Vagueness in Context

VAGUENESS IN CONTEXT

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Vagueness in Context STEWART SHAPIRO

CLARENDON PRESS OXFORD


AC

Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York # Stewart Shapiro 2006 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2006 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 Shapiro, Stewart, 1951– Vagueness in context / Stewart Shapiro. p. cm. 1. Vagueness (Philosophy) 2. Semantics (Philosophy) 3. Language and languages— Philosophy. I. Title. B105.V33S53 2006 121 0 .68—dc22 2005026129 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk ISBN 0–19–928039–8 978–0–19–928039–1 1 3 5 7 9 10 8 6 4 2

To my mother, Florence Feldman Shapiro. There is no vagueness in her devotion and strength.

Preface The purpose of this work is to develop a philosophical and a formal, modeltheoretic account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. The essay has a dialectical structure, typical of much work in logic and formal semantics. The first chapter provides a simplified, perhaps naive, account of how vagueness arises in language, and is manifest in the use of language. It is a commonplace that the extensions of vague terms vary with such contextual factors as the comparison class and paradigm cases. A person can be tall with respect to male accountants and not tall (or even short) with respect to professional basketball players. A person can be wealthy with respect to local business tycoons, but not wealthy with respect to CEOs of major software companies. The main feature of the present account is that the extensions (and anti-extensions) of vague terms also vary in the course of a conversation, even after the external contextual features, such as the comparison class, are fixed. A central thesis of the view is that, in some cases, a competent speaker of the language can go either way in the borderline area of a vague predicate without sinning against the meaning of the words and the non-linguistic facts. I call this open-texture, borrowing the term from Friedrich Waismann. It is perhaps getting common to use the term ‘‘contextualist’’ for views that the meanings of the terms in question—vague ones in this case—vary from context to context. No such claim is made here. It seems absurd to say that the meaning of a vague term changes every time a borderline case is decided in the course of a conversation, but I do not have settled views on what counts as ‘‘meaning’’. I continue to use the term ‘‘contextualist’’ to characterize the informal, philosophical elaboration of the present account. Nothing turns on this, however, and I will be glad to give up the term. Chapter 2 is a short introduction on the purpose and function of formal logic. I adopt a perspective that model theory provides a mathematical model of certain aspects of the correct use of certain terms, in roughly the same sense as the Bohr model is a model of the atom, and a point-mass is a model of an extended physical object. Truth in a model is a model of truth, or, to be a bit more precise, a model of the truth conditions of natural language sentences. Mathematical models are rarely, if ever, perfect matches of what they are models of. In the present case, artifacts are introduced from the fact that the model theory takes place in set theory. Set theory, of course, is not vague, but it can be used to model the semantic and pragmatic behavior of vague terms. In such cases, one must be careful not to draw conclusions about the phenomena being modeled from artifacts of the model.

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Chapter 3 begins to provide a technical model theory, using the resources of mathematical logic, for vagueness in some formalized languages. The system has a similar structure to the supervaluationist approach, employing the notion of a sharpening (or precisification) of a base interpretation. In line with the philosophical account, however, the notion of super-truth does not play a central role in the development of validity. The model theory is much like that of the Kripke semantics for intuitionistic logic, except that both extensions and anti-extensions vary, in concert with each other, throughout a given frame. And the system does not rely on completely sharp interpretations. In Ch. 4, various connectives and quantifiers are defined, and we settle on a local notion of validity. The ultimate goal, of course, is to delimit a plausible notion of logical consequence, and to explore what happens with the sorites paradox. It might be noted that the notion of ‘‘context’’ does not appear, as such, in the formal development. Several features of the deployment of vague terms can go under that name, and I see no need to call one of them the context of utterance. This might attenuate any misunderstandings of the philosophical term ‘‘contextualist’’. Problems with the simplified account and some unnatural features of the model theory lead to refinements of the philosophical account, the modeltheoretic semantics, and perhaps the logic as well. Chapter 5 deals with what passes for higher-order vagueness—vagueness in the notions of ‘‘determinacy’’ and ‘‘borderline’’. The philosophical picture is developed, by extending and modifying the account presented in Ch. 1. This is followed with the required modifications to the model theory, and the central meta-theorems. Up to that point in the book, the treatment concerns only predicates (or properties). Chapter 6 deals with singular terms (or objects). Some physical objects, such as the North Sea, seem to have fuzzy boundaries. If you push things, perhaps every physical object has fuzzy boundaries. A second concern of this chapter is with apparent indeterminacies in how to count certain objects. Do we have one cloud or two in the sky? One person or two in a given room? A third item of interest are abstract terms, such as income groups and heights, that also seem to be subject to sorites, since the underlying relation is not transitive. As with Ch. 5, the philosophical account is modified to accommodate the items in question—vague singular terms—and then the needed additions and changes to the model theory are sketched. Chapter 7 begins with the question of whether vagueness is (merely) a linguistic phenomenon, concerning how we describe the world, or whether it is the world itself that is vague. To be frank, I express frustration at this issue of metaphysical vagueness, rather than adjudicating it one way or the other. As I see it, vagueness is a linguistic phenomenon, due to kinds of languages that humans speak. But vagueness is also due to the world we find ourselves in, as we try to communicate features of it to each other. Vagueness is also due to the kinds of beings we are. I see no need to blame any one of these features for the pervasive phenomenon of

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vagueness. The second half of Ch. 7 concerns the objectivity of vague discourse. Perhaps this is what is at stake in typical discussion of metaphysical vagueness. The book closes with a brief Appendix on Friedrich Waismann’s account of open-texture and analyticity. His account of the openness of language lends perspective to the present account of vagueness. I have been working on this project, off and on, for several years. Along the way, I incurred many intellectual debts, and am certain to have forgotten some of them. But this does not excuse me from acknowledging at least some of them. I was first introduced to the issues and some of the literature on vagueness when I served on the Dissertation committees of Elizabeth Cohen and Rick DeWitt. The former defended supervaluation and the latter a many-valued logic. Intense conversations with them, and with their supervisor, George Schumm, and watching the three of them interact, got me to see the strengths and, more importantly, the weaknesses of those two approaches. Although each was clearly onto something, the solution had to lie elsewhere. I have learned a lot from conversations with George over the years since. My largest debt is to my colleague and friend Diana Raffman. Her original (contextualist?) resolution of the sorites paradox shows how psychological and pragmatic features go into fixing the extension of vague terms on any given utterance (Raffman 1994, 1996). The overall structure of her resolution inspired the present philosophical account, which is formulated in terms of conversations rather than psychological states, and it suggested the concomitant model theory. We taught two graduate seminars on vagueness together. One of these focused on the more or less standard literature (at the time), and the other delved into our own evolving accounts. In both cases I benefited immensely from Diana’s presentations and from student participation. Diana is a wonderful critic, helping me to see where I have gone wrong and, more importantly, helping me to see how to put things. Even when she thinks I am fundamentally mistaken, she has the ability to delve into how I see things (or am trying to see things) and help me explore the terrain. I could not have written this book without her, and I only hope that I have been as useful to her own work as she has been to mine. Turning to my second academic home, the Arche´ Research Centre at the University of St Andrews has been extremely useful in this, and many other projects. Over the past few years, I gave three or four series of seminars on this project (depending on how to count such things). Even though I went over the same material several times, as it was refined, the audiences never tired of delving into the view, finding shortcomings, and helping me to formulate things properly. One cannot underestimate the value of a dedicated group of colleagues who become familiar with the basic view, and the nuances of its model theory, so that we can explore its extensions together. I owe much to Crispin Wright. Even though we approached the subject of vagueness from rather different perspectives, our views have evolved together

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over the years, to the extent that we now agree more than we disagree. Crispin’s influence on my view derives both from his voluminous and insightful publications on vagueness, and from countless hours of conversation, both in seminar and in private. This will become apparent as the reader works through the present book. My former student Roy Cook, who spent four years at Arche´ as a postdoctoral fellow in the philosophy of mathematics project, helped me to formulate and elaborate the ‘‘logic as model’’ approach, and he helped me to formulate and then clean up the model theory, preventing some embarrassing errors. The other post-doctoral fellow in the Arche´ mathematics project, Agustı´n Rayo, has been the most persistent critic of this project. He pointed out many shortcomings and infelicities in the project, and helped me to formulate it more forcefully. He also helped me to see where I agreed and where I disagreed with various key claims in the literature, thus preventing many potential misunderstandings. I gave talks on the basic view, without the model theory, at a number of conferences and colloquia. These include the University of Maryland, the University of Florida, Union College, Hebrew University of Jerusalem, University of London, the Buffalo Logic Colloquium, and the conference ‘‘Liars and Heaps’’, held at the University of Connecticut. I am much indebted to the audiences for helping me to sharpen the view, and to avoid errors. An early version of Ch. 1 appeared as ‘‘Vagueness and Conversation’’ in the proceedings, Liars and Heaps (edited by JC Beall, Oxford, Oxford University Press, 2003). Thanks especially to my commentator on that occasion, Rosanna Keefe. Her remarks (published in the proceedings) helped me to reformulate some key theses. Thanks also to Rosanna for many hours of conversation and correspondence over the past few years, concerning both this chapter and much of the rest of the book. A version of the beginning of Ch. 5, on higher-order vagueness, is the subject of a Joint Session session, and appears in the supplement to the Proceedings of the Aristotelian Society. Thanks to the respondent, my St Andrews colleague and friend Patrick Greenough. Patrick was instrumental in the development of this book, both in the aforementioned Arche´ seminars and in countless hours of conversation. I have benefited from conversation and correspondence with many others over the years. The list includes Louise Antony, Jack Arnold, Julius Barbanel, Julian Cole, Mark Colyvan, Neil Cooper, Richard Dietz, Haim Gaifman, Katherine Hawley, Robert Kraut, Joseph Levine, Kirk Ludwig, Fraser MacBride, Michael Morreau, Sebastiano Moruzzi, Carl Posy, Graham Priest, Greg Ray, Georges Rey, Stephen Schiffer, Keith Simmons, Nicholas J. Smith, Mark Steiner, Jamie Tappenden, Neil Tennant, Brian Weatherson, Timothy Williamson, Elia Zardini, and several anonymous referees for Oxford University Press. I apologize to those

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I may have neglected to mention. My memory never was good, and it is getting worse. Thanks also to Eric Carter for helping with the proof reading and index, to Peter Momtchiloff, of Oxford University Press, for encouraging me to pursue this project, in book form, and to Rupert Cousens and Rebecca Bryant for seeing it through the process. This book is dedicated to my mother. Her underlying spiritual strength and devotion to family is an inspiration to all who know her. Nothing vague about that.

‘‘I don’t know what you mean by ‘glory’ ’’, Alice said. Humpty Dumpty smiled contemptuously. ‘‘Of course you don’t—till I tell you. I meant ‘there’s a nice knock-down argument for you!’ ’’ ‘‘But ‘glory’ doesn’t mean ‘a nice knock-down argument’ ’’, Alice objected. ‘‘When I use a word’’, Humpty Dumpty said in rather a scornful tone, ‘‘it means just what I choose it to mean—neither more nor less’’. ‘‘The question is’’, said Alice, ‘‘whether you can make words mean so many different things’’. ‘‘The question is’’, said Humpty Dumpty, ‘‘which is to be master—that’s all’’. Lewis Carroll, Through the Looking Glass . . . give me another word for it, you who are so good with words, and at keeping things vague. Because I need some of that vagueness now; it’s all come back too clearly. Joan Baez, ‘‘Diamonds and Rust’’

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Contents 1. The Nature of Vagueness: Humpty Dumpty Gets His Due

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2. Interlude: The Place and Role of Model Theory

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3. A Start on Model Theory

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4. Connectives, Quantifiers, Logic

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5. Refinements and Extensions I: So-Called ‘‘Higher-Order Vagueness’’

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6. Refinements and Extensions II: Objects, Identity, and Abstracts

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7. Metaphysical Matters: Language, the World, and Objectivity

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APPENDIX: Waismann on Open-Texture and Analyticity

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References Index

216 221

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1 The Nature of Vagueness: Humpty Dumpty Gets His Due Are there not shades of colour . . . which it is permissible to describe as ‘‘red’’—such a description would not be determinately wrong—but also permissible not to, since that description would not be determinately right either? Such a view of vague statements ought to be controversial. It’s tempting to say . . . that a statement’s possessing (one kind of ) vagueness just consists in the fact that, under certain circumstances, cognitively lucid, fully informed and properly functioning subjects may faultlessly differ about it. Crispin Wright [1992: 97, 144]

1. A N EUTRAL CHARACTERIZATION? WHAT ARE WE TALKING ABOUT? It is not easy to say what the phenomenon of vagueness is without begging the question against some philosophical account or other. One typically begins with examples of vague terms, such as ‘‘heap’’, ‘‘bald’’, or color words such as ‘‘red’’. A word or property is vague if it is like one of those. Although this seems to beg no questions, it highlights the question: In what way are the examples alike? Discussions often begin with an implicit or explicit statement that vague terms have, or might have, borderline cases.1 An author might give a purported borderline case of a vague predicate, wondering what to say or think about it. The reader might be invited to consider an object whose color is midway between red and pink, a supposed borderline case of ‘‘red’’ (and ‘‘pink’’). Or one might begin with a person for whom it is not quite correct to say that she is tall, and not quite correct to say that she is short, or even not quite correct to say that she fails to be tall. Controversy arises when we try to say what it is to be a borderline case, or what it is to be ‘‘not quite correct’’. 1 This is not to say that only vague terms have borderline cases. To adapt an example of Kit Fine [1975], define a natural number n to be ‘‘nice’’ if n > 15, ‘‘non-nice’’ if n < 13, and ‘‘borderlinenice’’ otherwise. Intuitively, ‘‘nice’’ is not vague, since the three categories are sharply bounded.

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Let P be a monadic predicate and let a be a singular term that denotes an object in the range of applicability for P. So P might be ‘‘bald’’ and a the name of a man, or P could be ‘‘red’’ and a the name of a colored object. As a first attempt, let us say that the object denoted by a is a borderline case of the predicate P if it is not true that Pa nor is it true that :Pa. Apologies for any use-mention conflations. Although some philosophers use a definition like this, others point out that it leads to contradiction when coupled with the following platitude concerning truth, sometimes called the T-scheme: It is true that F, if and only if F. The contrapositive of an instance of the T-scheme entails that if it is not true that Pa, then :Pa. The contrapositive of another instance entails that if it is not true that :Pa then : :Pa. So it follows that if the object denoted by a fits our first definition of a borderline case of P, then :Pa&: :Pa, which is an outright contradiction. The reasoning here is quite straightforward, using only intuitionistically correct inferences. So an advocate of the first definition of a borderline case must either advocate some radical changes to logic, or else demur from, or at least modify, the T-scheme. For example, the theorist might argue that the contrapositive of the T-scheme does not hold, perhaps because of the nature of the embedded conditional. A second attempt to define ‘‘borderline’’, and thus vagueness, invokes a ‘‘definitely’’ operator in the object language. Accordingly, the object denoted by a is a borderline case of P if it is not definitely true that Pa nor is it definitely true that :Pa. The focus now turns to what the word ‘‘definitely’’ means. Each theorist has his or her own definition of definiteness, and the various concepts have little in common. There seems to be no way to make further progress in defining ‘‘borderline case’’ or ‘‘definitely’’ without begging the question against some view or other. Most claim that vagueness involves ignorance. If the object denoted by a is a borderline case of P, then no one knows (and, perhaps, no one can know) whether Pa or :Pa is true. The competing accounts of vagueness then dispute what this ignorance comes to. Epistemicists, such as Timothy Williamson [1994] and Roy Sorenson [1988, 2001], say that with borderline cases we are ignorant of unknowable facts. The object denoted by a does or does not satisfy P, but it is not possible to know which. A supervaluationist (Kit Fine 1975, Rosanna Keefe 2000) or other indeterminist (e.g. Michael Tye 1994) says that were are ignorant because the sentence Pa is neither true nor false. In other words, there is nothing to know. The advocate of a many-valued treatment of vagueness (Dorothy Edgington 1992, 1997; Kenton Machina 1976) traces ignorance to the presence of an intermediate truth value. We do not know the sentence because it is not completely true, and we do not know its negation since the sentence is not

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completely false. An incoherentist (Michael Dummett 1975; Terrence Horgan 1994b) says that we do not know because our language is inconsistent. Of course, it will not do to characterize vagueness as ignorance, even for the epistemicist, since there are plenty of things we are ignorant of—and plenty of things that we are necessarily ignorant of—other than borderline cases of vague predicates. Moreover, there is no consensus that ignorance is necessary for vagueness. Contextualists, such as Diana Raffman [1994, 1996], Delia Graff [2000], Scott Soames [1999: ch. 3], Hans Kamp [1981], Haim Gaifmann [2005], and the present account, hold that we are not (always) ignorant of the truth values of borderline statements. It is just that the truth values of the sentences in question shift with context.2 Some vague terms are associated with (actual or possible) sorites series. In general, a sorites series for a predicate P is a sequence of objects a0, a1, . . . , an, where Pa0, :Pan, and it is at least prima facie plausible that for each i < n, if Pai then Pai þ 1. There are a number of ways to express this last, inductive principle. One can formulate it as a single sentence, with variables ranging over the natural numbers (or the natural numbers less than n), using essentially the same words as those above. In this case, some principles of (very) elementary arithmetic serve as extra, suppressed premises. In other versions of the sorites, the inductive principle consists of n different premises, each with the form ‘‘if Pai then Pai þ1’’. In either case, the premises are jointly inconsistent, on any transitive consequence relation that sanctions modus ponens and universal instantiation. Here are some paradigms: (1) Consider a series of 2,000 men. The fellow on the left is Yul Brynner, who, we will assume, has no hair at all, and so is clearly bald. Every other man in the series has only a small amount of hair more than the man to his left, and their hair is arranged in roughly the same fashion. The last man, Jerry Garcia in his prime, sports a fine head of hair, and so is not bald. This will be our standard series. (2) Envision a series of piles of sand, all with roughly the same shape. The one on the left is clearly a heap, and every other one has only one less grain than the one to its left. The pile on the right has no sand at all (the null pile) and so is clearly not a heap. (3) Envision a series of colored cards. The leftmost one is red, and every other card is ever so slightly less red and more orange than the card to its left. We can stipulate that each card is visually indistinguishable from the card to its left. A normal observer simply cannot tell them apart. The last card is clearly orange, and so not red. It is beyond dispute that the man with no hair is bald, that the null pile is not a heap, and that the first card in the series is red. It is also beyond dispute that Jerry Garcia, in his prime, is not bald, that the last pile of sand is not a heap, and that the last patch is not red. One is not going to question the basic mathematical principles involved, nor can one reasonably challenge modus ponens, universal 2 In some circles, ‘‘contextualism’’ is a view that the meanings of certain words shift with context. No such claim is in play here. The present ‘‘contextualists’’ speak only of the extensions of vague predicates.

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instantiation, or the transitivity of deduction (Tennant 1987 notwithstanding). So it does not seem reasonable to tinker with the logic. Discussion of sorites arguments thus inevitably turns on the inductive principle. For our first series, it is at least plausible that if two men differ only slightly in the number and arrangement of their hair, and if one is bald, then so is the other: there is no sharp cut-off between the bald ones and the non-bald ones. For the third series, if two cards are visually indistinguishable, and if one is red, then surely the other is, right? There is no sharp cut-off between the red ones and the orange (or non-red) ones. Some authors accept the inductive premise and accept the contradiction, arguing that vagueness induces incoherence (Dummett 1975; Horgan 1994a; 1994b). The most common theme, however, is to reject the inductive principle, and try to show why we thought it was plausible. And there is no consensus as to how this goes in detail. Perhaps everyone can agree, at least, that the presence of an actual or possible sorites series is sufficient for vagueness. But given that most theorists do not endorse the inductive principle, it may not be all that clear what we mean by a sorites series. Presumably, a sorites series is one in which it seems that the inductive principle is true. But once we see what is going on (by adopting the correct theory), we will not find the inductive principle plausible, and so it will not seem that the premises are all true. Moreover, I am not sure that the presence of a sorites series is necessary for vagueness, and thus whether we can use sorites as part of a neutral characterization of the phenomenon. There may be vague predicates for which we cannot easily envision a sorites series. For example, we often hear it said that synonymy is vague (in response to attacks on analyticity, for example). It seems that one can coherently maintain this view, without coming up with a stable notion of ‘‘differing marginally in meaning’’ and a sequence of words in a fixed language to construct a sorites series. Similarly, some predicates such as ‘‘religion’’ and ‘‘mansion’’ do not owe their vagueness to a single dimension, and so it may not be straightforward to define, say, ‘‘differs marginally concerning religionhood’’. It is a curious situation. There are legions of competing accounts of vagueness in the philosophical literature, and yet there appears to be no neutral description of what it is that these theories are accounts of. Is it conceivable that all these theories are at cross-purposes, each being an account of a different linguistic, semantic, or metaphysical phenomenon that the author calls ‘‘vagueness’’? Fine’s and Keefe’s supervaluation approach would be an account of one phenomenon (or two phenomena), Edgington’s and Machina’s degree-theoretic approaches would be accounts of something else entirely, and Williamson’s and Sorenson’s epistemicism focus on yet other features of language. And so on. This strains intuitions. Surely, these accounts are rivals. They are competing theories of one and the same thing. Indeed, the authors spend a great deal of time criticizing each other, and they tend to focus on the same examples: men who are or are not bald,

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chunks of sand that may or may not be heaps, etc. But what exactly are these theories rival accounts of? To get the present project started, I will fall back on the original ostensive definition, as unsatisfactory as it may be. A word is vague if it is relevantly similar to ‘‘bald’’, ‘‘heap’’, and ‘‘red’’, and I trust that enough has been said, here and elsewhere, to give the reader the idea of what it is to be relevantly similar. We may not be able to go much beyond announcing a Wittgensteinian family resemblance before entering a substantial and controversial theory of what this arresting family resemblance comes to. I propose here an account of how idealized versions of such predicates function in language. The interest of the project turns on how widespread the phenomenon of vagueness is. It has been said that just about every (nonmathematical) predicate of a natural language such as English is vague. If so, then an account of vagueness is central to the daunting project of understanding what language is and how it functions. 2. OK, BUT WHAT I S VAGUENESS? The account developed here draws on and benefits from much of the major work on vagueness in the philosophical literature, but it differs sharply from most of the items drawn upon. I have attempted a synthesis, drawing on the best and most plausible aspects of various accounts of vagueness. This section deals with the notions of determinacy and tolerance, at least in a preliminary way, and this introduces the concept of open-texture, the centerpiece of the present account.

2.1. Determinacy The present theory of how vague terms arise in natural language, and how their meaning is determined, begins roughly like that of Vann McGee and Brian McLaughlin [1994] (see also McGee 1991). I take it to be a truism that the competent users of a language determine the meaning of its words and phrases. As indicated in the first epigraph to this book, Lewis Carroll’s Humpty Dumpty says that meaning turns on a question of ‘‘which is to be master’’. When it comes to meaning, the proverbial ‘‘we’’—the community of competent, sincere, honest, well-functioning speakers—are master. And presumably, ‘‘we’’ determine the meanings of expressions by what we say, think, and do. To be sure, I have no special insights to bring as to how words acquire their meaning. There is also a nice question concerning who, exactly, are the competent speakers of a natural language such as English. Or, perhaps better, what is it to be a (competent) speaker of English? Who are ‘‘we’’? Wittgensteinian issues about rule following, language acquisition, etc., loom large. As will become evident, these matters cannot be easily dismissed in

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the usual way, as outside the scope of this work, and there may be no non-circular way to define ‘‘we’’, the competent users of a language. I do not have anything to say that resembles principled resolutions of these issues, or anything in the way of deep (or even shallow) insight, but I will at least gesture toward some desiderata and, perhaps, requirements on answers later, when we turn to so-called ‘‘higher-order vagueness’’ in Ch. 5. The question of who are ‘‘we’’ looms large. The truism that competent users of a language determine meaning is not meant to preclude content externalism, or just about any substantive theory of meaning. The truism is consistent with a claim that meaning supervenes on use, but it does not entail such a thesis. For some philosophers (including me), contemplations about twin earth and the like show that the world has a role to play in fixing the meaning of some of our terms. Even so, this meaning is fixed by the actions and thoughts of speakers of the language in the world in which they live. The word ‘‘water’’ refers to water in English because of the activities of speakers of English in this world. The fact (if it is a fact) that the same linguistic use would yield another meaning in another environment does not undermine the truism in question here. On twin earth, speakers of twin English fix the meaning of their terms by what they say, think, and do. There may be a problem of providing a rigorous, detailed formulation of the truism in such a way that it is clear that it is a truism, and does not beg any questions against substantive accounts of meaning. But I’ll duck that here. Let F be a monadic (possibly complex) predicate in a natural language. McGee and McLaughlin [1994: x2] introduce a technical term ‘‘definitely’’ thus: ‘‘to say that an object a is definitely an F means that the thoughts and practices of speakers of the language determine conditions of application for . . . F, and the facts about a determine that these conditions are met’’. In a note, they add that the word ‘‘determinately’’ would be better, since it captures the underlying idea. I will use the latter term here, especially since the word ‘‘definitely’’ is rather overworked (although ‘‘determinately’’ is also overworked). In general, when ‘‘a sentence is [determinately] true, our thoughts and practices in using the language have established truth conditions for the sentence’’, at least in the actual world, and the presumably ‘‘non-linguistic facts have determined that these conditions are met’’. Since we are still at the level of truisms and definitions, nothing should be all that controversial so far. McGee and McLaughlin are free to define technical terms as it pleases them, and I am free to follow suit (if in fact I am following suit). One point of contention is whether or not ‘‘determinately’’, so defined, is heterological.3 Is the notion of determinateness itself sufficiently determinate? I will have a bit more to say on this by way of further articulation (e.g. x7 below), 3 Following the lead of Coffa [1991: 10], I seek chances to use (as opposed to mention) the word ‘‘heterological’’.

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but will just assume for now that we have hold of at least a vague notion here, which can at least give us a start on theorizing. One point of contention is McGee and McLaughlin’s claim that the ‘‘determinately’’ operator is not, or may not be, compositional. It may be that a sentence of the form ‘‘F _ C’’ is determinately true even if it is not the case that F is determinately true, nor is it the case that C is determinately true. More notably, it may be that a sentence F fails to be determinately true without :F being determinately true. This, I believe, is the source of vagueness. To focus on an example, it may be determinately true that a given color patch is either red or pink, but not determinately true that it is red, and not determinately true that it is pink. Intuitions on this vary. Suppose that a new predicate P is introduced with the stipulation that P holds of a few indicated paradigm cases, a1, a2, and to all sufficiently similar things, and that it does not apply to b1, b2, and to all sufficiently similar things. Now suppose that c is sort of midway between a1 and b1 in the relevant respect. Then perhaps the statements Pc and :Pc both fail to be determinately true. I am not saying that real predicates in natural language are introduced in such a simple-minded manner. The example is only meant to illustrate how the determinately operator might not be compositional. To return to McGee and McLaughlin, define an object a to be a borderline case of a predicate F if Fa is ‘‘unsettled’’, i.e. if a is not determinately an F, nor is a determinately non-F. I take this to be the introduction of a technical term, and not an analysis of the English word ‘‘borderline’’ (see Ch. 5). So it should not be controversial. A second point of contention is whether there are any borderline cases—as the term ‘‘borderline’’ is defined here in terms of determinacy. Famously, the epistemicist maintains that the thoughts and practices of language users fix precise extensions, and complementary anti-extensions, for every predicate of the language. Consider, for example, the sentence ‘‘Josh arrived around noon’’. According to epistemicism, the established use of language determines a real number n such that if Josh arrived exactly n seconds after noon, then the sentence is true, but if he arrived n þ .0000001 seconds after noon, the sentence is false. Our epistemicist would thus have little use for the above definition of ‘‘borderline’’, since on his view, there simply are no ‘‘unsettled’’ cases. Williamson defines the word ‘‘borderline’’ in terms of what can be known, rather than in terms of truth-conditions, as above. Here, of course, I insist on the above definition, conceding that the property might be empty. Williamson’s and Sorenson’s carefully articulated and defended views have met with vigorous opposition, and even more incredulous stares. Here I do not muster a sustained argument against that view, and it is not polite to stare. Williamson claims that views other than his meet with deep difficulties. I am content to have the present view stand or fall with its overall plausibility, including any expressions of incredulity it generates.

