Vagueness and Degrees of Truth

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Vagueness and Degrees of Truth

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Vagueness and Degrees of Truth Nicholas J. J. Smith

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Great Clarendon Street, Oxford   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  Nicholas J. J. Smith 2008 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2008 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 the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Laserwords Private Limited, Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk ISBN 978–0–19–923300–7 10 9 8 7 6 5 4 3 2 1

For my parents, Sybille and Vivian Smith, with love and gratitude

Acknowledgements Thanks to Huw Price, who since I was an undergraduate has been an unfailing source of support, philosophical insight, and good advice. For valuable comments, discussions, and encouragement, I thank David Armstrong, Paul Benacerraf, David Braddon-Mitchell, Bob Brandom, John P. Burgess, Keith Campbell, Mark Colyvan, John Cusbert, Kit Fine, Alan H´ajek, Amitavo Islam, Jenann Ismael, the late Dick Jeffrey, Mark Johnston, Ed Mares, Ken Perszyk, Graham Priest, Gideon Rosen, and Scott Soames. Thanks to Peter Momtchiloff and Kate Walker of Oxford University Press for their guidance, to the three readers for their very useful comments, and to Jean van Altena for her helpful copy-editing. Thanks to my wife, Cath Vidler—for everything. Parts of Smith (2003), (2004), and (2005b) show up in reworked form in §5.1, §§5.2 and 5.6, and Chapters 3 and 4 respectively. Thanks to Oxford University Press, Springer, and Taylor & Francis for permission to reuse this material.

Contents Introduction

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Part I. Foundations . Beginnings 1.1 Toolkit 1.2 The Classical Semantic Picture . The Space of Possible Theories of Vagueness 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Epistemicism (Deny Nothing) Additional Truth Values (Deny 1a) Truth Gaps (Deny 1b) Supervaluationism (Deny 1c) Plurivaluationism (Deny 2) Contextualism Intuitionism (Assert Nothing)

15 17 24 33 34 50 71 76 98 113 122

Part II. Vagueness . What is Vagueness? 3.1 What Should We Want from a Deﬁnition of Vagueness? 3.2 Existing Deﬁnitions of Vagueness 3.3 Closeness 3.4 Vagueness as Closeness 3.5 The Advantages of Closeness . Accommodating Vagueness 4.1 Epistemicism 4.2 Additional Truth Values and Truth Gaps

127 127 133 140 145 159 175 175 186

viii

 4.3 Supervaluationism 4.4 Plurivaluationism 4.5 Contextualism

191 197 200

Part III. Degrees of Truth . Who’s Afraid of Degrees of Truth? 5.1 5.2 5.3 5.4 5.5 5.6 5.7

209

On the Very Idea of Degrees of Truth Classical Logic A Gated Community in Theory Space? Truth and Assertibility Truth-Functionality Denying Bivalence Different Senses of ‘Fuzzy Logic’

210 220 224 248 251 274 275

. Worldly Vagueness and Semantic Indeterminacy

277

6.1 Artiﬁcial Precision 6.2 Sharp Boundaries

277 304

Conclusion

317

References Index

321 333

Introduction Ordinarily we say that persons are vague if they regularly misplace their car keys, stare blankly into the fridge having forgotten why they opened it, and so on. In philosophy, however, the term ‘vague’ has a different use. It applies primarily (although not exclusively) to predicates.¹ Amongst all predicates, the vague ones are usually singled out in one or more of three ways: Borderline cases. There are some persons to whom the predicate ‘is tall’ clearly applies (e.g. most professional basketball players) and some to whom it clearly does not apply (e.g. most professional jockeys), but then there are other persons to whom it is unclear whether or not the predicate applies (I’m sure you know some of them). When asked whether such a person is tall, we tend to react with some sort of hedging response: ‘‘sort of’’, a shrug and a certain sort of scowl or exhalation of breath, a blank look, etc. Call these persons borderline cases for ‘tall’. Other predicates—for example, ‘is a prime number’—do not have borderline cases. Julius Caesar is not a prime number, seven is a prime number, eight is not a prime number, and so on: there is nothing at all of which it is unclear whether it is a prime number.² One standard way of drawing the distinction between vague and non-vague (or precise) predicates is to say that vague predicates have borderline cases, while precise predicates do not.³ Blurred boundaries. Imagine a line drawn around all the things to which a given predicate applies. One typical characteristic of vague predicates is that ¹ Predicates are items of language—e.g. ‘is tall’, ‘is happy’, ‘is running’—which go together with names—e.g. ‘Bill’, ‘Ben’, ‘Alice’—or deﬁnite descriptions—e.g. ‘the tallest woman in the room’, ‘the inventor of post-it notes’, ‘the man who threw the egg’—to form sentences—e.g. ‘Bill is tall’, ‘The inventor of post-it notes is happy’, ‘Alice is running’. ² For large numbers, we may suppose that we can use a computer to help us determine the answer. ³ The borderline case characterization can be traced at least as far as Peirce 1902.

2  we get a blurry, ill-deﬁned line. For example, there is no sharp boundary between the things to which ‘is heavy’ applies and the things to which it does not apply. For a precise predicate, by contrast, we get a sharp line, cleanly separating the things to which the predicate applies from the rest.⁴ Sorites paradoxes. Consider a hirsute person. He is not bald. Were we to remove a single one of his hairs, he still would not be bald. So too if we removed another, and so on—because one hair cannot ever make the difference between baldness and lack thereof. But our man has only a ﬁnite number of hairs, so eventually he will have none left. By our reasoning, he will still not be bald—whereas of course a person with no hairs on his head is bald. So something has gone wrong somewhere. This is an instance of the Sorites paradox. Note that the problem does not arise for all predicates. Instead of ‘is bald’, take the predicate ‘has ten or more hairs on his head’. Consider a hirsute person. He has ten or more hairs on his head. Were we to remove a single one of his hairs, he would still have ten or more hairs on his head. So too if we removed another, and another—but not ‘and so on and on’, because one hair can make the difference between having ten or more hairs, and having nine or less. In this case we get no paradox. One standard way of drawing the distinction between vague and precise predicates is to say that vague predicates generate Sorites paradoxes, while precise predicates do not. That is, where ‘is P’ is vague, we will typically be able to imagine a so-called Sorites series of objects, ranging from one which is clearly P to one which is clearly not P, but where we also feel strongly inclined to say that for any object in the series, if it is P, then so are its neighbours.⁵ So we are interested in predicates which admit of borderline cases, which draw blurred boundaries, and which generate Sorites paradoxes. Why are we interested in such predicates? For at least two reasons. First, vagueness is ubiquitous. Most of the predicates in our language are vague. Indeed, it is very hard to think of a predicate which is not at all vague. Some predicates have a wider class of borderline cases than others, but it is quite a challenge to think of predicates outside mathematics and the hard sciences which ⁴ The blurred boundaries characterization can be traced at least as far as Frege’s statement that if we represent concepts in extension by areas on a plane, then vague concepts do not have sharp boundaries, but rather fade off into the background (Grundgesetze, ii, §56; trans. in Beaney 1997, 259). ⁵ More precise characterizations of Sorites series and Sorites paradoxes will be presented below in §3.5.4.

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admit of no possibility of borderline cases—whose boundaries are not in the slightest bit blurry. (Try it now!) So vagueness is something we cannot avoid. If we want to understand our language and our world, we need to understand vagueness. Second, there is immense potential for many other research programmes to be signiﬁcantly advanced by progress on vagueness. When a key concept in a certain area is vague—and thanks to the ubiquity of vagueness, key concepts in many areas are vague—understanding how its vagueness works can have crucial payoffs. So we know which predicates we want to study, and why we want to study them. What exactly do we hope to gain from the study? What is our goal? What we ultimately want to gain from the study of vagueness is an account of the semantics of vague predicates—an account of what the meaning of such a predicate consists in, and of how (if at all) it differs from the meaning of a precise predicate. The development of modern logic (also known as classical logic) and model theory in the nineteenth and twentieth centuries led to immense progress in our understanding of language. However, the pioneers of these developments—notably Gottlob Frege and Alfred Tarski—were primarily concerned with mathematical language, and they ignored vagueness. Frege in particular set vagueness to one side as not worthy—or not susceptible—of treatment. He considered vagueness a defect of natural language, to be banished from a logically perfect scientiﬁc language. The problem with this attitude is that mathematical language is a special case: it is completely precise. Outside mathematics, virtually all of our language is vague, to a greater or lesser extent. Beginning in real earnest in the second half of the twentieth century, this realization that vagueness is ubiquitous in natural language led to a desire to approach ordinary language in the same spirit in which Frege approached mathematical language—in a bid not to eliminate vagueness, but to understand and adequately represent it. This led to the development of a large number of non-classical logics, and associated theories of vagueness (where by a theory of vagueness, I mean a formal logical core, together with a package of motivating and explanatory philosophical views). There are now many non-classical logics, and associated theories of vagueness, on the market. So in one sense we have advanced greatly since Frege’s time: we now have many systems of semantics designed speciﬁcally to accommodate vague language. In another sense, however, we are still no closer to our goal of obtaining an account of what the meaning of a vague predicate consists in—for we have

4  many accounts, and no apparent way of deciding which (if any) of them is correct.

Part I: Foundations As a ﬁrst step towards rectifying this situation, in Chapter 2—building on basic preliminary material which it is the purpose of Chapter 1 to present—I give an overview of the space of possible theories of vagueness, and show where existing theories live in this space. For we cannot begin to select a theory of vagueness until we fully understand how the different theories differ from one another: not just over particular matters of detail, but at a more fundamental and illuminating level of analysis. Accordingly, Chapter 2 provides a conceptual map of theories of vagueness, setting out the important axes along which different theories vary, and then giving the coordinates of existing and future theories on these axes. At the centre of the map is the classical view of vagueness, according to which vague language is—from a semantic point of view—exactly the same as precise language. This view applies the standard model theory for precise mathematical language to vague language. I begin by isolating the key elements of this classical model theory: these key elements give the axes of the space. With the classical view at the origin, other theories are located according to which of the key elements they modify, and then, at a ﬁner level of analysis, according to how they modify them. It is in this chapter that a distinction which plays a central role in this book ﬁrst emerges: the distinction between worldly vagueness and semantic indeterminacy. Here is a quick sketch of how it emerges. Model theory involves two aspects: a part which represents a language, and a part—a model —which represents the world. The process whereby items of language gain meaning is represented by matching up a language with a model—which is called giving an interpretation of the language. The underlying thought is that our words gain meaning when we use them to talk about the world. But language is versatile, and words could mean many things. Consider the sentence ‘The house is on the hill’. The symbol ‘house’ could have been used to mean what the symbol ‘cat’ actually means, and the symbol ‘hill’ could have been used to mean what the symbol ‘mat’ actually means, so there is an interpretation of the sentence on which it means



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that the cat is on the mat. But the interpretation on which the sentence means that the house is on the hill is privileged: it corresponds to what the sentence actually means. Such an interpretation is called the intended interpretation. In isolating the key elements of the classical model theory for precise language, I make a broad distinction between the internal nature of classical models, and the external part of the classical story, according to which each discourse has a unique intended interpretation. The classical picture tells us on the one hand that the world is of such and such a sort (it has a structure which can be represented by a classical model, and classical models have such and such internal features), and on the other hand that the meaning of anything we say is correctly speciﬁed by giving a single such model. This distinction yields two broad ways in which a theory of vagueness can differ from the classical view: it can replace classical models with non-classical models with different internal features, or it can alter the external story according to which only one model comes into play in describing what some utterance actually means. This distinction between two ways of departing from classical model theory underlies the distinction between worldly vagueness and semantic indeterminacy. Vagueness and indeterminacy are ruled out of the classical picture in two places. First, there is no vagueness in the world. Classical models are entirely precise: they represent the world as a crisp set of objects and properties such that for each property and each object, the object either deﬁnitely possesses the property, or deﬁnitely does not possess it. Second, there is no semantic indeterminacy—no vagueness in the relationship between language and the world. Each sentence has a unique intended model: so there is no vagueness about what any sentence means. A view which differs from the classical picture in the internal way (i.e. replaces classical models with non-classical models with different internal features) opens up the possibility of worldly vagueness, while a view which differs in the external way (i.e. denies that only one model comes into play in describing what some utterance actually means) opens up the possibility of semantic indeterminacy. For example, the view of vagueness built on fuzzy set theory (see §2.2.1) countenances worldly vagueness. It tells us that when we utter a vague predicate, we pick out a unique property in the world—the extension of our predicate on the unique intended interpretation of our utterance—but this property is inherently vague, in the sense that some objects possess it, some do not possess it, and others possess it to various intermediate degrees. This is a

6  matter of how things are out there in the world (on this view)—it is not a matter of the relationship between language and the world. In contrast, semantic indeterminacy enters when we have a view which denies that an utterance has a unique intended interpretation. On this sort of view, when we utter a vague predicate, there are several properties we might be picking out—the extensions of this predicate on the various equally intended (or equally not-unintended) interpretations—and there is no fact of the matter as to which, in particular, we mean. Here we have indeterminacy in the relationship between language and the world.⁶ Some surprising and signiﬁcant results emerge from the survey of theories of vagueness. One is that two quite different views have been conﬂated in the literature under the heading ‘supervaluationism’. I call these two views ‘supervaluationism’ and ‘plurivaluationism’. This distinction is particularly important, because ‘supervaluationism’ is widely regarded as the front runner amongst existing theories of vagueness—but once we clearly distinguish the two views that have been run together under this heading, we will be in a much better position to discern the real disadvantages of each.

Part II: Vagueness With a clear overview of theories of vagueness in hand at the end of Chapter 2, we still face the question of deciding which theory is correct. The literature abounds with objections to particular theories, and these have led to the outright rejection of a number of views—but still, several broad types of theory remain viable (for example, epistemicism and degreetheoretic approaches, to name just two). Within each type, there is general agreement that some particular versions are better than others—but there are no widely accepted criteria which can decide between the types themselves. Theorists of vagueness have thus found themselves divided into ⁶ In order to draw substantive conclusions about the location of vagueness—in the world itself, or in the relationship between language and the world—from a system of model theory, we must take a literal attitude towards model theory. That is, we must regard a model theory for (part of ) a language as giving a literal (although not necessarily complete) description of the relationship between that language and the world. I present reasons for taking such a literal attitude towards model theories for vague language (in the context of a study such as this one) in §2.1.3.1.

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camps. This is a very unsatisfactory situation. Given that the different types of view present completely different pictures of the relationship between vague language and the world, they cannot all be correct. We need some way of deciding which type of theory is right. We need some criteria for determining what the general form of the correct theory of vagueness should be. There is another loose end that also cries out to be tied up at this point. We have three informal characterizations of vague predicates: they give rise to borderline cases, their extensions have blurry boundaries, and they generate Sorites paradoxes. These three characterizations seem to be quite closely related to one another—yet not so closely related that they are merely three ways of saying essentially the same thing. So what is the relationship between them? Rather than three piecemeal characterizations of vagueness, it would be desirable to have a proper deﬁnition of vagueness: a crisp statement of what is of the essence of vagueness—of what vagueness ultimately consists in—given which, we can see why vague predicates have borderline cases, generate Sorites paradoxes, and draw blurred boundaries. My strategy is to link these two tasks: the task of ﬁnding the correct theory of vagueness and the task of ﬁnding an adequate deﬁnition of vagueness. Chapter 3 is concerned with the latter task. In that chapter, I discuss what we should expect from a deﬁnition of vagueness, critically discuss existing deﬁnitions, and then present a deﬁnition of vagueness according to which—roughly—a predicate F is vague just in case for any objects a and b, if a and b are very similar in respects relevant to the application of F, then the sentences Fa and Fb are very similar in respect of truth. So, for example, ‘is tall’ will be vague just in case for any persons a and b, if a and b are very similar in height, then the sentences ‘a is tall’ and ‘b is tall’ are very similar in respect of truth. The import of this deﬁnition can be grasped by comparing it with the claim that vague predicates are tolerant. A predicate F is tolerant with respect to φ if there is some positive degree of change in respect of φ that things may undergo, which is ‘‘insufﬁcient ever to affect the justice with which F is applied to a particular case’’ (Wright 1975, 334). So, for example, ‘is tall’ will be tolerant with respect to height just in case for any persons a and b, if a and b are very similar in height, then there is no difference in the applicability of the predicate ‘is tall’ to a and b—that is, the sentences ‘a is tall’ and ‘b is tall’ are exactly the same in respect of truth. The great problem with

8  the claim that any predicate F is tolerant is that, when conjoined with the claim that we can construct a Sorites series for the predicate F, it leads to contradiction—in particular, to the claim that each object in the Sorites series both is and is not F. My deﬁnition of vagueness is a weakening of the claim that vague predicates are tolerant: if a and b are very similar in height, then there need not be no difference in the applicability of the predicate ‘is tall’ to a and b—that is, the sentences ‘a is tall’ and ‘b is tall’ need not be exactly the same in respect of truth—but there cannot be much difference in the applicability of the predicate ‘is tall’ to a and b, and so the sentences ‘a is tall’ and ‘b is tall’ must be very similar in respect of truth. In the remainder of Chapter 3 I discuss the question of extending this deﬁnition to cover vagueness of many-place predicates, of properties and relations, and of objects, and I then explore the advantages of this deﬁnition, some of the most important of which are that it captures the intuitions which motivate the thought that vague predicates are tolerant, without leading to contradiction, and that it yields a clear understanding of the relationships between Sorites-susceptibility, blurred boundaries, and borderline cases. In Chapter 4, I turn to the task of determining what type of theory of vagueness we need. Having clearly presented the different types of theory in Chapter 2, and having presented a crisp deﬁnition of vagueness in Chapter 3, my strategy is to ask which types of theory can accommodate vague predicates—that is, predicates which satisfy the deﬁnition given in Chapter 3. It is here, then, that the link is made between the task of ﬁnding an adequate deﬁnition of vagueness and the task of ﬁnding the correct theory of vagueness. When vagueness is characterized informally in terms of borderline cases, blurred boundaries, and Sorites-susceptibility, all the main existing types of theory of vagueness can be seen as accommodating vagueness—simply because the informal characterizations are so loose. This leads to the ‘too many theories’ problem: the existing theories cannot all be right, as they conﬂict with one another; yet how to choose between them, if they all accommodate vagueness perfectly well? The situation changes, however, once we have to hand a sharp deﬁnition of the core property underlying the various surface phenomena standardly used to characterize vagueness. When we now ask whether each type of theory allows for the existence of predicates possessing the feature isolated in Chapter 3 as being of the essence of vagueness, it turns out that the answer



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is No. Only one type of theory does: the type which countenances degrees of truth. The basic idea of degrees of truth is that while some sentences are true and some are false, others possess intermediate truth values: they are truer than the false sentences, but not as true as the true ones. So, for example, if we line up a series of persons, ranging from one who is 7 feet in height to one who is 4 feet in height, in increments of a fraction of an inch, and then move along the series saying of each person in turn, ‘This person is tall’, the idea is that our statements start out quite true, then gradually get less and less true, until they end up quite false. Thus, from the project of deﬁning vagueness, an answer emerges to the question of what the general form of the correct theory of vagueness must be: it must be one which countenances degrees of truth.⁷ In order to reach the overall conclusion of Chapter 4—that we need a theory of vagueness that countenances degrees of truth—I need to do several things in the course of the chapter. First, I show that of the types of theory of vagueness distinguished in Chapter 2, only those which countenance degrees of truth can accommodate predicates which satisfy the deﬁnition of vagueness proposed and defended in Chapter 3. Second, I consider and reject two different strategies which non-degree theorists might employ to avoid my conclusion. The ﬁrst strategy is to propose an error theory of vagueness. Non-degree theorists might agree with the deﬁnition of Chapter 3, but still maintain the correctness of their semantic theory. The resulting position would be as follows: ‘‘For a predicate to be vague, it has to satisfy the deﬁnition of Chapter 3; my semantic theory does not allow for the existence of predicates that satisfy the deﬁnition of Chapter 3; but my semantic theory is correct; therefore there are no vague predicates.’’ I offer reasons for rejecting such an error theory of vagueness. The second strategy is to reject the deﬁnition of Chapter 3, to propose in its place an alternative deﬁnition which is compatible with a given non-degree theory, and to argue that this alternative deﬁnition has all the advantages which I claim for my deﬁnition in Chapter 3. I consider, and give reasons for rejecting, several proposals along these lines, including the proposal that a ⁷ Clearly, in order for the deﬁnition of vagueness to play the role of deciding between theories of vagueness, we must reject the widespread idea that a deﬁnition should be theory-neutral—that is, that the deﬁnition of P must be something on which all the candidate theories of P can agree. I discuss this issue in Ch. 3.

10  predicate F is vague just in case for any objects a and b, if a and b are very similar in respects relevant to the application of F, then the sentences Fa and Fb are very similar in respect of assertibility (rather than truth).

Part III: Degrees of Truth The upshot at this point is that we need a theory which countenances degrees of truth. This tells us the type of theory we need, but does not lead us to a particular theory. The best-known degree theory is the one based on fuzzy logic and set theory. It replaces the two classical truth values True and False with inﬁnitely many degrees of truth, which may be thought of as percentage values: thus a sentence might be only 45 per cent true, or 98 per cent true, and so on. In Chapters 5 and 6, I consider objections to the fuzzy view of vagueness in particular, and to degree-theoretic treatments of vagueness in general. In some cases, I argue that the objections do not carry weight. In other cases, I propose modiﬁcations and/or additions to the standard fuzzy view, in order to overcome the objections. The main objections covered in Chapter 5 are as follows. The very idea of truth coming in degrees is in some way a mistake! I show that it is not. The fuzzy theory involves an objectionable violation of classical logic! In response to this objection, I propose a new account of logical consequence in the fuzzy setting that allows us to derive a classical consequence relation from fuzzy semantics. That is, while the semantics for vagueness that I propose is non-classical, it validates classical logic. Degrees of truth cannot be integrated with key developments elsewhere in philosophy of language, outside the study of vagueness! I show that this is not correct. As part of my response to this objection, I propose a detailed new account of the relationship between degrees of truth and degrees of belief. Degree theories, such as the fuzzy theory, which treat the logical connectives as truth functions, cannot account for ordinary usage of sentences about borderline cases! I show that this objection is mistaken. Finally, denying bivalence—the view that every sentence is true or false, with no middle way (or ways, in the case of degrees of truth)—leads to contradiction! I show that it does not. In Chapter 6, I turn to what is arguably the major objection to the fuzzy view: namely, that it ‘‘imposes artiﬁcial precision . . . though one is

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not obliged to require that a predicate either deﬁnitely applies or deﬁnitely does not apply, one is obliged to require that a predicate deﬁnitely applies to such-and-such, rather than to such-and-such other, degree (e.g. that a man 5 ft 10 in. tall belongs to tall to degree 0.6 rather than 0.5)’’ (Haack 1979, 443). In response to this objection, I propose a new view—which I call fuzzy plurivaluationism—which departs from the standard fuzzy view by denying that each discourse has a unique intended (fuzzy) interpretation. This view thus incorporates both worldly vagueness (in the fuzzy models) and semantic indeterminacy (in the lack of a unique intended interpretation). The overall conclusion of the book is that fuzzy plurivaluationism is the correct theory of vagueness.

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P A RT I

Foundations

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1 Beginnings The purpose of Part I of this book is to survey the ﬁeld of theories of vagueness. Good surveys already exist—most notably Williamson (1994) and Keefe (2000)—so why do we need a new one? Partly because some theories, such as contextualism, have come to prominence in the years since Williamson’s and Keefe’s books appeared. However, the chief thing that distinguishes my survey and gives it its point is not what theories I cover, but the way I cover them. Existing surveys spend a relatively small amount of time on the formal details of each theory, and a relatively large amount of time looking at objections and replies; furthermore, the formal details that are given are generally presented in the manner in which the theory in question was ﬁrst presented—whatever that happens to be—i.e. no attempt is made to place the various theories within a common overarching theoretical framework. In contrast, my survey focuses on the formal details of the various theories, and it presents these details in a thematically uniﬁed way. More speciﬁcally, I focus on model theory, and on how each theory of vagueness differs from the semantic picture given by classical model theory. I thus aim to provide not a catalogue, but a conceptual map of theories of vagueness. At the centre of the map is the classical view of vagueness, according to which the semantics of vague language is exactly the same as the semantics of precise language. This view applies the standard model theory for precise mathematical language to vague language. The key elements of this classical model theory then give the axes of the space of possible theories of vagueness. With the classical view at the origin, other theories are located according to which of the key elements they modify, and then, at a ﬁner level of analysis, according to how they modify them. There are two reasons why a survey of this sort is needed. First, I think that this uniﬁed approach is genuinely illuminating: it enables us to

16  understand clearly how the different theories of vagueness differ from one another, not just over particular matters of detail, but at a fundamental level—and conceptual clariﬁcation and illumination of this sort is an end in itself in philosophy. Second, just as Williamson’s and Keefe’s dialectics require them to survey objections and replies comprehensively—for they each argue for their favoured theory by way of some form of cost/beneﬁt analysis of all the available theories (see pp. 129–30 below)—my dialectic requires me to explore carefully the inner workings of each theory. As discussed in the Introduction, my strategy for selecting amongst theories of vagueness is to ask of each theory whether it can allow for the existence of vague predicates—that is, ﬁrst to present a fundamental deﬁnition of vagueness, and then to ask of each theory whether it can allow that there are any predicates which satisfy this deﬁnition. In order to pursue this strategy, we need not only a very clear deﬁnition of vagueness, but a very clear understanding of the workings of each theory. Hence the need to focus on formal details, rather than surveying all known objections and replies to each theory. That said, I do note what I regard as the major objections and replies to each theory, in order to give a feel for these views to newcomers to the vagueness literature—and of course I explore objections and replies to degree-theoretic treatments of vagueness in great detail, in Part III. My emphasis on formal details in the foregoing should not trigger alarm bells in readers with little or no background in logic. The kind of reader that I had in mind when writing the book was one with no formal background other than a ﬁrst introductory logic course of one of the standard sorts. Indeed, the book is self-contained, and someone without even this much logic will—with a little more effort—be able to follow the discussion.¹ (Readers with more formal background might, accordingly, ﬁnd that I am labouring certain points, and should of course skim forwards at such places.) Chapter 2 maps the space of possible theories of vagueness. The present chapter deals with preliminaries. §1.2 presents the classical semantic picture which will be located at the origin of the map. Prior to that, §1.1 introduces some basic formal tools. ¹ I have taught several honours/postgraduate seminars based on drafts of this book: most of the students had taken no more than a ﬁrst course in logic, and some had no formal background at all.

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17

1.1 Toolkit This section can be read through as a ﬁrst introduction to the notions discussed, suitable for readers with little or no formal background. Alternatively, it can be treated as a glossary, to be skipped over entirely, and then consulted only in the event that one encounters a notion later on with which one is unfamiliar. 1.1.1 Sets A set is a bunch of objects; these objects are said to be members or elements of the set. We denote the set containing a, b, and c as members as follows: {a, b, c}. We denote the set of all things such that some condition C holds as follows: {x : C}. For example, the set of all red things is denoted {x : x is red}. We use the symbol ∈ (epsilon) to denote membership, as in a ∈ {a, b, c}. To say that something is not a member of a set we use the symbol ∈, / as in d ∈ / {a, b, c}. Sets are individuated by their members. That is, if ‘two’ sets have exactly the same members, they are identical; i.e. they are in fact one and the same set. So, for example, {a, b} = {b, a} (where = means is identical to, i.e. is one and the same object as). Note that sets are objects. The set containing my pen, my retractable pencil, my packet of polo mints, and my gonk is another object in its own right, distinct from the four objects in it. That is why sets can be members of other sets: the set just mentioned is, for example, a member of the set of all four-membered sets. The set just mentioned is not a visible, kickable object—unlike my retractable pencil or the other objects in the set. For this reason, sets are often referred to as ‘abstract objects’. But this does not make them any less objects: they are just objects which are not visible or kickable or located in space or time. This is the guiding idea of set theory: to treat a bunch of objects as an object in its own right, which we can then do things with—for example, make it the argument or value of a function (see below), put it in another set, and so on. 1.1.2 Subsets We say a set S is a subset of a set T (in symbols, S ⊂ T or S ⊆ T) iff every member of S is a member of T. Note that this leaves open whether or not

18  S = T: this depends upon whether there is anything in T which is not in S. If there is nothing in T which is not in S, i.e. if T ⊂ S as well as S ⊂ T, then S = T. If there is something in T which is not in S, i.e. S ⊂ T but not T ⊂ S, then we say S is a proper subset of T, sometimes symbolized S T. Note that every set is a subset of itself, but no set is a proper subset of itself. There is a set called the empty set or null set, symbolized ∅, which has no elements. We say that the null set is a subset of every set. A set containing just one element, for example {1}, is called a singleton, or unit set. Note that 1 is an element of the set {1, 2, 3} but is not a subset of it, whereas {1} is a subset of the set {1, 2, 3} but is not an element of it. Sometimes we are given a set S and we want to consider a set of subsets of S. Suppose we have the set S = {1, 2, 3, 4}, and we want to consider the set S2 of all two-membered subsets of this set: S2 = { {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} } (Note that {2, 3} = {3, 2}, so we do not list {3, 2} separately. Similarly for {2, 1}, etc.) Note: {1, 2} ⊂ S {1, 2} ∈ S2 That is, an element of S2 is a subset of S. One very important set of subsets of any set X is the power set of X —the set of all subsets of X —symbolized PX. For example, where X = {1, 2, 3}, PX = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} } 1.1.3 Set Operations and Relations When we are talking about some sets, we will generally be talking about some subsets of a given ‘background’ set—that is, we will be talking about some (maybe all) of the elements of the power set of this background set. We might not mention the background set explicitly, but it should be obvious from the context what it is. (If this ever isn’t obvious, take the background set to be some set that is sufﬁciently large to include all the things we are talking about in the context.) The complement of a set A, denoted A, is the set of all things which are not in A. (Here it is important that we are restricting ourselves to the content of some background set: A



19

contains everything in the background set that is not in A, not everything at all that is not in A.) In symbols: A = {x : x ∈ / A} The union of two sets A and B, denoted A ∪ B, contains everything which is in either A or B (or both): A ∪ B = {x : x ∈ A or x ∈ B} The intersection of two sets A and B, denoted A ∩ B, contains everything which is in both A and B: A ∩ B = {x : x ∈ A and x ∈ B} Two sets A and B are said to be disjoint if they have no members in common, i.e. if A ∩ B = ∅. The set A\B is the set of things which are in A but not in B: A\B = {x : x ∈ A and x ∈ / B} = A ∩ B If B ⊂ A, it is common to write A − B instead of A\B. 1.1.4 Ordered Pairs An ordered pair consists of two objects, given in a particular order: one ﬁrst, the other second. The ordered pair consisting of a ﬁrst and b second is represented as (a, b). (It is also common to use angle brackets to denote ordered pairs, i.e. a, b.) Order matters here, thus (a, b) = (b, a). In general, an ordered n-tuple (triple, quadruple, etc.) consists of n objects, given in a particular order: 1st, 2nd, . . . , nth; for example (a, b, c) and (a, c, d, e, b). (Again, it is also common to use angle brackets to denote ordered n-tuples, i.e. a, b, c, a, c, d, e, b.) For any sets X and Y , their Cartesian product X × Y is the set of all ordered pairs whose ﬁrst member is an element of X and whose second member is an element of Y . For example if X = {1, 2} and Y = {3, 4} then X × Y = {(1, 3), (1, 4), (2, 3), (2, 4)} Where X and Y are the same set, the Cartesian product X × X is denoted X 2 . The set of all ordered triples of elements of X is denoted X 3 , and in general the set of all ordered n-tuples of elements of X is denoted X n .

