. Such predicates are no doubt still vague.
318 (P0) (P1) (C1) (P2) (C2) . . . (P1,000,000) (C1,000,000)
Max K¨olbel P0 is not rich. If P0 is not rich then P1 is not rich. P1 is not rich. If P1 is not rich then P2 is not rich. P2 is not rich.
If P999,999 is not rich then P1,000,000 , is not rich. P1,000,000 , is not rich.
Now, on an indexical contextualist approach, each of the constituent modus ponens arguments is valid in the sense that if its premises are true in a context c, then the conclusion is also true in c. However, there is no context c such that all the nonconditional premises (C1)–(C1,000,000) are true in it, and there is no context such that all conditional premises (P1)–(P1,000,000) are true in it. This is because considering more and more people in the series will accumulatively change the context until at some point the context undergoes a sudden reversal (cf. Raffman’s 1994 ‘gestalt switch’ and Kamp’s 1981 ‘incoherent’ context). If someone were to begin pronouncing the entire argument, then in each of the premise pairs (Pn)/(Cn), ‘rich’ would express a slightly different property, until suddenly it would express a significantly different property. This means that the corresponding generalized conditional premise (GP) For all x, y: if x is not rich and y has only ¤1 more than x, then y is not rich either. [There are no x, y, such that x is not rich, y has only ¤1 more than x and y is rich.] is false in every context. Nevertheless, there is no context in which a counterexample of the form (B) a is not rich, b has only ¤0.01 more than a, and b is rich. could be uttered and be true at that context. This explains (GP)’s appearance of truth. If sentences containing vague predicates express vague propositions in the sense outlined above, and if the normative significance of propositional truth is given by a type (2) completion of (VP), then a structurally analogous response to the sorites is available. On the non-indexical approach, the premises of the sorites argument are not indexical, but express the same propositions in all contexts of use. However, the propositions expressed vary in truth value with the sharpening parameter in the circumstances of evaluation. The question we are now considering concerns the normative significance of these relative truth values, e.g. under what conditions it is correct to assert or believe such a proposition (i.e. how to complete (AP)). According to response (2), the sharpening or sharpenings relevant for evaluating an assertion will vary as a subject is marched along the sorites series. At the beginning of the series,
Vagueness as Semantic
319
when we are asking ourselves whether some Pn is rich, we’ll have to say that she is not, because we have just previously ruled that Pn−1 is not rich. The sharpening(s) relevant for the correctness of an utterance in a context c obey the following constraint: if two individuals x and y are relevantly similar (e.g. they differ only by ¤1), and their similarity is salient²² in c, then the sharpening(s) relevant for judging correct assertability (believability) in c will not classify differently the proposition that x is not rich and the proposition that y is not rich. As the subject moves further and further along the series, however, there will come a point at which the context undergoes a sudden leap (perhaps just because proximity to clear cases of rich is becoming all too obvious). This explains why (GP) is not correctly assertable (believable) in any of the contexts, yet each of its instances is. This in turn explains the deceptive pull exerted by (GP) despite its unacceptability. Despite emulating some aspects of indexical contextualism, the non-indexical approach here proposed clearly differs in other respects. According to the indexical approach, ‘rich’ expresses a different property, and concept, at each stage of the march through the sorites series, whereas on the relativist approach, the property and concept expressed by ‘rich’ typically remain constant as a subject is moving along a sorites series. It is merely the correctness of calling an individual ‘rich’ and the correctness of believing an individual to be rich that varies as we move along the series. A comparison with the more familiar case of contingent properties and propositions may be illuminating: in the actual situation it is correct to call Modotti (or believe her to be) a photographer. In some non-actual situations, it would not have been correct to call her (believe her to be) a photographer. Nevertheless, the property ascribed to her in the different situations is the same: the property of being a photographer. It’s just that that property has an extension that varies from one possible world to another. Similarly, the proposition accepted in each case is the same, it is merely the truth value of that proposition that changes from one world to another. The proposal is that we treat the variability in the range of things to which ‘rich’ can be correctly applied analogously with this variability in the extension of ‘is a photographer’. In summary, when completing (VP), we should make assertability and believability depend on a variable (range of ) sharpening(s): (VP) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at S(c). where ‘S’ is some contextual function that will be further described in the next section. 17.9
BORDERLINE CASES
It remains to argue that in completing (VP) we should privilege a range of sharpenings rather than an individual one, and then to superevaluate. The motivation for this ²² I here go with Fara’s (2000) ‘saliently similar’ rather than with Soames’s (1999) ‘similar and salient’.
320
Max K¨olbel
comes from our intuitions about borderline cases. There are three obvious options for construing S: (VPa) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at the sharpening determined by c. (VPb) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at all sharpenings in the range determined by c. (VPc) An assertion (belief ) that p occurring in context c is correct only if the proposition that p is true at some sharpenings in the range determined by c.²³ The consequence of (VPa) would be that vague propositions are, in any context, either correctly assertable or correctly deniable, and never both (where correct deniability of p is equivalent to correct assertability of not-p). This goes against all intuitions: against the intuition that in borderline cases of a predicate one may neither assert nor deny and also against the intuition that in borderline cases one may both assert and deny. Thus, I believe, (VPa) can be discarded. As for the remaining two options: it seems that there are two ways of thinking about borderline cases. According to one view (I believe the majority’s), borderline cases of richness are cases where it is neither correct to affirm nor to deny richness. Thus, for some n, it may neither be correct to call Pn rich, nor to call her not rich, at least in certain contexts (not, for example, when one has just judged Pn−1 to be not rich). Option (VPb) is the way to make room for this intuition.²⁴ Some have argued that borderline cases are cases where both verdicts are permissible (e.g. Wright 2003). According to them, in a borderline case it is both correct to assert and deny the property in question. A theorist supporting this view would naturally opt for (VPc). However, I am persuaded by the more common conception of borderline case. There is a close structural similarity, then, between the characterization of borderline cases adopted here and the supervaluationist position. So it will be worth pointing out the differences. Supervaluationists typically claim that truth is super-truth and that falsity is super-falsity. Thus, supervaluationist semantics involves the claim that some utterances are neither true nor false. The relativist here described, however, does not superevaluate in the semantics: the semantics does not specify super-truth conditions. Rather, the relativist superevaluates at the pragmatic level, when it comes to spelling out the normative significance of the semantic properties of expressions. One of the difficulties of supervaluationism is that it is committed to the truth of the negation of the general premise in the Sorites: (¬GP) For some x, y: x is not rich, y has only ¤1 more than x, and y is rich. ²³ These three options are clearly not exhaustive. For example we might replace ‘all’ in (VPb) with ‘most’, ‘many’, ‘a few’, or even with ‘twenty’. However, I do not see any reason to think that any of these options is promising. ²⁴ It s worth noting that even an epistemicist like Williamson can accept this characterization of borderline cases as cases where it is neither correct to assert (believe) nor to deny (disbelieve). For according to Williamson, correct assertability requires knowledge, and belief that is not knowledge is ‘botched’. See Williamson 2000.
Vagueness as Semantic
321
For (¬GP) is supertrue, i.e. true on each admissible sharpening. This seems to be a problem because there does not seem to be a true instance of (¬GP). It might seem that the current proposal is similarly committed to (¬GP) being correctly assertable. However, this is not so because, as pointed out above, the range of sharpenings which are relevant for assertability vary with the context. A similarity constraint will ensure that (¬GP) is not assertable in any context, while (GP) is assertable in every context. The upshot, then, is that a principle along the lines of (VPb) states the normative significance of propositional truth. This explains the seductiveness of the sorites and makes good sense of borderline cases without in any way departing from standard semantics. What is new is the pragmatics, i.e. the role truth plays in assessing assertions and beliefs for correctness. 17.10
H I G H E R - O R D E R VAG U E N E S S
There are at least two ways in which higher-order vagueness might arise on the current proposal. First, the notion of an (admissible) sharpening might be vague, and secondly, correct assertability (believability) may be vague, due to the contextual determination relation mentioned in (VPb) being vague. I shall discuss these in turn.
17.10.1 Is ‘sharpening’ vague? According to the semantic account of vagueness here proposed vague predicates vary in their extension with a sharpening parameter in the circumstances of evaluation. I likened this parameter to the sharpenings or precisifications familiar from supervaluationism: they are ways in which all concepts could be made precise consistently with clear cases and certain a priori principles. George Soros and Anita Roddick, for example, are clear cases of richness. An example of an a priori principle is the principle that if one person, A, is richer than another, B, then it cannot be that B is rich and A is not. Thus, according to my rough exposition of the relativist semantics, a precisification that does not count Carlos Slim as rich would not qualify as an admissible sharpening in any context, nor would a precisification according to which a non-rich person has more money than some rich person. Higher-order vagueness can arise in connection with the former issue: does a precisification qualify as a sharpening if it counts someone with possessions worth 10 thousand Euros as rich? What about one Euro less? It looks like the border between admissible and inadmissible precisifications in a given context is fuzzy. There are at least two ways of dealing with this. One is to accept that the notion of an admissible sharpening, as it figures in the semantic meta-language, is a vague notion. This, I believe, is in principle unproblematic. However, it is important to notice that admitting this form of higher-order vagueness is not required to make room for the phenomena of higher-order vagueness. For example, it is a phenomenon of higher-order vagueness that there does not seem to be a clear cut-off point between those who, in some context, may be called rich (or believed to be so) and those who
322
Max K¨olbel
are on the borderline between the two, i.e. those of whom it is neither correct to say (or believe) that they are rich nor that they are not. This phenomenon, in the current account will be accounted for by the vagueness of the notion of correct assertability (believability), which in turn derives from vagueness in the relation of determination between contexts of utterance and relevant sharpenings. I shall say more about this form of higher-order vagueness in a moment. What then is at stake in the question whether the notion of an admissible sharpening is vague? Consider a proposal according to which it is not vague. We might say that any precisification that respects the relevant ordering principles (e.g. ‘a non-rich person cannot have possessions worth more than some rich person’s’) is an admissible sharpening. In the case of supervaluationism, this would lead to the unwanted consequence that everyone is a borderline case of ‘rich’. However, the current approach characterizes borderline cases at the level of correct assertability or believability, which in turn requires truth in all sharpenings in the range determined by the context. As long as that range is occasionally restricted, we avoid the trivializing result that everything is borderline. What is at stake in the question whether ‘sharpening’ is vague is something quite different. According to the proposal that ‘sharpening’ is not vague, ‘rich’ varies in extension with various sharpenings, and for every person there is a sharpening that classifies him or her as a member of the extension and for every person there is a sharpening that classifies him or her as a non-member. Most of these sharpenings are pragmatically irrelevant because there is no context in which they are determined as relevant. So, even if everyone is classified as ‘rich’ in some sharpening, not everyone can correctly be called ‘rich’ or correctly be believed to be rich. However, a side-effect of this is that the semantic content of ‘rich’ will not differ in the expected way from that of ‘very rich’; ‘small’ not from ‘tiny’, ‘large’ not from ‘huge’ etc. The difference between ‘tiny’ and ‘small’ will not be that the extension of the former is less comprehensive than that of the latter. The differences between these concepts will show up only in the sharpenings that are determined as relevant by context.
17.10.2
Is correct assertability/believability vague?
Higher-order vagueness in the usual sense is, on this account, an entirely pragmatic phenomenon, in the sense that it concerns correct assertability and believability. Typical vague concepts have borderline cases: objects of which it is neither correct to assert (believe) nor to deny (disbelieve) the concept. However, correct assertability (believability) seems itself to be subject to tolerance constraints that lead to vagueness. Just as there seems to be no n such that Pn is not rich and Pn+1 is, there also seems to be no n such that it is correct to assert that Pn is not rich and not correct to assert that Pn+1 is not rich. Given the analysis of correct assertability proposed above, this form of higher-order vagueness could come about in two ways. First, the range of sharpenings determined by a context to be relevant to adjudicating the correctness of an assertion (belief ) might be vague. Thus, in a given context c, there is no n such that Pn is in the extension of ‘rich’ relative to all c-relevant sharpenings and Pn−1 is not. Thus, the
Vagueness as Semantic
323
determination relation that determines for each context of use a range of relevant sharpenings is itself a vague relation. Secondly, it may be that in each context a precise range of sharpenings is determined as relevant for correct assertability. Thus, for any context c, there is an n such that Pn is in the extension of ‘rich’ relative to all c-relevant sharpenings and Pn−1 is not. In that case it may still be true that there is no n such that it is correct to assert that Pn is not rich and not correct to assert that Pn+1 is not rich, i.e. higher-order vagueness may be present in this sense. However, this is only the result of the context changing when different Pn are under discussion. Hold any a context c fixed, and there will be an n such that it is correct to assert in c that Pn is not rich and not correct to assert in c that Pn+1 is not rich. The second view, I believe, faces the challenge of explaining how the predicates in question acquire the pragmatic features that determine a sharp borderline of correct assertability in a context, given that we manifestly have no idea where that borderline is located. This is analogous to the challenge facing epistemicists and indexical contextualists in explaining how vague predicates acquire their precise extensions (in contexts of use). I therefore prefer the first account of higher-order vagueness. 17.11
C O N C LU S I O N
I have shown how a standard semantic framework along the lines of those proposed by Kamp and Lewis for modal indexical languages can accommodate vague predicates, conceived of as extensionally indeterminate. I discussed three ways in which standard semantics makes room for extensional indeterminacy of predicates: ambiguity, indexicality and sensitivity to circumstances of evaluation (e.g. contingency). After discussing the prospects for treating vagueness as a phenomenon of ambiguity or indexicality, I moved on to develop an account of vagueness that assimilates the extensional indeterminacy of vague predicates to that of contingent predicates. Vague predicates, on this view, vary their extension with an additional parameter in the circumstances of evaluation, a ‘sharpening’. On this approach the semantics remains absolutely standard, and it is only the way in which the semantic notion of truth figures in pragmatic norms, the norms of assertion and belief, that requires some modification. The account bears similarities with both contextualism and supervaluationism, so it will be worth once more to point out the differences. The difference between viewing vagueness as a form of indexicality and the relativist view here proposed is clear. Indexical contextualists claim that the extensional indeterminacy of vague predicates is owed to their content being context-sensitive. Vague predicates (qua vague predicates) express different properties and concepts in different contexts of use, and vague sentences (qua vague sentences) express different propositions or contents in different contexts. The non-indexical view I proposed claims that the contents of vague predicates and sentences (qua vague predicates and sentences) are invariant, and that it is merely their extension that varies with circumstances of evaluation.
324
Max K¨olbel
This difference is, I believe, sufficiently clear. What is not completely clear is whether well-known contextualists about vagueness should be read as putting forward a version of the indexicality view here described. As far as I can tell, much of what contextualists say is indeterminate between what I have called indexicality and sensitivity to circumstantial parameters. In so far as this is true, the account here outlined should be taken as a contribution to developing further the views of contextualists about vagueness. In so far as this not true, i.e. in so far as contextualists about vagueness subscribe to an indexical view, this chapter should be taken as a proposal for modifying contextualism about vagueness.²⁵ ²⁵ Soames is the only clear case of an indexical contextualist about vagueness because in his 2002 he makes it explicit. Kamp comes at times very close to being an explicit indexicalist, though since the framework treated here as standard was only emerging at that time (with Kamp one of the pioneers), the terminology may well be misleading here. Even though Kamp’s contexts seem clearly intended as contexts of use, Kamp considers only one narrow aspect of the context of use, namely sentences that have previously been uttered. Raffman’s account (1994, 1996) is closely related to Kamp’s. The way she discusses the various aspects of context in her 1994, 64, suggests strongly that she thinks of the context as an utterance context, the quote by Kamp suggests it especially. But literally and strictly, what she says is compatible with both an indexical and a non-indexical reading of contextualism. For she does not usually consider the question whether the property or proposition expressed by a predicate or sentence varies with the context, but only whether the extension thus varies. For this reason, her account seems to be undecided between an indexical and a non-indexical reading. In her 2005 she explicitly distances herself from the indexicality view, though in the context it is not clear whether she here intends ‘indexical’ to mean ‘pure indexical’ (in Kaplan’s sense) or indexical in the wider Kaplanian sense of ‘the content varies with the context of use’. Fara (2000) is a complicated case because she considers many kinds of context-sensitivity of vague predicates. ‘is tall’ means roughly the same as ‘is significantly (x) taller than is typical (y) for (z)’. The extension of ‘tall’ varies with a comparison class (z), a norm of what’s typical by way of tallness for the comparison class (y), and also with standards of significance (x). The latter, she repeatedly says, is interest-relative. All these seem clearly to be intended to be aspects of the context of utterance, and the phenomenon one of indexicality. On this view, the property expressed by a vague predicate changes with the context of utterance. However, she insists on 64 and 75, that at least the interest factor is not to be understood in this way: ‘the property attributed to John by a particular utterance of ‘‘John is tall’’—that is, once all contextual elements are fixed—is still a property the extension of which may vary even as the heights of everything remain stable, since the extension of the property may vary as the interests of the relevant parties vary, that is, as different differences become more or less significant as different similarities become more and less salient.’ (75) This is puzzling without any further explanation. The idea is the property expressed by ‘tall’ in a given context remains invariant but its extension still varies. On this view of a property, differences in extension are not sufficient for differences in property, and therefore what Fara had in mind in her 2000 may well have been something akin to what I have been proposing in this chapter. Fara 2008 makes this more explicit: here she speaks explicitly of interest-relative properties and interest-relative propositions. Stanley 2003 claims that on Fara’s view, vague predicates are not indexicals because they express invariant properties, just as ‘is a US citizen’, which always expresses the same property, but that property’s extension varies with time. Thus, ‘is tall for a British male’ expresses the same property, namely that of being significantly taller than is typical for a British male, but of course that property changes its extension not just with time but also with interests.
Vagueness as Semantic
325
What is the difference between the current relativistic account and supervaluationism? First, the relativist does not accept the ‘supervaluationist’s slogan’ (Keefe 2000, 202) that truth is super-truth.²⁶ The semantic truth-conditions of a sentence (even of a sentence in a context) are relativized truth conditions, not conditions of super-truth. Supervaluation comes at the level of assessing assertions or beliefs for correctness in their context, not at the level of assessing the truth of the contents of such assertions of beliefs. Another difference is that the current relativistic account makes the range of sharpenings over which we superevaluate (when assessing correctness) contextually variable. This is not usually part of supervaluationism. Re f e re n c e s Burns, Linda (1991), Vagueness: An Investigation into Natural Languages and the Sorites Paradox, Dordrecht, Kluwer. Cappelen, Herman (2008), ‘Content relativism and semantic blindness’ in Manuel Garc´ıaCarpintero and Max K¨olbel (eds.), Relative Truth, Oxford, Oxford University Press, 265–86. Cappelen, Herman and Ernie Lepore (2004), Insensitive Semantics, Oxford, Blackwell. Ellis, Jonathan (2004), ‘Context, indexicals and the sorites’, Analysis 64, 362–4. Fara, Delia Graff (2000), ‘Shifting sands: An interest-relative theory of vagueness’, Philosophical Topics 28, 45–81. (2008), ‘Profiling interest relativity’, Analysis 68, 326–35. Evans, Gareth (1979), ‘Does tense logic rest on a mistake?’ in his Collected Papers (1985), 341–63, Oxford, Clarendon Press. Fine, Kit (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. Kamp, Hans (1981), ‘The paradox of the heap’ in U. M¨onnich, ed., Aspects of Philosophical Logic, Dordrecht, Reidel, 225–77. Kaplan, David (1977), ‘Demonstratives’ in Almog et al., eds, Themes from Kaplan, Oxford, Clarendon Press 1989. Keefe, Rosanna (2000), Theories of Vagueness, Cambridge, Cambridge University Press. King, Geoffrey (2003), ‘Tense, modality and semantic value’, Philosophical Perspectives 17, 195–245. K¨olbel, Max (2008), ‘Motivations for relativism’ in Manuel Garc´ıa-Carpintero and Max K¨olbel, eds., Relative Truth, Oxford, Oxford University Press, 1–38. (forthcoming), ‘Assertion, intention and convention’ forthcoming in Sarah Sawyer, ed, New Waves in Philosophy of Language, Hampshire, Palgrave Macmillan. Lewis, David (1975), ‘Languages and language’ in Minnesota Studies in the Philosophy of Language 7, 3–35. Reprinted in Lewis (1983). However, Stanley also attributes to Fara the view that one of the effects of this interest-relativity is that it will depend on the interests of an utterer which proposition is expressed by the utterance of a vague sentence. This suggests that on the wider Kaplanian sense of ‘indexical’ (the one I have been using earlier in this chapter), vague sentences are indexical in the sense that they express different propositions in different contexts of use, and that predicates are indexical if the content they contribute to the proposition expressed depends on the context of use. ²⁶ Though McGee and McLaughlin 1995, despite being supervaluationists, do not accept the slogan.
326
Max K¨olbel
Lewis, David (1983), Philosophical Papers, vol. 1, Oxford, Oxford University Press. ¨ (1980), ‘Index, context, and content’ in Stig Kanger and Sven Ohman, eds., Philosophy and Grammar, Dordrecht, Reidel. Reprinted in Lewis, Papers in Philosophical Logic, Cambridge, Cambridge University Press 1998. McGee, Vann and Brian McLaughlin (1995), ‘Distinctions without a difference’, Southern Journal of Philosophy 33, 203–51. Pagin, Peter (2009), ‘Central gap domain restriction’. This volume. Pagin, Peter and Jeff Pelletier (2007), ‘Content, context and composition’ in G. Peter and G. Preyer, eds., Content and Context. Essays on Semantics and Pragmatics, Oxford, Oxford University Press, 25–62. Prior, Arthur (1959), ‘Thank goodness that’s over’, Philosophy 34, 12–17. (1967), Past, Present and Future, Oxford, Clarendon Press. Raffman, Diana (1994), ‘Vagueness without paradox’, Philosophical Review 103, 41–74. (1996), ‘Vagueness and context relativity’, Philosophical Studies 81, 175–92. (2005), ‘How to understand contextualism about vagueness: Reply to Stanley’, Analysis 65, 244–8. Recanati, Franc¸ois (2007), Perspectival Thought, Oxford, Oxford University Press. Richard, Mark (1981), ‘Temporalism and Eternalism’, Philosophical Studies 39, 1–13. Sainsbury, Mark (2001), ‘Two ways to smoke a cigarette’, Ratio 14, 386–406. Shapiro, Stewart (2003), ‘Vagueness and conversation’, in Jc Beall, ed., Liars and Heaps, Oxford, Oxford University Press. Soames, Scott (1999), Understanding Truth, Oxford, Oxford University Press. (2002), ‘Replies’, Philosophy and Phenomenological Research 62, 429–52. (2003), Beyond Rigidity, Oxford, Oxford University Press. Sorensen, Roy (1988), Blindspots, Oxford, Clarendon. Stanley, Jason (2000), ‘Context and logical form’, Linguistics and Philosophy 23, 391–434. (2003), ‘Context, interest-relativity and the sorites’, Analysis 63, 269–80. Travis, Charles (1985), ‘On what is strictly speaking true’, Canadian Journal of Philosophy 15, 187–229. (1996), ‘Meaning’s role in truth’, Mind 105, 451–66. Tye, Michael (1989), ‘Supervaluationism and excluded middle’, Analysis 49, 141–3. Williamson, Timothy (1994), Vagueness, London, Routledge. (2000), Knowledge and its Limits, Oxford, Oxford University Press. Wright, Crispin (2003), ‘Vagueness: A fifth column approach’ in Jc Beall, ed., Liars and Heaps, Oxford, Clarendon Press.
18 How to Respond to Borderline Cases Dan L´opez de Sa
It seems that Hannah and her wife Sarah may disagree as to whether Homer Simpson is funny, without either of them being at fault. This is an (almost) uncontroversial case of apparent faultless disagreement. More cases are arguably provided in other philosophically interesting domains: predicates of personal taste, evaluative predicates in general, epistemic modals, and knowledge attributions. With respect to any of these, it is held, it seems that there could be contrasting judgements without fault on the part of any of the participants. Some philosophers seem to think that vagueness should be included in the list above: borderline cases provide further cases of apparent faultless disagreement. My aim here is to argue against such a suggestion. After elaborating briefly on the notion of apparent faultless disagreement, I present the case for my main claim: with respect to borderline cases, people typically do not respond by taking a view—in contrast to what is the case in genuine cases of apparent faultless disagreement (section 18.1). The status of this kind of claim, both descriptive (of paradigm cases, at least) and normative—though familiar in many other domains, such as the theory of meaning, decision theory, or moral psychology—is likely to raise suspicion. The main part of this chapter is devoted to alleviating such suspicion. I argue that my claim is indeed respected and actually accounted for by paradigm cases of semantic and epistemic views on the nature of vagueness (section 18.2). And I also argue that my claim turns out to be, initial appearances notwithstanding, compatible with other claims in the literature—to the effect that, in appropriate circumstances, there are indeed, or there might well be, ‘macho,’ admissible, forced, and hesitant responses to borderline cases (section 18.3). Earlier versions were presented at the 7th Arch´e Vagueness Workshop and the LOGOS Seminar. Thanks to the audiences then, and in particular to Richard Dietz, Manuel Garc´ıa-Carpintero, Max K¨olbel, Sebastiano Moruzzi, Diana Raffman, Sven Rosenkranz, Stewart Shapiro, Crispin Wright, and Elia Zardini, for very helpful objections and suggestions, and to anonymous referees for Oxford University Press. Research partially funded by projects HUM2004-05609-C02-01 and FFI200806153/FISO, and a GenCat-Fulbright Postdoctoral Fellowship. Thanks to Mike Maudsley for his linguistic revision.