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2.2. Tolerance and Open-Texture Crispin Wright [1976: x2] defines a predicate F to be tolerant with respect to a concept f, ‘‘if there is . . . some positive degree of change in respect of f insufficient ever to affect the justice with which F applies to a particular case’’. Vague predicates, or at least some vague predicates, are said to be tolerant. Wright’s language suggests that the issue concerns the proper application of a predicate, or perhaps proper judgment. I take this to be an important insight. Consider the predicate ‘‘bald’’, which is arguably tolerant with respect to number and arrangement of head hair. Suppose that two men h, h 0 differ only marginally in the amount and arrangements of their head hair. Wright’s principle seems to say that if someone competently judges h to be bald, then she must judge h 0 to be bald too—either both are bald or neither are. Similarly, the predicate ‘‘red’’ is tolerant with respect to small, or at least indistinguishable, differences in color. If two colored patches p, p 0 are visually indistinguishable, and if someone judges p to be red, she must judge p 0 to be red as well. As plausible as this interpretation of tolerance may seem at first, especially in the case of the colored patches, it leads to absurdity in the now familiar manner. In general, a sorites series arises when we have a (prima facie) tolerant predicate P, and a series of objects running from a clear (or determinate) instance of P to a clear non-instance of P, with each differing marginally from its neighbors. I propose, instead, this principle of tolerance: Suppose that two objects a, a 0 in the field of P differ only marginally in the relevant respect (on which P is tolerant). Then if one competently judges a to have P, then she cannot competently judge a 0 in any other manner. This, I submit, is the key to avoiding contradiction. This chapter and the next three develop a detailed semantics in which this tolerance principle can be maintained coherently, as part of a larger sketch of the meaning of vague terms. For now, suppose that h and h 0 differ only marginally in the amount and arrangements of their head-hair. It is compatible with the principle of tolerance that someone judge h to be bald and leave the bald-state of h 0 unjudged (one way or the other). Note, however, that it does violate tolerance, in the present sense, if a subject judges h to be bald and decides to leave h 0 unjudged. The decision to leave the case unjudged is itself a decision, and a judgment (as paradoxical as this sounds). To decide to leave h 0 unjudged is to judge h 0 different from h. The observation here is that the subject does not violate tolerance if he has not considered the state of h 0 . The principle of tolerance demands that if this person competently judges h to be bald and is then asked, or forced, to judge h 0 , and does not change his mind about h, then he must judge h 0 to be bald. Suppose that our subject judges h to be bald and we then force a judgment about h 0 . He can satisfy the principle of tolerance by judging h 0 to be bald, and he can satisfy tolerance by judging h 0 to be not bald (or in some borderline category, or in no category) and

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retracting his previous judgment that h is bald. I suggest that in the cases of interest, the meaning of the word ‘‘bald’’ and the semantic and non-semantic facts allow this option.4 Stay tuned. This is not to say that tolerance can never be violated. There are some situations where it must be, and a speaker is not incompetent just because she violates tolerance in such a situation. To follow an example from Sainsbury [1990], consider an art shop that has the red paints on one shelf, marked ‘‘red’’, and the orange ones on another, labeled ‘‘orange’’. The proprietor would not be judged incompetent concerning English, or otherwise in error, if no one can distinguish the last jar on the ‘‘orange’’ shelf from the first one on the ‘‘red’’ shelf. Gaifman [2005] calls situations like this ‘‘infeasible’’; Kamp [1981] calls them ‘‘incoherent’’ (of which more later). Because of situations like this, I am reluctant to say that tolerance is part of the very meaning of vague predicates. The more modest claim is that vague predicates are tolerant in most circumstances. Paint shops, and the like, are rare. My goal is to show how the predicate can be deployed in situations in which tolerance is in force, and, eventually, to provide a model theory for and delimit the logic of such situations. Let P be a tolerant predicate for which there is a sorites series: a list a1, . . . an, where it is determinate that Pa1, it is determinate that :Pan, and for each i < n, the difference between ai and ai þ 1 is insufficient to affect the justice with which P applies, to paraphrase Wright [1976: x2]. As noted, I take it as a premise (without much in the way of argument) that somewhere in the series, say aj, there is a borderline case of P. That is, Paj is neither determinately true nor determinately false. Since P is a predicate of a natural language such as English, and aj is in its range of applicability, our ‘‘thoughts and practices in using the language have established truth conditions for’’ the sentence Paj. My premise is that despite this, the (presumably) non-linguistic facts have not determined that these conditions are met. Moreover, language users have established truth conditions for :Paj, and those conditions have not been met either. As far as the language has evolved to date, Paj is still open. The sentence is ‘‘unsettled’’. Recall that the negation operator need not commute with the determinately operator. A sentence can fail to be determinate without its negation being determinate. To be sure, this is not to say that a given competent speaker, when asked about Paj (under normal conditions, whatever those are) realizes that the sentence is indeterminate, or has the phenomenological feel that there is nothing to say or that she can go either way. Some competent speakers may feel that way, but other competent speakers may be inclined to assert Paj, while still others might be 4 Compare Sainsbury [1990: x6]: ‘‘A boundaryless concept is one which . . . never makes it mandatory to apply the concept to one member of [a marginally different pair] and withhold it from the other . . . ’’; and Scott Soames’s [1999: 215] principle P2 : ‘‘For any two patches of color x and y that are perceptually indistinguishable to competent speakers under normal conditions, if someone who is presented with x characterizes the predicate looks green as applying to it, then that person is thereby committed to a standard that counts the predicate as applying to y as well.’’

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inclined to assert :Paj —all under the same circumstances. The speakers are not thereby incompetent, nor are they necessarily in error in their assertions or judgments. These are delicate matters that we will take up presently. The claim now is that if Paj is borderline, then the meanings of the words, the non-linguistic facts, etc. do not determine a verdict. The presence of ‘‘unsettled’’ cases is precisely the sort of situation that suggests a supervaluational semantics, much like the one provided by McGee and McLaughlin [1994] (and Fine 1975). The idea is to introduce a notion of an admissible sharpening, or precisification, of the language. In the case at hand, a sharpening of the language would contain a sharp (i.e. non-vague) predicate P  such that for all x in the range of P, if Px is determinately true, then P  x, and if :Px is determinately true, then :P  x. In addition, an admissible sharpening decides all of the borderline cases, presumably in an admissible manner. A sentence F is super-true if it comes out true under all admissible sharpenings of the language. So super-truth coincides with determinate truth, as that notion is defined above. Unlike McGee and McLaughlin, some supervaluationists identify truth with super-truth and thus with determinate truth. Keefe [2000: 202] writes that ‘‘ ‘Truth is super-truth’ can be the supervaluationist’s slogan’’. I agree that the overall supervaluational framework is natural and helpful here—eventually—but there are features of the use (and the meaning) of vague terms that need to be incorporated into it. Since we encounter, and effectively and competently decide, borderline cases regularly, there is more to truth than super-truth, and there is more to validity than the necessary preservation of super-truth. To introduce and motivate the further structure, I speculate on the role and function of vague terms in ordinary discourse. Suppose, again, that a is a borderline case of P. I take it as another premise that, in at least some situations, a speaker is free to assert Pa and free to assert :Pa, without offending against the meanings of the terms, or against any other rule of language use. Unsettled entails open. The rules of language use, as they are fixed by what we say and do, allow someone to go either way. Let us call this the open-texture thesis.5 Wright [1987: 244] seems to endorse the open-texture thesis (at least parenthetically): ‘‘Borderline cases are . . . cases about which competent speakers are allowed to differ.’’ See also the epigraph of this chapter. And Sainsbury 5 The term is borrowed from Friedrich Waismann [1968]. For Waismann, a concept or word exhibits open-texture if the linguistic rules do not provide for every possible case. That is, there are possible cases in which established meanings and non-linguistic facts would allow either verdict (or no verdict at all). Waismann held that just about every empirical concept is open-textured in this sense, and that a concept is vague if there are, in fact, unlegislated (or open) cases. As far as I know, Waismann did not endorse the present idea that when a concept is open, competent speakers can go either way in different contexts without altering the meaning of the term, but much of his work points in that direction. His main application was with the notion of analyticity. See the Appendix to this book for more on Waismann’s views.

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[1990: x9]: ‘‘Given the nature of boundarylessness, semantics give freedom. There is some number of minutes such that the nature of the concept of a person, together with the nature of the world, makes it neither mandatory nor impermissible to apply the concept to a foetus of that age in minutes.’’ The account of vagueness sketched in Soames [1999: ch. 7] is also built around open-texture. He holds that a typical vague predicate, like ‘‘bald’’ is associated with two distinct and easily distinguishable classes of individuals: those to which it determinately applies and those to which it determinately does not apply. These two classes are mutually exclusive, though not jointly exhaustive. Individuals in neither class are those about whom the semantic rules of the language governing the predicate issue no verdict. However, this does not mean that the predicate can never correctly be used to characterize them. Rather, there is a realm of discretion reserved for individual speakers and hearers. If on a particular occasion one wishes to characterize an individual x in the intermediate range as bald, one is free to do so provided that others in the conversation are prepared to accept this characterization. (p. 210)

See also Kamp [1981] and Gaifman [2005]. As I see it, open-texture is a more or less empirical claim about the proper use of vague terms in language. However, I have very little in the way of empirical evidence to offer for it from this armchair, beyond a few intuitions. That’s why I call it a ‘‘premise’’. My purpose is to provide a formal framework that accords with open-texture, and let the entire package stand on its merits. I realize that with the assertion of open-texture, I am leaving the comfort of consensus and truism (if it has not been left already). Some supervaluationists claim that if a is a borderline case of P, then a lies outside the extension of P, and so it is (determinately) incorrect to assert Pa. It is supposedly a platitude that one should assert only truths (of which, see x7 below). And, again, the supervaluationist slogan is that truth is super-truth. Since, by hypothesis, Pa is not super-true, it is not true and so it is not correct to assert it—or so argue these supervaluationists. Similarly, those inclined toward a many-valued approach claim that if a is a borderline case of P, then Pa is less than completely true. Since, strictly speaking, one should assert only full truths, these theorists hold that in strict circumstances, it is not correct to assert Pa, in agreement with our supervaluationists.6 The fuzzy logicians might concede that in less strict situations, one can assert near truths, but presumably, one should only assert near truths even then. In contrast, I would take the premise of open-texture to apply generally, even to borderline cases that are not ‘‘near truths’’. I will not pause to offer arguments against these opposing views, or the intuitions that lie behind them, but will rest content to let the entire framework being developed here serve as a rival account. Let us be clear as to what the open-texture premise is, and what it is not. Suppose that a is a borderline case of P. The open-texture thesis is that in some 6 On one approach, called ‘‘sub-valuationism’’, borderline cases are both true and false (adopting a paraconsistent logic, of course). See e.g. Hyde [1997]. On that view, one can (and should) assert Pa, when a is a borderline case of P, since the sentence is true. One can also assert :Pa.

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circumstances a competent speaker can, in fact, go either way without offending against the meaning of the terms, the non-linguistic facts, and the like. As above, the open-texture thesis does not entail that he will always be conscious of the fact that he can go either way. Second, even if a is a borderline case of P, it is not true that the rules for language use allow a speaker to assert Pa in any situation whatsoever. For example, one is not free to assert Pa if one has just asserted (and does not retract) :Pa. This would offend against logic (dialetheism notwithstanding). Similarly, and more controversially, one is not normally free to assert Pa if one has just asserted (and does not retract) :Pa 0 , where a 0 is only marginally different from a. That would offend against tolerance. To sum up, if a sentence F is determinate, then one can correctly assert it at any time and can never correctly deny it. If :F is determinate, then one can correctly deny F at any time, and can never assert it. If F is unsettled, then it depends on the conversational situation one is in. We need a mechanism to track the features of a situation that allow and disallow certain moves. This I borrow from David Lewis’s influential ‘‘Scorekeeping in a Language Game’’ [1979]. 3. CONVERSATIONAL SCORE It is often noted that the truth values of instances of some vague predicates are relative to a comparison class. A short NBA player is not a short business executive. A person can be wealthy in some nation, or neighborhood, while someone else with exactly the same resources is not wealthy in other situations. Similarly, the truth value of an instance of a vague predicate can vary with an instance to which it is compared or contrasted. An income may be paltry with respect to Bill Gates, but not so with respect to George W. Bush. Or, to follow Graff [2000: x3], a man can be bald with respect to Yul Brynner and not so with respect to Mikhail Gorbachev (even though the comparison class is the same in both cases, all men). It is a well-known phenomenon that a colored patch may look orange with respect to a pink one but not with respect to a red one. The present program introduces another relativity, to a conversational context. This includes not only the environment of the conversation, what it is about, and the implicit or explicit comparison class or paradigm or contrasting examples, but also what has already been said in the course of the conversation. For this purpose, I employ the notion of a conversational score, or conversational record, invoked by some linguists interested in pragmatics. Lewis [1979] develops the idea in sufficient detail for present purposes. The record, or score, is a local version of common knowledge. During a conversation, the score contains the assumptions, presuppositions, and other items implicitly or explicitly agreed to. For example, the conversational score contains the range of quantifiers like ‘‘everyone’’ and the denotata of proper names like ‘‘Louise’’ and ‘‘Joe’’. It also settles the relevant comparison class, paradigm cases, and/or contrasting cases for

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predicates like ‘‘tall’’ (NBA players, executives, etc.) and ‘‘bald’’ (Brynner, Gorbachev). The score also contains propositions that have been (implicitly or explicitly) agreed to, and are not up for dispute or discussion, at least for the moment. Among these are the ‘‘presuppositions’’ to the conversation.7 The conversational record is a sort of running database. It is continually updated, in that items are put on the record and, notably here, removed from it in the course of a conversation. Items get removed when the topic changes, when what is agreed to or some presupposition comes into question, or when some of the participants change their minds about items that are on the record. Since the participants in a conversation may be mistaken about what has been agreed to and what has not, they may be mistaken about what is on the conversational record. Normally, this sort of thing gets cleared up in due course, if it turns out to be important, except perhaps in some intractable philosophical disputes. In most cases, once confusion about the score ensues, either consensus on some presuppositions is reached, or the participants take the disputed items off the record. The conversational record is a theoretical posit, to help make sense of ordinary discourse. Lewis [1979: 345] delimits some features of the database, most of which are relevant here: (1) . . . the components of a conversational score at a given stage are abstract entities. They may not be numbers, but they are other set-theoretic constructs: sets of presupposed propositions, boundaries between permissible and impermissible courses of action, or the like. (2) What play is correct depends on the score. Sentences depend for their truth value, or for their acceptability in other respects, on the components of the conversational score at the stage of the conversation when they are uttered . . . [T]he constituents of an uttered sentence—subsentences, names, predicates, etc.—may depend on the score for their intension or extension. (3) Score evolves in a more-or-less rule-governed way. There are rules that specify the kinematics of score: If at time t the conversational score is s, and if between time t and time t 0 , the course of the conversation is c, then at time t 0 . . . the score is some member of the class S of possible scores, where S is determined in some way by s and c. (4) The conversationalists may conform to directives, or may simply desire, that they strive to steer components of the conversational score in certain directions . . . (5) To the extent that conversational score is determined, given the history of the conversation and the rules that specify its kinematics, these rules can be regarded as constitutive rules akin to definitions.

Something like (2) is invoked by Kamp [1981: 242]: ‘‘One kind of contribution which the context can make to the evaluation of an utterance is mediated by the background that context provides, i.e., by the information that is taken for granted by the participants in the discourse of which the utterance is a part.’’ 7 Robert Stalnaker’s [1999] detailed and insightful account of presupposition and context is largely compatible with the present sketch.

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Lewis points out that unlike most games, conversations tend to be cooperative. Indeed, that is usually the point of having a conversation in the first place. So ‘‘rules of accommodation . . . figure prominently among the rules governing the kinematics of conversational score’’ ([1979: 347]). I presume that accommodation comes under item (4). The idea is that the conversational record tends to evolve in such a way that, other things equal, whatever is said will be construed as to count as correct, if this is possible. The record will be updated to make this so. Suppose, for example, that someone utters a sentence such as ‘‘Harry no longer smokes unfiltered cigarettes’’ which has the presupposition that Harry used to smoke unfiltered cigarettes. Unless someone objects, the presupposition goes on the record. To be sure, cooperation is ‘‘not inevitable, but only a tendency’’, as Lewis puts it. And, of course, presuppositions can be retracted later. Lewis goes on to use this notion of conversational score to illuminate six diverse features of the semantics and pragmatics of natural language. The list includes definite descriptions, performatives, and a painfully brief treatment of our present topic, vagueness ([1979: 351–4]). That discussion begins by noting that with a predicate such as ‘‘bald’’, ‘‘nothing in our use of language’’ fixes a sharp border between the bald and the non-bald. This much echoes the above McGee–McLaughlin theme, and like them, Lewis gestures toward the framework of supervaluation. Suppose that Fred is a borderline case of ‘‘bald’’. Then whether the sentence ‘‘Fred is bald’’ is true depends on ‘‘where you draw the line’’. Relative to ‘‘some perfectly reasonable ways of drawing a precise boundary between bald and non-bald, the sentence is true. Relative to other delineations, no less reasonable, it is false’’. Philosophers and linguists cannot settle on a single, precise border, since a number of such borders would do. Instead, they ‘‘must consider the entire range of reasonable delineations’’. Lewis says that if ‘‘a sentence is true over the entire range, true no matter how we draw the line, surely we are entitled to treat it simply as true’’. In other words, the supervaluationist notion of super-truth is sufficient for simple truth. But this is truth only in a strict sense. Lewis goes on to recognize a looser use of language: ‘‘But also we treat a sentence more or less as if it is simply true, if it is true over a large enough part of the range of delineations of vagueness. (For short, if it is true enough.) If a sentence is true enough (according to our beliefs) we are willing to assert it, assent to it without qualification . . . ’’ He notes that we usually do not get into any trouble with this loose use of language. Note, however, that we can get in such trouble and sometimes we do—witness sorites paradoxes. Lewis uses the notion of conversational score to resolve an issue that concerns this notion of ‘‘true enough’’: When is a sentence true enough? Which are the ‘‘large enough’’ parts of the range of delineations of vagueness? This is itself a vague matter. More important for our present purposes, it is something that depends on context. What is true enough on one occasion is not true enough on another. The standards of precision in force are different from one

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conversation to another, and may change in the course of a single conversation. Austin’s ‘‘France is hexagonal’’ is a good example of a sentence that is true enough for many contexts, but not true enough for many others. Under low standards of precision it is acceptable. Raise the standards and it loses it’s acceptability.

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

4. OPEN-TEXTURE, CONVERSATION, AND SORITES Unfortunately, this is the extent of the treatment of vagueness in Lewis [1979], and I am not aware of any use of conversational score with respect to vagueness in

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his other writings (including his extensive writing on vagueness).8 In particular, he does not invoke conversational score to deal with sorites. Indeed, the only (possible) reference to the paradox is a remark that loose thinking sometimes leads to trouble. Given the framework he proposes, I presume that Lewis accepts the standard supervaluational resolution, or at least he did in 1979. Recall that the premises of the arithmetic version of the sorites consists of Pa1, :Pan and the induction principle: for each i < n, if Pai then Paiþ1. If P is precise, then there will be at least one number m < n such that Pam and :Pamþ1. Therefore, on each sharpening of the language (whether it is acceptable or not), the inductive premise is false. So, in Lewis’s terminology, the inductive premise is ‘‘simply false’’. Although this blocks the contradiction, the supervaluationist still has to explain why we thought that the inductive premise is plausible. Why was there ever an issue? Notice that each instance, if Paj then Pajþ1, of the induction principle is true in all but one acceptable sharpening, and so each instance is ‘‘true enough’’, as Lewis might put it. But the inductive premise itself—the universal generalization of the (true enough) instances—is super-false. So this version of the inductive principle is a false universal generalization over a bunch of propositions, all of which are at least true enough. Nothing paradoxical about that. As noted above, there is a version of the sorites reasoning which does not use the single, inductive premise. Rather, it invokes each instance as a separate premise, and uses modus ponens over and over: :Pan Pa1 Pa1 ! Pa2 Pa2 Pa2 ! Pa3 Pa3 ... Pan1 ! Pan Pan contradiction

premise premise premise modus ponens premise modus ponens premise modus ponens

In Lewis’s terminology, some of the conditionals in this argument are simply true (i.e. the first bunch and the last bunch), and, as above, all of them are true 8 Lewis’s main work on vagueness occurs as part of his general semantic theory. See e.g. Lewis [1969, 1970, 1975]. Keefe [2000: ch. 6] provides a nice summary, calling it the ‘‘pragmatic’’ account of vagueness. It has much of the flavor of a supervaluation account.

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enough. Thus we see that a (classically or intuitionistically) valid argument can go from premises that are true enough to a conclusion that is simply false, if the argument is long enough. Something analogous happens with the same argument when evaluated in terms of fuzzy logic: we go from premises that are either completely true or almost completely true to a conclusion that is completely false. Of course, this is not surprising, nor is it an original analysis (at least at this point in time). Notice that on any of the canvassed views, we still lack a guide on how we reason, or how we ought to reason, from premises that are (only) ‘‘true enough’’. What is the proper logic for reasoning with vague predicates? Classical and intuitionistic reasoning leads to trouble—unless we stick to premises and are determinately true, simply true, super-true, etc., or else we only make very short deductions. If we use more than one premise that is only true enough, then we risk coming to a conclusion that is not true enough. The more ‘‘less than fully true’’ premises we invoke, the greater the danger of coming to a conclusion that is not ‘‘true enough’’. And if we use enough ‘‘less than fully true’’ premises, we might validly reason our way to an outright falsehood. In the following chapters, I propose a model for reasoning with vague predicates, based on a model theory. It is motivated by the features of vagueness invoked above. Recall our premise of open-texture: borderline cases of vague predicates can go either way in some conversational contexts. Once a borderline case is ‘‘resolved’’ in the course of a conversation, that information goes on the conversational record, and will remain on the record unless it is implicitly or explicitly retracted. To develop a fully general account of reasoning with a conversational score, we would need a paraconsistent logic, since it is possible for conflicting propositions to be put on a record, without the group retracting any of them. The participants may not realize they have contradicted themselves. Even if they do realize the contradiction, they may not want to retract the statements, since they remain attracted to each of them individually, and do not yet know which one(s) to give up. Such is paradox. The presence of a contradiction on the conversational record (whether detected or not) does not, or at least should not, commit the group to every proposition whatsoever. For present purposes, however, we need not invoke paraconsistency, since our focus is on admittedly artificial cases in which consistency is easily enforced. I take it that consistency is a regulative ideal for a conversational record, in the sense that the participants try to achieve consistency, retracting questionable items from the score as needed (Priest 1987 notwithstanding). Let us begin with a conversational variant of Horgan [1994a]’s ‘‘forced march’’ sorites. Suppose we have 2,000 men lined up in a row. The first is a mature Yul Brynner, who is clearly bald—he has no hair at all (or so we will assume). The last man is Jerry Garcia, in his prime. I take this as a paradigm case of a non-bald man. The hair of each man in the series (who has hair) is arranged

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in roughly the same way as his immediate predecessor in the series. After the first, each man differs from the one before by having only slightly more hair, perhaps imperceptibly more. Now suppose that the participants in a conversation start asking themselves about the baldness-state of each man in the series, starting with Yul Brynner, and they insist on a communal verdict in each case. As each question in the form ‘‘Is man n bald?’’ is put, they are to provide an answer of ‘‘yes’’ or ‘‘no’’. This forced-bivalence is only a convenience. The situation would be essentially the same if we gave them the option to answer ‘‘borderline bald’’, ‘‘unsettled’’, ‘‘no fact of the matter’’, etc. It would also be the same if we gave them the option to agree on silence. To paraphrase Raffman [1994: 41 n. 1], it isn’t merely that there is tolerance between (determinate) cases of baldness and (determinate) cases of non-baldness. There is tolerance between baldness ‘‘and any other category— even a ‘borderline’ category’’. Or Wright [1976: x1]: ‘‘no sharp distinction may be drawn between cases where it is definitely correct to apply [a vague] predicate and cases of any other sort’’. What matters for present purposes is that the participants in our conversation must answer (or refuse to answer) by consensus— whatever the allowed answers may be. If, at any point, they want to stop answering, they must agree to this silence. We do not allow them to stop the march by simply failing to agree on a verdict. So let us allow the simplifying assumption that only two answers are allowed: ‘‘bald’’ and ‘‘not bald’’. Being competent speakers of English, the conversationalists all agree that Yul Brynner is bald, that the second man is bald, etc. Eventually, they will move into the borderline area, and encounter cases whose baldness state is ‘‘unsettled’’, as in x2.1 above. Again, the thoughts and practices in using the language have established truth conditions for statements about baldness, and truth conditions for statements about non-baldness. In the borderline region, the non-linguistic facts have not determined that either of these truth conditions are met. Nevertheless, the conversationalists in this exercise will probably continue to call the men bald as they move through the borderline area—for a while. If they call man n bald, they will probably call man n þ 1 bald as well, since by hypothesis, they can barely tell the two heads apart in the relevant respect. This is all right, so far as their language competence goes. Given our opentexture premise, borderline cases can go either way (without offending against meaning and the facts), and the participants in this conversation are just going one way rather than the other as they enter the borderline area from this direction. As above, it does not matter whether they realize that they can go either way in any of these borderline cases. For some of the conversationalists and for some of the cases, it certainly won’t feel like they can go either way. Following their instructions, they just call ’em as they see ’em. And as they start to move into the borderline cases, they continue to see ’em to be bald—for a bit.

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This puts such propositions as ‘‘man 923 is bald’’ and ‘‘man 924 is bald’’ on the conversational score (assuming that those are borderline cases). Now recall the principle of tolerance: Suppose that two objects a, a 0 in the field of P differ only marginally in the relevant respect (on which P is tolerant). Then if one competently judges a to have P, then she cannot judge a 0 in any other manner. Let us assume that this is in force in the present conversation. In particular, if we were to present any adjacent pair from the series to the conversationalists at the same time, they would agree that either both of the men are bald or that neither are (or that both are borderline, or that they do not know what to say about them, etc., if we suspend the forced-bivalence). This explains why they are likely to see man 924 as bald if they have just declared that man 923 is bald. In discussing a similar series, Soames [1999: 213] notes that ‘‘each time we move one step further along the scale, the . . . extension . . . of the predicate changes, and therefore the standards for further application . . . of the predicate do too’’. Nevertheless, since the participants in the exercise are competent speakers of English, we can be sure that they will not blindly go through the entire series, and call #2000, Jerry Garcia, bald. That way lies madness (or gross perceptual error, or incompetence). Eventually, a few of the participants will demur from calling one of the men bald. If this group finds themselves in a small minority, they will go along with the majority, to maintain the consensus—for a while. As the group proceeds through the borderline area, more and more of the participants will demur, or feel like demurring. At some point, enough of them will demur that the consensus on ‘‘this man is bald’’ will break down. Given the ‘‘forced march’’ instruction, they will eventually agree to call one of the men ‘‘not bald’’. Kamp [1981] considers a very similar thought experiment, involving a single person viewing and judging colored patches ranging from green to yellow. He articulates a principle of tolerance which he calls EOI—equivalence of observationally indistinguishable entities: ‘‘Suppose the objects a and b are observationally indistinguishable in the respects relevant to [a predicate] P; then either a and b both satisfy P or else neither of them does’’ (pp. 237–8). In the forced march, you are subject to two forces which eventually work against each other. On the one hand, there is a commitment to EOI which requires you to keep answering ‘‘green’’, as it never seems quite right to suddenly stop saying ‘‘green’’, after you have already said ‘‘green’’ to the previous square—after all, the two squares look just the same. On the other hand there are the indubitably green squares on the extreme left and the indubitably yellow squares on the extreme right. These serve as it were as ‘‘anchor points’’. And there comes a time when the square about which you are asked looks as much like the yellow squares on the right as it looks like the green ones on the left. At that point, or shortly afterwards, the second force comes to dominate the first, and you either suspend or switch judgment.