20  1.1.5 Relations A relation from a set X to a set Y is a subset of X × Y , that is, a set of ordered pairs whose ﬁrst elements are in X and whose second elements are in Y . For example, if X is the set of females {Alice, Ethel} and Y is the set of males {Bill, Charles, Dennis}, then if Alice, Bill, and Charles are siblings, and Dennis and Ethel are siblings, the ‘brother of’ relation from X to Y is {(Alice, Bill), (Alice, Charles), (Ethel, Dennis)} and the ‘sister of’ relation from Y to X is {(Bill, Alice), (Charles, Alice), (Dennis, Ethel)}. Where X and Y are the same set, what I have called a relation from X to Y is usually called a binary relation on X, i.e. a subset of X 2 . If R is a binary relation on X, and x and y are elements of X, then it is common to express the fact that x stands in the relation R to y in any of the following ways: (x, y) ∈ R, R(x, y), Rxy, xRy. A ternary relation on X is a subset of X 3 (i.e. a set of ordered triples of elements of X), and in general an n-place relation on X is a subset of X n . A binary relation R on X is said to be reﬂexive if it satisﬁes the condition that ∀x ∈ X(xRx); transitive if ∀x, y, z ∈ X(xRy ∧ yRz → xRz); symmetric if ∀x, y ∈ X(xRy → yRx); antisymmetric if ∀x, y ∈ X(xRy ∧ yRx → x = y); and connected if ∀x, y ∈ X(xRy ∨ yRx). If R is reﬂexive, symmetric, and transitive, it is said to be an equivalence relation. 1.1.6 Functions A function (aka map, mapping, operation) f from a set A to a set B, written f :A→B assigns particular objects in B to objects in A. A is called the domain of the function, and B the codomain. The essential feature of a function is that it never assigns more than one object in B to any given object in A. If x is a member of A, f (x) is the object in B which the function f assigns to x. We say that f (x) is the value of the function f for the argument x, or is the value at x; we also say x is sent to f (x) and that f (x) is hit by x. Note that A and B may be the same set. For example, the ‘mother of’ function assigns to each person the person who is his or her mother; the ‘successor’ function assigns to each natural number the number which comes directly after it in the sequence of natural numbers. A function f : A → B is commonly identiﬁed with the set of all ordered pairs (x, f (x)), where x is an object in A which is sent to some object in



A

f

21

B

Figure 1.1. Picturing a function as a bunch of arrows.

B by f , and f (x) is the object in B to which x is sent. For example the successor function is the set of ordered pairs {(0, 1), (1, 2), (2, 3), . . .}. The crucial feature of a function—that is, that it never assigns more than one object in B to any given object in A—emerges here as the requirement that no element of A appears more than once as the ﬁrst element of an ordered pair in the set. Another useful way to picture a function f : A → B is as a bunch of arrows, pointing from objects x in A to objects f (x) in B (Fig. 1.1). Binary relations can also usefully be pictured as bunches of arrows. In this depiction, functions are distinguished from relations in general by the requirement on functions that no object has more than one arrow departing from it. A function f : A → B is said to be total if it satisﬁes the condition that every member of A gets sent to some member of B. A function which is not total is called partial. Such a function assigns nothing at all to some member(s) of A. In the representation of a function as a set of ordered pairs, to say that a ∈ A is assigned no value by the partial function f : A → B is to say that a does not appear as the ﬁrst element of any ordered pair in the set; in the representation of a function as a bunch of arrows, it is to say that a has no arrow leading from it (it does not have an arrow leading to some special object, The Nothing!). A function f : A → B is said to be onto (aka surjective, a surjection) if it satisﬁes the condition that every member of B gets hit at least once; one–one (aka one-to-one, into, injective, an injection) if no member of B gets hit more than once; and a correspondence (aka bijective, a bijection) if it is total, onto, and one–one. Given a function f : A → B and a function g : B → C, the composite function g ◦ f (read this as ‘‘g after f ’’) from A to C is deﬁned thus: for

22  every x in A, (g ◦ f )(x) = g( f (x)). Visually, you get the composite function by taking each f arrow from an object x in A to an object y in B and extending it so that it hits whatever object z in C the g arrow from y hits. (If there is no g arrow from y, then in the composite function there is no arrow from x.) Given a function f : A → B, that is, a set of ordered pairs/a bunch of arrows, we can invert the function (i.e. switch the ﬁrst and second members of each ordered pair/make each arrow point the opposite way). If the result of this process is a function (from B to A), this resulting function is called the inverse function of f and is denoted f −1 . So far we have considered functions which take a single object as argument and assign to it an object as value. What about functions such as addition or multiplication, which take two objects as arguments, and assign to them an object as value? In general, what about functions which take n = 1 objects as values? A standard move is to treat an apparently n-place function from A to B as a function from the set of n-tuples of members of A to B. So, for example, the addition function, commonly thought of as a two-place function from the set of integers to the set of integers, can be thought of as a 1-place function from the set of ordered pairs of integers to the set of integers. 1.1.7 Structures and Algebras A structure is a set of objects together with some relations on that set. The relations are required to satisfy some conditions: the particular conditions imposed determine the type of structure in question. An algebra is a structure where the relations are functions.² Examples follow. A poset (aka partial order, or partially ordered set) (X, ≤) is a set X together with an ordering ≤ (i.e. a binary relation on X) which is reﬂexive, transitive, and antisymmetric. A linearly ordered set (aka linear order) is a partially ordered set where, in addition, ≤ is connected. A lattice may be deﬁned in two equivalent ways. According to the ﬁrst, order-theoretic deﬁnition, a lattice (X, ≤) is a poset in which every pair of elements of X has a supremum and an inﬁmum in X, where: •

y ∈ X is the supremum of (a, b) iff a, b ≤ y, and y ≤ z for any z with a, b ≤ z ² This usage of the terms ‘structure’ and ‘algebra’ is neither non-standard nor universal.

 •

23

y ∈ X is the inﬁmum of (a, b) iff y ≤ a, b, and z ≤ y for any z with z ≤ a, b

According to the second, algebraic deﬁnition, a lattice (X, ∨, ∧) is a set X together with two binary operations ∨ and ∧ on X which satisfy the following conditions: 1. 2. 3. 4.

idempotence: x ∨ x = x and x ∧ x = x commutativity: x ∨ y = y ∨ x and x ∧ y = y ∧ x associativity: x ∨ (y ∨ z) = (x ∨ y) ∨ z and x ∧ (y ∧ z) = (x ∧ y) ∧ z absorption: x ∨ (x ∧ y) = x and x ∧ (x ∨ y) = x

It is easily veriﬁed that a lattice (X, ≤) yields a lattice (X, ∨, ∧) if we set x ∨ y equal to the supremum of x and y with respect to ≤, and set x ∧ y equal to the inﬁmum of x and y with respect to ≤; and that a lattice (X, ∨, ∧) yields a lattice (X, ≤) if we set x ≤ y iff x ∧ y = x iff x ∨ y = y. The following are some additional properties which a lattice may or may not have: 1. There is an identity for ∨, i.e. an element 0 ∈ X such that 0 ∨ x = x for all x ∈ X; and there is an identity for ∧, that is an element 1 ∈ X such that 1 ∧ x = x for all x ∈ X. Equivalently: the lattice has a maximum element 1, with x ≤ 1 for all x ∈ X, and a minimum element 0, with 0 ≤ x for all x ∈ X. A lattice satisfying this condition is said to be bounded. 2. The lattice has identities, and each element x ∈ X has a complement, i.e. an element x ∈ X with x ∨ x = 1 and x ∧ x = 0. A lattice satisfying this condition is said to be complemented. 3. The operations ∨ and ∧ satisfy the distributive laws: x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) and x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). A lattice satisfying this condition is said to be distributive. 4. Each nonempty subset Y of X has a supremum (written Y ) and an inﬁmum (written Y ) in X, where: • y ∈ X is the supremum of Y iff (i) ∀x ∈ Y , x ≤ y and (ii) for any z that satisﬁes (i), i.e. ∀x ∈ Y , x ≤ z, we have y ≤ z • y ∈ X is the inﬁmum of Y iff (i) ∀x ∈ Y , y ≤ x and (ii) for any z that satisﬁes (i), i.e. ∀x ∈ Y , z ≤ x, we have z ≤ y A lattice satisfying this condition is said to be complete. A Boolean algebra is a bounded, distributive, complemented lattice. An equivalent deﬁnition is that a Boolean algebra is a nonempty set together

24  with a unary operation and two binary operations ∨ and ∧, satisfying the following identities: 1. 2. 3. 4. 5.

x ∨ y = y ∨ x and x ∧ y = y ∧ x x ∨ (y ∨ z) = (x ∨ y) ∨ z and x ∧ (y ∧ z) = (x ∧ y) ∧ z x ∨ (x ∧ y) = x and x ∧ (x ∨ y) = x x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) and x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) x ∨ (y ∧ y ) = x and x ∧ (y ∨ y ) = x

A unary operation on a bounded lattice which satisﬁes the following conditions is an involution (or duality): • •

(x ) = x if x ≤ y then y ≤ x

If is an involution, then the following identities—the De Morgan laws— may or may not hold: • •

(x ∨ y) = x ∧ y (x ∧ y) = x ∨ y

A bounded distributive lattice with an involution that satisﬁes the De Morgan laws is a De Morgan algebra; if the underlying lattice is complete, it is a complete De Morgan algebra. If a De Morgan algebra satisﬁes the following condition: •

x ∧ x ≤ y ∨ y

then it is a Kleene algebra; if the underlying lattice is complete, it is a complete Kleene algebra.

1.2 The Classical Semantic Picture Classical logic, as taught in introductory logic classes, can seem to be a melting-pot of miscellaneous tools, from a host of connectives each with its own truth table, through to quantiﬁers with strange introduction and elimination rules, whose semantics are given in terms of models rather than truth tables. There is, however, a unifying perspective which grounds all this diversity in a single simple structure. This structure—the fundamental structure underlying classical logic and set theory—is the set {0, 1}, or



25

{False, True}, of classical truth values, together with certain operations deﬁned on this set. The clearest way to think of these operations is in terms of their relations to the fundamental ordering of the two truth values. This ordering is arrived at by ﬁrst considering under what conditions one sentence is at least as true as another sentence. First, any sentence, whether it is true or false, is at least as true as itself. Second, suppose sentence S is false and sentence T is true; then S is strictly less true than T (i.e. T is at least as true as S, and S is not at least as true as T). Now in all the cases just considered, what the sentences involved mean is not important: the same relationships would hold between any sentences which were respectively both true, both false, or true and false. So we may transfer our ordering of sentences in respect of truth onto the truth values that sentences possess. The result is 0 ≤ 0, 0 ≤ 1, 1 ≤ 1.³ Now with this ordering in place, we want to deﬁne some operations on {0, 1}. The basic set of operations is , , and : the idea is that these will be used to deﬁne, respectively, logical disjunction (‘or’) and set-theoretic union, logical conjunction (‘and’) and set-theoretic intersection, and logical negation (‘not’) and set-theoretic complementation. This is the ultimate point of deﬁning the operations—but at this stage we are deﬁning operations on {0, 1}, the set of truth values. Now, if S and T have the same truth value, then we want ‘S or T’ to have that truth value also, and if S and T have different truth values, then we want ‘S or T’ to have the greater of these truth values. Thus we want to say that for any truth values x and y in {0, 1}, x y is the supremum of x and y with respect to our ordering of {0, 1}. For conjunction, if S and T have the same truth value, then we want ‘S and T’ to have that truth value also, and if S and T have different truth values, then we want ‘S and T’ to have the lesser of these truth values. Thus we want to say that for any truth values x and y in {0, 1}, x y is the inﬁmum of x and y with respect to our ordering of {0, 1}. For negation, if S is less true than T, then we want ‘not T’ to be less true than ‘not S’, and we also want the negation of the negation of S to have the same truth value as S; thus we set 0 = 1 and 1 = 0. We now have an algebra ({0, 1}, , , ), consisting of a set together with some operations on that set. It is a straightforward matter to verify that ³ This is exactly the ordering we get on {0, 1} if we restrict the standard ordering of the real numbers to the subset {0, 1}—and this is in fact one reason why we use 0 to represent falsity and 1 to represent truth.

26  this is a Boolean algebra. This Boolean algebra is the fundamental structure underlying classical logic and set theory. Let us look ﬁrst at classical propositional logic. We have a language whose vocabulary consists in a set P of propositional constants {p, q, r, . . .} and three logical connectives ∨ (or), ∧ (and), and ¬ (not), as well as the parentheses ( and ). The set F of well-formed formulae (wfs) is deﬁned as follows: • • •

Propositional constants are wfs. If A and B are wfs, then so are (A ∨ B ), (A ∧ B ), and (¬A ).⁴ Nothing else is a wf.

Thus, we consider a sentence (wf) to be a sequence of symbols. Such a sequence gets a meaning through being given an interpretation. An interpretation of the language consists in an assignment of a truth value [p] to every propositional constant p ∈ P. Given an interpretation, every wf A ∈ F is assigned a truth value [A ] by the following recursive deﬁnition: • • •

[A ∨ B ] = [A ] [B ] [A ∧ B ] = [A ] [B ] [¬A ] = [A ]

Thus we are saying that the truth value of the wf A ∨ B —the object assigned to this sequence of symbols on the interpretation in question—is found by locating the object assigned to A and the object assigned to B , and then performing the operation on these two objects. The result of this operation is an object in our algebra of truth values—and this object is the truth value of A ∨ B . Note that the foregoing information is often presented by means of truth tables (see Table 1.1). These tables combine our earlier deﬁnitions of the operations , , and on the set {0, 1} of truth values (these now occur in the body of the table—for example, the deﬁnition of is in the ﬁnal column) with our recursive deﬁnition, in the previous paragraph, of the truth values of compound propositions. I think it is more perspicuous to keep the two deﬁnitions apart. First, it makes us less likely to confuse the symbols of our formal language (∨, ∧, and ¬) with the operations on ⁴ Parentheses will be omitted when this will cause no confusion.



27

Table 1.1. Truth tables for classical propositional logic A

B

(A ∨ B )

(A ∧ B )

¬A

1

1

1

1

0

1

0

1

0

0

1

1

0

0

0

0

0

1

our set of truth values (, , and ).⁵ Second, it will allow us to see more clearly the differences between certain theories of vagueness which we shall examine below—for example, between three-valued views (§2.2) and the supervaluationist view (§2.4). Two wfs A and B are logically equivalent (written A ≡ B ) if they have the same truth value on every interpretation. ≡ is an equivalence relation on F, and F/≡ is the set of equivalence classes of F under ≡. Letting |A | be the equivalence class containing the wf A , we may set: • • •

|A | ∨ |B | = |A ∨ B | |A | ∧ |B | = |A ∧ B | ¬|A | = |¬A |

Note that the occurrences of ∨, ∧, and ¬ on the right of these identities represent the sentential connectives, while the occurrences on the left denote newly deﬁned algebraic operations on F/≡. It is a straightforward matter to check that these operations are well deﬁned, and that F/≡ together with these operations is a Boolean algebra.⁶ It is not uncommon to refer to these equivalence classes of wfs as propositions, and to this Boolean algebra of equivalence classes as the classical propositional calculus. What about the conditional? That is, what are we to say of sentences of the form A → B ? There are two strategies: both yield the result that A → B is equivalent to ¬A ∨ B and to ¬(A ∧ ¬B ), but via different ⁵ Such confusion is easier if—as is standard practice—we use the same symbols (∨, ∧, and ¬) to represent both symbols of the language and operations on the truth values. To avoid this confusion, I have used different symbols: ∨, ∧, and ¬ for the symbols of the language, and , , and for the operations on the truth values. Later on—once we have the distinction between symbols of the formal language and operations on the set of truth values clearly in mind—I will revert to the standard practice of using the same symbols for both and letting context disambiguate. ⁶ For more details see Halmos and Givant 1998. This algebra of equivalence classes is often called the Lindenbaum algebra.

28  routes. One strategy is to regard → as a deﬁned symbol, with A → B abbreviating either ¬A ∨ B or ¬(A ∧ ¬B ), rather than being a genuine wf in its own right. The other strategy is to regard → as a primitive connective, and then (i) add a new clause to our deﬁnition of a wf, saying that if A and B are wfs, then so is (A → B ), (ii) add a new quotient operation / to our algebra of truth values, subject to the identities f /g = f g = ( f g ) , and (iii) add a clause to our truth deﬁnition which says that [A → B ] = [A ]/[B ]. I think it will be clearer in what follows if we keep our algebra of truth values as uncluttered as possible, so I shall adopt the former strategy. Let us look next at classical set theory. It is a well-known fact that for any set X, the power set PX of X is a Boolean algebra under the operations of union, intersection, and complementation—or equivalently, with the set-inclusion or subset relation ⊆ as the ordering. The connection with the Boolean algebra of classical truth values becomes transparent if we identify each subset of X with its characteristic function.⁷ For any subsets (i.e. characteristic functions) f and g, and any x ∈ X, we may then deﬁne: • • • •

( f ∪ g)(x) = f (x) g(x) ( f ∩ g)(x) = f (x) g(x) f (x) = f (x) f ⊆ g ⇔ ∀x ∈ X, f (x) ≤ g(x)

Thus we are saying that to ﬁnd out whether a given x is in the union of f and g —that is, to ﬁnd out whether this union set (considered as a characteristic function) assigns x 1 or 0—we ﬁnd the object (1 or 0) which f assigns to x, and the object (1 or 0) which g assigns to x, and then perform the operation on this pair of objects. The result of this operation is an object in our algebra of truth values—1 or 0—and this object is the value assigned to x by the union set f ∪ g. If it is 1, then x is in the union; if it is 0, then x is not in the union. By asking this question of every x ∈ X, we build up a complete picture of the union set f ∪ g. It is a straightforward matter to verify that these deﬁnitions yield a Boolean algebra of subsets. It is also clear that the operations we have just deﬁned are indeed the ordinary set-theoretic operations of union, ⁷ The characteristic function fS of a subset S of X (also known as the indicator function) is a function from X to the set {0, 1} of truth values, which assigns 1 to every member of X which is in S, and 0 to every member of X which is not in S.

 intersection, and complementation. With , , and ‘and’, and ‘not’, we have: • • •

29

interpreting ‘or’,

x is in the union of f and g iff x is in f or x is in g x is in the intersection of f and g iff x is in f and x is in g x is in the complement of f iff x is not in f .

As well as unions and intersections of pairs of sets f and g, we can also deﬁne unions and intersections of arbitrary families of sets {fi } (the above deﬁnitions for pairs of sets are just special cases of these more general deﬁnitions): • {f }(x) = {fi (x)} i • {fi }(x) = {fi (x)} Finally, let us consider classical predicate logic. In addition to the propositional constants, the connectives, and the punctuation marks, we now have individual variables x, y, z, . . . , individual constants a, b, c, . . . , n-ary predicate letters P n , Qn , Rn , . . . for each n ≥ 1,⁸ and quantiﬁers ∃ and ∀. Terms and well-formed formulae are deﬁned as follows: • • • • •

Variables and individual constants are terms; nothing else is a term. Propositional constants are wfs. If t1 , . . . , tn are terms and P is an n-ary predicate letter, then P(t1 , . . . , tn ) is a wf. If A and B are wfs and x is a variable, then (¬A ), (A ∨ B ), (A ∧ B ), ((∃x)A ), and ((∀x)A ) are wfs. Nothing else is a wf.

Thus, as before, a wf is a sequence of symbols. Such a sequence gets a meaning through being given an interpretation. An interpretation M = (M, I) of the language consists in a nonempty set M (the domain), together with a function I which assigns: • • •

a truth value [p] to every propositional constant p; an object I(a) in M to every individual constant a; an n-ary relation I(P) on M to every n-ary predicate P.

⁸ The superscript indicating the arity (number of places) of the predicate will be omitted when it is obvious from the context.

30  In relation to the ﬁnal clause, note that an n-ary relation on M is a set of n-tuples of members of M. Identifying this set of n-tuples with its characteristic function, we regard an n-ary relation on M as a function from the set M n of all n-tuples of members of M to our set of truth values {0, 1}. Given an interpretation M, every closed wf A is assigned a truth value [A ]M on that interpretation by a recursive deﬁnition.⁹ (Where the interpretation in question is obvious from the context, or is irrelevant, the subscript M is omitted.) The clauses for non-quantiﬁed compound wfs are just as before. For an atomic wf P(a1 , . . . , an ), which consists of an n-ary predicate P followed by n individual constants a1 , . . . , an , the truth deﬁnition is as follows: [P(a1 , . . . , an )] = I(P)(I(a1 ), . . . , I(an )) The idea here is this. As discussed, each individual constant a is assigned an object I(a) in the domain, and P is assigned a function I(P) from the set of all n-tuples of members of M to our set of truth values {0, 1}. Whatever value this function assigns to the n-tuple (I(a1 ), . . . , I(an )), this value is the truth value of the sentence P(a1 , . . . , an ). Thus ‘Bob is bald’ is true just in case Bob is in the extension of ‘is bald’; ‘Bill loves Ben’ is true just in case Bill stands in the loving relation to Ben; and so on. For quantiﬁed sentences, saying that everything is P is treated as saying that this thing is P, and this thing is P, and this thing is P, . . . for everything in the domain; and saying that something is P is treated as saying that this thing is P, or this thing is P, or this thing is P, . . . for everything in the domain. This basic idea of universal quantiﬁcation as generalized conjunction, and existential quantiﬁcation as generalized disjunction, is made precise in the following way. Let Ax a be the sentence obtained by writing a in place of all free occurrences of x in A , a being some constant that does not occur in A ; and given an interpretation M with domain M, let Mao be the interpretation which is just like M except that in it the constant a is assigned the denotation o. Then: • [∃xA ]M = {[A a] a : o ∈ M} x Mo • [∀xA ]M = {[Ax a]Mao : o ∈ M} ⁹ Looking at the deﬁnition of a wf, we see that the only way a quantiﬁer can get into a wf is by being stuck on the front of a wf A , as in ((∃x)A ) or ((∀x)A ). This wf A is called the scope of the quantiﬁer. A variable x occurring within the scope of a quantiﬁer (∀x) or (∃x) is said to be bound. ( The variable x in (∀x) or (∃x) itself is also said to be bound.) A variable x which is not within the scope of any quantiﬁer (∀x) or (∃x) is said to be free. A closed wf is one containing no free variables.



31

The idea here is that to ﬁnd out the truth value of, say, ∀xPx, we consider the sentence Pa, and we ask what truth value it would have if everything about our interpretation were the same, except that a denoted this object in the domain; we note this truth value. Then we ask what truth value Pa would have if everything about our interpretation were the same, except that a denoted this other object in the domain; we note this truth value. And so on, for all objects in the domain. We now have a set of truth values—the ones we noted down along the way. To get the truth value of ∀xPx, we apply our inﬁmum operation to this set of truth values, yielding another truth value. The analogy with conjunction enters in the fact that is the generalization to arbitrary sets of truth values of our operation on pairs of truth values. We have now ﬁnished our tour through classical logic (propositional and predicate) and set theory. We have seen that the Boolean algebra of classical truth values is at the heart of the story. The set-theoretic relations between, and operations on, sets, and the logical relations between, and operations on, sentences, are just the images projected into the worlds of sets and of sentences respectively by the algebraic operations on our set of truth values. If we unplugged the classical algebra of truth values, and plugged in a different one, and then told the rest of the ensuing story of set theory, propositional logic, and predicate logic in the same way as above, we would get different images appearing in the worlds of sets and sentences—i.e. different set-theoretic relations between, and operations on, sets, and different logical relations between, and operations on, sentences—but once we have the recipe for telling the story of logic and set theory given an initial algebra of truth values, how the story ends is completely determined by our initial choice of algebra. Later on we shall indeed vary the choice of algebra, and look at the results for set theory and logic. The point for the moment is simply that once we have the underlying algebra, the rest of the story comes in a natural progression. One thing has been left out of our story so far: something that is essential if we wish to apply the classical semantic picture to ordinary language. Typically we want to know whether a sentence is true (simpliciter), not just that it is true on such-and-such interpretations and false on others. What is going on here is that when we utter a sentence, we mean something —some particular thing—by the sentence we utter. That is, we utter it relative to a particular interpretation. The particular interpretation that is relevant in a

32  given case has been called many things, including intended (Putnam 1983 [1977]; Merrill 1980; Lewis 1999 [1984], 1993; Weiner 2004), correct (Lepore 1983; Davidson 2005 [1986]; van Cleve 1992; Islam 1996; Field 2000), actual (Lepore 1983), real (Hájek 1999), standard (Abbott 1997), proper (Przełe¸cki 1976), and the interpretation which accords with the semantics of the language being spoken (Field 1974, 210–11). Whatever we call the interpretation in question,¹⁰ the ﬁnal part of the classical semantic picture is the supposition that every time you utter a sentence, you invoke a particular interpretation of the language, and utter the sentence relative to that interpretation; that is, it is this special or designated interpretation that gives the actual meaning of what you say on that occasion. An utterance of a sentence is then true simpliciter if the sentence uttered is true on the intended interpretation (as invoked by that utterance of it), or in other words, if it is an utterance of a sentence relative to an interpretation on which that sentence is true. In a slogan: truth simpliciter is truth on the intended interpretation.¹¹ ¹⁰ In this book, I shall regard all these terms—and some others besides, such as ‘designated interpretation’ and ‘special interpretation’—as interchangeable. The term ‘intended interpretation’ might be felt to carry the connotation that it is the speaker’s intentions that make a certain interpretation the one relative to which she is speaking. I do not intend any such connotation to go with my use of this term. The question of what does determine the semantic facts will be discussed below: see in particular §2.1.1 and §6.1.1. ¹¹ Cf. e.g. Field 1974, 212; Przełe¸cki 1976, 376; Lepore 1983, 182; Menzel 1990, 358; Islam 1996; and Weiner 2004, 165. Thanks to Amitavo Islam for helpful discussions here.

2 The Space of Possible Theories of Vagueness In this chapter, I present the main existing theories of vagueness, as well as some possible theories which have not yet found advocates. As discussed at the beginning of Chapter 1, I present each theory not necessarily as it was ﬁrst presented in the literature, but rather as it looks from the point of view of the classical semantic picture. That is, we shall examine whether, and if so how exactly, different theories of vagueness depart from that picture. The classical picture was presented in §1.2. We will be aided in what follows if we have before us a checklist of its main components: 1. Interpretations of the language have the following features: (a) They employ a two-element Boolean algebra of truth values (which serve both as the truth values of sentences and as the values of the characteristic functions of sets). (b) The interpretation function (which assigns truth values to simple sentences, elements of the domain to names, and n-ary relations on the domain to n-ary predicates) is total, and the characteristic functions of sets are total. (c) The truth values of compound wfs built up from simpler wfs by means of the connectives ∨, ∧, and ¬ are determined in a recursive fashion from the truth values of their components. 2. Each discourse has a unique intended interpretation. We shall structure our examination of theories of vagueness according to which item(s) on this checklist each theory denies. Each theory will thereby

34  be located in a space of possible theories which differ from the classical view in one or more of these four possible ways.¹

2.1 Epistemicism (Deny Nothing) The epistemicist about vagueness denies nothing on our checklist: she buys the classical semantic picture holus-bolus.² Thus, according to the epistemicist, when I say something vague—for example, ‘Bob is tall’—I speak relative to a particular intended classical interpretation of my language. Each name in the language is assigned some object in the domain of this interpretation (its referent), and each one-place predicate is assigned a subset of this domain (its extension). The sentence ‘Bob is bald’ is true simpliciter just in case the referent of the name ‘Bob’ is in the extension of the predicate ‘is bald’ on this intended interpretation. But a subset of the domain divides it into two parts: those in and those out. There is no remainder. Thus, my sentence ‘Bob is bald’—and indeed every sentence of the language—is either true simpliciter or false simpliciter. There is no other possibility. Rather than focusing on a particular sentence (‘Bob is bald’), focus on a particular predicate, say ‘is a heap’. On the epistemicist story, this predicate has a particular extension on the intended interpretation of the language. This extension is a classical subset of the domain. Thus, for some number n, piles of n or more grains are in the set and piles of n − 1 or less grains are out.³ We need not know what number it is; but there is such a number nevertheless. One key challenge for the epistemicist will be to explain how it is that this line between the heaps and the non-heaps gets to be where it is—especially given that we might have no knowledge of where it is. I discuss this issue in §2.1.1. ¹ One thing we assume throughout is that our language never changes: it is always our standard ﬁrstorder language. In other words our syntax never changes—we only consider changes to our semantics. ² Advocates of epistemic theories of vagueness include Cargile 1997 [1969], Campbell 1974, Sorensen 1988, ch. 6; 2001, Williamson 1997 [1992]; 1994, chs. 7–8, and Horwich 1998. ³ Note that the classical semantic picture, all by itself, does not require that the set of heaps be so ‘nicely’ organized; i.e. as far as classical semantics alone is concerned, it would be ﬁne if the set of heaps consisted of piles with an even number of grains—thus as we remove grains from our heap, we would go ‘heap’, ‘non-heap’, ‘heap’, ‘non-heap’, etc. All that classical semantics, by itself, requires, is that ‘is a heap’ have a crisp set of objects as its extension—i.e. each object is either in the set, or is out, with no fuzziness or blurriness at the edges of the set.