327
328
Dan L´opez de Sa 18.1
T H E PH E N O M E N O N O F A P PA R E N T FAU LT L E S S D I S AG R E E M E N T
We are quite familiar with situations such as that of Hannah and Sarah regarding whether Homer Simpson is funny: disputes as to whether spinach is tasty or disgusting, or as to whether getting to the party late is cool or lame, or as to whether Brad Pitt or Uma Thurman are really sexy or rather overrated. In cases like these, people often take views on the matter, even strong ones, which sometimes issue in (long) discussions and arguments. Still, it seems that none of the parties need be mistaken with regard to their views, after all. Whether such an appearance of faultless disagreement is to be endorsed—or even whether it could be endorsed—is, of course, a matter of controversy. Following the lead of Crispin Wright 1992, one can conceive of relativism in general as precisely the attempt to so endorse the appearances of faultless disagreement, in the different domains—for such needs to involve, in one way or other, some relativity to contrasting features of the subjects in question. The different sources the relativity might be held to have are what give rise to the different relativisms. Moderate relativism has it that such an endorsement can be done within the general Kaplan–Lewis–Stalnaker two-dimensional framework, in which the basic semantic notion is that of a sentence s being true at a context c at the index i.¹ It may in effect be the case that s is true at c (at its index ic ) but false at c ∗ (at ic∗ )—due to the content of sentence s at c being different from that of s at c ∗ (indexical contextualism); or, even if the content is the same, due to relevant differences in the indices ic and ic∗ determined by c and c ∗ (non-indexical contextualism). Radical relativism, by contrast, claims that appropriately endorsing appearances of faultless disagreement requires departing from the two-dimensional framework, in that s at the very context c can be true from a certain perspective but false from another—where perspectives are to be thought of as the same sort of thing as contexts, but representing a location from where a sentence, as said in a (possibly different) location, could be viewed or assessed.² ¹ The jargon I adopt is from Lewis (1980). A context is a location—time, place, and possible world, or centered world for short—where a sentence could be said. It has countless features, determined by the character of the location. An index is an n-tuple of features of context, but not necessarily features that go together in any possible context. Thus an index might consist of a speaker, a time before his birth, a world where he never lived at all, and so on. The coordinates of an index are features that can be shifted independently, unlike those of a context, and thus serve to represent the contribution of sentences embedded under sentence operators, such as ‘possibly’ or, more controversially, ‘somewhere,’ ‘strictly speaking,’ and so on. Given a context c, however, there is the index of the context, ic : that index having coordinates that match the appropriate features of c. Given this uniqueness, the basic two-dimensional relation can be abbreviated in this special case: sentence s is true at context c iff s is true at context c at index ic . ² I propose to use ‘perspectives’ instead of MacFarlane’s ‘contexts of assessment,’ see his 2003, 2005. I think this terminology helps to avoid confusions with ‘context of use/utterance’ (‘context’ here) and, more importantly, with ‘circumstance/point of evaluation’ (‘index’ here). My taxonomy is greatly indebted to—and some of the labels due to—John MacFarlane. I elaborate on the details in L´opez de Sa (2009b).
How to Respond to Borderline Cases
329
Whether the appearances of faultless disagreement are (can be) endorsed is, as I said, controversial. But that such appearances exist is, I take it, a datum for nonrelativists and relativist alike—appearances that are to be explained away, if they are not endorsed. Hannah may have a judgement she might naturally express in an ordinary context by uttering ‘Homer Simpson is funny’ with its literal meaning; whereas Sarah may have a judgement she might naturally express in an ordinary context by uttering ‘Homer Simpson is not funny.’ And these contrasting judgements need not involve, apparently, any error on the part of Hannah nor Sarah. Similarly in some other philosophically interesting cases—including future contingents, predicates of personal taste, evaluative predicates in general, epistemic modals, and knowledge attributions—there can be contrasting judgements about an issue in the domain that do not seem to involve fault on the part of any of the participants: they all involve cases of (at least) apparent faultless disagreement.³ Some philosophers seem to think that vagueness should be included in the list above: borderline cases provide further cases of apparent faultless disagreement. However, this does not seem to be so. Take Jason and his husband Justin, and consider a borderline green towel.⁴ Typically, I submit, they would not respond to it by taking a view as to whether the towel is green or not. They would simply lack the judgements that they would naturally express in an ordinary context by asserting ‘The towel is green’ or ‘The towel is not green’ with its literal meaning: rather, if questioned about it, they would easily converge in something like that ‘it sort of is and sort of isn’t,’ ‘it’s greenish,’ etc.—and they would be rational in so doing. But then they would lack the building blocks for the appearance of faultless disagreement clearly present in the other cases considered above: the (contrasting) judgements. Hannah and Sarah do typically form polar opinions with respect to issues such as whether Homer Simpson is funny; Jason and Justin typically do not form such verdicts with respect to issues such as whether the towel is green. So this is in essence why I think that vagueness does not provide further cases of apparent faultless disagreement: with respect to borderline cases, people typically do not respond by taking a view—in contrast to what is the case in genuine cases of apparent faultless disagreement. The status of this kind of claim, however, is likely to raise suspicion. I am submitting Jason and Justin as paradigmatic with respect to people’s actual ways of responding to borderline cases. And I am also suggesting the normative view that it is indeed rational for them so to respond. I take it we are familiar with this kind of situation—claims that are submitted as both descriptive (of ³ Some use ‘faultless disagreement’ in a more restricted sense, requiring that there be a single content or proposition which is contrastingly judged, see for instance K¨olbel (2003). According to this more restricted sense, it cannot just be taken as a datum for relativists and non-relativists alike that there are apparent faultless disagreements, nor do all versions of relativism endorse that there are in effect faultless disagreements in the relevant domains. These I take to favor my more liberal usage. ⁴ To provide an adequate characterization of what it is for something to be a borderline case is of course part of what is at stake. For present purposes, however, it suffices to point to cases with respect to which the different views as to how to respond to them differ.
330
Dan L´opez de Sa
paradigm cases, at least) and normative—in many other domains, such as the theory of meaning, decision theory, or moral psychology. But this familiarity by itself need not remove suspicion. As I said at the beginning, what follows is my best attempt to alleviate such suspicion in the case at hand. I will argue that my claim is indeed respected and actually accounted for by paradigm cases of semantic and epistemic views on the nature of vagueness. And besides, I will also argue that my claim turns out to be, initial appearances notwithstanding, compatible with other claims in the literature—to the effect that, in appropriate circumstances, there are indeed, or there might well be, ‘macho,’ admissible, forced, and hesitant responses to borderline cases.
18.2
N O R M S O F A S S E RT I O N A N D T H E N AT U R E O F VAG U E N E S S
One of the main views on the nature of vagueness has it that vagueness is a phenomenon of semantic indecision: (roughly) whatever it is that in the thoughts, experiences and practices of language users determines the meaning of expressions, it fails to determine, for vague expressions, any single one from a given range of similarly natural candidate references. Each way of (‘arbitrarily’) fixing what is left semantically indeterminate gives rise to a precisification or sharpening of the original vague expression. Although all such sharpenings are, by essence, arbitrary to a certain extent, not all of them are admissible. In the case of predicates, admissible ones should preserve clear cases, both of application and of non-application—Yul Brynner should count for ‘is bald,’ while Andy Garc´ıa cannot—, and they should also preserve penumbral connections —‘Whoever is bald is bald,’ ‘If someone is bald, then so is anyone who is balder,’ and so on—.⁵ What one says by means of a vague expression is true, according to this view, if it would be true however one (admissibly) precisifies it—or, as I will put it, if it counts as true according to all admissible sharpenings. And it is false if it counts as false according to all admissible sharpenings. Otherwise, if there are admissible ways of precisifying it which give rise to truths, but also admissible ways of precisifying it which give rise to falsehoods, the vague sentence is indeterminate: neither true nor false. That is indeed the situation with respect to borderline cases, as the view has it. Take Harry, a borderline case with respect to ‘is bald,’ having exactly 3,833 hairs on his scalp. Whatever it is that in the thoughts, experiences and practices of language users determines the meaning of expressions, it fails to determine whether someone with this very number of hairs does or does not fall under ‘is bald.’ Thus ‘is bald’ can be admissibly precisified by (let us assume) ‘has at most 3,832 hairs on his scalp,’ but ⁵ Thus sharpenings are, strictly speaking, of the language as a whole, and not of isolated expressions, see Fine (1975). How to characterize in an explicit satisfactory way the notion of admissible constituted by these connections (possibly among others) would of course be crucial for a full defense of the view of vagueness as semantic indecision. Notice that ‘is admissible’ is, of course, itself vague: this is arguably part of what accounts, in this framework, for the phenomenon of ‘higher-order’ vagueness. Complications arising from this will be set aside here.
How to Respond to Borderline Cases
331
also by ‘has at most 3,834 hairs on his scalp.’ Hence, ‘Harry is bald’ fails to be true, given that ‘Harry has at most 3,832 hairs on his scalp’ is false, but it also fails to be false, given that ‘Harry has at most 3,834 hairs on his scalp’ is true.⁶ As most of its critics also acknowledge, the view of vagueness as semantic indecision is certainly—at least initially—intuitively very plausible. But it is at odds with the claim that borderline cases exhibit apparent faultless disagreement: it predicts that people would typically not (and should not) take a view with respect to borderline cases, as the relevant statements would lack a truth value and thus would not be true. In this way, I hold, the view provides further support to my main claim. Let me elaborate. As we have just seen, the phenomenon of faultless disagreement requires that people do typically form judgements on the matter (which may be contrasting while apparently fault-free). Judgements like these are typically manifested by people’s asserting the relevant statements in question, at their respective contexts. Thus, as we saw, Hannah could perfectly well express her judgement by asserting ‘Homer is funny’ at her context, and Sarah by asserting its negation at hers. Assertions are (arguably) acts governed by norms. The weakest sensible norm for assertion, most would agree,⁷ is the truth rule: One must: assert s at c only if s at c (at ic ) is true. The truth rule forbids untrue assertions.⁸ Borderline cases exhibiting the phenomenon of apparent faultless disagreement would require things like (say) Jason forming a view to the effect that the towel is green, and Justin forming a contrasting view to the effect that the towel is not green. But according to the view of vagueness as semantic indecision, ‘The towel is green’ is not true at Jason’s context, nor is it true ‘The towel is not green’ at Justin’s. Thus Jason should not assert ‘The towel is green,’ nor Justin assert its negation. And this is, as we saw, in clear contrast with the case of Hannah and Sarah. To the extent to which it is sensible to assume that people’s actions typically conform to their characteristic norms (at least in paradigmatic instances of the relevant action types), the view of vagueness as semantic indecision also accounts for why people typically do not take a view with respect to borderline cases. I have argued that the view of vagueness as semantic indecision, which is certainly (at least initially) plausible from an intuitive point of view, respects and accounts for my main claim and thus provides support to the contention that borderline cases do not exhibit the phenomenon of apparent faultless disagreement. The same is true, ⁶ Thus the characteristic denial of the principle of bivalence: not everything that says something is either true or false, as borderline cases are indeterminate. Williamson (1994) contains an argument for the incompatibility of this feature with Tarskian views about truth and falsity, which apparently convinced most people in the field. In my view, however, Andjelkovi´c and Williamson (2000) contains the key elements for resisting it: see for elaboration and further discussion L´opez de Sa (2009a). ⁷ As also noticed by defenders of radical relativism, their view is committed to depart from standard ones on this (related) count as well. ⁸ See (Williamson, 2000, ch. 11) for further (critical) discussion. He says: ‘The truth rule forbids false assertions’ 2000, 242, my emphasis, which is just a proper consequence—in the absence of the (independent) principle of bivalence.
332
Dan L´opez de Sa
I now claim, about one of the other main rival views on the nature of vagueness: epistemicism as defended by Williamson (1994)—provided that some epistemic norm for assertion holds. Were Jason to form a view to the effect that the towel is green, this would not constitute knowledge. And neither could Justin knowledgeably form a contrasting view to the effect that the towel is not green, for the same reasons. Most people would agree: if it is borderline, then there is no knowing that the towel is green, and no knowing that it is not green. According to defenders of the view of vagueness as semantic indecision, the explanation of this is straightforward (and shows why it would be misleading to label the situation as one of ignorance): there is no knowing because, as noted above, there is no truth there to be known. By contrast, according to epistemicism as defended by Williamson (1994), we may suppose, either ‘The towel is green’ is true at Jason’s context, or it is true ‘The towel is not green’ at Justin’s.⁹ Still, the epistemicist holds, neither of the judgements that Jason or Justin could naturally express in their respective contexts would constitute knowledge.¹⁰ Now, although admittedly more controversial, a case has been made for assertions being acts governed by the (stronger) knowledge rule, see (Williamson, 2000, ch. 8): One must: assert s at c only if one knows p, where p is the content of s at c. But if this holds, we have a corresponding support for my main claim, even according to epistemicism. For again borderline cases exhibiting the phenomenon of apparent faultless disagreement would require things like (say) Jason forming a view to the effect that the towel is green, and Justin forming a contrasting view to the effect that the towel is not green. But according to epistemicism, neither could knowledgably form such judgements. Thus again Jason should not assert ‘The towel is green,’ nor should Justin assert its negation, in clear contrast with the case of Hannah and Sarah. To the extent to which it is sensible to assume that people’s actions typically conform to their characteristic norms (at least in paradigmatic instances of the relevant action types), the epistemicist view of vagueness also accounts for why people typically do not take a view with respect to borderline cases.
18.3
F U RT H E R R E S P O N S E S TO B O R D E R L I N E C A S E S
I have submitted that the case of Jason and Justin regarding whether borderline green towel is green is intuitively very different from that of Hannah and Sarah regarding whether Homer Simpson is funny. The former typically won’t (and shouldn’t) take a view on the matter, thus lacking the judgements that are the building blocks for the phenomenon of apparent faultless disagreement, present in the latter. I have also ⁹ The main positive argument offered by Williamson (1994) in favor of epistemicism is precisely the one in favor of the principle of bivalence mentioned in footnote 6. ¹⁰ The explanation of this given the presence of a truth on the issue, is much more complex: see (Williamson, 1994, ch. 8) and (Williamson, 2000, ch. 5)
How to Respond to Borderline Cases
333
argued that this claim is indeed respected and actually accounted for by paradigm cases of semantic and epistemic views on the nature of vagueness, provided that certain plausible norms of assertion hold. In the remainder of this chapter I will try to show that my claim turns out to be, initial appearances notwithstanding, compatible with other claims in the literature concerning various responses to borderline cases.
18.3.1
‘Macho’ responses
In his discussion of Williamson (1994), Paul Horwich contends that the essence of [the phenomenon of vagueness] is not that borderline predications cannot be known to be correct. The essence of it . . . is that in certain cases we are normally unwilling to apply the predicate, unwilling to deny that it applies, and confident that no further investigation could yield a decision. The problem of knowledge in such circunstantes is a result of this paralysis of judgement. (Horwich, 1997, 931)
I would not say that the issue of how to respond to borderline cases constitutes the essence of vagueness, as opposed to being part of its characteristic manifestation. As implied above, I take the different views of the nature of vagueness to be views such as the view of vagueness as semantic indecision and epistemicism, which account for the manifestation of vagueness in how to respond to borderline cases. As a result, pace Horwich, I do not take the contention that a certain kind of ‘paralysis of judgement’ with respect to borderline cases is a characteristic manifestation of vagueness to be in tension with an epistemic view on the nature of vagueness. On the contrary, as I have just argued, epistemicism seems to account nicely for this, provided that certain epistemic norms for assertion hold. My main claim, however, that with respect to borderline cases people typically don’t and shouldn’t form categorical judgements—in contrast with what is clearly the case in domains that exhibit apparent faultless disagreement—seems clearly in tune with Horwich’s contention that a certain kind of ‘paralysis in judgement’ with respect to borderline cases is indeed a characteristic manifestation of vagueness: Jason and Justin would typically not respond to a borderline green towel by taking a view as to whether it is green or not. It is important to observe that this claim is compatible with the possibility of Jason and Justin being, on occasions, more opinionated and ‘macho’ than one typically is (and should be), and thus with them forming the judgements as to whether the towel is green or not. After all, we have all discovered ourselves, on occasions, in discussions as to whether something is or is not a certain way, just to realize that the case in question was simply borderline. My claim has it, however, that giving such ‘macho’ responses is not the way we typically respond to borderline cases, nor of course the way we should respond to them, and that this is clearly in contrast with situations such as that of Hannah and Sarah regarding whether Homer Simpson is funny. In his intriguing reply to Horwich, however, Williamson says: Horwich notwithstanding, paralysis in judgement is quite unnecessary for vagueness. Consider an opinionated macho community, in which everyone applies the term ‘bald’ or its negation
334
Dan L´opez de Sa
confidently and unhesitatingly on the basis of impressions gained from causal observations whenever the issue arises. There is no appeal to precise necessary and sufficient conditions. Speakers accept that the application of ‘bald’ depends only on the exact distribution of hairs on someone’s scalp, but vagaries of mood and perception cause them often to apply the term ‘bald’ when the distribution is exactly the same as in a previous case which they classified as ‘not bald.’ When they disagree, each dogmatically insist that the other is clearly wrong. When inconsistencies are pointed out in a single speaker’s application of the term, they are denied ad hoc (‘I never said that!’, ‘His hair has grown since then!’). On Horwich account, ‘bald’ is not vague in the language of this community, because there are no cases in which speakers ‘are normally unwilling to apply the predicate, unwilling to deny that it applies, and confident that no further investigation could yield a decision.’ But ‘bald’ is vague in the language of this community. (Williamson, 1997, 945–6)
I think there are two ways of conceiving of such a community. If Jason and Justin can on occasions give such ‘macho’ responses, we can conceive of them as always giving them—and we can also conceive of the rest of the population being similarly ‘macho.’ On this way of conceiving the community, it is certainly the case that the relevant expressions are still vague in the language of the community, as Williamson contends. But this being so does not contradict Horwich’s contention about ‘paralysis of judgement,’ at least understood along the lines of my main claim. For it would still be the case that, in the relevant sense, this would be the conceiving of a community in which people typically wouldn’t and shouldn’t give such ‘macho’ responses: on this way of conceiving the community, people are disposed as we are, it is just that we imagine the conditions to be such that they do not manifest their dispositions. Many other domains, such as the theory of meaning, decision theory, or moral psychology—or indeed basic dispositions to judge that the lines in the M¨uller–Lyer illusion are the same length—provide situations that are structurally analogous. If, by contrast, we conceive of a situation in which people simply lack the relevant dispositions to manifest ‘paralysis of judgement’ with respect to borderline cases at all, then I submit we no longer have the intuition that the relevant expressions are, indeed, vague —as ours indisputably are.
18.3.2
Admissible responses
On the face of it, my main claim that, with respect to borderline cases, people typically don’t (and shouldn’t) take a view seems in tension with the idea that, with respect to borderline cases, people can ‘go either way.’ Here is Stewart Shapiro’s recent statement of this idea: Suppose . . . that a is a borderline case of P. I take it as another premise that, in some situations, a speaker is free to assert Pa and free to assert ¬Pa, without offending against the meanings of the terms, or against any other rule of language use. Unsettled entails open. The rules of language use, as they are fixed by what we say and do, allow someone to go either way in the borderline region. Let us call this the open-texture thesis. (Shapiro, 2003, 43)
How to Respond to Borderline Cases
335
However, I do not think that the tension is genuine. For the idea seems to amount to the thought that, in certain conversations and given the presence of certain particular knowledge, purposes, etc., participants are capable of altering the standards of precision prevalent in that conversation, with the effect of producing local (explicit, or more plausibly, implicit) stipulations that give rise precisely to a precisification of the relevant vague expression—at least, for the purpose of the conversation at hand. The presence of the mechanism can be motivated independently, in a straightforward enough way within the framework of the view of vagueness as semantic indecision. I assume that similar moves might be available to the friend of epistemicism, although I will not attempt to adapt the consideration here. In his ‘Scorekeeping in a Language Game,’ David Lewis 1979 famously introduced the figure of a conversational score, whose kinematics—including prominently the rules of accommodation—he precisely illustrated with, among others, the case of vagueness. If Fred is a borderline case of baldness, the sentence ‘Fred is bald’ may have no determinate truth value. Whether it is true depends on where you draw the line. Relative to some perfectly reasonable ways of drawing a precise boundary between bald and non-bald, the sentence is true. Relative to other delineations, no less reasonable, it is false. Nothing in our use of language makes one of these delineations right and all the others wrong. We cannot pick a delineation once and for all (not if we are interested in ordinary language), but must consider the entire range of reasonable delineations. If a sentence is true over the entire range, true no matter how we draw the line, surely we are entitle to treat it simply as true. But we also treat a sentence more or less as if it is simply true, if it is true over a large enough part of the range of delineations of its vagueness. (In short: if it is true enough). . . . When is a sentence true enough? Which are the ‘large enough’ parts of the range of delineations of its vagueness? This is itself a vague matter. More important for our present purposes, it is something that depends on context. What is true enough on one occasion is not true enough on another. The standards of precision in force are different from one conversation to another, and may change in the course of a single conversation. Austin’s ‘France is hexagonal’ is a good example of a sentence that is true enough for many contexts, but not true enough for many others. Under low standards of precision it is acceptable. Rise the standards and it loses its acceptability. (Lewis, 1979, 244–5)
As I suggested, the idea that, with respect to borderline cases, people can ‘go either way,’ as I understand it, can indeed be seen as providing further ways in which accommodation can alter the standards of precisions in force in a given conversation. For consider the following conversation between Jason and Justin: [Jason has just finished having his shower. In their bathroom, there is both the borderline green towel and another, white one.] —Justin, please, pass me the towel, would you? —Which one you want? —I don’t know . . . whichever . . . Just give me the green one, but please hurry up, I’m freezing!
336
Dan L´opez de Sa
I submit that, in a conversation such as this, participants would accommodate by relaxing standards of precision dramatically, so as to count sentences as ‘The towel is green’ at the context as true, regardless the fact that the part of the range of reasonable delineations which do so may be not large at all. But all this is, I take it, compatible with my main claim that, with respect to borderline cases, people typically don’t (and shouldn’t) take a view: in the absence of the particular knowledge, purposes, etc., that we naturally imagined partly informing the conversational score, one would regard an unqualified utterance of ‘The towel is green’ by Jason at his context to be simply a ‘macho’ response.
18.3.3 Forced responses Following Diana Raffman 1994, Shapiro considers the responses in a ‘forced march’ scenario—where subject are asked to say (say) ‘yes’ or otherwise to the question ‘Is this towel green?’ concerning items that conform a sorites series—with respect to (among others) borderline cases. In her response, Rosanna Keefe wonders: Is it reasonable to draw any significant conclusions from the response subjects are driven to make when they are marched through a Sorites series and forced to judge each case either one way or the other? . . . [S]uppose you make subjects respond with ‘yes’ or ‘no’ to questions involving unfulfilled presuppositions; e.g. you ask them ‘Have you stopped φ-ing?’ when they’ve never φ-ed. They may be reluctant to answer yes or no—both answers are misleading—but they may nonetheless choose one of those answers when forced. Surely their choice in that situation should not be taken as deeply significant, nor as helping to illuminate the semantics of sentences involving unfulfilled presuppositions. . . . Second analogy: reading too much into the response to forced march paradoxes seems rather like forcing someone to guess the weight of something and then taking that guess to reveal that the subject believes that the weight is exactly that. (Keefe, 2003, 79)
I do not want to assess here whether Keefe is right in these comments, nor how this would affect the tenability of the contextualist proposals of Raffman and Shapiro. For my present purposes, a much weaker and rather uncontroversial remark is pertinent. Whichever way one conceives of the relevance of these forced responses, the fact that in situations like those envisaged people are asked to issue them is compatible with my main claim. For, indeed, it would seem that awareness of the past items in the series, of the likely future one, and their respective similarity in the relevant respects, among other things, gives rise to a peculiar conversational score in which the forced responses in question can be regarded as admissible. But this is so even if, typically, one would not (and should not) issue them.