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you are subject to two increasingly conflicting semantic principles. The second principle eventually wins—but not without leaving you with the feeling that there is some sort of inconsistency in the sequence of your responses. (p. 241)

Kamp concludes that these sorites arguments ‘‘show that a substantial class of terms which our language contains, and which it could hardly do without, are ruled by semantic principles which are strictly incompatible’’ (pp. 241–2). When the switch to ‘‘yellow’’ occurs, the context has become ‘‘incoherent’’, since the speaker cannot satisfy both semantic rules, namely that squares that are indistinguishable from those that were correctly called ‘‘green’’ are themselves green, and if a square looks enough like the paradigm ‘‘yellow’’, then it is yellow (and the truism that no green square is yellow). Kamp goes on to develop a complex contextualist semantics for such predicates, that turns on the notion of a coherent context (which seems to be similar to what Gaifman [2005] calls a ‘‘feasible’’ context). This would be where tolerance fails. I submit that we need not postulate such incoherent or infeasible contexts. Tolerance can remain in force, and, if so, the conversational record will track this. Let us return to our communal thought experiment with the 2,000 men in various states of baldness. They call the first few men bald, of course, and continue until they jump to ‘‘not bald’’ (perhaps noting the malice on our part). Suppose this happens with man #975. Recall that the conversationalists have just agreed, perhaps reluctantly for many of them, that #974 is bald, and so they put ‘‘Man 974 is bald’’ on the conversational score. Of course, when they then said that #975 is not bald, they did not contradict themselves. Nor have they violated the tolerance principle (or Kamp’s EOI). At that point, tolerance applies in reverse—we take the contrapositive. In declaring man 975 to be not bald, they implicitly deny that man 974 is bald, and so ‘‘Man 974 is bald’’ is removed from the conversational record. It is similar to what happens when any presupposition is challenged (or contradicted) in the course of a conversation. The event in this forced march is similar to the outcome in one of Lewis’s scenarios, recounted in the previous section. In that story, the participants in a conversation first agree to a ‘‘low’’ standard (for ‘‘true enough’’) when they accept ‘‘France is hexagonal’’. Later, when they demur from ‘‘Italy is boot-shaped’’, the standard is raised, and so ‘‘France is hexagonal’’ is implicitly removed from the record, since it does not meet the raised standard. Similarly, if somebody in a conversation points out that Harry never smoked unfiltered cigarettes, and no one challenges this, then the presupposition that he did is removed from the score, if it was there previously. In short, the conversational score is the device that enforces tolerance, and consistency. Just as ‘‘Man 974 is bald’’ comes off the record, so does ‘‘Man 973 is bald’’; ditto for a few more of their recent pronouncements. We cannot and thankfully need not set a precise boundary as to how many sentences are removed from the

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record once they jump. Exactly what is and what is not on the conversational record is itself a vague matter.9 Typically, however, borderline cases of ‘‘what is on the record’’ do not interfere with a conversation. If a question arises about a specific case—say #967—the participants can ask each other about that fellow, and so the status of that case can be made explicit. Let us continue the thought experiment by reversing the order of query. We assume that #975 is the first ‘‘jump’’, where our participants shift and deny that the man is bald. Suppose that we explicitly ask them about #974 again, after reminding them that they just called him ‘‘bald’’, and that they can barely distinguish #974 from #975 (if at all). Although I don’t intend to apply for a grant to confirm this empirical hypothesis, I’d speculate that they would retract that judgment, saying that #974 is not bald (and thus put ‘‘Man 974 is not bald’’ on the record). Suppose that we then ask them about #973. They would retract that judgment as well. Then we can ask about #972. Of course, they will not move all the way back down the series, and end up denying that #1, Yul Brynner is bald. After all, he has no hair whatsoever. At some point, they will jump again, and declare a certain fellow to be bald—suppose it is man 864. This again will result in the removal of certain items from the conversational record, such as the denial that man 865 is bald. If our conversationalists do not lose patience with us, we can then go back up the series, asking them about man 865 and 866. The conversationalists would declare men they encounter to be bald for a while, eventually jumping. In general, they will move backward and forward through the borderline area. Tolerance is enforced at every stage, by removing judgments from the conversational record whenever a jump occurs. Notice, incidentally, that pending empirical research, there is no reason to think that the participants will always jump at the same places as they move back and forth through the middle part of the series. I return to this matter in Section 6 of this chapter. With empirical matters like this, of course, I cannot count on competent speakers always cooperating with the details of the philosophical thought experiment. Nor is it crucial to the present account of vagueness that competent speakers always behave as indicated in the thought experiment, that is, by running backward and forward through the series, revising their judgments as they 9 For this reason, the notion of conversational score is not a component of the formal account of vagueness developed in the following chapters. For that matter, the notion of context, which is also vague (and itself context sensitive) is not part of the formal development. Those notions are central parts of the philosophical account that motivates the formal development. Since the meta-language for the model theory is ordinary set theory which, I assume, has no vagueness, all meta-linguistic devices and properties are likewise precise. See the next chapter for an account of the role of metatheory. In the formal account, however, there are various analogs of the notions of ‘‘score’’ and ‘‘context’’ that more or less correspond to their informal counterparts. The technical notions of ‘‘partial interpretation’’ and ‘‘forcing’’, for example, are rough analogs of context and conversational score, with emphasis on the word ‘‘rough’’.

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go.10 I take it as given that come hell or high water, on any given run up or down the series, conversationalists will eventually jump, well before they call Jerry Garcia bald and well before they deny that Yul Brynner is bald—assuming of course that they remain competent speakers of English and do not go batty as a result of this experiment. This much is quite independent of any hypothesis about tolerance or incoherence. Suppose, again, that the first jump occurs with man 975. Once the conversationalists judge this man to be not bald, we immediately ponder the status of 974. Given tolerance, they cannot judge man 974 to be bald and 975 to be not bald at the same time (i.e. in the same context), so long as tolerance is in force. So it follows that the status of 974 has changed with the jump. So it is natural to ask about that fellow, to confirm this, and then proceed down the series. I submit that the back and forth phenomenon is just about forced by tolerance (and the obvious facts that Yul Brynner is bald and Jerry Garcia is not). In some situations, however, the conversationalists may simply violate tolerance. After one or two runs up and down, or perhaps at the first jump, they may call a fellow not bald and stick to their previous judgment that his neighbor is bald. They may get fed up with the experimenters, or think that such an answer is what they are supposed to give. I noted above (x2.2), that there are situations in which tolerance is not in force (situations that Gaifman calls infeasible and Kamp calls incoherent). Our forced march may turn into such a situation either eventually or even right from the start. It is a highly artificial scenario, even more so than Sainsbury’s paint shop. If tolerance is indeed violated, then there is no problem with the sorites. The inductive premise is false. The point here is that the sorites paradox can be resolved even if tolerance remains in force, and the conversationalists continue up and down the series as depicted above. Consider the version of the paradox that uses a single, inductive premise: for each i < n, if Pai then Paiþ1. It is outright false that for each i < 2,000, if our conversationalists judge man i to be bald, they will then judge man i þ 1 to be bald. As they go through the forced march, sooner or later they will jump. And this jump does not undermine their competence as speakers of English. Indeed, if they did not jump, and went on to call man 2,000, Jerry Garcia, bald, they would thereby display incompetence. 10 My colleague Diana Raffman is working with psychologists to test similar hypotheses (largely concerning color predicates). Preliminary results are favorable. It might make a difference if the forced march is set up in different ways. For example, it might matter whether the questions are put phenomenally or verbally. That is, the subjects may be instructed to look at balding men or colored patches, or else they might be given questions like ‘‘Is an income of $35,000 paltry for residents of Ohio?’’ or ‘‘Is 200 seconds after noon noonish?’’ (as in Sorenson 2001: 58). It also might make a difference if the subjects are explicitly made aware of their previous answers as they proceed, or if they can see the previously judged cases as they judge the new ones. Thanks to Brian Weatherson for pointing out the complexities of the forced march, and for getting me to consider that the empirical situation is not crucial for the overall account of vagueness.

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Moreover, as we have seen, a jump need not violate tolerance, since it comes with a retraction of items from the conversational record. Something similar happens with the version of sorites that avoids a single, inductive premise in favor of each instance of it. Whenever the conversationalists jump from ‘‘bald’’ to ‘‘not bald’’, they thereby undermine one of the premises in the long argument. Although the inductive premise is never true (in full splendor), notice that so long as tolerance is in force, the conversational score never contains what may be called a ‘‘strong counterexample’’ to the inductive premise.11 That is, there is never a number n, such that ‘‘man n is bald’’ and ‘‘man n þ 1 is not bald’’ are both on the record at the same time. If they jump to ‘‘not bald’’ at #n þ 1, then ‘‘man n is bald’’ is retracted, and removed from the score. In the scenario as envisioned, tolerance is enforced on the conversational record (unless and until tolerance is given up). This may just be a feature of the forced march, however. Suppose instead that the participants consider the men in the series in random order. Then they very well might judge man n to be bald and sometime later judge man n þ 1 to be not bald without explicitly or even implicitly retracting the first judgment. Indeed, they might even judge man n to be not bald and later judge man n þ 1 to be bald—violating a penumbral connection. In fact, if they are not careful, they might even judge man n to be bald and a bit later judge the same man to be not bald. In such cases, they have not noticed the violation of tolerance, penumbral connection, or consistency. But once the violation is pointed out, they will retract at least one of the judgments, and so one of the offending propositions will go off the record. For present purposes, one nice feature of the forced march is that tolerance, penumbral connection, and consistency are easily enforced. Not so in general. In the forced march, there will surely be cases m, where, for example, ‘‘man m is bald’’ is on the record and ‘‘man m þ 1 is bald’’ is not on the record. But that does not undermine tolerance, as that notion is defined above. Since man m and man m þ 1 differ only marginally, the participants cannot judge them differently at the same time, but there is nothing to prevent them from judging one and not the other. As noted in x2.2 above, it would violate tolerance if the conversationalists judge man n to be bald, and decide to leave man n þ 1 unjudged. But there is no violation if man n þ 1 is simply not judged one way or any other. As Soames [1999: 216–17] puts it: Often there will be a sharp line separating things that are F, according to a given conversational standard, from things for which, according to the standard, the predicate is undefined . . . Attempts to display that sharp dividing line for inspection will result in a change of standards, according to which the pair of adjacent items in the progression come to be characterized in the same way (with respect to the predicate) 11

This observation suggests a non-bivalent semantics, which is provided in subsequent chapters.

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It follows, incidentally, that the conversationalists cannot simultaneously judge every member of the series and remain competent in the use of the term ‘‘bald’’, assuming that tolerance remains in force. This is just a matter of logic and the meaning of ‘‘tolerance’’, as it is stipulated here. That is, the participants cannot have a judgment for every man on the conversational record all at once. If they did, they would either violate tolerance, or else they would call Yul Brynner not bald, or call Jerry Garcia bald. That’s life. At any time, some of the baldness states will have to be left unjudged. Perhaps they can get to a point where they have done a maximal amount of judging. Suppose, for example, that they judge every man from #1 to #974 to be bald, and they judge every man from #979 to #2,000 to be not bald. Moreover, suppose that they cannot make a judgment about any of #975 through #978 without either violating tolerance or jumping and thus retracting some of their previous judgments. Such is the nature of the series.

5. R AF FMA N ON VAG U ENES S The present resolution of the sorites paradox (when tolerance is enforced) has the same structure as that of Raffman [1994, 1996], and I acknowledge a deep debt to that penetrating work. Raffman envisions a single subject who is marched through a sorites series consisting of color patches, but she is explicit that her account is to apply to sorites-prone terms generally. To adapt her framework to our series above, the subject would begin by calling the first few fellows bald, and continue that way a bit into (what I call) the borderline area.12 But our subject eventually jumps and calls one of the men not bald. Again, suppose that this first occurs at #975. Raffman says that at that point, the subject has changed to a different psychological state. In the new state, it is not true that, say, #974 is bald. Once again, it is a commonplace that many predicates—vague or otherwise— are relative to context. For example, a lightweight internal lineman among high school football players would be an extremely heavy jockey. That is what Raffman calls an ‘‘external’’ relativity to a comparison class. For vague predicates generally, Raffman proposes that there is an additional, internal relativity, to a psychological state. Patches look red, or are red only relative to the state of a person judging them to look red (perhaps with some stipulation on the conditions). Similarly, a man is bald, or not, only relative to the state of a competent person judging whether he is bald. Raffman argues that (something like the tolerance principle entails that) a competent subject cannot give different verdicts to ‘‘man 974 is bald’’ and ‘‘man 975 is bald’’ at the same time—and while in the same state. As above, if the 12 Raffman and I disagree on the proper definition of ‘‘borderline’’, but that is not relevant here. Nor is it a substantial issue. The reader is invited to use a different word. See Raffman [2005].

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subject is presented with any adjacent pair and asked to judge them both, he will say either that they are both bald or neither is bald: ‘‘at no time are adjacent [members] of the series simultaneously category-different’’ ([1994: 53]). So it is never the case that an instance of the inductive premise is false. So long as the subject remains in the same state, he will not differ in his judgment concerning any adjacent pair of men in the series. According to Raffman [1994: 50], when the subject jumps during the forced march, he moves into a new psychological state. There is a ‘‘category shift’’ in which the judged category—‘‘not bald’’ in this case—‘‘spreads backward’’ along a string of the preceding men in the series. In the new state, the extension of nonbald ‘‘expands backward, instantaneously’’, to include some of the men that formerly fell in the the extension of ‘‘bald’’. To adapt Raffman’s narrative to the present example: My hypothesis is that, at the moment of judging [#975], the speaker undergoes a kind of Gestalt shift that embraces [#974] (and probably some of [his] predecessors) as well as [#975]. At the moment of shift to [‘not bald’], the speaker is disposed to judge both [#974] and [#975] (plus some [men] on both sides) as being [not bald], thereby allowing for a change in kind while preserving the effective continuity of the series. Intuitively speaking, a string of [men] shift their [baldness state] simultaneously, so that [#974] and [#975] never differ . . . at the same time. Like the duck-rabbit and Necker cube, these [men] can ‘‘go either way’’: they can be ‘‘seen as [bald]’’ or ‘‘seen as [not bald]’’—now one way, now the other . . . If asked to reverse direction and retrace his steps down the series toward #1, the speaker would now judge [as not bald] some [men] that he previously judged [bald]. At some point, of course, he would shift back to [‘bald’]; for example he might judge [#974] through [#903] [not bald], but then undergo a Gestalt switch back to [‘bald’] at [#902]. And so forth . . . (Raffman 1996: 178) My claim is that whenever marginally different items are assigned incompatible predicates relative to the same external context, a Gestalt-like shift has occurred so that those predicates are assigned relative to different internal contexts. (Raffman 1996: 180)

The metaphor with Gestalt shifts is suggestive. A subject’s perception of a Necker cube, for example, changes without any changes in the drawing itself. With a vague predicate, the judgment shifts with a small (and perhaps imperceptible) change in the item being judged.13 Raffman makes a distinction between a ‘‘categorical judgment’’, which consists of a subject considering a single case, and a ‘‘comparative judgment’’, where the subject deals with two (or more) instances at once. Raffman illustrates this, and the relativity, by envisioning a forced march in which the items are presented in pairs. Let us return to our series of 2,000 men. We first ask the subject, or the participants, to judge the pair h#1,#2i. Both are judged to be bald. Then we present the pair h#2,#3i. Both bald. Then h#3,#4i, and so on. Since the pairs are marginally different, perhaps indistinguishable, the subject(s) will always judge 13

Thanks to an anonymous referee.

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each pair alike. But, again, the subject(s) will eventually jump. They won’t go on to call Jerry Garcia and his predecessor both bald. Suppose that the pair h#887,#888i are judged to be bald, but the pair h#888,#889i are judged to be not bald a few seconds later. If we do not allow for a contextual shift of some sort—be it a change in psychological state, a change in the conversational record, or something else—the subjects will have contradicted themselves at that point. Since man 888 is in both pairs, he is judged to be bald and then judged to be not bald a moment later. There are not many options for interpreting this scenario. The theorist can conclude that vague predicates such as ‘‘bald’’ have some contradictory cases, or that any language (such as English) with vague predicates is incoherent generally (following Dummett 1975, Horgan 1994b, and perhaps even Kamp 1981). Or the theorist can find fault with the scenario. Perhaps there is something illegitimate in demanding a forced march through a sorites series.14 According to an epistemicist, for example, the subjects should refuse to answer when they get near the (precise) border, despite their instructions. Since they do not and cannot know where the border is, they do not and cannot know the baldness state of the men near the border. Thus, they have no business venturing an opinion on those cases. Horgan’s [1994a] council is similar (for different reasons). Or the theorist can claim that the subject(s) are incompetent, or have made some sort of mistake with their answers. But what sort of mistake? The subjects are told to call them as they see them. What did they do wrong? Or the theorist can follow Raffman and the present program and say that a shift of context has occurred. Man 988 is bald in one context and not bald in another. Raffman’s contextualist framework complements the present one. She sees the extensions of vague predicates as varying with the psychological states of (competent) speakers of the language, while I see the extensions as varying with conversational records among (competent) speakers. In one sense, her thought experiment is a special case of the present one. We might think of hers as a limiting case of a ‘‘conversation’’ consisting of only one participant. The ‘‘conversational score’’ would contain features of the psychological state of the subject. Of course, a conversation consists of its individual members, and Raffman’s notion of psychological state is intended to explain what happens in the mind of each of those individuals. The communal, conversational analog of Raffman’s gestalt-shift is what I call a ‘‘jump’’, the combination of a breakdown of consensus among the members of the conversation, and the forging of a new consensus on a new verdict. A consensus breaks down only if enough of the individual participants demur from the communal judgment, and if Raffman is correct this happens when enough of them have made the relevant gestalt-shift 14 Kamp [1981: 242] notes that ‘‘one almost feels that the man who conducts such an experiment and leads the guileless subject to the embarrassing contradictions in which he will inevitably get entangled, may be accused of malice’’. Of course, this moral judgment does not imply that there is something semantically or logically wrong with the experiment.

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(and refuse to switch back even for the sake of conversational harmony). So if Raffman’s account can be sustained, it is more basic than mine. I have little to contribute to the articulation of the relevant notion of psychological state, nor to the psychological explanation of individuals as they judge vague predicates (unless the model-theoretic framework presented in the next few chapters proves helpful to that enterprise). I return to this briefly in the next section. The present account focuses instead on the communal aspects of language use, noting the role of language in public communication. Surely, it is individuals who communicate, and these individuals are in different psychological states at different times. I do not have much to say by way of a detailed account of how this works. What is the bridge between the different psychological states of individual communicators and the public side of communication? The cooperative features of conversations that Lewis [1979] sketches and, in particular, the defeasible drive for consensus, feed into the role of conversational record in handling borderline cases of vague predicates. They suggest the more communal sort of relativity that the present work focuses upon.

6 . UN D E R GR O U N D I N TH E BO R D E R L I N E AR E A Let us return to our standard sorites series consisting of 2,000 men ranging from Yul Brynner to Jerry Garcia in his prime. We envisioned a forced march conversation through this series, starting with Yul Brynner. Given their instructions, and assuming their competence in English and normal powers of observation, the participants to the conversation will eventually jump and call one of the men not bald. Again, suppose that this first happens with man 975. If we were to run the same series (in the same order) with the same group of speakers on another occasion, they might jump at a different place, say #967 or #984. They would almost certainly jump at a different place if they were marched through the series in the reverse order, starting with Jerry Garcia, and moving down. Presumably, something causes the jumps when they occur. The events are too macroscopic for quantum randomness to be involved (or are they?). The point here is that so long as the man being judged for baldness is in the borderline area, the meanings of the word ‘‘bald’’, the external contextual factors, and the non-linguistic facts about the amount and arrangement of his hair do not themselves determine a correct response. So as far as the normativity of meaning goes, the participants can go either way. Raffman [1996] emphasizes that the exact location of the jump carries ‘‘no normative force’’; it underwrites ‘‘no distinction between correct and incorrect usage’’.15 A different group, or the 15 I would say instead that the exact location of a jump causes no distinction between correct and incorrect usage outside the present (internal) context. Once the conversationalists declare a certain man to be bald, it then becomes incorrect (in that context) to deny that he is bald, and it usually becomes incorrect to deny that a man marginally different from him is bald. Logic is always in force

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same group on another occasion, that jumps at a different place is not thereby in error. Raffman elaborates further: ‘‘there is no reason to [jump], hence no justification for [jumping], at any particular #n in the series (as opposed to #(n  1) or #(n þ 1)). If there were a reason, then either the predicate in question would not be vague or the differences between adjacent items in the series would not be marginal in the sense required to generate a paradox’’. By ‘‘reason’’, Raffman presumably means a ‘‘semantic reason’’. It is virtually analytic that if semantic rules (and non-linguistic facts) always determine when the jump is supposed to occur, then the predicate is not vague after all—there are no borderline cases.16 The situation is (vaguely) reminiscent of the old saw about Buridan’s Ass. The animal finds itself exactly midway between two bales of hay. Since he does not have any reason to walk toward the first bale (as opposed to the second), he does not approach that bale. Since he does not have any reason to go to the second bale (as opposed to the first), he does not approach that bale either. So the poor animal remains where it is, and starves to death. An unfortunate and irrational outcome. He really should have gone one way or the other. Similarly, our conversationalists do not have a (semantically compelling) reason to jump at man 974, nor do they have a reason to jump at #975, nor do they have a reason to jump at #976, etc. But they are required to jump somewhere, and they become semantically incompetent if they do not. Kamp [1981: 272] comes to a similar conclusion, on a structurally similar situation: the truth value gap of the context predicate . . . can . . . be resolved in any one of a number of ways. How it is to be resolved is a matter for decision. Sometimes such decisions must be made; and then somebody has to make them. That is what you just did. The justification for your decision lies, as it must in the case of any genuine decision, in the circumstance that a decision had to be made, rather than the superiority of the option you chose over its possible alternatives.

Raffman [1994: 46] writes: Now one thing we know is that at some point on each (complete) run of judgments along the series . . . a [jump] just does occur: the subject’s slide down the slippery slope is broken . . . Just where the shift does occur on any given run will depend on a constellation of factors, including the direction in which the subject proceeds along the series, where in the series he begins his judgments, his perceptual state at the time, and so forth . . . But occur it does, on every run.

The same goes for the present, social conversational framework. and tolerance usually is. If the speakers go on to deny that this fellow (or a neighbor in the series) is bald, they have thereby shifted to another context. 16 An epistemicist such as Williamson [1994] holds that semantic rules and non-linguistic facts do always determine when the jump is supposed to occur. It is just that we cannot tell where that point is. As noted above, he thus holds that there are no borderline cases, in the present sense of ‘‘borderline’’.

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In a given case, the exact location of the jump is a ‘‘brute mechanical’’ matter (Raffman 1994: 65). Presumably, it will depend on how tired or attentive the subjects are, what else is on their minds at the time, their emotional state, and countless other factors. To be sure, these factors have nothing to do with the amount and arrangement of hair on the heads of the men in the series, and so are irrelevant to the meaning and thus the proper application of ‘‘bald’’. But they are most relevant to its extension of that predicate in the given context. They are relevant to what counts as having little or no hair on the scalp at the time. Raffman [1994: 53] adds that what ‘‘will or will not trigger the [jump] is not something to which we, as judging subjects, have access; so far as the subject is concerned, the [jump] simply occurs’’. Presumably, the members of the conversation can feel the shift coming as they march through the sorites series. When they move through the borderline area, they will find themselves less and less comfortable with their judgments that the men are bald, and the group will find it harder and harder to maintain consensus on baldness. If they are astute to the proceedings, they can see what is coming. So the jump is not a blind process, in the sense that they have no idea when it will occur. After all, they are in charge of the situation (or, to paraphrase Humpty Dumpty, they are master). But the jump is blind in the sense that the subjects need not be aware of the factors that actually trigger the shift—i.e. when enough is enough. An adaption of an example from Wright [1987] illustrates the situation. Consider a digital tachometer connected to a motor. This is a ‘‘brute mechanical’’ device if anything is. Suppose the device registers in units of 10 rpm, and suppose that the motor is running smoothly at 300 rpm. The tachometer dutifully reads ‘‘300’’. Now we slowly increase the speed of the motor in increments of 0.1 rpm. Of course, the tachometer will jump to 310 somewhere. Say it happens when the motor hits 306.2 rpm. If we speed the motor up a bit more, and then start slowing it down, the tachometer will jump back to 300 somewhere. But there is no reason to think that this jump back will occur at 306.2. In a typical case, the tachometer will keep its holding at 310 for a bit, and not jump back to 300 until later, say 305.7 rpm. If we did the experiment on another occasion—the next day, say—we might get slightly different results. It might first jump to 310 at 306.9 rpm and it might jump back to 300 at 306.2. On each occasion, the exact location of the tachometer’s jump from 300 to 310, or back, depends on a number of factors, such as the ambient temperature, relative humidity, wear and tear on the parts of the tachometer, slight but normal fluctuations in the power source, etc. Of course, these things have nothing to do with the speed of the motor. However, for the practical purposes at hand, this does not matter. So long as the random fluctuations affect the reading only in the ‘‘borderline’’ area—say roughly 302.5 to 307.5 rpm—they will not keep the tachometer from operating correctly. When the motor is running at or near a ‘‘clear’’ case, say 309.5 rpm, then the fluctuations do not affect the reading (assuming that the device is in good working order, of course).

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The analogy with our conversationalists on a forced march is straightforward. When they are in the borderline area, they can go either way without impugning their competence, and which way they do go in that area depends on a number of factors irrelevant to the baldness state of the men in the series. It depends on where they are in the series, what judgments they previously made, how attentive they are, how much sleep they got the night before, how annoyed they are at their fellow human beings, etc. When they are considering a man who is not in the borderline area, then these side factors do not affect them, and they judge correctly (if they are competent and observing properly). With the tachometer and the sorites series for baldness (or redness), the exact location of the jumps does seem to be completely arbitrary, in that it really does not matter at all where the jumps occur (so long as they occur in the borderline area). With some sorites series, however, the jump point is not quite arbitrary. Consider a sorites that goes from a spatially separated sperm-egg pair to a resulting 2-year-old baby. The location of the borderline of the predicate ‘‘human being’’, or ‘‘being with human rights’’, has obvious moral ramifications. People who put the border in one place (say viability) rather than another (conception) are not acting arbitrarily. They may well have compelling reasons for what they do. The point here is that there are no linguistic reasons for putting the border in one place rather than another. Presumably, both sides of the abortion debate are speaking the same language. If I consistently put the border in the region of viability, and a mob comes and burns down my house as a result, they will not be accusing me of linguistic incompetence (whatever else they may accuse me of ).17 Returning to our paradigm cases, for both the conversation about baldness and the tachometer, the jump-point is a ‘‘brute mechanical’’ process, and there is no principle of tolerance for it. Nature does have sharp jumps (or at least apparent discontinuities). Raffman [1994: 56] writes that ‘‘we have no intuition that if [#874] does not trigger . . . a [jump] then neither will [#875]’’, just as we have no intuition that if raising the motor to 306.1 rpm does not cause the tachometer to jump to 310, then neither will 306.2 rpm. To be sure, if our conversationalists jump too close to the clear cases, their competence as speakers of English (or their powers of observation) will be compromised. Similarly, if our tachometer jumps at the wrong place—say it reads 310 when the motor is running at 300.2 rpm—its accuracy will be compromised, and possibly also its usefulness (depending on what it is to be used for). Although there is no principle of tolerance in the operation of a mechanical device, the usefulness of such a device is a context-dependent, and probably also a judgment-dependent matter. How accurate the tachometer need be to be useful is a vague matter, and a principle of tolerance does hold for that. 17 I am indebted to Carl Posy here. Notice that, in any case, there will still be some arbitrariness in the ‘‘jump’’ point. Viability and, to a lesser extent, conception are themselves vague matters.