      

35

Assuming that the classical picture is the correct picture of precise discourse, this means that from a semantic point of view, there is, according to the epistemicist, no difference between vague discourse and precise discourse. But of course there is some difference between the two sorts of discourse. After all, we are tempted by Sorites arguments involving ‘tall’, but not by Sorites arguments involving ‘is greater than or equal to 1800 mm in height’. According to the epistemicist, the difference is an epistemological one: vagueness is a matter of (necessary) ignorance.⁴ When F is vague and a is a borderline case of F, the epistemicist thinks that Fa is either true, or false—but we cannot know which. The idea is that, from the logical and semantic points of view, vague predicates are just like precise predicates, but from the epistemological point of view, they are signiﬁcantly different. Thus, where other theorists of vagueness will need to spend time motivating and explaining their non-classical semantic frameworks, the epistemicist needs to spend time motivating and explaining an epistemological theory which explains why, even though there is a sharp boundary between the things which are (say) bald and the things which are not, we cannot know where this boundary is. I discuss this issue—the one which gives epistemicism its name—in §2.1.2. How does epistemicism solve the Sorites paradox? The idea is that there is a precise cut-off between the heaps and the non-heaps, the bald persons and the non-bald persons, the tall persons and the non-tall persons, and so on. Thus the Sorites paradox is solved by denying the inductive premiss: the premiss which says that if anything in the Sorites series is a heap, then so is the next thing in the series. This premiss is false: there is a pair of adjacent things in the series such that one is a heap and one is not. But this is only part of the solution to the Sorites paradox: we need to say not only where the mistake is, but also why we were taken in, and did not see the mistake instantly. The paradox involves premisses which seem obviously true and reasoning which seems obviously valid. So to solve it, we need to show that actually the reasoning is not valid, or that actually one or more of the premisses is false. But we need to do something else as well. We have a paradox because very plausible starting points (i.e. premisses and reasoning) lead us to an unacceptable conclusion. To solve the paradox, we cannot ⁴ Cf. Williamson 1994, 202: ‘‘ignorance is the real essence of the phenomenon ostensively identiﬁed as vagueness’’, and Williamson 1996b, 327: ‘‘This is not to deny that vagueness exists; it is to assert that its underlying nature is epistemic.’’

36  just show that the starting points are not acceptable: we already know they cannot all be acceptable, because they lead to a conclusion which we cannot accept. We have to explain why we thought they were acceptable: we have to explain why we were taken in. Only then do we have a satisfying solution to the paradox.⁵ The second part of the epistemicist’s story is that we cannot know where the cut-off is. This is turned into an explanation of why we were taken in by the Sorites, as follows. We cannot know where the cut-off is, so we mistakenly think it isn’t anywhere. Our ignorance of where the cut-off is makes us think there is no cut-off at all. That is why we are inclined to accept the inductive premiss, even though, according to the epistemicist, the inductive premiss is actually false. 2.1.1 The Location Problem The epistemicist thinks that there is a particular number of hairs dividing the last bald man from the ﬁrst non-bald man, a particular number of grains of sand dividing the last heap from the ﬁrst non-heap, and so on. What determines these numbers? What ﬁxes the locations of the boundaries drawn by vague terms? Suppose that the relevant number of grains of sand is 23. Why not 22? What makes 23 the correct number here? How did 23, rather than 22 or 24, get singled out as correct? How did the boundary between heaps and non-heaps come to be located just here?⁶ It seems to most people who have thought about these matters that language is essentially a human artefact, and so there should be some sort of connection between meaning and use.⁷ The basic thought is that the sounds we make mean what they do because of the kinds of situations in which we, and earlier speakers, have made those sounds. So, for example, had we always used the word ‘dog’ where we in fact used ‘cat’, and vice versa, then ‘dog’ would have meant what ‘cat’ in fact means, and vice versa. Here is a prima facie plausible principle which connects meaning and use (for the case of predicates): ⁵ Thanks to Delia Graff Fara for ﬁrst helping me to appreciate this point. Cf. Stalnaker 1999, 74; Fara 2000, 50; and Schiffer 2000 233 ff. ⁶ Cf. e.g. Wright 1995, 2003 and Schiffer 1999, to name just two of the critics of epistemicism who have focused on this issue. ⁷ Cf. e.g. the opening sentence of Lewis 1992: ‘‘Surely it is our use of language that somehow determines meaning’’, or Williamson 1994, 205: ‘‘Words mean what they do because we use them as we do’’, or Shapiro 2006, 5: ‘‘I take it to be a truism that the competent users of a language determine the meaning of its words and phrases.’’

      

37

(MU) The claim Pa is true if and only if most competent speakers would conﬁdently assent if presented with a in normal conditions and asked whether it was P, and is false if and only if most competent speakers would conﬁdently dissent if presented with a in normal conditions and asked whether it was P. The biconditionals in (MU) connect meaning (i.e. extension on the intended interpretation, which determines truth and falsity) with use (actual and counterfactual) only in the weak sense of saying that they co-vary. (MU) is compatible with, but does not require, the view that usage directly determines meaning—that is, the view that a certain object is in the extension of P because we do or would apply P to this thing. (MU) is also compatible with the view that we apply words in certain ways because doing so allows us to speak truthfully; in conjunction with this view, MU merely requires that competent speakers in general have access to the meanings of their terms—in keeping with the guiding idea that language is a (useful) human tool. The epistemicist must reject (MU), for the following reason. Take a clear case of ‘is bald’ (say Yul Brynner). ‘Yul Brynner is bald’ is true, and also, everyone would assent to it, so here meaning and use match up. Take a clear non-case of ‘is bald’ (say Fabio). ‘Fabio is bald’ is false, and also, everyone would dissent from it, so here again meaning and use match up. So far so good. But what about the borderline cases? According to the epistemicist, for any borderline case x of baldness, ‘x is bald’ is either true, or false. Yet most competent speakers would neither assent to nor dissent from ‘x is bald’: we shrug our shoulders, we say ‘he is and he isn’t’ or ‘he’s sort of bald’, we say nothing either way, etc. So within the borderline cases, there is—given the epistemicist view—a failure of match-up between meaning and use. We can picture the situation—say for the predicate ‘is tall’—as in Figure 2.1. What we see is that epistemicism entails a mismatch between use (on the left) and meaning (on the right). There are standard counterexamples to (MU). For example, consider the predicate ‘is a lump of gold’. Most ordinary speakers would classify a lump of fools’ gold as a lump of gold, and yet we do not think that ‘is a lump of gold’ truly applies to a lump of fools’ gold. In other words, fools’ gold is not gold, even though most of us would classify it as such: most of us are just wrong. The mechanism which makes ‘is a lump of gold’ fail to have

38  use

meaning 8′

assert true

6′ hedge

4′

false

deny

2′ persons

Figure 2.1. Use and meaning.

any lumps of fools’ gold in its extension, even though we would assert of many lumps of fools’ gold that they are lumps of gold, is as follows. We originally applied our term ‘gold’ to some instances of a certain natural kind. From there, the natural kind did the work of determining the extension of our term. The term truly applies to all instances of the natural kind, even if we would not co-classify them with the original samples, and it fails to apply to any non-instance of the natural kind, even if we would co-classify it with the original samples. By latching our term onto a natural kind, we take away the rest of the job of determining its extension (i.e. beyond the initial instances to which we applied the term) from usage, and hand it over to the world.⁸ This sort of counterexample to (MU) will not help the epistemicist, however—precisely because of the essential role it gives to natural kinds. For with vague predicates, there are no natural kinds in the vicinity to bridge the gap between use and meaning. Consider the predicate ‘is tall’. ⁸ See Kripke 1980 and Putnam 1975.

      

39

We have applied this term to people who are 6 ft 4 in., 7 ft, 6 ft 2 in., . . . How do we get from this to a particular height such that the predicate truly applies to everyone greater than (or equal to) this height, and no one else? That is the epistemicist’s problem. If there was a particular height x such that being of that height or greater was a natural kind, our problem would be solved. But there is no such height. There is nothing more natural about being 6 ft or more than about being 5 ft 11 in. or more or being 6 ft 1 in. or more, or indeed being x foot or more for any x at all. Thus the epistemicist violates (MU) without being able to posit anything else, apart from usage—such as a natural kind—to determine meaning. It may seem at this point as though the epistemicist is forced to say that meaning facts are primitive, that it is just a brute fact that the extension of ‘is tall’ is some particular precise set—an option which may seem simply unacceptable.⁹ However, there is in fact another option: for despite the fact that she has to deny (MU), the epistemicist does not have to deny that use determines meaning. Recall Figure 2.1. (MU) tells us that the left-hand side and the right-hand side must match. The claim that use determines meaning is the claim that the left-hand side determines the right-hand side. Now to claim that one set of facts—meaning facts—are determined by another set of facts—facts about usage—is to claim that meaning supervenes on use, that there can be no difference in meaning without a difference in use; that if the meaning were different, the use would be too. Thus the claim that use determines meaning becomes the claim that if the right-hand side of the picture were different, the left-hand side would be different too. Now there is nothing inconsistent about claiming both that the left-hand side and the right-hand side do not match, and that the right-hand side could not be different without the left-hand side being different. This is the sort of line taken by Williamson (1997 [1992], §4; 1994, §7.5).¹⁰ His claim is that use does determine meaning—just not in a straightforward way. There is a relationship between use and truth conditions such that the former determines the latter—but it is complex, and we do not know what it is. Certainly use does not determine meaning in the simple way mentioned ⁹ For further discussion see pp. 282–3 below. ¹⁰ This line bears strong afﬁnities to Soames’s 1997 response to Kripkenstein’s sceptical puzzle. Soames distinguishes a second sense of ‘determines’, in which the A facts determine the B facts if we can, in principle, deduce (a priori) the B facts from the A facts. His point is that facts about usage (plus facts about the environment and our mental states) do determine the meaning facts in the sense discussed in the text above, but not in the ‘a priori derivability’ sense.

40  above (immediately after the introduction of (MU)), according to which an object is in the extension of P because we do or would apply P to this thing. We have no idea of the mechanisms intervening between use and meaning: we do not know how (if at all) the right-hand side of our picture would alter if we twiddled this or that bit of usage on the left-hand side. Yet we can still maintain that the right-hand side could not be different unless the left-hand side was too. This is a consistent response to the problem of how precise extensions for vague predicates are determined, but it is by nature unsatisfying: we are told merely that something is the case (meaning is determined by use), when we want to know how it could possibly be the case (assuming, with the epistemicist, that meaning is cleanly bipartite, while usage is fuzzily tripartite).¹¹ Indeed, a residual dissatisfaction with the idea that our usage does determine sharp lines between the bald and the non-bald, the heaps and the non-heaps, and so on (combined with a dislike of primitive meaning facts) is part of the motivation for some of the other approaches to vagueness that we shall examine below. Williamson does at one point suggest a more informative account of the relationship between use and meaning. He considers the following objection to his view: Suppose . . . that in normal perceptual conditions any competent speaker of English refuses to classify me as thin and refuses to classify me as not thin. How could the truth or falsity of ‘[Williamson] is thin’ possibly supervene on that pattern of use? (1994, 207)

and responds as follows: The . . . worry concerns the apparent symmetry of the situation. However, the concepts of truth and falsity are not symmetrical. . . . The epistemic theorist can see things this way: if everything is symmetrical at the level of use, then the utterance fails to be true, and is false in virtue of that failure . . . In that sense, truth is primary. (1994, 208) ¹¹ On a related note, Burgess 2001, 513 argues that ‘‘the critic [of epistemicism] is misrepresented if he or she is taken to be looking primarily for a supervenience thesis. The critic . . . wants to know, primarily, of particular borderline examples, what makes them heaps, bald men, red objects, or whatever, when other objects, perhaps indiscriminable from them by the standard methods of testing performed under suitable conditions, are not correctly so describable.’’ See also Keefe 2000, 80–3.

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Burgess (2001, 519) sees here a ‘parasitic’ strategy for saying how use determines meaning; as Weatherson (2003a, 277) puts it: ‘‘wait for the indeterminist to offer a theory of when sentences are true, accept that part of the indeterminist theory, and say all other sentences that express propositions are false’’.¹² In our terms, we can say that the idea is to accept the ﬁrst biconditional in (MU) (the one connecting assent and truth), while providing a principled reason (viz. the asymmetry of truth and falsity) for rejecting the second biconditional (the one connecting dissent and falsity).¹³ We can see, however, that the parasitic strategy will not work as part of the overall epistemicist view. The epistemicist wants to explain our hedging over Fa in borderline cases as a manifestation of our ignorance as to whether a is or is not F, rather than in terms of its being neither true nor false that a is F.¹⁴ As we shall see in the next section, the best account of why we would be ignorant in borderline cases—when in fact, in such cases, either a is F, or a is not F, according to the epistemicist—employs the idea of knowledge requiring a margin for error, which ensures that we cannot know of objects in the vicinity of the non-F/F boundary whether or not they are F. As Williamson (1997 [1992], 278) puts it: ‘‘an utterance of ‘[Williamson] is thin’ is an expression of knowledge only if I am some way from the boundary of ‘thin’ ’’. Thus, this part of the epistemicist picture requires that the hedging region cover the false/true boundary. But now consider the parasitic strategy. In accordance with (MU), the indeterminist locates false/neither-true-nor-false and neither-true-nor-false/true boundaries in the same places as the deny/hedge and hedge/assert boundaries provided by usage. The epistemicist who follows the parasitic strategy will then end up with a false/true boundary which coincides with the hedge/assert boundary—thus conﬂicting with the idea that the hedging region must cover the false/true boundary.¹⁵ ¹² The ‘indeterminist’ here is the theorist who thinks, in opposition to the classical semantic picture which the epistemicist accepts, that statements about borderline cases are neither true nor false. ¹³ Thanks to Robbie Williams for helpful comments here. ¹⁴ See e.g. Williamson 1994, 3. ¹⁵ Apart from this problem with the parasitic strategy, I also have deep misgivings about the underlying idea that there is a fundamental asymmetry between truth and falsity—but discussing this issue would take us far aﬁeld. In addition to these worries, Burgess 2001 argues that the parasitic strategy yields the wrong truth values in some cases; Weatherson 2003a correctly points out that Burgess is wrong about these cases, but substitutes other cases in their place.

42  2.1.2 Why Can’t We Know Where the Lines Are? Supposing that there is a precise dividing line between the bald and the non-bald (etc.), why can’t we know where it is? (Recall that the claim that we cannot know where the cut-offs are is a key plank in the epistemicist’s resolution of the Sorites paradox: it provides the explanation of why we are taken in by the paradox in the ﬁrst place.) The epistemicist who has made the most progress towards answering this epistemological question is Williamson (1997 [1992], §6; 1992; 1994, ch. 8). Williamson argues that where our capacities of discrimination are limited, knowledge requires a margin for error. For example, suppose that you walk into a football stadium which in fact contains 30,000 people, and say ‘There are 30,000 people here’. Your claim is true—but did you know there were 30,000 people there, or were you just right by luck? That depends upon your powers of discrimination. If you can detect very ﬁne differences in crowd size—if, had there been one less, or one more, person there, you would have noticed a difference, and would not have claimed there were 30,000 people there (in that counterfactual situation)—then it seems right to say you do know there are 30,000 people there (in the actual situation). But suppose your powers of discrimination are coarser: that you would only begin to notice a difference in the crowd if 100 or more people were removed, or added. Then, if there had been one less, or one more, person there, you would not have noticed any difference, and would still have claimed there were 30,000 people there (in that counterfactual situation)—and so it seems right to say you do not know there are 30,000 people there (in the actual situation). Given your powers of discrimination, you could (in the actual situation) know a claim such as ‘there are at least 29,900 people present’. For if we were to alter the situation in a way you can not detect–by adding or subtracting up to 100 people–then you would not notice any difference, and would still claim there are at least 29,900 people present; but you would be right (in these counterfactual situations). The general moral is that for you to know something in the actual situation, it must be true not only in the actual situation, but also in counterfactual situations which are similar enough to the actual one that you would not notice the difference (so, how similar is similar enough depends upon your powers of discrimination):

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(ME) ‘S knows that P’ is true in a situation T only if ‘P’ is true in situations that are sufﬁciently similar to T. Williamson calls this a margin for error principle for P.¹⁶ It might seem obvious how to apply this idea to the case of vagueness. Let’s assume the epistemicist is right about the semantics of vague predicates, and let’s suppose, for the sake of argument, that as a matter of fact the cut-off between tall and not-tall comes at 1.8 m (i.e. those whose height is greater than or equal to 1.8 m are tall). If Bob is 1.6 m in height, then I can know he is not tall, because in similar situations—where his height is slightly different—it’s still true that he is not tall. However if he is 1.8 m in height, then I cannot know that he is tall, because while this claim is true, had his height been ever so slightly less, the claim would have been false, and yet I would not have detected any difference. This clearly does not get to the heart of the matter, however. For the Sorites paradox for ‘is tall’ does not turn on our being faced with an actual line of men, whose height we cannot determine with absolute precision. It is simply stipulated that we have a line of men, ranging in height from four feet to seven feet, in nanometre increments. So the height of each man is given to us—and yet we feel there is a paradox. Thus if ignorance is supposed to explain the feeling that there is a paradox, then this ignorance must be of something other than the exact heights of the persons in the Sorites series. Indeed it is, according to Williamson. There must be a margin for error around the actual situation (in which Bob is 1.8 m in height, and the cut-off for tall is at 1.8 m), if we are to know, in that situation, that Bob is tall. We initially supposed this margin to consist of cases in which the cut-off for tall is still at 1.8 m, but Bob’s height is slightly different. This was, as we saw, a mistake. Instead, the margin consists of cases in which Bob is still 1.8 m in height, but the cut-off for tall—i.e. the extension of ‘is tall’ on the intended interpretation—is slightly different. We know from the previous section that Williamson’s view is that for the extension of ‘is tall’ to be different, community usage of ‘is tall’ must be different. So, we are to imagine counterfactual situations in which usage is slightly different, but Bob’s height is the same. Consider ﬁrst the case in ¹⁶ The view that knowledge requires a margin for error has obvious afﬁnities with both reliabilist and tracking theories of knowledge. For more on the surrounding epistemological issues, see Williamson 2000.

44  which in the actual situation, Bob is 1.6 m in height, and we claim he is not tall. Community usage would have to be very different for the cut-off for tall to be shifted below 1.6 m: that is a big change from the actual cut-off of 1.8 m. Assuming I would recognize such a difference in usage and not say in that situation that Bob is not tall, I do know in the actual situation that Bob is not tall: I am safely within my margin for error. But consider the case in which in the actual situation Bob is 1.8 m tall. We might suppose that for the cut-off to be shifted just above 1.8 m, usage would not have to be very different: just a few people here or there would have to have different dispositions regarding the use of ‘is tall’ to make such a tiny shift in its extension.¹⁷ Plausibly, were such a change in usage to take place, I would be none the wiser: so I do not know in the actual situation that Bob is tall, for I have no margin for error. If this is correct, then we can see why we cannot know where the cut-off for tall is. In order to know that the cut-off between tall and not-tall is at 1.8 m, I would have to know both that a person 1.8 m in height is tall, and that a person arbitrarily less than 1.8 m in height is not tall. But we have just seen that I cannot know such things, because when we are so close to the cut-off that the amount of change in usage required to push the cut-off over the point we are considering is small enough for us not to be able to notice it, there is no margin for error. Thus I cannot know where the cut-off between tall and not-tall is—and similarly for other vague predicates. 2.1.3 Worldly Vagueness? A question which I shall be asking about each of the theories we examine is ‘Where does the theory locate vagueness: in the relationship between language and the world, or in the world itself?’ In other words, does the theory see vagueness as a semantic phenomenon, or as a metaphysical phenomenon? The epistemicist—unlike proponents of any of the other theories we shall examine—locates vagueness in neither of these places. She sees vagueness as an epistemic phenomenon. Vagueness and indeterminacy are ruled out of the classical semantic picture—which the epistemicist buys holus-bolus—in two places. First, there is no vagueness in the world. ¹⁷ In fact, it is hard to see how Williamson could be entitled to claims of this sort, given his view (discussed in the previous section) that ‘‘Meaning may supervene on use in an unsurveyably chaotic way’’ (1994, 209).

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Classical models are entirely precise: they represent the world as a crisp set of objects and properties such that for each property and each object, the object either deﬁnitely possesses the property, or deﬁnitely does not possess it. Second, there is no semantic indeterminacy—no vagueness in the relationship between language and the world. Each discourse has a unique intended model: so there is no vagueness or indeterminacy concerning what any name, predicate, or sentence means. For the epistemicist, then, vagueness is neither a metaphysical nor a semantic phenomenon. It is an epistemic phenomenon. Although the world itself is completely precise, and all items of our language have a unique, precise meaning, some of these meanings are unknowable. For example, ‘is tall’ picks out a particular crisp set of objects in the world, but we cannot know the full details of the membership of this set—and it is in this necessary ignorance that the vagueness of ‘is tall’ is located. 2.1.3.1 Semantic Realism I just asked whether epistemicism locates vagueness in the relationship between language and the world, or in the world itself—and in the other sections of this chapter entitled ‘Worldly Vagueness?’ I shall be asking the same question about the other theories of vagueness to be examined. It is worth making explicit a presupposition of the question, and the methodology I employ in answering it. Taking the second point ﬁrst: my method is to read off the answer directly from the system of model theory employed by the theory of vagueness in question. Underlying this method is a literal attitude towards model theories for vague language. That is, I regard a model theory for (part of) a language as giving a literal (although not necessarily complete) description of the relationship between that language and the world: a system of model theory as a whole tells us about the kinds of relationships that a language may have to a world; what is going on in the intended interpretation(s) of a particular discourse tells us the actual relationship between that discourse and the world. Of course this is not always the appropriate attitude to take towards model theory. A logician is perfectly entitled, for example, to employ some system of model theory in order to establish results about provability in some formal system: for this, she needs only relevant soundness or completeness results; it does not matter whether the model theory in any sense gives the real meaning of the sentences in question. Likewise, a philosopher might, for example, employ a model theory for a system of modal logic involving

46  possible worlds and merely possible objects, but not take this model theory literally in my sense: when his model theory tells him that ‘Possibly P’ is true iff there is a possible world in which P is true, he does not regard this as a literal description of what goes on when I say ‘Possibly P’; that is, he does not think that I stand to a bunch of possible worlds, which between them do or do not make my statement true, in the same way that the teacher who says ‘Someone threw that apple core’ stands to the collection of students in the room, who between them do or do not make that statement true. Rather, he might treat his model theory in an instrumental fashion, regarding it as a calculus for keeping track of our modal commitments.¹⁸ This is all quite right, but in the present context—i.e. in the various sections of this chapter entitled ‘Worldly Vagueness?’—a literal attitude towards model theory is appropriate. This is because a presupposition of the question being asked in these sections is semantic realism, and where one is a semantic realist about an area of discourse, one should be a model-theoretic literalist. Let me explain. Semantic realism, in the sense I mean it here, is the view that there are genuine semantic properties and relations: in particular, relations of reference holding between sub-sentential or sub-propositional expressions (names, predicates, relation symbols) and parts of the world (objects, properties or sets of objects, relations or sets of n-tuples of objects), and properties of truth possessed by (some) sentences or propositions.¹⁹ Note that while semantic realism implies that there are real relations of reference, it does ¹⁸ This non-literal attitude towards modal model theory is to be contrasted with the ersatzist perspective, which involves taking some form of Kripke-style modal model theory employing possible worlds literally, but denying that the possible worlds and mere possibilia which ﬁgure in the model theory are to be regarded as concrete things on a par with the actual world and its inhabitants. In relation to the distinction between literal and non-literal attitudes towards model theory, cf. the discussions in the literature on modal logic of the distinction between pure and applied (aka depraved) semantics. The original distinction is due to Plantinga 1974, 126–8. For some recent discussion see Gregory 2005; Zimmerman 2005, 414 ff.; and Divers 2006. My ﬁrst introduction to the idea of seeing (classical) model theory as providing an account of the relationship between natural language and the world was in helpful discussions with Amitavo Islam. ¹⁹ Within the class of semantic realists—characterized by adherence to the view that both reference and truth are real—we might furthermore distinguish those who hold that reference is primary, and those who hold that truth is primary. On the former view (which I call ‘reference-ﬁrst semantic realism’), the truth status (e.g. true, false, no truth value, an intermediate truth value) of sentences is explicable in terms of the referents of their parts. For example, in the sentence ‘Maisy is hungry’, the name ‘Maisy’ refers to a certain dog, and the predicate ‘is hungry’ picks out a certain property (or set of things), and it is because Maisy in fact has this property (is a member of this set of things) that the sentence is true. On the latter view (which I call ‘truth-ﬁrst semantic realism’), what ﬁxes or determines reference is the truth conditions of certain sentences. For example, one might hold that the reference of some term is whatever it has to be in order to make all the sentences in some overarching theory true.

      

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not imply that every sub-sentential expression has a unique or determinate reference.²⁰ It also does not imply that all sentences are true or false: the properties of truth countenanced by the semantic realist might include truth statuses such as neither true nor false, true to such-and-such degree, and so on. Semantic realism contrasts with semantic antirealist views which deny the existence of real reference relations, or real truth properties, or both. Advocating the reality of reference while denying the reality of truth seems bizarre, and to my knowledge this position remains unoccupied; there are those, however, who deny the reality of truth and reference, and those who deny the reality of reference but not of truth. In the former camp we ﬁnd, for example, the antirepresentationalists. According to their ‘push-and-pull’ or ‘he said–she said’ view of language, our use of language is to be understood as a game or practice, and sentences and other utterances are to be assessed in terms of their appropriateness, like moves in a game—that is, in terms of their ‘horizontal’ connections with other moves, rather than in terms of their ‘vertical’ (i.e. referential or representational) relations with extra-linguistic reality.²¹ In the latter camp we ﬁnd, for example, Davidson (1984 [1977]), who combines an analogy between semantics and physics—in physics, ‘‘we explain macroscopic phenomena by postulating an unobserved ﬁne structure. But the theory is tested at the macroscopic level’’ (p. 222), and in semantics, on Davidson’s analogy, truth of sentences plays the role of the macroscopic phenomena, and reference of sub-sentential expressions plays the role of the underlying ﬁne structure—with an instrumentalist rather than realist attitude towards the postulated underlying machinery. Thus on Davidson’s approach it ‘‘makes no sense . . . to complain that a theory comes up with the right truth conditions time after time’’, but gets the references of sub-sentential expressions wrong (p. 223).²² ²⁰ Indeed, this follows from the fact (n. 19) that semantic realism is compatible with the view that the truth conditions of sentences determine all other semantic facts, together with the now well-known fact (emphasized by Quine, Wallace, and Putnam, amongst others—see e.g. Wallace 1977 and Putnam 1981, 33), that in general the truth conditions of whole sentences do not determine unique references for their parts. ²¹ Abbott 1997, 128–30 has a useful categorization of views of reference, which distinguishes two other views besides semantic realism and antirepresentationalism (which Abbott calls ‘bold antirepresentationalism’). ²² Referential antirealism of this sort must be distinguished from truth-ﬁrst semantic realism. The latter view agrees with Davidson on the primacy in semantics of the truth conditions of whole sentences, but disagrees that there are no real relations of reference between sub-sentential expressions and (parts of ) the world.

48  The question asked about each of our theories of vagueness in the sections of this chapter entitled ‘Worldly Vagueness?’ presupposes semantic realism. The idea is that when we speak vaguely, there is our language on the one hand, the world on the other hand, and semantic relations between the two. The question is then: where is the vagueness? Does the theory in question locate vagueness in the world itself, or in the relationships between language and the world? Clearly, from a semantic antirealist perspective, the question is misguided. That leaves two points to discuss. First, by what right do I presuppose semantic realism in these sections? Second, given this presupposition, why should we be model-theoretic literalists (in the context of answering the question posed in these sections)? On the ﬁrst question: Semantic realism is the obvious, default view of language. For instance, it certainly seems that the thing that distinguishes language from a game such as chess or draughts is precisely that words refer to objects in the world, so that combining words in different ways on paper or in speech allows us to say things about the world, to represent it as being thus or so, whereas chess or draughts pieces do not refer to objects in the external world, and moving them and combining them in different ways on a board does not represent anything about goings-on external to the board. Even Davidson (an opponent of semantic realism), refers to the ‘Building-Block theory’—a version of reference-ﬁrst semantic realism—as ‘‘an old [view] and a natural one’’ (1984 [1977], 220). We are therefore entitled to assume a broadly semantic realist perspective unless this is shown to be untenable. But that has never been done: none of the arguments against semantic realism, put forward by antirepresentationalists, referential antirealists, and others, is convincing. I shall not defend this claim, however: to do so properly would require a book-length treatment, and it seems more appropriate to save that for another occasion than to load it onto the present work, for two related reasons. First, most of the vagueness literature assumes semantic realism as a guiding viewpoint (although see §2.7 for one exception); second, the debate between semantic realisms and antirealisms is of very broad signiﬁcance, far beyond the study of vagueness. Semantic realism will, then, be an unargued presupposition. On the second question: Given semantic realism, model-theoretic literalism follows almost immediately. For if we approach vague language from a semantic realist standpoint—if we believe in referential relations between

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linguistic items and items of extra-linguistic reality, in truth properties of sentences, and that truth and reference are related in certain fundamental ways (e.g. we might believe that ‘Maisy is tired’ is true just in case the referent of ‘Maisy’ is in the extension of ‘is tired’)—then we would be foolish not to adopt a model-theoretic framework for exploring different particular views on the reference and truth of vague expressions. For model theory is purpose-built for discussing such matters: it enables the discussion to proceed with a far greater level of clarity and precision than would otherwise be possible, and provides a unifying perspective from which the differences between rival views of vagueness can be perceived with great accuracy. But then once we have adopted a model-theoretic framework for these reasons, there is simply no room left for us to turn around and take a non-literal attitude towards it. For the whole idea was that we want to know what is going on when we use vague language—what the world we talk about is like, and how our language is related to it—and we adopted a model-theoretic framework precisely because of the clarity and precision of the answers to these questions that such a framework can be seen as providing.²³ Summing up: In the various sections of this chapter entitled ‘Worldly Vagueness?’, I presuppose a broadly semantic realist perspective, according to which when we use vague language, we refer to and/or represent goingson in extra-linguistic reality. I am not assuming that we determinately refer to unique parts of the world: I am assuming only reality—not determinacy—of reference. More generally, I am not, at the outset, assuming any particular account of the relationship between vague language and the world—I am not assuming of any particular model theory for vague language that it is the correct one. The task is to determine exactly what the relationship is between vague language and reality—and part of this task involves determining what the reality that vague language talks about is like in itself. Among other things, this will give us an answer to the question whether the world can be regarded as precise, with vagueness being a matter of the ²³ Of course, as I have already noted, there may well be other contexts in which we wish to employ some system of model theory for different reasons and in which a literal attitude to that model theory would not be appropriate. This also means that when I say, for example, that many-valued theories of vagueness locate vagueness in the world (§2.2.2), this cannot be read as saying that previous advocates of many-valued semantics think that vagueness is a worldly, not a semantic, phenomenon: for while an author may have advocated a many-valued model theory for vague language, she may not have taken her model theory literally in my sense.