18.3.4 Hesitant responses Third Possibility can be seen as the generic view that, if sentence s at context c is borderline, then it has some kind of third status incompatible with each of the poles,
How to Respond to Borderline Cases
337
truth and falsity—be it lacking a truth value, possessing a third value, or what have you. The view of vagueness as semantic indecision is a paradigm case of Third Possibility. Call verdict a judgement to the effect that something is F , or that it is not F —where the question of whether something is or is not F might be borderline. Verdict Exclusion says that with respect to borderline cases no such verdict constitutes knowledge. Both the view of vagueness as semantic indecision and epistemicism are paradigm cases of Verdict Exclusion. One consequence of my main claim in this chapter might be a plea for Verdict Exclusion. As Crispin Wright—to whom these labels are due—says: According to Verdict Exclusion, one ought, all things considered, to offer no verdict about a borderline case and to have no opinion which could be expressed in such a verdict. (Wright, 2003, 92)
In effect, I have claimed, intuitively, and in sharp contrast with the case of Hannah and Sarah regarding whether Homer is funny, Jason and Justin would typically not, and should not, offer a verdict as to whether the towel is green, nor have any opinion which could be expressed in such a verdict. This is accounted for by two of the main views about the nature of vagueness, and is compatible with their issuing ‘macho,’ admissible, forced, and, as we are now about to see, with their issuing hesitant responses. As Wright points out, The manifestation of vagueness, in the kinds of case we are concerned with, is not a consensus on certain cases as borderline—not if that is to be a status which undercuts both polar verdicts. Rather, the impression of a case as borderline goes along with a readiness to tolerate other’s taking a positive or negative view—provided, at least, that their view is suitably hesitant and qualified and marked by a respect for one’s unwillingness to advance a verdict. (Wright, 2003, 92–3, my emphasis)
I think, however, that the defender of Third View and Verdict Exclusion can—with a qualification to come—fully appreciate this insight. For what her view excludes, as we have seen, is that people typically offer—non-hesitant, unqualified—verdicts concerning borderline cases. Jason and Justin can indeed give such responses, as in Williamson’s ‘macho’ opinionated community. But they would typically not do so, nor should do so. This does not mean that they should issue no response at all, refusing to form any opinion on the matter whatsoever. They may eventually refuse to do so, but in most contexts it would be more natural (and rational) for them precisely to issue the suitably hesitant and qualified responses: ‘yeah . . . it’s kind of green,’ ‘sort of is’n sort of isn’t,’ ‘ . . . greenish . . . ,’ ‘it’s more green than blue, I guess’—or even ‘it’s green,’ which the appropriate gestural and/or intonational vagueifying markers. So, the defender of Third View and Verdict Exclusion can, it seems, fully appreciate the insight contained in the second part of Wright’s quote. She would probably resist, and this is the qualification announced above, the remark in the first part. For, she might hold, the predicted consensus on certain cases as borderline can indeed take the form of people precisely issuing the suitably hesitant and qualified opinions—not necessarily confining themselves to an aseptic agnostic silence. What I
338
Dan L´opez de Sa
am in effect suggesting is that Third View and Verdict Exclusion can indeed turn out to be compatible with the contention that there is a characteristic psychological attitude of the sort advocated by Wright 2003 himself—see also, for a related proposal, Schiffer (2003).¹¹ (Of course, one may hold that vagueness is characteristically manifested by a certain way of responding to borderline cases and still hold that the nature of borderline cases has to do with semantic indecision, irremovable ignorance, and so on.) Substantiating this suggestion of mine is something I am not in a position to do here.¹² Fortunately, defending my main claim does not require it. For my main claim has been simply that people typically do not and should not respond to borderline cases by forming—non-hesitant, unqualified—verdicts concerning them. This is compatible with their forming an opinion—provided they are suitably hesitant and qualified. 18.4
C O N C LU S I O N
It seems that Hannah and Sarah may disagree as to whether Homer Simpson is funny, without either of them being at fault. They may typically form (non-hesitant, unqualified) judgements on the matter, and it is not clear at all that they should not: hence the appearance of faultnessness in their disagreement, which most people are, in the case at hand, inclined to endorse. By contrast, Jason and Justin do not typically form (non-hesitant, unqualified) judgements on whether the (borderline green) towel is green or not. That this is so is not only the intuitive view but also indeed respected and actually accounted for by paradigm cases of semantic and epistemic views on the nature of vagueness. And it turns out to be compatible with their issuing ‘macho,’ admissible, forced, and hesitant responses with respect to borderline cases. Thus Jason and Justin just lack the (eventually contrasting) judgements, which are the building blocks of apparent faultless disagreements. Borderline cases do not provide further cases thereof. Re f e re n c e s Andjelkovi´c, M. and Williamson, T. (2000), ‘Truth, falsity, and borderline cases’, Philosophical Topics 28, 211–43. Fine, K. (1975), ‘Vagueness, truth and logic’, Synth`ese 30, 265–300. Horwich, P. (1997), ‘The nature of vagueness’, Philosophy and Phenomenological Research 62, 929–35. ¹¹ Wright explicitly notes that the quoted reflections on the characteristic manifestation of vagueness ‘are, to stress, strictly inconsistent neither with Third Possibility nor, therefore, with Verdict Exclusion.’ 2003, 93 He nonetheless adds: ‘What they are inconsistent with is our knowing that either of those proposals correctly characterizes borderline cases—or better, if someone insist that either is a correct characterization, with there being any definite (known) borderline case in the sense of the characterization.’ 2003, 93. I am suggesting that they can be consistent in the latter case as well. ¹² I hope to attempt this elsewhere.
How to Respond to Borderline Cases
339
Keefe, R. (2003), ‘Context, vagueness, and the sorites’, in Jc Beall, ed., Liars and Heaps, Oxford University Press, Oxford. K¨olbel, M. (2003), ‘Faultless disagreement’, Proceedings of the Aristotelian Society 104, 53–73. Lewis, D. (1979), ‘Scorekeeping in a language game’, Journal of Philosophical Logic 3, 339–59. Reprinted in his Philosophical Papers vol. 1, Oxford University Press, 1983 (q.v.). ¨ (1980), ‘Index, context, and content,’ in S. Kanger and S. Ohman, eds, ‘Philosophy and Grammar’, Reidel, Dordrecht. Reprinted in Papers in Philosophical Logic, Cambridge University Press, 1998 (q.v.). L´opez de Sa, D. (2009a), ‘Can one get bivalence from (Tarskian) truth and falsity?’, Canadian Journal of Philosophy 39, 273–82. (2009b), ‘The many relativisms: Index, contex, and beyond’ in S. D. Hales, ed., The Blackwell Companion to Relativism, Blackwell, forthcoming. MacFarlane, J. (2003), ‘Future contingent and relative truth’, Philosophical Quarterly 53, 321. (2005), ‘Making sense of relative truth’, Proceedings of the Aristotelian Society 105, 321–39. Raffman, D. (1994), ‘Vagueness without paradox’, Philosophical Review 103, 41–74. Shapiro, S. (2003), ‘Vagueness and conversation’ in Jc Beall, ed., Liars and Heaps, Oxford University Press, Oxford. Schiffer, S. (2003), The Things We Mean, Oxford University Press, Oxford. Williamson, T. (1994), Vagueness, Routledge, London. (1997), ‘Reply to commentators’, Philosophy and Phenomenological Research 62, 945–53. (2000), Knowledge and Its Limits, Harvard University Press. Wright, C. (1992), Truth and Objectivity, Harvard University Press, Cambridge. (2003), ‘Vagueness: A fifth column approach’ in Jc Beall, ed., Liars and Heaps, Oxford University Press, Oxford.
This page intentionally left blank
PA RT I I T H E LO G I C O F VAG U E N E S S
This page intentionally left blank
V Supervaluationism
This page intentionally left blank
19 Supervaluationism and the Report of Vague Contents Manuel Garc´ıa-Carpintero
In two recent papers, Schiffer (1998, 196–8; 2000, 246–8) advances an argument against supervaluationist accounts of vagueness, based on reports of vague contents. Suppose that Al tells Bob ‘Ben was there’, pointing to a certain place, and later Bob says, ‘Al said that Ben was there’, pointing in the same direction. According to supervaluationist semantics, Schiffer contends, both Al’s and Bob’s utterances of ‘there’ indeterminately refer to myriad precise regions of space; Al’s utterance is true just in case Ben was in any of those precisely bounded regions of space, and Bob’s is true just in case Al said of each of them that it is where Ben was. However, while the supervaluationist truth-conditions for Al’s utterance might be satisfied, those for Bob’s cannot; for Al didn’t say, of any of those precisely delimited regions of space, that it is where Ben was. From a perspective more congenial to supervaluationism than Schiffer’s, McGee and McLaughlin (2000, at 139–7) pose a related problem about de re ascriptions of propositional attitudes and indirect discourse. The same difficulty is gestured at in this argument: ‘there are additional concerns about the ability of supervaluational proposals to track our intuitions concerning the extension of ‘‘true’’ among statements involving vague vocabulary: ‘‘No one can knowledgeably identify a precise boundary between those who are tall and those who are not’’ is plausibly a true claim which is not true under any admissible way of making ‘‘tall’’ precise’ (Wright 2004, 88). In an earlier version of the material that I will present here (Garc´ıa-Carpintero 2000) I replied to Schiffer’s argument that supervaluationism has an independently well-motivated defense. The response is essentially based on the point that the occurrence of ‘there’ in Bob’s utterance (and of ‘tall’ in Wright’s argument) occurs An earlier version of this chapter was presented at talks at the university of Navarra and Arch´e, St Andrews; I thank the audience for criticisms and suggestions. My work has benefited from comments by Pablo Cobreros, Richard Dietz, Cian Dorr, Dan L´opez de Sa, Josep Maci`a, Daniel Nolan, Manuel P´erez Otero, Timothy Williamson, and Crispin Wright. Thanks also to Michael Maudsley for his grammatical revision. Financial support was provided by the research project HUM2006–08236, funded by the CICYT, Spanish Government, and by a Distinci´o de Recerca de la Generalitat, Investigadors Reconeguts 2002–2008, DURSI, Generalitat de Catalunya.
345
346
Manuel Garc´ıa-Carpintero
in indirect discourse, and supervaluationists may allow that it shifts its referent there.¹ Schiffer’s (2000b) reply to this response shows that it was not made sufficiently clear.² In this chapter I will try to improve on that score. In his more recent reply, Schiffer (2000b, 325) dismisses a proposal like the one I will make, mainly because it ‘undermines . . . a leading virtue of supervaluationism . . . its implication that vagueness is . . . not a feature of the world.’ I will argue that my reply does not undermine the fundamental contentions of the supervaluationist account. Suppose that, in a context where the size of a given rod is being discussed, Alex utters (1) while placing his symmetrically extended hands one opposing the other at a certain distance: (1) The rod was this length. In uttering (1) Alex makes an assertion, the kind of speech act that we routinely classify as true or false and has therefore truth-conditions, which illustrates the sort of data that theories of vagueness attempt to account for. The basic datum, put in a way as neutral as possible among possible potentially conflicting accounts, is this: the facts about the rod that Alex wanted to report might be such that it is indeterminate whether (1) is true, and it is indeterminate whether (1) is false; the size of the rod being discussed in the context might be a borderline case of the type of length that Alex signified with the predicate ‘was this length’. Call this ‘DV’, the datum of vagueness. Supervaluationism is an account of vagueness that upholds certain claims for which DV poses a problem requiring theoretical elucidation. Or, rather, it is not supervaluationism per se that provides the account. Supervaluationism is a mathematical model-theoretic technique, and, as McGee says (1998, 156): ‘It has been thought that the model theory provides a deep explanation of the way we use vague language; specifically, it has been thought to explain the fact that we are able to use classical logic even in the face of semantic indeterminacy. But that can’t be right. Model theory is just mathematics, and, as such, it can’t explain anything about language use.’ The explanation is provided by a philosophical account that applies the model theory. It concerns the nature of vague language, illustrated by (1), and distinguishes itself from others by upholding those intuitions. Following David Lewis, I will refer to the explanatory philosophical theory as vagueness as semantic indecision,‘VSI’. A first claim with which DV is prima facie in conflict is the correspondence claim. Language and thought are representational at their root: some expressions are semantically substantively related to objective, mind- and language-independent objects. Consider (1). As we said, it is used to make an assertion, assessable as true or false, and has therefore a certain truth-condition such that, together with the facts of the actual world, determines (1)’s truth value (and, together with the facts of ¹ After the reference shift, the term still refers indeterminately because of higher-order vagueness. The arguments here discussed differ from objections to supervaluationism based on higher-order vagueness; hence, for the sake of simplicity I will ignore it here. ² While he rejects my answer, he proposes ‘to stay with the topic’ so as ‘to consider a supervaluationist response I was too quick to dismiss’ (Schiffer 2000b, 322); the response is in substance the one I was intending to convey.
Supervaluationism and the Report of Vague Contents
347
other possible worlds, its truth value across possible worlds). This truth-condition is compositionally determined by (1)’s logical form, its semantically relevant syntactical composition out of lexical units and phrases formed from them. The correspondence claim is that (1)’s truth-condition is such that, if it is met and (1) is true, there is a mind- and language-independent truth-maker in the actual world making it so, on which (1)’s truth would then depend. In particular, ‘this length’ in (1) contributes to (1)’s truth-condition a mind- and language-independent object constituting that truth-maker, a specific length. Secondly, there is the clear-cut world claim; this is the contention that the objective, mind- and language-independent world does not include vague objects, kinds or properties. D. Lewis provides a compelling rationale for it: ‘I doubt that I have any correct conception of a vague object. How, for instance, shall I think of an object that is vague in its spatial extent? The closest I can come is to superimpose three pictures. There is the multiplicity picture, in which the vague object gives way to differences between precisifications, and the vagueness of the object gives way to differences between precisifications. There is the ignorance picture, in which the object has some definite but secret extent. And there is the fadeaway picture, in which the presence of the object admits of degree, in much the way that the presence of a spot of illumination admits of degree, and the degree diminishes as a function of the distance from the region where the object is most intensely present. None of the three pictures is right. Each one in its own way replaces the alleged vagueness of the object by precision. But if I cannot think of a vague object except by juggling these mistaken pictures, I have no correct conception’ (Lewis 1993, 27).³ Finally, we have the claim that the logical validity of our ordinary arguments is to be accounted for ultimately on the basis of the classical, Tarskian model-theoretic validity of arguments, by formalizing them in the languages devised by logicians. Now, relative to our illustrative case (1), we can see how the three claims create a difficulty in the presence of DV. For given the third, a predicate like ‘was this length’ in (1) should signify a subset of a domain of discourse, a class of lengths; given the first and the second, this should be a class containing a precisely delimited length (one thus to which any given length either belongs or does not belong, tertium non datur). This conflicts with DV, unless we could account for it on epistemic grounds; but supervaluationists assume that this is excluded by the notion that semantic properties in general, and the truth-conditions in particular of speech acts and thoughts, depend on their role in rational activities in which conscious, potentially reflective beings like us engage, and that as a result such vagueness as it is illustrated by (1) is not a matter of ignorance.⁴ It is here that VSI, vagueness as semantic indecision, together with the supervaluationist technique, comes to the rescue, reconciling the claims with DV. As Williamson ³ Elaborating on suggestions from Evans, McGee (1998) provides an argument against the view that a term like ‘Kilimanjaro’, intending to refer to a mind- and language-independent mountain, refers to a vague object. ⁴ See, for instance, Horgan (1997) as an expression of this well-known form of skepticism about Williamson’s (1994) epistemic theory of vagueness.
348
Manuel Garc´ıa-Carpintero
(1994, 142) aptly puts the idea:⁵ ‘the vagueness of a language consists in its capacity in principle to be made precise in more than one way. Not every substitution of precise meanings for vague ones counts as making the language precise . . . vague meanings are conceived as incomplete specifications of reference. To make the language precise is to complete these specifications without contradicting anything in the original content.’ As required by our three claims, the intended models for our discourse are the sharp models for, say, a logician’s first-order language; they allow for classical, bivalent definitions of truth in a model. Vagueness is due to the fact that ‘our thoughts and practices do not pick out a unique model as the actual model. They pick out a class of models’ (McGee 1998, 154). As McGee puts it, the fundamental hypothesis of VSI is that the semantics of a vague language can be described by singling out an appropriate class of models such that a sentence is determinately true if and only if it is true in every model in the class. According to VSI, there are two notions of truth required to account for DV while validating the claims. There is the fundamental notion involved in stating the truth-conditions of our assertions and judgments, given the representational character of language and thought. This is the fundamental non-bivalent determinate truth or super-truth, which comes handin-hand with a related correspondence notion of reference; the adjustment required by the correspondence claim in view of vague sentences such as (1), according to VSI, is that they do not just represent a unique truth-maker, but a plurality thereof. And there is, in addition, the semantically ancillary notion of truth, the bivalent truth in a model, and the related notion of reference.⁶ Let us consider now the original problem based on indirect discourse posed by Schiffer (1998, 197): ‘Suppose that in uttering ‘‘Harry is bald’’, Renata said that Harry was bald. Then the sentence ‘‘Renata said that Harry was bald’’ is true. But the supervaluationist must say that it wouldn’t be true if the that-clause in [it], ‘‘that Harry was bald’’. . . indeterminately referred . . . to various precise propositions . . . not one of those precisifications will be true, since, even taking into account the vagueness of ‘‘say’’, Renata obviously didn’t say any precise proposition . . . Evidently, then, the supervaluationist must say that [its] that-clause refers to the vague proposition that Harry is bald’. In my reply, I relied on the following theoretical basis: ‘propositional attitude verbs . . . express relations between agents and interpreted logical forms (ILFs). ILFs are annotated constituency graphs or phrase-markers whose nodes pair terminal and non-terminal symbols with a semantic value’ (Larson and Ludlow 1993, 305). Larson and Ludlow’s semantic values are classical semantic values: objects for terms, sets for predicates, truth values for sentences. On an alternative version (Pietroski, 1996), symbols are paired with Fregean senses in ILFs (which, in their turn, determine semantic values). ILFs, under either of those proposals, are the sort of entity that can be vague, in the sense that they admit different precisifications, and admit thereby a supervaluationist treatment. On Larson and Ludlow’s version, vague ILFs can be ⁵ Williamson is far from accepting it, of course. ⁶ This corresponds to the distinction by McGee and McLaughlin (1995) between the senses of truth answering, respectively, a ‘correspondence’ and a ‘disquotational’ conception.
Supervaluationism and the Report of Vague Contents
349
neither true nor false as a result of the fact that (ignoring higher-order vagueness) at least some terminal node (say, the one corresponding to ‘bald’ in Schiffer’s example) is paired, not with an appropriate semantic value, but with a class of them (its admissible precisifications). On Pietroski’s version, the same obtains if the mode of presentation with which the symbol is paired does not determine a unique semantic value, but a class of admissible ones. Before moving on to the problem posed by de re ascriptions which will be the main focus of the present chapter, let me emphasize that the proposal so far substantially agrees with Schiffer’s diagnosis, quoted two paragraphs back. To put it impressionistically, the supervaluationist agrees in accepting, besides the precise truth-makers indeterminately represented in vague sentences, some ‘vague entities’: i.e. vague contents, modeled along the ILF accounts. But, far from being incompatible with VSI, this is taken to be a crucial aspect of it. What matters is that truth and falsity (in their fundamental, non-ancillary senses linked to the correspondence claim) are ultimately determined relative to the class of precisifications.⁷ To make this more vivid, consider the following Schifferian argument. Supervaluationism treats all vague expressions as indeterminately referring to precise referents. In particular, supervaluationism treats ‘this length’ in (2) as indeterminately referring to precise lengths in a given class. However, none of those lengths is an observable property, if by ‘observable’ we understand something like discriminable by the naked eye. Thus, for any of the lengths to which ‘this length’ indeterminately refers, (2) is false. Hence, (2) should be superfalse, against compelling intuitions: (2) This length is an observable property. To provide an adequate response to this argument, it is enough to characterize a prima facie plausible way to reject it, compatible with VSI. The response could legitimately rely on contentious philosophical views, if they can be defended independently of the present issue. We do not need to go further into the details of a well-argued defense of the proposal; we do not need to defend the contentious philosophical assumptions. For we will have already shown that Schiffer has at most established a conditional: supervaluationism is false, unless such-and-such philosophical view is correct. A response of this kind to the Schifferian argument goes like this. A first premise is that the very same expression (‘this length’) that in a given context (its occurrence in (1)) refers, albeit indeterminately, to the precise lengths constituting the objective world, in a different context (its occurrence in (2)) refers to something else. The second premise is that, in addition to containing precise types of lengths, the world also contains what, in a manner of speaking, can be intelligibly called ‘imprecise lengths’, of which it is not determinate of all lengths in the first group ⁷ Garc´ıa-Carpintero (2007) elaborates on this, on the basis of more detailed considerations on the nature of truth and its relation to what is said. Keefe (2008)—a nice presentation of the main ideas defining supervaluationism—also emphasizes the centrality of quantification over precisifications to the account, and its compatibility with ‘vague entities’ of some such representational sort.
350
Manuel Garc´ıa-Carpintero
whether or not they instantiate them. Combining these two premises, we can counter the Schifferian argument by saying that, although the occurrence of ‘this length’ in (1) should be semantically treated following the supervaluationist guidelines (it indeterminately refers to lengths of the precise kind), the one in (2) somehow shifts its reference, determinately denoting instead a length of the imprecise sort.⁸ The second premise might superficially appear to be incompatible with the philosophical motivation we have provided for supervaluationism. However, let us reflect more carefully on it, in order to clarify the qualification ‘in a manner of speaking’. Does it follow from the three claims that VSI tries to accommodate that there are no vague entities in the world at all? It is not just that the answer to this is negative; it should be clear that VSI rather requires that the world include vague entities. VSI only assumes that the objective, mind- and language-independent world does not contain vague entities, and that truth and falsity is ultimately to be accounted for on the basis of supervaluationist quantification over those entities. However, VSI assumes that representational facts create vagueness, and representational facts are, of course, facts (albeit obviously not mind- and language-independent facts). They induce (in a manner of speaking) new properties and kinds instantiated by the precise objects constituting the objective world, which are (in a manner of speaking) in their turn new objects, potential objects of reference, which can in a clear sense be called ‘imprecise’.⁹ The first premise is also in good philosophical standing. It has it that the very same expression that in a context refers indeterminately to entities in the mind- and language-independent world, in a different context might refer (whether determinately or indeterminately, depending on the issue of higher-order vagueness, which we are putting aside here) to the indeterminately instantiated objects induced by the representational fact involving indeterminate reference in the previous context. Fregean theories assert the existence of this kind of systematic ambiguity to account for quotation and direct discourse in general, and for indirect discourse.¹⁰ This is what, on the present suggestion, happens to ‘this length’ in (2). It does not refer indeterminately to a length, but (as it were) to a new kind of ‘lengths’, a length as referred to by a demonstrative expression with the contextual help of a certain way of grasping lengths. This way is constituted by perceptual experiences of the same kind as that on which the speaker is relying, and takes his audience to be relying, in the context of his utterance (2). This way of grasping lengths is distinguished by its not being able to discriminate among a given set of (precise, as there are no others in the ⁸ Remember that, for the sake of simplicity, we are ignoring higher-order vagueness. A more realistic treatment should also use the supervaluationist strategy with respect to ‘this length’ in (2), allowing that it indeterminately refers to a length of the imprecise variety. ⁹ The mechanism though which ‘precise’ and ‘imprecise’ acquire a new sense is the same creating metonymies and other cases of ‘semantic transfer’; this is why I qualify my claims with ‘in a manner of speaking’. All these apparent references to, and quantification over, imprecise ‘objects’ should at a fundamental level be subjected to a fictionalist explanation; see Garc´ıa-Carpintero forthcoming. ¹⁰ As I indicated in my original contribution, we do not need to have recourse to a strictly speaking Fregean theory to justify this; a theory which attributes the shift in reference to the implicit presence in the utterance of a ‘hidden-indexical’ could serve as well, and in fact my own Davidsonian sympathies when it comes to the account of quotation suggest that much.
Supervaluationism and the Report of Vague Contents
351
mind- and language-independent world) lengths. It induces a mind-dependent sort of (mind-independent, precise) length, an ‘imprecise’ kind of length which is not just one precise length instantiate, but several—a sort that counts thereby as imprecise.¹¹ Thus, for the present purposes, we can take the semantic value of ‘this length’ in (2), in contrast to (1), to be a length of such a sort. These contentions involve no violation of the fundamental assumptions of VSI; on the contrary, they are to be properly justified ultimately on their basis. For these imprecise lengths are in effect representational entities, entities constituted by their role in representational activities; and the supervaluationist apparatus of precisifications is still required to obtain the truth-conditions of utterances—and mental states—whose content they help characterize, such as our original (1). This puts me in agreement with the main claim in Merricks (2001), that VSI is either a form of metaphysical vagueness, or a form of epistemic vagueness, by my embracing the first disjunct. Notice, however, that this is only because, in characterizing metaphysical vagueness, Merricks does not distinguish, as I have done, among entities in general, those responsible for fundamentally accounting for the semantic values of expressions, in particular the truth values of assertions and judgments. Metaphysical vagueness just consists for him in that ‘for some object and some property, there is no determinate fact of the matter whether the object exemplifies the property’ (145); properties are understood here in a fully liberal, ‘abundant’ sense. Merricks then considers a proposal like the one I have made concerning the sentence ‘Bald’ applies to Harry, with ‘Harry’ denoting a borderline case of baldness. Against the perhaps more orthodox supervaluationist line, on which such a sentence signifies many different precise propositions, I have granted that there is a sense in which such a sentence expresses a vague proposition, one ascribing to Harry the vague property λx(‘Bald’ applies to x): think of ‘Bald’, as previously suggested, as referring to a semantically individuated word. However, whether or not an object exemplifies such vague representational property is to be accounted for, at a fundamental level, relative to supervaluationist quantification over precise properties (as there are no others at the fundamental level). Merricks (op. cit, 155–6) is right, however, that supervaluationist arguments against metaphysical vagueness, such as the one by Lewis quoted before, do not mention any distinction between fundamental and non-fundamental entities of the VSI account. But it is not that difficult to insert adverbs such as ‘fundamentally’ at the proper places, and it seems to me more charitable to do so. Thus, what is unintelligible is not that there is a vague ‘object’, with an indeterminate spatial extent—we have already envisaged vague ‘kinds’ (vague sorts of lengths), and presently we will be considering vague ‘‘particulars’’ (vague locations). What is unintelligible is rather that such objects have an explanatory fundamental role in accounting for the truthconditions of our assertions and judgments. And, on the present view, they don’t; those are explained in terms of supervaluationist quantification over precise entities.¹² ¹¹ At the risk of boring the reader, I should insist that I am ignoring higher-order vagueness. ¹² Williams (2008) diagnoses a loophole in the usual semantic ways of dealing with Evans (1978) infamous argument against vague objects, which question the λ–conversion step. That
352
Manuel Garc´ıa-Carpintero
Let us now move on to arguments involving de re ascriptions. Schiffer argues that a proposal along the previous lines cannot work in cases involving singular terms, such as (3) below, or ‘there’ in (5), taken as a report of Al’s utterance of (4): (3) (4) (5) (6) (7)
Alex said that the rod was this length. Ben is at that mountain. Al said that Ben was there. ∃x(x is where Al said Ben was). There is where Al said Ben was.