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So how close to the border is ‘‘too close’’ for competent speakers to jump? How inaccurate can the tachometer be without being ‘‘too inaccurate’’ for a given purpose? Since these are also vague matters, one might think that we have just postponed (or moved) the problem of this work. We will deal with this matter later, when we get to so-called ‘‘higher-order vagueness’’ (Ch. 5). 7. DETERMINACY REVISITED, TRUTH, AND CLASSICAL SEMANTICS The time has come to further refine the foregoing notion of determinacy, to indicate the contextual elements that are relevant to its extension.18 This will further articulate the present account of vagueness. Let F be a monadic (possibly complex) predicate in a natural language. Recall the above definition of determinacy, quoted from McGee and McLaughlin [1994: x2]: ‘‘to say that an object a is [determinately] an F means that the thoughts and practices of speakers of the language determine conditions of application for . . . F, and the facts about a determine that these conditions are met’’. To generalize a bit, a first and hasty interpretation of this is that a sentence S is determinately true if and only if the (linguistic) meaning of the words in S and the non-linguistic facts alone guarantee that S is true. In short, S is determinate in this sense if its truth supervenes on meaning and non-linguistic fact. Since English is a public language, it seems plausible that the meaning of its words is at least somewhat independent of the particular context of utterance (content externalism, ambiguity, and considerations like those in Davidson [1986] notwithstanding). Otherwise, we are left with the specter of Humpty Dumpty’s claim to Alice, and no real communication. And, of course, non-linguistic facts such as the amount and arrangement of hair on a person’s head are also largely independent of the context of utterance. Notice that this first and hasty interpretation of determinacy allows virtually no contextual elements into the notion. Suppose, for example, that Harry just went to see a baseball game, the only game in town at the time, and that he saw the entire game. No vagueness about that (or none that matters here). Still, the sentence ‘‘Harry went to the game’’ would not be determinately true, since none of the relevant facts about Harry, the players, etc., or the meaning of the words determines that this sentence is true. The (linguistic) meaning of the phrase ‘‘went to the game’’ does not fix which game we are talking about. Similarly, on the present interpretation, the sentence ‘‘Everyone is present and accounted for’’ would not be determinately true in any context of utterance, since the meanings of the words by themselves do not fix who we are talking about. The sentence, ‘‘I am hungry’’, would also fail to be determinately true in any context, since meaning and relevant facts do not determine the referent of ‘‘I’’. 18

I am indebted to Agustin Rayo here.

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McGee and McLaughlin did not have such an austere notion of determinacy in mind. Surely, ‘‘the thoughts and practices of speakers of the language’’ manage to fix some referents and extensions differently in different contexts of utterance. If a hungry speaker says ‘‘I am hungry’’, then what she says is determinately true, on the McGee and McLaughlin notion. Meaning and the context of utterance fixes the referent of ‘‘I’’ and the non-linguistic fact does the rest. In the first example, the members of the conversation would determine which game they are talking about, by what they say, think, and do. So ‘‘Harry went to the game’’ is determinately true in that context of utterance. Similarly, in a given context, the sentence ‘‘everyone is present and accounted for’’ would be determinately true or determinately false (unless there is some vagueness involved). The members of the conversation would determine the extent of ‘‘everyone’’, and this information would be on the conversational score. I conclude, then, that at least some contextual elements should be included in the analysis of determinacy. When it comes to vagueness, determinacy is sensitive to the comparison class, the paradigms, or the contrasting cases. Suppose, for example, that during a conversation about NBA players, somebody says that a player who is 6 feet, 2 inches tall is short. That sentence is determinately true in that context. The conversational record would indicate the comparison class, and given that class, the sentence is true. So far, I presume that everything is in line with the notion of determinacy invoked by McGee and McLaughlin [1994]. I suspect that I am about to part company with them. I propose that the borderline cases of a vague predicate that have been decided in the course of a conversation not be included in what fixes determinate truth. Suppose, for example, that even after the relevant comparison class, paradigm cases, or contrasting cases have been fixed, Harry remains a borderline case of baldness. According to the foregoing open-texture thesis, in some situations, speakers are free to assert ‘‘Harry is bald’’ and they are free to assert that ‘‘Harry is not bald’’ without undermining their competence. So suppose that someone asserts ‘‘Harry is bald’’, in the course of a conversation, and this assertion goes unchallenged. Then the sentence (or proposition) ‘‘Harry is bald’’ goes on the conversational score. One might take it that ‘‘Harry is bald’’ would then be determinately true in the context after the utterance, at least until the assertion is retracted (in which case the context changes). After all, the thoughts and practices of those speakers have fixed a truth value for this sentence. Nevertheless, I propose that ‘‘Harry is bald’’ is not determinately true in the context of this conversation. As noted in x5 above, Raffman [1996] distinguishes the ‘‘external’’ context of a judgment (concerning a vague predicate), which fixes the comparison class and the like, from the internal context, consisting of the subject’s psychological state. Just about everyone accepts that the extensions of vague predicates vary with the external context. Raffman’s thesis is that these extensions also vary with the internal, psychological context. I invoke a similar distinction here, in the broader

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perspective of public conversations. The present proposal is that ‘‘external’’ factors—comparison class, paradigm cases, contrasting cases, etc.—are included in what fixes determinate truth and determinate falsehood, but internal factors, and in particular, the decisions within (what I call) the borderline area, do not modify the extension of determinate truth and determinate falsehood. On occasion, when it matters, I will use the phrase ‘‘externally-determinate’’, or ‘‘e-determinate’’, for the present notion, to distinguish it from the broader McGee–McLaughlin notion, where all contextually fixed factors, including decided borderline cases, are included. However, for the most part, I’ll use the more common term ‘‘determinacy’’ for the present notion, and use ‘‘established’’ for the broader notion.19 The proposal here is not meant to be a substantial philosophical or analytical thesis concerning a word in use. The notion of ‘‘determinacy’’ is a technical term of art in philosophical treatments, and I presume that I am within my rights to use it as I please, so long as the definition is coherent (even if vague). However, the reader deserves some albeit pragmatic justification for this particular definition. First a word in favor of the opposing conception. One advantage of the broader McGee–McLaughlin notion of ‘‘established’’ is that it allows, or may allow, the imposition of ordinary, classical semantics. That is, within a fixed context, one can coherently maintain that vague predicates have precise extensions, and complementary anti-extensions. Raffman, for example, uses counterfactual conditionals to fix the extensions of vague terms in a fixed psychological context. Consider, for example, our paradigm sorites series, consisting of 2,000 men ranging from Yul Brynner to Jerry Garcia, and consider a single competent subject in a particular psychological state s. A given man m in the series is in the extension of ‘‘bald’’ in s if it is the case that the subject would judge m to be bald when in state s, and m is in the anti-extension of ‘‘bald’’ otherwise. There is no middle ground, no indeterminacy. To be sure, at any given time, neither the observing subject nor an outside examiner can learn the baldness statuses of all of the men relative to the psychological state s, or at least not in the straightforward manner. It is crucial to Raffman’s view that for some of the men in (what I call) the borderline area, if the subject actually did judge them, she would no longer be in the state s. Some judgments trigger a gestalt-like shift to a new psychological state. Once that happens, we are no longer able to determine the status of the other men in the original state s, unless we somehow get the subject back into that state. Nevertheless, on Raffman’s view, for each external context and each psychological state, there is a fixed and precise extension for each vague predicate in that state. Raffman’s proposal can be imported full blown into the present conversational framework. Let us define a man in our series to be in the extension of bald 19 Soames [1999] uses the word ‘‘determinate’’ for the broader notion, which I call ‘‘established’’. The closest he comes to external-determinacy is his notion of ‘‘default determinate-extension’’ and ‘‘default determinate-anti-extension’’.

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at a given stage in a conversation, if the conversationalists would judge him to be bald, if they were asked that question. In the limit, however, we can only discover the status of some of the men that way. Once we ask the conversationalists about any of the men in the borderline area, and get an answer, we thereby change the context. We are then at a new stage in the conversation, and all bets are off concerning the original stage. Nevertheless, the status of each man in the series is fixed at any given time by how the subjects would respond, if queried about that man. Our inability to determine the status of every man in every state is a merely epistemic limitation. Another prominent contextualist, Delia Graff [2000], holds that at any given time, there are sharp borders in the sorites series. It is just that the borders shift with the focus of the subjects. The reason a subject never sees or notices the border is that it is never located at the place where she is looking—the border is never salient. Such is the nature of contextually determined vague predicates. With the imposition of precise extensions (and complementary antiextensions) in each context, the foregoing contextualists can invoke an ordinary, classical model-theoretic semantics. With that comes classical logic, a neat package. To accomplish this, however, the Raffman framework assumes that the counterfactuals are all well defined, and that each has a unique truth value. That is, for each psychological state s and each man m in the series, there must be a fact of the matter concerning how a subject in state s would respond if queried about m. If there is any vagueness, or indeterminacy, in the counterfactuals, the imposition of classical semantics might fail. The same goes for the potentially similar treatment in the present conversational context. There must be a determinate fact of the matter, true or false, concerning how the conversationalists would respond to each query. Similarly, Graff ’s imposition of classical semantics stands or falls with the assumption that at any given time, there are indeed precise, but unknowable borders that lie outside the range of consideration. I do not intend to challenge these assumptions of my fellow contextualists. Let me simply register skeptical agnosticism concerning the presuppositions, and then concede, for the sake of argument, that we can indeed achieve classical semantics, and classical logic, if we invoke the broad notion of determinacy (i.e. establishment). We can and should reason classically, provided that both the external context and the internal psychological state or the conversational context are held fixed. Note, however, that the relevant internal context can change very rapidly. In Raffman’s case, it is an empirical question exactly when, and how often, a given subject’s psychological state changes. At a minimum, the state changes whenever one of the counterfactuals: if asked about man #i, the subject would respond ‘‘bald’’

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changes its truth value. Given how delicate things can get in the borderline area (see x6 above), it is possible that this happens every moment, or every few moments. In the present conversational framework, the context changes every time something is added to the score, which happens every time a new borderline case is called. For Graff, the extension of vague predicates presumably change every time the subject’s attention shifts, however slightly. In our sorites series, for example, the extension of ‘‘bald’’ changes as the subject looks a bit to the left or to the right in the series of men. This rapid and silent change in extension is to be expected, of course, and it helps explain why the sorites reasoning seems so tempting, and yet is incorrect. One can reason classically only so long as the references of the terms and the extensions of the predicates in an argument remain fixed. We keep reminding students in introductory logic classes of this, noting the ancient fallacy of four terms and the like. Nevertheless, I submit that these rapid changes in extension put a damper on the usefulness of classical logic when reasoning with vague predicates. The supposedly precise extensions of the predicates can change in the very act of our considering them as we go through an argument, trying to reason in an ordinary situation. We might call this a Heraclitus problem. Just as the river changes every time we step into it, the extensions of vague predicates change momentarily, literally right before our eyes. An argument may be valid, and its premises might be true at a given moment, but by the time a human reasoner has concluded something on the basis of them, the extension of the terms may have changed, and along with that, some of the sentences may no longer be true. It is small comfort to learn that the conclusion was true. It seems to me that it is plausible to hold the external context fixed for a period of time, since that does not change quite so rapidly, nor need it change without notice. Again, the external context fixes the comparison class, the paradigm cases, and/or the contrasting cases (as well as other things, such as the extensions of proper names and pronouns such as ‘‘everyone’’). I am interested in developing a notion of correct reasoning, and a definition of validity, that handle internal contextual shifts in stride. I submit that this is the normal background for reasoning with vague predicates, and I am interested in the norms of correct reasoning in such contexts. Let the actual extensions of vague predicates vary within a fixed external context, as we reason within it. The current notion of e-determinacy is important for developing this ‘‘logic’’ for vague predicates. It plays a central philosophical role in the model theory developed in the following chapters. It is clear that e-determinate truth is not the same thing as truth simpliciter. Suppose, again, that external factors are fixed and that Harry remains a borderline case of baldness. And suppose that a given competent subject judges Harry to be bald, in a given conversational context. Then it is true in that context that Harry is bald. Nevertheless, it is not e-determinately true that Harry is bald

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in that context. As far as e-determinacy goes, Harry remains a borderline case. I realize that this may be a somewhat nonstandard use of ‘‘determinately true’’, clashing with intuitions apparently shared by many insightful, careful thinkers who write about vagueness, including many who do not adopt the supervaluationist slogan that truth is super-truth.20 For example, Wright [1987: x5]: ‘‘I have heard it argued that the introduction of a [definiteness or determinacy] operator can serve no point since there is no apparent way whereby a statement could be true without being [determinately] so. That is undeniable . . .’’ I have no desire to deny the undeniable, and so I just note that the present usage of ‘‘determinate’’ differs from that of other authors. The burden of the rest of this book is to show that the present notion of e-determinacy is an insightful notion to use when explicating correct reasoning for vague statements, despite the demurral from the slogan that there is no difference between truth and determinate truth. There is, I presume, a norm of conversation that one should assert only truths (other things equal, of course). This norm is not that one should assert only e-determinate truths. That would preclude us from deciding any borderline cases, even temporarily. On the present account, truth varies with context, especially in the borderline area of vague predicates. If, in fact, Harry is borderline bald, and nothing on the conversational score (thus far) precludes judging Harry to be bald (e.g. it is not the case that ‘‘Harry is not bald’’ is on the score), and nothing precludes judging Harry to be not bald, then a speaker cannot help satisfying the norm in asserting or denying Harry’s baldness. The acceptance of ‘‘Harry is bald’’ onto the conversational score thereby makes the sentence true in that context, and thus automatically satisfies the norm. Of course, this holds only for items in the borderline area. Anyone who calls Jerry Garcia bald or denies that Yul Brynner is bald has violated the norm.

8. WHAT DOES ALL THIS CONVERSATIONAL STUFF HAVE TO DO WITH SEMANTICS AND LOGIC? A common, perhaps natural, response to the present program (and to Raffman’s) is that it focuses on the pragmatics (or psychology) of the use of vague terms, and has nothing to do with semantics, and thus nothing to do with logic. The underlying objection seems to be that the correct resolution of the sorites paradox 20 There are exceptions. On the supervaluationist account developed in McGee and McLaughlin [1994], it is possible for a sentence to be true without being determinately true. Tim Williamson [1994] allows for a difference between ‘‘true’’ and ‘‘definitely true’’, but his notions and thus his reasons are very different from those developed here. In Williamson’s epistemicist framework, ‘‘definitely true’’ comes to ‘‘knowably true’’. Sentences with vague terms can be true without being knowable (and thus ‘‘definite’’).

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should turn exclusively on the meaning of the terms. After all, logic flows from meaning. This is a false dilemma. Raffman [1994: 43] writes that her ‘‘story is at bottom a psychological one, resting on a hypothesis about the mental representations that underlie our usage of vague words’’ and so ‘‘where vague predicates are concerned, logic and semantics are more intimately entwined with psychology than might have otherwise been supposed’’. The present story turns more on the pragmatics of conversations involving vague terms, and so I would conclude instead (or, better, in addition) that where vague predicates are concerned, logic, if not semantics, is intimately entwined with pragmatics. Along similar lines (perhaps), Wright [1987: 277] proposes that we seek a semantics according to which linguistic competence is understood ‘‘on the model of a practical skill, comparable to the ability to . . . ride a bicycle’’. Of course, unlike bicycle riding, human beings exhibit the skill of language mastery in concert with each other (more or less). Each language user is beholden to the group, and his or her use must at least largely conform to that of other language users. Otherwise, there would be no communication which is, after all, the point of this enterprise (or at least one of the central points of it). The extensions and perhaps even the meanings of vague terms are tied up with the proper display of this skill, and with the ability of speakers to coordinate their usages to each other in conversations (cf. Lewis’s rules of accommodation, x3 above). As Humpty Dumpty ought to have put it, competent users of the language are its masters. I submit that the correct use of vague terms is bound up with psychology and pragmatics. The notion of open-texture (see x2.2 above) illustrates how this goes. It may require revisions of some widely held views on semantics, meaning, and extensions. Raffman [1994: 69–70] articulates the following biconditional for vague predicates: (B) [A]n item lies in a given category if and only if the relevant competent subject(s) would judge it to lie in that category. See also Wright [1976, 1987]. For most, or perhaps all predicates of natural language, the biconditional surely has exceptions. No matter how competent they are, the relevant subjects sometimes get things wrong—unless we just define ‘‘competent’’ as ‘‘infallible’’ and thus make (B) true by definition. If nothing else, there are occasional performance slips, where a subject might declare that a determinately hairy man is bald, for example. Although Raffman restricts the biconditional to vague predicates, I suggest that for a vast range of predicates—vague or not—in public languages, (B) is at least largely correct. A typical piece of metal is gold if and only if most, or typical relevant competent subjects would judge it to be gold (if asked); for the most part, an animal is a marsupial if and only if most, or typical relevant competent subjects would judge it to be a marsupial; a smallish natural number is prime if

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and only if most, or typical relevant competent subjects would judge it to be prime. And a man is bald if and only if most, or typical relevant competent subjects would judge him to be bald. In what follows, I will leave out hedges like ‘‘most or typical’’, assuming them to be understood. Of course, I do not claim that (B) is even close to true for every predicate of natural language. Moreover, my suggestion that many cases of (B) are true depends crucially on what it is to be a ‘‘relevant competent subject’’. Admittedly, that phrase is doing a lot of work. The claim, so far, is only that in a lot of cases, it is possible to specify what it is to be a relevant competent subject (in a plausible, non-ad-hoc manner) in such a way that (B) comes out true, with the hedges. The biconditional in (B) is material. In each case, after the class of relevant competent subjects is specified, so that (B) is largely true (when it is), there is a question as to which is the chicken and which the egg. It is an instance of what Wright [1992: 108–40] calls the Euthyphro contrast. To adapt one of Wright’s examples, consider this instance of (B): A story is funny if and only if competent subjects would judge it to be funny. In this case, competent subjects are those with normal senses of humor (who also understand the language), and I presume that with this stipulation, there is no controversy over the truth of this instance of (B). It is plausible that in this case, the judgments of competent judges are somehow constitutive of funniness. What makes a story funny is that people tend to find it funny. Plausibly, humor is response-dependent, or, to paraphrase Raffman, judgment-dependent. Let us call this the Euthyphro reading of (B). Contrast this with another case of (B): A number is prime if and only if competent subjects would judge it to be prime. In this case, the relevant competent subjects are those good at arithmetic, or perhaps (some) mathematicians. With this stipulation, the material biconditional is true, with the stipulated hedges, at least for smallish numbers. But in this case, of course, it is not the judgments of these competent subjects that make the number prime.21 A number is prime if it has exactly two (distinct) divisors: itself and 1. A subject is competent here only if she gets it right most of the time, other things equal. In this case, competent judges track the truth, their judgments do not constitute it. Call this the Socrates reading of (B). A Euthyphro, judgment-dependent reading of an instance of (B) is a thesis about the extension of the predicate in question (‘‘funny’’). A Socrates reading is a thesis about what it is to be competent. To be sure, I do not claim that this distinction is a sharp one, nor that it is an all or nothing matter. There may be no 21 What is at stake here is the objectivity of the relevant discourse (see Wright 1992). Readers sympathetic to (semantic) anti-realist or idealist accounts of mathematics should pick another example. I return to objectivity in Ch. 7 below.

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purely Euthyphronic predicates.22 Even judgments of humor, by people with normal senses of humor under normal conditions, are defeasible, sometimes for objective-sounding reasons: ‘‘No, that’s not funny. It is crass and racist.’’ ‘‘Yes, I know it isn’t funny. I just can’t help laughing’’ (see Wright 1992: Ch. 3). Suppose that a predicate P is at least predominantly judgment-dependent, or Euthyphronic, so that its extension is largely constituted by the judgments of competent speakers (under normal circumstances). Suppose also that there are cases on which there is no consensus among competent speakers. I presume that for most, perhaps nearly all, judgment-dependent predicates, there are such cases. Socrates himself asked Euthyphro about the piety of acts that are pleasing to some gods and not to others. There are stories that many people find funny and many don’t; there are sensations that many people find painful and others don’t; there are wines that many people find smooth and many people don’t; etc. Humans are sufficiently diverse that one should not expect complete agreement on their responses and judgments. This is bound to happen with tolerant predicates. There is enough variation in the responses of a single person over time that there will not be complete consistency in her own judgments, even under normal conditions (see x6 above). In sum, an adequate semantics for Euthyphronic, judgment-dependent predicates has to allow for cases in which judgments vary, and there is nothing even approaching consensus. One natural option is that the extensions of judgment-dependent predicates vary with the context of judgments—either the psychological state of the individual judge or the conversational context of an utterance. If there is rough uniformity over the majority of cases, communication will not be impaired very often, and the predicate remains useful. Notice, however, that some words, such as ‘‘groovy’’ or ‘‘cute’’, are arguably judgment-dependent throughout their range of applicability, and yet there seems to be no (or very few) cases on which there is consensus among competent users. It would take us too far afield to explore the role of such predicates in communication. OK, what of thesis (B) on our current example?: A man is bald if and only if competent subjects would judge him to be bald. In this case, the specification of ‘‘competent’’ is rather broad. A subject is competent if she understands the language, and is accurately perceiving the man in question under normal conditions (so she can see at least roughly how much hair each man has and how it is arranged). Raffman [1994: 70] writes that thesis ‘‘B is true with respect to borderline cases because our competent judgments of borderline cases determine their category memberships; conversely B is true with respect to clear cases because clear cases determine what counts as competent judgment.’’ In present terms, the first of these statements is that if the predicate in question is vague, then (B) is to be read 22

I am indebted to Samuel Wheeler here.

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with Euthyphro for its borderline cases (in the present sense of ‘‘borderline’’). This is correct. Since, by open-texture, borderline cases can go either way, the judgments of otherwise competent subjects determine whether the man is bald in the relevant conversational (or psychological) context. As Raffman [1994: 44] puts it early on, ‘‘an adequate treatment of vague predicates and their sorites puzzles must appeal to the character of our judgments about the items in the series’’. Every vague predicate is judgment-dependent in its borderline area (at least). What of the clear, or e-determinate cases? Are those judgment-dependent as well? Raffman’s statement that ‘‘clear cases determine what counts as competent judgment’’ suggests that vague predicates are to be read with Socrates for their e-determinate cases. Frankly, I am not sure. It depends on the predicate. Some vague predicates are arguably (predominantly) judgment-dependent throughout their range of applicability. Consider, for example, the predicate ‘‘looks red’’, or perhaps the predicate ‘‘is painful’’. In those cases, the e-determinate cases will be those on which there is consensus (or near consensus) among those judges who understand the words, are sincere, not terribly confused, etc. Suppose, for example, that we administer a sharp blow to the mid-section of someone who is not in particularly good physical shape. He then screams, and falls on the ground moaning. Suppose we then ask him if he is in pain, and he says ‘‘no’’ (between moans). The most natural conclusion is that he did not understand what we asked him (perhaps because he was in so much pain), and so is incompetent, at least for the time being.23 Along similar lines, the predicate ‘‘looks bald’’ is arguably judgmentdependent throughout its range of applicability. We need not make a definitive claim about our pet predicate ‘‘bald’’, and others like ‘‘red’’ or ‘‘heap’’. For present purposes, it does not matter whether (B) is read with Socrates, or with Euthyphro, or as some combination, for e-determinate cases. On all readings of (B), if someone asserts that Yul Brynner is not bald, or that Jerry Garcia in his prime is bald, we would regard her as at least temporarily or locally incompetent—either as not understanding the meaning of ‘‘bald’’, or not looking carefully (or at all), or misperceiving. Once again, the central claim here—the one that matters—is that when it comes to borderline cases, (B) is to be read with Euthyphro. We need not adjudicate the status of the e-determinate cases of (B). The predicates ‘‘borderline bald’’ and ‘‘competent speaker’’ may themselves be vague. If so, it may not be determinate in a given case whether (B) is to be read with Euthyphro or with Socrates. And it may be indeterminate whether a given instance of the biconditional is used to fix the extension of ‘‘competent speaker’’ or to fix the extension of ‘‘bald’’, or perhaps a combination of both. This matter will be dealt with in due course, when we get to so-called higher-order vagueness (in Ch. 5). 23 Of course, this conclusion is not infallible. It may be that something strange is happening in the person’s brain, so that even though he screams and moans, he is not really in agony.

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Sorenson [2001: 32–3] accuses Raffman of ‘‘confusing speaker meaning as statement meaning’’. If this charge sticks, then it applies to me as well. Sorenson invites us to consider a ‘‘sorites’’ ‘‘concocted with the perfectly precise predicate ‘bald barber who shaves all and only those who do not shave themselves’.’’ He points out, no doubt correctly, that some ‘‘logically naive’’ speakers may undergo some sort of gestalt switch when running through such a series, displaying backward spread and the like. But they would be mistaken. Since there are no barbers who shave all and only those who do not shave themselves, then there are no such bald barbers. The predicate involving our bald shaving barber ‘‘is logically empty regardless of whether people believe it to be empty’’. Although this observation is correct, it does not undermine the account of vagueness presented here, or the account presented by Raffman. The predicate ‘‘bald barber who shaves all and only those who do not shave themselves’’ is constructed from a judgment-dependent predicate, ‘‘bald’’. But that part gets trumped by the rest of the predicate. The combined predicate is not judgment-dependent, and it has no borderline cases. Consequently, there is no issue of sorites with it. The present view is that vagueness turns on judgmentdependence. If there is no judgment-dependence, then there is no vagueness.24 Sorenson makes a second observation: Many vague phrases are too complex to ever be thought about. Iterating ‘‘the mother of ’’ a thousand times yields a grammatical predicate of English. No one will have [a] gestalt switch concerning ‘‘grand998-mother’’. Yet we know there are sorites arguments that use ‘‘grand998-mother’’ as the inductive predicate. The infinite class of sorites argument that are beyond our limits of memory and attention cannot owe their existence to a human penchant . . . Human beings cannot . . . even grasp those arguments.

There is an interesting, general issue concerning what to make of a predicate that is composed of judgment-dependent parts, but is itself too long and complex to be judged by anyone. We have examples like ‘‘Sorenson’s grand998-mother was a good cook’’. Or consider increasingly complex modifications of judgmentdependent predicates, like ‘‘such and such a scene is sort of very interestingly sort of weirdly . . . funny’’. Or let T be a very long Boolean combination whose components are all of the form ‘‘such and such a scene is funny’’, ‘‘this and so wine is tasty’’, ‘‘this sensation is painful’’, or, for present purposes, borderline instances of vague predicates like ‘‘bald’’, ‘‘red’’, and ‘‘human being’’. Suppose that the combination is in fact contingent, and so we cannot fall back on the previous maneuver concerning the contradictory bald barber. Now consider the sentence, ‘‘Roughly, T.’’ I do not see how any of these complex sentences can be understood 24 One might think that the complex predicate ‘‘bald barber who shaves all and only those who do not shave themselves’’ is vague, since it has a vague component. On the present view, it does not count as vague, since it can have no borderline cases, but this is only a terminological matter. If the reader wants to count the complex predicate as vague, much of the text will have to be rewritten, but the changes will be routine. Similar complex predicates are treated in x2 of Ch. 7, on the topic of objectivity.