50  referential and representational relationships between vague language and the world, or whether we must ultimately locate some or all vagueness in the world itself. Because of the advantages that model theory provides in terms of clarity and precision, these questions will be pursued within a modeltheoretic framework—and because this is the reason for employing such a framework, the model theory will be taken literally: that is, I will regard what is going on in the intended model(s) of vague discourse, according to a particular model theory for vague language, as a (proposed, but of course not necessarily correct) literal account of the relationship between vague talk and reality. Granted this perspective of model-theoretic literalism, the claims that I make about worldly vagueness and semantic indeterminacy in the various sections of this chapter entitled ‘Worldly Vagueness?’—which might seem contentious given other possible perspectives on the use of model theory—are obvious and uncontroversial.

2.2 Additional Truth Values (Deny 1a) Our story in §1.2 began with the Boolean algebra ({0, 1}, ∨, ∧, ) of classical truth values.²⁴ We saw that with this algebra in hand, classical propositional logic, set theory, and predicate logic come in a natural progression. We then added the idea of there being an intended interpretation of any given discourse, to complete our presentation of the classical picture. If we take this story and keep everything the same, except that we unplug the algebra of classical truth values, and instead plug in a different algebra of truth values, then we get a family of different semantic pictures—one member of the family for each underlying algebra. In the cases where the algebra of truth values that we plug in has more than two elements, what we get is a many-valued logic. The simplest case is where we have three truth values, leading to versions of three-valued logic. This idea has a natural application in the case of vagueness—for it is a fairly natural thought about vague sentences such as ‘Bob is bald’, where Bob is a borderline case of baldness, that they ²⁴ In §1.2, I used different symbols for the algebraic operations on the set of truth values and for the connectives in our language, in order to make this distinction very clear. As foreshadowed in ch. 1 n. 5, I now revert to the standard practice of using the same symbols for both, and letting context disambiguate.

      

51

are neither true nor false, but have a third truth value.²⁵ One particularly nice aspect of this approach is that it allows us to maintain an attractive parallelism between meaning and use, of the sort that the epistemicist was forced to reject. We can now say that a vague sentence Pa is true if most competent speakers would conﬁdently classify a as being P, is false if most competent speakers would conﬁdently deny that a is P, and is neither true nor false if most competent speakers would hedge over whether a is P. The epistemicist had only two semantic statuses for sentences—true and false—whereas ordinary speakers have at least three pragmatic stances which they can adopt towards a sentence—assertion, denial, or hedging. The three-valued approach posits a new semantic status, thereby allowing a parallelism between meaning and use to be maintained.²⁶ In developing any three-valued view, our ﬁrst step is to specify an algebra ({0, 1, ∗}, ∨, ∧, ) which has three truth values, 1, 0, and the new truth value ∗, and three operations, ∨, ∧, and . Consider Table 2.1. Every different way of ﬁlling in the blank cells with a 1, 0, or ∗ yields a different algebra of truth values. (A cell marked with the ‘ditto’ symbol must be ﬁlled with whatever is in the cell above it.) There are twenty-one empty cells, and three possible ﬁllings for each; thus we have 321 = 10, 460, 353, 203 possibilities. Thus, our family of three-valued logics is large! Some members of the family are more interesting than others. For a start, it is natural to be more interested in those operations ∨, ∧, and which, when fed only 1’s and 0’s, output what the classical operations would output. This ﬁxes every row which does not have a ∗ in it, reducing the number of empty cells to eleven (and thus leaving 311 = 177, 147 possible algebras remaining). It is also natural to want the truth values of A ∨ B and A ∧ B to be the same as those of B ∨ A and B ∧ A respectively—thus we want our binary operations ∨ and ∧ to be symmetric. Introducing ²⁵ Other common motivations for introducing three-valued logics include: Future contingents. Consider a statement about the future, such as ‘Bob will eat pizza for dinner next Tuesday’. You might think that as of right now, this statement is neither true nor false: it’s up to Bob, and he has not decided yet. Semantic paradoxes. Consider the sentence: ‘This sentence is false.’ Suppose it is true; then what it says is indeed the case; but what it says is that it is false. So if it’s true, it’s false, so it’s not true. So what if it is false? Then what it says is not the case. What it says is that it is false; so it is not false, i.e. it is true. So if it is false, it is true, so it is not false. So it seems that the sentence can be neither true nor false. It might reasonably be thought that these motivations lead most naturally not to the idea of a third truth value, but to the idea that some sentences lack a truth value altogether. We shall come to this idea in §2.3. ²⁶ For further discussion of this issue see p. 58 below.

52  Table 2.1. Framework for algebras of three truth values ∨

∧

1

1

1

∗

1

0

∗

1

∗

∗

∗

0

0

1

0

∗

0

0

the convention that a cell marked with the symbol 2 must be ﬁlled with whatever is in the cell above the cell above it, this leaves us with seven empty cells (see Table 2.2). Finally, we are naturally more interested in those algebras where ∨ and ∧ are duals (with respect to ).²⁷ Thus, once we Table 2.2. Partially constrained framework for algebras of three truth values 1

1

1

∗

1

0

∨

∧

1

1

0

1

0

2

2

∗

1

∗

∗

∗

0

0

1

1

0

1

0

∗

2

2

0

0

0

0

²⁷ See p. 69 for an explanation of this notion of duality.

      

53

have ﬁxed one of these operations, the other is ﬁxed too. That removes three more cells. The number of algebras left is still 34 = 81. Of these, I shall mention only two! The ﬁrst is arrived at via the idea that any compound proposition with a component that has the value ∗ should itself have the value ∗. Given that, once we have ﬁxed the algebra of truth values, we are simply going to run the rest of the classical story without further changes—thus, for example, the truth value of A ∨ B will be that truth value to which the operation ∨ in our algebra of truth values maps the pair (the truth value of A , the truth value of B )—this means that our operations must be as in Table 2.3 (where every blank cell in Table 2.2 has been ﬁlled with a ∗).²⁸ A second rationale for ﬁlling in the blank cells in Table 2.2 is as follows. Suppose the ∗ were a 1, and calculate the value of the corresponding classical operation, and suppose the ∗ were a 0, and calculate the value of the corresponding classical operation; if you get 1 both times, the value of the new operation that you are deﬁning is 1; if you get 0 both times, the value of the new operation is 0; if you get 1 once and 0 once, then the value of the new operation is ∗. (Where you are trying to determine what the Table 2.3. Algebra of three truth values: ﬁrst rationale ∨

∧

1

1

1

1

0

1

∗

∗

∗

1

0

1

0

∗

1

∗

∗

∗

∗

∗

∗

∗

∗

0

∗

∗

0

1

1

0

1

0

∗

∗

∗

0

0

0

0

²⁸ This gives us something analogous to Kleene’s 1952 weak truth tables. It does not give us Kleene’s weak truth tables themselves, because what I have just given is not a truth table (recall the discussion on p. 26).

54  new operation should assign to a pair of ∗’s, you need to calculate all four possibilities where each ∗ is replaced by a 1 or a 0.) Following this rationale through, we get Table 2.4.²⁹ The rationale used to deﬁne the latest set of operations may have sounded like supervaluationism, but the outcome is in fact very different. The three-valued view just presented—along with every other view constructed from the classical view by replacing the algebra of classical truth values with an alternative algebra—is truthfunctional, which is to say that the truth value of a compound sentence is determined solely by the truth values of its components. Truth-functionality is ensured by the part of the picture which says that [A ∨ B ] = [A ] ∨ [B ], [A ∧ B ] = [A ] ∧ [B ] and [¬A ] = [A ] (see p. 26 above). On the other hand, as we shall see when we discuss it in §2.4, the supervaluationist view is not truth-functional. A third rationale for ﬁlling in the blank cells in Table 2.2 is as follows. Recall that at the beginning of our discussion of the classical picture, we considered the appropriate ordering of our two classical truth values 1 and 0. Let us do this again, with our three truth values 1, 0, and ∗. We may ask, under what assignments of truth values to sentences S and T is S at least as true as T? Obviously, if S and T have the same truth value, then S ≥ T Table 2.4. Algebra of three truth values: second rationale ∨

∧

1

1

1

1

0

1

∗

1

∗

1

0

1

0

∗

1

1

∗

∗

∗

∗

∗

∗

∗

0

∗

0

0

1

1

0

1

0

∗

∗

0

0

0

0

0

²⁹ This is Łukasiewicz’s three-valued logic (minus his treatment of the conditional, which is the three-valued analogue of the fuzzy Łukasiewicz conditional, to be introduced in §2.2.1.1). It is also analogous to Kleene’s 1952 strong truth tables (cf. n. 28).

      

55

(letting ≥ mean ‘at least as true as’), and if S is true and T is false, then S is truer than T (i.e. S ≥ T and T S). But what if S has the value ∗ and T does not? One natural thought is that if T is false, then S is truer than T, and that if T is true, then T is truer than S. This yields the following lattice of truth values: 1 | ∗ | 0 Setting x ∨ y equal to the supremum of x and y relative to this ordering, x ∧ y equal to the inﬁmum of x and y relative to this ordering, and setting 0 = 1, ∗ = ∗, and 1 = 0, yields an algebra of truth values. As is easily veriﬁed, it is in fact the same as our previous algebra. We wanted to replace the algebra of classical truth values with a different structure. We have seen that we have many choices here. Making this choice is our only substantive step. After making it, we then run the rest of the classical story just as before. Thus, when it comes to propositional logic, the truth value [A ∧ B ] of the proposition A ∧ B will be [A ] ∧ [B ], where [A ] is the truth value of A , [B ] is the truth value of B , and ∧ is the operation in our new algebra of truth values that replaces the classical operation, symbolized by the same symbol, in our original Boolean algebra of truth values.³⁰ When it comes to set theory, we have a ready-made ³⁰ Of course, not all the details of the resulting story of propositional logic will be the same as in the classical story: what we are holding ﬁxed is the method of generating the story, given an algebra of truth values as input; when we vary that algebra, the details will, in general, change. The changes are, however—and this is the point—perfectly predictable. Thus, e.g., the classical algebra of truth values is a Boolean algebra, and so all the laws of Boolean algebras reappear, in different guise, as laws of classical propositional logic. For example, mirroring law 5 of Boolean algebras on p. 24, two formulas A and A ∨ (B ∧ ¬B ) of classical propositional logic always have the same truth value. Our second algebra of three truth values is not a Boolean algebra—for example, in relation to law 5 just considered, 0 ∨ (∗ ∧ ∗ ) = ∗ = 0—and so, correspondingly, two formulas A and A ∨ (B ∧ ¬B ) will have different truth values when A has the value 0 and B has the value ∗. Our ﬁrst algebra of truth values (unlike our second) is not even a lattice—for example, in relation to law 4 of lattices on p. 23, 1 ∨ (1 ∧ ∗) = ∗ = 1—and so, correspondingly, two formulas A and A ∨ (A ∧ B ) will have different truth values when A has the value 1 and B has the value ∗—unlike in classical propositional logic (and the logic resulting from our second algebra of truth values), where such formulae always have the same truth value. However, the point is that the laws of each logic do correspond perfectly to the laws of its own underlying algebra of truth values—thanks to the uniform method of generating the logic from the algebra.

56  notion of a three-valued set: a set whose characteristic function maps objects to our three-membered set of truth values {0, 1, ∗}. Just as some sentence might be neither true nor false, but instead have truth value ∗, so too some object might be neither in a set S (i.e. mapped to 1 by S’s characteristic function), nor not in S (i.e. mapped to 0 by S’s characteristic function)—this object will be mapped to ∗ by S’s characteristic function. Then, supposing, for example, that x is in S, but neither in nor out of T, what is x’s status with respect to S ∪ T? The relevant part of the classical story tells us that (S ∪ T)(x) = S(x) ∨ T(x), where ∨ is now the operation in our new algebra of truth values that replaces the classical operation, symbolized by the same symbol, in our original Boolean algebra of truth values. So we need to see to what value this operation maps the pair (1, ∗). In the ﬁrst algebra we looked at, the answer is ∗—i.e. x is neither in nor out of S ∪ T. In the second algebra, the answer is 1—i.e. x is in S ∪ T. In order to handle unions and intersections of arbitrary numbers of sets in the way we handled them in the classical story, we need operations and . But as long as our operations ∨ and ∧ are symmetric, associative, and idempotent—as they are on the proposals we have examined—these are easily obtained. Where x, y, z are any elements of our algebra of truth values, we just set {x, y, z} = (x ∨ y) ∨ z = x ∨ (y ∨ z) = (x ∨ z) ∨ y = x ∨ (z ∨ y) = (z ∨ x) ∨ y = z ∨ (x ∨ y) = . . . , and likewise for and ∧. The link between logic and set theory in the classical picture is also maintained in the new three-valued picture. If, on a certain interpretation, ‘Bob’ is assigned x as its referent, and ‘is bald’ is assigned the set S as its extension, then the truth value of ‘Bob is bald’ will be the value (1, 0, or ∗) to which S maps x. Quantiﬁed wfs are handled via the operations and just mentioned, in a way that is perfectly analogous to their classical treatment. Finally, ‘Bob is bald’ will be true simpliciter if it gets the value 1 on the intended interpretation. How will three-valued views handle the Sorites paradox? Consider a Sorites series for, say, ‘is tall’. To make things simpler, let’s number the people in the series from 0 to 2000. 0 is 1 m tall, 1 is 1 mm taller, 2 is 1 mm taller again, and so on, until we get to 2000, who is 3 m tall. We want to say that ‘0 is tall’ is false, that ‘2000 is tall’ is true, and that for each person n, if n is tall, then so is (n − 1)—that is, that the following Sorites conditionals are all true: ‘if 2000 is tall, then 1999 is tall’; ‘if 1999 is tall, then 1998 is tall’; . . . ; ‘if 2 is tall, then 1 is tall’; ‘if 1 is tall, then 0 is tall.’ The

      

57

paradox is that these things that we want to say cannot all be true (at least in the classical picture). One obvious strategy for a three-valued response to this situation—illustrated with respect to the second of the two particular three-valued views introduced above—is as follows. Where Bob is a clear case of tallness, ‘Bob is tall’ has the value 1; where Bob is a borderline case of tallness, ‘Bob is tall’ has the value ∗; and where Bob is a clear countercase of tallness, ‘Bob is tall’ has the value 0. Going from the 2000 end to the 0 end, our Sorites series begins with people who are clear cases of tallness and ends with people who are clear countercases of tallness, and in between contains people who are borderline cases of tallness. So now consider our Sorites conditionals, after recalling the part of the picture which says that A → B is an abbreviation of ¬A ∨ B , and then noting that 1 ∨ 1 = 1, 1 ∨ ∗ = ∗, ∗ ∨ ∗ = ∗, ∗ ∨ 0 = ∗, and 0 ∨ 0 = 1.³¹ The upshot is that some of the Sorites conditionals are not true. So that’s the error in the paradoxical reasoning. Why, then, do we get taken in by the paradox? Because none of the conditionals is false, either. The only way for ‘if P then Q’ to have the value 0 is for P to have the value 1 and Q the value 0. This does not happen in our series: between the persons who are tall (i.e. are mapped to 1 by the set of tall things) and those who are not tall (i.e. are mapped to 0 by the set of tall things) are the borderline cases—that is, the persons who are neither tall nor not tall (i.e. are mapped to ∗ by the set of tall things).³² There are two main objections to three-valued approaches to vagueness. One of them focuses on the fact that the three-valued approach is truthfunctional. I shall postpone discussion of it until p. 85.³³ The other objection is that the three-valued approach falls foul of ‘the problem of higher-order vagueness’, by imposing sharp cut-offs between the objects to which a predicate does not apply and its borderline cases, and between the borderline cases and the objects to which the predicate applies. In fact we ³¹ Alternatively, we could regard A → B as an abbreviation of ¬(A ∧ ¬B ). This makes no difference, because (1 ∧ 1 ) = 1, (1 ∧ ∗ ) = ∗, etc. ³² In general in this chapter, when I say how a given view of vagueness will handle the Sorites paradox, I do so for the purpose of giving a feel for the view to newcomers to the literature, not for the purpose of assessing the view (recall the discussion at the beginning of Ch. 1). Thus in the present case, for example, I do not mean to suggest that the approach to the Sorites just laid out would withstand thorough scrutiny, nor that no other response to the Sorites could be available to proponents of three-valued views of vagueness. ³³ The discussion there is framed in terms of truth gaps, rather than additional truth values, but as far as this objection is concerned, that difference is irrelevant.

58  need to distinguish two potential problems here—problems which are, in my view, conﬂated in the literature under the one heading ‘the problem of higher-order vagueness’. One problem is that the view does not allow for a gradual transition between the clear cases and the clear countercases. I shall postpone discussion of this jolt problem (as I call it) until Chapter 4 (see also §3.5.5). The other problem is the location problem discussed in connection with epistemicism. Recall our claim, at the beginning of this section, that one particularly nice aspect of the three-valued approach is that it allows us to maintain an attractive parallelism between meaning and use, of the sort that the epistemicist was forced to reject. We can now say that a vague sentence Pa is true if most competent speakers would conﬁdently classify a as being P, is false if most competent speakers would conﬁdently deny that a is P, and is neither true nor false (i.e. has the value ∗) if most competent speakers would hedge over whether a is P. Let’s reconsider this claim. Certainly, the three-valued approach does better than epistemicism in maintaining a broad link between use and meaning—at least we now have a tripartite distinction on both sides. However, at a ﬁner level of detail it seems that the three-valued picture is not so nice after all. Suppose that we are faced with a series of men ranging in height from 4 feet to 8 feet, in very small increments. We are to go down the series, classifying each man as tall, or not. At the beginning, all competent speakers will conﬁdently deny tallness; by the end, all will conﬁdently assert it. But different speakers will become less conﬁdent, and start to hedge, at different points, and then later will cease to hedge, and become more conﬁdent, at different points. Thus, on the use side we seem to have blurred boundaries between denial and hedging, and hedging and assertion, whereas on the semantic side, we have—on the three-valued account—perfectly sharp divisions between the men mapped to 0 by the characteristic function of the extension (on the intended interpretation) of ‘is tall’ and the men mapped to ∗, and between the men mapped to ∗ and the men mapped to 1. I do not ﬁnd this objection to the three-valued account anywhere near as potent as the corresponding objection to the epistemicist. In the latter case, we could see no plausible route whatsoever from usage to the epistemicist’s semantic picture. In the present case, however, I think we can. We can say that the set of men mapped to 1 by the characteristic function of the extension (on the intended interpretation) of ‘is tall’ is the set of men which every competent speaker would conﬁdently classify as tall; the set

      

59

of men mapped to 0 is the set of men of which every competent speaker would conﬁdently deny tallness; and the set of men mapped to ∗ is all the rest. Given this story, the divergences between individuals concerning where they start and stop hedging do not threaten the parallelism between meaning and use—where ‘use’ here means what it obviously should mean in this context: namely, use of the speech community as a whole. That’s a brief statement of a response to the location problem as it faces the three-valued view, and there is more to say about this issue. I shall not say it here, however, for I present a more detailed response along the same lines to a related problem in §6.2.2.³⁴ 2.2.1 The Fuzzy Picture Continuing our examination of the many-valued approach, we ask now: what if we have more than three truth values? The situation will be the same in all cases: we replace the Boolean algebra of classical truth values with a different structure, and then proceed as we did in the classical picture. One natural family of ﬁnitely many-valued logics comprises those built on an underlying algebra of n truth values whose elements 1 , . . . , n−2 are {0, n−1 n−1 , 1} for some natural number n ≥ 2.³⁵ Ordering these truth values in accordance with the standard ordering of the rational numbers yields a bounded lattice, and (echoing the third rationale discussed above in connection with three-valued logic), for all truth values x and y in our chosen set of values, we may then set x ∨ y equal to the supremum of x and y, x ∧ y equal to the inﬁmum of x and y, and we may set x equal to 1 − x. However, while four-valued logics have been studied in some detail, they have not ﬁgured prominently in discussions of vagueness. Nor have many-valued logics with a larger ﬁnite number of truth values. Moving to inﬁnitely many truth values, the ﬁrst natural stopping point (for reasons to be discussed in n. 40 below) is to have uncountably many truth values—or more concretely, to take as our set of truth values all real ³⁴ Note that I do not think that there is no problem for the three-valued view stemming from its positing of perfectly sharp divisions between the men mapped to 0 by the characteristic function of the extension of ‘is tall’ and the men mapped to ∗, and between the men mapped to ∗ and the men mapped to 1. In fact I think there is a devastating problem here: the jolt problem, to be discussed in Ch. 4. My point at present is that there is not a big problem concerning how our usage—as a speech community—could determine the position of these sharp divisions. ³⁵ The ﬁnitely many-valued logics of Łukasiewicz take this form; see Łukasiewicz and Tarski 1970 [1930], 141 or Malinowski 1993, 36.

60  numbers between 0 and 1 inclusive. (You can think of these as percentage values—so a sentence can be 0.3 true, i.e. 30 per cent true, or 0.56 true, i.e. 56 per cent true, etc.) This takes us to the fuzzy picture.³⁶ In this section I shall present the fuzzy view in some detail. All that I will be doing is spelling out, step by step, that the fuzzy picture arises from the classical picture simply by replacing the Boolean algebra of classical truth values with a different algebra of truth values, which has as its elements the real numbers between 0 and 1 inclusive. After making this change, we simply follow through the classical story, step by step. I shall nevertheless spell this out in detail for two reasons. First, there is some suspicion of the fuzzy view in the literature. There is a perception that it is built on sand—that there is something suspect, from the logico-mathematical point of view, about its foundations. This impression is mistaken—and I hope that this section will make this clear. Second, discussions in the vagueness literature of the fuzzy approach typically focus on the fuzzy logical calculus. We are introduced to various sentential connectives, and told how to calculate the truth values of compound sentences given the truth values of their components. A deﬁnition of validity or consequence is introduced, and it is explained how a typical Sorites argument is invalid or unsound (depending upon the particular version of fuzzy logic in question). This is all very well, as far as it goes, but simply being taught how to calculate with fuzzy truth values in this way is not very illuminating. We can gain a much clearer understanding of the fuzzy position if we also look at fuzzy sets, and at fuzzy model theory, which is where fuzzy logic meets fuzzy sets. In this section I want to convey not just a facility with certain techniques, but a whole semantic picture —as rich and detailed as the classical picture, and (as I shall argue later) a worthy rival to it. The classical truth values 1 and 0 can be thought of as Yes and No. In the classical picture, when we ask whether a sentence is true, or whether a given object is in a given set, the answer can only be Yes or No. The basic thought behind the fuzzy view is that while the classical picture is ﬁne in the realm for which it was developed—mathematics—it is inadequate when ³⁶ The study of inﬁnite-valued logics begins with Łukasiewicz; see e.g. Łukasiewicz and Tarski 1970 [1930]. Fuzzy set theory was born in Zadeh 1965; a related earlier idea is Post’s notion of an n-valued set (Malinowski 1993, 47, 98). The history of both branches of the fuzzy view (i.e. logic and set theory) is rich; see Hájek 1998, ch. 10, and Novák et al. 1999, ch. 8, for overviews. For technical introductions to fuzzy logic and set theory, see e.g. Klir and Yuan 1995; Hájek 1998; Novák et al. 1999; and Nguyen and Walker 2000. The chief sources for the fuzzy view of vagueness are Goguen 1968–9; Lakoff 1973; and Machina 1976; Black 1997 [1937] is a relevant earlier work.

      

61

it comes to modelling vague language and the properties expressed therein. For example, neither Yes nor No will do as an answer to the question as to whether Balding Bob is bald. We need a richer set of options: we need degrees of truth and of set-membership. Standard fuzzy practice is to take [0, 1], the set of all real numbers between 0 and 1 inclusive, as the set of truth values, in place of the classical set {0, 1}. 0 is thought of as No and 1 as Yes, but now we have many other options as well: a number between 0 and 1 represents a tentative Yes or No; the higher the number, the less tentative a Yes (and the more tentative a No) it is. Just as there is a natural ordering of the classical truth values, so there is a natural ordering of the fuzzy truth values, arrived at in an analogous manner. Suppose, for example, that S is 0.7 true and T is 0.6 true; then T is strictly less true than S. Now, what S and T mean is not important here: the same relationship would hold between any sentences which were, respectively, 0.7 true and 0.6 true. So we may transfer the truth-ordering of sentences onto the truth values that sentences possess—and overall, the result turns out to be exactly the ordering we get when we restrict the standard ordering of the real numbers to the interval [0, 1]. With this ordering in place, we want to deﬁne some operations on [0, 1]. The basic set of operations is ∨, ∧, and : as before, the idea is that these will be used to deﬁne, respectively, logical disjunction and set-theoretic union, logical conjunction and set-theoretic intersection, and logical negation and set-theoretic complementation. Now, just as before, if S and T have the same truth value, then we want ‘S or T’ to have that truth value also, and if S and T have different truth values, then we want ‘S or T’ to have the greater of these truth values. Thus we want to say that for any truth values x and y in [0, 1], x ∨ y is the supremum of x and y with respect to our ordering of [0, 1]. For conjunction, if S and T have the same truth value, then we want ‘S and T’ to have that truth value also, and if S and T have different truth values, then we want ‘S and T’ to have the lesser of these truth values. Thus we want to say that for any truth values x and y in [0, 1], x ∧ y is the inﬁmum of x and y with respect to our ordering of [0, 1]. For negation, if S is less true than T, then we want ‘not T’ to be less true than ‘not S’, and we also want the negation of the negation of S to have the same truth value as S. It is customary to set x = 1 − x, for each x ∈ [0, 1], although in fact (as we shall see in §2.2.1.1) this is not the only deﬁnition that satisﬁes our two requirements.

62  We now have an algebra ([0, 1], ∨, ∧, ), consisting of a set of truth values together with some operations on that set. Unlike in the classical case, this is not a Boolean algebra, for it is not complemented.³⁷ It is, however, a Kleene algebra. Just as the Boolean algebra of classical truth values is the fundamental structure underlying classical logic and set theory, this Kleene algebra of fuzzy truth values is the fundamental structure underlying fuzzy logic and fuzzy set theory. Let us begin with fuzzy propositional logic. We have the same language as in the classical case: what changes is its interpretation. As before, an interpretation of the language consists in an assignment of a truth value [p] to every propositional constant p—only this time, the truth values are elements of [0, 1], not {0, 1}. Given an interpretation, every wf A is assigned a truth value [A ] by the following recursive deﬁnition: • • •

[A ∨ B ] = [A ] ∨ [B ] [A ∧ B ] = [A ] ∧ [B ] [¬A ] = [A ]

Note that the occurrences of ∨ and ∧ on the left of these identities represent the sentential connectives, while the occurrences on the right denote the operations on our Kleene algebra of truth values. (The situation in the third case is quite clear, as we have two separate symbols, ¬ and .) Thus we are saying that the truth value of the wf A ∨ B —the object assigned to this sequence of symbols on the interpretation in question—is found by locating the object assigned to A and the object assigned to B , and then performing the ∨ operation on these two objects. The result of this operation is an object in our algebra of truth values—and this object is the truth value of A ∨ B . Two wfs A and B are logically equivalent (written A ≡ B ) if they have the same truth value on every interpretation. ≡ is an equivalence relation on F, and F/≡ is the set of equivalence classes of F under ≡. Letting |A | be the equivalence class containing the wf A , we may set: • • •

|A | ∨ |B | = |A ∨ B | |A | ∧ |B | = |A ∧ B | ¬|A | = |¬A | ³⁷ For details see Novák et al. 1999, 23.

      

63

Note that the occurrences of ∨, ∧, and ¬ on the right of these identities represent the sentential connectives, while the occurrences on the left denote newly deﬁned operations on F/≡. It is a straightforward matter to check that these operations are well deﬁned, and that F/≡ together with these operations is a Kleene algebra.³⁸ As in the classical case (p. 27 above), it is not uncommon to refer to these equivalence classes of wfs as propositions, and to this Kleene algebra of equivalence classes as the fuzzy propositional calculus. An interesting fact here is that this Kleene algebra of fuzzy propositions is identical to the algebra of propositions yielded by Łukasiewicz’s three-valued logic.³⁹ Let us look next at fuzzy set theory. Given any ordinary (classical, crisp) set X, we may identify each fuzzy subset of X with its characteristic function, which is a function from X to [0, 1]. The number assigned by this function to x ∈ X represents x’s degree of membership of the subset in question. For any subsets (i.e. characteristic functions) f and g, and any x ∈ X, we deﬁne: • • • •

( f ∪ g)(x) = f (x) ∨ g(x) ( f ∩ g)(x) = f (x) ∧ g(x) f (x) = f (x) f ⊆ g ⇔ ∀x ∈ X, f (x) ≤ g(x)

Note that the occurrences of ∨, ∧, and on the right of these identities denote the operations on our Kleene algebra of truth values, and the occurrence of ≤ denotes our ordering of the truth values. Thus we are saying that to ﬁnd out to what degree a given x is in the union of f and g, we ﬁnd the object (in our algebra of truth values) which f assigns to x, and the object which g assigns to x, and then perform the ∨ operation on this pair of objects. The result of this operation is an object in our algebra of truth values, and this object is the value assigned to x by the union set ( f ∪ g); that is, the degree of membership of x in the union set ( f ∪ g). By doing this for every x ∈ X, we build up a complete picture of the union set ( f ∪ g). It is a straightforward matter to verify that these deﬁnitions yield a Kleene algebra of subsets. It is also clear that the operations we have just deﬁned are the obvious fuzzy analogues of the classical set-theoretic ³⁸ See Nguyen and Walker 2000, 68. ³⁹ For details see Nguyen and Walker 2000, 66–70. Recall that (the non-conditional fragment of ) Łukasiewicz’s three-valued logic was introduced in §2.2.

64  operations of union, intersection, and complementation. As in the classical case, as well as unions and intersections of pairs of sets f and g, we also deﬁne unions and intersections of arbitrary families of sets {fi } (and again, the above deﬁnitions for pairs of sets are just special cases of these more general deﬁnitions): • {fi }(x) = {fi (x)} • {fi }(x) = {fi (x)} The crucial thing to note here is that for these deﬁnitions to work, S and S need to be well-deﬁned for an arbitrary set S of truth values—in other words, the lattice of truth values must be complete. The lattice of classical truth values contains only two elements, so it is automatically complete—but when an ordered set has more than two elements, it may be a lattice without being a complete lattice. However, ([0, 1], ≤) is indeed a complete lattice.⁴⁰ Finally, let us consider fuzzy predicate logic. Again, we have the same language as in the classical case: what changes is its interpretation. An interpretation M = (M, I) of the language consists in a nonempty set M (the domain), together with a function I which assigns: • • •

a truth value [p] to every propositional constant p; an object I(a) in M to every individual constant a; an n-ary fuzzy relation I(P) on M to every n-ary predicate P.