Notice that both ‘this length’ in (3) and ‘there’ in (5) are de re at least in that, say, (5) entails (6) and, (‘by demonstrative specification’), (7). ‘Here . . . the supervaluationist evidently has to take her standard line: in a sentence of the form ‘‘There is such-and-such’’, ‘‘there’’ must be taken to indeterminately (or partially) refer to each member of a set of precise places, the set of places that can be used to give the supervaluationist truth-conditions of the sentence in which the demonstrative occurs’ (op. cit., 198). This is how Schiffer’s argument goes: ‘There was no problem initially in the idea that the that-clause in ‘‘Renata said that Harry was bald’’ referred to a vague proposition, because there was no problem initially in the idea that ‘‘bald’’ in that that-clause expressed a vague property, a property with a penumbra. The problem with (5) comes when we try to make sense of the idea of there being a vague place to which ‘‘there’’ might refer. What could possibly be both a place, a region of space, and fail to have precise boundaries? It might be thought that the supervaluationist could take a vague proposition to be a set of precise propositions, those used to give the supervaluationist truth conditions of the vague proposition. Then the reference of ‘‘there’’ can be taken to be a set of precise places. But I don’t think this will work . . . A set of places is not a place. The problem is that the occurrence of ‘‘there’’ in (5) is de re and thus occurs as a demonstrative seeking to refer to a place’ (op. cit., 198).¹³ step cannot be validly instantiated with referentially indeterminate expressions; but their referential indeterminacy could be the result of ontic vagueness, and not its cause. Thus, the argument does not after all dispose of ontic vagueness, even granting the controversial assumptions it requires. Williams goes on to provide a model for ontic vagueness, based on an ersatzist conception of possible worlds. On such a view, worlds are abstract maximal properties that the one and only Reality could have; ‘the’ actual world is one more abstract property, and not Reality itself. This allows that there is not just one ‘actuality’, if for w to be actualized is for w not to be determinately uninstantiated. On this view, propositions understood as sets of worlds—properties predicated of Reality, in assertions and judgments—are themselves indeterminately instantiated, and thus vague; by Merricks’s lights the view counts as propounding metaphysical vagueness. However, the truth or falsehood of assertions and judgments expressing those vague propositions is ultimately explained on the basis of supervaluationist quantification over precise propositions, and thus, to the extent that I find this view intelligible, it is just a form of VSI. ¹³ The problem that Weatherson (2003, 482) takes to be Schiffer’s, and for which he offers a solution, is that for (4) to be true, Al must have said of every candidate-mountain that Ben was there; but Al ‘could not have said all those things’. But this does not distinguish between the problem posed by predicates, as in Schiffer’s original example with ‘Harry is bald’, previously discussed, and the problem posed by singular terms, as Schiffer does here. The problem Schiffer poses is not that supervaluationism has Al saying too many things, if (4) is to be true, but that none of those
Supervaluationism and the Report of Vague Contents
353
But Schiffer’s conclusion does not follow. Let us take Kaplan’s (‘Quantifying In’) account of the truth-conditions of de re ascriptions, as in ‘Joan does not know that her best friend betrays her’, which we could formalize (only for the sake of the discussion, for a full account should be more complicated) as (8), where ‘R’ stands for an appropriate representational relation, one (involving acquaintance, or what have you) sustaining correct de re ascriptions between constituents in ILF and their semantic values, ‘VPA’ for any verb ascribing propositional attitudes, and the Greek variables such as ‘α’ range over modes of presentation (in Kaplan’s original presentation), over parts of ILF on the view of attitude ascriptions I am assuming here: (8) ∃α(R(α, τ , S) ∧ S VPA σ (α)) On such a view, we describe the ILF in indefinite terms, existentially quantifying over some of its nodes, by indicating only its semantic value—the omitted complication consists in that typically some additional information about the mode of presentation α is given in de re ascriptions, such as that it is a mode of presentation of a mountain, it is demonstrative, or, indeed, it is (im)precise. Where α is a constituent of a vague ILF, the simplest theory is that an instance of this schema obtains just in the case that τ is one of the several semantic values with which the vague term in α is paired (one of the semantic values determined by its paired sense). Under this interpretation, the ascription (5) may well be true. It is misleading to object, as Schiffer does, that ‘Al didn’t say, of any precise place, that it was where Ben was’; for this rings true only by contextually suggesting that, under the proposal, the truth of (5) requires Al to have expressed a precise thought (one with a precise ILF); the omitted complication would properly deal with this, if it is explicitly specified that α was indexical and vague.¹⁴ The present proposal rejects this claim by McGee: ‘In order for us to have de re beliefs, at least on our usual understanding of them, our thoughts and practices have to pick out one particular thing as the object the belief is about’ (1998, 147); for (5) is things, being precise, are good candidates for reporting what Al said; moving to saying-relations to imprecise contents is OK when we only consider predicates, as in ‘Renata said that Harry was bald’, but de re ascriptions, according to Schiffer, make this move irrelevant. Unlike Weatherson’s proposal, mine properly deals with the problem posed by de re ascriptions which I take Schiffer to be raising here. I will come back later to Weatherson’s views. ¹⁴ In her contribution to this volume, Rosanna Keefe (2009) provides a more orthodox reply, which avoids vague entities by assuming only the penumbral coordination of the precisification of the embedded sentence in an attitude report, and that of the reported sentence or mental item. She discusses an objection: ‘Someone might object to the above solution that ‘‘Renata said that Harry is bald1 ’’ should come out determinately false (where bald1 is a precisification of ‘‘bald’’), whereas on the above treatment, it comes out indeterminate. (Schiffer, 2000, 248, suggests something like this objection.) But, this intuition, if there is one, is far less strong than the intuition that ‘‘Renata said that Harry is bald1 ’’ should not be determinately true.’ On my account, however, we could say more, if we take the ‘suggestion’ that the representational device used by Renata was a precise one, produced by the use of a precise device in the ascription, to go into the truth-conditions of the report; for, if so, the ascription would turn out to be determinately false after all. (Otherwise, we could appeal to a pragmatic explanation of the incorrect impression that the ascription is false.) I take it that the possibility of thus capturing the intuition, even if it is a weak one, is an advantage of my proposal.
354
Manuel Garc´ıa-Carpintero
a correct de re report of a de re assertion, but neither the reported asserter, nor the reporting utterer, need in any way to have been in a position to ‘pick out’ one in particular of the several precise mountains that could legitimately be invoked in order to precisify the utterances. The proposal so far does not require that precisifications are coordinated so as to assign the same candidate-mountain to ‘that mountain’ in (4) and ‘there’ in (5), as in Weatherson’s (2003, 482–3) and Keefe’s (2009) more orthodox replies to Schiffer. (4) is (super-) true just in case, for each admissible value for ‘that mountain’, (4) is true; (5) is true just in case, for each admissible value for ‘there’, Al was in the proper R-relation with it through whatever corresponds in the thought he expressed to ‘that mountain’ in his utterance (4). But perhaps the proposal does require such coordination among precisifications implicitly, in the conditions for a candidate-mountain to be an admissible value for ‘there’ in (5). For in specific contexts, it may be part of the intended meaning of those singular terms in de re ascriptions that they are in a sort of anaphoric relation with corresponding ‘singular terms’ in the vehicle for the reported propositional attitude.¹⁵ I will conclude by discussing a different, but related objection, made by McGee and McLaughlin (2000, 145–6). They consider an atom at or around the base of Kilimanjaro, called Sparky, and define Kilimanjaro(+) ‘to be the body of land constituted . . . by the atoms that make up Kilimanjaro together with Sparky [and] Kilimanjaro(−) [to] be the body of land constituted . . . by the atoms that make up Kilimanjaro other than Sparky’ (2000, 129); and they argue as follows, about someone like Al in the previous example: ‘In fact, there isn’t anything, either in his mental state or in his neural state or in his causal relations with his environment that would make one of Kilimanjaro(+) and Kilimanjaro(−), rather than the other, the thing [Al’s assertion] is about. [The thought he expressed] can with equal justice be imagined to be the singular proposition obtained from the propositional function described by the English open sentence ‘‘that it is the snow-capped mountain within sight of the equator where Ben is’’ by supplying Kilimanjaro(+) as argument and the proposition obtained by supplying Kilimanjaro(−) as argument. But exactly one of those propositions is true. The possibility that [Al said] all of the countless billions of singular propositions obtained by supplying Kilimanjaro candidates as arguments of the proposition function can be readily dismissed, for it implies that, no matter how careful and knowledgeable a geographer [Al] may be, his every true [thought] about Kilimanjaro is accompanied by countless billions of false [thoughts]’. In discussing this argument, we need to keep in mind a warning made by McGee himself (‘Kilimanjaro’, 152): ‘Just to make sense of the attachment of the word ¹⁵ As Keefe (2009) points out, Weatherson’s proposal that precisifications should be given wholesale, for every word in the language, is no modification of VSI, for penumbral connections, a fundamental ingredient of the supervaluationist account, are holistic in that way. That precisifications should be given in this holistic way not just for words, but for tokens thereof (or words-in-context), as Weatherson rightly insists, is a consequence of context-dependence in general, such as the long-term discourse anaphoric relations envisaged in the main text in particular. Keefe is nonetheless right that this coordination of token-precisifications raises further problems, which do not depend specifically on issues of vagueness.
Supervaluationism and the Report of Vague Contents
355
‘‘determinately’’ to an open sentence containing free variables is a bit of a stretch, since we primarily think of determinacy as an attribute of sentences. A sentence is determinately true, determinately false, or unsettled. We need to go beyond this familiar usage if we want to say of an object that it either determinately satisfies, determinately fails to satisfy, or is indeterminate with respect to an open sentence with one free variable’. This is a warning that Williams (2006) ignores, in providing an argument in support of Lewis’s (1993) supervaluationist solution to the problem of the many, a solution which differs from the equally supervaluationist one I prefer; let me elaborate, in order to provide a useful background for the discussion of McGee and McLaughlin’s argument. The supervaluationist solution I prefer has it that on every way of making the language precise, exactly one of the many candidates for being the referent of ‘Kilimanjaro’ will count as a mountain. Williams (2006, 415) argues that, given what he takes to be the ‘standard’ treatment of ‘Definitely’ as applied to open sentences, this solution entails the falsity of (9): (9) ∃x Definitely (x is a mountain) Williams argues that this is bad news for supervaluationists, because it conflicts with their standard ‘confusion’ explanation for our intuitions regarding the mayor premises in sorites arguments. The standard explanation is that we confuse ‘Definitely ∃x . . .’, truly stating that there is a cut-off point in every precisification, with ‘∃x Definitely . . .’, falsely asserting that there is a definite such cut-off; i.e. we read existentially quantified claims in terms of the ‘∃x Definitely . . .’ scope relations, not the other way around. If this account is generally correct, he contends, we should read ‘there are mountains’ as in (9), and therefore (given the proposal to deal with the problem of the many we are assuming) judge it false, against what our intuitions in fact tell us. To preserve the confusion explanation, Williams proposes to adopt instead Lewis’s solution to the Problem of the Many, according to which all mountain-candidates are indeed in the extension of ‘mountain’,¹⁶ and thus (9) turns out to be true, assuming that ‘standard’ interpretation of the interaction of ‘Definitively’ with open sentences. But this will not do, because, generalizing the confusion explanation in the same way, we would read ‘there is exactly one snow-capped mountain within sight of the Equator’ as: ∃!x Definitely (x is a snow-capped mountain within sight of the Equator), and, assuming now the Lewis solution that Williams is arguing for, judge it false— which is not what we in fact do. Instead of arguing on the basis of claims about our intuitions very difficult to uphold in this area, it is in my view preferable to rethink the interpretation of the interaction of ‘Definitely’ with (what intuitively corresponds to) open sentences, attending to McGee’s own warning. Consistently with my account so far, I propose to appeal to a representational relation R (with a contextual parameter C instead of the subject parameter S), generalizing the previous proposal to interpret de re locutions in ascriptions of propositional attitudes. Thus, I propose to ¹⁶ Lewis deals with the problem by appealing to flexible standards for counting.
356
Manuel Garc´ıa-Carpintero
analyze sentences involving ‘quantifying in’ the Definitely-operator, such as (9), along these lines: (10) ∃x ∃α(R(α, x, C) ∧ Definitely σ (α)) Thus: there is an ILF-part α, which in the context C represents x, such that the logical form consisting in plugging α in the frame σ (x) is supertrue. Given my preferred, standard supervaluationist solution to the Problem of the Many, in any particular precisification only one of the many Kilimanjaro-candidates will be in the extension of ‘mountain’, and will therefore be an acceptable candidate for being represented in the context C by the relevant instance of α. This proposal is therefore compatible with the confusion explanation of sorites reasoning.¹⁷ This proposal allows us to diagnose the problem with McGee and McLaughlin’s claims about the singular propositions obtained by supplying Kilimanjaro(+) or Kilimanjaro(−) as arguments for the propositional function described by the English open sentence ‘that it is the snow-capped mountain within sight of the equator where Ben is’—in particular, the claim that ‘exactly one of [them] is true’. What is true is only that, assuming the supervaluationist solution to the problem of the many, in each precisification the English open clause ‘that it is the snow-capped mountain within sight of the equator where Ben is’ is made true at most by assigning to ‘‘it’’ as value one of Kilimanjaro(+) or Kilimanjaro(−). But the issue is what follows from this with respect to the correctness of de re ascriptions like (11): (11) Al said of n that it is the snow-capped mountain within sight of the equator where Ben is. Given the previous proposal, what McGee and McLaughlin’s considerations— including the assumption that Al is a careful and knowledgeable geographer—show is only that the conditions on a candidate-mountain to be an admissible value for the referential expression that Al used (the one on which ‘it’ in (11) is ultimately anaphoric), in particular the condition required to deal with the problem of the many that there is at most one snow-capped mountain saliently within sight of the Equator, will extend to the conditions a candidate-mountain should meet to be an acceptable value for ‘n’ in (11) in each precisification. Weatherson (2003, 488) is right in assuming that ‘there is a penumbral connection between the subject of [Al’s assertion] . . . ¹⁷ Williams (op. cit., 415) argues that on the standard supervaluationist solution to the problem of the many ‘the ability of the confusion hypothesis to explain intuitions about the sorites premise is undermined. To illustrate this, let us put the explanatory challenge in the following contrastive form. (a) In the original case presented above, where we have a range of emanations from Kilimanjaro to Glastonbury Tor, decreasing in height by a few metres from one to the next, we have strong ‘‘no cut-off’’ intuitions. (b) Consider a new range, which consists in Kilimanjaro standing next to Glastonbury Tor. In this scenario, we have strong intuitions that there is a cut-off: a mountain standing next to a non-mountain. The datum to be explained is the contrasting intuitions in the two cases (a) and (b).’ The present proposal to understand the interaction of ‘Definitively’ and intuitively open sentences accounts for this datum. In both cases, we read the existential quantifier outside the scope of the definitely-operator; in (a), we get a falsehood (but the narrow-scope reading is in the vicinity, which explains our confusion); in (b), understood as I have proposed, in terms of (10), we get a truth.
Supervaluationism and the Report of Vague Contents
357
and the word ‘‘mountain’’ ’.¹⁸ Thus, in a sense Al said ‘‘all of the countless billions of singular propositions’’ (the same sense in which, against Schiffer, Al did say, of every relevant (precise) location, that it was where Ben was). But this does not imply at all that ‘‘his every true [thought] about Kilimanjaro is accompanied by countless billions of false [thoughts]’’. Reckoning by the more intuitive counting in terms of vague thoughts, Al just had one true de re thought about Kilimanjaro. Let me summarize. Both in his original paper and in his reply to my original criticism, Schiffer argues against the kind of proposal I have made, by contending (correctly in my view, as I have said) that claims like (3) are de re in that they entail reports like (12): (12) This length is such that Alex said that the rod was it But in view of our discussion about (2), this does not pose new problems. That the report (3) is de re, as shown by the fact that the inference to (12) holds, only requires that ‘this length’ in it—as in (2)—still refers to a type of lengths, an imprecise kind instantiated by particular lengths. This is compatible with its referring to a kind individuated in part by mind-dependent matters (a perceptual way of grasping lengths), which accounts for its being an imprecise kind in accord with the intuitions that VSI tries to support.¹⁹ In his reply, Schiffer (2000b, 322) asked me to characterize ‘the nature of the modes of presentation’ under which Alex said something about a myriad precisely delimited lengths; to say how a set of lengths, which is not a property, can be a property of modes of presentation; and what the truthconditions of statements like (3) are. These requests are well taken, but I think I have met them here. The modes of presentation at stake are in part types of contextually salient perceptual experiences. The relevant property is the property of being a type of perceptual experience presenting any length in the given set to a perceiver experiencing it.²⁰ The truth-conditions can be given (with some licenses, mostly in the metaphoric reference to parts of contents) as follows: (3) is true iff Alex made an assertion whose propositional content ‘consist’ of a ‘part’ contributed by ‘the rod was’ and another ‘part’ signifying the imprecise kind of length determinately referred to by ‘this length’. As mentioned before, we may or may not additionally assume that a contextual indication of the sort of perceptual experience on which Alex contextually relied to refer to a length is part of the full characterization of that propositional content. ¹⁸ But this solution has nothing to do with Weatherson’s previous appeal to naturalness. I find it difficult to understand how, although ‘in reality Kilimanjaro(+) is no more natural than Kilimanjaro(−)’, nevertheless ‘according to any precisification, one of them will be more natural than the other, for precisifications determine content by determining relative naturalness.’ I cannot see how precisifications, which are arbitrary reinterpretations of the language, can determine naturalness; rather, naturalness and other facts about the language such as penumbral connections determine which of them are acceptable. ¹⁹ That is to say, it is compatible with the occurrence of ‘this length’ in (3) being only weakly de re, in terms of the distinction I made in my original reply to Schiffer, Garc´ıa-Carpintero (2000). ²⁰ By referring to sets we can of course refer to the properties determining them, when we are not in a fastidious mood.
358
Manuel Garc´ıa-Carpintero
Schiffer (2000b, 325) criticizes accounts of this sort on three counts: ‘First, it requires finding truth-conditional ambiguity’ in forms like (1) and (2), ‘when in fact those forms don’t seem ambiguous in any relevant way. Second, the move looks suspiciously like saying that the supervaluationism is to be limited to those cases that aren’t clear counterexamples to it. Third, it undermines what some will have thought was a leading virtue of supervaluationism—namely, its implication that vagueness is either not a feature of the world at all, but of our ways of describing it, or, failing that, a feature of the world that is wholly reducible to, a construct out of, non-vague features of the world. Evidently, the supervaluationist theory that survives doesn’t have this ‘‘virtue,’’ since it recognizes that vague objects and properties may have features not possessed by the precisifications of those objects and properties.’ As regards the third and main point, as we have seen, properly understood supervaluationism is not only compatible with this consequence, but actually requires it. The relevant claim that VSI tries to validate is only about the mind- and languageindependent world; the leading virtue of supervaluationism lies in its capacity to buttress this claim. The account, however, entails (rather than being incompatible with it) that our representational activities induce imprecise particulars, properties and kinds, possessing distinctive properties of their own; and these induced particulars, properties and kinds are, of course, also part of the wider world.²¹ Against what Schiffer says, this is compatible with their being ‘constructed out of’ the precise objects, at least in a sense which can be precisely explicated in terms of some form of supervenience: no difference in the imprecise objects, without a corresponding difference in precise objects. Schiffer’s main criticism is thus shown to depend on a misleading characterization of supervaluationism’s ‘leading virtue.’ As regards the first and second points, the claim of ambiguity, as I have suggested, can be motivated on independent, Fregean-like considerations. Schiffer only has validly argued for a conditional: if referential expressions never shift their referents in the way suggested by Fregean-like theories, then supervaluationism is wrong. However, the reader should only realize how wide-ranging ‘Fregean-like’ is in the antecedent of this conditional, to appreciate the extent to which its falsity is probable. As I said, even theories that explain the shift of reference attributing it to other expressions implicitly or explicitly present in the utterance (hidden-indexicals, or other expressions in the sentence) count, for present purposes, as Fregean-like. I conclude that Schiffer has not given us a compelling new argument against VSI. Weatherson (2003) and Keefe (2009) offer alternative solutions to Schiffer’s challenge, on which I have made some critical remarks before. The main difference is that they do not countenance vague entities, such as the vague representational items my proposal envisages. I think that in that way they miss what I see as its main virtue, that it allows us to capture the sense in which, as Schiffer insists, Al didn’t say, of any precisely delimited regions of space, that it is where Ben was; or the corresponding sense in which Wright’s claim in the quotation provided in the first paragraph is correct. ²¹ As I said before, I take this reference to imprecise entities to be amenable to a fictionalist treatment; but we do not need to go into this for present purposes.
Supervaluationism and the Report of Vague Contents
359
Re f e re n c e s Garc´ıa-Carpintero, Manuel (2000), ‘Vagueness and indirect discourse,’ Philosophical Issues 10, E. Villanueva, ed., Boston, Blackwell, 258–70. (2007), ‘Bivalence and what is said’, Dialectica 61, 167–90. (forthcoming), ‘Fictional entities, theoretical models and figurative truth’ in Frigg, R, and Hunter, M., eds., Beyond Mimesis and Convention—Representation in Art and Science, Springer. Horgan, Terence (1997), ‘Deep ignorance, brute supervenience, and the problem of the many’ in Philosophical Issues 8: Truth, E. Villanueva (ed.), Ridgeview, Atascadero, CA, 229–36. Keefe, Rosanna (2008), ‘Vagueness: supervaluationism,’ Philosophy Compass 3 (2), 315–24. (2009), ‘Supervaluationism, indirect speech reports and demonstratives’, this volume. Larson, Richard, and Ludlow, Peter (1993), ‘Interpreted logical forms,’ Synthese 95, 305–55. Lewis, David (1993), ‘Many, but almost one,’ in Ontology, Causality and Mind, J. Bacon, K. Campbell and Ll. Reinhardt, eds., Cambridge, Cambridge University Press, 23–38. McGee, Vann (1998), ‘Kilimanjaro’, Canadian Journal of Philosophy: Meaning and Reference, supp. vol. 23, A. Kazmi, ed., 141–63. McGee, Vann and McLaughlin, Brian (1995), ‘Distinctions without a difference’, Southern Journal of Philosophy, supp. vol. 33, 203–51. (2000), ‘The lessons of the many’, Philosophical Topics 28, 129–51. Merricks, Trenton (2001), ‘Varieties of vagueness,’ Philosophy and Phenomenological Research 62, 145–57. Pietroski, Paul (1996), ‘Fregean innocence,’ Mind and Language 11, 338–70. Schiffer, Stephen (1998), ‘Two issues of vagueness’. The Monist 81, 193–214. (2000a), ‘Vagueness and partial belief,’ Philosophical Issues 10, E. Villanueva, ed., Boston: Blackwell, 220–57. (2000b), ‘Replies,’ Philosophical Issues 10, E. Villanueva, ed., Boston: Blackwell, 321–43. Weatherson, Brian (2003), ‘Many many problems’, Philosophical Quarterly 53, 481–501. Williams, J. Robert (2006), ‘An argument for the many’, Proceedings of the Aristotelian Society 106, 409–17. (2008), ‘Multiple actualities and ontically vague identity’, Philosophical Quarterly 58, 134–54. Williamson, Timothy (1994), Vagueness, London: Routledge. Wright, Crispin (2004), ‘Vagueness: A fifth column approach’ in Jc Beall, Liars and Heaps, Oxford, Oxford University Press, 84–105.