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as Socratic. Their components are all either judgment-dependent predicates, logical operators like ‘‘and’’, ‘‘or’’, and ‘‘not’’, or modifiers that induce vagueness. The complex sentences are grammatical, and thus it seems that they must be judgment-dependent, and thus Euthyphronic. But how is this possible if the sentences, or even the individual predicates in them, are too complex to be parsed, let alone judged? We can get some mileage here with idealizing counterfactuals. If the predicates are not terribly complex, then perhaps we can tie the truth values of the sentences (in a given context) to how a subject would respond if various limitations on memory and attention span were removed. This is a common move in mathematics concerning notions like computability and derivability. I suspect, however, that if the predicates are sufficiently complex, then the counterfactuals in question will themselves be indeterminate. There simply is no fact of the matter concerning how an idealized version of myself would respond to one of the above sentences. As we speculated (x6 above), in the borderline area of a vague predicate, judgments one way or another can depend on such factors as fatigue and attention span, the very things we are trying to idealize out in dealing with these complex predicates. So be it. If the counterfactuals are themselves indeterminate, then so are the complex sentences. There is no fact of the matter whether they are true or false. But this indeterminacy is not due to vagueness. It applies generally to any overly complex predicate that is judgment-dependent, whether it is vague or not. Since vagueness is not involved, the present account does not apply.25 In any case, on either a Socratic or a Euthyphro reading, thesis (B) is not, by itself, a statement about the meaning of the predicate in question. Raffman [1994: 58] writes that the ‘‘sorites . . . is solved independently of any particular meaning analysis of the predicate . . . On the contrary, all that is required to solve the puzzle is a claim about the correct application or extension . . . of the predicate at issue.’’ Sorenson’s point also turns on the extensions of the predicates at issue, not their meaning. A bit later, Raffman elaborates the point: I do not claim that the meaning analysis or intension of a vague predicate includes a judgmental element. For instance, I do not claim that in calling an object red one means or is saying, in either the ‘‘speaker’’ or ‘‘semantic’’ sense, that the object is merely redrelative-to-me-now or red-relative-to-such-and-such-a-context . . . Rather, I claim that the extension of ‘‘red’’—the class of objects that satisfy the predicate—is always relativized to certain psychological (and nonpsychological) contexts. The sorites is a puzzle about the correct application of vague predicates, and that is all my story addresses. (Raffman 1994: 66)

25 It might be a vague matter whether a given sentence is indeterminate due to judgmentdependence (and thus to vagueness) or whether it is of the kind under discussion here, where the counterfactuals lose their determinacy because they are too complex and we have to idealize away from the very factors that normally fix extensions.

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Arguably, the very meaning of some vague predicates, such as ‘‘looks red’’ or ‘‘looks bald’’, does include a judgmental element. But not every vague predicate is like this. The first entry in Random House Webster’s Unabridged Dictionary for ‘‘bald’’ reads ‘‘having little or no hair on the scalp’’. The relevant entry in the Oxford English Dictionary reads in part ‘‘having no hair on some part of the head where it would naturally grow’’. There is no reason to challenge these definitions, which make no mention of—or even implicit reference to—the judgments of competent users of the term. However, it is clear that what counts as having little hair on the scalp depends on what Raffman calls the ‘‘external’’ context of the use of the term. Is it a sales meeting of a company that makes shampoo, or a sales meeting of a company that makes toupees, or a sales meeting of a company that makes sun-screen? And if the foregoing account is correct, what counts as having little hair can also vary with the shifting internal context of a conversation (or, on Raffman’s account, it varies with psychological state). When Alice objected to Humpty Dumpty’s claim that his words mean just what he chooses them to mean, he replied that it is a question of who ‘‘is to be master’’. Surely, our sympathies here are with Alice. Individual speakers are not ‘‘masters’’ of the meanings (or intensions) of the words they use, independently of the thoughts and actions of the wider community of language users (up to the compelling points in Davidson 1986). But in the borderline region, individual, competent speakers are indeed ‘‘masters’’ of the extensions of vague terms. Humpty Dumpty is right in this limited domain. Ordinary model theory deals with the extensions of predicates (in each interpretation), and only indirectly with their meaning. The underlying theme, or the received view, is that meaning plus context determines extension. If the foregoing account of vagueness is correct, then pragmatic, conversational features figure in how vague words get their extensions in various contexts. The account says something about the sorts of extensions that such words have, and how those extensions are best modeled formally. This is what I meant by the statement at the start of this section that logic is bound up with pragmatics. 9. SUMMARY AND CONCLUSION The foregoing account was developed and illustrated via a forced march sorites series involving a single predicate. Although this is clearly an artificial situation, the story is meant to be general account of vagueness. Let R be a relation and a1, . . . , an be a sequence of objects in its range of applicability. Then the sequence ha1, . . . , ani is a borderline case of R if neither Ra1 . . . an nor :Ra1 . . . an is e-determinate or, in the terminology of McGee and McLaughlin [1994], both are ‘‘unsettled’’. I do not claim that having unsettled cases is sufficient for vagueness. There are some perhaps artificial predicates that are completely sharp but also have borderline cases, in the present sense of

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‘‘borderline’’ (see n. 1 above). I do take the presence of borderline cases to be necessary for vagueness.26 A principle of tolerance is an essential component of sorites. Indeed, a sorites series just is a sequence of objects in which the difference between adjacent members is ‘‘insufficient . . . to affect the justice with which’’ a given predicate applies (again, following Wright 1976: x2). The sequence begins with a determinate case of the predicate and ends with a determinate case of an incompatible predicate. Being prone to sorites is sufficient for vagueness. After all, the ancient paradox of the heap is what got this industry going. However, as noted in x1 above, sorites is not necessary for vagueness. The present account applies to all vague terms, whether they are subject to sorites or not. The main thesis of the present account is that of open-texture. The idea is that in some contexts, a speaker can go either way with a borderline case of a vague predicate without compromising her competence in using the predicate. The slogans are that ‘‘borderline’’ is ‘‘unsettled’’ and that ‘‘unsettled’’ entails ‘‘open’’. The notion of context, and conversational score, are invoked to make the needed distinctions, showing how the use of vague predicates is coherent. Open-texture concerns competent use of predicates in question, and validity turns on extensions. Competent use and extension are connected in this chapter via the thesis that vague predicates are predominantly judgment-dependent, at least in their borderline areas. To at least some extent, a given object, or sequence of objects, is in the extension of a predicate just because competent users of the language judge it to be in the extension. In the borderline area, competent users do not come close to consensus. There may, of course, be indeterminacy concerning chunks of language that does not turn on judgment-dependence, breakdown of consensus, or the like. As noted a few times above, some counterfactuals may be indeterminate in this way. And perhaps some of this indeterminacy is tied to something which deserves the name ‘‘vagueness’’. But if so, the present account does not address it. I do claim to have an account of vague terms to the extent that they are predominantly judgment-dependent in their borderline areas (at least), and I claim that typical vague predicates fit this mold. We now have enough of the picture available and can turn to model theory.

26

Chapter 6 below contains an account of vague objects.

2 Interlude: The Place and Role of Model Theory The next two chapters take a first (and second) pass at a model-theoretic semantics for a formal language containing vague terms. The system, which is based on the foregoing philosophical account, is further refined in subsequent chapters. Although the overall framework is that of the supervaluation theory, it has some unique aspects and angles. The main break with that tradition is that the notion of super-truth does not play a major role. Part of the reason for this is that, following Ch. 1 above, I do not believe that super-truth (or determinate truth, or definite truth) is the key notion when we reason with vague predicates. With McGee and McLaughlin [1994], I thus reject the supervaluationist thesis that truth is super-truth. With that, I also reject the definition of validity as the necessary preservation of super-truth. The relevant notions of truth and validity are more ‘‘local’’. Another break with the supervaluationist treatment is that the present treatment does not require complete sharpenings, models in which all vagueness has been eliminated. The treatment is self-contained, in that I do not presuppose a prior understanding of the supervaluational framework. However, the reader should have some familiarity with formal logic. A first graduate course in meta-theory is more than sufficient. I call the present theory a ‘‘first pass’’ because it has artificial and unwanted features. One of these is that the first account does not address so-called ‘‘higherorder vagueness’’. For each predicate, the model theory has sharp boundaries between the determinate cases and the borderline cases—even though there may be no such sharp boundary for vague predicates of natural language. This feature of the semantics is justified because, for the present, we are not interested in what happens in the neighborhood of those borders. The goal is to provide a formal account of the pragmatic, semantic, and normative features of cases that occur further inside the borderline area—the range where competent users can and do go either way. So-called ‘‘higher-order vagueness’’, or what passes for that, will be taken up in Ch. 5, and the model theory will be extended to deal with this— using the account here as a basis. Another item not treated in the first pass is the existence of vague objects (if such there be). Issues surrounding these are treated

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in Ch. 6, and the model theory is extended to cover the phenomena. But we have to walk before we can run. This chapter briefly presents a philosophical orientation toward the enterprise of logic and formal semantics. As indicated by the title, the key item of interest is the role and function of model theory. This will determine criteria that the present account should meet, and will help forestall some objections that would be at cross-purposes. I then present an interesting dilemma for a logician trying to use a model-theoretic semantics to shed light on correct reasoning for a chunk of natural language. What is to be the logic for the meta-theory, the theory in which we develop the semantics in order to fix the logic of the target theory? 1. LOGIC AS MODEL A typical logic consists of a formal language plus a deductive system and/or a model-theoretic semantics. I presume that we know what these things are, from our education, as philosophers, logicians, mathematicians, linguists, computer scientists . . . But what do formal languages, deductive systems, and model-theoretic semantics do? What are they for? A typical elementary logic course or text begins with the slogan that logic is the study of correct reasoning. Since reasoning is one of the ways that we acquire knowledge, it would seem that logic is a branch of epistemology. So far, so good. But what does all this mean, and what does it have to do with the actual material of the course or text? Once the preliminary slogans are out of the way, the typical student finds himself working with mathematically defined formal languages, rigorous deductive systems, and set-theoretic model theory. He does translations to and from natural language, exercises in deduction, and, depending on the course, proves some meta-theorems. Our question is this: what do the formal symbols produced after the first day of the term have to do with correct reasoning, the advertised goal of our subject? This is an instance of a more general question concerning the application of mathematics to non-mathematical reality. Here, of course, it is an application to a specifically philosophical, normative topic, the nature of correct reasoning. Formal languages lie at the heart of contemporary logic, and so we will begin there with the issues at hand, as a sort of case study. A formal language is a recursively defined collection of strings on a fixed alphabet. This mathematical object somehow corresponds—or is supposed to correspond—to whatever the medium of reasoning is. Well, what is the medium of (correct) reasoning? Is it a natural language, a realm of propositions, a language of thought, or something else entirely? This, of course, is a deep and far-reaching issue in metaphysics and the philosophy of mind, and I plan to be as neutral as possible. Since the notions treated in mathematical logic all turn on the syntax and semantics of formal languages, the enterprise seems to presuppose that

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the medium of reasoning has a syntax and a semantics. That is, the medium of reasoning is much like a language. To be a bit more careful, if there is such a thing as a completely non-linguistic type of reasoning, logic does not deal with it. Apparently, the presupposition of logical study is that the medium of reasoning somehow represents the world in much the same way that a natural language does. So, for convenience, I assume that the medium of reasoning under study just is a natural language, like English, or at least the medium is sufficiently like English for us to use the latter as an example. Not much is changed if the reader substitutes ‘‘realm of propositions’’ or ‘‘language of thought’’ for ‘‘natural language’’ or ‘‘English’’ in what follows. Vagueness seems to be a linguistic phenomenon, at least in the first instance.1 It certainly is on the present account. So what is the relationship between formal languages and natural languages? Why even use the word ‘‘language’’ for both? It is not particularly helpful to wonder whether the set of declarative sentences of English is a recursively defined collection of strings on the familiar alphabet. The issues lie elsewhere. Going back to Aristotle, the mantra of logic is that validity is a matter of form. One of the main staples of our discipline is the thesis that if an argument is valid, then so is any other argument in the same form. This is what makes the intense mathematical work possible. How do we translate this insight, if this is what it is, to natural languages, or whatever the medium of correct reasoning may be? Some writers (e.g. Richard Montague 1974, Donald Davidson 1984, William Lycan 1984) argue that declarative sentences of natural languages themselves have logical structure. Each such sentence has a (perhaps unique) underlying logical form. The purpose of formal language is to display these forms in its formulas (see also the papers in Davidson and Harman 1972). These are realist positions in that they entail that a given formal language is correct to the extent that it displays the relevant formal structure underlying whatever is expressed by declarative sentences of natural languages. The formal language accurately or inaccurately describes aspects of the corresponding natural languages. A close cousin of such views is the thesis that formal languages provide the forms of propositions expressed by natural language sentences. The very same proposition can be expressed by sentences in different languages, and synonymous sentences express the same proposition. Gottlob Frege held that there is an objective realm of ‘‘thoughts’’, and that sentences of a respectable, scientific language should reflect the logical composition of the thoughts they express. On the present topic, he wrote: I can best make the relation of my ideography to ordinary language clear if I compare it to that which the microscope has to the ordinary eye . . . Considered as an optical instrument, [the eye] exhibits many imperfections, which ordinarily remain unnoticed . . . [A]s 1 Merricks [2001] notwithstanding. I take up the question of whether vagueness is only linguistic in Ch. 7.

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soon as scientific goals demand great sharpness of resolution, the eye proves to be insufficient. (Frege 1879: Preface)

This metaphor at least suggests a realist interpretation to the formal language. Just as a microscope reveals what is already there on the slide, but not visible to the naked eye, the formal language reveals what is already there in the natural language sentences. And what is ‘‘already there’’ is the logical makeup of the sentence/proposition/thought, its form. The present study is agnostic on this realist, or descriptive orientation to formal languages. It does not matter much here whether sentences have unique logical forms or not. If there are forms, or propositions, or Fregean thoughts, our question concerns the extent to which they accommodate vagueness, or the extent to which vagueness is a result of some slippage between natural languages and the forms. But that issue can be left unresolved (at least until Ch. 7, when we get to metaphysical vagueness). Frege was perhaps not completely consistent on the relationship between formal languages and natural languages. Following Leibniz, he held that natural languages should be replaced by formal languages, to rid the former of such undesirable features as ambiguity, non-denoting terms, and, dare we say it, vagueness.2 A similar view, held by Quine (e.g. 1960, 1986), is that a natural language should be regimented, cleaned up for serious scientific and metaphysical work. One desideratum is that the logical structures in the regimented language should be transparent. It should be easy to ‘‘read off ’’ the logical properties of each sentence. A regimented language is similar to a formal language regarding, for example, the explicitly presented rigor of its syntax, its truth conditions, and its rules of inference. And, again, vagueness is something that is to be cleaned up (Quine 1981). Call this the normative orientation toward formal languages. The founders of modern mathematical logic—Frege, George Boole, Alfred Tarski, Alonzo Church, et al.—focus almost exclusively on mathematics. Their target was correct reasoning in mathematics. Since there is no vagueness in mathematics (or so it seems), it is no surprise that the manifestly successful systems developed around the turn of the previous century did not deal with vagueness. On the contemporary scene, the extensive industry on vagueness, ambiguity, intensional operators, modal operators, and the like shows that it is not at all straightforward to extend standard logical systems to capture these features. It is natural that some logicians do not like vagueness—for much the same reason that carpenters do not like working with wet wood. However, it does not follow that vagueness is an undesirable feature of natural languages (or whatever the medium of correct reasoning is), nor that a logician 2 One of the purposes of a Begriffschrift is that each sentence express a single thought, and that it be clear in each case just which thought this is (see Jeshion 2001). Ambiguous sentences express more than one thought. As far as I know, Frege did not have a developed view of vagueness. I do not know if, for Frege, there are any vague thoughts.

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can safely ignore it. We reason with vague predicates all the time and, presumably, there are norms behind such reasoning. To be sure, we can, and do, sharpen aspects of the language, as needed. Indeed, on the present view, sharpening is a central component of the actual deployment of vague terms. Borderline cases get decided on a case by case basis, in the course of a conversation. But we rarely need, and even more rarely achieve, full precision. Hans Kamp [1981: 230–1] wrote: Vagueness is typically experienced as a flaw. Often when we come upon an object a and our criteria for the predicate P do not give us a clear verdict as to whether a satisfies P or not; and when, moreover, it appears that this is not because of a lack of empirical information about a, or some other incidental inaptitude on our part at applying the criteria, then we feel a certain pressure to modify the criteria so that they will decide the case which confronts us . . . Generally we shall wish to extend or sharpen the criteria . . . In this way we make the associated predicate more precise, its truth-value gap has been narrowed . . . It seems unreasonable to suppose that truth-value gaps generally are, or even that they could be, completely eliminated through a single modification.

Like it or not, vagueness is here to stay. It is part of the way we deal with the world via language. Given the sorts of beings we are, we have to deal with the world this way (see e.g. Wright 1976, and Ch. 7). Thus, there is not much point in trying to use a precise language instead of English, or to pretend that one is not using a vague language, or to look forward to the day when we no longer use such a language. I thus reject the normative orientation, at least concerning vagueness. The present claim is that a formal language is a mathematical model of a natural language, in roughly the same sense as, say, a Turing machine is a model of calculation, a collection of point masses is a model of a system of physical objects, and the Bohr construction is a model of an atom. In other words, a formal language displays certain features of natural languages, or idealizations thereof, while ignoring or simplifying other features. The approach to logic sketched here is developed further in Corcoran [1973], Shapiro [1998, 2001], and Cook [2000, 2002]. M. Sa´nchez-Miguel [1993] also speaks of a formal language as a mathematical model, or what he calls a ‘‘mock-up’’, of natural language. In general, a mathematical model is a tool that is easy to work with, and easy to study. Of course, a model should also provide insight into the phenomenon being modeled—the logical features of natural languages in the case at hand. To do this, the model should be ‘‘realistic’’ in that some of its features do correspond, more or less, to features of what it is a model of. Insight can be achieved if there is a balance between simplicity and closeness of fit. The logic-as-model approach is perhaps closer to the descriptive than the normative orientation, since it does hold that a good model is ‘‘realistic’’, corresponding in some way with actual features of whatever it is a model of. However, with mathematical models generally, there is typically no question of ‘‘getting it exactly right’’. For a given purpose, there may be bad models—models

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that are clearly incorrect—and there may be good models, but it is unlikely that one can speak of the one and only correct model. There is almost always a gap between a model and what it is a model of. One can sometimes make a model more ‘‘realistic’’ (i.e. more correct) at the cost of making it more cumbersome and difficult to study and use—as happens, for example, when volumes are added and friction is considered in models of physical systems. The more accurate models may be less useful, just because they are too complex, and there need be no competition between simple and realistic models. The models may be serving different purposes. From this perspective, ordinary first- and higher-order formal languages with classical (and precise) model-theoretic semantics are good models of natural languages. But vagueness is idealized away in these models. One agenda of this book is to produce a more realistic model without sacrificing any more simplicity and tractability than is necessary. We proceed in stages, starting with so-called first-order vagueness concerning predicates. Someone who deploys the logic-as-model approach has a task not shared by advocates of the descriptive and normative orientations to logic. Typically, some features of a given mathematical model correspond to features of the reality that is being modeled and some do not. Let us call the former the representors and the latter the artifacts of the model. In a point-mass model of a physical system, the coordinate system, the units of measure, and the notation for numbers are artifacts of the model. They do not correspond to anything in real physical systems. The metric and the various relationships between forces and distances, like the inverse-square principle, are representors. Notice that, in a given case, it may not be clear what is representor and what is artifact, and perhaps the boundary is not sharp. A scientific revolution revealed that within the Bohr model, spheres with sharp boundaries are, at best, artifacts. One can also view the controversies over absolute space as debates over the extent to which parts of the coordinate system are artifacts. It seems that different theorists can agree that a model is good while disagreeing about which parts of the model are representors and which are artifacts. In the case of logic, it must be determined which features of formal languages correspond to relevant features of correct reasoning in natural language, and which features do not. Otherwise, there is a danger of inferring something about the target—reasoning with vagueness in this case—on the basis of an artifact of an otherwise good model. Typically, a formal language consists of variables, connectives, quantifiers, parentheses, and schematic letters of various types, together with formation rules. Clearly, parentheses are artifacts, serving to make the formulas unambiguous. Natural languages have devices for disambiguating, but there is no reason to think that these devices correspond to parentheses in formal languages, nor is ambiguity under study here. We are interested in vagueness. With formal languages, variables, connectives, and quantifiers do have counterparts in natural languages, more or less, and so they are representors. Their deductive

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and semantic behavior is at the core of the various formal concepts of consequence. Moving on to the other aspects of formal logic, deductive systems are mathematical models of actual or possible chains of reasoning in whatever is modeled by the formal language. At the very least, the notion of ‘‘deductive consequence’’ should be a representor. A conclusion in natural language should be deducible from some premises if (and perhaps only if ) the corresponding formal argument is deductively valid in the formal system. Otherwise, it is hard to see how a deductive system models anything. Some authors claim more on behalf of various deductive systems, that various components of various deductive systems have counterparts concerning (correct) reasoning. The ‘‘natural’’ in ‘‘natural deduction’’ suggests that the deductions themselves represent actual, correct chains of deductive reasoning. If so, then the notion of ‘‘deduction’’ is a representor. Ideally, a chain of reasoning in natural language should be correct if and only if it is represented by a deduction in the given deductive system, or perhaps if there is a deduction that represents an expansion of the chain of reasoning, plugging any gaps. One might argue that rules of inference represent ‘‘primitive’’ or ‘‘minimal’’ steps in a chain of reasoning, steps whose correctness is obvious, or have no gaps, or cannot profitably be broken down into parts. If so, rules of inference are representors and not mere artifacts. Some authors, like Ian Hacking [1979] and Neil Tennant [1987], claim that logical terms, such as connectives and quantifiers, are characterized by conservative introduction and elimination rules of natural deduction systems. They argue that such rules somehow represent the way that the ‘‘logical terminology’’ of natural language is (or could be, or should be) learned. This would make introduction and elimination rules representors of the actual acquisition of logical terminology. But we need not take sides on this contentious matter here. The strongest claim we need to make is that individual deductions represent actual, correct chains of reasoning. One often neglected item in formal treatments of vagueness is the extent to which a natural deduction system reflects how one reasons, or how one ought to reason. I take it as a strong desideratum that a formal semantics should make sense of natural deduction systems as theories of how one can or should reason with the language. In effect, the desideratum is that deductions themselves be representors. As shown in the next chapter, typical supervaluational approaches fail to do this, and they rest content with a claim that deductive systems reproduce the valid premise-conclusion pairs. Classical logic is usually sanctioned by the supervaluational semantics; and so a classical natural deduction system will be sound as far as overall logical consequence is concerned. However, its rules may not correspond to proper reasoning patterns. The issue concerns the status of premises and assumptions, and how one reasons under assumptions. So for the supervaluationist, the deductions themselves are mere artifacts, and do not correspond to anything concerning correct reasoning. I think we can and should do better.

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What of model theory, the third and usually most complex component of a formal logic? What, if anything, do the various interpretations or models represent? In other words, what, if anything, are the models (of model theory) models of ? One austere position is that models are not models of anything—they are just artifacts of the logical system. Model theory would be only an elaborate technical device, a tool to study the deductive system. On the other hand, there is more to deductive reasoning than giving deductions. There is also a practice of challenging or refuting purported inferences. Typically this is done by giving a counter-argument, an argument in the same form as the one put forward, but with true premises and false conclusion. The realm of models may be thought of as representing the source of counter-models in natural language reasoning. For better or worse, it is common (but not universal) among logicians and philosophers to think of the model-theoretic notion of consequence as primary. A deductive system must answer to the model theory, not the other way around. It is the deductive system that is said to be ‘‘sound’’ or ‘‘complete’’ visa`-vis the model theory. On such views, presumably, the interpretations of model theory correspond to something concerning correct reasoning, and this something is related to whatever it is that natural language deductions must answer to. As an approximation, I propose that the satisfaction relation between interpretations and formulas of formal languages corresponds, more or less, to whatever it is that makes declarative sentences of natural languages true or false, i.e. to truth conditions. To borrow an elegant phrase from Harold Hodes [1984], truth in a model is a model of truth. This recapitulates the intuition that valid arguments are ‘‘truth preserving’’. On this approach, the relation of satisfaction is a representor—it represents truth or truth conditions. I will not go further into the relevant distinctions concerning ordinary model theory, since it would be too much of a side-track. The point of the present model-theoretic treatment is to develop a system that accommodates or at least models the foregoing notion of determinacy and open-texture regarding vague predicates. So the ‘‘models’’ or interpretations, are more complex than the models of ordinary, first-order model theory. But, hopefully, they are more ‘‘realistic’’. A brief foray into another treatment of vagueness will help illustrate some of the issues concerning the logic-as-model approach. The many-valued account begins with a natural and plausible thought that the attribution of a predicate to a borderline case may be less than fully true and more than fully false. This suggests that there are different truth values, intermediate between complete truth and complete falsehood. The most common accounts use real numbers between 0 and 1 for these truth values. So an ordinary statement such as ‘‘Seymore is borderline tall’’ or ‘‘Seymore is more or less tall’’ becomes something like ‘‘ ‘Seymore is tall’ is true to degree .764’’. But do we have to attribute such precise

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intermediate values to accommodate vagueness? Why .764 and not .682? It would not advance matters to say, instead, that ‘‘ ‘Seymore is tall’ is true to about degree .764’’. Along similar lines, in a many-valued interpretation of a sorites series, there is a first instance in which the predication is less than fully true. This does not jibe with the common intuition that there should be no such boundary. Michael Tye [1994: 14] writes: One serious objection to [the many-valued approach] is that it really replaces vagueness with the most refined and incredible precision. Set membership, as viewed by the degrees of truth theorist, comes in precise degrees, as does predicate application and truth. The result is a commitment to precise dividing lines that is not only unbelievable but also thoroughly contrary to what I [call] ‘‘robust’’ or ‘‘resilient’’ vagueness. For . . . it seems an essential part of the resilient vagueness of ordinary terms such as ‘‘bald’’, ‘‘tall’’, and ‘‘overweight’’ that in Sorites sequences . . . there is indeterminacy with respect to the division between the conditionals that have the value 1, and those that have the next highest value, whatever it might be. It is this central feature of vagueness which the degrees of truth approach, in its standard form, fails to accommodate, regardless of how many truth-values it introduces.