In relation to the ﬁnal clause, note that an n-ary fuzzy relation on M is a fuzzy set of n-tuples of members of M. Identifying this fuzzy set of n-tuples with its characteristic function, we regard an n-ary fuzzy relation on M as a function from the set of all n-tuples of members of M to our set of truth values [0, 1]. Given an interpretation M, every closed wf A is assigned a truth value [A ]M on that interpretation by a recursive deﬁnition. (Again, where the interpretation in question is obvious from the context, or is irrelevant, the subscript M is omitted.) The clauses for non-quantiﬁed compound wfs ⁴⁰ So are the lattices of ﬁnitely many truth values introduced at the beginning of §2.2.1. However, the set of all rational numbers between 0 and 1 inclusive, under the standard ordering, yields a lattice which is not complete (e.g., the set of all such rationals which are less/greater than √12 has no supremum/inﬁmum). That is why we do not consider taking the set of all rational numbers between 0 and 1 inclusive as our set of truth values, but instead move straight from ﬁnitely many truth values to continuum-many.

      

65

are just as before. For an atomic wf P(a1 , . . . , an ), which consists of an n-ary predicate P followed by n individual constants a1 , . . . , an , the truth deﬁnition is as follows: [P(a1 , . . . , an )] = I(P)(I(a1 ), . . . , I(an )) The idea here is this. As discussed, each individual constant a is assigned an object I(a) in the domain, and P is assigned a function I(P) from the set of all n-tuples of members of M to our set of truth values [0, 1]. Whatever value this function assigns to the n-tuple (I(a1 ), . . . , I(an )), this value is the truth value of the sentence P(a1 , . . . , an ). Thus ‘Bob is bald’ is true to the degree to which Bob is in the extension of ‘is bald’; ‘Bill loves Ben’ is true to the degree to which Bill stands in the loving relation to Ben; etc. For quantiﬁed sentences, saying that everything is P is treated as saying that this thing is P, and this thing is P, and this thing is P, . . . for everything in the domain; and saying that something is P is treated as saying that this thing is P, or this thing is P, or this thing is P, . . . for everything in the domain. This basic idea of universal quantiﬁcation as generalized conjunction, and existential quantiﬁcation as generalized disjunction, is made precise in the following way. Let Ax a be the sentence obtained by writing a in place of all free occurrences of x in A , a being some constant that does not occur in A ; and given an interpretation M with domain M, let Mao be the interpretation which is just like M except that in it the constant a is assigned the denotation o. Then: • [∃xA ]M = {[A a] a : o ∈ M} x Mo • [∀xA ]M = {[Ax a]Mao : o ∈ M} The idea here is that to ﬁnd out the truth value of, say, ∀xPx, we consider the sentence Pa, and we ask what truth value it would have if everything about our interpretation were the same, except that a denoted this object in the domain; we note this truth value. Then we ask what truth value Pa would have if everything about our interpretation were the same, except that a denoted this other object in the domain; we note this truth value. And so on, for all objects in the domain. We now have a set of truth values—the ones we noted down along the way. To get the truth value of ∀xPx, we apply our inﬁmum operation to this set of truth values, yielding another truth value. The analogy with conjunction enters in the

66  fact that is the generalization to arbitrary sets of truth values of our ∧ operation on pairs of truth values.⁴¹ In sum: starting from the algebra of fuzzy truth values, we gain a useful perspective from which to see the links between fuzzy logic, set theory, and model theory. As in the classical case, we see that with this algebra in hand, everything else comes in a natural progression. Furthermore, it is now clear that although we start from different algebras of truth values, the way in which we proceed from our starting point is the same in both cases. Finally, both pictures also agree in their addition at the end—in order to yield a notion of truth simpliciter, given only a notion of truth on an interpretation—of the idea of there being a unique intended interpretation of any discourse. Recall our checklist of the main elements of the classical view. All our many-valued views differ from the classical view—and from each other—only over point 1a: they replace the classical algebra of truth values with an alternative algebra —with different alternatives in the case of different many-valued theories. How does the fuzzy view solve the Sorites paradox? What are the main objections to the fuzzy picture? I shall discuss these questions, and many more, in Part III. But before moving to our next approach to vagueness, I shall ﬁll out my presentation of the fuzzy picture with a discussion of one further matter. ⁴¹ One might think that as in the case of the deﬁnitions of union and intersection for arbitrary families offuzzy sets, in order for these truth deﬁnitions for quantiﬁed sentences to work, we need S and S need to be well-deﬁned for an arbitrary set S of truth values—in other words, the lattice of truth values must be complete. This time, however, completeness is a sufﬁcient not necessary but condition for our deﬁnitions to work. In order for the deﬁnitions to work, S and S need not be deﬁned for every set S of truth values: they need only be deﬁned for every set S of truth values which we might arrive at by taking a closed wf ∀xA or ∃xA , stripping off the quantiﬁer, replacing any free occurrences of x by a new constant a, and then putting one truth value, viz. [Ax a]Mao , into S for eachobject o in the domain. Thus the number of sets S of truth values for which we need S and S to be deﬁned will not be greater than the number of wfs of our language. (NB: We are talking about the number of sets of truth values, not the number of truth values in any of these sets.) If (as is standardly the case, and is the case in my presentation here) there are countably many wfs, then (given that we have uncountably many fuzzy truth values) the number of sets S of truth values for which we need S and S to be deﬁnedwill be less than the cardinality of the power set of the set of truth values, and thus we do not need S and S to be well-deﬁned for arbitrary sets S of truth values. Thus, lattice completeness gives us more than we actually need. Getting by on just what we do need, without the completeness requirement, requires some effort (see Restall 1994, §4.1, who draws on Brady 1988). I shall not go into the details, because we need our truth values not just for logic, but also for set theory—and there, as we have seen, we do need the lattice of truth values to be complete.

      

67

2.2.1.1 Operations on Fuzzy Truth Values I have so far mentioned the basic operations ∨, ∧, and on the fuzzy truth values, where x ∨ y is the supremum of x and y, x ∧ y is the inﬁmum of x and y, and x = 1 − x. Let us now consider some other operations. First, negations. A fuzzy negation c is a function from [0, 1] to [0, 1]. There are four noteworthy properties that such a function might possess: 1. 2. 3. 4.

c(0) = 1 and c(1) = 0 If x ≤ y, then c(x) ≥ c(y) c is continuous c(c(x)) = x

Earlier (p. 61) I mentioned that properties 2 and 4 are desirable; as a matter of fact, if a function c : [0, 1] → [0, 1] possesses these properties, then it possesses the other two as well (Klir and Yuan 1995, 52). In the fuzzy literature, conditions 1 and 2 are taken as minimal, and negations which satisfy only these two conditions are studied; notable examples are negations of the threshold type, of which there is one for each value of the parameter t ∈ [0, 1): 1 for x ≤ t c(x) = 0 for x > t As for negations which satisfy all four conditions, the standard fuzzy negation is not the only example. Other notable examples are Sugeno negations, of which there is one for each value of the parameter λ ∈ (−1, ∞): cλ (x) =

1−x 1 + λx

and Yager negations, of which there is one for each value of the parameter w ∈ (0, ∞): cw (x) = (1 − xw )1/w Another notable negation is the Gödel negation:⁴² 1 if x = 0 c(x) = 0 otherwise ⁴² See e.g. Malinowski 1993, 89, and Hájek 1998, 31. For some more details on fuzzy negations, see Klir and Yuan 1995, 51–61, and Nguyen and Walker 2000, 100–8.

68  Fuzzy conjunctions and disjunctions are functions i and u from [0, 1] × [0, 1] to [0, 1]. Some basic properties which we would like them to have are: 1. 2. 3. 4.

i(x, 1) = x and u(x, 0) = x i(x, y) = i(y, x) and u(x, y) = u(y, x) i(x, i(y, z)) = i(i(x, y), z) and u(x, u(y, z)) = u(u(x, y), z) If y ≤ z, then i(x, y) ≤ i(x, z) and u(x, y) ≤ u(x, z)

It turns out that functions with these properties had already been studied in connection with the theory of statistical metric spaces, where they were referred to as t-norms (or triangular norms) and t-conorms respectively.⁴³ There are further conditions which t-norms and t-conorms may or may not satisfy. One condition which was mentioned earlier is idempotence: i(x, x) = x and u(x, x) = x. It turns out that the standard fuzzy conjunction is the only idempotent t-norm, and the standard fuzzy disjunction is the only idempotent t-conorm (Klir and Yuan 1995, 63, 77). Nevertheless, other t-norms and t-conorms are studied in the literature; notable examples are Yager t-norms, of which there is one for each value of the parameter w ∈ (0, ∞): iw (x, y) = 1 − min(1, [(1 − x)w + (1 − y)w ]1/w ) and Yager t-conorms, of which there is one for each value of the parameter w ∈ (0, ∞): uw (x, y) = min(1, (xw + yw )1/w ) Other notable conjunctions are the Łukasiewicz conjunction:⁴⁴ i(x, y) = max(0, x + y − 1) and the product conjunction:⁴⁵ i(x, y) = xy Fuzzy conditionals are functions m from [0, 1] × [0, 1] to [0, 1]. Notable examples include the Łukasiewicz conditional: 1 if x ≤ y m(x, y) = 1 − x + y otherwise ⁴³ This literature begins with Menger 1942. ⁴⁴ The terminology is from Hájek (1998, 28 and ch. 3); as Hájek (1998, 277) notes, ‘‘what we call ‘Łukasiewicz conjunction’ seems never to appear explicitly in [Łukasiewicz’s] writings’’. ⁴⁵ See e.g. Goguen 1968–9, 346–7, and Hájek 1998, 28 and ch. 4. For some more details on fuzzy conjunctions and disjunctions, see Klir and Yuan 1995, 61–83, and Nguyen and Walker 2000, 83–110.

       the Gödel conditional:⁴⁶ m(x, y) = and the Goguen conditional: m(x, y) =

69

1 if x ≤ y y

otherwise

y/x if x ≥ y and x > 0 1

otherwise

I shall discuss the conditional further in §5.5.1. We have now considered separately a number of operations on the fuzzy truth values which might be regarded as candidate interpretations of the logical connectives (negation, conjunction, disjunction, and the conditional). Some groupings of these operations hang together more naturally than others. There are two main types of coherence that a group of operations might have. First, a trio of a negation c, a conjunction i, and a disjunction u hang together in one sense if i and u are duals with respect to c, i.e. the De Morgan laws hold: c(i(a, b)) = u(c(a), c(b)) c(u(a, b)) = i(c(a), c(b)) Our basic operations ∨, ∧, and hang together in this sense.⁴⁷ Second, given a conjunction i, the conditional m is said to result from i by residuation (or to be a residuum of i) if: m(x, y) = {z : i(x, z) ≤ y} and where m is a residuum of i, the negation c is said to be a precomplement of i if: c(x) = m(x, 0) Notable trios of operations where the conditional and the negation are, respectively, the residuum and the precomplement of the conjunction are:⁴⁸ (Łukasiewicz conjunction, Łukasiewicz conditional, ) (∧, Gödel conditional, Gödel negation) (product conjunction, Goguen conditional, Gödel negation) ⁴⁶ See e.g. Rescher 1969, 44, and Malinowski 1993, 89, 100. ⁴⁷ This trio of operations is not unique in this respect; see e.g. Klir and Yuan 1995, 83–7. ⁴⁸ See Hájek 1998, 27–32.

70  Given a set of two truth values, there are only 22 = 4 unary operations 2 on that set, and only 2(2 ) = 16 binary operations. On the set of fuzzy truth values [0, 1], by contrast, there are uncountably many n-place operations for each n ≥ 0.⁴⁹ This structural richness makes fuzzy logic an interesting object of study from a technical point of view. From the point of view of studying vagueness, on the other hand, while I would not want to rule out the possibility that further down the track, distinctions amongst different connectives might shed light on important vagueness-related issues, nevertheless I cannot see, at the present stage of enquiry, that there is anything to be gained by moving beyond the basic fuzzy operations ∨, ∧, and . Therefore, with the exception of the discussion of the conditional in §5.5.1, I shall not discuss other operations further in this book. 2.2.2 Worldly Vagueness? Many-valued approaches locate vagueness not in the relationship between language and the world, but in the world itself. The idea is that when I speak vaguely, I invoke a unique intended interpretation, and on this interpretation, each name in my language has a unique referent, each predicate has a unique extension, and so on. The vagueness is all within this intended model. For example, the particular property that my predicate ‘is red’ picks out is inherently indeterminate: some objects possess it, some do not possess it, and others neither possess it nor fail to possess it (i.e. its characteristic function maps these objects to an intermediate truth value—to ∗ on the three-valued approach, or to a value in (0, 1), on the fuzzy approach). On this approach, vagueness is found in the world itself, not in the relationship between language and the world. Well, I really need to be more subtle here. In fact, I do not think that the three-valued approach accommodates vagueness at all, whereas I do think that the fuzzy approach does. I shall argue for this claim in §4.2. What I should say at this point is that all the many-valued approaches keep the referential links between language and the world as determinate as they are on the classical view. Where they differ from the classical view is in their picture of the world itself, which now contains indeﬁnite properties. Whether this indeﬁniteness is enough to warrant calling these properties ⁴⁹ Cf. Novák 1998, 80.

      

71

vague is a matter to which we shall come. The point for now is that the divergence from the classical picture comes on the world side, not in the relationship between language and the world. Forbes (1983, 245) writes: ‘‘Note that on [the fuzzy] approach, vagueness resides entirely in concepts. The objects in [the domain of the fuzzy membership function] are perfectly determinate and the fuzzy sets themselves also have exact identity conditions: two such sets are the same iff the same things are members of each to the same degree.’’ Forbes is clearly assuming that worldly vagueness requires vague objects and/or vague identity. I disagree. If the fuzzy view is correct, there exist vague properties and relations alongside the precise ones (indeed, the precise ones are those special cases of the vague ones that map all n-tuples to 0 or 1)—and this is one perfectly genuine sort of worldly or metaphysical vagueness. In order to describe the world, it is not enough just to list the objects it contains: one must also describe their properties and the relations they bear to one another. Metaphysical vagueness is often taken to mean vagueness in objects—but if there is vagueness in properties or relations, this will also be a kind of metaphysical or worldly vagueness. On the fuzzy semantic picture, there are properties and relations in the world that are vague. Contra Forbes, this is not vagueness in concepts.

2.3 Truth Gaps (Deny 1b) We have been examining one way of departing from the classical picture: unplug the algebra of classical truth values, plug in a different algebra, and then proceed as before. We now turn to a second way of departing from the classical view. We leave the classical algebra of truth values untouched, and depart from the classical picture at two other points. First, we allow the propositional part of the interpretation function—the part which assigns a truth value to each simple sentence—to be partial (rather than total). That is, some simple sentences get assigned no truth value at all. (We still say that the interpretation function is total when restricted to names—i.e. every name gets a referent—and when restricted to predicates—i.e. every predicate is assigned an extension.) Second, we allow the characteristic functions of sets to be partial. Recall that if the characteristic function of the set S assigns the object x the value 1, this means x is in S, and if it

72  assigns x the value 0, this means x is not in S. We now allow that S might assign x no value at all—which we interpret as meaning that x is neither in S nor not in S. Thus, we now have truth value gaps for simple sentences, and we have gappy sets.⁵⁰ Given these changes, we now want to run the rest of the classical story with no further changes—or, where this is not possible, with only the minimal changes required to accommodate the two basic changes which we have just made. One immediate upshot is that we get truth gaps for atomic sentences, as well as for simple sentences. For recall the classical story, in which the truth deﬁnition for an atomic wf P(a1 , . . . , an ), which consists of an n-ary predicate P followed by n individual constants a1 , . . . , an , is as follows: [P(a1 , . . . , an )] = I(P)(I(a1 ), . . . , I(an )) Recall the idea here: each individual constant a is assigned an object I(a) in the domain, and P is assigned an n-ary relation on the domain, that is a function I(P) from the set of all n-tuples of members of M to our set of truth values {0, 1}. Whatever value this function assigns to the n-tuple (I(a1 ), . . . , I(an )), this value is the truth value of the sentence P(a1 , . . . , an ). This function may now be partial. So suppose it assigns nothing to the n-tuple (I(a1 ), . . . , I(an )); then the atomic sentence P(a1 , . . . , an ) will have no truth value. More concretely, ‘Bob is bald’ is true if Bob is in the extension of ‘is bald’ (i.e. the characteristic function of the extension of ‘is bald’ assigns 1 to the referent of ‘Bob’), is false if Bob is not in the extension of ‘is bald’ (i.e. the characteristic function of the extension of ‘is bald’ assigns 0 to the referent of ‘Bob’), and is neither true nor false if Bob is neither in nor out of the extension of ‘is bald’ (i.e. the characteristic function of the extension of ‘is bald’ assigns nothing to the referent of ‘Bob’). This idea has a natural application in the case of vagueness—for it is a fairly natural thought about vague sentences such as ‘Bob is bald’, where Bob is a borderline case for baldness, that they are neither true nor false. As in the case of the three-valued logicians discussed in §2.2, this allows us to maintain an attractive parallelism between meaning and use, of the sort that the epistemicist was forced to reject. The difference this ⁵⁰ In principle, we could separate these aspects: we could, for example, consider a view according to which all simple sentences have a truth value, but some sets are partial. In practice, however, the two aspects go together very naturally, and we shall not explore the option of separating them.

      

73

time is that instead of supposing—with the three-valued logicians—that sentences about borderline cases have a third truth value, we now suppose that they simply have no truth value. Instead of positing extra truth values which some sentences might have, we stick with our two truth values 1 and 0, but we suppose that some sentences might possess neither of them.⁵¹ There is an open question remaining. The classical story about propositional logic said that • • •

[A ∨ B ] = [A ] ∨ [B ] [A ∧ B ] = [A ] ∧ [B ] [¬A ] = [A ]

And the story about set theory said that • • •

( f ∪ g)(x) = f (x) ∨ g(x) ( f ∩ g)(x) = f (x) ∧ g(x) f (x) = f (x)

Now, what are we to say about the truth value of a compound wf, say A ∨ B , if one or both of A or B lacks a truth value? And similarly, what are we to say concerning what the characteristic function of, say, S ∪ T assigns to x, if one or both of the characteristic functions of S and T assigns x nothing? We shall examine two different ways of approaching this issue—one in §2.3.1, the other in §2.4. We shall focus on the problem of assigning truth values to compound sentences, rather than on the analogous problem for set theory, because this has been the focus in the literature on vagueness. It should be noted, however, that for each of the approaches we shall discuss, perfectly analogous things could be said in regards to the set-theoretic problem. ⁵¹ In the literature, many writers who take this sort of approach implement it by saying that a predicate, instead of being assigned an extension (as on the classical view), is assigned a pair of an extension and an antiextension, which need not exhaust the domain (Kripke 1975; Soames 1999). I say instead that a predicate is assigned a set as its extension, both on the new view and on the classical view, but the difference is that sets may now be partial, that is, their characteristic functions may be partial, rather than total (as on the classical view). Given the way that I presented the classical picture, my formulation is smoother. The two formulations are, however, equivalent. The set of things assigned 1 by the characteristic function of my extension is their extension; the set of things assigned 0 by the characteristic function of my extension is their antiextension; and the set of things assigned nothing by the characteristic function of my extension is the set of things which are in neither their extension nor their antiextension.

74  2.3.1 The Recursive Approach One core part of the classical view—item 1c on our checklist—was the idea that the truth values of wfs of the forms A ∨ B , A ∧ B , and ¬A are determined solely by the truth values of A and B . The ﬁrst way of handling our problem sticks as closely as possible to this idea, given that, now, A and B might not have truth values at all. What we need to do is give a recursive deﬁnition of the truth values (or lack thereof) of such compounds, given the truth values (or lack thereof) of their components. More speciﬁcally, the way this is usually done is to give truth tables which tell us the values that compound sentences are to have, given the values of their components. Now, all the same considerations arise here as arose in connection with the search for three-element algebras of truth values in §2.2. Thus, one idea is that any compound proposition with a component that has no truth value should itself have no truth value. This leads to Kleene’s (1952) weak truth tables (see Table 2.5). Note that Table 2.5 is not incomplete. I have used blanks, rather than a symbol such as ∗, to indicate truth value gaps, in order to mark the difference between what we are doing here and what we did in §2.2. In §2.2 we were proposing a new algebra of truth values, with a third object ∗ in it, as well as 1 and 0. Here, the classical algebra of truth values is unchanged. What changes is the assignment of truth values to compound wfs. For we can no longer say Table 2.5. Kleene’s weak truth tables A

B

A ∨B

A ∧B

¬A

1

1

1

1

0

1 1

0

1

0

1

0 0

1

1

0

0 0

1

0

0

0

      

75

that [A ∨ B ] = [A ] ∨ [B ] etc.—because while the algebraic operation on the right is unchanged, some sentences do not get assigned a value in the algebra at all, and where this happens, the clause just stated tells us nothing about the value which should be assigned to compounds made up from such sentences. The second idea is as follows. If a compound sentence has components which lack truth values, one should look at all possible ways of assigning truth values to those components, and in each case calculate the truth value of the compound sentence (using the classical rules: remember, we have not altered the classical algebra of truth values, we have simply supposed that some sentences might have no truth value; but where every sentence does have a truth value, we proceed classically). If the answer is 1 every time, the compound sentence gets the value 1; if the answer is 0 every time, the compound sentence gets the value 0; if the answer is 1 sometimes and 0 at other times, the compound sentence gets no truth value. This leads to Kleene’s (1952) strong truth tables (see Table 2.6). Once again, the rationale just used may sound like supervaluationism, but the outcome is in fact very different. As was desired at the outset, the view just presented has the property that the truth values of wfs of the forms A ∨ B , A ∧ B , and ¬A are determined recursively, given the truth values of A and B . The supervaluationist assignment of truth values to compounds does not have this recursive structure, as we shall see in §2.4. Table 2.6. Kleene’s strong truth tables A

B

A ∨B

A ∧B

¬A

1

1

1

1

0

1 1

1 0

1

1

1

0

0 0

1

1

0 0

0

0

0

0

1

0

0

76  The third idea is to consider the ordering of sentences with respect to truth, and to say that a sentence with no truth value is truer than a false sentence and less true than a true sentence. As we would now expect, this rationale also leads to Kleene’s strong truth tables. There are many more views of this sort. For there are as many different gappy truth tables for compound wfs based on the one underlying classical algebra of truth values as there are three-membered algebras of truth values which agree with the classical algebra where only 1 and 0 are involved—and as we saw, there are 311 = 177, 147 of the latter. We shall not consider any more of them in detail. The remaining parts of the gappy story—for example, the treatment of quantiﬁers—are perfectly parallel to the corresponding parts of the three-valued story. The response to the Sorites paradox on the part of the recursive truth gap view is perfectly parallel to the three-valued story. Likewise, the major objections to the recursive truth gap view are perfectly parallel to the major objections to the three-valued view (and similarly for the responses thereto). So too when it comes to the question of locating vagueness in the relationship between language and the world, or in the world itself: here again, the comments made about the three-valued view apply also to the truth gap view (see §2.4.3 for further discussion).

2.4 Supervaluationism (Deny 1c) The view that the truth values of compound sentences are not determined in a recursive way from the truth values of simpler sentences (i.e. the denial of 1c) can be combined naturally either with the supposition of additional truth values (i.e. the denial of 1a) or with the supposition of truth gaps (i.e. the denial of 1b). In this section we look at one very prominent strategy for denying 1c: supervaluationism. We start by presenting this strategy in combination with truth gaps. Let us return to the point in the development of the gappy account at which we allow the interpretation function to assign nothing to simple sentences, and allow the characteristic functions of sets to be partial. That is, we have a partial interpretation Mp = (M, I p ) of our standard ﬁrst-order language, which consists in:

       • •

77

a nonempty set M (the domain) a partial function I p which assigns: – a truth value (in our classical algebra of truth values) to zero or more of the propositional constants in our language – an object in M to every individual constant in our language – an n-ary relation on M to every n-ary predicate in our language—but the n-ary relation assigned may have a characteristic function which is partial.

We now face the issue of what to say about the truth values of compound sentences. We have examined one type of approach to this question—the recursive approach. We now examine an alternative type of approach: the supervaluational approach. We shall say that a classical interpretation extends a partial interpretation if it agrees with all the assignments that the partial interpretation makes, and also makes an assignment everywhere that the partial interpretation leaves a gap. More precisely, the classical interpretation M = (M , I) extends the partial interpretation Mp = (M, I p ) just in case all of the following conditions hold: • • • •

M = M (i.e. the classical and the partial interpretation have the same domain). For every propositional constant q in the language, if I p assigns q a truth value, then I assigns q that same truth value. For every individual constant a in the language, I(a) = I p (a). For every n-ary relation R in the language and every n-tuple x in M n , if the characteristic function of I p (R) assigns x a value, then the characteristic function of I(R) assigns x that same value.

We can now give a statement of our basic strategy for assigning truth values to compound sentences in our partial interpretation Mp (one important subtlety will be added to the basic strategy below). A compound sentence gets assigned the value 1 on Mp if it comes out as having the value 1 on every classical interpretation which extends Mp ; it gets assigned the value 0 on Mp if it comes out as having the value 0 on every classical interpretation which extends Mp ; and it gets assigned no truth value on Mp if it comes out as having the value 1 on some classical interpretations which extend Mp and 0 on others. The function just described—which assigns truth values

78  to sentences on our partial interpretation, according to the truth values they receive on classical extensions of our partial interpretation—is called a supervaluation. So far, I have mentioned only simple sentences and compound sentences—but what about atomic sentences, i.e. sentences formed from an n-place predicate followed by n names? There are two options here. First, we may regard atomic sentences as getting their truth values directly from the partial interpretation, without any role being played by the classical extensions: the truth value of P(a1 , . . . , an ) on the partial interpretation Mp is the value assigned by the function I p (P) to the n-tuple (I p (a1 ), . . . , I p (an )); thus, in particular, P(a1 , . . . , an ) will have no truth value where I p (P) assigns nothing to this n-tuple. Second, we may regard atomic sentences as getting their truth values from the supervaluation, just as compound sentences do: the truth value of P(a1 , . . . , an ) on the partial interpretation Mp is 1 (0) if the truth value of this wf is 1 (0) on every classical interpretation which extends Mp ; P(a1 , . . . , an ) has no truth value on the partial interpretation Mp if it has the value 1 on some classical extensions and the value 0 on others. It is not hard to see that these two methods of assigning truth values to atomic sentences yield exactly the same truth value assignments. It may therefore seem as though we can pick whichever option takes our fancy, and that nothing hangs on the fact that two different options are available. In fact I think consideration of the two options leads to interesting insights into supervaluationism—but we will in a position to see this more clearly once some other material has been covered, and so I shall postpone further discussion of this issue until the end of §2.4.1. The reader may notice that the supervaluational procedure just outlined is superﬁcially similar to the rationale behind the strong Kleene truth tables. But as I mentioned earlier, the outcome is in fact very different—for the supervaluational method of assigning truth values to compound sentences is not recursive. For example, suppose that the simple sentence q has no truth value on our partial interpretation. There is at least one classical interpretation which extends our partial interpretation on which q is true, and at least one on which q is false. On the former, ¬q is false, and on the latter, ¬q is true. Thus, by the foregoing account, ¬q is neither true nor false on the partial interpretation. Consider q ∨ ¬q. This is a classical tautology, i.e. true on every classical interpretation. Thus, in particular it is true on every classical interpretation which extends our partial interpretation. Thus, by the

      

79

foregoing account, q ∨ ¬q is true on the partial interpretation. Suppose that, like q, the simple sentence r has no truth value on our partial interpretation. There is, then, at least one classical extension of our partial interpretation on which q and r are both true, one on which they are both false, one on which q is true and r is false, and one on which q is false and r is true. Now consider q ∨ r. It will be true on the ﬁrst, third, and fourth of these four types of classical extension, and false on the second. Thus by the foregoing account, q ∨ r is neither true nor false on the partial interpretation. But now compare q ∨ ¬q and q ∨ r. Both have the logical form of a disjunction, with each disjunct neither true nor false on our partial interpretation. Yet the former is true on our partial interpretation, and the latter is neither true nor false. Thus, on the supervaluationist account, the truth values of compound sentences are not determined solely by the truth values of their components. The formal technique just described for assigning truth values to compound sentences on a partial interpretation is due to van Fraassen (1966). Given an extra twist, it has a natural application in the context of vagueness, as Fine (1997 [1975]) showed.⁵² The extra twist is the notion of an admissible extension of a given partial interpretation. We approach this notion via the idea of a precisiﬁcation of vague language. We often need to precisify some part of vague language. For example, we decide that an adult shall be someone over 18 years old, for the purposes of the law; we decide that a reasonable catch of ﬁsh shall be ten or less ﬁsh (per person per day) for purposes of setting recreational ﬁshing quotas; we decide that an individual (with no dependents) shall be in poverty if he has an annual income of less than $10,488, for purposes of compiling statistics and setting social security beneﬁts; and so on. In general, different precisiﬁcations of the same vague term may seem equally legitimate: the point is that we need some precisiﬁcation, not any particular one. For example, some jurisdictions set the cut-off for adulthood at 18 years of age, some at 21. However, none set the borders of the set of adults in anything like the following ways: •

A person is an adult if and only if his or her age is greater than or equal to 18 and divisible by 2 (i.e. the adults are those of ages 18, 20, 22, . . .).

⁵² Apart from Fine, presentations of supervaluationist approaches to vagueness include Mehlberg 1997 [1958]; Dummett 1997 [1975]; Kamp 1975; Pinkal 1995; Bennett 1998; and Keefe 2000, chs. 7–8.

80  • •

A person is an adult if and only if his or her age is greater than or equal to 18 and his or her favourite colour is green. A person is an adult if and only if either his or her age is greater than or equal to 21 and it is a weekday, or his or her age is greater than or equal to 18 and it is a weekend.

Yet note that if we have an initial partial interpretation which assigns ‘adult’ a gappy extension corresponding to our ordinary vague use of that term, then for each of the deﬁnitions just given, there is a classical extension of this partial interpretation which closes the gaps in the application of ‘adult’ in accordance with that deﬁnition. For all that we have required of a classical extension is that it must close the gaps in the original partial interpretation: we have imposed no further constraints on how the gaps must be closed.⁵³ Yet the foregoing example shows clearly that not any way of closing the gaps gives us something that we would ordinarily regard as a legitimate precisiﬁcation of vague language. An admissible extension of a given partial interpretation, then, will be one which corresponds to a legitimate precisiﬁcation of the vague language whose intended interpretation was given by that partial interpretation. There are various constraints on legitimate precisiﬁcation. One obvious one is that we have leeway over how to decide the borderline cases, but it is not admissible to stipulate, for example, that a man with no hair is not bald, or that a man with lots of hair is bald. This constraint is already built into the idea of a classical interpretation being an extension of a partial interpretation: it makes assignments where the partial interpretation makes none; it does not disrupt any of the assignments which the partial interpretation does make. But there are other constraints which are not built into the very idea of a classical extension; we stipulate that an extension is admissible if it satisﬁes these further constraints. For example: •

•

Suppose that x is a borderline case of both ‘red’ and ‘orange’. If we precisify ‘red’ in such a way that x comes out as red, then we must not precisify ‘orange’ in such a way that x comes out as orange. Suppose that x and y are borderline cases of ‘tall’, and y is taller than x. If we precisify ‘tall’ in such a way that x comes out as tall, then y must also come out as tall. ⁵³ Cf. n. 3 above.