20 Supervaluationism, Indirect Speech Reports, and Demonstratives Rosanna Keefe
According to the supervaluationist theory of vagueness, a vague sentence such as ‘Bob is tall’ is true iff it is true on every way of making it precise. In general, the truthconditions of sentences containing vague terms involve quantification over different ways of making the various vague components of the sentence precise.¹ In this chapter I consider whether, as Stephen Schiffer argues, this popular theory of vagueness is undermined by considerations about indirect speech reports. As a very brief summary of the potential problem, consider the sentence ‘Carla said that Bob is tall’. The worry is that this sentence will be true only if it is true on all ways of making precise its vague terms, including ‘tall’, resulting in the condition that the speech report is true only if Carla said that Bob was over 6.0001 feet tall and said that he was over 6.0002 feet tall etc. But, the objection goes, she clearly didn’t say any, let alone all, of those things. I will argue that the supervaluationist can satisfactorily deal with indirect speech reports in general, and I will offer solutions to the various problems raised by cases such as Schiffer’s. The most interesting cases involve demonstratives, I will argue, but these can be handled by the supervaluationist as well. 20.1
SCHIFFER’S OBJECTIONS
In his 1998, Schiffer sets up the problem as a dilemma.² With a vague sentence such as An earlier draft of this chapter was presented to the Third Navarra Workshop on Vagueness in Granada: I am very grateful to the participants and organizers, especially Elia Zardini, Pablo Cobreros and Maria Cerezo. For comments and advice on other drafts of this chapter, thanks to Jenny Saul, Sebastiano Moruzzi, and Richard Dietz. I am also very grateful to the Arts and Humanities Research Council who funded a period of leave during which this chapter was originally written. ¹ This is subject to penumbral constraints: we consider acceptable ways of making the whole language precise at once, respecting relations between different vague terms, such as the fact that nothing will count as both red and orange on a precisification of those two terms. On the supervaluationist theory of vagueness, see Fine 1975 and Keefe 2000. ² See also his 2000a, 246–8 and 2000b, 321–6, where the problems are presented differently.
360
Supervaluationism, Indirect Speech Reports, and Demonstratives
361
[a] Harry is bald, supervaluationists have to say one of two things about the propositional content expressed by that sentence. They can either say that propositions are precise and it is indeterminate which proposition is expressed by [a], or they can maintain that there are vague propositions and that [a] expresses one of them. If Harry is borderline tall then [a] will turn out neither true nor false on both options, either because of the divergence in truth value of the precise propositions that [a] indeterminately expresses, or because the vague proposition that it (determinately) expresses is itself neither true nor false in the context. The problems Schiffer identifies for both options involve indirect speech reports. Take [b] Renata said that Harry was bald The first option (involving indeterminate reference to many precise propositions) supposedly comes to grief because ‘in order for [b] to be true, according to the supervaluationist, it must be true under every way of precisifying the reference of its that-clause. Yet not one of those precisifications will be true, since, even taking into account the vagueness of ‘‘say’’, Renata obviously didn’t say any precise proposition, let alone all of the precise propositions to which the that-clause partially, or indeterminately referred.’ (Schiffer 1998, 197). In relation to the second option (involving vague propositions), Schiffer focuses on ‘Al said that Ben was there’, where this can also be reported as ‘there is where Al said Ben was’, which, he says, is surely not true for each precisification of ‘there’.³ We would equally expect this problem to arise with ‘Everest is what Al said Ben climbed’: the key feature is that the relevant singular term has wide scope, so I shall call this the de re problem. Before proceeding, I will clarify why the above argument ignores the vagueness of ‘says’. Like most other English expressions, ‘says’ is vague: there will be instances of ‘S says that p’ that intuitively are borderline due to the vagueness of ‘says’; for example, it might not be clear whether or not S has said that p, because he was muttering. Schiffer’s objections, however, can be run on a case where there is seemingly no unclarity about whether, say, Renata said that Harry was bald (for example, she uttered the very words ‘Harry is bald’ in appropriate circumstances). Then, assuming we are right about that intuitive classification of the report, it should be the case that, no matter how we make ‘says’ precise, the indirect speech report comes out true. The construction of the argument as a dilemma looks misplaced, however. For the truth-conditions of [b] turn on the truth values that result when we make precise those components, whatever we decide about the nature of propositions. Even if ‘Harry is bald’ determinately refers to a unique vague proposition, quantification over ³ He also objects that we cannot make sense of a vague place to be the referent of ‘there’, but I think we can pass quickly over this problem. Indeed, Schiffer himself has dropped it by his subsequent presentations of the problem (2000a and 2000b), arguing that the greater Boston metropolitan area could be a vague place. Note that on a Fregean conception of propositions, there would be no need for a vague place as a component of the proposition.
362
Rosanna Keefe
precisifications is still needed.⁴ Similarly, the de re problem is equally a problem for the supervaluationist who rejects vague propositions. The objection in relation to ‘there is where Al said Ben was’ is that the supervaluationist must maintain that Al said of each of the relevant range of precise places that Ben was there. And this is so whether or not they accept vague propositions. In section 20.3, I will argue that the general de re problem can be solved by the supervaluationist. I will then go on to ask whether there is a particular problem with demonstratives in relation to indirect speech reports (though not specifically de re ones). 20.2
I N D I R E C T S PE E C H R E P O RTS
The heart of Schiffer’s objection involving the sentence ‘Renata said that Harry was bald’, is summarized in the following quotation: ‘Renata obviously didn’t say any precise proposition, let alone all of the precise propositions to which the that-clause partially, or indeterminately, referred.’ (1998, 197). Supervaluationists must indeed say that ‘Renata said that Harry was bald’ is true on each precisification if it is to count as true simpliciter, as intuitively it should. But, as we’ll see, that doesn’t commit them to saying that she said any precise proposition. Consider a precisification, s1 , according to which ‘bald’ means bald1 , for some precisification of bald, and ‘Harry is bald’ says that p1 (for some precise proposition p1 ). When I use ‘bald’ in reporting Renata’s utterance, then according to precisification s1 , I mean bald1 by ‘bald’. Now, Renata uttered the words that, according to s1 , mean p1 , so surely according to that precisification, she did say that p1 and my report to that effect is true on that precisification. More generally, according to precisification si , Renata said that pi (where pi is precise) and according to the same precisification I report her as having said pi . According to different precisifications she said different precise things and is reported as having said different precise things. But it isn’t true (i.e. true simpliciter) that she said p for any precise p, for there is no precise proposition that, according to all precisifications she said. What she said differs according to the precisifications. So, it certainly doesn’t need to be the case that she said all of the precise propositions: that is clearly false on all precisifications, so false simpliciter. If A says ‘a is F’, and B says ‘A said that a is F’, then B’s report is true iff it is true on all precisifications. But, to put it somewhat loosely, whether B’s utterance is true on a precisification depends on what, according to that precisification, A says. To demand, for the truth of the report, that it is true (so true on all precisifications) that A said all of the precise things is, in effect, to recognize the variation between precisifications over what B is reporting A as having said, without acknowledging the corresponding variation over the actual content of A’s utterance according to those precisifications. ⁴ By analogy, compare an unusual supervaluationist who maintained that there were vague properties, and that ‘bald’ determinately referred to one of them, but that something counts as having that vague property iff it has all the precise properties appropriately related to it. Supervaluationist quantification is needed when there is vagueness, whether it is an indeterminacy of reference or determinate reference to a somehow vague entity.
Supervaluationism, Indirect Speech Reports, and Demonstratives
363
Analysing the supervaluationist’s truth-conditions of indirect speech reports in further detail may require commitment to some particular account of indirect speech reports (which is typically determined by one’s account of propositional attitudes more generally). Alternative such accounts include sententialist accounts—according to which the truth of the speech report turns on whether the speaker uttered a sentence appropriately related to (e.g. saying the same as), the sentence attributed to them—and accounts of indirect speech reports as relations to propositions, where these may be Fregean, Russellian, sets of possible worlds or various other possibilities. I maintain, though I will not argue it here, that the above solution to the problem is available on any of these accounts.⁵ For example, ‘A said that a is F’ may express a relation between A and a different precise Russellian proposition on different precisifications (differing as to the precise property picked out by ‘F’ on that precisification). But for each of those precisifications, the proposition which is the relata of this relation is also the content of A’s utterance according to that precisification. So the speech report can be true on all precisifications. What the case of indirect speech reports brings out is that sometimes the truth value of a sentence on a precisification depends on the values of other sentences on that precisification. This is a kind of penumbral connection and is unproblematic for the supervaluationist.⁶ Someone might object to the above solution that ‘Renata said that Harry is bald1 ’ should come out determinately false (where bald1 is a precisification of ‘bald’), whereas on the above treatment, it comes out indeterminate. (Schiffer, 2000, 248, suggests something like this objection.) But, this intuition, if there is one, is far less strong than the intuition that ‘Renata said that Harry is bald1 ’ should not be determinately true. If it is indeterminate whether ‘Harry is bald’ means that Harry is bald1 , it is reasonable to maintain that it is indeterminate whether Renata said that Harry is bald1 , when she uttered ‘Harry is bald’. It might then be thought that since according to each precisification, there is some precise p such that Renata says that p, (albeit a different one according to different precisifications), then it will come out true simpliciter (since true on all precisifications) that she says something precise. But this putative consequence does not in fact follow, given the supervaluationist treatment of sentences involving ‘precise’, ‘vague’ etc. A sentence such as ‘ ‘‘bald’’ is precise’ does not come out true on all, or indeed on any precisifications, despite the fact that ‘bald’ receives a precise interpretation on all those precisifications. For ‘ ‘‘bald’’ is precise’ is a metalinguistic claim and whether it is true on a precisification depends on what is true on other precisifications, not ⁵ Garc´ıa-Carpintero (2000) defends supervaluationism against Schiffer’s objection by adopting what he calls a syncretic account of propositions, in which modes of presentation play a role even though the constituents of propositions are entities, as on a Russellian picture. See Schiffer (2000b) for his response, where he argues that adopting this account of propositions does not solve the problem. See Garc´ıa-Carpintero (2009) for further discussion. For a brief argument that Schiffer’s problem does not arise on Davidson’s paratactic account of indirect speech (Davidson 1968), see Keefe (2000, 158). ⁶ Weatherson (2003) tackles the de re problem and similarly appeals to penumbral connections. For more discussion of Weatherson, see below.
364
Rosanna Keefe
just at the precisification in hand (see Keefe 2000, 186–7; and compare the way that the truth of a modal sentence at a world depends on the truth of sentences at other worlds). Similarly, then, for a sentence such as ‘Renata said something precise’: the differences in what she said on the different precisifications are enough to ensure that this comes out false, just as the differences in the values of p at different worlds makes ‘p is contingent’ come out true. So, Schiffer’s objection fails and the supervaluationist can accept normal, vague speech reports without being committed to the absurd consequences he claims. It needn’t be determinately true that Renata said any of the relevant range of precise propositions for the speech report to be true. As an analogy to Schiffer’s objection, consider an objection to supervaluationism centring on the compelling claim ‘ ‘‘Harry is bald’’ means that Harry is bald’. On each precisification the second ‘Harry is bald’ gets some precise interpretation (e.g. Harry has less than 2003 hairs on his head). We can parallel Schiffer’s objection as follows: ‘Harry is bald’ does not mean any of these precise things, let alone all of them. Again, this objection would be misguided. It needn’t be true simpliciter that ‘Harry is bald’ has any of the relevant precise meanings for the meaning claim to be true. Rather, it is true according to each precisification that it has some such meaning. On p1 , ‘Harry is bald’ means Harry is bald1 , while on p2 , it means Harry is bald2 . So, according to each precisification, ‘Harry is bald’ has some precise meaning, but there is no precise meaning that it is true that this sentence has. Next, consider the situation with propositional attitude reports. Consider ‘Simon believes that Harry is bald’. On a given precisification, ‘bald’ will get a well-defined extension and, putting it loosely, that extension will figure in the content of Simon’s belief according to that precisification. Now, it might seem strange that well-defined extensions get into the content of beliefs, even on precisifications. An opponent might argue as follows: why should the content of Simon’s beliefs depend on how a particular expression is made precise? To say that the belief ascription is true on a given precisification, s1 —where ‘Harry is bald’ means p1 , say—is to claim that it is true that Simon believes that p1 according to s1 (ignoring precisifications of ‘believes’). And, the opponent might continue, Simon’s belief—his state of mind—does not change with change in how expressions are made precise, so it seems as if it should thus also be true on precisification s2 that Simon believes that p1 . This would mean that if it is to be true on all precisifications that he believes that p, then it must be true (i.e. true on all precisifications) that he believes all the precisifications of p. To see how the supervaluationist can reply, consider first what the epistemicist says about Simon’s belief. According to epistemicism, there will be a single precise interpretation of ‘Harry is bald’ and Simon’s belief will have the content given by that interpretation. The contents of our beliefs, according to the theory, depend on the meanings of our words, which are, in turn, determined by the use of those words in the community.⁷ This means that the content of Simon’s belief could have been slightly different if the meaning of ‘bald’ had been different, and Simon may not have noticed this. The situation for the supervaluationist can be similar in relation to each ⁷ See Williamson 1994, ch. 7.
Supervaluationism, Indirect Speech Reports, and Demonstratives
365
precisification. What Simon counts as believing on s1 depends on the meaning of ‘bald’ according to s1 , and if ‘Harry is bald’ means that p1 on s1 , then on s1 it is true that Harry believes that p1 and this does not imply that on s2 he believes that p1 . So, the belief report can be true without it being true at all precisifications that he believes each precisification of ‘Harry is bald’. We will return to the comparison between the supervaluationist and epistemicist later.⁸ Next, I turn to the de re problem.
20.3
VAG U E S I N G U L A R T E R M S A N D D E M O N S T R AT I V E S
Recall the de re problem with ‘there is where Al said Ben was’. Since I will argue that there are distinctive issues surrounding demonstratives, I shall start by considering the example ‘Everest is what Al said Ben climbed’. The worry, recall, is that the supervaluationist will have to say that it is true of all precise delimitations of Everest that Al said that Ben climbed them. I will argue that the supervaluationist can solve the alleged problem with this example in the same way that the previous version of the problem was solved. On precisification s1 , ‘Everest’ names O1 , say. Al utters the words ‘Ben climbed Everest’ and on s1 this is true iff Ben climbed O1 . On s1 , then, it is true that O1 is what Al said Ben climbed. On s2 , though, it is true that O2 is what Al said Ben climbed and false that O1 is what Al said Ben climbed. So, the sense in which it is true of all precise delimitations of Everest that Al said that Ben climbed them, is that of each of those objects, according to some precisification, Al said that Ben climbed it. And this does not have the unwanted consequence that Al said something about a huge quantity of precise objects. There is no precise object, Oi , of which it is true that Al said Ben climbed it, for of no such object is this true on all precisifications. The de re presentation of the issue poses no additional problems. This solution seems to turn on the way in which the reference of ‘Everest’ in ‘Everest is what Al said Ben climbed’ is guaranteed to be the same on a precisification as the reference of that name in Al’s report. This guarantee is provided simply by the use of the same expression. Whatever the reference is on a precisification will be the reference for any occurrence of the name. Next, consider a case where Al says ‘Ben climbed the highest mountain in the world’ and, again, I report this with ‘Everest is what Al said Ben climbed’. Vagueness aside, there would be disagreement over whether this will be true—whether this change in manner of picking out the mountain is ⁸ One kind of influential objection to the epistemicist has centred on the question ‘how are the exact extensions to our vague predicates determined’, where the thought is that ‘they are determined by use’ is not sufficiently specific and remains problematic (see e.g. Keefe 2000, 76–83). Now, the opponent might suggest that since the supervaluationist needs to say exactly the same about extensions on particular precisifications, then they face the same objection with respect to each precisification (so they face it many times over!). But, the supervaluationist, unlike the epistemicist, can still maintain that use does not determine a unique well-defined extension to a vague term: for the supervaluationist, each of the precisifications is compatible with use and there is nothing that selects between them. We don’t have to pick out a precise interpretation and what is true at it: no unique one is singled out by our use. So, the supervaluationist, here and below, is not simply appealing to an unattractive feature of epistemicism to solve their problem.
366
Rosanna Keefe
compatible with the report being accurate. ⁹ But such debates should be independent of one’s theory of vagueness. And the truth and falsity of the report could each be accommodated by the supervaluationist, if other features of one’s views on indirect speech reports allow it. For the reference on a precisification of my use of ‘Everest’ to be guaranteed to match that of Al’s use of ‘the highest mountain in the world’, there must be some kind of penumbral connection. For example, Everest is definitely a mountain, and so any simultaneous precisification of ‘mountain’ and ‘Everest’ must make that true. On a given precisification, there is only one object in the right vicinity for Everest that counts as a mountain, and that is the same object that counts as Everest on that precisification. So, the use of different expressions in the speech report from those used in the reported speech act need not deliver the kind of problem Schiffer highlights. The details of the required penumbral connections have not been drawn out here, but such a story is needed to accommodate the truth of various other compelling sentences such as ‘Everest is the tallest mountain in the world’. Although on different precisifications the statement will be about different precise objects, the penumbral connection guarantees that the two sides of the identity claim refer to the same thing. What about the cases with demonstratives? Again, to account for the truth of a speech report containing a demonstrative, the aim is to establish a connection on each precisification between the referent of the speakers’ demonstrative and the referent of the reporter’s term (or, for the epistemicist, a connection between the actual referent of each). Here there are two kinds of cases. Suppose Al points at Everest and says ‘Ben climbed that’ and the reporter similarly points at Everest and says ‘that is what Al said Ben climbed’. We can reasonably take ‘that’ to mean ‘that mountain’ and the case can be solved. On any precisification, there will be but one object that counts as the mountain in the vicinity, and on that precisification, the same object will count as the mountain in assessing both Al’s assertion and the reporter’s report. So, the report will be true on all precisifications. (And, if your view of indirect speech reports allows the truth of ‘that is what Al said Ben climbed’ when Al has said ‘Ben climbed Everest’, then this can again be accommodated by the supervaluationist by taking into account the penumbral connections between ‘Everest’ and ‘mountain’.) For the first kind of case, then, the demonstrative is coupled (perhaps implicitly) with some sortal that does the job of ensuring a treatment of the above kind will work.¹⁰ The second kind of case involves a bare demonstrative, where Schiffer’s ‘there’ is one of the most forceful examples. Being a bare demonstrative, there is no sortal that ⁹ Cappelen and Lepore (2004), among others, would allow that the report can be true in this case—at least assuming that all parties involved know that Everest is the highest mountain in the world—since it is something a reasonable person might endorse as a correct report. ¹⁰ In fact, this may not work on all accounts of complex demonstratives. On a minimal theory, with ‘this F’ the sortal, F, does not play a semantic role in determining the content of what is said, just a pragmatic role in helping the hearer pick up on the speaker’s referent (see, e.g. Larson and Segal 1995). On such a theory, there may be no significant difference between the cases just discussed and the cases of bare demonstratives discussed below, depending on the exact details of the story.
Supervaluationism, Indirect Speech Reports, and Demonstratives
367
could play the role ‘mountain’ played in the previous example in ensuring co-reference across the utterances. Even if ‘there’ means ‘that place’, ‘place’ is not the right kind of sortal to do the above job, since, for example, ‘place’ will not get precisified in the way that ‘mountain’ does such as to ensure that on a given precisification there are no two substantially overlapping places. Appeal to the community’s use of the chosen expression (e.g. ‘there’ or ‘that’) will not help here either, given that such demonstratives can be used to refer to almost anything.¹¹ But appeal to use could still help, if we focus on the individual’s use. Consider the problem in relation to the epistemicist again. What, for the epistemicist, could make it the case that I refer to place P1 with ‘there’ rather than a very similar, but slightly differently delineated precise place, P2 ? As usual, the epistemicist will surely say that this is determined by features of use (in particular the use by the speaker, but perhaps relevant utterances from other people). How this occurs is mysterious, but an epistemicist like Williamson allows that ‘meaning may supervene on use in an unsurveyably chaotic way’ (1994, 209). If you then report my utterance, also using the word ‘there’, then the epistemicist can say that it is features of your use of that expression that determine its reference. A key feature of your use of ‘there’ in that speech report, is that you intend it to have the same reference as my use of the expression. Perhaps that intention is enough (in suitable circumstances) to guarantee that it does have the same reference. If so, your speech report will come out true. Now, the supervaluationist can say exactly the same in relation to a given precisification without having to say that there is some unique precise referent for Al’s term. According to some particular precisification, s1 , when Al says ‘Ben was there’, he refers to a particular precise place. But my use of ‘there’ when I report ‘there is where Al said Ben was’ gets to pick out the same place, due to the key feature of my use (which will hold for all precisifications) that I intend to pick out the same place as Al. My intentions ensure the existence of penumbral connections. This is, then, at least the beginning of a way out of the apparent problem with demonstratives for both the epistemicist and the supervaluationist. But the viability of this solution depends on the treatment of demonstratives offered, and that is questionable. In general, for my use of a demonstrative to refer to the same thing as yours, it is not enough that I intend it to. For, I can have several referential intentions which conflict. For example, suppose I point to John and say ‘you said he was F’ intending to refer to the same person you were talking about when you pointed to Mark and said ‘he is F’. In Kaplan’s terminology, my ‘directing intention’ here picks out John, and it is plausible to contend that this intention trumps my intention to co-refer with you and that I thereby incorrectly report you as having said something about John. ¹¹ The above treatment could, however, be used for a case where the speaker and the reporter each seek to refer with ‘there’ to a vague place which can be picked out independently, for example the greater Boston metropolitan area. Here, the relevant place will be a different precise area on different precisifications, but each speaker’s use of ‘there’ will pick out, on a precisification, whatever is the referent of the associated term on that precisification. Note that the problem with demonstratives here is not dependent on the de re formulation of the reporter’s utterance: it would equally arise if I say ‘Al said Ben was there’.
368
Rosanna Keefe
Once directing intentions are considered central, the above solution is questionable: in the reporter’s context, ‘there’ will be associated with some set of precise interpretations and in the speaker’s it will be associated with another set and there will be no penumbral connection between them.¹² (Or for the epistemicist, the reporter and reportee will not count as picking out exactly the same place.) In other words, the reporter’s intentions cannot piggyback on the speaker’s to guarantee co-reference, as was assumed in the solution above.¹³ I find this general story about demonstratives rather compelling. But vagueness poses a problem. It isn’t merely supervaluationism that faces it: it may be that no theory of vagueness is any better placed. Consider the Epistemic View. Al’s intentions determine an exact referent of his utterance of ‘there’, but this is by no means guaranteed to coincide with what my intentions determine as the referent of my use of ‘there’ when I report his speech. It is thus highly likely that my utterance of ‘there is where Al said Ben was’ will be false. A theory such as a Degree Theory or other many-valued theory surely offers no new way out of the problem. Allowing degrees of truth for sentences is of no help in guaranteeing the truth of the various reports. Schiffer himself, who advocates a view of vagueness involving vague partial beliefs that come in degrees, does not tackle speech reports involving demonstratives within his own framework, and the way of dealing with them is equally unclear. There is no advantage for the currently popular contextualist theories of vagueness either. The context clearly changes between the reportee’s original utterance and the reporter’s report, and with it the referent of terms like ‘there’ is liable to change. (For wider problems with indirect speech reports facing the contextualist, see Keefe 2007.) It is tempting to say that this is not a problem of vagueness; it is a problem of a reporter matching demonstrative reference with the reportee, or of giving an account of demonstratives that accounts for this. Given the prevalence of vagueness and other necessary features of the example, it is hard to come up with a suitable problematic example in which there is no vagueness, but that doesn’t make it a task for a theory of vagueness. Is there anything other than intentions that could guarantee co-reference between the speaker’s and the reporter’s use of ‘there’, thereby ensuring the truth of the speech report ‘Al said Ben was there’? Weatherson (2003) offers a proposal that would fill the gap. He draws on Lewis’s notion of naturalness (e.g. Lewis 1983). In the case of some terms—natural kind terms, for example—the referent or extension of our term is determined by our use of the term in conjunction with the world. Our term gets ¹² See also Bach (1992), who maintains that you refer to the thing that you intend and expect the audience to recognize as your referent. When I say ‘there is where Al said Ben was’, my intention to refer to whatever Al referred to cannot be the intention by which I intend my audience to pick up on my reference, since they have no independent grasp on that—they are expected to realize what I am referring to from my demonstration. ¹³ Alternatively, the fact that a speaker has conflicting referential intentions, such as in the John/Mark case above could be taken to render the report neither true-nor-false since there is no unique thing being talked about. On that picture, then, the vague demonstrative case is equally messy and we should not be discouraged by a verdict of neither true nor false in such a case, seeing this as a problem which is not due to vagueness.