The complaint is that the many-valued approach introduces many super-precise borders, when really there should be none. Suppose that the many-valued approach to vagueness is understood on the descriptive orientation of mathematical logic. That is, suppose that the model theory is meant as an accurate description of the semantics of vague languages. Then Tye’s objection strikes home. It is indeed fantastic to claim that truth values for natural language sentences are that precise. But the objection is misguided against the logic-as-model approach to logic. From that perspective, the many-valued system is only a model, and the super-precise truth values, along with the sharp borders, are artifacts of this model. We use the mathematical precision of the continuum to model the fuzzy slide from truth to falsity as we go through a sorites series. There is nothing untoward about using a precise structure to model a vague one (see also Gaifman 2005). The model theory developed here has this feature as well. Rosanna Keefe [2000: Ch. 2] points out that this response, natural as it is, will not do by itself. When it comes to vague languages, the advocate of a given model must show just which features of the model are artifacts and which are representors, and show how the sorites paradox is resolved in terms of the representors. It is not fair to present a model that supposedly resolves the sorites paradox, but then not accept the very features of the model that resolve the paradox. This, I believe, is a fair challenge to any logic-as-model approach, many-valued or otherwise. Roy Cook [2002] provides the start on a direct response to this charge, on behalf of the many-valued approach. That semantics, however, is not our concern here. I only mention the Tye-Keefe-Cook exchange to illustrate the burden on the logician who follows the logic-as-model

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approach for vagueness. It is a burden that I fully accept, and will discharge as we pursue the development.3

2. THE LOGICIANS’ DILEMMA The general plan of mathematical logic is to use a model-theoretic semantics to determine, or at least shed light on, the notion of logical consequence for a given area or type of discourse. The main success story, of course, is the development of ordinary model theory for classical logic. As noted above, the historical focus of this study—the item to be modeled—is the extensional, precise language of mathematics. The success of this work spawned attempts to capture the logic of other discourses, and to understand the semantics behind other, alternate logics for mathematical discourse. The list includes Kripke structures and topological models for intuitionistic logic, possible worlds semantics for modal logic, and, of course, supervaluational model theory, fuzzy set theory, and the like for vague languages. All of these accounts build on ordinary model theory, taking it as a sort of base. Most of the studies enforce a now familiar distinction between the object language, whose logic is under study, and the meta-language in which the study is done. But this immediately raises a question concerning the logic of the metatheory. What is the logic we are to use when trying to delimit the logic of a given object language? Call this the logician’s dilemma. Sometimes the dilemma is potentially serious, and sometimes it isn’t. We can live with different logics for the object language and the meta-language, provided that the two languages are somehow different from each other, and that we already know, or at least assume that we know, what the logic of the meta-theory is. The left hand of the meta-theory keeps track of the right hand of the object language. For example, the meta-language in which possible worlds semantics is executed is usually extensional—it is ordinary set theory, in fact— and we presuppose that classical logic is correct for such a language. We then use this extensional language, with its presumed logic, in order to shed light on the intensional, modal languages. To be sure, there are controversies over the accessibility relation, and other details of various proposals, and such controversies affect the logic of the modalized object language. However, there is general agreement over the logic of the meta-theory. Prima facie, at least, the meta-theory does not employ a modal logic. 3 For example, the initial model theory, beginning in the next chapter, has sharp boundaries between the determinate cases and the borderline cases of a vague predicate. This may be an artifact of the system, depending on the status of the notion of determinacy, what passes for higher-order vagueness (see Ch. 5). Nevertheless, the sorites is resolved independently of this artifact. The relevant action takes place sufficiently far from the sharp borders in each interpretation.

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Intuitionism is a more interesting case. It is typical for an advocate of classical logic to use Kripke structures or topological models to sharply delimit and study the logical consequence relation for intuitionism. For example, to show that an argument is not intuitionistically valid, one finds a Kripke structure or topological model that satisfies the premises and fails to satisfy the conclusion. And the classical logician uses classical logic for the meta-theory—the theory of Kripke structures or topological models. For the classical mathematician, then, intuitionism and modal logic are on a par. A classical meta-theory is used to shed light on the logic of an object language. Of course, this approach will not do for the thoroughgoing intuitionist, who claims that intuitionistic logic is the proper logic for all mathematics. The metatheory of formal languages, and the theories of Kripke structures and topological models are, of course, themselves parts of mathematics, and so subject to intuitionistic logic. If this logician wishes to use Kripke structures to determine the correct logic for intuitionistic languages, she must first know what the logic of the meta-language is supposed to be. But that is the very thing which she is trying to determine. If there is something illegitimate about classical logic for mathematics, then we cannot very well use such logic in the mathematical metatheory. In this case, there is at least some internal harmony. One can show (in the meta-theory) that Kripke structures are sound for the Heyting predicate calculus by using only inferences sanctioned by the Heyting predicate calculus. Moreover, one can construct structures that do not satisfy excluded middle, double negation elimination, etc. without using any disputed inferences in the meta-theory. In short, if we first assume or presuppose that we already know what intuitionistic logic is (namely, the Heyting predicate calculus), we can show that Kripke semantics is sound for that very logic for the object language. And so the intuitionistic assumption (that the Heyting predicate calculus is correct) is at least partially confirmed in the Kripke semantics. This is a good result for the intuitionist, even if it will not move the classical mathematician. Something analogous happens with ordinary model theory for classical logic. The soundness and completeness results for the object language use classical logic in the meta-theory. We thus find that the object language has the very same classical logic that we use in the meta-theory, and so the classical logician achieves harmony. To be sure, the soundness and completeness theorems of ordinary, first-order logic are not intended to adjudicate the dispute between advocates of classical and intuitionist logic. If they were, the question would be begged in a rather gross manner, since we use that classical logic for the set-theoretic meta-language, assuming it to be the correct logic for that language. In particular, to sanction excluded middle in the object language, we have to use it (or something equivalent) in the meta-theory. The same goes for just about all the axioms and rules of inference. If we used only intuitionistic logic for the meta-language, we would

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validate the rules for Heyting predicate calculus, but we would not validate excluded middle. In this case, we would have a soundness proof for intuitionistic logic.4 We have learned to live with this situation. We assume we already know what the correct logic for the meta-theory is, and settle for the internal harmony, and perhaps the thought that we cannot do any better. The dispute between advocates of classical logic and intuitionists has to be settled some other way. With our present topic, vagueness, we have no reason to be even partially sanguine. For one thing, as noted above, classical logic was developed with the languages of mathematics in mind, languages which seem to have no vagueness. So we cannot take classical logic to be the default, or even the first guess, for vague languages. We cannot transfer the success of classical semantics for mathematics to vague languages, since vagueness was idealized out in the founding work that led to classical model theory and logic. We have no initial reason to think that excluded middle holds. Perhaps it is the direct opposite. Borderline cases at least seem to be failures of bivalence, which underlies classical semantics. Moreover, bivalence is connected with excluded middle via the truth schemes: It is true that F if and only if F. It is false that F if and only if :F. Using uncontroversial inferences, it follows from these that F _ :F is equivalent to bivalence. There are, of course, ways around this in the literature on vagueness. For now, however, I submit that one cannot even count it as a strength for a given model-theoretic account of vague languages that it preserves classical logic—not until we have some independent reason to think that the classical logic is correct for such languages. It is in fact quite contentious what the correct logic for vague languages is. In light of sorites, it is not even clear that intuitionistic reasoning is valid. One cannot assume that the correct logic for the meta-language is classical—unless the meta-language is itself completely precise, purged of any vagueness. On some accounts it is, and on others it isn’t. For the supervaluationist, at least, there may be a short route to the conclusion we want. This theorist defines a sentence to be super-true if it is true under all acceptable sharpenings of the language (see the next chapter). Sharpenings are classical models, in that they lack borderline cases. So we notice that every classical logical truth—including excluded middle—is true under every sharpening, and so every classical logical truth is super-true. QED. But even this is too fast. The theorist encounters (so-called) higher-order vagueness. Exactly which sharpenings are acceptable? It is usually conceded that 4 We cannot prove completeness, however, since the usual Lindenbaum construction uses excluded middle essentially. See McCarty and Tennant [1987] and Shapiro [1988].

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this is itself a vague matter. Acceptability is a vague property of sharpenings, since some sharpenings are borderline acceptable. A common response to this problem is to insist that the meta-language is vague (see Keefe 2000). We thus use a vague model theory, or at least a model theory with a vague predicate, to shed light on vague object languages, via the supervaluational framework. Fair enough, perhaps. But now we cannot just assume that the logic for the meta-language is classical, without begging the question. In sum, we are out to determine or at least shed light on the logic for vague languages. It is genuinely open (or at least controversial) what the logic for vague languages is to be. So if the meta-language of the supervaluational framework is itself vague, we cannot just assume we know what its logic is. The harmony that we found with classical logic and classical model theory is hollow here, due to the controversy over the correct logic for vague languages—just as the harmony is ineffective in the dispute between classical logic and intuitionism. At this point, our theorist might introduce a meta-meta-theory, but this just pushes the problem up to that level. To echo Wittgenstein, eventually we must encounter a language that we just use without the benefit of an additional metameta . . . theory. If that language is vague, then we cannot assume that its logic is classical. Mark Sainsbury [1990: x7] comes to a similar conclusion: the homophonic approach fails us in connection with logic . . . The problem . . . is to say something worthwhile about the logic of the object language. There are two obstacles. First, we do not know what our actual logic is . . . We do not know, for example, whether every instance of P or not P is counted true in our language and thought, and one pertinent reason for this doubt stems from vagueness. Second, even if we knew what our actual logic is, we could not uncritically reuse it in a semantic project, for the existence of sorites reasoning casts doubt upon whether we are right to subscribe to the logic to which we actually subscribe.

A fellow contextualist, Kamp [1981], is also acutely ware of the present issue. He notes that any account of vagueness that employs classical set theory as its metalanguage will encounter sharp boundaries in unwanted places: any semantic account of a vague predicate P according to which at least some objects are definitely P and others are either definitely not P or [belong] to the truth value gap of P, will produce . . . a sharp distinction if the language in which this account is given contains only sharply defined predicates and the apparatus of classical logic and set theory . . . [S]uch is the classical theory of sets.

The obvious way out, of course, is to adopt a vague meta-language that itself employs the proper logic for vague expressions. But what logic is that? Kamp continues: The only escape from this predicament lies in the adoption of a non-classical metalanguage: I do not know . . . of any proposal of this sort that has been worked out in full detail . . . I believe that any such attempt would be premature. For the value of any semantic analysis depends on the adequacy of the logic which underlies the meta-language

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in which the analysis is carried out. But what is the logic of a language that contains vague concepts? This is precisely the question which the paradox of the heap poses with so much force. The purpose of this essay is to elucidate that question. To adopt a vague metalanguage would mean either that we put our cart before the horse or else that by defaulting on the theorist’s obligation to make the logic of his meta-language explicit, fail to attach the cart to the horse in any way whatsoever. (Kamp 1981: 254–5)

For what it is worth, the present logic-as-model approach does not have this particular problem. The systems developed in the subsequent chapters of this book employ a precise meta-theory. Indeed, the background framework is ordinary set theory, the very same meta-theory used for classical model theory. I assume, without argument, that the logic for this theory is classical, with apologies to my intuitionistic friends and opponents. As Kamp put it, ‘‘such is the classical theory of sets’’. This precise, classical framework is employed to model the semantics and logic of a vague object language. I realize that by doing it this way, I cannot get things exactly right. There is almost always a gap between model and modeled. The tools we have are sharp, and I want to go as far as I can with them in modeling the less-than-sharp phenomenon of vagueness. I thus agree with Kamp that no ‘‘intuitive significance can be attached to the sharp boundary [in the semantics] between the truth value gap of a vague predicate and its positive, or negative extension’’ (p. 253). The logic-as-model approach does suffer from an attenuated version of the logician’s dilemma. As noted in the previous section, we have to make sure that the logic that we attribute to the object language (via the semantics) does not turn on any artifacts of the system. Suppose, for example, that in the course of the treatment, we show that the object language has a certain logic L. To claim that this logic is the correct logic for vague languages, we have to coherently argue that the theorem that sanctions L depends only on features of the model theory that are genuine representors. We have to make sure that artifacts of the model theory do not have any untoward ramifications for the notion of correct inference attributed to the object language. The situation may be tractable. As noted above, the initial model theory proposed here does not account for so-called higher-order vagueness (if there is such a thing). The sharp border between clear cases and borderline cases may in fact be an artifact of the system (see Ch. 5). So we have to be wary of features of the attributed logic that turn on this sharp border. As will be seen, however, so long as there is no determinacy operator in the object language, this feature of the system is not a factor in fixing the logic for the object language. Our interest focuses only on what happens (well) inside the borderline area. Once an operator for determinateness is introduced, however, the situation is different. Then we must be more careful. But we are getting ahead of ourselves. In Kamp’s formal development, the logician’s dilemma comes up in another place, and, by his own admission, leads to horrendous complications. As we saw in Ch. 1, Kamp invokes a notion much like that of conversational score, or

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record. Suppose that conversationalists commit themselves to some sentences G, and that another sentence, Pb, is a classical consequence of G. One would think that the conversationalists are committed to Pb. But does Pb really follow from [G]? The inferences are clearly valid in classical logic; but we have as yet no good reason to expect that the logic which our semantic analysis will generate will turn out to be that logic . . . Even so, our analysis must generate some logic or other, and whatever can be inferred according to that logic from sentences which explicitly belong to the [conversational score] must count among the statements to which the context commits us. (Kamp 1981: 251)

Kamp resolves the dilemma by introducing a parameter ‘ for the background logic and, as he says, we then ‘‘hope for the best’’ (p. 257). The consequence relation is defined in terms of this parameter, via a model-theoretic notion of truth-in-context. In effect, Kamp defines a function from possible background logics to consequence relations. If such-and-such is the logic of the background assumptions (i.e. the conversational score) then so-and-so is the consequence relation for the language. The next step is to look for fixed points of this function. That is, we seek values for the parameter ‘, such that if we start with this as the background logic, we get the same logic for the consequence relation. Under certain assumptions (subject to challenge), Kamp shows that there are such fixed points, but he found none that have the properties one would expect a genuine logic to have. He writes: I cannot emphasize enough . . . that virtually all meta-mathematical work in the area of which this paper has tried to produce a first, sadly incomplete chart, is yet to be done. The scope for such work appears, from the modest efforts I have so far made in this direction, almost unlimited; yet anyone who launches himself into the labyrinth of possibilities easily succumbs to doubts whether the work deserves to be done at all. For it appears less than likely that any reasonable logic could be found there. (Kamp 1981: 270–1)

Kamp then envisions a ‘‘negative result’’ which would show that a semantic theory of the sort he develops ‘‘cannot generate a reasonable logic’’. This would show the depth of the difficulty with sorites. I hope that even this limited pessimism is not warranted. The present model theory is, admittedly, a more simplified model than Kamp’s. The notion of ‘‘context’’ or ‘‘conversational score’’ are not official components of the system, although the system does contain some rough counterparts to those notions. Thus, there is no need to seek fixed points that have the proper features. Several different consequence relations, and indeed, several different ‘‘truth’’ predicates, can be defined, and, I believe, some light can be shed on the logic and pragmatics of vague expressions. It is now time to consider details.

3 A Start on Model Theory To establish notation, let us begin with an extremely brief review of ordinary model theory, restricting attention to first-order languages. An interpretation M is a pair hd,I i in which d is a non-empty set (the domain of discourse of M), and I is a function that gives appropriate denotations and extensions to the non-logical terminology. In particular, (1) If c is a constant, then Ic [ d. (2) If f is an n-place function letter, then If is a function from d n to d. (3) If R is an n-place relation letter, then IR is a subset of d n. A variable-assignment s on M is a function that assigns a member of d to each variable. The denotation function for M,s from terms to d is defined in the usual manner: the denotation of a constant c is Ic; the denotation of a variable u is su, and if f is an n-place function letter and t1, . . . , tn are terms, then the denotation of ft1 . . . tn is the result of applying If to the sequence hm1, . . . , mni, where each mi is the denotation of the corresponding ti under M,s. The satisfaction relation, written M,s  F, between interpretations M, assignments s, and formulas F of the given formal language is defined recursively as follows: If a and b are terms, then M,s  a ¼ b if and only if the denotation of a in M,s is identical to the denotation of b in M,s. If R is an n-place predicate letter, and t1, . . . , tn are terms, then M,s  Rt1 . . . tn if and only if hm1, . . . , mni is in IA, where for each i, mi is the denotation of ti under M,s. M,s  :F if and only if it is not the case that M,s  F. M,s  F _ C if and only if either M,s  F or M,s  C. M,s  F & C if and only if both M,s  F and M,s  C. M,s  F ! C if and only if either M,s  C or it is not the case that M,s  F. M,s  9x F if and only if there is at least one variable-assignment s 0 on M that agrees with s on every variable except possibly x such that M,s 0  F. M,s  Vx F if and only if M,s 0  F for every variable-assignment s 0 on M that agrees with s on every variable except possibly x.

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It is, of course, customary to use only two connectives and one quantifier, and to define the others in terms of those. For example, sometimes 9x F is an abbreviation of :Vx :F. However, the correctness of the inter-definitions depends on the classical interpretations of the logical terminology. Since we do not presuppose classical logic for vague languages, we give separate clauses for each connective and quantifier.

1. A CRASH COURSE IN THE F RAMEWORK OF S U P E R V AL UA T I O N M O D E L T H E O R Y In model theory, the extension of each predicate is a set, with sharp boundaries. So unless an epistemicist story is correct, standard model theory does not accommodate vagueness. Indeed, an advantage cited for epistemicism, perhaps its main theoretical advantage, is that it allows for ordinary, classical model theory. Indeed, something in the neighborhood of epistemicism is an almost inevitable consequence of imposing classical model-theoretic semantics on languages with vague terms. As noted in the previous chapter, the purpose of the model theory presented here is to model the semantics and use of vague expressions. Let a be a borderline case of a predicate P. A main theme of the present philosophical account of vagueness is open-texture, the thesis that in some circumstances, competent speakers can go either way with borderline cases of vague predicates. That is, there are circumstances in which a speaker can judge Pa and she can judge :Pa without compromising her competence, or getting the facts wrong (even if we hold the external context fixed). And if she judges Pa in such a context, then this sentence thereby becomes true, in that context. Similarly, if she judges :Pa, then that sentence becomes true. Of course, no one can competently judge Pa and :Pa at the same time—in the same context. And, when tolerance is in force, one cannot judge Pa and :Pa 0 at the same time, if a and a 0 differ only marginally in the relevant respect. One key desideratum of the present enterprise is that the unit of the model theory, an interpretation, should represent a possible state of a conversation among competent speakers with vague predicates. Interpretations thus should allow for the shifting of truth-values of borderline sentences. Moreover, if a is a borderline case of P, and if conversationalists have not somehow committed themselves to Pa or to its negation in a given conversation, then Pa is neither true nor false in that context. Thus, we need interpretations in which bivalence fails. The overall framework of supervaluation accounts of vagueness will serve our purposes. So we start there. The first modification to the classical framework is the notion of a partial interpretation. This is a pair hd,I i, where, again, d is a non-empty set—the

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domain of discourse—and I is a function. Clause (1) and (2) in the above definition of ‘‘interpretation’’ carry over: (1) if c is a constant, then Ic [ d. (2) if f is an n-place function letter, then If is a function from d n to d. This amounts to a decision, for now, not to deal with vague functions or with ‘‘vague objects’’, such as clouds and mountains. We take those up in Ch. 6. The ‘‘partial’’ of partial interpretation concerns the treatment of predicates and relations. Let R be an n-place relation letter. In the partial interpretation M ¼ hd,I i, IR is a pair h p,qi of sets such that p  dn, q  dn, and p is disjoint from q. The idea is that p is the extension of R in the partial interpretation, the ntuples of objects that the relation applies to there; q is the anti-extension of R, the n-tuples of objects that the relation fails to apply to. If IR ¼ h p,qi, then define IR þ ¼ p and IR  ¼ q. From now on, we write ‘‘interpretation’’ for ‘‘partial interpretation’’. Any n-tuples from the domain of discourse d that are in neither IR þ nor IR  are borderline cases of R in the partial interpretation M. If IR þ [ IR  ¼ dn, then R has no borderline cases in M, in which case we say that R is sharp in M. A partial interpretation M is completely sharp if every relation in the language is sharp in M. A completely sharp interpretation corresponds to a classical interpretation since, in that case, the anti-extension of each predicate is just the complement of the extension, just as in classical model theory. We introduce a ‘‘three-valued’’ semantics on partial interpretations. The ‘‘values’’ are t (truth), f (falsehood), and i (indeterminate). Although it does not make much difference formally, it is convenient if we do not think of i, or ‘‘indeterminate’’, as a truth-value on a par with truth and falsehood. Rather, we take i to indicate the lack of a (standard) truth-value in the given partial interpretation.1 First the atomic cases: If a and b are terms, then a ¼ b is true in M,s if and only if the denotation of a in M,s is identical to the denotation of b in M,s; a ¼ b is false in M,s otherwise (and so a ¼ b is never indeterminate, as above). Let R be an n-place predicate letter, and t1, . . . , tn terms. For each i, let mi be the denotation of ti under M,s. Then Rt1 . . . tn is true in M,s if hm1, . . . , mni is in IR þ; Rt1 . . . tn is false in M,s if hm1, . . . , mni is in IR  ; and Rt1 . . . tn is indeterminate in M,s otherwise. Any sentence letters in the language are taken to be 0-place predicates. If P is such a letter, then P is true in M,s if IP þ is non-empty, false in M,s if IP  is non-empty, and indeterminate in M,s otherwise. 1 We could get by without introducing the ‘‘value’’ i at all, by giving necessary and sufficient conditions for a formula to have value t and necessary and sufficient conditions for a formula to have value f. The indeterminate cases would be those not covered by either set of conditions. I find the equivalent ‘‘three-valued’’ system more convenient for exposition.

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We use the strong, internal negation. Its truth-table is: F t f i

:F f t i

Once again, open-texture entails that indeterminate or borderline cases ‘‘can go either way’’ without offending against meaning or the non-linguistic facts. It seems that if F can go either way (in a given context), then :F can also go either way (in the same context). So if F is indeterminate, then so is :F. For some purposes one can introduce a weak, external (Boolean) negation: F t f i

F f t f

The idea here, of course, is that if F is indeterminate, then F is not true, and so F is true. In a sense, :F says that F is determinately not true, while F only says that F is something other than (fully) true, that is, that F is not true. Officially, the external negation ‘’ is not in the object language, and I submit that it does not belong there. The introduction of an operator like this would bring in unwanted precision. The underlying idea behind the vagueness of, say, baldness is that there are no sharp boundaries between the bald cases and those of any incompatible category. To quote Crispin Wright [1976: x1], ‘‘no sharp distinction may be drawn between cases where it is definitely correct to apply [a vague] predicate and cases of any other sort’’. Given simple logical principles, the external negation forces precise boundaries where there should not be any. Wright [1992: 135] comes to a similar conclusion: ‘‘it is essential to lack the expressive resources of the sort of broad [i.e. weak, external] negation operator . . . which always generates a polar [i.e. true or false] sentence, no matter what the status of the sentence it operates on.’’ Wright argues that the ‘‘arrival’’ of this negation operator ‘‘marks the demise of any hope of a satisfactory characterization of higher-order vagueness’’ (see Ch. 5). Even without external negation, the unwanted precise boundaries are definable in the meta-theory. Given that we invoke a precise meta-language, this is inevitable. In each partial interpretation, there is a sharp boundary between the extension of each predicate R and its complement (i.e. the union of the anti-extension of R and the borderline cases). After all, IR þ is a set, as is IR. So we can express the content of the external negation in the meta-language: Ra1 . . . an holds under M,s just in case Ra1 . . . an is not in the extension of R. In general, F holds in M,s if F is either f or i. We make much use of the expressive resources in the meta-language in the formal development below—to

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assess various properties of the system. In the terminology of the previous chapter however, the sharp boundary between the extension and the borderline cases is, or at least may be, an artifact of the system, a feature that does not represent anything in the semantics of natural languages. If we introduced the external negation into the object language, then that language would have this flaw, and this would notably detract from the usefulness of the semantics as a mathematical model of vagueness. We return to this issue when we introduce a determinately operator into the object language (x3 below), and in more detail in Ch. 5, where the phenomenon of higher-order vagueness is treated. For the other connectives, we employ the strong Kleene truth-tables: F&C t f i

t t f i

f f f f

i i f i

F!C t f i

t t t t

f f t i

i i t i

F_C t f i

t t t t

f t f i

i t i i

These truth-tables reflect the open-texture of vague predicates. Suppose, for example, that C is indeterminate, and so could go either way in a given context, and suppose that F is true in that context. Then F _ C cannot go either way in that context. The truth of F guarantees the truth of the disjunction, no matter how C might come out. So disjunctions are true if either conjunct is. On the other hand, if C is indeterminate and w is false, then the disjunction w _ C would also be indeterminate. It will come out true if C comes out true, and false if C comes out false, as reflected in the Kleene truth-table. Something similar holds for the other connectives. The quantifiers are interpreted similarly: Vx F is true in M,s if and only if F is true in M,s 0 for every assignment s 0 that agrees with s at every variable except possibly x; Vx F is false in M,s if and only if there is an assignment s 0 that agrees with s except possibly at x such that F is false in M,s 0 ; Vx F is indeterminate in M,s otherwise. 9x F is true in M,s if and only if there is an assignment s 0 that agrees with s except possibly at x such that F is true in M,s 0 ; 9x F is false in M,s if and only if F is false in M,s 0 for every assignment s 0 that agrees with s at every variable except possibly x; 9x F is indeterminate in M,s otherwise. Suppose, for example, that F is true for at least one of its instances. Then 9x F is also true—it cannot go either way—no matter what happens with the other instances. In a sense, the true instance makes the other instances irrelevant. However, if F is indeterminate for some of its instances, and not true for any,

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then 9x F can itself go either way. If at least one of the instances ends up true, then so does 9x F, and if they all end up false, then so does 9x F. Recall that a partial interpretation M is completely sharp if every relation in the language is sharp in M. A straightforward induction shows that a completely sharp interpretation is equivalent to the corresponding classical interpretation (obtained by ignoring the anti-extensions). So far, we do not have a plausible model for a semantics of vagueness. Suppose, for example, that we have a patch whose color is on the borderline between red and orange. To express this state of affairs, we would make the statement Rt that the patch is red indeterminate; and make the statement Ot that the patch is orange indeterminate. In such a partial interpretation, :(Rt & Ot) would also be indeterminate. But this seems implausible. Even if it is indeterminate that the patch is red and it is indeterminate that it is orange, it surely true that it is not both red and orange—nothing can be red (all over) and orange (all over) at the same time. Again, think of open-texture. The statement that the patch is red can go either way, and the statement that the patch is orange can go either way, but, surely, there is no context in which the patch is both red and orange. Calling it one way precludes the other way. Similarly, the sentence Rt ! Rt is also indeterminate in the given interpretation. Even if Rt can go either way, it surely is determinately true that if t is red, then t is red. For a second example, consider a pair of men, c, d. Both are borderline bald, but c has a bit more hair than d (arranged in the same way). A partial interpretation that reflects this situation would have Bc and Bd both indeterminate, and so Bc ! Bd is indeterminate. But surely, if conversationalists judge Bc to be true (as they can, presumably), then they would thereby make Bd true, since d has even less hair than c. So, intuitively, the conditional Bc ! Bd should come out true, and not indeterminate.2 There is no way that Bc can be true without Bd also being true. The framework of supervaluation goes some way toward remedying these defects of the simple three-valued system. Let M1 ¼ hd1,I1i and M2 ¼ hd2,I2i be partial interpretations. Say that M1  M2 if: (1) d1 ¼ d2; (2) the interpretation functions I1 and I2 agree on each constant and function letter; and (3) for each relation letter R, I1R þ  I2R þ and I1R   I2R  . The idea is that if M1  M2 then the two interpretations have the same domain and agree on the constants and function letters (as per clauses (1) and (2)), and the two interpretations agree on the clear (non-borderline) cases of M1 (as per 2 Hans Kamp [1981: x9] suggests that a contextualist account of vagueness requires a ‘‘nonstandard account of the conditional’’. The idea is that the consequent is interpreted in contexts in which the antecedent is true (if there are any). The ‘‘intuitionistic’’ conditional introduced in the next chapter has a similar feature.