       •

81

Suppose that Bill is 16, Ben is 18, and Bob is 20. If we precisify ‘juvenile’ and ‘adult’, it must turn out that each of Bill, Ben, and Bob falls in exactly one of these categories, and it must not be the case that Bill and Bob fall in one category, while Ben falls in the other.

The ﬁrst constraint concerns how one object should be classiﬁed relative to several predicates; the second constraint concerns how several objects should be classiﬁed relative to one predicate; and the third constraint concerns how several objects should be classiﬁed relative to several predicates. This is just a tiny sample of the constraints on admissible precisiﬁcation, not a full list. But it should be enough to give the idea. Now suppose we have a partial interpretation which corresponds to our actual use of a vague language. A classical extension of this partial interpretation will (to repeat) be admissible just in case it corresponds to a legitimate precisiﬁcation of that vague language. Note that we have deﬁned the notion of an admissible classical extension of a given partial interpretation. Do not call the admissible extension an admissible interpretation of the given vague language—this will lead to confusion later. Our delineation of a certain class of classical interpretations as admissible extensions is always relative to a given partial interpretation, which is assumed to correspond to a body of usage of vague language. That is, there is a unique intended partial interpretation of some vague language, and then it has many admissible classical extensions. These correspond to legitimate precisiﬁcations of the vague language whose intended interpretation was given by the original partial interpretation—they are not themselves intended interpretations of the vague language. (They could not be, because they remove all its vagueness! But more on this below—see especially §4.4.) Returning to supervaluationism as applied to vagueness, the strategy for assigning truth values to compound sentences in our partial interpretation Mp is now as follows. A compound sentence gets assigned the value 1 on Mp if it comes out as having the value 1 on every admissible classical extension of Mp ; it gets assigned the value 0 on Mp if it comes out as having the value 0 on every admissible classical extension of Mp ; and it gets assigned no truth value on Mp if it comes out as having the value 1 on some admissible classical extensions of Mp and 0 on others. The intuitive gloss on this is that a vague sentence is true simpliciter if it would be true no matter how its vagueness were removed; it is false simpliciter if it would be false no matter how its vagueness were removed; and it is

82  neither true simpliciter nor false simpliciter if there are legitimate ways of removing its vagueness that would render it true, and others that would render it false. One point of interest about the supervaluationist approach is that, although its semantics is non-classical, it yields classical logic. That is, any classical tautology is a supervaluationist tautology, and vice versa, and any inference which is valid in classical logic is valid according to the supervaluationist, and vice versa. Let’s be more precise about this. I am assuming here—as throughout this chapter—that we are dealing with our standard ﬁrst-order language; if we enrich the language in certain ways, then the following result does not hold.⁵⁴ Where W is the set of wfs of our language, a consequence relation Cx is a subset of PW × W . Where is a set of wfs and A is a wf, we say |=x A iff (, A ) ∈ Cx . Any A for which (∅, A ) ∈ Cx is called an x tautology. Now, we deﬁne the classical consequence relation Cclass thus: (, A ) ∈ Cclass iff there is no classical interpretation on which all the members of are true and A is false. We deﬁne the supervaluationist consequence relation Csval thus: (, A ) ∈ Csval iff there is no partial interpretation on which all the members of are true and A is not true (i.e. is false, or is neither true nor false).⁵⁵ It then turns out that Cclass = Csval .⁵⁶ ⁵⁴ As Hyde 1997, 652 emphasizes, the result also does not hold—even for our standard ﬁrstorder language—if we consider the multiple-conclusion consequence relation, in place of the usual multiple-premiss-single-conclusion consequence relation considered in what follows. See §2.4.2 below. ⁵⁵ It has been said that supervaluationists have a choice as to how deﬁne validity (cf. Dummett 1997 [1975], 108, and Fine 1997 [1975], 137, and for discussion see e.g. Williamson 1994, 147–8): they can say what we have just said (‘global validity’) or they can say that an argument is valid just in case there is no classical extension of any partial interpretation on which all the members of are true and A is not true (‘local validity’). It should be obvious that the latter deﬁnition in fact has no plausibility at all—i.e. our deﬁnition is obviously the correct one. However, we shall see in §2.5 that there is a view quite different from supervaluationism, which I call plurivaluationism—and for this view, the obvious deﬁnition of validity corresponds to the latter idea. Plurivaluationism and supervaluationism have been conﬂated in the literature, leading to the illusion that there is one view of vagueness which has two natural options concerning how to deﬁne validity. ⁵⁶ Proof. (i) Suppose (, A ) ∈ Cclass . Then (A) there is no classical interpretation on which all the members of are true and A is false. Now suppose we have an arbitrary partial interpretation on which every member of is true. This means that on every extension of our partial interpretation, every member of is true. Then by (A), on every extension of our partial interpretation, A is true. Hence A is true on our partial interpretation. Thus on any partial interpretation on which all the members of are true, A is true; i.e. (, A ) ∈ Csval . (ii) Suppose (, A ) ∈ / Cclass . Then there is a classical interpretation on which all the members of are true and A is false. But note that a classical interpretation is a partial interpretation: it is a special case of the latter, where the only classical interpretation which extends it is itself. So, there is a partial interpretation on which all the members of are true and A is false; hence (, A ) ∈ / Csval .

      

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How will the supervaluationist view handle the Sorites paradox? Suppose we remove one grain at a time from a 10,000-grain pile of sand, until we have one grain left. Call the pile with n grains ‘pile n’. Our ordinary, vague usage of the term ‘heap’ corresponds to a partial interpretation on which, let us say, piles 100 through 10,000 are assigned the value 1 by the characteristic function of the extension of ‘is a heap’, piles 1 through 10 are assigned 0, and piles 11 through 99 are assigned no value. (I am talking here about the usage of the whole community of speakers—recall the discussion on pp. 58–9.) Now, for each 11 ≤ n ≤ 100, there is an admissible extension—which we shall call extension N —of this interpretation on which piles 1 through n − 1 are assigned 0, and piles n through 10,000 are assigned 1. (There are also extensions on which, for example, piles 1 through 10, 35, 39, and 42 are assigned 0, and the rest are assigned 1, but these are not admissible.) Let the sentence ‘the cut-off is at n’ mean that piles 1 through n − 1 are non-heaps, and piles n through 10,000 are heaps. Consider the inductive premiss of the Sorites argument associated with this setup: ‘For every 2 ≤ n ≤ 10,000, if n is a heap, then so is n − 1.’ This sentence is false on each of our admissible extensions: on interpretation N, where the cut-off is at n, the corresponding instance of this universal claim is false, and hence the universal claim itself is false. Thus, according to the supervaluationist approach, the inductive premiss gets the value 0 in our partial interpretation. This, then, is the mistake in the paradoxical reasoning: the inductive premiss is false. So why is the paradox compelling? Consider the claim ‘The cut-off is at n’. For each 11 ≤ n ≤ 100, this claim is true on exactly one of our admissible extensions. Thus, in our partial interpretation, it is neither true nor false. There is, then, no point n such that we can truly say ‘The cut-off is at n’. So far, so good. But here, says the supervaluationist, is where we make a natural mistake, and so get drawn into the paradox. We conclude from the foregoing that ‘There is no cut-off’ is true—or in other words, we conclude that the inductive premiss is true. But here we are mistaken. ‘There is no cut-off’ is false on every admissible extension (for each extension puts the cut-off somewhere—just at a different point on different extensions), and hence—according to the supervaluationist—false simpliciter. Thus, the paradox is compelling because we tend to assume that because the cut-off is not here, or here, or here, . . . through all the possible positions where the cut-off might be, it follows that ‘There is no cut-off’ is true—that is, that the inductive premiss is true. On the supervaluationist

84  account, this does not follow. There is no point n such that we can truly say ‘The cut-off is at n’, and yet ‘There is a cut-off’ is true simpliciter —and so the inductive premiss, which denies this, is false simpliciter. The major objection unique to supervaluationism arises directly from the solution to the Sorites just presented. I call it the problem of missing witnesses and counterexamples, or just the ‘problem of missing witnesses’ for short. Intuitively, if a universal claim is false, then there must be a counterexample which makes it false, and if an existential claim is true, then there must be a witness which makes it true. For example, if ‘Everyone is female’ is false, there must be some particular person who is not female, and if ‘Someone is female’ is true, there must be some particular person who is female. But on the supervaluationist view, these relationships between quantiﬁed claims and instances do not hold. Consider the inductive premiss of a typical Sorites argument: For every x in the series, if Fx then Fx . As we have seen, the supervaluationist account has this come out false. This would naturally lead us to suppose that one of the following sentences is true, to give us a counterexample to the universal claim: Fx1 but not Fx2 Fx2 but not Fx3 .. . Fxn−1 but not Fxn However, according to the supervaluationist, none of these is true. Consider the sentence ‘Fx but not Fx ’. There are several possible cases: •

•

If x and x are both clear cases of F, or if x and x are both clear countercases of F, then the sentence is false on every admissible extension and hence false in the partial interpretation. If x is a clear case of F and x is a borderline case, or if x is a borderline case of F and x is a clear countercase, or if x and x are both borderline cases of F, then the sentence is true on some admissible extensions and false on others, and hence neither true nor false in the partial interpretation.

Thus the sentence will be either false, or neither true nor false, but will never be true. So we have a false universal claim without a counterexample

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to make it false. Likewise, if we assert ‘There is an x such that Fx but not Fx ’, this is true in the partial interpretation, because it is true on every admissible extension; yet, as we have seen, there is no sharp line to make it true. Thus the supervaluationist view does not accord with our ordinary use of quantiﬁers: in particular, it violates expected relationships between quantiﬁed claims and their instances.⁵⁷ The major objection unique to the recursive version of the truth gap view is the truth-functionality objection. The objection has been pushed most strongly by Fine, and has also been endorsed by Williamson (1994, 135–8), Urquhart (1986, 113), and others. Fine took as the chief motivation for his supervaluationist theory, as against the recursive approach, the idea that while (for example) it may be unclear whether a particular point on the rainbow is red or orange—so that neither ‘Point x is red’ nor ‘Point x is orange’ is clearly true—it is certainly the case that ‘Point x is red or point x is orange’ is true, and it is certainly the case that ‘Point x is red and point x is orange’ is false. This is one example of a general phenomenon which he called ‘penumbral connection’: ‘‘the possibility that logical relations hold among indeﬁnite sentences’’ (Fine, 1997 [1975], 124). Fine argued that the phenomenon of penumbral connection poses a problem for the recursive approach. Consider three sentences: ‘Bob is bald’, ‘Point x is red’, and ‘Point x is orange’. Suppose that each of them lacks a truth value (or possesses a third truth value: the argument applies equally to the recursive version of the three-valued approach). Then, according to a recursive account, the following two sentences have the same truth value (or lack thereof): ‘Point x is red and Bob is bald’ and ‘Point x is red and point x is orange’. The problem, Fine thinks, is that this is just obviously wrong: the ﬁrst sentence should lack a truth value, but the second should be false. Similarly, where Bob is a borderline case of baldness—i.e. ‘Bob is bald’ lacks a truth value—the sentence ‘Bob is bald or Bob is not bald’ will lack a truth value on the recursive approaches we have examined, whereas on the supervaluationist approach it will be true. Again, Fine thinks that this supervaluationist assignment is just obviously, intuitively correct, while the recursive assignment is clearly wrong. ⁵⁷ Cf. e.g. Sanford 1976; Rolf 1984, 232; and Tappenden 1993, 564. An analogous problem arises at the level of sentential connectives (which is not surprising, given the analogies between disjunction and existential quantiﬁcation, and conjunction and universal quantiﬁcation): a disjunction can be true without either disjunct being true, and a conjunction can be false without either conjunct being false.

86  I have always found this puzzling. Consider ‘red’. If one indicates a point on a rainbow midway between clear red and clear orange and asks an ordinary speaker the following questions, then in my experience the responses are along the lines indicated: • • • •

‘‘Is the point red?’’ Umm, well, sort of. ‘‘Is the point orange?’’ Umm, well, sort of. ‘‘But it’s certainly not red and orange, right?’’ Well, no, it sort of is red and orange. ‘‘OK, well it’s deﬁnitely red or orange, right?’’ No, that’s what I’ve been saying, it’s a bit of both, the colours blend into one another.

These reactions ﬁt with the recursive assignments of truth values, not the supervaluationist assignments. Ordinary speakers hedge over ‘x is red’ and ‘x is not red’ when x is a borderline case of redness, and they hedge in just the same way over ‘x is or isn’t red’ and ‘x is and isn’t red’. I have found these reactions to be robust, over a range of examples. Where Bob is a borderline case of baldness, ordinary speakers do not think that ‘Bob is bald or he is not bald’ is clearly true, and they do not think that ‘Bob is bald and he is not bald’ is clearly false: they react to these two sentences with just the sort of hesitancy with which they react to the sentence ‘Bob is bald’; they regard all three sentences as equally dubious. Similarly for ‘heap’, ‘tall’, and so on.⁵⁸ We thus have two sets of intuitions on the table, one of which ﬁts naturally with the supervaluationist assignments of truth values, and one of which ﬁts naturally with the recursive assignments. But at least one other set of intuitions has also been reported in the literature.⁵⁹ According to this third set, when a is a borderline case of F, ‘a is F or a is not F’ is not assertible (which is, prima facie, in conﬂict with the supervaluationist view), but ‘It is not the case that a is F and that a is not F’ is assertible (which is, prima facie, in conﬂict with the recursive view). What are we to make of all this? There is much to say, but I will save this discussion for §5.5. For now, I think that any fair-minded person who steps back from her own favoured set of intuitions and even-handedly surveys the literature would have to agree—given ﬁrst that there are ⁵⁸ Machina 1976 has similar intuitions to mine. ⁵⁹ See Burgess and Humberstone 1987, 199–200; cf. also Tappenden 1993.

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conﬂicting (reports of the) data, and second that some of the data do not ﬁt (easily) with the supervaluationist view, some do not ﬁt (easily) with the recursive view, and some do not ﬁt (easily) with either view—that there is no simple, knockdown intuitive argument here either way (i.e. in favour of supervaluationism over the recursive approach, or vice versa). This is still news: for in some quarters it is thought that the supervaluationist’s penumbral connection argument against recursive approaches is a simple, knockdown argument from intuition. In light of the foregoing, I consider that position untenable—and even more so once we notice that the feature of the supervaluationist semantics which leads to the assignment of True to ‘Bob is or isn’t bald’ even when ‘Bob is bald’ and ‘Bob isn’t bald’ are assigned no truth value is the very same feature which leads to the missing witness problem. It is hard to claim—in the context of discussing truth-functionality—that this feature of supervaluationism gives it a clear intuitive advantage over recursive approaches, when it is the very feature that leads to the missing witness problem—the latter being a great intuitive disadvantage of supervaluationism, and one which the recursive approach does not share. The upshot, then, is that the debate between supervaluationist and recursive approaches will need to be settled either by a much more detailed argument from data about the assertibility of sentences about borderline cases, based on proper empirical foundations (see §5.5), or else by other means altogether.⁶⁰ 2.4.1 Supervaluationism and Additional Truth Values As mentioned at the outset, one can combine a supervaluationist approach to compound sentences with a many-valued—rather than two-valued but partial—approach to the interpretation of simple sentences and to the characteristic functions of sets. The three-valued case is straightforward: just as the recursive three-valued approach is perfectly analogous to the recursive truth gap approach, so too the supervaluationist three-valued approach is perfectly analogous to the supervaluationist truth gap approach.⁶¹ If we ⁶⁰ As far as the dialectic of this book is concerned, I present an argument against all non-degree theories of vagueness (Part II) and a number of arguments against supervaluationist degree theories (§2.4.1)—thereby leading us to recursive degree theories. I then return to a detailed discussion of the truth-functionality objection to recursive degree theories in §5.5. ⁶¹ We start with a three-valued interpretation, which consists in a nonempty domain M together with a (total) interpretation function I which assigns a truth value (1, 0, or ∗) to each propositional

88  have more than three truth values, however, then things can get more interesting on supervaluationist versions of the many-valued approach than they can on supervaluationist versions of the partial two-valued approach. In particular, suppose we have a fuzzy interpretation—i.e. a many-valued interpretation where the underlying algebra of truth values is the Kleene algebra [0, 1] discussed in §2.2.1. Instead of extending it to a fuzzy model of the entire language in the recursive way examined in §2.2.1, how might we extend it to a model which assigns an element of [0, 1] to each sentence as its degree of truth, but where the assignments to compound sentences are not a function of the assignments to simple sentences? One way would be to say that a compound sentence gets assigned the value 1 if it comes out as having the value 1 on every classical extension; it gets assigned the value 0 if it comes out as having the value 0 on every classical extension; and it gets assigned the value 0.5 in all other cases. This is not very attractive, however: the assignment of 0.5 in the third case is arbitrary, and the proposal allows simple and atomic sentences to have any degree of truth, while restricting other types of sentence to the values 1, 0, and 0.5, which seems odd.⁶² How, then, might we allow compound sentences to have the full range of intermediate degrees of truth, but in a non-recursive way? One idea is to introduce a further piece of machinery: a measure deﬁned over the set of classical extensions of our fuzzy interpretation.⁶³ We stipulate that the measure is normalized, and we say that the degree of truth of a compound sentence is equal to the measure of the set of classical extensions on which it is true. A classical contradiction is true on no classical interpretation constant, an object in M to each individual constant, and an n-ary relation on M (the characteristic function of which is a total function from M n to our set of three truth values) to each n-ary predicate. Instead of taking the recursive approach from this point on, we say that a classical interpretation extends a many-valued interpretation if it agrees with all the assignments of classical truth values (i.e. 0 and 1) that the many-valued interpretation makes (this can be made precise by making the obvious changes to our deﬁnition of a classical interpretation extending a partial interpretation); we introduce the notion of an admissible extension of a many-valued interpretation (in a way perfectly analogous to our introduction of the notion of an admissible extension of a partial interpretation); and we then follow the supervaluationist route: a compound sentence gets assigned the value 1 on our many-valued interpretation if it comes out as having the value 1 on every admissible extension, etc. ⁶² I am assuming here that atomic sentences get their truth values directly from the base interpretation, rather than from the supervaluation—recall the two options regarding the assignment of truth values to atomic sentences discussed on p. 78 above, and see below for further discussion of this point. ⁶³ A measure over a set S is a function μ from the power set of S to the non-negative real numbers, such that μ(∅) = 0, and for any disjoint subsets A and B of S, μ(A ∪ B) = μ(A) + μ(B). Intuitively, the measure μ assigns to each subset of S a size (a real number). The measure is said to be normalized if μ(S) = 1.

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whatsoever; so the set of classical extensions on which it comes out true is the empty set; hence on this proposal, it comes out as having degree of truth 0. A classical tautology is true on every classical interpretation whatsoever; so the set of classical extensions on which it comes out true is the entire set of classical extensions; hence on this proposal, it comes out as having degree of truth 1. If we restrict ourselves to admissible extensions, the constraints on precisiﬁcation will generate other sentences with degree of truth 0 and 1—for example, ‘x is red and x is yellow’ will get degree of truth 0, because there will be no admissible extension on which both ‘x is red’ and ‘x is yellow’ come out true.⁶⁴ This may sound like a neat idea, but it faces a number of problems. First is the problem of selecting a measure. Where we have only ﬁnitely many admissible extensions, there is a natural choice. Where A is the set of admissible extensions on which A is true, and is the set of all admissible extensions, the measure of A, and hence the degree of truth of A , is simply |A| || .⁶⁵ So, if there are 100 admissible extensions, and A is true on 54 of them, then the degree of truth of A is 0.54. That’s ﬁne—but the ﬁnite case is a very special one.⁶⁶ In realistic cases, there will be inﬁnitely many—indeed, uncountably many—admissible extensions. For example, we can set the cut-off for tallness anywhere between, say, 1.8 m and 1.9 m. And in the inﬁnite case, there is no non-arbitrary, natural choice of measure.⁶⁷ So the degree form of supervaluationism is incomplete: we either need to be told how to ﬁnd a unique, non-arbitrary, natural measure on a set of admissible extensions—which seems utterly hopeless—or we need to be ⁶⁴ For views in the ballpark of degree-theoretic supervaluationism as just presented, see Kamp 1975; 1981, 234–5; and Lewis 1983 [1970], 228–9; 1983 [1976], 69–70. (Sanford 1993, 225 presents a different sort of view, in which the admissible valuations are fuzzy rather than classical, and the supervaluation assigns a sentence not a single degree of truth, but a range of values.) Note that, instead of our fuzzy base interpretation, we could take the degree supervaluationist’s treatment of compound wfs and apply it over a three-valued, or partial two-valued, base interpretation. But then we would face the opposite problem to that faced by the idea of applying the original supervaluationist’s treatment of compound wfs over a fuzzy base interpretation: we would allow compound sentences to have any degree of truth, while simple and atomic sentences could only have the values 1, 0, or ∗. ⁶⁵ |X| is the cardinality of the set X. ⁶⁶ Thus Akiba 2004, 422 moves too fast when he says ‘‘suppose that we assign real numbers from 0 to 1 as degrees of Bruce’s baldness. If you accept . . . supervaluationism, such an assignment is very easy to obtain: you may just assign the ratio that the number of . . . interpretations in which ‘Bruce is bald’ is true bears to the number of all [interpretations].’’ This will work only where there are ﬁnitely many interpretations to consider. ⁶⁷ Our set of admissible extensions is not like a set of points in physical space, or in a space such as Rn , where we have a natural notion of the size of a subset.

90  told how to handle a multiplicity of acceptable measures. We already have a multiplicity of admissible extensions of a given many-valued interpretation, bringing with them a multiplicity of truth values for each sentence (one on each extension). Now we need something different: a way of processing a multiplicity of measures over the set of admissible extensions, bringing with them a multiplicity of degrees of truth of sentences in the one intended base interpretation. No such proposals currently exist. Second, once we acknowledge uncountable sets of admissible extensions, we have no guarantee that there will be any measure function which assigns a size to every set of admissible extensions. This means we could end up with compound sentences with no degree of truth. For if A is true on the set A of admissible extensions, and A has no measure, then A has no degree of truth. To remedy this situation, we would need to do one of three things. We might show that we will never encounter non-measurable sets of admissible extensions. Failing that, we might show that even though there might be such sets, none of them will ever be the set of extensions on which some sentence of our language is true—so they will not lead to sentences which have no degree of truth. Failing that, we might propose methods for handling both degrees of truth and degree gaps (together). Again, however, no proposals along any of these lines exist. So the degree form of supervaluationism needs a lot of work. But there are even deeper problems which suggest that it is not worth the effort—that the whole idea of degree supervaluationism is misguided. We have seen that degree supervaluationism gives us a view which assigns the full range of degrees of truth to all sentences, but which—unlike the fuzzy view—is not recursive. However, on closer inspection, it turns out that something rather odd is going on. True, all sentences can have the full range of degrees of truth—but nevertheless, simple and atomic sentences, on the one hand, and compound sentences, on the other, get their degrees of truth in fundamentally different ways. Thus, saying that ‘Bob is bald’ has degree of truth 0.3, or that ‘John is tall’ has degree of truth 0.5, really means something quite different from saying that ‘Bob is bald and John is tall’ has degree of truth 0.25. The ﬁrst sentence has an intermediate degree of truth because as a matter of fact, out there in the world, Bob neither completely possesses the property of baldness, nor completely fails to possess it—he possesses it to degree 0.3. Similarly for the second sentence. But the third sentence has an intermediate degree of truth for a completely

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different reason: because one quarter of the classical extensions of our fuzzy interpretation assign it the value 1. That is, on one-quarter of the admissible ways of precisifying our language, this sentence comes out (completely, utterly) true. I think it is misleading to have these two completely different routes to truth disguised under the same description (i.e. an attribution of a degree of truth between 0 and 1 inclusive). But this is not the real problem—for we could gain facility with this framework, and never be confused as to the origin or meaning of a particular sentence’s degree of truth. The real problem is that the rules for compound sentences are different from those for atomic and simple sentences, in a way that seems arbitrary, unmotivated, and downright odd. If you say ‘Bob is bald’ and I say ‘John is tall’, each of our statements is judged by looking at the world, ﬁnding the extensions of our predicates and the referents of our names, and seeing how they lie with respect to one another. However if, rather than each of us saying one of these things, you (or I) say both of these things, in the form ‘Bob is bald and John is tall’, then this claim is assessed in a completely different way. Now we look at all the possible precisiﬁcations of our language, and determine which of them make this sentence true; its degree of truth is then the proportion of precisiﬁcations on which it comes out true. But why should the assessment of two individual sentences be so totally different from the assessment of their conjunction? It’s as if we were to assess individual runners by their time around the track, and assess relay teams not by their cumulative time around the track, but by how colour-coordinated their clothes are. I mentioned earlier (p. 78) that there are two options for the supervaluationist as regards the assignment of truth values to atomic sentences. First, he may regard atomic sentences as getting their truth values directly from the base interpretation, without any role being played by the classical extensions: the truth value of P(a1 , . . . , an ) on the base interpretation M = (M, I) is the value assigned by I(P) to the n-tuple (I(a1 ), . . . , I(an )). Second, he may regard atomic sentences as getting their truth values from the supervaluation, just as compound sentences do. In the present discussion of supervaluationism as applied to many-valued base models, I have so far been assuming the ﬁrst option (cf. n. 62 above). We can avoid the problem just encountered by moving to the second option, according to which atomic sentences—like compound sentences—are judged according to the proportion of admissible extensions which make them true. Thus the

92  base interpretation assigns an atomic sentence the truth value 0.3 just in case 30 per cent of the admissible extensions of the base interpretation assign this sentence the value 1. So now we avoid the strange situation in which atomic sentences and compound sentences get their degrees of truth by completely different means. Nevertheless, this second option faces severe problems of its own. The view under consideration is that even though the intended base interpretation assigns ‘is bald’ a function f as its extension, and ‘Bob’ an object x as its referent, and f (x) = 0.3, still it does not follow from this that ‘Bob is bald’ is 0.3 true. Rather, to determine the truth of ‘Bob is bald’ we have to see what proportion of admissible classical extensions of the base interpretation make this sentence true. But when we think about it, this is a bizarre view. It ﬂies in the face of the most basic intuitions about truth: in particular, about the way truth is determined by the meanings of our words together with the way the world is. Let’s think it through: the view is that ‘Bob’ means this man and ‘is bald’ means that property—and this man possesses that property to degree 0.3—and yet this does not make it the case that ‘Bob is bald’ is 0.3 true. Rather, what makes the sentence 0.3 true is the fact that 30 per cent of the admissible classical extensions of the base interpretation make this sentence true. One wants to say at this point that what this view is calling the ‘truth values’ of atomic sentences have very little to do with truth as we know it—i.e. with truth as something that is jointly determined by meanings and the way the world is. We have thus arrived at a situation whereby atomic and compound sentences get their ‘truth values’ in the same way only by calling something the ‘truth value’ of the atomic sentence which in fact has very little to do with truth. Here is a way to bring out the worry. I noted in the original case of supervaluationism built over a base of partial two-valued models that our two methods of assigning truth values to atomic sentences always yield the same results. In the case currently under discussion—i.e. supervaluationism built over a base of fuzzy models—we have no such guarantee: there is no reason to expect that I(P)(I(a1 ), . . . , I(an )) = x in the base model just in case the measure over classical extensions assigns x to the set of models in which P(a1 , . . . , an ) is true. So let us suppose for a moment that in the base model, the extension of ‘is bald’ assigns the referent of ‘Bob’ the value 0.3, while ‘Bob is bald’ is true in 40 per cent of the classical extensions of the base model. Now let’s ask: how true do we want to say the sentence ‘Bob is bald’ is? Is it 0.3 true, or 0.4 true? It seems clear that the value assigned

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by the base model is the one that captures the sentence’s degree of truth, and that the other value—the one assigned by the measure over classical extensions of the base model—while it may (or may not) be interesting in some other way, simply doesn’t have much to do with truth as we know it. That much seems very clear when the values assigned by the base model and by the measure over extensions diverge—but of course, the moral carries over to the case where the values are the same (whether by luck or because we impose it as a constraint on acceptable measures that I(P)(I(a1 ), . . . , I(an )) = x in the base model just in case the measure assigns x to the set of models in which P(a1 , . . . , an ) is true). Thinking otherwise would be like thinking that the tachometer in my car measures the frequency to which my car radio is tuned—as long as (whether by accident or by rigid disciplining of my driving and/or listening habits) the read-out on the tachometer is the same as the read-out on the radio tuner. Thus, while degree supervaluationism seems like a neat idea at ﬁrst sight, it is ultimately a deeply unhappy one. Furthermore, the moral carries over to the original form of supervaluationism (in which both the base model and the supervaluation are partial or three-valued). The supervaluationist faces a choice regarding the assignment of truth values to atomic sentences: this can be done by the base model, or by the supervaluation. When we originally noted this fact, we said that it might seem to be of little import, because the two methods are guaranteed (in the case of the original form of supervaluationism) to give the same results. But we can now see that neither choice is satisfactory. We cannot have the truth values of atomic sentences being assigned by the supervaluation, for this violates our most basic intuitions about truth—in particular, about the way truth is determined by the meanings of our words together with the way the world is. But then if the truth values of atomic sentences are assigned by the base model, while those of compound sentences are assigned by the supervaluation, we have the bizarre situation in which (for example) the individual truth assessments of two atomic sentences operate in a completely different way from the truth assessment of their conjunction. Thus, the core idea of supervaluationism—of any kind—is an unhappy one. 2.4.2 Subvaluationism Given a base interpretation—many-valued, or partial two-valued—we have seen two non-recursive ways of assigning truth values to compound

94  wfs on that interpretation: the original supervaluationist way and the degree supervaluationist way (the degree of truth of a sentence on the base interpretation is the size of the set of admissible extensions on which it is true). There are other ways of proceeding. One notable one is the dual of the original supervaluationist way: we say that a sentence is true/false on the base interpretation if it is true/false on some admissible extension (as opposed to all admissible extensions, as in the original supervaluationist view). This is called subvaluationism.⁶⁸ Clearly, some sentences may now end up true and false in the base interpretation (viz. those which are true in some admissible extensions and false in others). How are we to handle this? There are two obvious ways: one corresponding to (the basic form of) supervaluationism built over three-valued base models, and one corresponding to (the basic form of) supervaluationism built over partial two-valued base models. In the three-valued case, we retain all the machinery as is, and simply reinterpret the third value ∗ as ‘both true and false’ rather than ‘neither true nor false’ (when considered as a truth value) and as ‘both in and out of the set’ rather than ‘neither in nor out of the set’ (when considered as the value of a characteristic function). In the two-valued case, instead of allowing the interpretation function and the characteristic functions of sets to be partial, we allow them to be non-functional (but total) relations—i.e. they may relate some arguments to more than one value (but they do relate every argument to at least one value). Either way, a subvaluationist semantics leads most naturally to a paraconsistent logic.⁶⁹ Hyde (1997) presents a subvaluationist view where the logic that results is Ja´skowski’s discursive logic.⁷⁰ Those who object to the very idea of a sentence being true and false will reject subvaluationism immediately. Aside from such general worries, is there any reason to favour supervaluationism over subvaluationism as a view about vagueness? A notable feature of discursive logic is the failure ⁶⁸ A subvaluationist approach to vagueness is presented in Hyde 1997; for some further discussion see Akiba 1999; Hyde 1999; Beall and Colyvan 2001; and Hyde 2001. ⁶⁹ A logical consequence relation |= is explosive just in case for all A and B, {A, ¬A} |= B; it is paraconsistent just in case it is not explosive (sometimes this is called weak paraconsistency, with strong paraconsistency consisting in the fact that A ∧ ¬A B). A logic is paraconsistent just in case its logical consequence relation is paraconsistent. A theory T is inconsistent just in case for some A, {A, ¬A} ⊆ T; it is trivial just in case for all B, B ∈ T. Thus, paraconsistent logics provide the basis for inconsistent but non-trivial theories. See e.g. Priest and Routley 1989. ⁷⁰ This was the ﬁrst formal system of paraconsistent logic (originally presented in 1948). See Hyde 1997, 648, and Ja´skowski 1969.