Supervaluationism, Indirect Speech Reports, and Demonstratives
369
to pick out the most natural candidate that is compatible with its use. Now, with a vague predicate such as ‘bald’, there is no most natural candidate property to be its referent. All the various candidate precisifications are equally natural or unnatural and nature does nothing to choose between them. Weatherson’s suggestion is that associated with each precisification is a complete ordering with respect to naturalness, so that there is always an answer to which of two properties is most natural. Then, according to a given precisification, ‘bald’ picks out whatever is the most natural candidate according to the naturalness well-ordering associated with that precisification. And different precisifications will have different well-orderings and so different chosen extensions for ‘bald’. The same, Weatherson assumes, will go for objects or places or whatever are the referents of singular terms, including demonstratives: the naturalness ordering for each precisification selects, for example, some exact area of space as the referent of an utterance of ‘there’. The thought is that this can then explain the penumbral connections—between the reportee’s and reporter’s use of ‘there’, for example—which will guarantee the truth of the kinds of indirect speech reports in question. For, on a given precisification, Al’s use of ‘there’ will pick out what counts as the most natural of the candidate regions according to that precisification. And in reporting Al’s utterance, my use of ‘there’ will pick out that same region, since that will be the most natural of the candidates again. Can Lewisian naturalness really be employed for this purpose? One problem is that naturalness is, for Lewis, a feature of properties, while Weatherson needs it equally to be a feature of whatever entities are the referents of singular terms and other terms, including objects and regions of space etc. For Lewis, the feature of naturalness lines up with qualitative duplication between things sharing the property: perfect qualitative duplicates share all their perfectly natural properties. So, for example, two chairs can be qualitatively the same without sharing the non-natural property of being my favourite chair. Perhaps the naturalness of an object o can be smuggled in by considering the naturalness of the property ‘is identical to o’. But this will not do, partly because all instances of any property of the form ‘is identical to x’ will automatically completely resemble each other in the relevant respect, so all such properties seem to be on a par as regards to naturalness. A second problem for Weatherson’s purposes is that Lewisian naturalness is not a vague notion subject to complete sharpening via imposition of a complete ordering. Even if there is some vagueness that could be resolved in different ways on different precisifications, no precisification should result in a complete ordering. For, there are ties for naturalness, such as ties between all the perfectly natural properties, of which none are more natural than the others. Imposing a complete ordering on candidates for naturalness is not simply resolving vagueness in the idea of naturalness, but revising the idea in more radical ways. Dropping the requirement of a complete ordering and allowing ties within the ordering will not do either. For, then there would be no guarantee that there was just one most natural candidate region to be the referent of Al’s utterance of ‘there’, for example. As long as several candidates can be equally natural, the problems will all re-emerge. Now, perhaps Weatherson can accept the deviation from Lewis’s own notion of naturalness and deny that his precisifications of the language are giving a precisification of ‘natural’ in the ordering they are committed to. We could just stipulate that
370
Rosanna Keefe
there is a complete ordering of entities on each precisification, which preserves certain relations in the naturalness ordering. But then it isn’t clear what we are being offered in addition to a claim that the required penumbral connections do exist. It is simply a picture according to which on precisifications, truth-conditions are given as if certain objects (or regions of space etc.) are privileged, independently of the intentions of speakers. Certain exact regions of space are singled out above others to be semantic values on a given precisification, for example, and they will be the privileged one among the candidates for various different, apparently independent uses of ‘there’. This will generate penumbral connections where you might think there should be none; e.g. where there are two completely unrelated uses of ‘there’ pointing in roughly the same direction. Consider whether an approach like Weatherson’s would help out the epistemicist, where there would be a unique complete ordering of naturalness among objects, properties and other entities. Nature surely doesn’t pick out a precise privileged area to the referent of Al’s ‘there’, and it is natural to think that it is entirely Al’s intentions that determines such an area if, as the epistemicist maintains, one such is determined. On the Weatherson approach, however, nature does determine the referent, in conjunction with the speakers’ intentions. That seems, at best, very surprising and properties such as ‘big animal’ and ‘zebra’ come out as much more on a par than expected. It was an advantage of the details of Williamson’s epistemicism that he could explain the existence of sharp boundaries to our vague predicates without metaphysical commitment to implausible privileged boundaries in nature; but this advantage is lost on the Weathersonian approach in question. As argued above, the approach cannot merely maintain that it is simply employing a notion of Lewisian naturalness which is sharp but about which we are ignorant, so the epistemicist would seem to be saddled with a highly significant metaphysical commitment to an implausible counterpart to Lewisian naturalness. Is there another approach available to deal with demonstratives in indirect speech reports? When Al says ‘Ben was there’ and I say ‘Al said Ben was there’, on some precisifications, the place Al denotes will diverge from the one I denote, if there is no penumbral connection between them. But on other precisifications, the referents will coincide, and the reporter’s utterance will come out true. Being true on some precisifications and false on others, the utterance will count as neither true nor false overall. Maybe this is an acceptable consequence. We might informally describe the case as one where the original speaker and the reporter have each picked out a rough place, where they were only roughly the same place. There may then be a further pragmatic story to be told about why such indirect speech reports appear true and/or are useful to make—assuming they do and are—but I won’t enter into this in detail here.¹⁴ Of course, a response to the problem that trumpets the fact that the problematic reports will come out neither true nor false, rather than false, is not available to the ¹⁴ According to the standards required for truth on various accounts of indirect speech reports, many such apparently true reports are strictly false (e.g. if the reporter uses a different but coextensive expression). Such accounts will naturally be combined with a pragmatic explanation of such speech reports, and many of the devices used there could be transposed for our purposes here.
Supervaluationism, Indirect Speech Reports, and Demonstratives
371
epistemicist. If the reporter’s directing intention is all that is relevant to determining the referent of the demonstrative, then the (unknown) exact referent for the original speaker and the reporter is likely to be different (albeit only slightly). The report would thereby come out false rather than neither true nor false. On the other hand, a pragmatic story may be shared, where for the epistemicist, this would be a matter of explaining why a false—not indeterminate—utterance is useful or compelling. On the vexed issue of demonstratives within reported speech (where vagueness is not identified as the key issue), see Sainsbury 2004 and Altham 2004. One aspect of the problem, as they see it, is that accurately to report speech involving demonstratives, it is not enough to indicate the same thing: it must be indicated in the same way. So, considering a case where Jill says ‘there’s a bird on that post’, Altham writes, ‘Tom cannot properly report what Jill said in indirect speech by saying ‘‘Jill said that there was a bird on that post’’ . . . ‘‘that post’’ . . . refers from Tom’s perspective rather than Jill’s and so, even if it refers to the same post, does not do so as Jill did’ (2004, 237). This suggests that reports of speech involving demonstratives will typically not be true, regardless of vagueness and that vagueness raises no new problems that didn’t already face a theory of demonstratives. To summarize: supervaluationism can allow the truth of most intuitively compelling indirect speech reports, whether they involve vague predicates or vague singular terms, de dicto or de re. For the reports can be true on all precisifications because of penumbral connections with the reportee’s terms. The most challenging cases involve demonstratives, specifically bare demonstratives. The account of these cases turns more on one’s account of demonstratives and indirect speech reports than on one’s theory of vagueness. For example, if one’s account of demonstratives allows a reporter’s reference to be determined by his/her intention to co-refer with the reportee, then the reports come out true. Or if one requires that a speech report involving a demonstrative exactly matches the perspective of the reportee, then the truth of such speech reports will be hard to come by, regardless of vagueness. In the hard cases, a supervaluationist may accept that speech reports are strictly neither true nor false, where apparent truth can be explained. There is, then, a range of options compatible with supervaluationism; and the options are also compatible with other theories of vagueness, for which the issues surrounding demonstratives within speech reports arise as much—or as little—as for supervaluationism. Re f e re n c e s Altham, J. E. (2004), ‘Reporting indexicals’ in Studies in the Philosophy of Logic and Knowledge, T. R. Baldwin and T. J. Smiley, eds., Oxford University Press, Oxford. Bach, K. (1992), ‘Intentions and demonstrations’, Analysis 52. Cappelen, H. and Lepore, E. (2004), Insensitive Semantics: In Defense of Semantic Minimalism and Speech Act Pluralism, Basil Blackwell, Oxford. Davidson, D. (1968), ‘On saying that’, Synthese 19. Fine, K. (1975), ‘Vagueness, truth and logic’, Synthese 30. Reprinted in Vagueness: A Reader, R. Keefe and P. Smith, eds., MIT Press, Cambridge MA. Garc´ıa-Carpintero, M. (2000), ‘Vagueness and indirect discourse’, Philosophical Issues 10.
372
Rosanna Keefe
Garc´ıa-Carpintero, M. (2009), ‘Supervaluationism and the report of vague terms’, in this volume. Keefe, R. (2000), Theories of Vagueness, Cambridge University Press, Cambridge. (2007), ‘Vagueness without context change’, Mind 116. Larson, R. and Segal, G. (1995), Knowledge of Meaning, MIT Press, Cambridge MA. Lewis, D. (1983), ‘New work for a theory of universals’, Australasian Journal of Philosophy 61. Sainsbury, R. M. (2004), ‘Indexicals and reported speech’ in Studies in the Philosophy of Logic and Knowledge, T. R. Baldwin and T. J. Smiley, eds., Oxford University Press, Oxford. Schiffer, S. (1998), ‘Two issues of vagueness’, Monist 81. (2000a), ‘Vagueness and partial belief ’, Philosophical Issues 10. (2000b), ‘Replies’, Philosophical Issues 10. Weatherson, B. (2003), ‘Many many problems’, Philosophical Quarterly 53. Williamson, T. (1994), Vagueness, Routledge, London.
21 Scope Confusions and Unsatisfiable Disjuncts: Two Problems for Supervaluationism Delia Graff Fara
Supervaluationism as a theory of vagueness has its advantages and its disadvantages. Most of the advantages of supervaluationism are ones that favor it over those of its rivals that also reject bivalence. Since I believe in bivalence, those advantages do not hold much sway over me. I think that the disadvantages of supervaluationism far outweigh its advantages. But I offer no cost-benefit analysis here. Rather, I want to provide some detailed discussion of a couple of the disadvantages. The view discussed here is canonical supervaluationism, which I’ll take to be the view presented by Kit Fine in his 1975 article and defended by Rosanna Keefe in her more recent book (Keefe 2000).¹ On this view, a claim is supervaluationally true (or just true) when it is true-according-to-classical-semantics on all of the different admissible ways of collectively ‘precisifying’ the vague expressions in its language.² The same goes for falsity: a claim is supervaluationally false (or just false) when it is false on all of the different admissible ways of collectively precisifying the vague expressions in its language. Since some claims involving vague expressions are true on some admissible precisifications but false on others, there are claims containing vague expressions that are deemed by the supervaluationist to be neither true nor false. This is what it is to reject bivalence. A predicate is vague on the canonical view only if its extension gap is non-empty, only, that is, if there are objects of which it is neither true nor false. To precisify a vague predicate is to assign all of the objects in its extension gap to one or the other of its extension (the things of which it’s true) or anti-extension (the things of which it’s false). What makes some precisifications of the vague expressions jointly admissible?³ Those precisifications must not alter the truth value of any claim that ¹ But see also Hans Kamp (1975) and Dominic Hyde (1997) for very different applications of the supervaluationist techniques to some problems of vagueness. ² By a claim, I mean an utterance with the following properties: it’s made in a particular language, it’s made in a particular context, and it says something (it ‘expresses a proposition’). When I say that a claim contains a certain expression, I mean that the sentence of which that claim is an utterance contains that expression. ³ The precisifications must be done collectively, since some admissible precisification of ‘small’, for example, will render a number of precisifications of ‘tiny’ inadmissible: everything tiny is small,
373
374
Delia Graff Fara
already has a truth value. Some of these claims will relate different vague predicates to one another (‘no one huge is tiny’); others will relate vague predicates to associated relational expressions (‘everyone larger than someone huge is also huge’); still others will relate vague predicates to their clear cases (‘anyone under one meter tall is tiny’).⁴ These claims are said to express ‘penumbral connections’—ones that hold even when the quantifiers in them range over borderline cases of the vague predicates in them: over the things in the predicate’s ‘penumbral region’.⁵ It would be mistaken to describe these constraints by saying that the supervaluationist deems some precisifications admissible whenever they’re compatible with our current usage of vague terms, since on her view a lack of extension gap is not compatible with vagueness. ( This turns out to be the underlying source of the focus of Section 21.2, below) After a brief inventory in Section 21.1 of four advantages and disadvantages of canonical supervaluationism, we will focus on two of the disadvantages: (Section 21.2) canonical supervaluationism allows not merely for true disjunctions with no true disjuncts, but also for true disjunctions with no satisfiable disjuncts; (Section 21.3) supervaluationists have yet to provide us with any convincing answer to the question of how we could ever find the false premise of a sorites argument to be as appealing as we find it to be.
21.1
B E N E F I TS A N D C O S TS
The main advantages of supervaluational treatments of vagueness are these: a. Their account—in terms of truth value gaps—of what it is to be a borderline case of a vague predicate is satisfying to most philosophers not wedded to bivalence (which unfortunately may well be most philosophers). b. They preserve the truth of truisms expressing penumbral connections, such as ‘anyone shorter than a short person is short’ or ‘everything tiny is small’, whereas rival theories that also ditch bivalence do not. c. They account for the fallacy involved in soritical reasoning: in deeming false the inductive premise of a sorites argument (e.g. ‘any man just one nanometer taller than a short man is himself also short’), they declare the argument unsound. d. They preserve classical logic, for the most part anyway.⁶ Not surprisingly, these advantages come with their disadvantages. (No theory is perfect.) The chief ones of these are: and there are some small things that aren’t tiny; in semantic talk, all, but not only, things in the extension of ‘tiny’ on a precisification must be in the extension of ‘small’ on that precisification. ⁴ Which cases are clear cases may vary with context. If ‘tiny’ means ‘tiny for an adult’, then one meter is a clear case; though not if it means ‘tiny for a two-year old’. ⁵ The terminology is adapted from Bertrand Russell (1923). ⁶ See Timothy Williamson (1994) and Delia Graff Fara (2004, §2) for arguments that the preservation of classical logic is incomplete in important ways.
Scope Confusions and Unsatisfiable Disjuncts
375
a. a commitment, despite the rejection of bivalence, to vague predicates’ having sharp boundaries—in the sense of there being a least tall height, one nanometer below which renders a person not tall,⁷ a reddest red, a thinnest fat person, etc; b. a rejection of standard forms of reasoning, such as contraposition of arguments and reductio ad absurdum; c. a failure to provide a satisfactory explanation of why we’re mistakenly tempted by sorites reasoning; and d. an assignment of extremely counter-intuitive truth conditions to the logical particles other than negation.
21.2
T RU E D I S J U N C T I O N S W I T H U N S AT I S F I A B L E D I S J U N C TS
Supervaluation yields truth conditions that are extremely counter-intuitive for the logical particles other than negation. It is often criticized, and justly so, for allowing there to be true disjunctions or true existential generalizations that have no true disjunct or instance; and, correspondingly, false conjunctions or false universal generalizations that have no false conjunct or instance. These failures of the classical truth conditions are alleged to arise due to what Fine called a ‘truth value shift’. In the case of disjunctions and existentials, different disjuncts or instances can be the verifying ones on different admissible precisifications; but as long as there is at least one verifier on any given admissible precisification, the disjunction or the existential will be true simpliciter. The main problem with this as a justification, as I will discuss shortly, is that a sentence can be true on an admissible precisification without its even being possible for the sentence to be true simpliciter. This makes room for there being true disjunctions both of whose disjuncts could not possibly be true. That there could be a shift in truth from disjunct to disjunct when neither disjunct could be true is repellent. I say: ‘Someone in this room is the shortest tall person’. Supervaluationists say: ‘We agree, but there needn’t be a correct answer to the question Who?’ And I say: ‘Either Juan or Carlos is the shortest tall person in this room’. Supervaluationists say: ‘We agree, but there needn’t be a correct answer to the question Which one?’ And supervaluation allows all statements made in the following dialog to be correct: YOUR CHAIR: Don’t be late for the 4:00 meeting. YOU: OK, what’s the latest I can be without being late? CHAIR: A short time after the hour. ⁷ This isn’t exactly accurate; the commitment is rather to there being either a shortest tall height or a tallest non-tall height, and likewise for the other cases mentioned. When variation occurs along a continuous scale, a region can be bounded without itself containing that boundary.
376
Delia Graff Fara YOU: I figured that, but what time? CHAIR: I couldn’t truly tell you any particular time.
This leaves me staring incredulously, as it should you. In fact, it’s such a strange position to hold that one often hears this said of supervaluationists: They think that (i) there’s a shortest tall man, even though they don’t think that (ii) someone in particular is such that he is the shortest tall man.
But of course, supervaluationists do believe both of these things; like all of us, they regard these claims as trivially equivalent. What they do believe, which the mistaken attributer is trying to capture, is that there is a shortest tall man, even though no one in particular is such that it’s true that he is the shortest tall man. Supervaluationism is so discordant with the way we actually speak that there are philosophers who understand the view but who aren’t yet fluent in the language we would be speaking if it were correct. But these philosophers are fluent in the language we’re in fact speaking. Whatever prescriptive merits supervaluationism might have, it is not descriptively correct. The following supervaluationist explanation of the anomaly has significant appeal. The reason ‘It’s either pink or red’ can be true even when neither disjunct is true results from the vagueness of ‘pink’ and ‘red’, and in particular from the fact that the object in question is a borderline case of both predicates. We can tell by looking at the thing that it falls somewhere on the spectrum between pink and red, which explains why the disjunction is true. But it’s indeterminate which of pink or red it is, since there are different ways of drawing a boundary between pink and red that count as admissible precisifications of our vague usage of these words, and the object falls on different sides of that boundary on different ones of these ways. And that is why neither disjunct is true simpliciter.
The problem with the explanation is that it does not sit at all well with the following fact about supervaluational semantics. It allows not only for true disjunctions where neither disjunct is true, it allows also for true disjunctions where neither disjunct could be true. I assume that the supervaluationist must say that a sentence could not be true when there is no supervaluational model in which it is true. Let us use s in the following way: s φ is true on a precisification just in case is true in some supervaluational model.⁸ s is the supervaluational satisfiability operator. Then for an appropriately chosen and , supervaluationists allow for the truth of the following: ( ∨ ) ∧ ¬ s ∧ ¬ s . ⁸ The connection with genuine possibility should be apparent. One would think that possibility should amount to truth in some possible world, where on the supervaluationist’s conception each possible world would correspond to some supervaluational model, namely that model that verified and falsified exactly the same sentences as it. Depending on the supervaluationist’s treatment of modality, the diamond used here might not be the diamond of modal logic, since it remains open for the supervaluationist to say that for the modal operator , is true if is true in some possible world, but false only if is false in every possible world. In that case, since our (and too) will be untrue in every supervaluational model, but also neither true nor false in some supervaluational models, would be neither true nor false, while s is false.
Scope Confusions and Unsatisfiable Disjuncts
377
The disjunction of this appropriately chosen and is true because at least one of them is true on every precisification, even though each disjunct is truth valueless, and even though neither disjunct could be true. If we are to accept the claim of the form ‘ ∨ , but it’s indeterminate which’ on the grounds that things could go either way depending on how you drew precise boundaries for the predicates involved, then it should be indeed that things could go either way. But in the case to be presented, it is not. So what’s the example? It’s very simple. Suppose we have something that’s a borderline case of ‘pink’. Let’s represent the claim that it is pink as p. This claim is indeterminate—i.e. neither true nor false according to supervaluationism. We’ll use B to stand for the borderline-case operator. So Bp is true. Supervaluation verifies every instance of excluded middle, even for indeterminate propositions like p. Take the true disjunction p ∨ ¬p; conjoin each of its disjuncts with the true claim Bp; then the result, (Bp ∧ p) ∨ (Bp ∧ ¬p), is true.⁹ But neither of its disjuncts could be true. To see that neither could be true, note that when Bp is true, neither p nor ¬p can be true: we’ll have p true on some admissible precisifications of ‘pink’, ¬p on others. But Bp manages to be true on all of these precisifications, for its truth at a precisification depends on the values p takes over the whole space of precisifications: Bp is true on a precisification when p is true on some admissible precisifications, but false on others of them.¹⁰ This is not something that can vary from precisification to precisification, so Bp is true simpliciter. But Bp is incompatible with p, and also with ¬p. For the truth of Bp requires precisely that neither of these claims be true on every precisification, that each be true on some, false on others. So, we have the supervaluational truth of the following, when Bp is true: ((Bp ∧ p) ∨ (Bp ∧ ¬p)) ∧ ¬ s (Bp ∧ p) ∧ ¬ s (Bp ∧ ¬p). We have substituted (Bp ∧ p) for and (Bp ∧ ¬p) for in the schema given earlier. The underlying source of the difficulty is that supervaluationists (i) think that the existence of borderline cases is not compatible with precision, understood as no gap in truth value; yet they (ii) supply a semantics on which claims of borderline status remain true upon complete precisification of the vague expressions in the language. They are constrained to supply such a semantics by their thoughts that (iii) there are claims of borderline status that are true simpliciter while (iv) truth simpliciter is to be identified with truth-on-every-precisification.¹¹ Putting these views together requires ⁹ Another way to see the commitment is to note that the conjunction of two true claims is true. So Bp ∧ (p ∨ ¬p) is true. By classical distribution laws, (Bp ∧ p) ∨ (Bp ∧ ¬p) is true. ¹⁰ The truth conditions associated with the B operator are structurally like those associated with contingency in modal logic. Those for the B operator involve quantification over admissible precisifications, while those for contingency quantify over possible worlds. ¹¹ Cf. Keefe (2000, 186f). Here Keefe discusses whether supervaluationists face the problem that many of their own theoretical claims (e.g. ‘a predicate is vague only if it has an extension gap’) are not true on any precisification, much less on all of them. She argues that the best supervaluational response is to supply a semantics of the kind described as feature (ii) above. On that point, I agree with her.
378
Delia Graff Fara
there to be claims of borderline status (ones of the form B) that are true on each precisification, even though it must be that either or its negation is true on that precisification. I take (i–iv) to be at least partly constitutive of of the view I’m here criticizing; to reject any of these in light of my criticism is to concede defeat. In particular, to give up (i), the incompatibility of precision with the existence of borderline cases, is to allow for bivalence in spite of vagueness; while to refuse (ii) is to reject at least one of (iii) and (iv). To reject (iii) is to reject vagueness. To reject (iv), meanwhile, is to reject supervaluationism at its core. I would like to say that if a sentence cannot be true, then that suffices for its being impossible. The supervaluationist might object to this philosophical platitude, but if I were to help myself to it, I could put the main point of this section this way. Supervaluationists think that a single impossibility need not be false, and that the disjunction of two impossibilities can be true. This is because they think that it’s impossible for something to be both borderline pink and pink.¹² Yet, for anything that is a borderline case of pink, the claim that it’s both borderline pink and pink can only be indeterminate, hence not false, according to their neither-true-nor-false account of indeterminacy.
21.3
S C O PE C O N F U S I O N A N D T H E P S YC H O LO G I C A L QU E S T I O N
Supervaluation falsifies the inductive premises of sorites arguments, but does not directly provide an answer to the question of why we’re mistakenly inclined to believe them in the first place. Kamp was prompted to spurn his (1975) supervaluational theory of vagueness for just this reason. Fine, who published an independentlydeveloped supervaluational account of vagueness that same year, did offer some explanation (1975, 286). I have argued elsewhere that that explanation was unsatisfactory (2000, 50–2). This was primarily on the grounds that it implicitly required that we tend to equate truth with non-falsity, something which an opponent of bivalence cannot do without undermining his own theory. Keefe, one of the most prominent current supervaluationists, has offered a supervaluationist answer to this ‘psychological question,’ as I have called it (2000, 50). The psychological question is the question why we are so inclined to believe a sentence with the form of (U) in many cases where it is in fact false. I know of only two places where a supervaluationist answer to this psychological question is offered or defended: in Fine’s article (1975, 286) and in Keefe’s book (2000, 183–6). (U) ∀x(x → x ). Here is some sorites-susceptible predicate and x is the successor of x on some sorites series for that predicate. ¹² This is not the case on the typical view that affirms bivalence, although Raffman’s (2005) pro-bivalence view is a prominent exception.
Scope Confusions and Unsatisfiable Disjuncts
379
By a supervaluationist answer to the question, I mean one that derives from supervaluationism per se, not from supervaluationism as it might be supplemented with some further view better designed to answer the psychological question, such as a contextualist theory or some boundary-shifting theory more generally.¹³ The main flaw in both of Fine’s and Keefe’s explanations is an implicit reliance on our tending to equate truth with non-falsity. This is at best in serious tension with the supervaluationist rejection of bivalence. In the remainder of this section I argue against the success of Keefe’s answer in particular to the psychological question.¹⁴
21.3.1 Keefe’s scope-confusion explanation of our mistake Keefe’s explanation of why we believe (U)—when we do, which as she points out (183f ), may not be always—‘turns on the fact that supervaluationism can distinguish between the falsity of (U) and its having a false instance, and correspondingly between (E)’s being true and its having a true instance’ (184). ((E) is an equivalent of the negation of (U).) (E) ∃x(x ∧ ¬x ), ( TE) true: ∃x(x ∧ ¬x ), (ET ) ∃x true: (x ∧ ¬x ). ‘true:’ here represents the operator ‘it is true that’. We mistakenly confuse (ET) with (TE) (and hence with (E), an equivalent of the latter), so that our correct denial of (ET) mutates into an incorrect denial of (E).
21.3.2 The problem with the explanation The first question to address is how exactly a confusion of (ET) with (E), and a concomitant confusion of the denial of (ET) with the denial of (E), explains our incorrect attitude towards an entirely different sentence, (U). It behooves us to spell out the chain of reasoning. We begin with the correct denial of (ET), which, due to a putative scope confusion, leads to an incorrect denial of (TE). Then a chain of good logical reasoning leads us to the incorrect affirmation of (U). The connecting links are spelled out below. In the ‘Reason’ column, I cite the justification for the judgement made in that row on the basis of the judgement made in the immediately preceding row. In the ‘Correctness’ column I indicate whether the judgement made in that row is good, not whether the reasoning is good. Each step of reasoning is good, except for the scope confusion made at the outset. What’s missing, as I’ve indicated, is an explanation of why we move from step (4) to step (5). Why would we go from denying a certain ¹³ Boundary-shifting theories—whether they be contextualist ((Kamp, 1981), (Raffman, 1994, 1996), (Soames 1999) or invariantist (Fara, 2000, 2008))—are not in competition with any particular account of what it is to be a borderline case of a vague predicate, where these latter accounts include epistemicism, degree theories, supervaluationism, or truth value gap theories more generally. ¹⁴ I have argued against Fine’s elsewhere (Fara, 2000, 50–2).