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clause (3)). But M2 may decide some borderline cases of M1. That is, M2 might give a truth-value to indeterminate atomic sentences of M1. If M1  M2, we say that M2 extends, or sharpens M1. Kit Fine [1975] calls M2 a ‘‘precisification’’ of M1, but I prefer the less cumbersome term ‘‘sharpening’’. Since M1  M1, we say that each M1 sharpens itself, following a customary abuse of language. A partial interpretation represents a state of a conversation, and a sharpening of that partial interpretation represents a possible future state of the conversation. In particular, a sharpening represents a situation in which the conversationalists may have decided some borderline cases that were open previously, but have not withdrawn or taken back any such judgments. In the terminology of Ch. 1, a sharpening represents a continuation of the conversation before there are any jumps. The first result is that the sharpening relation preserves truth and falsehood: Theorem 1. Suppose that M1  M2, and let s be an assignment to the variables (over the common domain). If a formula F is true (resp. false) under s in M1, then F is true (resp. false) under s in M2. The proof of Theorem 1 is a straightforward induction on the complexity of F. Recall my suggestion to not think of i or ‘‘indeterminate’’ as a truth-value, but rather as the lack of a (standard) truth-value. In these terms, it follows from Theorem 1 that the sharpening relation is monotonic. As we sharpen, we do not change the truth-values of formulas that have truth-values; we can only give truth-values to formulas that previously lacked them. Since the proof of Theorem 1 goes by induction on the complexity of formulas, it depends on the particular expressive resources in the object language. In particular, Theorem 1 fails if we add an external negation to the language. Indeed, suppose that in M1, the object denoted by a is in neither the extension nor the anti-extension of a monadic predicate P. Then in M1, Pa is indeterminate (i), and so Pa is true in M1. Let M2 be a sharpening of M1 in which the object denoted by a is in the extension of P. Then Pa is true in M2, and so Pa is false in M2. So the semantics of the extended language is not monotonic: Pa goes from being true in M1 to being false in the sharpening M2. As we shall soon see (x3), something similar happens when we add a crude determinately operator. So far, things are pretty rigorous. Now we wax intuitive for a bit. Not every sharpening is legitimate, or true to the meanings of the terms being modeled. Suppose that we have a predicate for ‘‘rich’’ and that there are two members of the domain, Jon and Joe, who are borderline rich in a given partial interpretation (so that neither fall in the extension or the anti-extension of ‘‘rich’’ in that interpretation). Suppose that Joe’s total worth is a bit greater than the total worth of Jon. A sharpening that declares Jon rich and fails to declare Joe rich (or declares that Joe is not rich) would clearly be unacceptable. It is incompatible with the meaning of ‘‘rich’’. Similarly, a sharpening that declares a man bald and

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declares a man with less hair (arranged similarly) to be not bald is likewise unacceptable. Fine [1975] uses the term ‘‘penumbral connection’’ for analytic (or all but analytic) truths for the terms in question.3 An example of a penumbral connection is that if someone is rich then (other things equal) anyone with more money is rich. Another penumbral connection is that if someone is not bald then someone with more hair (arranged the same way) is also not bald. Another is that nothing is completely red and completely orange. Of course, formal languages do not have such analytic (or almost analytic) truths, other than logical truths. The formulas are just sequences of characters, and have no meaning beyond the stated truth conditions for the logical terminology and perhaps any non-logical premises and axioms that are explicitly added. So, as a first approximation, we assume a collection of conditions on partial interpretations. Suppose, for example, that the object language has a predicate B, to represent the English word ‘‘bald’’, and a binary relation R, such that Rab represents the statement that a is less hairy than b. Then a partial interpretation M is not acceptable if there are objects m,n in the domain of discourse such that m is in the extension of B, M satisfies Rnm, but n is not in the extension of B. Similarly, M is not acceptable if m is in the anti-extension of B, M satisfies Rmn and n is not in the anti-extension of B. Define a partial interpretation M to be acceptable if it satisfies all of its penumbral connections, and say that a partial interpretation M2 is an acceptable sharpening of a partial interpretation M1 if M1  M2, and both are acceptable on the same set of penumbral connections. Notice that penumbral connections are given here in the meta-theory: they explicitly refer to extensions, anti-extensions, and the like. One might try instead to specify penumbral connections in the object language, and declare that a partial interpretation M is acceptable only if it makes such sentences true. This would make penumbral connections like non-logical axioms. However, this does not work here. In the case at hand, the ‘‘analytic’’ truths would include: Vx Vy ( (Bx&Ryx) ! By), and Vx Vy ( (:Bx&Rxy) ! :By) The problem is that if B has a few borderline cases in a partial interpretation M, then these sentences are indeterminate in M, not true. The result would be that a partial interpretation is acceptable only if it is completely sharp. This may be the proper formulation for supervaluationists who focus on completely sharp interpretations, but we do not do so here (see x2 below). Thus, we are left with the meta-theoretic specification of the penumbral connections, for the time being. 3 Readers who do not like the notion of analyticity are free to substitute another term, such as ‘‘obvious’’. There is no need to broach Quinean issues here (but we briefly revisit analyticity in Ch. 7).

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I regard this as a (perhaps minor) defect of the system. Penumbral connections are important truths about vague predicates, and it would be good to have an object language that can express their role as penumbral connections, or at least make them true. In the next chapter, we will go some way toward meeting this goal, by expanding the object language. Let F be a formula and let M be a partial interpretation and s a variableassignment. Say that F is super-true in M,s if (1) there is no acceptable sharpening M 0 of M such that F is false in M 0 ,s and (2) there is at least one acceptable sharpening M 0 of M such that F is true in M 0 ,s. Similarly, define F to be superfalse in M,s if (1) there is no acceptable sharpening M 0 of M such that F is true in M 0 ,s and (2) there is at least one acceptable sharpening M 0 of M such that F is false in M 0 ,s. It is immediate that F is super-false in M,s if and only if :F is super-true in M,s. The idea, of course, is that F is super-true if it comes out true under every acceptable way of making the predicates sharp. That is, if we sharpen the predicates in the course of a conversation, then F comes out true, if it gets a truthvalue at all (and F does get a truth-value in at least one acceptable sharpening). Because of the present desire not to rely on completely sharp interpretations, the above definitions of super-truth and super-falsity are more complicated than usual. Let F be a sentence, and assume, for now, that each acceptable partial interpretation has at least one acceptable, completely sharp sharpening. This is a version of Fine’s [1975] completability requirement. Then, in light of monotonicity (Theorem 1), F is super-true if and only if F is true in every acceptable, completely sharp sharpening of M. So it follows from the completability requirement that the notion of super-truth (and super-falsity) depends only on the semantics of completely sharp interpretations, and, in effect, those are ordinary, classical interpretations. So if the completability requirement holds, then the notion of super-truth does not involve the three-valued semantics above. All the action takes place in the completely sharp interpretations. But, as we shall soon see, this is a big ‘‘if ’’. Many defenders of the supervaluation approach insist that the primary (or perhaps the only) notion of truth is super-truth. Rosanna Keefe [2000: 202] writes that ‘‘ ‘Truth is super-truth’ can be the supervaluationist’s slogan’’.4 Accordingly, they define validity as necessary preservation of super-truth: Let G be a set of sentences and F a sentence. Then G S F if for every partial interpretation M, if every member of G is super-true in M, then F is supertrue in M. Theorem 2. Assume that every acceptable partial interpretation has an acceptable, completely sharp sharpening. Let G be a set of sentences and F a sentence. Then G S F if and only if F is a classical consequence of G. 4 Not every supervaluationist agrees. McGee and McLaughlin [1994] distinguish determinate truth from a more deflationary truth, arguing that the latter is the primary notion.

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Proof: Suppose that G S F, and let M be a classical interpretation that makes every member of G true. Then M corresponds to a completely sharp partial interpretation. We stipulate that this partial interpretation has no penumbral connections. So M is an acceptable, completely sharp sharpening of itself, and since it is completely sharp, it has no other sharpenings. So every member of G is super-true in M, and thus F is super-true in M. Since M is classical, it is bivalent: every sentence is either true in M or false in M. A fortiori, F is true in M. Conversely, suppose that F is a classical consequence of G, and let M be a partial interpretation such that every member of G is super-true in M. Let M 0 be an acceptable, completely sharp sharpening of M. Then each member of G is true in M 0 . So each member of G is true in the classical interpretation that corresponds to M 0 . Since F is a classical consequence of G, F is true in that interpretation, and so F is true in M 0 . Since M 0 is arbitrary, F is super-true in M. Corollary 3. For any sentence F, SF if and only if F is a classical logical truth. Given the completability requirement, classical consequence corresponds to supervaluational validity, and so the supervaluationist accepts classical logic for vague expressions, provided that the language stays as it is.

2. TOLERANCE: FEATURE AND BUG Once again, for a supervaluationist, a sentence is super-true if it is true in every acceptable, completely sharp sharpening of the language, and a sharpening is acceptable if it respects all the penumbral connections for the terms in question. For the supervaluationist, completely sharp interpretations are only a technical device used to define super-truth. They need not correspond to actual or even possible uses of the predicates. In the present model-theoretic account, however, the partial interpretations are to represent possible states of conversations among competent speakers of the language. In particular, the extensions (and anti-extensions) of predicate and relation letters stand for possible extensions (and anti-extensions) of vague predicates in various conversational contexts. In Ch. 1 above, I suggested that vague predicates normally satisfy a principle of tolerance (following Wright 1976). If we transfer the idea to the present model-theoretic context, a strict version would be the following requirement: ( ) Suppose a predicate P is tolerant, and that two objects a, a 0 in the field of P differ only marginally in the relevant respect (on which P is tolerant). Then if Pa then Pa 0 . Of course, if something like this were a penumbral connection, then no partial interpretation could represent a non-trivial sorites series (even partially). Indeed,

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suppose that M ¼ hd,Ii satisfies ( ) and that there are objects a1, . . . , an in d such that for each i < n, M satisfies the statement that ai differs only marginally from aiþ1. Then if a1 in the extension of P (so that Pa1 is true in M), then an is also in the extension of P. Such is sorites. Recall that we rejected such a strong principle of tolerance (on grounds of consistency), in favor of the more plausible: Suppose a predicate P is tolerant, and that two objects a, a 0 in the field of P differ only marginally in the relevant respect (on which P is tolerant). Then if one competently judges a to have P, then she cannot judge a 0 in any other way. This is readily transferred to the present formal framework. Let us say that a partial interpretation M ¼ hd,Ii is tolerant with respect to a predicate P if the following holds of M: (T) If M satisfies a formula representing a statement that two objects a, a 0 in d differ only marginally with respect to P, and if Pa is true in M, then it is not the case that :Pa 0 is true in M. Notice that (T) is formulated in the meta-language. There is, in general, no sentence F such that (T) holds in a partial interpretation M if and only if F is true in M. If we had a weak, external negation operator in the object language, we could remedy this. Let IND(x,y) be a statement that x differs marginally from y. Then (T) holds in M if and only if M satisfies VxVy((IND(x,y)&Px) ! :Py). However, I argued above that an external negation is not appropriate in the object language, and that matter stands. A partial interpretation M for which (T) holds can represent a non-trivial sorites series. Let a1, . . . , an be objects in the domain such that for each i < n, M satisfies the statement that ai differs only marginally from aiþ1. And suppose that a1 is in the extension of P and an is in the anti-extension of P. It follows that there is at least one j such that aj is neither in the extension of P in M nor the antiextension of P in M. That is, aj is borderline, and Paj is indeterminate in M. As expected. It follows, of course, that there can be no partial interpretation M that satisfies (T) such that P is sharp in M and there is a sorites series for P represented in M. So if we think of (T) as a penumbral connection, then it precludes complete sharpenings for P. And I submit that (T) should be a penumbral connection. Although, as noted in Ch. 1, there are some artificial situations in which tolerance is violated,5 in normal contexts tolerance seems as essential to the proper deployment of the term as the more usual penumbral connections. In the present model theory, unlike that of the supervaluationist, this tells against Fine’s [1975]

5

Haim Gaifman [2005] calls such contexts ‘‘infeasible’’.

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completability requirement, stating that every acceptable partial interpretation has an acceptable, completely sharp sharpening. I do not want to go so far as to claim that if we sharpen our terms to eliminate the vagueness altogether, we would irrevocably change the meaning and function of the words. The common but artificial situations in which tolerance is violated do not constitute a violation of the meaning of the terms. The supervaluationist Keefe [2000: 188] gets it right when she writes that ‘‘various general considerations seem to show that tolerance is essential to the whole point of our vague predicates’’.6 This insight, transferred to the present model theory, precludes acceptable, completely sharp interpretations, at least if we are to model the common situations in which tolerance is in force. So part of the motivation for the present program is to see how far we can go without invoking completely sharp interpretations. This avoidance is consonant with the philosophical view of vagueness developed in Ch. 1. As indicated, we (the competent speakers of the language) regularly do sharpen vague predicates in the course of conversations, but we do not (usually) completely sharpen them. As we saw in the previous chapter, Hans Kamp [1981: 231] agrees, writing that generally, ‘‘we shall wish to extend or sharpen the criteria [for a vague term]. In this way we make the associated predicate more precise, its truth-value gap has been narrowed . . . It seems unreasonable to suppose that truth-value gaps generally are, or even that they could be, completely eliminated through a single modification.’’ Keefe [2000: 190] concedes that complete sharpenings ‘‘do indeed fail to capture all features of the meanings of our predicates’’. Given her commitment to defining truth as truth in all acceptable, complete sharpenings, she cannot hold that tolerance (T) is a penumbral connection, for then it would hold in all acceptable sharpenings. In a footnote she writes: we cannot straightforwardly read off cases of penumbral truths from our unreformed intuitions, for the sorites inductive premise would then count as a penumbral truth . . . We cannot start off with all sentences that are intuitively true (both atomic predications and compound sentences) and then construct the structure of [acceptable sharpenings] so as to respect all these truths by ensuring that they are true in all [sharpenings]. The super-truth proposal that truth is truth on all complete and admissible [sharpenings] is incompatible with including sorites premises among the repository of truths, since predicates must have sharp boundaries in complete [sharpenings] and so the premises involving them must be false. (Keefe 2000: 193 n. 14)

Amen. See also Tappenden [1993] and Burns [1991]. A few pages later, in another note, Keefe agrees that meaning is not preserved through sharpening, 6 See Fodor and LePore [1996]. This point is related to a criticism of supervaluationist accounts mounted by Tye [1989] and Sanford [1976]. They argue that the truth of a sentence involving a vague predicate cannot be determined by the truth-value of other sentences using other (sharp) predicates.

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and adds that she disagrees with Fodor and LePore [1966: 522] that ‘‘the very motivation that leads supervaluation theorists to postulate ‘penumbral connections’ is to allow that conceptual truths are respected by all classical models, including classical valuations’’. According to Keefe, what matters instead is ‘‘respecting truths of meaning in the supervaluationary model as a whole’’. In the overall dialectic, this is a fair point. The supervaluationist need not claim that completely sharp interpretations are consistent with the meaning of vague terms. The main idea is that completely sharp interpretations are a tool in figuring the truth conditions of vague sentences.7 In the terminology of Ch. 2, completely sharp interpretations are an artifact of the supervaluation system. And a useful artifact at that, since it produces a familiar, clean model that is easy to work with. The completely sharp interpretations allow the preservation of classical logic (assuming, of course, that the preservation of classical logic is what we want). Nevertheless, I think we can get a better model. In this work, I show that we can have our cake and eat it too, developing a supervaluation-type of system that does not embrace completely sharp sharpenings, at least when tolerance is in force. Every partial interpretation in the system represents a possible state of a conversation, respecting all penumbral connections and tolerance (when it is in effect). As noted in the previous chapter, classical logic is not presupposed, nor should it be. Classical logic was developed with mathematical languages in mind. The logic of vague expressions will be what it will. That is one of the things we are trying to figure out. Recall that Theorem 2 and Corollary 3 above depend on Fine’s completability requirement. If there is a partial interpretation which has no acceptable, completely sharp sharpenings (along the lines of (T)), then these results no longer hold. Indeed, suppose that M is a sharpening that represents a non-trivial sorites series for a predicate P, as above. Let M 0 be a sharpening that has the denotata of a1, . . . , a492 in the extension of P, and the denotata of a494, . . . , a1,000 in the anti-extension. Moreover, the denotation of a493 cannot go into either the extension or the anti-extension if P without violating tolerance (T). Then there is no acceptable sharpening of M 0 in which Pa493 _ :Pa493 is true. Thus, Pa493 _ :Pa493 is not super-true in M 0 . So at least one classical logical truth fails to be super-true. Keefe [2000] raises another objection to the traditional supervaluationist account that turns on its play with completely sharp sharpenings. Suppose we have a partial interpretation whose language has a word for ‘‘tall’’, and assume that the completability requirement is in effect, so that there is at least one acceptable, completely sharp sharpening. Then the following comes out as 7 I am indebted here to Michael Morreau, Georges Rey, and a referee for Oxford University Press, although I do not know how much they agree with me or even if they would accept the way I just put the point.

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super-true in that partial interpretation: (H) There is a height x such that people of height x are tall while people 0.01 inches shorter are not tall. I take it that pre-theoretic intuitions declare this sentence to be untrue, if not outright false. Keefe [2000: 183] concedes that the super-truth of sentences like (H) is ‘‘one of the least appealing aspects of the theory’’. Nevertheless, she accepts the traditional supervaluational framework in light of its compensating virtues, writing that the ‘‘costs are easily worth paying given the advantages of the theory’’. Well, this depends on the value of the coin of the realm, and the available alternatives.

3. T H E W R O N G N O T IO N O F D E T E R M I N A C Y In the supervaluational framework, a sentence is determinately (or definitely, or clearly) true if it is super-true. It is thus natural, perhaps, to add a determinateness operator DET0 to the object language. Given the slogan that truth is super-truth, DET0 is just a truth operator.8 The following would then be added to the recursive definition of satisfaction: Let M be a partial interpretation and let s be a variable-assignment on the domain of discourse. If F is a formula, DET0(F) is true in M,s if F is super-true in M,s, i.e. if there is no acceptable sharpening M 0 , such that F is false in M 0 ,s, and there is at least one acceptable sharpening M 0 of M such that F is true in M 0 ,s. DET0(F) is false in M,s if it is not the case that F is super-true in M. Notice that if DET0(F) is false in M,s, it does not follow that F is super-false in M,s. The super-falsity of F in M,s corresponds to the truth of DET0(:F) in M,s. If the meta-language is precise (as we assume here), then for any formula F and any partial interpretation M and variable-assignment s, it is not the case that DET0(F) is indeterminate (i) in M,s: DET0(F) is either true or false. So the determinateness operator forces precision into the object language. In this sense, the DET0 operator is like weak or external negation, allowing us to express a precise meta-linguistic property in the object language. As with external negation, this precision is unwanted (depending on how we resolve the issue of higher-order vagueness). Notice that we can define external negation in terms of the DET0 operator: F  df :DET0(F). Along similar lines, DET0(F) is true in M,s if and only if DET0(F) is super-true in M,s, if and only if DET0(DET0(F)) is (super-)true in M,s. If the DET0 operator were added, the semantics would no longer be monotonic (and so Theorem 1 above would fail). Indeed, let M be a partial 8

Since DET0 is an operator and not a predicate, we are not flirting with paradox.

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interpretation in which an atomic sentence F is indeterminate (i). Then DET0(F) is false in M. Let M 0 be a sharpening of M in which F is true. But then DET0(F) is true in M 0 . That is, DET0(F) is false in a partial interpretation and true in a sharpening of the interpretation. With the expanded language, Theorem 2 and its corollary no longer hold, and so the connection to classical logic is broken. For example, {DET0(F)} S F, but F is not a classical consequence of {DET0(F)}, for the trivial and boring reason that DET0 is not a classical logical operator. Similarly S (DET0(F) ! F), but (DET0(F) ! F) is not a classical logical truth. To be sure, it follows from the definitions that (DET0(F) ! F) is super-true. So, in a sense, (DET0(F) ! F) is an analytic truth, true by stipulation. So the break with classical logic is rather minimal thus far. But it gets worse. Consider the converse implication, from F to DET0(F). Notice, first, that we have {F} S DET0(F). Indeed, let M be a partial interpretation such that F is super-true in M. Then DET0(F) is true in M, and in any sharpening of M. So DET0(F) is super-true in M. Intuitively, the inference from F to DET0(F) seems correct—in light of the definition of validity (S) as the necessary preservation of super-truth. In contrast, the sentence (F ! DET0(F)) is not super-true in every partial interpretation. Let M be a partial interpretation in which the object denoted by a constant a is neither in the extension nor in the anti-extension of a predicate P. Then Pa is indeterminate in M, and so :DET0(Pa) is true in M. Let M 0 be a sharpening of M in which the object denoted by a is in the extension of P. So Pa is true in M 0 and in every sharpening of M 0 . So DET0(Pa) is true in M 0 . So :DET0(Pa) is false in M 0 . Thus DET0(:DET0(C)) is false in M. So, putting it all together, :DET0(Pa) ! DET0(:DET0(Pa)) is false in M. The upshot of this example is that the rule of arrow-introduction (or the deduction theorem) is not valid in the expanded language. We have an instance of the scheme {F} S C in which S (F ! C) fails. Incidentally, we do have that if F is an atomic sentence, then (F ! DET0(F)) is super-true, and S (F ! DET0(F)). Consider, for example, a sentence in the form Pa. Suppose that Pa is true in a partial interpretation M. So the object denoted by a is in the extension of P in M. But then a is also in the extension of P in any sharpening of M. So Pa is super-true in M and so DET0(F) is true in M. So (Pa ! DET0(Pa)) is true in M. Since M is arbitrary, (Pa ! DET0(Pa)) is true in every partial interpretation and, in particular, in every sharpening of M. So (Pa ! DET0(Pa)) is super-true. In light of the monotonicity of the original language (Theorem 1 above), this result generalizes a bit. Let F be any sentence in which the DET0 operator does not occur. Then (F ! DET0(F)) is super-true.

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I submit that this is a counterintuitive feature of the semantics. The problem lies with the loss of monotonicity, and perhaps with the slogan that truth is super-truth. Suppose that an atomic sentence F is indeterminate in a partial interpretation M, in which case DET0(F) is false in M. If we sharpen things, and move to a partial interpretation M 0 in which F is true, we find that DET0(F) is true in M 0 as well! That is, as we sharpen to make F true, we thereby make F super-true. So the DET0 operator, and the notion of super-truth, correspond to the broad, McGee-McLaughlin notion of determinacy, which takes all contextually fixed factors, including decided borderline cases, into account (see Ch. 1: x7). In this sense, once a borderline case is called, the corresponding sentence is determinately true. In x7 of Ch. 1, I proposed a more stable notion, which I called e-determinacy. A sentence is e-determinately true if the meaning of the terms, the non-linguistic facts, and the comparison class and other external contextual factors make it true. Putting so-called higher-order vagueness aside, e-determinate truth is independent of any decisions made in the course of conversation among competent speakers of the language. In the model theory, the idea is that although the truthvalue of an indeterminate sentence F can vary from sharpening to sharpening, the truth-value of DET(F) should remain fixed. A sentence can be true in a sharpening without being determinately true in that sharpening. Our formal framework, as defined so far, does not allow this. To model e-determinacy, we need to introduce some sort of index, so that when we envision a sharpening that makes F true, we somehow still evaluate DET(F) from the perspective of the original partial interpretation. This will restore monotonicity, but it will drive a wedge between truth and super-truth. The proper DET operator will also undermine the above notion of validity—the necessary preservation of super-truth. But I get ahead of myself. For now, note that we do not officially add the DET0 operator to the language. 4. FRAMES, OPEN-TEXTURE, AND THE FORCED MARCH I propose to add a Kripke-structure-style structure to the system. It will model the open-texture view of vagueness, and the concomitant notion of e-determinacy, developed in Ch. 1, better than the straight supervaluationist framework sketched just above. Define a frame F to be a structure hW,M i in which W is a collection of partial interpretations, M [ W, and for every partial interpretation N in W, M  N (so that all of the partial interpretations in W have the same domain). The designated partial interpretation M is the base of the frame F. The decision to have every partial interpretation in a frame have the same domain is only a convenience, to help the model theory get started. In general,

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as conversations evolve, more (or fewer) objects may come under consideration. We will go a little way toward relaxing this requirement in Ch. 6, on (so-called) vague objects.9 Burgess and Humberstone [1987] invoke a similar notion, but without a designated base. A frame is what Fine [1975] calls a ‘‘specification space’’, with each N in W being a ‘‘specification point’’. However, as noted above, Fine’s account of vagueness makes essential use of completely sharp sharpenings. All the relevant action takes place in those interpretations. In light of the completability requirement, a specification space might as well consist of a partial interpretation together with a set of completely sharp sharpenings of it. The other ‘‘points’’ play no role. Here, of course, we have no completability requirement. The sentences that are true in the base M of a frame F ¼ hW, M i represent e-determinate truths, and the sentences false at the base represent e-determinate falsehoods. The other sentences are indeterminate (in that frame). The main indeterminacies, of course, are those due to borderline cases of vague predicates, although we will briefly use the framework to model other sorts of indeterminacies, such as future contingents (x6 below) and Brouwerian choice sequences (at the end of the next chapter). Each sharpening in the frame represents one way that some indeterminacies can turn out, consistent with the meaning of the predicates, the non-linguistic facts, and the externally fixed contextual factors (such the comparison class or paradigm cases of vague predicates). Let me illustrate the role of a frame with the forced march sorites series introduced in Ch. 1 (along the lines of Horgan 1994a). We have a series of 2,000 men lined up. The first is Yul Brynner, who has no hair at all, and the last is Jerry Garcia in his prime, who we will assume is as hairy as can be. After the first, each man in the series has slightly more hair (arranged in roughly the same way) than the one just before. To model the situation, consider a formal language that has a predicate B, for baldness, and names m1, m2, . . . , m2,000 for the men in the series. It also has a binary predicate Rxy for ‘‘x has only slightly less hair than y, arranged in the same way’’. Consider a frame F ¼ hW,M i. The domain of discourse for each partial interpretation in W consists of the men in the series. In the base partial interpretation M, the extension of B consists of the men that are determinately bald. Thus every partial interpretation in W also has these men in the extension of B. Similarly, the anti-extension of B consists of the men that are determinately not bald. For each i < 2,000, the pair hmi, miþ1i is in the extension of R in M (and so in every partial interpretation in W ). The other partial interpretations in W represent various ways that statements can be put on the 9 The intuitive idea is that the external context is held fixed throughout a frame, and the various partial interpretations in it track changes in the internal context—the conversational record for example. When it comes to a vague term, such as ‘‘bald’’, the domain of discourse might include the entire comparison class. Changing that would change the external context. However, there are other situations in which a change in domain would not alter the external context. The present account cannot handle those. Thanks to a referee for this point.

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conversational record consistent with the meaning of ‘‘bald’’and the nonlinguistic facts (as well as the determinate truths). That is, the various partial interpretations in W mark what competent speakers of the language (who perceive things accurately) can hold. The conversationalists envisioned in Ch. 1 were asked about the baldness-state of each man in the series, starting with the first, and we insisted on a communal verdict in each case. This keeps them in a single context at each moment. Being competent speakers of English, they agree that the first man is bald, that the second is bald, etc. Eventually, they move into the borderline area, and encounter cases whose baldness state is ‘‘unsettled’’. Nevertheless, the participants to this language exercise will probably continue to call the men bald as they move through the borderline area—for a while. This puts propositions like ‘‘man 923 is bald’’ and ‘‘man 924 is bald’’ on the conversational record. Each time they declare a borderline case to be bald, they explicitly sharpen the interpretation. For example, when they declare #902 bald, they move from a sharpening M1 in which m901 is in the extension of B and m902 is not, to a (further) sharpening M2 in which m902 is in the extension of B (assuming that m902 is a borderline case). We have M  M1  M2. So long as the conversationalists are competent speakers of English (and perceive things accurately), they will not go through the entire series, and call #2,000, Jerry Garcia, bald. At some point, the consensus on ‘‘this man is bald’’ will break down. Given the ‘‘forced march’’ instruction, they will eventually agree to call one of the men ‘‘not bald’’. Suppose this first happens with #975. Note that they had just agreed that #974 is bald, and so at that point, they put ‘‘Man 974 is bald’’ on the conversational score. This corresponds to a further sharpening, say M74 of M2. Recall the principle of tolerance: Suppose a predicate P is tolerant, and that two objects a, a 0 in the field of P differ only marginally in the relevant respect (on which P is tolerant). Then if one competently judges a to have P, then she cannot judge a 0 in any other way. We assume that tolerance remains in force during the conversation. Recall that in the partial interpretation M1, m901 is in the extension of B and m902 is not. This is not a violation of tolerance. In the conversational situation corresponding to this partial interpretation, the baldness state of man 902 has not been judged at all, one way or the other. The conversationalists violate tolerance if they judge the two men differently, but not if they have judged one and not the other.10

10 As in Ch. 1, it would violate tolerance if the conversationalists judge man 902 to be borderline bald, or if they decided to leave man 902 unjudged. This last is a judgment, of sorts. For simplicity, we give the conversationalists only two verdicts: bald and not bald. This artificial simplicity is reflected in the present model theory, recalling our decision to not think of the ‘‘value’’ i as a third truth-value, on a par with truth t and falsity f.