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of adjunction: A, B A ∧ B. This and other deviations from the classical consequence relation have led commentators such as Keefe (2000, 198) to compare subvaluationism unfavourably with supervaluationism. However, as Hyde points out, if we widen our view from the customary focus on a (multiple-premiss) single-conclusion consequence relation, to consider a (multiple-premiss) multiple-conclusion consequence relation, the symmetry between subvaluationism and supervaluationism is restored.⁷¹ In particular, corresponding to the failure of adjunction in subvaluationism, we have the failure in supervaluationism of (the classically valid principle) subjunction: A ∨ B {A, B}. We see, then, that the failure of adjunction in subvaluationism—and the corresponding objection that subvaluationism simply gets the meaning of conjunction wrong—is the dual of (one manifestation of) the missing witness problem for supervaluationism. Consider the sentences A, ¬A, A ∨ ¬A, and A ∧ ¬A. Suppose that A and ¬A have the value ∗ in the base model: in supervaluationism, this means that they are neither true nor false; in subvaluationism, it means that they are both true and false.⁷² A ∨ ¬A is true in every classical extension and A ∧ ¬A is false in every classical extension, and so in both supervaluationism and subvaluationism, A ∨ ¬A has the value 1 in the base model, and A ∧ ¬A has the value 0. Thus, the supervaluationist allows a disjunction to be true when neither disjunct is, and allows a conjunction to be false when neither conjunct is, while the subvaluationist allows a disjunction to be true (and not false) when both disjuncts are false (and true), and allows a conjunction to be false (and not true) when both conjuncts are true (and false). Arguably, all of these claims run counter to ordinary intuitions about the truth conditions of disjunctions and conjunctions—but it is hard to see how the subvaluationist’s claims could be thought worse than the supervaluationist’s, especially once we appreciate the perfect symmetry of the semantic pictures that underlie these claims. Thus, even if we agree with Keefe (2000, 198 n. 24) that single-premiss consequence is more important ⁷¹ Multiple-conclusion consequence may be thought of as follows. Where and are sets of wfs, |= just in case in every model in which every member of is true, some member of is true. Where contains only one wf, this reduces to the usual notion of single-conclusion consequence. (In the case of singleton sets of wfs, we typically just write the wf, without set-brackets around it; for example we write {A, ¬A} |= B rather than {A, ¬A} |= {B}.) ⁷² In the alternative way of proceeding, where the base model is two-valued, suppose that A and ¬A are assigned neither value in the supervaluationist case, and are assigned both values in the subvaluationist case.

96  than multiple-premiss consequence (at least in the context of discussing vague natural language), in the ﬁnal analysis the supervaluationist’s preservation of classical single-conclusion consequence merely provides a coat of paint to hide the underlying rust (i.e. the missing witness problem). Supervaluationism may thus appear more attractive than subvaluationism at ﬁrst sight, but on closer inspection, the supervaluationist’s corrosion problem seems no less worrying than the (unpainted, fully exposed) rust in subvaluationism (e.g. the failure of the truth of the conjuncts to ensure the truth of the conjunction). 2.4.3 Worldly Vagueness? Both the recursive and supervaluationist forms of the truth gap view have a common base: the idea that each vague discourse has a unique intended partial interpretation. There is no vagueness here in the relationship between language and the world. We are not saying that ‘is bald’ has several possible extensions; we are saying it has one unique extension—there is one property out there which it uniquely picks out—but this property is inherently gappy, in that some things in the domain neither possess it nor fail to possess it. Now from here, our two versions diverge over how they assign truth values to compound sentences on a partial interpretation—but none of these subsequent developments either removes the vagueness inherent in the partial interpretation or introduces any further semantic indeterminacy. In the recursive case this is obvious. It is just as true of the supervaluationist approach. This may be surprising, because supervaluationism is widely touted as a purely linguistic or semantic theory of vagueness—as a theory which, precisely, locates vagueness solely in the relationship between language and the world. But this is a mistake, born of conﬂating supervaluationism—as presented above—with an entirely different theory of vagueness, which does locate vagueness solely in the relationship between language and the world. This is the theory I call plurivaluationism; we shall examine it in the next section. Plurivaluationism countenances only classical interpretations: it has no truck with partial interpretations, or non-recursive rules for assigning truth values to sentences. Instead, it says that each discourse has many admissible interpretations, not one correct one. As we shall see, this view locates vagueness solely in the relationship between language and the world. But this is a different view entirely. In the supervaluationist picture, the admissible

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extensions of our unique intended partial interpretation do not have the semantic status they have on the plurivaluationist view—they are not candidate meanings for vague expressions. The supervaluationist’s semantic picture consists of a partial interpretation—i.e. one partial interpretation, the intended/correct/etc. one—which assigns certain truth values to compound sentences. The classical interpretations which extend this partial interpretation are not part of the supervaluationist’s semantic picture as such. They are auxiliary calculating devices for determining the truth values of compound sentences on the partial interpretation. They are, as it were, written on scratch paper—not part of the ﬁnal answer; they are a ladder which we kick away once we have completed our determination of the truth values of compound wfs on our partial interpretation. Once we have completed this determination, the partial interpretation alone stands as our account of the meaning of vague language. In sum, then, we can say of the supervaluationist version of the truth gap approach exactly what we said of the recursive version of the truth gap approach, which is exactly what we said of the three-valued approach. The idea is that when I speak vaguely, I invoke a unique intended interpretation, and on this interpretation, each name in my language has a unique referent, each predicate has a unique extension, and so on. The vagueness is all within this model. For example, the particular property that my predicate ‘is red’ picks out is inherently indeterminate: some objects possess it, some do not possess it, and others neither possess it nor fail to possess it (i.e. its characteristic function does not send these objects anywhere). On this approach, vagueness (or at least indeterminacy—recall the discussion on pp. 70–1) is found in the world itself, not in the relationship between language and the world. The point generalizes. We have considered base interpretations which are partial two-valued, relational two-valued, three-valued, and fuzzy, and we have considered, as non-recursive strategies for assigning values to compound wfs on these base interpretations, the original supervaluationist proposal, the degree supervaluationist proposal, and the subvaluationist proposal. In all cases, when it comes to the question of worldly vagueness, the base interpretation is what matters, and the strategy for assigning truth values to compound wfs is irrelevant. Thus, a recursive approach over a base interpretation of type X and one of our non-recursive approaches over the same base will not differ in respect of locating vagueness (or indeterminacy) in the world, as opposed to in the relationship between

98  language and the world. All these views deny any indeterminacy in the relationship between language and the world; all the indeterminacy is in the world itself. This is because they posit a unique intended base interpretation which has some form of inherent indeterminacy (different forms in the different kinds of base interpretation we have considered—but always some form). True, they also posit a multiplicity of classical models which extend this base interpretation—but these classical models are not admissible interpretations of the language. Rather, they are admissible extensions of the uniquely correct base interpretation of the language. They play an auxiliary role in calculating the semantic value of compound wfs on the base interpretation—they have no direct semantic signiﬁcance. In short, the multiplicity of admissible extensions does not introduce semantic indeterminacy. This is in sharp contrast to the plurivaluationist picture, to which we turn now.

2.5 Plurivaluationism (Deny 2) The ﬁnal component of the classical picture which we isolated in §1.2 was claim 2: each discourse has a unique intended interpretation. We look now at a view which denies this: plurivaluationism. We are to accept all the other parts of the classical picture, apart from claim 2: so our interpretations are perfectly classical, and we retain almost all our classical machinery and notions. What we need to revisit is the idea of a sentence being true simpliciter, as opposed to true on this or that interpretation. For we said that a sentence is true simpliciter if it is true on the intended interpretation. If there is no unique intended interpretation, then when will a sentence count as, simply, true? Well, if we jettisoned the very notion of a correct interpretation entirely, then all we would be able to say about a sentence was whether it was true on any particular interpretation. Given this, we could draw a distinction between sentences which are true on every interpretation (i.e. logically true) and those which are not, but we would not be able to distinguish amongst the latter between those sentences which are actually (in fact, as a matter of how things are, etc.) true and those which are actually false. In turn, we would not be able to distinguish the sound arguments as a subset of the valid arguments (i.e. the ones whose premisses

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are as a matter of fact true). But plurivaluationists do not go this far. They do not jettison the idea of correct or intended interpretations altogether: they just say that in general there is not a unique correct or intended interpretation. The way that a given speech community uses a language places constraints on what is an acceptable interpretation of that language. For example, we apply the name ‘Helen Clark’ to a certain person, so on acceptable interpretations, this person is the referent of that name; we apply the term ‘apple’ to certain objects, so on acceptable interpretations, these objects are in the extension of that predicate; and so on. Now the classical view holds that the set of acceptable interpretations has one member, which we called the intended/correct/etc. interpretation. The plurivaluationist denies this—but retains the notion of an acceptable interpretation. Thus, we have a privileged class of interpretations, but not a single interpretation privileged above all others. One natural motivation for plurivaluationism would be a fondness for classical interpretations, coupled with a conviction that epistemicism cannot adequately answer the location problem. The plurivaluationist can regard a unique intended classical interpretation as a goal which we never actually reach: our usage rules out certain interpretations as incorrect, but never manages to narrow down the remaining set of acceptable interpretations to just one member. Now to return to our question about truth simpliciter. We have a class of acceptable interpretations. Suppose that a given sentence is true on all of them. Then in an obvious sense it does not matter that we have many acceptable interpretations and not just one: we can still say that our sentence is true. Similarly, if our sentence is false on all the acceptable interpretations, then again it does not make any difference that we have many acceptable interpretations and not just one, and we can say that our sentence is false. But if our sentence is true on some acceptable interpretations and false on others, then it does matter that we have many acceptable interpretations and not just one: we can say neither that our sentence is true simpliciter, nor that it is false simpliciter. This may sound like supervaluationism: we have a range of acceptable classical models, and our sentence comes out true/false simpliciter if it is true/false on all of them. Plurivaluationism is, however, crucially different from supervaluationism. This has not been appreciated in the literature: the

100  two positions have never been distinguished.⁷³ Contrast the following two quotations: Broadly speaking, supervaluationism tells us two things. The ﬁrst is that the semantics of our language is not fully determinate, and that statements in this language are open to a variety of interpretations each of which is compatible with our ordinary linguistic practices. The second thing is that when the multiplicity of interpretations turns out to be irrelevant, we should ignore it. If what we say is true under all the admissible interpretations of our words, then there is no need to bother being more precise. (Varzi 2003b, 14) In van Fraassen’s original terminology, for each acceptable U-model U, the function that assigns to each sentence its truth value in U is called a classical valuation, while the function that assigns the value true to those sentences true in every acceptable U-model is called a supervaluation. (McGee 1997, 154 n. 12)

My usage of the term ‘supervaluationism’ agrees with that of McGee and, before him, van Fraassen. Varzi, by contrast, is using the term ‘supervaluationism’ to describe a very different view: what I am calling plurivaluationism.⁷⁴ In order to avoid confusion, we need to have separate terms for separate theories. As the inventor of the term ‘supervaluation’, van Fraassen may be given priority here—i.e. his term will be applied to his view. That means that the other view will need a different name—hence my introduction of the term ‘plurivaluationism’.⁷⁵ ⁷³ Even Varzi 2007, §1, which seeks to lay out all the options, does not make this distinction. ⁷⁴ This is not intended as a criticism of Varzi: his usage is not out of line with the vagueness literature. I have singled out Varzi only because the passage quoted gives such a clear description of one of the two views that I want to distinguish. ⁷⁵ Because supervaluationism and plurivaluationism have not been distinguished in the literature, it is impossible to say of previous authors who espouse views in the ballpark of either of these positions whether they should be seen as advocating supervaluationism, plurivaluationism, or some (possibly confused) mixture of the two. Hence I shall make no attempt to classify existing ‘supervaluationist’ views with respect to this distinction. Three views should be mentioned in connection with plurivaluationism, however. One is the partial denotation view of Field 1973, 1974. Field clearly espouses the idea (central to plurivaluationism) that the various classical models in his view play the role of equally acceptable interpretations of the language. (This contrasts with the supervaluationist view of classical models as admissible extensions of a unique intended non-classical interpretation.) However, Field does not draw the distinction between regarding the truth condition ‘true if true on all models’ as a mere manner of speaking (as in the plurivaluationist view), as opposed to a further piece of semantic machinery (as in the supervaluationist view), and so ultimately it is not clear where to classify Field’s view. The second view which should be mentioned is that of Przełe¸cki 1976. Przełe¸cki is explicit that (as in

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But how exactly are the views different? Many views countenance a range of classical models: this is common to supervaluationism, plurivaluationism, and contextualism (see §2.6), among others. What distinguishes each of these views is the role it gives to these classical models, in conjunction with any further semantic machinery that the view introduces. Consider supervaluationism. As I was at pains to point out, the classical models are admissible extensions of a unique intended partial interpretation of the language. They do not give the content of the language in its actual, unprecisiﬁed state: they interpret the language as it would be were it precisiﬁed. They do not stand in a direct interpretational relationship to the language. Rather, they serve as calculating devices which yield an assignment of truth values to (actual, vague) sentences of the language. This assignment of truth values, together with the other assignments made in the base interpretation—not in its classical extensions—play the interpretational role: they are what constitute a description of the actual semantic state of the language. In the supervaluationist view, each sentence is in a unique semantic state, arrived at via a supervaluation over many classical models. The truth value of a sentence—i.e. the one assigned to it, on the base interpretation, by the supervaluation (as opposed to any of its truth values on the various classical extensions of this base interpretation)—represents the one actual semantic state of the vague sentence. This truth value merely happens to be arrived at via a description of idealized precisiﬁcations which misrepresent the actual, vague state of the language. Plurivaluationism, by contrast, has no further semantic machinery, besides the classical models. In the ﬁnal story about the actual semantic state of the language, there is no other, non-classical model: there are only the classical plurivaluationism, not supervaluationism) his view involves no non-classical semantic machinery, only a multiplicity of classical models, and he makes the point (crucial to plurivaluationism: see §2.5.1—which was written before I read Przełe¸cki’s paper) that on his view ‘‘reality is sharp—composed of exact structures. It is the connection between language and reality that is fuzzy—linking language not to a single exact structure, but to a whole class of such structures’’ (pp. 379–80). The classiﬁcation of Przełe¸cki as a plurivaluationist is rendered problematic, however, by his claim (p. 380 n. 3) that Fine’s (supervaluationist) view is similar to his: whether this foils the classiﬁcation depends upon just how similar he thinks Fine’s view is to his own. The third view is the account of abstract mathematical discourse presented by Islam 1996. Like Field, Islam makes it clear that the various classical models in his view play the role of equally correct interpretations of the language, and, like Przełe¸cki, he makes it clear that his view involves no non-classical semantic machinery. Unlike the plurivaluationist view of vagueness, however, it is no part of Islam’s view that a multiplicity of correct interpretations goes hand in hand with speakers failing to specify exactly what their terms are to mean: on his view, talking relative to multiple interpretations is often exactly what we want to do.

102  ones. Together, they give the content of the language in its actual, vague state. Each of them is an interpretation of the language, not a mere extension of the one uniquely correct interpretation. (This is why I have used different terms for the classical models in the context of the different views: admissible extensions versus acceptable interpretations.) In the plurivaluationist view, there is no further semantic story, beyond that told by the many acceptable interpretations. Here we have indeterminacy of meaning: there are many acceptable interpretations, and that’s all there is to say. Thus each sentence is not in a unique semantic state: semantic states are individuated by interpretations, and each sentence has many of them. On this view, the actual semantic state of the sentence is indeterminacy of truth value: there are many acceptable assignments of truth value to the sentence, and nothing to decide between them. That is a complete and accurate description of the actual semantic state of the sentence. Rather than indeterminacy, we might just as well describe this as plurality or multiplicity of meaning. The essential point remains: each sentence is not in a unique semantic state, as semantic states are individuated by interpretations, and each sentence has many of them, with nothing to decide between them. Figure 2.2 is an attempt to give visual form to the distinction between supervaluationism and plurivaluationism. Suppose the request is made to describe the semantic state of a given sentence. The plurivaluationist can say only: it is true on this interpretation, false on this one, and so on, through all the acceptable interpretations. This is because these interpretations are the only pieces of semantic machinery she has. Her picture is of a language which simultaneously has many classical interpretations, and whose semantic state is thereby indeterminate (or equivalently, which is thereby in many semantic states simultaneously). The supervaluationist, by contrast, will describe the sentence as being in a unique, determinate semantic state. For in her picture, she has further semantic machinery, over and above the classical models: partial interpretations, which assign truth values to compound sentences via supervaluations over classical models. For her, the sentence is not in an indeterminate ﬁrst-order (classical) semantic state—it is in a perfectly determinate higher-order (non-classical) semantic state. Note that while there are important analogies between plurivaluationism, supervaluationism, and possible-worlds semantics for modal languages—analogies which have, for example, enabled formal results from modal logic to be applied in discussions of supervaluationism in the

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supervaluationism

plurivaluationism

language

language

key: classical model

non-classical model

interprets/ gives the meaning

extends

Figure 2.2. Plurivaluationism and supervaluationism.

literature—the possible-worlds semantic picture is not the same as either the plurivaluationist or the supervaluationist picture. Figure 2.3 shows the possible-worlds semantic picture. In this picture, the modal language has a unique intended interpretation, which (in a standard version of this kind of semantics—details vary in other versions, but not in any way that matters to the discussion here) consists in a set of possible worlds, a binary accessibility relation on this set, a domain (i.e. set of objects) for each possible world, and a valuation function which assigns to each name a referent at each world, and to each predicate an extension at each world. We can, then, regard the overall model as an arrangement of worlds, and we can regard each world as a classical model (for each world comes with a domain of objects and a referent/extension for each name/predicate). Now we can look at things this way if we wish: but we must remember that the role which these classical models (i.e. worlds)

104 

language

key: possible world/ classical model

modal model

interprets/ relative possibility/ gives the meaning accessibility

Figure 2.3. Possible-worlds semantics.

play in the possible-worlds semantic picture is quite different from the role that classical models play in either the plurivaluationist picture or the supervaluationist picture. In the modal picture, unlike in the other two pictures, each classical model is not the kind of thing that could, all by itself, provide an interpretation for the entire language. (Note that the language in this case is a modal language, rather than the ﬁrst-order language considered in relation to supervaluationism and plurivaluationism.) In the plurivaluationist picture, each classical model interprets the entire language; as there are many classical models in play, we end up with many interpretations of the language. In the modal case, by contrast, all the classical models put together go to make up one interpretation of the entire (modal) language. In the modal case, therefore, the classical models are not in any sense rival interpretations (not of the language as it is, as in the plurivaluationist case, or of the language as it would be were it

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precisiﬁed, as in the supervaluationist case); rather, they are components of a single (non-classical) interpretation. There is a strong intellectual urge to move from indeterminacy or multiplicity with respect to ﬁrst-order states to determinacy and uniqueness with respect to higher-order states. Thus, for example, it is very natural to think of plurivaluationism as follows. The semantic state of a sentence (for example) is a sequence of truth values, indexed by the acceptable interpretations—say (T, F, F, T, T, F, . . .). We might call this sequence the sentence’s truth proﬁle. But note that we have now moved from plurivaluationism to a different view. On this view, the sentence is in a unique semantic state, represented by the assignment to it of a unique sequence of truth values. The sentence’s semantic state is given by its truth proﬁle, and it has a unique one of these. On the plurivaluationist view, by contrast, the sentence is in an indeterminate state (or a plurality of states), represented by many assignments of truth values to it simultaneously. Again, to take another example, it is very natural to think of plurivaluationism as follows. In modal semantics, we assign predicates intensions instead of extensions. But as we have just discussed, possible worlds are like classical models, so we might now proceed to assign a predicate an ‘intension’ in our plurivaluationist approach to vagueness. This ‘intension’ is just the obvious analogue for predicates of the truth proﬁle of a sentence. But again, we have now moved from plurivaluationism to a different view. On this view, the predicate is in a unique semantic state, represented by its ‘intension’. On the plurivaluationist view, by contrast, the predicate is in an indeterminate state (or a plurality of states), represented by many assignments of extensions to it simultaneously. 2.5.1 Worldly Vagueness? We thus have a distinction between the semantic state of a sentence (or other linguistic item) being indeterminate—because the only semantic states there are, are classical ones, and no unique one of these is privileged—and the semantic state being determinate—because as well as classical semantic states, we have non-classical semantic states, and a unique one of these is privileged for each sentence. This distinction is very important, because only the plurivaluationist view warrants a number of descriptions which have often been attached to ‘supervaluationism’ in the literature. I am thinking of all those claims which talk of a purely linguistic theory of vagueness, of

106  vagueness as semantic indecision, of vagueness as being conﬁned solely to the relationship between language and the world, and so on. The classic quote of this sort is from Lewis: I regard vagueness as semantic indecision: where we speak vaguely, we have not troubled to settle which of some range of precise meanings our words are meant to express. (1986a, 244 n. 32)⁷⁶

The crucial point to note is that this view of vagueness ﬁts with plurivaluationism, not supervaluationism. We saw in §2.4.3 that on the supervaluationist approach, vagueness—or at least indeterminacy—is found in the world itself, not in the relationship between language and the world. As far as the issue of worldly vagueness is concerned, the supervaluationist approach over a certain sort of base interpretation is in exactly the same boat as the recursive approach over that sort of base interpretation: the base interpretation brings with it worldly indeterminacy (e.g. partial properties), and the superstructure cannot then take it away. The supervaluational approach differs from the recursive approach in positing worldly indeterminacy and a ‘non-local’ (non-truth-functional) account of how compound sentences get their truth values. The plurivaluationist, by contrast, locates vagueness entirely in the relationship between language and the world. The plurivaluationist model theory gives us a picture on which each n-place predicate is associated with many n-place relations (one in each acceptable interpretation), each of which is inherently precise. The idea is that when we use a vague predicate, we do not refer to a unique relation, but all the relations we simultaneously refer to are inherently precise. Suppose ‘Bob is tall’ comes out true on some acceptable interpretations of ‘tall’ and false on others. Then we can say neither that ‘Bob is tall’ is true nor that it is false. But, unlike on the supervaluationist approach, this does not mean that the property of tallness is inherently indeterminate. The point is rather that there is no unique property of tallness. There are many properties, each corresponding to an acceptable extension of ‘tall’, and all precise, in the sense that each object either deﬁnitely possesses or deﬁnitely fails to possess each property. ⁷⁶ Cf. Lewis 1986a, 212.

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When we speak vaguely, we fail to single out a unique such property. So vagueness is solely a matter of the relation between language and the world. The world itself is precise. To cover this ground one more time: the supervaluationist’s classical extensions represent ways in which vague predicates (and other vague parts of language) could be precisiﬁed. What the predicate ‘tall’ actually means is this one, unique partial property which it is assigned in the intended base interpretation. There is no indeterminacy as to what the predicate means: it means this property and no other. However, there are many possibilities as to how this property could be sharpened —one way corresponding to each extension which the predicate might get assigned on an extension of the base interpretation. So we have complete determinacy of meaning. But the thing that is uniquely meant—the property tallness—is inherently indeterminate (because partial, or three-valued, etc.). Contrast the plurivaluationist view, where there are no indeterminate properties (no partial models, no threevalued models, etc.). There are only the perfectly precise classical extensions. But there are many of them, and nothing to decide between them. The indeterminacy here concerns which—out of all these candidate precise properties—is meant by ‘tall’. This indeterminacy is purely semantic. 2.5.2 More on Plurivaluationism How will the plurivaluationist view handle the Sorites paradox? Suppose we remove one grain at a time from a 10,000-grain pile of sand, until we have 1 grain left. Call the pile with n grains ‘pile n’. Our ordinary, vague usage of the term ‘heap’ corresponds to a family of acceptable classical interpretations, one for each 11 ≤ n ≤ 100, on which piles 1 through n − 1 are assigned 0, and piles n through 10,000 are assigned 1. (I am talking here about the usage of the whole community of speakers—recall the discussion on pp. 58–9.) Let the sentence ‘The cut-off is at n’ mean that piles 1 through n − 1 are non-heaps, and piles n through 10,000 are heaps. Consider the inductive premiss of the Sorites argument associated with this setup: ‘For every 2 ≤ n ≤ 10, 000, if n is a heap, then so is n − 1.’ This sentence is false on each of our acceptable interpretations: on an interpretation where the cut-off is at n, the corresponding instance of this universal claim is false, and hence the universal claim itself is false. Thus, according to the plurivaluationist approach, the inductive premiss can be said to be, simply,

108  false: it is false on every acceptable interpretation, so the fact that we have more than one acceptable interpretation becomes irrelevant here. This, then, is the mistake in the paradoxical reasoning: the inductive premiss is false. So why is the paradox compelling? Consider the claim ‘The cut-off is at n’. For each 11 ≤ n ≤ 100, this claim is true on exactly one of our acceptable interpretations. Thus we can say neither that it is true nor that it is false: the fact that we have more than one acceptable interpretation is relevant here. There is, then, no point n such that we can say ‘The cut-off is at n’. So far so good. But here, says the plurivaluationist, is where we make a natural mistake, and so get drawn into the paradox. We conclude from the foregoing that ‘There is no cut-off’ is simply true—or in other words, we conclude that the inductive premiss is true. But here we are mistaken. ‘There is no cut-off’ is false on every acceptable interpretation (for each puts the cut-off somewhere—just at a different point on different interpretations), and hence—according to the plurivaluationist—we can simply say that it is false. Thus, the paradox is compelling because we tend to assume that because we cannot say that the cut-off is here, or here, or here, . . . through all the possible positions where the cut-off might be, it follows that we can say ‘There is no cut-off’—that is, that the inductive premiss is true. On the plurivaluationist account, this does not follow. There is no point n such that we can truly say ‘The cut-off is at n’, and yet we can simply say that ‘There is a cut-off’ is true—and so we can simply say that the inductive premiss, which denies this, is false. This is, of course, analogous to what the supervaluationist said about the paradox—but it is not exactly the same as what the supervaluationist said. The crucial difference is as follows. The supervaluationist has an extra piece of non-classical semantic machinery (a supervaluation) which assigns the sentence ‘There is a cut-off’ the value True, and which assigns the sentences ‘The cut-off is at n’ no truth value (or the value ∗). The plurivaluationist, on the other hand, has only a way of talking. On both views, we can say that ‘There is a cut-off’ is true, while we cannot say that ‘The cut-off is at n’ is true or that it is false—but what this really amounts to differs on the two views. On the supervaluationist view, it corresponds directly to the semantic facts. On the plurivaluationist view, it is a mere way of talking. The genuine semantic facts are that ‘There is a cut-off’ is true on this interpretation, and this one, and so on. We can sum this up by simply saying it is true, but we are not thereby describing some other level of

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semantic reality, beyond what is going on in each classical interpretation. We are merely summing up the actual, long description of the semantic facts (i.e. of what is going on in each classical interpretation) in a short phrase. (Analogy: If each person on the street believes that the government will lose the election, we can say ‘The man on the street thinks the government will be voted out’. Here we talk as if there is a unique person on the street, who believes that the government will be voted out, but this is just a manner of speech: really we are not talking about a particular person—we believe neither that one amongst all the actual persons on the street is privileged, nor that there is an additional special person, over and above the multiplicity of real persons on the street; we are saying that each of the many persons on the street believes that the government will be voted out. When it is not the case that each of the many persons on the street believes the same thing, we cannot use this manner of speaking.)⁷⁷ This difference means that whereas the supervaluationist response to the Sorites led to the missing witness problem, the plurivaluationist—who offers something analogous to the supervaluationist response to the Sorites—faces only an analogue of the missing witness problem. The problem, recall, was that ‘There is a P’ can come out true, even though ‘This is P’ is true of nothing. On the plurivaluationist view, we have the outcome that ‘There is a cut-off’ may be said to be true, even though no sentence ‘The cut-off is at n’ may be said to be true. Note the ‘may be said to be’s. The plurivaluationist does not say that ‘There is a cut-off’ actually is true even while each sentence ‘The cut-off is at n’ actually is not true. On the plurivaluationist picture—as opposed to the supervaluationist one—there is no level of semantic fact at which an existential claim is assigned the value true while none of its instances is. For the only semantic facts concern what happens on each acceptable interpretation—and these are entirely classical. There is an overall level of talk at which we say that some existential sentence is true, while not saying that any of its instances is true; but this talk is, from the semantic point of view, epiphenomenal. It is mere talk. So we do not have the missing witness problem as such: a sentence being made true while none of its instances is. The problem we do have is that ⁷⁷ In fact, we employ this way of speaking even if most persons believe the government will be voted out: it does not matter if a few don’t. To get the analogy with plurivaluationism right, we need to suppose for a moment that we say ‘‘The man on the street ’s’’ only when all persons on the street . Cf. n. 78 below.