380
Delia Graff Fara
Table 21.1.
(ET) (TE) (3) (4) (5) (6)
Sentence
Judgement
Justification
Correct?
∃x TRUE: (Fx ∧ ¬Fx ) TRUE: ∃x(Fx ∧ ¬Fx ) TRUE: ¬∀x(Fx → Fx ) FALSE: ∀x(Fx → Fx ) TRUE: ∀x(Fx → Fx ) ∀x(Fx → Fx )
Deny Deny Deny Deny Affirm Affirm
Good Judgement Scope Confusion Substitution of Equivalents FALSE: ≡ TRUE: ¬ ???? True: ≡
Correct Incorrect Incorrect Incorrect Incorrect Incorrect
falsity ascription to affirming the truth of its embedded clause? For given the supervaluationist rejection of bivalence, there are two ways for a claim to fail to be false. It could be true or it could be indefinite. Two further questions are pressing. First, why is our good judgement directed at (ET) rather than (TE); why don’t we correctly affirm (TE) and then because of scope confusion incorrectly affirm (ET), leading us eventually to the correct conclusion that the inductive premise is false?¹⁵ If anything, our attitude to (TE) should be the dominant one, since of the two sentences, only it (according to the supervaluationist) is equivalent to a simple non-metalinguistic claim. Second, why do we tend toward confusion of the relative scopes of a quantifier and a truth value operator only in the case of ∃ and ‘true:’? If we ignore quantifiers other than the ones typically appearing in formal first-order languages, there are four possible scope confusions to be considered. Table 21.2 represents the only even remotely plausible position the supervaluationist could take on the chance of our making any of these scope confusions. It is telling that the only case in which we’d be at all inclined to confuse the relative scopes of a truth value operator and a quantifier is the one where the supervaluational opinion about equivalence differs from the classical opinion. Table 21.2. Equivalence? ?
∃x TRUE: x ≡ TRUE: ∃xx ?
∃x FALSE: x ≡ FALSE: ∃xx ?
∀x FALSE: x ≡ FALSE: ∀xx ?
∀x TRUE: x ≡ TRUE: ∀xx
Chance for Mistake?
Classical?
Supervaluational?
Yes
Yes
No
No
No
No
No
No
No
No
Yes
Yes
Table 21.2 displays, incidentally, that something like the reasoning in table 21.1 must be involved in our inference to the wrong conclusion in the case of (U)—if that ¹⁵ I have in mind this chain of inference: Correct affirmation of true: ∃x (x ∧ ¬x ) ⇒ Incorrect affirmation of ∃x true: (x ∧ ¬x ) ⇒ incorrect affirmation of ∃x true: ¬(x → x ) ⇒ incorrect affirmation of ∃x false: (x → x ) ⇒ correct affirmation of false: ∀x(x → x ).
Scope Confusions and Unsatisfiable Disjuncts
381
inference is to involve a scope confusion. There is no other illegitimate scope confusion of the relevant kind which we’re in danger of making. Keefe does allege that that we are in danger of confusing the relative scopes of ‘true:’ and ∃ in at least one other case. I don’t find the example convincing. She writes: [The confusion of (E) and (ET)] is thus like a confusion between saying that it is true that someone ought to do X and saying that it is true of someone that they ought to do X : the latter may be false while the former is true. We would run the two together if we thought the only way that ‘someone ought to do X ’ could be true was if there was someone, y who ought to do X . But . . . the former could hold because X being done is a right of z’s and so it ought to be done by someone, though it is no individual’s duty to do it (185).
In my assessment, we should use a deontic operator ‘ought:’ to represent the conflated sentences as follows. (OE) true: ought: ∃yXy, (EO) ∃y true: ought: Xy. There is a scope confusion all right. But it is one between ought: and ∃, not true: and ∃, and can be represented without involving truth at all: (OE*) ought: ∃yXy, (EO*) ∃y ought: Xy. Let me summarize the points made in this section. The scope-confusion explanation of our failure to recognize (U) as false fails for the following reasons: first, the scope explanation succeeds only if we’re apt to confuse non-falsity with truth, which we should not be if supervaluational semantics were correct; second, scope confusion is symmetric, but the explanation on offer requires an unexplained and unlikely asymmetry in the inferential order of our judgements; third, there is no good explanation for why we might make such a scope confusion, since we’re not at all in danger of doing so in any of the relevantly similar cases (table 21.2); while fourth, the only supposedly clear example of the confusion in question is in fact an example of an unrelated sort of scope confusion.
21.4
C O N C LU S I O N
No plausible or satisfying supervaluationist answer to the psychological question has yet been offered. I probably phrased the question somewhat badly at the outset, however. But it is the question so phrased that Keefe, and Fine before her, answered it. The problem with the phrasing is that it demands an explanation of our attitude to a false universal generalization, whereas what is really needed is an explanation of why we’re inclined to believe each instance of the generalization given that we have overwhelming evidence that not all of its instances are true. Without such an explanation, we have no answer to the question why we’re inclined to accept sorites reasoning
382
Delia Graff Fara
when its premises do not include a universal generalization but rather, in the place of one, a series of its instances—a series of claims about adjacent pairs of minimally differing objects, e.g. ‘these are either both tall or both not’ or ‘this one is tall if that one is’. Combined with this, ideally, would be an explanation of why we’re unable to locate a shift in any kind of status along a sorites series, not only from truth to falsity, but also, e.g. from truth to truth valuelessness, from clear cases to borderline cases, or even from clear cases to cases about which nothing relevant could truly be said (perhaps, not even that). A supervaluationist might remedy the deficit by supplementing her view with some complementary contextualist or boundary-shifting answer to these questions. The considerations in Section 21.2, however, suggest that we should be leery of the prospects for ultimate success. Since supervaluationists allow for true disjunctions with only unsatisfiable disjuncts, they cannot legitimately appeal to ‘truth value shift’ in order to explain away the strangeness of the truth conditions they assign to disjunctions, conjunctions, etc. Re f e re n c e s Fara, Delia Graff (2000), ‘Shifting sands: An interest-relative theory of vagueness’, Philosophical Topics 28(1), 45–81. Published under the name ‘Delia Graff’. (2004), ‘Gap principles, penumbral consequence and infinitely higher-order vagueness’ in Jc. Beall, ed., Liars and Heaps: New Essays on Paradox, Oxford University Press, 195–221. Published under the name ‘Delia Graff’. (2008), ‘Profiling interest relativity’, Analysis 68(300), 326–35. Fine, Kit (1975), ‘Vagueness, truth and logic’, Synthese 30, 265–300. Hyde, Dominic (1997), ‘From heaps and gaps to heaps of gluts’, Mind 106(424), 641–60. James, E., Slater, J. et al., eds. (1983–), The Collected Papers of Bertrand Russell, Allen and Unwin/Unwin Hyman, London. Kamp, Hans (1975), ‘Two theories about adjectives’ in E. L. Keenan, ed., Formal Semantics of Natural Language, Cambridge University Press, Cambridge, 123–55. (1981), ‘The Paradox of the heap’ in U. M¨onnich, ed., Aspects of Philosophical Logic, D. Reidel, Dordrecht, 225–77. Keefe, Rosanna (2000), Theories of Vagueness, Cambridge University Press, Cambridge. Raffman, Diana (1994), ‘Vagueness without paradox’, Philosophical Review 103(1), 41–74. (1996), ‘Vagueness and context-relativity’, Philosophical Studies 81, 175–92. (2005), ‘Borderline cases and bivalence’, Philosophical Review 114(1), 1–31. Russell, Bertrand (1923), ‘Vagueness’, Australasian Journal of Philosophy and Psychology 1, 84–92. Page references are to reprint in E. James, J. Slater et al., eds. 1983–. Soames, Scott (1999), Understanding Truth, Oxford University Press, New York. Williamson, Timothy (1994), Vagueness, Routledge, London.
VI Paraconsistent Logics
This page intentionally left blank
22 The Prospects of a Paraconsistent Response to Vagueness Dominic Hyde
Might the challenge to logic and semantics presented by the sorites paradox, and vagueness more generally, be met with a paraconsistent response? Some have thought so but the majority view has been that a paracomplete response¹—roughly, a truth value gap response—is as radical a departure from classical semantics as is necessary (and, of course, there are those, e.g. epistemicists, who see no need to depart from classical semantics at all). Paraconsistency is often thought to represent a revision of logical theory that is too radical to be defensible. For those theorists convinced that vagueness is a semantic, rather than merely epistemic, phenomenon the current best contender appears to be that paracomplete response known as supervaluationism. The logic of supervaluationism, SpV , is not only taken to be more conservative than a paraconsistent response by virtue of its paracomplete, gappy, semantics but it also is commonly said to ‘preserve classical logic’ in spite of its non-classical semantics and this too is taken to speak in its favour in respect of theory choice. The view then is that SpV is superior to any paraconsistent theory both at the level of semantics and logic. A paraconsistent response will require the abandonment of classical logic and requires a more radical departure from classical semantics. The choice is seen by many then to be clear. I have argued elsewhere, however, that the choice is far from clear.² The paraconsistent logic SbV , or subvaluationism, is no less conservative than SpV nor more so. In defence of SpV , Keefe (2000) responds by suggesting that paraconsistency is objectionable per se and, more interestingly I think, that SbV is indeed less conservative than its rival. The debate throws up a range of interesting issues but in the end, I shall argue, such issues only serve to reinforce both the conservatism of subvaluationism and the radicalism of supervaluationism. In the end both logics offer equally compelling theoretical approaches to vagueness. Each approach is, I shall argue, equally objectionable with neither providing an adequate account of vagueness but this criticism arises from a feature shared by each approach that is independent of their paracompleteness or paraconsistency per se. ¹ More exactly defined below.
² See Hyde (1997), (1999), (2001).
385
386
Dominic Hyde
For all that has been said, a paraconsistent approach and the associated recourse to truth value gluts remains a contender in accounting for vagueness.
22.1
S U PE RVA LUAT I O N I S M
Let us turn, firstly, to briefly describe salient aspects of that revision of classical twovalued logic advocated by supervaluationists in response to vagueness, the logic SpV .³ ( Those familiar with Hyde (1997) can simply skip the next section.)
22.1.1
Describing supervaluationism
The theory begins from the observation that vague predicates are such that their determinate extension and anti-extension are not exhaustive; there are objects which are in neither. And thus, assuming a full complement of names in the language, there are sentences for which the possibility arises of their being neither determinately true nor determinately false. The supervaluationist then equates determinate truth with truth simpliciter, or ‘supertruth’, thus defining a concept of truth for which bivalence fails and in terms of which we may describe borderline cases as giving rise to truth value gaps. On this view, where a is a borderline case of P, the indeterminacy of Pa amounts to its being neither true nor false.⁴ The non-bivalent logic SpV is thus an example of an incomplete logic. Let us say that a logic is complete if and only if, for any valuation or model of any contradictory pair of sentences A and ∼A, one or the other must be true in the valuation or model. i.e. A, ∼A.⁵ Classical logic is a paradigm of a complete logic. A logic will count as incomplete just in case it is not complete—i.e. for some sentence A, neither it nor its negation need be evaluated as true. i.e. A, ∼A. Non-bivalent approaches to the ³ An informal account appears to have first been proposed in Mehlberg (1958). Interestingly, Mehlberg was a former student of the Lvov–Warsaw School of philosophy, itself a well-known centre of logical innovation and the origin of the subvaluationist paraconsistent approach to vagueness proposed a decade earlier by Ja´skowski. Despite its early advocacy by Mehlberg, supervaluationism as applied to the phenomenon of vagueness is generally considered a reinterpretation of the ‘presuppositional languages’ of van Fraassen, formally described in his (1966). It is most extensively described and defended in Keefe (2000). ⁴ Note that the semantics for the precise fragment of natural language is usually taken to be classical—truth and falsity are considered exclusive and exhaustive. This assumption is not essential. A supervaluational model structure could equally well be built upon an underlying semantics that was nonclassical, e.g. intuitionist, relevant, etc. In this sense a supervaluationist approach merely aims to provide a non-bivalent semantic superstructure sensitive to vagueness which collapses to one’s preferred underlying semantics where vagueness does not arise. However, since it is traditionally a development of a non-classical semantics from a classical base, and this tradition has circumscribed the ensuing issues, problems and debate, supervaluationism as it is discussed and debated is now synonymous with this classically oriented theory—classical supervaluationism. We shall continue in this tradition and take classical supervaluationism as our object of focus, referring to it simply as supervaluationism. ⁵ ‘’ represents the generalized, multiple-conclusion consequence relation. Given a set of sentences (the multiple-premise set) and a set (the multiple-conclusion set) we shall say that if and only if whenever all the members of are true then some member of is true.
The Prospects of a Paraconsistent Response to Vagueness
387
problem of vagueness like SpV that postulate truth value gaps exemplify logics which are incomplete since, given the shared assumption that a sentence is false if and only if its negation is true, to admit sentences which are neither true nor false is to admit sentences which fail to be true whilst their negations also fail to be true. Thus: SpV A, ∼A. Now there is obviously a trivial sense in which a logic might be incomplete—namely, if whenever a sentence and its negation fail to be true every sentence fails to be true. This is not the sense of incompleteness to which truth value gap approaches in general, and SpV in particular, are committed. Such approaches countenance quarantined gaps by rejecting the spread-principle, B A, ∼A, according to which if there are truth value gaps anywhere then they are everywhere. That is, they accept that some sentence B can be true whilst not every sentence or its negation is. Gaps do not implode everywhere—the logic is non-implosive. So in addition to incompleteness there is also a commitment to the non-triviality of the incompleteness. i.e. B A, ∼A. Let us say that a logic which is non-trivially incomplete is paracomplete. The cornerstone of mainstream responses to the logical and semantic problems posed by vagueness, and SpV in particular, amounts to the view that vagueness necessitates a paracomplete response at worst. Thus: B SpV A, ∼A. Given this non-classical constraint on the semantics of vague expressions we may wonder to what extent classical logic remains intact. For instance, if a sentence A and its negation ∼A are indeterminate what of their conjunction and disjunction? Are they likewise indeterminate? Are the classical laws of excluded middle and noncontradiction still theorems? Do inferences like modus ponens or proof by cases remain valid in such semantics? What supervaluationists aim to do in developing a logic of vagueness is to admit truth value gaps whilst respecting what they describe as ‘penumbral connections’. To paraphrase Fine (1975, 269f ), suppose that a certain blob is a borderline case of ‘red’ and let S be the sentence ‘the blob is red’. Though we may agree that S is indeterminate as is its negation, ∼S, nonetheless their conjunction should count as false since they are contradictories. The boundary of the one shifts, as it were, with the boundary of the other. Similarly, since S and ∼S are complementary over the given colour range, their disjunction, S ∨ ∼S, is true. Penumbral connection is the possibility that just such logical relations hold among indeterminate sentences. The supervaluationist’s claim then is that penumbral truths must be respected (and, as a consequence, some non-truth-functional approach must be sought). They are insisting, in effect, that classical theorems, in so far as they reflect these supposed penumbral connections, must be respected despite the failure of bivalence in accommodating vagueness. It is this which makes classical supervaluationism especially interesting. Consequently, the paracomplete logic that is sought differs crucially from other paracomplete logics that have been proposed to deal with vagueness, for example Łukasiewicz’s three-valued logic or Kleene’s popular strong three-valued logic, where
388
Dominic Hyde
lem fails. The logic that is sought is what, following Arruda (1989), we may describe as a weakly paracomplete logic. That is to say, though it admits of non-trivial incomplete valuations which do not make true either A or ∼A (i.e. B A, ∼A), nonetheless all such non-trivial valuations make true A ∨ ∼A. (By contrast, strongly paracomplete logics do not distinguish between the non-truth of A ∨ ∼A and the non-truth of both disjuncts, and consequently such theories do not contain A ∨ ∼A yet do not implode and may contain some truths, thus being non-trivial.) Thus: B SpV A ∨ ∼A. The supervaluationist model of vagueness attempts to deliver just such a weakly paracomplete semantics. In the now familiar way, supervaluationism defines supertruth as truth in all admissible valuations, identifies supertruth with truth simpliciter, and subsequently defines SpV -consequence in the obvious way: SpV if and only if whenever all the members of are (determinately) true then some member of is (determinately) true. Such an account of consequence will indeed establish all classical (CL) theorems as theorems of SpV , as desired, since in the special case where = ∅ and is a singleton set, it is easily shown that: (I) SpV A if and only if CL A.⁶ This confirms, for example, that though the principle of bivalence is rejected (i.e. despite the failure of the logical theory to be complete) the law of excluded middle remains valid. Thus: lem: SpV A ∨ ∼ A. Where a restriction is placed on the consequence relation to the effect that be a singleton set, the multiple-conclusion consequence relation narrows to the more commonly studied single-conclusion consequence relation. This relation is coextensive with classical consequence and admits as valid all and only those inferences that are classically valid. Thus: (II) SpV A if and only if CL A. It is not hard to show then that rules like modus ponens, contraposition, conditional proof, proof by cases and reductio are all SpV -valid. Notably, as Williamson points out, all these principles fail in a language extended to include a determinacy operator, ‘D’. But even in the unextended language currently under consideration, cracks are already evident in the supposedly conservative logical veneer of SpV . Multipleconclusion consequence already deviates from its classical counterpart. As a result of being paracomplete but only weakly so, SpV fails subjunction: A ∨ B SpV A, B.⁷ ⁶ See Williamson (1994, ch. 5, §3). ⁷ Since, were subjunction to hold, substituting ∼A for B, lem would mandate completeness (i.e. one of A and ∼A would have to be true).
The Prospects of a Paraconsistent Response to Vagueness
389
This non-classical feature of SpV manifests the non-truth-functional account of disjunction required to underwrite a weakly paracomplete theory of vagueness and, as we shall see shortly, is a major source of concern when evaluating the system’s adequacy. How can it make sense to deny the truth of two unacceptable claims considered separately whilst accepting them jointly by accepting their disjunction? The anomaly is evidence of the fact that SpV ’s preservation of classical single-conclusion consequence incurs a correlative cost at the level of classical multiple-conclusion consequence more generally. Classical multiple-conclusion consequence is preserved in SpV only in the following restricted sense: (III) SpV A1 ∨ A2 ∨ . . . ∨ An
22.1.2
if and only if CL A1 , A2 , . . . , An .
Defending supervaluationism
The conservatism of the supervaluationist approach expressed through (I) and (II) above is frequently cited as a major virtue. Retaining classical logic to that extent whilst accommodating vagueness would be a significant achievement if, indeed, it is achievable.⁸ There are, however, problems with such an approach—problems attend the resolution of the sorites paradox and the retention of classical laws. Consider, firstly, how the paradox is resolved. In particular, consider the following version: A man with 1 hair on his head is bald. For any n, if a man with n hairs on his head is bald then a man with n + 1 hairs on his head is bald. ∴
A man with a million hairs on his head is bald.
The conclusion is deemed unacceptable yet the reasoning is valid by classical lights. By (II) above then the SpV theorist accepts the argument as valid, but deems it unsound. Despite the prima facie truth of the premises, supervaluationists deny the truth of the second, universally quantified conditional premise. Of course, the mere non-truth of a premise does not entail its falsity given the failure of bivalence, nonetheless for the SpV theorist the paradox in this form does indeed have a false premise. The universally quantified conditional premise is false and the following is accepted: (1) It is true that there is some n such that a man with n hairs on his head is bald whilst a man with n + 1 hairs on his head is not bald. However, the acceptability of (1) is not to be confused with another, very similar claim which in SpV is nonetheless quite distinct: namely, that there is some n for which it is true that a man with n hairs on his head is bald and a man with n + 1 hairs on his head is not bald. The truth of there being a hair-splitting n no more entails ⁸ More generally, retaining one’s preferred underlying logic—classical or otherwise—to that extent in the face of vagueness would be a virtue if achievable. (Recall that we are considering what I earlier termed classical supervaluationism, thus the default logic is assumed to be classical.)
390
Dominic Hyde
there being an n of which it is true that it is hair-splitting than the truth of A ∨ ∼A entails the truth of A or the truth of ∼A. The following is rejected: (2) For some n, it is true that a man with n hairs on his head is bald whilst a man with n + 1 hairs on his head is not bald. In an attempt to chart a course between the acceptance of paradox and the rejection of vagueness, it is recommended that one accept (1) whilst rejecting (2). Acceptance of (1) is considered reasonable (vagueness is not so ‘serious’ as to warrant its rejection) and does not preclude the vagueness of the relevant predicate, whereas acceptance of (2) would do so for it would amount to a rejection of the truth value shift, so one should reject (2). In response to this counterintuitive recommendation Varzi (2001) makes explicit one line of defence that remarks of Fine (1975, 285) gesture at, namely a justification by appeal to an associated metaphysical thesis taken to underlie the formal theory of SpV . Vagueness is a real enough phenomenon to warrant logical reform but is not serious enough to be ontologically grounded—it is purely representational, and more particularly in the case of vague language it is purely semantic. The Varzi thesis that the non-standard analysis of the quantifier is mandated by the purely semantic nature of vagueness deserves closer scrutiny. However, my own view is that even if the anomalous behaviour can be shown to follow from a purely semantic account, such an account should be rejected anyway. Keefe opts for a quite different response. Rather than any metaphysical defence, the choice of theory is made on pragmatic grounds. ‘I advocate the indirect argument that we should accept the phenomenon because of its role in an altogether successful theory of vagueness’ (2000, 182). Agreeing that (2) precludes the vagueness of the predicate involved, it is indeed to be rejected. Nonetheless (1) is true, thus undermining the soundness of the sorites paradox above. This acceptance of (1) is admittedly counterintuitive, however the costs are said to be offset by the overall benefits of the SpV theory—‘any costs [that accrue by virtue of the acceptance of (1)] are easily worth paying given the advantages of the theory’ (2000, 183). Attempts to mitigate the costs though are seriously flawed. Firstly, claims that the major premise of the paradox would not always be assented to, even if plausible, hardly serve to show the plausibility of the truth of its negation, i.e. the truth of (1). Of course, they would, if successful, serve to show that those who object to (1) on the grounds that the major premise is true are on shaky ground; the truth of the premise is not beyond question. But in the absence of the Principle of Bivalence one might agree with Keefe on the non-truth of the major premise and nonetheless object to its being counted false, as (1) demands. Arguably, the most compelling (and frequently cited) reason for objecting to (1) is that already discussed earlier—namely, its apparently untoward consequence in the form of (2). On this point Keefe seeks to redress the balance in favour of (1) by pointing to the fact that in SpV the untoward consequence simply doesn’t follow. ‘The rest of my defence turns on the fact that supervaluationism can distinguish between [(1) and (2)]’ (2000, 184). But this is not itself a non-question-begging defence unless the distinction, whose maintenance
The Prospects of a Paraconsistent Response to Vagueness
391
depends on the failure of the inference from (1) to (2), is itself independently plausible. And it is not, at least not without further argument, for as we have seen it requires an interpretation of existential quantification at odds with our ordinary understanding. The costs of accepting (1) are not mitigated, but rather, the attempted mitigation simply shifts the costs of accepting (1) onto the semantic analysis of quantification. Not surprisingly then, the apparent anomaly surrounding quantification is one which supervaluationists are keen to dispel. They sometimes appeal to facts about the behaviour of the quantifiers in other contexts and Keefe is no different in this respect. The change of scope involved in the inference from (1) to (2) is compared to syntactically similar scope changes involving the existential quantifier. The generally agreed unacceptability of the latter is then offered as evidence for the (required) unacceptability of the former. For example, most, if not all, people would agree that its being true that someone ought to perform action X does not entail there being someone of whom it is true that they ought to perform action X. Similarly, it seems plain that its being true that some seat has been promised to me on my booked flight to destination Y does not entail there being any seat of which it is true that it has been promised to me. In each case, the change in scope of the quantifier is deemed illicit. Obligation and promising do not distribute over ‘there is’. Similarly, claims the supervaluationist, it is to be expected that truth does not distribute over ‘there is’ and objections to the theory based on an assumption to the contrary are misguided. The claims to similarity however do not obtain in the relevant sense. To be sure, there is a syntactic similarity to all three cases considered. But this is irrelevant. The claim underlying the objection is not that truth ought to distribute over the quantifier since any operator ought to do so. Clearly some operators do not. (Examples to do with obligation and promising are cases in point.) The claim is that truth ought to distribute by virtue of what ‘there is’ means. Given our ordinary understanding of the meaning of existential quantification an existential claim is true if and only if it has some true instance. Truth and the existential quantifier interact in this way given a proper semantic understanding of the quantifier. There is no analogous reason for thinking that operators representing obligation and promising should behave similarly, moreover their not doing so reveals nothing salient concerning the meaning of ‘there is’ and so nothing salient concerning the illegitimacy of the change of scope involved in the inference from (1) to (2). We are left with no reason for rejecting the strong inclination that the inference is valid and consequently SpV is strongly counterintuitive in this regard. So much for the resolution of the paradox. Analogous problems confront the SpV account of disjunction—problems which highlight the tension between vagueness and classical laws. After all, isn’t it the case, as many seem to suppose, that the presence of vague language within the scope of logic threatens the validity of some classical laws, e.g. the law of excluded middle? Not according to SpV , and nor should it according to supervaluationists. When the objector argues that, for vague A, A ∨ ∼A fails because neither A nor ∼A—e.g. ‘Tim is tall or Tim is not tall’ fails because Tim is neither tall nor not tall—the SpV theorist responds by admitting that neither A nor its negation are (super)true, and so
392
Dominic Hyde
were ‘∨’ truth-functional A ∨ ∼A would fail to be (super)true but, just as with ‘∃’, truth does not distribute over ‘∨’, i.e. the connective is not truth-functional and, as noted earlier, subjunction fails. Though the SpV theorist accepts failures of bivalence, this would only entail the failure of lem were disjunction to be truth-functional, which it is not. On this view the objection, like that which arose in response to the SpV solution to the sorites above, is based on the mistaken supposition that acceptance of lem commits one to semantic precision (if everything’s red or it isn’t then ‘red’ is precise), however, one should not confuse lem with the Principle of Bivalence. One should not confuse the claim that: (3) It is true that A ∨ ∼A with the claim that: (4) It is true that A or it is true that ∼A. SpV accepts the former whilst denying the latter. The latter is denied by virtue of the vagueness of A (just as (2) was denied by virtue of the vagueness of the predicate B) yet (3) is accepted (just as (1) was).⁹ As with (1), it is the acceptance of (3) that gives rise to misgivings and leads to doubts concerning the adequacy of SpV . Unlike (1) though, where the supervaluationist simply bit the bullet and sought to minimize the damage done by undermining the supposedly untoward consequences, (3) has been defended not as a cost worth paying but, rather, as a claim that all theories should embrace. (3) is straightforwardly true and follows from the logical truth of excluded-middle claims which themselves can be seen to be mandatory despite some peoples’ intuitions to the contrary. In the case of (3), objectors are quite simply wrong. Costs attend its rejection, not its retention. This is because (3) reflects supposed ‘penumbral connections’. Of course, to defer to talk of ‘penumbral connections’ as a means of defending a commitment to (3) is only successful to the extent that the existence of penumbral connections is independently defensible. Obviously then, to defend claims for the existence of penumbral connections it would simply beg the question to cite the need to retain classical laws. That is the very issue in question here, so argument independent of classical laws must be presented. And indeed it is. Keefe cites Edgington’s arguments for the non-truth-functionality of disjunction as evidence of such ‘penumbral connections’ (though Edgington is intent on developing an alternative, non-truth-functional logic to SpV ). Edgington presents a number of arguments for non-truth-functional disjunction. Initially, ‘ ‘‘Sibling’’ means the same as ‘‘brother or sister’’. There are sex changes; and they are not instantaneous. ⁹ Just as earlier concerns centering on the existential quantifier extend to the universal quantifier, so too here claims centering on disjunction extend to conjunction in the obvious way. Despite the non-falsehood of both A and ∼A, their conjunction is nonetheless always false, thus guaranteeing the validity of the law of non-contradiction. We will return to the issue of vagueness and lnc later.
The Prospects of a Paraconsistent Response to Vagueness
393
Therefore, at times, while someone is definitely a sibling, it is indeterminate whether they are a brother, or a sister. Therefore, someone can be definitely a brother or a sister, without being definitely a brother, or definitely a sister: a disjunction can be definitely true without either disjunct being definitely true’ (1997, 310). Problems attend the first assertion though. Arguably ‘sibling’ does not in fact mean the same as ‘brother or sister’. In the absence of borderline cases they are extensionally equivalent, however, ‘sibling’ simply means ‘having (at least one of ) ones’ parents in common’ and thus does not name a simple disjunctive category, but, rather, spans the categories of ‘brother’ or ‘sister’, or anything in-between. Edgington offers a further argument for non-truth-functional disjunction, one which would generalize to a defence of lem in the context of vagueness, and which seeks to explicitly force a disjunctive reading of the key term. ‘A library book can be such that it is not clear whether it should be classified as Philosophy of Language or Philosophy of Logic; but if we have a joint category for books of either kind, it clearly belongs there. It is not unusual for a term in one language to require a disjunctive translation in another. Suppose a language trivially different from English which has one word ‘‘bleen’’, for ‘‘blue or green’’. Something can be definitely bleen, but neither definitely blue nor definitely green. Therefore, something can be definitely blue or green, while neither definitely blue, nor definitely green’ (1997, 310). The argument is similarly unsuccessful. If one means by ‘a joint category for books of either kind’ a category that includes all those in the category Philosophy of Language and all those in the category Philosophy of Logic then the book in question is not clearly in this simple disjunctive category. ( Just imagine the librarian moving books onto the new shelves purchased to house books in the new category. Any book from either of the older categories is placed there. Is it clear that the contested book should be placed there? Surely not.) Of course, if the ‘joint category’ is one which spans books from Philosophy of Language, Philosophy of Logic and all in-between then the book will clearly belong in this category but this is no longer the simple disjunctive category required for the argument to establish the existence of a definitely true disjunction with admittedly indeterminate disjuncts. More generally, terms with ‘disjunctive translations’ like ‘bleen’ do definitely apply to objects which admittedly do not definitely satisfy either of the disjuncts if the ‘disjunctive translation’ names a span that covers each of the two disjunct categories and all in-between. But where the term’s ‘disjunctive translation’ is a mere disjunction and names a simple disjunctive category, as required for the argument to succeed, it is unclear whether the term definitely applies in cases where neither of the disjuncts definitely apply. Only by equivocating on exactly what one means by a disjunctive translation can the argument succeed. Both Edgington and Keefe have failed to establish (3) and have failed, more generally, to establish the acceptability of a non-truth-functional analysis of disjunction or the acceptability of penumbral connections. Opting for a weakly paracomplete response as opposed to a strongly paracomplete one that both abandons lem, restores subjunction and endorses a truth-functional analysis of disjunction, remains a costly option to pursue from a purely pragmatic point of view.
394
Dominic Hyde
As with concerns above arising from the retention of classical consequence, the retention of the classical laws themselves, especially lem, in conjunction with the resulting non-truth-functional account of disjunction might again be defended by claiming that such a logic follows from a representational account of vagueness. Fine seems to respond in this way, offering representationalism as a defence. Since we cannot precisely describe the precise world we cannot in general say precisely how it is, though we can say that A ∨ ∼A is the case if we can say that the world is precise. lem is indeed counter-intuitive in the context of vagueness, but the merely semantic nature of vagueness does not impugn lem. ‘Suppose I press my hand against my eyes and ‘‘see stars’’. Then lem should hold for the sentence S = ‘‘I see many stars’’, if it is taken as a vague description of a precise experience’ (1975, 285). If vagueness is merely semantic, as the representationalist takes it to be, then lem is prescribed and so defensible. SpV is again defended by appeal to a substantive theory of vagueness. Again, as with the defence of classical consequence and the behaviour of the existential quantifier, such an appeal needs careful elaboration and subsequent scrutiny. My own view is that, even if merely semantic vagueness is capable of justifying such anomalies, such an account is itself unacceptable. The supposed virtues of the conservatism encapsulated in (I) and (II) are overplayed.
22.1.3 Assertion, denial, and logical consequence In addition to overplaying the supposed virtues of the conservatism inherent in (I) and (II), the non-conservatism of the supervaluationist approach expressed through (III) is underplayed. Reflection on the nature of logical consequence ought convince us that (III) represents SpV as departing significantly from classical, regulative, logical principles. The non-coextensiveness of CL multiple-conclusion consequence and SpV multiple-conclusion consequence reflects a difference in respect of a relation that matters. Let me explain. Supervaluationism is a theory which explicitly recognizes truth value gaps, abandoning the Principle of Bivalence in the face of incompleteness as evidenced by applications of predicates to their borderline cases. There will be a sentence A which is not true and so too its negation. Given the plausible view that we should deny what is not true it follows that supervaluationists should deny both A and ∼A. Moreover, given that one ought not assert what one also denies, and thus ought assert only truths, supervaluationists should refrain from asserting either of A and ∼A. It follows then that the denial of A must be separated from the assertion of its negation, unless supervaluationists be required both to assert ∼A (by way of denying A) and not assert ∼A (since it is not true).¹⁰ Restall (2005) presents additional arguments for their separation independent of considerations of truth value gaps, but the weaker claim that ¹⁰ To be sure, as Keefe points out (2000, 155, fn. 1), denial of A may sometimes be expressed by a locution ‘which is hard to distinguish in practice from the assertion of the negation’—‘not A’ with an emphasis on ‘not’—nonetheless they are distinct. Keefe is ambivalent on the matter, suspecting they may amount to one and the same thing and viewing any resulting incoherence on what she sees as possible conflict arising from the absence of
The Prospects of a Paraconsistent Response to Vagueness
395
supervaluationists, at least, ought to separate the two notions is all that is relevant for current purposes. The distinctness of denial and assertion means that we cannot characterize constraints on denial by simply appealing to constraints on assertion. Constraints on denial must be independently characterized along with constraints on assertion. Such constraints are captured by an adequate theory of logical consequence. As Restall (2005, 191–3) argues, ‘[i]t is common ground that logical consequence—whatever it amounts to—has some kind of grip on assertion and denial, [the speech-acts associated with] acceptance and rejection. . . . Logical notions are nothing if they have no applicability to regulate cognitive states of agents like us, and the content of such states. . . . If an agent’s cognitive state, in part, is measured in terms of those things she accepts and those she rejects, then valid arguments constrain those combinations of acceptance and rejection. . . . [A valid] one-premise, one-conclusion argument from A to B constrains acceptance/rejection by ruling out accepting A and rejecting B’. More generally, a multi-premise entailment such as A constrains acceptance and rejection in the obvious way, ruling out accepting all of while rejecting A. As Restall goes on to point out, this understanding of the role of logical consequence ‘has the advantage of symmetry’. It does not privilege acceptance over rejection, assertion over denial. Recognition of the entailment equally mandates that those accepting all of cannot, on pain of cognitive incoherence, go on to reject A and that those rejecting A cannot, on pain of incoherence, go on to accept all of . This view of consequence and its role in regulating patterns of acceptance and rejection does, however, have repercussions for our understanding of logical consequence. Suppose an agent, a supervaluationist say, rejects both A and B. Can such an agent coherently accept their disjunction, A ∨ B? That, of course, depends on whether or not the multiple-conclusion consequence relation A ∨ B A, B holds—i.e. it depends on whether subjunction is accepted. To think that it is renders incoherent the acceptance of the disjunction while rejecting each of the disjuncts. Since supervaluationists, as we have seen, do accept some disjunctions (e.g. A ∨ ∼A) while rejecting each disjunct, they must, as we have also seen, fail subjunction. The generalized consequence relation describes the logical constraint that supervaluationists are forced to reject. Now, were denial the assertion of negation and rejection the acceptance of negation then rejection of each of A and B would be equivalent to acceptance of∼A and ∼B, and the coherence of accepting A ∨ B while rejecting both disjuncts would simply be equivalent to the coherence of accepting A ∨ B, ∼A a clear-cut rule for assertion. However she fails to heed the full force of the reductio argument just presented, and treats denial of non-truths and assertion of only truths as distinct and competing rules for assertion which may conflict rather than seeing the latter rule as directly following from the former. Parsons (2000, 20), on the other hand, goes on to treat denial as assertion of an alternative ‘exclusion negation’, ¬, but this will lead to an infinite regress. Given higher-order vagueness, there will be borderline cases between A and ¬A where denial of both is appropriate, and so we should refrain from asserting either. Denial of A then will need to be distinguished from the assertion of ¬A and attempts to cast denial as the assertion of a negation will require yet another species of negation. And so on.
396
Dominic Hyde
and ∼B. And that is representable without recourse to a multiple-conclusion consequence relation as A ∨ B, ∼A, ∼B ∅. Given the non-equivalence of denial and asserted negation though, there is no avoiding the generalized multiple-conclusion consequence relation as the means for representing the point at issue. Whether or not inference as ordinarily understood only ever takes one from premises to a single conclusion, as some seem to think (e.g. Keefe 2000, 198), it would be foolish to think that the logical notion associated with its evaluation—a single-conclusion consequence relation—was of paramount importance. Inference matters precisely because good inference generates constraints on acceptance and rejection, and this underlying value shows the importance of single- and multipleconclusion entailments equally. Just as multiple-premise, single-conclusion consequence establishes the incoherence of rejecting the conclusion while accepting all premises, single-premise, multiple-conclusion consequence establishes the incoherence of accepting the premise while rejecting all conclusions. The ‘symmetry’ associated with the cognitive-constraint account of logical consequence coupled with the relative independence of denial and negation thus points to another symmetry between premise-sets and conclusion-sets associated with logical consequence: each may be empty, a singleton, or many-membered. This having been said, those who, like Keefe, accept the failure of subjunction in SpV as an inevitable result of the acceptance of lem have, at least implicitly, manifested a preference for one logical principle (lem) over another (subjunction). Where it is explicitly acknowledged, the preference is justified by playing down the significance of the failure of subjunction on the grounds that it is not a principle employed in ‘ordinary life’. Keefe claims, for example that ‘ . . . [subjunction] fails according to supervaluationism . . . But we do not use multiple-conclusion arguments in ordinary life and it is reasoning in vague natural language that is in question’ (2000, 198, fn. 24). Even supposing that ordinary arguments are not multiple-conclusion arguments, we are now in a position to see that this focus on an asymmetric, single-conclusion consequence relation is not justified. There is a clear sense in which we do and must use multiple-conclusion consequence in ordinary life. The presumed relative irrelevance of subjunction as compared with principles capable of being represented by a single-conclusion consequence relation is illusory. Summing up then, a paracomplete response to vagueness must choose between lem and subjunction. Supervaluationists opt for a weakly paracomplete account, retaining lem. But we have seen their arguments for that theorem unconvincing or, in the case of the representationalist defence, as resting on an as yet untried assumption as to the merely semantic nature of vagueness. The supposed virtue of recognizing ‘penumbral connections’ is not established. Now we can also see that the cost of their recognition is significant indeed and the failure of subjunction would already represent a considerable departure from classical, regulative, logical principles. Unless a representationalist theory of vagueness can be defended (and even then, contra Fine and Varzi, we might doubt it sufficient to defend SpV ) a weakly paracomplete approach, in general—and supervaluationism, in particular—looks hard, if not impossible to defend.
The Prospects of a Paraconsistent Response to Vagueness 22.2
397
S U BVA LUAT I O N I S M
Closely allied to supervaluationism is another non-classical approach to vagueness—that revision of classical two-valued logic known as subvaluationism, SbV .¹¹ Let us briefly describe its key features. ( Those familiar with Hyde [1997] can simply skip the next section.)
22.2.1
Describing subvaluationism
Subvaluationism is the paraconsistent cousin of supervaluationism and SbV admits truth value gluts where SpV admits truth value gaps. Paraconsistency is, in fact, the dual of paracompleteness. A logic is said to be consistent if and only if, for any valuation or model of any contradictory pair of sentences A and ∼A, they cannot both be true in the valuation or model. i.e. A, ∼A . Classical logic is a paradigm of a consistent logic. A logic will then count as inconsistent just in case it is not consistent—i.e. for some sentence A, both it and its negation can be true together. i.e. A, ∼A . Assuming a sentence to be false if and only if its negation is true, approaches to the problem of vagueness like SbV that, as we shall see, postulate truth value gluts exemplify logics which are inconsistent. A, ∼A SbV . As with incompleteness, however, logics might only admit inconsistency in a trivial sense, so that whenever a sentence and its negation are both true in a theory every sentence and its negation is true. Not so here—in addition to the admission of inconsistency there is also a commitment to the non-triviality of the inconsistency. i.e. A, ∼A B. As is now standard, we shall say that a logic which admits non-trivial inconsistent theories is paraconsistent. The foundation of SbV as an approach to vagueness is that vagueness necessitates a paraconsistent response. Thus: A, ∼A SbV B. A paraconsistent response has been pointed to by a range of theorists in the past.¹² The main problem with many of these suggestions that vagueness warrants a paraconsistent analysis is that while they point in a paraconsistent direction they do not explain in any detail how vagueness is to be analysed from either a formal or philosophical point of view. Ja´skowski’s discussive logic reinterpreted as a dualization of supervaluationism presents us with both a formal analysis of vagueness and a philosophical interpretation as informative as its paracomplete rival.¹³ ¹¹ The formal system was first proposed as an account of vagueness in Ja´skowski (1948). Ja´skowski, a student of the Lvov–Warsaw School of philosophy, published his account a decade before Mehlberg, a former student of the same School, proposed the now popular supervaluationist account. ¹² See Hyde (2007, §4) for a discussion. ¹³ The formal duality between SpV and SbV was originally examined in detail in Varzi (1994). The suggestion that they can be seen as dual philosophical accounts of vagueness was presented in Hyde (1997).
398
Dominic Hyde
To begin, subvaluational semantics treat borderline cases for a vague predicate like ‘heap’ as cases to which both the predicate and its negation applies. That is, if a is a borderline case for ‘heap’ then ‘a is a heap’ is true and ‘a is not a heap’ is true (i.e. ‘a is a heap’ is false). Where supervaluational semantics defined truth simpliciter (or supertruth) as applying to a sentence just in case that sentence was true no matter how one admissibly precisified any vague constituents of the sentence (i.e. just in case the sentence was true for all admissible precisifications) subvaluational semantics defines truth simpliciter (or subtruth) as applying to a sentence just in case that sentence is true for some admissible precisification. Whilst determinate truths are still those sentences which remain true for all admissible precisifications, determinate falsehoods are still those which are false for all admissible precisifications, and indeterminate (vague) sentences still those which are true on some but not all admissible precisifications, this third class now consists of those sentences that are both true simpliciter and false simpliciter (as opposed to neither true nor false simpliciter). Indeterminate sentences take on both truth values. Unlike paracomplete responses to vagueness where indeterminacy is analysed as underdetermination, paraconsistent responses—and SbV in particular—analyse indeterminacy as overdetermination. Determinate truth is now considered a matter of truth only and determinate falsity a matter of falsity only. Since truth and falsity are taken to be exhaustive (i.e. completeness is assumed), indeterminate sentences are now considered neither true only nor false only but, rather, both true and false. It is easy to show that for such sentences, e.g. ‘a is a heap’, both it and its negation are true. Moreover, an evaluation which ascribes both truth values to such a sentence might nonetheless ascribe just the value ‘false’ to another sentence ‘b is a heap’. The logic is clearly paraconsistent. Like supervaluationism, subvaluationism seeks to minimize logical revision necessary to accommodate vagueness and so, against a classical background, will seek to preserve all classical tautologies. Subvaluationists thus require a paraconsistent logic of vagueness yet aim, in particular, to retain the law of non-contradiction (lnc). More particularly still, not only should contradictions therefore always be false (i.e. ∼(A & ∼A)) but they should also never be true (i.e. A & ∼A )—a non-trivial distinction in the current paraconsistent context. Consequently, the paraconsistent logic that is sought differs crucially from others that have been proposed to deal with the Liar Paradox, for example. There the most plausible candidate is the logic of Priest (1979), LP, where contradictions are sometimes true. The logic that is sought here, however, is what, following Arruda (1989), we may describe as a weakly paraconsistent logic; though it admits of non-trivial inconsistent theories which contain both A and ∼A (i.e. A, ∼A B), nonetheless no such non-trivial theory includes A & ∼A. (By contrast, strongly paraconsistent logics like LP do not distinguish between the truth of A & ∼A and the truth of both conjuncts, and consequently they admit of non-trivial valuations making A & ∼A true, i.e. A & ∼A B.) Thus: A & ∼A SbV B. Defining SbV -consequence in terms of preservation of truth simpliciter, i.e. subtruth, satisfies the foregoing constraints. Thus, SbV if and only if whenever all the
The Prospects of a Paraconsistent Response to Vagueness
399
members of are true then some member of is true (i.e. whenever all the members of are true in some admissible precisification then some member of is true in some admissible precisification). Or, equivalently for current purposes, it is impossible for all the members of to be true and all the members of to be not true. Such an account of logical consequence will obviously establish all classical (CL) theorems as theorems of SbV : (I )
SbV A if and only if CL A.
So, for example, though the principle governing the exclusivity of truth-values is rejected the law of non-contradiction is preserved in both of the following senses: lnc: SbV ∼(A & ∼A), and lnc : A & ∼A SbV . Where a restriction is placed on the consequence relation to the effect that the premise-set be a singleton set, the multiple-conclusion consequence relation is coextensive with classical consequence and admits as valid all and only those inferences that are classically valid. Thus: (II )
A SbV
if and only if A CL .¹⁴
But multiple-conclusion SbV -consequence, more generally considered, deviates from its classical counterpart. As a result of being paraconsistent but only weakly so, SbV fails adjunction: A, B SbV A & B. ¹⁵ This non-classical feature of SbV manifests the non-truth-functional account of conjunction required to underwrite a weakly paraconsistent theory of vagueness and is a major source of concern when evaluating the system’s adequacy. How can it make sense to accept two claims considered separately whilst rejecting them considered jointly? Classical multiple-conclusion consequence is preserved in SbV only in the following qualified sense: (III ) A1 & A2 & . . . & An SbV
if and only if A1 , A2 , . . . An CL .¹⁶
How then might such a paraconsistent approach to vagueness resolve the sorites paradox? To answer this question, consider the standard (i.e. many-conditionals) form of the paradox. A man with 1 hair on his head is bald. If a man with 1 hair on his head is bald then a man with 2 is. If a man with 2 hairs on his head is bald then a man with 3 is. ¹⁴ Proof follows from a simple generalization of the proof of (II) in Hyde (1997, 648). ¹⁵ Since, were adjunction to hold, substituting ∼A for B, lnc would mandate consistency (i.e. not both of A and ∼A could be true). ¹⁶ For a proof see Hyde (1997, 655).
400
Dominic Hyde .. . If a man with 9,999 hairs on his head is bald then a man with 10,000 is. ∴
A man with 10,000 hairs on his head is bald.
Since SbV does not preserve classical consequence unrestrictedly, the possibility arises of the paradox being discounted by virtue of its invalidity. One diagnosis available to the SbV theorist is exactly that—the premises, including conditional premises, are all true but modus ponens is not unrestrictedly valid. Consider the sentence ‘A pile of n grains of sand is a heap’ where a pile of n grains counts as a borderline case for ‘heap’. The sentence is true and false, so it is true. Since it is also false then the material conditional ‘If a pile of n grains of sand is a heap then a pile of n − 1 grains is a heap’ is true by virtue of the falsity of its antecedent. Nonetheless, a pile of n − 1 grains of sand might be determinately not a heap thus making the sentence ‘A pile of n − 1 grains of sand is a heap’ false.¹⁷ So: Heap(n), Heap(n) → Heap(n − 1) SbV Heap(n − 1), for some n. As is familiar from other paraconsistent logics then, modus ponens for material implication, i.e. disjunctive syllogism, is not valid in SbV . (Notice that what is claimed is that modus ponens is here denied for material implication—the implication relation typically assumed in modelling the sorites conditional. Beall and Colyvan (2001) are, of course, right to point out that this analysis of the sorites may hide the problem rather than solve it, since it assumes what many consider a very weak reading of conditionality. Stronger conditionals are definable and alternate responses are then available to the SbV theorist. Entailment is a much stronger conditional for which modus ponens clearly holds, but it does not provide an interpretation of the sorites premises which renders them true. A mid-strength connective, ‘ →’, just strong enough to satisfy modus ponens can be explicitly defined in SbV but it is easily shown that this does not provide an interpretation of the sorites-conditionals which renders all the premises true.¹⁸ There is, of course, an analogous problem lurking here for the supervaluationist who can equally be charged with assuming an overly strong reading of the conditional which is sufficiently strong to validate modus ponens but not weak enough to provide an interpretation of the sorites premises which renders them true.)¹⁹ ¹⁷ Higher-order vagueness may complicate matters here, but even the simplified approach will subsequently be found untenable. The simplification is thus harmless. Notice that the conditional premise will be counted both true and false in SbV under the conditions described, whereas SpV would count it neither true nor false. ¹⁸ Define ‘A → B’ as follows: A → B is true simpliciter iff either A is false in every admissible precisification or B. The aforementioned counterexample to modus ponens now renders the corresponding (ponendable) conditional ‘Heap(n) → Heap(n − 1)’ false. The newly defined connective is, in fact, just Ja´skowski’s discussive implication, ‘→D ’ suitably reinterpreted, which he explicitly introduced to recapture a conditional satisfying modus ponens. For further discussion of discussive implication see Priest and Routley (1989, 158f ). ¹⁹ See Hyde (2001).
The Prospects of a Paraconsistent Response to Vagueness
401
The foregoing simple paraconsistent reinterpretation of supervaluational semantics reproduces exactly the first formal system of paraconsistent logic—discussive logic—developed by Ja´skowski over fifty years ago, which already at the time was claimed to be applicable to vagueness. Ja´skowski’s long-standing proposal to treat vagueness from a paraconsistent perspective by means of a discussive logic is simply reinterpreted so as to be the dual of the dominant paracomplete supervaluationist approach. More exactly, where and are sets of sentences of the shared language of SpV and SbV and ∼ = df {∼A : for all A ∈ }: SpV
if and only if ∼ SbV ∼.²⁰
22.2.2 Defending subvaluationism Given the duality between supervaluationism and subvaluationism, it is unsurprising that subvaluationism faces objections that are the exact dual of those pressing