110  we are given a way of talking—‘say that a sentence is simply true if it is true on every acceptable interpretation’—that clashes with our ordinary use of quantiﬁers. In particular, it violates expected relationships between quantiﬁed claims and their instances. Similar comments apply to the penumbral connection/truth-functionality issue. The plurivaluationist will tell us that ‘This leaf is red’ and ‘This leaf is not red’ can be said neither to be simply true nor simply false, while ‘This leaf is red or not red’ can be said to be simply true, and ‘This leaf is red and not red’ can be said to be simply false. Yet, of course—for reasons that should now be familiar—we have no violation of truth-functionality here. There is no level of semantic fact at which a conjunction is assigned the value False, while neither of its conjuncts is, or at which a disjunction is assigned the value True, while neither of its disjuncts is. For the only semantic facts are the facts about what is happening in each acceptable interpretation—and these are entirely classical (hence truth-functional). What we have is just a level of talk laid on top of these semantic facts. The talk sounds non-truth-functional, but it is in fact epiphenomenal. Unlike on the supervaluationist view, it does not literally describe a non-truthfunctional semantic reality. The question remains, of course, whether this way of talking ﬁts or clashes with ordinary usage. 2.5.3 Other Forms of Plurivaluationism? In the discussion of supervaluationism we considered some variants of the basic view: for example, the degree form of supervaluationism and subvaluationism. Is there a parallel range of variants of plurivaluationism? Yes and no. Yes, in that we can distinguish plurivaluationists of the following kinds: 1. those who want to say that a sentence is neither true nor false when it is true on some acceptable interpretations and false on others (cf. the basic form of supervaluationism) 2. those who want to say that a sentence is half true when it is true on half of the acceptable interpretations and false on the other half (cf. the degree form of supervaluationism) 3. those who want to say that a sentence is both true and false when it is true on some acceptable interpretations and false on others (cf. subvaluationism).

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No, in that there is no semantic difference between any of these views—they merely employ different descriptions of, or ways of talking about, the very same semantic facts. We have seen that for the plurivaluationist, the semantic facts are completely encapsulated in and exhausted by the multiplicity of acceptable classical interpretations—i.e. in the facts about which out of all the possible classical interpretations of a language are acceptable, and the facts about the nature of these acceptable interpretations (i.e. what referent each assigns to each name, what extension each assigns to each predicate, which sentences each makes true, and so on). Now we can describe the way in which the truth of a sentence varies over the acceptable interpretations however we like—for example, in any of the three ways outlined above.⁷⁸ No speciﬁc descriptive mechanism is part of the plurivaluationist view (i.e. is part of the core semantic view which I am calling ‘plurivaluationism’). Any, or all, of them is OK (as long as we are all clear on how we are using words in a given context). The key point is that beyond the core semantic facts, anything further that the plurivaluationist says is mere talk, which does not add to or alter the semantic facts in any way. This is in contrast to supervaluationism, where there is a genuine difference in the semantic machinery posited by, for example, the basic supervaluationist, the degree supervaluationist, and the subvaluationist. 2.5.4 Pragmatism There are several approaches to vagueness to be found in the writings of David Lewis. One of them is distinctive (or at least apparently so—but see on) in that it locates vagueness not in semantics, but in pragmatics. Lewis sets up a framework in which a language (in one sense) is a function which assigns meanings to sentences and their components: the meaning of a sentence is a function from possible worlds to truth values; the meaning of a predicate is a function from possible worlds to sets of possible objects; and so on. Lewis then has a story to tell about how a particular population P comes to use a particular one of these languages £. ⁷⁸ There are many other possibilities besides these three. For example, we can also imagine a plurivaluationist who wants to say that a sentence is true/false when it is true/false on most acceptable interpretations. Cf. n. 77 above, and Lewis 1983 [1979], 244. Another view worth mentioning is that of Braun and Sider 2007, according to which a sentence is true just in case it expresses a unique proposition which is true. On this approach, we would say that a sentence which has many admissible interpretations is untrue, regardless of its truth values on its various admissible interpretations.

112  Brieﬂy, the connection between P and £ holds by virtue of a convention of truthfulness and trust in £ which prevails in P: members of P only make utterances which they believe to be true in £, and they proceed as if other members of P are doing this too. Now, Lewis’s languages are precise: a sentence is true in a given language at a given world, or it is false in that language at that world; there is no vagueness in the semantics of any language. Rather, vagueness enters into the relationship between the population P and the set of possible languages: it appears in the form of indeterminacy concerning which language a population uses. Lewis writes: our convention of language is not exactly a convention of truthfulness and trust in a single language, as I have said so far. Rather it is a convention of truthfulness and trust in whichever we please of some cluster of similar languages . . . The convention conﬁnes us to the cluster, but leaves us with indeterminacies whenever the languages of the cluster disagree. We are free to settle these indeterminacies however we like. Thus an ordinary, open-textured, imprecise language is a sort of blur of precise languages—a region, not a point, in the space of languages. (1983 [1975], 188)⁷⁹

A Lewis-type language is a function from syntactic items to meanings. If we hold the syntactic items ﬁxed, then considering different Lewis-type languages is just the same as considering different interpretations of one uninterpreted, syntactically speciﬁed formal language.⁸⁰ The pragmatist tells us that our practice does not serve to associate us with a unique Lewis-type language, but only with a cluster of such languages. This is just another ⁷⁹ See also Lewis 1969, 200–2: ‘‘I think we should conclude that a convention of truthfulness in a single possible language is a limiting case—never reached—of something else: a convention of truthfulness in whichever language we choose of a tight cluster of very similar possible languages. The languages of the cluster have exactly the same sentences and give them corresponding sets of interpretations; but sometimes there are slight differences in corresponding truth conditions . . . . Our actual language is like a resonance hybrid of the possible languages that make it up.’’ Lewis does not appear to be very attached to this pragmatic view. Elsewhere he writes: ‘‘we have so far been ignoring the vagueness of natural language. Perhaps we are right to ignore it, or rather to deport it from semantics to the theory of language-use. We could say, as I do elsewhere, that languages themselves are free of vagueness but that the linguistic conventions of a population, or the linguistic habits of a person, select not a point but a fuzzy region in the space of precise languages. However, it might prove better to treat vagueness within semantics, and we could do so as follows’’ (Lewis 1983 [1970], 228). Lewis then proceeds to present a degree-theoretic form of supervaluationism (see §2.4 above). Burns 1991, 1995, on the other hand, defends the pragmatic view as the correct theory of vagueness. ⁸⁰ ‘‘By a language, I mean an interpreted language. So what I call a language is what many logicians would call a language plus an interpretation for it. For me there cannot be two interpretations of the same language; but there can be two languages with the same sentences’’ (Lewis 1969, 162).

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way of saying that our practice does not ﬁx a unique correct interpretation of our (uninterpreted) language, but only a cluster of such interpretations. And that is plurivaluationism! Pragmatism, then, is simply a stylistic variant of plurivaluationism.⁸¹

2.6 Contextualism While they differ in many ways from one another, the views we have looked at so far can all be regarded as having one thing in common: they offer synchronic accounts of vagueness. They hold that in order to see what is distinctive of vague language, we need to look at two areas: the nature of its interpretations (classical, gappy, many-valued, etc.) and the number (one or many) of these interpretations which are singled out as correct or acceptable. No mention is made here of how (if at all) which interpretation(s) is intended changes over time or with context. No doubt the proponents of the views we have examined would be prepared to admit that such changes do, or at least might, occur: for example, it is plausible that ‘today’ uttered today has a different referent from the one it had when uttered yesterday. But the proponents of these views will not think that such changes have anything in particular to do with vagueness. This is where contextualism is distinctive. The basic idea behind contextualism is that vagueness is a diachronic phenomenon, which only emerges when we consider the semantic state of a language over time (or more generally, over multiple instances of interpretation). It is to be accounted for not (primarily) in terms of the nature and number of correct interpretations (at any time), but in terms of how the intended interpretation(s) changes over time. Hence the opening sentence of Tappenden (1993): ‘‘This paper develops one aspect of a program aimed at studying normative constraints on the evolution of language’’, and Soames (1999, 209): ‘‘We need a dynamic model of vagueness’’ (my emphases). ⁸¹ In fact, the interpretations in Lewis 1983 [1970] are more complicated than the classical interpretations countenanced by the plurivaluationist, so we need to be a little more subtle—but my basic point remains. The point can be more accurately stated thus: relative to a given syntax and a given sort of interpretation, plurivaluationism and pragmatism are simply stylistic variants of one another. Keefe 1998a argues (independently) that pragmatism either fails or collapses into supervaluationism (see also Keefe 2000, ch. 6). I agree with many of her points, but wish to stress that in my terms, pragmatism is the same as plurivaluationism, not supervaluationism.

114  Consider a vague predicate: say, ‘is tall’. Here is one way in which the extension of this predicate might be thought to vary with context. Suppose I say ‘Bill Bradley is tall’. If we are engaged in a discussion of current and former basketball players, my claim would seem to be false, because Bill Bradley is below the average height of current and former basketball players. If we are engaged in a discussion of past presidential candidates, my claim would seem to be true, because Bill Bradley is among the tallest of past presidential candidates. Plausibly, what is going on here is that ‘is tall’ has a different extension in the different contexts: in the ﬁrst context it excludes persons who are not of a height signiﬁcantly greater than the average height of current and former basketball players; in the second context it excludes persons who are not of a height signiﬁcantly greater than the average height of past presidential candidates. Many would agree with (something like) this story. Yet clearly this story does not get to the heart of the issue of the vagueness of ‘is tall’—if in fact it touches this issue at all. There are two ways of seeing this. First, the vagueness of ‘is tall’ emerges within a ﬁxed context of the sort just considered: for example, we can easily generate a Sorites series for the predicate ‘is tall (for a basketball player)’, and likewise for the predicate ‘is tall (for a presidential candidate)’. Second, the phenomenon under consideration affects precise terms, such as ‘act of parliament’. Assuming that the above story about ‘is tall’ is plausible, it is also plausible that ‘is an act of parliament’ has a different extension when uttered in a discussion of English law from the one it has when uttered in a discussion of Australian law. Yet there is no vagueness about whether something is an act of the English parliament or whether something is an act of the Australian parliament. It thus seems clear that ‘large-scale’ contextual variation of extension of the sort just discussed is different from vagueness, which arises even within a ﬁxed (large-scale) context. Accounts of vagueness should therefore kick in and do their distinctive work after the (large-scale) context has been ﬁxed. So contextualism about vagueness is not the view that ‘tall’ has a different extension when we are discussing basketball players from when we are discussing jockeys, that ‘bald’ has a different extension when we are looking for a male model for a shampoo commercial from when we are looking for a male model for a scalp oil commercial, and so on. That thesis—about what I am calling large-scale contextual variation—is not an alternative to any of the views we have discussed, but a near truism which proponents of

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all of them should accept. So what is contextualism about vagueness? As we shall see, we can have a contextualist version of any of the views we have so far considered.⁸² However, it will make for the simplest presentation of the contextualist view if we begin with contextualism built on a foundation of the strong Kleene recursive truth gap view (see §2.3.1).⁸³ We are thus to suppose that the semantic state of the language is just the way the strong Kleene recursive truth gap view says it is. A vague predicate such as ‘is tall’ is assigned as its extension a partial function from the domain to the set of classical truth values. The predicate is true of those objects sent to 1 by this function, false of those objects sent to 0 by this function, and neither true nor false of objects sent nowhere by this function. The truth values of compound sentences are determined recursively, via the strong Kleene truth tables. For the contextualist, however, this is just the beginning of the story: it is just the way things are initially, at the beginning of our conversation or discourse. At every stage of the discourse, the semantic state of the language will ﬁt this general description—but the speciﬁcs will change: which partial interpretation is intended will alter, and so, of which objects ‘is tall’ is true and false will change. How and why do these changes take place? The contextualist position is characterized by two theses, which I call Freedom and Power. Freedom tells us that if P is vague and a is a borderline case of P, then a competent speaker is free to assert Pa or to assert ¬Pa, without compromising her competence.⁸⁴ Power tells us that if a speaker does assert Pa in such a situation, then this assertion thereby becomes true. That is, if P is vague and a is a borderline case of P, and competent speakers judge that Pa, then a is in the extension of P just because of this judgement.⁸⁵ Note that Pa is neither true nor false, on the interpretation that was correct just before Pa was judged or uttered; but making the judgement or utterance changes which interpretation is correct, in such a way that the newly correct interpretation is one on which Pa is true. Now there is a further ⁸² Given (as discussed in the opening paragraph of this section) that contextualism differs from the views we have considered on a dimension orthogonal to any of the dimensions on which those views differ from one another, this is not surprising. ⁸³ Contextualist treatments of vagueness include Kamp 1981; Tappenden 1993; Raffman 1994, 1996; Soames 1999, ch. 7; Fara 2000; and Shapiro 2006. Tappenden, Soames, and Shapiro explore versions of the strong Kleene recursive truth gap approach, while Fara is sympathetic towards classical models as the basis for contextualism. ⁸⁴ This formulation is due to Shapiro 2006, who calls this thesis ‘open texture’. ⁸⁵ Shapiro 2006 calls this thesis ‘judgement-dependence’.

116  important aspect to the contextualist story. It is not only the object a that moves into the extension of P when a competent speaker judges that Pa; other objects move too, according to the rules of adjustment associated with P.⁸⁶ For example, if I decide for present purposes in the course of some conversation (for example, about what T-shirt sizes to order, where the options are S, M, L, and XL) that Bill is tall, then it thereby becomes true that Bill is tall, and it also becomes true of anyone who is negligibly different in height from Bill that he or she is tall (i.e. an interpretation which assigns an extension to ‘is tall’ which assigns 1 to all these people thereby becomes the intended interpretation of our language, at this point in the conversation or discourse).⁸⁷ Note that this further adjustment just happens: I do not have to explicitly consider these other people, and say that they are tall. I simply consider Bill, and say he is to count as tall; this affects the context, and we get a new correct interpretation in which not just Bill, but those negligibly different in height from him, are assigned 1 by the extension of ‘is tall’. It is as if Bill is bound to those who are negligibly different from him in height: when I move him into the positive extension of ‘is tall’ (i.e. into the set of things sent to 1 by the extension of ‘is tall’), those very similar to him move with him. The contextualist idea of deciding borderline cases in a partial interpretation one way or the other is related to, but importantly different from, the supervaluationist’s idea of extending a partial interpretation. For the supervaluationist, an extension decides all borderline cases of all predicates one way or the other, resulting in a classical interpretation. In the contextualist picture, we decide one borderline case of one predicate one way or the other. The rules of adjustment ensure that other borderline cases of this predicate thereby get decided too—but in general, not all the borderline cases of this predicate get decided, and in general, other predicates are not affected.⁸⁸ Thus after the adjustment, we still have a partial interpretation, not a classical one. What is the advantage of adding a contextualist story on top of the strong Kleene recursive truth gap view? I shall argue in Chapter 4 that a major ⁸⁶ This term is due to Soames 1999. ⁸⁷ Similarly, if I decide that Bill is not tall, then an interpretation which assigns an extension to ‘is tall’ which assigns 0 to Bill and those who are negligibly different from him in height thereby becomes the intended interpretation of our language, at this point in the conversation or discourse. ⁸⁸ With the exception of predicates whose meanings are related to those of the predicate directly affected—e.g. ‘is short’, in the case where we decide that Bob is to be tall.

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problem with all versions of the truth gap view (and of the three-valued view) is that they do not allow for a gradual or jolt-free transition between the clear cases and countercases: they posit two sharp semantic jumps in a Sorites series for a predicate F —one between the objects of which the predicate is true and those of which it is neither true nor false, and one between the objects of which the predicate is neither true nor false and those of which it is false.⁸⁹ One apparent advantage of contextualism over the (non-contextualist) strong Kleene recursive truth gap view is that it can explain why we do not think there are such sharp transitions (even though in fact there are, in any context).⁹⁰ Suppose that in the current context, Bob is the last tall man: in the interpretation which is correct at this stage of the conversation or discourse, Bob is sent to 1 by the extension of ‘is tall’, while everyone shorter than Bob is sent nowhere or to 0. If I wanted to identify the boundary between the tall and the borderline cases as lying between Bob and the next man, I would have to say ‘Bob is tall, and the next man is a borderline case’. But now we have a problem: when I say ‘Bob is tall’, the context shifts in such a way that everyone negligibly different from Bob is also sent to 1, and this includes the next man. So I generate a new context in which what I set out to say is not true: in the new context, Bob is tall, and so is the next man—he is no longer a borderline case. The rules of adjustment ensure that when I classify Bob as tall, the next man goes with him. Thus I can never state where the boundary between the positive and borderline cases (or negative and borderline cases) is without generating a new context in which the boundary is somewhere else. We can never nail down the borderline: just as we are about to get it in our sights, it shifts. The contextualist can thus explain why we do not think there are sharp semantic jumps in a Sorites series for a predicate F (even though there are such jumps, in any context, on this view), in a way that the basic truth gap view cannot. The response to the Sorites paradox on the part of the recursive truth gap view was as follows: the error in the paradoxical reasoning is that some of the Sorites conditionals are not true; the reason we are taken in by the paradox is that none of the conditionals is false, either. The contextualist ⁸⁹ As I have mentioned, this jolt problem is one of two problems which are commonly run together under the heading ‘higher-order vagueness’. The other one is the location problem: the problem of how the positions of these jumps could be determined by our usage of vague terms. This, as I have argued and will argue further, is much less of a problem for the truth gap and three-valued views. ⁹⁰ I say the advantage is apparent, because I shall argue in Ch. 4 that contextualism does not solve the jolt problem.

118  inherits this solution,⁹¹ but can add something more as well. First, not only are none of the Sorites conditionals false (in any context), but furthermore, if we think about any of the conditionals, it is liable to become true. Recall that if we say that Bill is tall, it thereby becomes true that Ben is tall also (where Ben is just after Bill in the Sorites series). Now of course if we merely entertain the conditional ‘If Bill is tall, then Ben is tall’, we do not make it true; but if we say for the sake of argument that Bill is tall, and then consider whether, given that Bill is tall, Ben is tall, we do make the conditional true. This provides an added reason why we tend to think that the Sorites conditionals are true, and hence get taken in by the paradox. Second, the contextualist has a response to the dynamic version of the Sorites paradox (also known as the ‘forced march’ Sorites paradox (Horgan 1994)). Suppose we are walked along our Sorites series for F, and asked of each object in the series whether it is F, and then walked back the other way, and asked the same question of each object again. It is very likely that the point at which we stopped saying ‘Yes’ on the way out would be further along the series than the point at which we started saying ‘Yes’ on the way back. This behaviour might seem rather difﬁcult to explain on the recursive truth gap view, but it is easily explained by the contextualist: as we classify an object as F, we thereby make it true that that object and all others very similar to it—including the next object in the series—are F. We thus push the boundary between the F’s and the borderline cases out before us as we go—and on the way back, we push the boundary back the other way.⁹² Let us now consider other forms of contextualism. If we take a nonrecursive truth gap view—say, supervaluationism—as our foundation, we get a very similar view to the one just presented. At any stage of the conversation or discourse, one partial interpretation is the intended one; speakers have the freedom to classify borderline cases of a predicate one way or the other, and so classifying them changes which interpretation is intended in such a way that the classiﬁcation is true. The only difference from ⁹¹ In fact, not all contextualists accept their inheritance. For example Soames 1999, 214 ff. offers a rather more complex solution to the regular Sorites paradox, based upon the solution to the dynamic Sorites paradox provided by his contextualism (see on). ⁹² There is clearly a problem looming here when we get all the way through the borderline cases: when we classify the last borderline case as P, we thereby push the next object in the series into the positive extension of P also. But the next object in the series was not a borderline case of P —it was a clear negative case—and Freedom and Power tell us only that we can make borderline cases into positive or negative cases. For an objection along these lines directed speciﬁcally at Soames’s contextualism, see Robertson 2000; for a reply see Soames 2002.

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the recursive view concerns what is said about compound sentences. In particular, then, the response to the Sorites paradox will be different—but the basic form of the response will be the same: the contextualist inherits a solution from the view which she takes as her starting point, and can then add to this a further explanation of why we ﬁnd the Sorites compelling, as well as an account of the dynamic Sorites paradox. What about many-valued contextualism? Recursive three-valued contextualism is perfectly analogous to recursive partial contextualism, and likewise for their non-recursive counterparts. With more than three truth values, the basic story is still the same: the only difference is that now, presumably, the rules of adjustment should say not only which other objects (apart from the one we are directly classifying) should get reclassiﬁed by the extension of the vague predicate in question, but how much they should be reclassiﬁed.⁹³ What if we were to build our contextualism on a foundation of classical interpretations? Recall that Freedom tells us that if P is vague and a is a borderline case of P, then a competent speaker is free to assert Pa or to assert ¬Pa, without compromising her competence. Thus far we have taken the borderline cases to be the objects which the extension of P sends nowhere. When we move to classical interpretations, there are no such objects, so we need to interpret ‘borderline case’ differently. An obvious move would be to borrow the epistemicist’s idea that a borderline case is one of which we cannot know whether it is P. Power then tells us that if a speaker does assert Pa or ¬Pa of such an object a, then which interpretation is intended changes in such a way that the assertion is true.⁹⁴ This is importantly different from the epistemicist view. On the latter view, while you might not show incompetence with P by asserting Pa when a is a borderline case which in fact is not in the extension of P, you would still say something false. On the contextualist, but not the epistemicist view, saying makes it so. ⁹³ Goguen 1968–69, 351 considers a view according to which the algebra of truth values may vary from context to context. This is much more radical than fuzzy contextualism in my sense, according to which the intended interpretation may vary from context to context, but is always a fuzzy interpretation—i.e. is always based on the same underlying algebra of fuzzy truth values. ⁹⁴ This might not in fact involve a change of intended interpretation if a already was P (unbeknownst to the speaker) and the speaker asserted Pa, or if a already was not P (unbeknownst to the speaker) and the speaker asserted ¬Pa. But even in these cases, it might still involve a change of interpretation, if a was very close to the border: the rules of adjustment might then shift the border, even though the border does not cross a.

120  Is there any advantage for a contextualist in going classical? Simplicity and conservativeness are the obvious attractions. There is also a big disadvantage: we run headlong into the location problem, the avoidance of which was an advantage of moving from the classical picture to partial (or many-valued) interpretations. The classical contextualist tells us that in any context, there is a unique intended classical interpretation of our language. She thus posits bipartite semantic distinctions (between the bald and the non-bald, etc.) where we have tripartite distinctions of usage (those we agree are bald, those we deny are bald, and those over whom we hedge). The problem then is how the dividing line on the semantic side could have got to be located where it is. Telling us that it will be located somewhere else in a minute does not help at all with this problem. Note that while classical contextualism, non-contextualist supervaluationism, and non-contextualist plurivaluationism all have something in common—namely, they consider a range of classical models—there are crucial differences between these three views. Classical contextualism and noncontextualist plurivaluationism are alike—and unlike non-contextualist supervaluationism—in that they regard the classical models they consider as genuine interpretations of the language (not as mere auxiliary calculating devices). Classical contextualism and plurivaluationism differ, in that the contextualist considers these models to be correct interpretations of the language one at a time, whereas the plurivaluationist considers these models to be acceptable interpretations of the language all at once (i.e. plurivaluationism is a synchronic account of vagueness, while contextualism is a diachronic account). Finally, there is a version of contextualism built on plurivaluationist foundations. Indeed, if our reasons for being a plurivaluationist are of the sort I mentioned earlier, then contextualism is a very happy addition to the view. The sort of plurivaluationist I am thinking of is one who regards a unique intended classical interpretation as a goal which we never actually reach: our usage rules out certain interpretations as incorrect, but does not narrow down the remaining set of acceptable interpretations to just one member. Contextualism sits easily with this view. As we newly classify borderline cases of P as positive/negative cases, we narrow down the set of acceptable interpretations in such a way that an interpretation on which Pa is false/true is no longer acceptable.

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I have not indicated in the title of this section how contextualism differs from the classical picture. Of course, contextualism built on a non-classical foundation differs from the classical picture in an obvious way—but what about contextualism built on a foundation of classical interpretations? One thing which is clear is that the classical picture does not demand that which interpretation of a given language is the intended one be ﬁxed for all time. One way of seeing this is to note that the classical picture is meant to give an account of mathematical language—and it is a commonplace that mathematicians use the same symbol to mean different things in different contexts. But this fact does not establish that contextualism is already part of the classical picture. We already saw that everyone should admit large-scale contextual variation. What is distinctive of contextualism is the positing of a particular mechanism for changing the intended interpretation, which applies in particular to vague predicates. The mechanism is the combination of Freedom (if P is vague and a is a borderline case of P, then a competent speaker is free to assert Pa or to assert ¬Pa, without compromising her competence) and Power (if a speaker does assert Pa in such a situation, then this assertion thereby becomes true). Now clearly Freedom and Power are not part of the classical picture—but also, they do not conﬂict with any part of it. The classical picture allows variation of the intended interpretation with context; contextualism posits a particular mechanism—speciﬁcally involving vague predicates—which generates such variation. So, unlike proponents of any of the other views we have examined—with the exception of epistemicism—the (classical) contextualist does not reject any part of the classical picture; but unlike the epistemicist, the contextualist does not simply accept the classical semantic story as is: she adds something to it. This addition—Freedom plus Power as a mechanism for changing the intended interpretation—can also be added to other sorts of semantic view, apart from the classical one: to many-valued views, partial views, and so on. That is why we can have a contextualist version of any of the views discussed in previous sections. 2.6.1 Worldly Vagueness? We turn ﬁnally to the question of whether the contextualist locates vagueness in the relationship between language and the world, or in the

122  world itself. The answer is that this depends entirely upon what sorts of interpretations are chosen as the basis for the contextualist view in question. Worldly vagueness is a matter of the internal nature of our models; semantic indeterminacy is a matter of the relationship between a language and its models. A model is a picture of what the world is like. Classical models, for example, tell us that all the properties and relations in the world are precise: for each property and each object in the world, the object either possesses that property (outright) or does not possess it (at all). Fuzzy models, for example, tell us that the world contains inherently vague properties and relations: there are objects and properties such that—out there in the world, quite independently of human thought and language—that object possesses that property to an intermediate degree. So if a contextualist builds her view on classical foundations, she posits no worldly vagueness; if she builds her view on fuzzy foundations, she does posit worldly vagueness; and so on for the other sorts of models she might take as the basis for her view. If there is worldly vagueness in the base models, the contextualist superstructure cannot take it away; if there is no worldly vagueness in the base models, the contextualist superstructure cannot introduce it. Semantic indeterminacy enters when a view tells us that which object or property is picked out by a name or predicate is indeterminate: there are multiple, equally correct answers. In other words, semantic indeterminacy enters when a view tells us that instead of one correct interpretation, a language has (at some time) many acceptable interpretations. This is what plurivaluationism tells us; it is not what contextualism tells us (except for contextualism built on a plurivaluationist foundation). The contextualist tells us that meaning is variable —it is one unique way (there is one intended interpretation) at any given time, but different ways at different times—not that it is indeterminate. (Contextualism built on plurivaluationist foundations tells us that meaning is both indeterminate—at a time—and variable—over time.)

2.7 Intuitionism (Assert Nothing) Putnam (1983b, 284–6) suggests that the best logical system for schematizing inferences that involve vague terms is intuitionism: we should ‘‘treat

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vague predicates (e.g. ‘bald’) just as undecidable predicates are treated in intuitionist logic’’. We accept as valid the argument: 1. ∀n (If a man with n hairs is bald, then a man with n + 1 hairs is bald). 2. A man with 0 hairs is bald. 3. ∴ ∀n (A man with n hairs is bald). We also accept that 2 is true and 3 is false. Thus we conclude that 1 is false, and we accept its negation. However, reasoning intuitionistically, we are not then forced to conclude: 4. ∃n (A man with n hairs is bald, and a man with n + 1 hairs is not bald) as we would be if we were reasoning classically. Thus we can defuse the Sorites paradox, without being committed to the highly unintuitive 4. Read and Wright (1985) argue that if intuitionist logic is combined with the standard intuitionist semantics for negation, then the Sorites paradox recurs in a modiﬁed form. Putnam (1985) replies that he ‘‘was proposing a logic for inferences involving vague predicates, not a ‘semantics’, so the attack on intuitionist semantics is another example of ignoratio elenchi’’.⁹⁵ Putnam’s, then, is an example of a ‘no semantics’ approach to vagueness. Such an approach differs from the classical picture not over this or that matter of detail—whether concerning the internal nature of the models countenanced or the external relationship between a language and its intended model(s)—but by not proposing any semantic picture at all. In §2.1.3.1, I contrasted semantic realism and antirealism, and said that the majority of current approaches to vagueness assume a semantic realist viewpoint. Here, however, we have a view of vagueness from an antirepresentationalist perspective: a view according to which we approach vague language not in terms of its referential and representational relationships to the world, but in terms of what one is permitted to assert at a given point. In much the way that model theory is the natural framework within which to discuss and develop views of vagueness from a semantic realist perspective, proof theory is the natural framework within which to discuss and develop views of vagueness from an antirepresentationalist perspective. Putnam’s ⁹⁵ For further discussion of Putnam’s view see Schwartz 1987, 1990; Rea 1989; Schwartz and Throop 1991; Putnam 1991; Williamson 1996a; and Chambers 1998. For a different intuitionist approach to vagueness, see Wright 2001.

124  is such a proof-theoretic, as opposed to model-theoretic, proposal.⁹⁶ It is worth noting the possibility of a proof-theoretic/antirepresentationalist as opposed to model-theoretic/semantic realist approach to vagueness, but I will not discuss the former sort of approach in this book. That is because the interesting issues here are all concerned with which sort of approach to meaning is better in general —rather than having anything in particular to do with vagueness—and because (relatedly) there simply is not the weight of detailed theories of vagueness on the proof-theoretic side that there is on the model-theoretic side, and so there is less to talk about on the former side, and less interesting connections and patterns to discern between different theories. Perhaps if the general debate between semantic realism and antirealism moves in a certain direction, a proof-theoretic approach to vagueness will become a serious contender—but at this stage the race is between rival model-theoretic approaches. ⁹⁶ Of course, proponents of model-theoretic approaches to vagueness may explore the question of the appropriate proof theory for reasoning involving vague language—but they will regard proof theory as semantically grounded. That is, they will regard a system of proof theory as correct if it always preserves truth, or some other important semantically deﬁned property. What I am calling the prooftheoretic approach, by contrast, regards proof theory as able to stand alone, without (model-theoretic) semantics to give it life.

P A RT II

Vagueness

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3 What is Vagueness? In the Introduction, I presented three standard characterizations of vague predicates: they give rise to borderline cases; their extensions have blurry boundaries; and they generate Sorites paradoxes. As noted, these three characterizations seem to be quite closely related to one another—yet not necessarily so closely related that they are simply three ways of saying the same thing. So what is the relationship between them? Rather than three piecemeal characterizations of vagueness, it would be desirable to have a fundamental deﬁni