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Men who are in neither the extension nor the anti-extension correspond to those that have no judgment at all. When the participants jump and declare man 975 to be not bald, they implicitly deny that man 974 is bald, and so ‘‘Man 974 is bald’’ is removed from the conversational record. This much is required if tolerance is in force. At the same time, ‘‘Man 973 is bald’’ also comes off the record, and so do a few more of their recent pronouncements. The situation after the jump corresponds to a partial interpretation N1 in which m975 is in the anti-extension of B. In light of tolerance, m974, m973, and a few more are not in the extension of B in N1. This is not to say that these men are also in the anti-extension of B in N1. What matters is that they are not in the extension. So N1 is not a sharpening of M74. When the participants jump, they implicitly ‘‘unsharpen’’ (some of the cases). This is the formal analog of items being removed from the conversational record. Of course, N1 is still a sharpening of the base interpretation M. It represents a possible state of a conversation, consistent with the meanings of the terms, etc. The situation in the frame follows the metaphor of ‘‘jumping’’. When the participants keep calling the men bald, they move out on a single branch of the frame. When they finally ‘‘jump’’ and call one of the men not bald, they leave that branch and ‘‘jump’’ to a different branch. At this point, we reverse the order of query. We asked the conversationalists about #974 again. I suggest that they would explicitly retract that judgment, saying that #974 is not bald (and thus put ‘‘Man 974 is not bald’’ on the record). This corresponds to a sharpening N2 of N1, in which m974 is in the anti-extension of B. Their subsequent denial that m973 is bald corresponds to a sharpening N3 of N2 (and N1). Of course, the conversationalists will not move all the way back down the series, and end up denying that Yul Brynner is bald. At some point, they will jump again, and declare a certain fellow to be bald—suppose it is man 864. This again will result in the removal of certain items from the conversational record, such as the denial that man 865 is bald. And this second jump would correspond to another unsharpening. The partial interpretation corresponding to this state of the conversation would not be a sharpening of the one just before, since items were removed from the conversational score. This corresponds to the removal of some items from the anti-extension of B; they ‘‘jump’’ to yet another branch in the frame. In general, our conversationalists will move backward and forward through the borderline area. Tolerance is enforced at every stage, by removing judgments from the conversational record—and the extension or antiextension of B—whenever a jump occurs. As discussed at the end of Ch. 1, there is no reason to think that the participants will always jump at the same place(s) as they move back and forth through the middle part of the series. We only know that on each run, they will eventually jump, well before they assert that the men near the end are bald and well before they deny that the men near the beginning are bald. So the set W of partial interpretations in the frame would be rather large, containing one partial

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interpretation for each batch of judgments that are jointly consistent with the meaning of the words and the non-linguistic facts. Given the meaning of ‘‘bald’’, it is natural to require if mi is in the extension of B in a partial interpretation N [ W and if j < i, then man mj is in the extension of B in N. Similarly, if mj is in the anti-extension of B in a partial interpretation N [ W and if j < i, then mi is in the anti-extension of B in N. This corresponds to what Fine [1975] calls a ‘‘penumbral connection’’. The principle of tolerance would entail that if mi is in the extension of B in a partial interpretation N [ W, then miþ1 is not in the anti-extension of B in N. This implies that there are no completely sharp interpretations in W. There is always a gap between the extension and the anti-extension of B, although the size and location of the gap varies from partial interpretation to partial interpretation. Competent speakers do sharpen, but they do not regularly completely sharpen, and they cannot completely sharpen so long as tolerance remains in force. In the remainder of this chapter, and the next, we add features to the system that allow us to model these features.

5. FORCING Let F ¼ hW,M i be a frame and let N [ W. Let s be a variable-assignment and F a formula. Say that F is forced at N under s if for each sharpening N1 in W such that N  N1, there is a sharpening N2 in W such that N1  N2 and F is true in N2 under s.11 Forcing is another analog, or model, of truth. It will prove most useful. Theorem 1 above is that the semantics is monotonic: if a formula is true at a partial interpretation (under a variable-assignment) then it remains true at every sharpening of that interpretation (under the same assignment), and if a formula is false at a partial interpretation (under a variable-assignment) then it remains false at every sharpening of that interpretation (under the same assignment). This feature is preserved in what follows, through all (official) additions and modifications to the system. In light of monotonicity, if a formula F is true under s in N, then F is forced at N under s in F. Notice that there is no sentence that is forced at every partial interpretation in every frame. To see this, consider the completely indeterminate partial interpretation O, in which every predicate has both an empty extension and an empty anti-extension. A straightforward induction shows that every formula is indeterminate in O, under every assignment. Let G be a frame in which O is the only partial interpretation. Then no sentence is forced at O in G. Forcing is roughly analogous to super-truth, but it is local to a partial interpretation within a frame. It also shares some features with the broad notion of 11 Burgess and Humberstone [1987] invoke this notion as well, under a different name. The family resemblance with forcing in set theory is more or less unintentional.

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determinacy, which I called ‘‘established’’ (see Ch. 1: x7 and McGee and McLaughlin 1994). If a formula F is forced at a sharpening in a frame, then no matter how things turn out concerning borderline cases (according to the frame), and no matter how other indeterminacies are resolved, there is always a further sharpening in which F is true. If an indeterminate sentence F is forced at a partial interpretation in a frame, then the frame and partial interpretation guarantee that F will become true. Suppose, for example, that F is indeterminate due to borderline cases of a vague predicate, and suppose that F is forced at a partial interpretation N in a frame that represents the possible extensions of some vague predicates. Someone who accepts the truths (and falsehoods) in the partial interpretation N is thus committed to the truth of F. Suppose that a disjunction F _ C is forced at a partial interpretation N in a frame F. This entails that no matter what decisions are made concerning borderline cases and no matter how other indeterminacies turn out, one or the other of F or C will end up true. For example, if an instance of excluded middle F _ :F is forced at a partial interpretation, then no matter how things turn out, come hell or high water, F will eventually get a (standard) truth-value, truth or falsity. A main theme of the rest of this book is that forcing is the right notion, or at least a good notion, for modeling correct reasoning involving vague predicates (and other indeterminacies). In discussing conversations involving vague predicates, Kamp [1981] puts forward the plausible proposal that, in present terms, participants in a conversation are committed to the logical consequences of whatever is explicitly put on the conversational record. The problem, of course, is that we do not know what the consequence relation is. That is one of the goals of the model theory. I suggest that the present notion of forcing is a decent model of logical consequence for this purpose. The sentences forced at a partial interpretation of a frame represent, in some sense, the conversational commitments in a state of a conversation. I can illustrate this with another grumble aimed at the supervaluationist mantra that truth is super-truth.

6. NATURAL DEDUCTION, WEAK FORCING, AND THE CONTINGENT FUTURE: TALE OF A BROKEN PROMISE Theorem 2 is that if the completability requirement holds, then classical logic is sanctioned by the supervaluational system. Even so, the framework does not make sense of ordinary, natural deduction systems, as models of how one correctly reasons with vague expressions.12 The issue concerns how one should 12 Williamson [1994: 151–2, 295–6 n. 10] provides some related criticisms of supervaluationist treatments with respect to natural deduction. Keefe [2000: ch. 7: x4] responds. Burgess and Humberstone [1987] treat natural deduction in a supervaluationist context, but their agenda is

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understand each line of a deduction. What are we thinking as we utter sentences in the course of reasoning? For example, when one says ‘‘assume F’’, what, exactly, are we asked to assume? In ordinary contexts, we assume that F is true. The slogan that truth is super-truth would suggest that in a deduction involving vague terms, the content of the assumption is that F is super-true. This does not make sense of some otherwise valid deductions. Consider an instance of _-elimination. Suppose we have reached a formula of the form F _ C, and we reason as follows: (125) F _ C (126) assume F ... (177) w (depends on (126)) (178) assume C ... (216) w (depends on (178)) (217) w (depends on (125), discharging (126) and (178)) According to the slogan that truth is super-truth, at line (125), we have determined that F _ C is super-true (perhaps depending on some premises). To start a _-elimination, we assume that F is super-true and then deduce that w is super-true from that (plus other assumptions). Then we assume that C is supertrue and deduce that w is super-true. Then we discharge the assumptions, concluding that w is super-true. But when understood this way, the reasoning does not seem warranted. On the interpretation in question, we have that F _ C is super-true (at line (125)). But, as is notorious with the supervaluational approach, F _ C can be super-true without either F being super-true or C being super-true. For example, we have that F _ :F is super-true, although it may be that neither F nor :F is supertrue. In the above reasoning, however, the only cases considered are that F is super-true (at line (126)) and that C is super-true (at (178)). So it seems that the reasoning does not take account of every possible case. Of course, the argument is valid nevertheless, since the relevant natural deduction systems is sound for classical logical consequence, and we have a nice meta-theorem (Theorem 2 above) that classical logic holds for supervaluational validity—provided that the completability requirement holds. But it would be good to make better sense of the actual reasoning we employ, or ought to employ, with vague predicates. That is what natural deduction is supposed to do. I propose to interpret the above deduction differently, in light of the account of vagueness developed in Ch. 1. The content of line (125) is not that F _ C is super-true (or e-determinately true), but rather that we have sharpened things somewhat different from ours. They begin with the intuition that excluded middle is not valid, but the law of non-contradiction is, and they mold the system to deliver this result. The present plan is to develop the system to model the open-texture of vague predicates, and then see what logic results.

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sufficiently to guarantee that F _ C. In other words, the sentence F _ C is on the conversational record, or is implied by something that is: F _ C is forced. This guarantees that as we further sharpen things, we will either make F true or make C true. The content of the assumption at line (126) is that we have gone on to sharpen things to make F true. We then conclude (at (177)) that w is true, on the basis of that assumption. At line (178), we assume instead that we have sharpened things to make C true, and we go on to show that w is true on the basis of that assumption. In light of line (125), we conclude that w holds in either case. Unlike the above interpretation, I suggest that this reconstruction does take all cases into account, and it captures the reasoning involved in a typical proofby-cases. But the reconstruction depends on driving a wedge between truth and super-truth. I propose that when we reason with vague predicates, we deal with sentences that have been stipulated as true in the course of a conversation, as in Ch. 1. Such sentences may not be e-determinately true (and thus not super-true). To illustrate this point further, and take another shot at the mantra that truth is super-truth, I submit that the forcing of a material conditional in appropriate frames makes for good models of conditional relations between vague statements and future contingents. Consider the following scenario: A father makes some promises to his children: ‘‘If we have nice weather on Sunday, we will go to a ball game; if we don’t have nice weather, we will go to a movie’’. The kids are delighted, and they reason that they will have a good time on Sunday, since they enjoy ball games and movies. As luck would have it, the weather on Sunday is borderline between nice and not nice; and the family does not do anything fun at all. The kids feel cheated, but the father insists that he told the truth, and did not break his promise. The antecedents of the conditionals (or promises) were not met. He says that if the weather had been (determinately) nice, they would indeed have gone to the ball game, and if it had been (determinately) not nice, they would indeed have gone to the movie. But, the father tells them, it is not super-true that the weather is nice, nor is it super-true that the weather is not nice. Thus, the father is not committed to do anything with them—or so he argues. I presume that the reader’s sympathies are with the children here. Mine are. Their grievance against the father is sanctioned by the present open-texture account of vagueness. On that view, once again, if the weather on Sunday is borderline nice, then the rules of language use, plus the non-linguistic facts on the ground, do not require them to hold that the weather is nice, and the said rules and facts do not require them to hold the weather is not nice. However, they are permitted either of those choices. And the father’s promise put the family into a conversational situation in which a choice is required. Given his promises, he simply does not have the option of leaving it undecided. There is a loose analogy with legal reasoning, under precedent. Suppose that in a difficult case, a judge must make a ruling as to whether a defendant is liable.

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A hard-nosed realist might think of the judge as attempting to discover the truth in this case, but it is more natural to say that in at least some cases, there is no (prior) fact of the matter to be determined, and the judges decision (defeasibly) constitutes such a fact, and establishes a precedent. Suppose that either ruling is consistent with previous precedents and the facts of the case. Nevertheless, the rules of the game do not give the judge the option to simply demur. She cannot simply declare that there is no fact of the matter. She must decide. Similarly, in light of the promises, the family (or the father) must decide whether the weather is nice. One decision will put ‘‘the weather is nice’’ on the record, and thus obligate them to go to the ball game. The other decision will put ‘‘the weather is not nice’’ on the record, and obligate them to go to the movie. One difference between the domestic and the legal scenarios, of course, is that the family’s decision about the weather need not set a precedent. They can decide the other way in the future, even under similar weather conditions. And, as above, items on the conversational record can be removed later. They can change their mind on the way to the ball game or movie, even if the weather conditions do not change on the way. The supervaluationist slogan that truth is super-truth gives conflicting results to the scenario, depending on how one approaches the situation. The overall philosophical view, and the slogan that truth is super-truth, might favor the father’s interpretation of his promises. The first promise was that if the weather is nice, then they will go to a ball game. The truth scheme entails that for any sentence ‘‘F is true’’, and F itself are equivalent. So the first promise is that if it is true that the weather is nice, then they will go to the ball game. According to the slogan, this comes to: if it is super-true that the weather is nice, then they will go to a ball game. And this, of course, is exactly how the father interpreted his promise. Since the antecedent of the conditional is not (super-)true, he is not obligated to the conclusion. Ditto for the other promise. One might think that if the supervaluationist slogan is correct, it should hold not only for sentences that are asserted by themselves, but also for sentences appearing in more complex sentences, such as the antecedents of conditionals. To be sure, advocates of supervaluationist approaches are aware of the distinction between sentences asserted standing alone and sentences that occur as parts of complex sentences. As noted above, a disjunction can be (super-)true without either disjunct being (super-)true. So the content of a sentence in the form F _ C is not that either F is super-true or C is super-true. So we were too quick to identify the antecedent of the promised conditionals with its super-truth. Keefe responds to an argument, due to Mark Sainsbury [1988: 40], in the same form as that of the children, which supposedly has a counterintuitive conclusion. The first premise of Sainsbury’s argument is an instance of excluded middle: either person a is an adult or a is not an adult. Second, if a is an adult, then watching a hard-core pornographic movie will do him no harm. Presumably, this

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is because he is mature enough to watch the film. Third, if a is not an adult, then watching a hard-core pornographic movie will do him no harm. This is because he is too young to know what is going on. So interpreted, these two premises thus sound plausible. With classical logic, the first premise is a logical truth. It follows from the second and third premises that watching the movie will do a no harm. Sainsbury suggests, however, that the conclusion is false in the case in which a is an adolescent, on the borderline of adulthood. Such a person might very well be harmed by the movie. Keefe [2000: 175] responds that to understand the argument, we have to take the premises themselves as super-true: this example shows instead that we must be careful about the acceptance of certain premises, in this case the second and third ones. Even if a is G whenever a is a clear, positive case of F, we must ask whether it is also true that ‘‘if Fa then Ga’’ is true however F is made precise. To assess this we need to ask of a borderline F whether counting it as F would mean that it would count as G as well. In Sainsbury’s problem case, the answer should be ‘‘no’’. (‘‘If a is definitely [an adult] then a is definitely [unharmed by the movie]’’ is true, but this must not be confused with Sainsbury’s premise.) It is not the form of reasoning that would lead us astray here but the mistaken acceptance of two of the premises.

We saw above that classical reasoning is valid in the supervaluationist framework, and the children in the present scenario did reason classically. Keefe’s reasoning shows where the father went wrong. If the family does not do anything fun, then the father failed to keep his promises—both of them. This is because we are to interpret him as promising that the two conditionals will be super-true. It also follows from Keefe’s analysis that making the two promises puts the father in a peculiar bind. He wakes up on Sunday and sees that there is an acceptable sharpening of the situation in which the weather is nice. So, on Keefe’s reading, he realizes that he must go to the ball game in order to keep the first promise—in order for the conditional to be super-true. But the father also sees that there is an acceptable sharpening in which the weather is not nice, and so he realizes that he must go to the movie in order to keep his second promise. If we assume that they cannot do both—perhaps because the movie and ball game occur at the same time—then he is in a bind. He must fail to keep at least one of his promises, no matter what the family does. We get the same result from the supervaluationist model theory. Let’s render the relevant promises thus: Ca ! B :Ca ! F, where a represents Sunday, Cx says that the weather on x is nice, B says that the family goes to a ball game, and F says that they go a movie. We put aside, for now, the (quite plausible) possibility that the father did not use a material conditional (although we will briefly return to the family in notes 2 and 4 of the

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next chapter). What matters is that proof by cases be valid. The father claims that both conditionals are, and remain (super-)true in the borderline situation. The kids then validly conclude B _ F (from Ca _ :Ca). We can assume that going to a ball game and going to a movie are sharp. Even if there are borderline cases of ball games and borderline cases of movies, they do not bear on the present situation. We can just stipulate that there are no such borderline events in the vicinity. Even if there were, the father would not discharge his responsibility by going to a borderline ball game/movie upon finding the weather to be borderline nice. So we can assume that the promises are conditionals that have vague antecedents and sharp consequents. Suppose M is a partial interpretation in which Ca ! B and :Ca ! F are both super-true, and in which neither Ca nor :Ca is super-true. This is the actual situation on Sunday morning. Let M1 be an acceptable sharpening of M in which Ca is true. There must be one, since :Ca is not super-true. By monotonicity, Ca ! B holds in M1. So B is true in M1. But remember that B is sharp, so it must have the same truth-value in M (before any sharpening). So B is already true in M. The family was destined to go to the ball game, no matter how the weather turned out. Similarly, let M2 be a sharpening in which :Ca is true. By monotonicity, :Ca ! F holds in M2, and so F holds in M2. Since F is sharp, it must also hold in M. So the only way for the father’s promises to be super-true (if the weather is indeterminate) is for the family to go to the ball game and go to the movie. The kids end up getting more than they bargained for. The scenario with the father and his children has untoward consequences for other views on vagueness. For an advocate of a many-valued approach to vagueness, the situation is much like that of the supervaluationist. Suppose, for example, that going to a movie and going to a ball game are sharp, and so each has truth-value 1 or 0. Suppose also that the truth-value of ‘‘the weather on Sunday is nice’’ is .6. Then at least one of the conditionals promised by the father is not true (and not even close to true)—unless the family goes to a ball game and goes to a movie. Suppose, for example, that they go to the ball game, and not the movie, since the weather is closer to nice than to not nice. Then, on perhaps the most popular way to handle negation and the material conditional, the second promise, :Ca ! F, has value .6. Not even a near truth. If they go to the movie instead, then the first conditional, Ca ! B, has value .4. According to epistemicism (e.g. Williamson 1994, Sorenson 2001), vague expressions have sharp, but unknowable boundaries. So if the weather on Sunday is borderline nice (i.e. near the sharp boundary), then either it is nice or it is not nice, but neither the father nor the children have any way of knowing which. Thus, the family will not know what they have to do in order to fulfil the father’s promise. For example, if they go to a ball game and the weather is, in fact, not nice, then the father has broken his second promise (given that they do not also go to the movie). And if they go to a movie and the weather is, in fact, nice, then he has broken his first promise. I conclude that according to epistemicism, no

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one should make a promise that has a potentially vague antecedent and a sharp consequent, since if the antecedent falls near the border, she will not know what to do (since she cannot know if the antecedent is true). This sort of situation comes up whenever we have a conditional with a vague antecedent and a sharp consequent (or a consequent whose vagueness is not tied to the vagueness of the antecedent). In fact, this is a very common situation: ‘‘if the weather is threatening, I’ll take my umbrella’’; ‘‘if a student asks too many dumb questions, I’ll fail him’’; ‘‘if the captain wakes up in a bad mood, there will be a sea battle tomorrow’’. I claim that the model-theoretic system introduced here handles such situations correctly. It treats proof by cases along the lines suggested above. I suggest that the father’s (fulfilled) promises are modeled with a frame F ¼ hW,M i in which those two material conditionals are forced at the base interpretation M. This guarantees that no matter how borderline cases of vague predicates are decided, and no matter how (other) future contingents turn out, there will always be a further sharpening in which Ca ! B and :Ca ! F are both true. The family wakes up on Sunday and finds that the weather is borderlinenice. This corresponds to a sharpening N of the base in which Ca is still indeterminate. Now suppose that the father refuses to let the family decide whether the weather is nice—that is, he refuses to further sharpen the predicate C to include a in its extension or in its anti-extension—and that they do not go to the ball game (so :B holds) and do not go to a movie (:F ). Then there is no sharpening of N in which either conditional is true, contrary to the assumption that the conditionals are forced at the base interpretation. Here is an easy exercise. Suppose that the two conditionals Ca ! B and :Ca ! F are both forced at the base of a frame, and assume that :(B&F ) is true (or at least forced) at the base, as seems plausible (since they cannot go to both the ball game and the movie). Then it follows that the instance of excluded middle, Ca _ :Ca, is also forced at the base of the frame. In other words, if the conditionals are forced at the base of the frame—as they should be if the father is to keep his word—then he does not have the option to leave the status of the weather indeterminate. The forcing of the promises entails that the family (or the father) must make a decision about the weather on Sunday (if it is borderline nice), and, of course, that decision will entail that they do something fun. The unrepentant father, or his attorney, might retort that if the family does not do anything fun, at least he did not break his promises, since there is no sharpening in which either conditional has a true antecedent and a false consequent. What would happen, he asks, if he had made only the first promise, Ca ! B, and they did not go to the ball game because the weather was either borderline nice or determinately bad on Sunday? He certainly would not be culpable in that case. His promise was conditional, and the condition was not met. So long as there is a sharpening of the Sunday situation in which the weather is bad, then they are under no obligation to go to the ball game.

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We might concede, perhaps, that the father did not break his promise, in some technical sense. Even so, this is small comfort to the children. If neither Ca ! B nor :Ca ! F is forced at the base interpretation, then he did not live up to his promises, and that is about as bad as actually breaking them in this case. A parent should set a better example of integrity. Still, we can make some space for the father’s take in the formalism. Let F ¼ hW,Mi be a frame and let N [ W. Let s be an assignment and F a formula. Say that F is weakly forced at N under s if there is no sharpening of N in W in which F is false. In other words, F is weakly forced at N under s if F is true under s in every N 0 in W such that N  N 0 and F has a truth-value in N 0 under s. In the scenario, the father might claim that the conditionals that expressed his promises are only weakly forced, and so he is not required to decide the borderline case of the weather on Sunday. I suggest below that forcing (and not weak forcing) is an appropriate notion to model reasoning with vague predicates (if not the logic of promising). However, weak forcing has an important role to play. Stay tuned. Given monotonicity, if F is forced at a partial interpretation N under an assignment in a frame F, then F is weakly forced at N under the same assignment. The following is straightforward: Theorem 4. A sentence F is weakly forced at every partial interpretation in every frame if and only if F is a classical logic truth. Proof: If F is a classical logical truth, then it cannot be false in any interpretation, and so it is weakly forced everywhere. If F is not logically true, then there is an ordinary, completely sharp interpretation M in which F is false. And so F is not weakly forced in the frame whose only interpretation is M. So we maintain at least this connection with classical logic, although it may be only an artifact of the background framework of classical model theory. If we had started instead with an intuitionistic background, weak forcing would not correspond to classical logical truth. With the overall framework in place, we now turn to a more detailed look at the connectives, quantifiers, and natural deduction. There is a lot of detail to be laid out, and much to be learned along the way.

4 Connectives, Quantifiers, Logic Let us briefly review the main technical notions introduced in the previous chapter. A partial interpretation is a pair hd,I i, where d is a non-empty set—the domain of discourse—and I is a function that gives the denotations and extensions of the non-logical terminology. In particular, if c is a constant, then Ic [ d; if f is an n-place function letter, then If is a function from dn to d; and if R is an n-place relation letter, then IR is a pair hIRþ, IRi of sets such that IRþ  dn, IR  dn, and IRþ is disjoint from IR. We say that IR þ is the extension of R in the partial interpretation and IR is the anti-extension of R in the partial interpretation. A three-valued satisfaction relation is defined, following the strong Kleene truth tables. The custom is not to think of the third value, i, as a truth-value on a par with truth (t) and falsehood (f). Rather, think of i as the lack of a (standard) truth-value. Intuitively, a partial interpretation represents the denotations and extensions of the non-logical constants at some point in a conversation in which the terms are deployed in accordance with their meaning, some fixed external contextual matters, and the non-linguistic facts. Let M1 ¼ hd1,I1i and M2 ¼ hd2,I2i be partial interpretations. We say that M1  M2 if d1 ¼ d2; the interpretation functions I1 and I2 agree on each constant and function letter; and for each relation letter R, I1þR  I2þR and I1R  I2R. In other words, if M1  M2 then the two interpretations have the same domain and agree on the constants, function letters, and the ‘‘decided’’ cases of M1. M2 may also extend the extension and anti-extension of each relation. That is, M2 may decide some of the indeterminate cases of M1. If M1  M2, we say that M2 extends, or sharpens M1. A frame F is a structure hW,M i in which W is a collection of partial interpretations, M [ W, and for every partial interpretation N in W, M  N (so that all of the partial interpretations in W have the same domain). The designated partial interpretation M is the base of the frame F. The intuitive idea is that the partial interpretations in a frame represent the range of possible extensions and anti-extensions of vague predicates, consistent with the meaning of the terms, the external contextual factors, and the non-linguistic facts. Let F ¼ hW,M i be a frame and let N [ W. Let s be a variable-assignment and F a formula. We say that F is forced at N under s if for each sharpening N1 in W such that N  N1, there is a sharpening N2 in W such that N1  N2 and F is true

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in N2 under s. Forcing is a rough but local analog of super-truth, representing a broad notion of determinacy. If F is forced at N under s, then F is eventually true. The formula F is weakly forced at N under s if there is no sharpening of N in W in which F is false. In other words, F is weakly forced at N under s if F is true under s in every N 0 in W such that N  N 0 and F has a truth-value in N 0 under s. We now define other useful connectives, quantifiers, and operators. These aid the system in its role as a model of reasoning with vague predicates, following the foregoing philosophical account. 1. CONDITIONALS In the previous chapter, we saw that the usual material conditional is rather awkward for expressing conditional relations among vague predicates. We kept resorting to the meta-language. As noted above, Hans Kamp [1981: 245] suggests that the deployment of vague terms in context requires a ‘‘non-standard account of the conditional’’. With the proper conditional, we do not evaluate antecedent and consequent in the context of utterance, and then determine the truth-value of the conditional from those evaluations. Rather, we are to evaluate the consequent in contexts that result when the antecedent is accepted as true (if there are any). As Kamp puts it, ‘‘A conditional ‘if f then c’ is true in a context c iff, provided the evaluation of f in c is positive the evaluation of c in the context modified by this evaluation is positive too’’ (p. 247, emphasis mine). In the present framework, this suggests a connective defined along the lines of Kripke semantics for intuitionistic languages. We now define several such conditionals, and, after comparing them, add one to the object language. Let F ¼ hW,Mi be a frame, let N [ W, and let s be a variable-assignment. Say that (F ) C) holds at N under s in F if for any N 0 in W such that N  N 0 , if F is true in N 0 under s, then C is true at N 0 under s. The idea is that (F ) C) holds just in case C comes out true once we sharpen things (or otherwise resolve indeterminacies) such that F becomes true. Say that (F ) C) holds at N under s in F if for any N 0 in W such that N  N 0 , if F is true in N 0 under s, then C is forced at N 0 under s. This is similar, except that we only require that C is forced (rather than true) once we sharpen things such that F becomes true. Say that (F ) C) holds at N under s in F if for any N 0 in W such that N  N 0 , if F is forced in N 0 under s, then C is forced at N 0 under s. This makes for enough conditionals to keep us occupied.1 Fortunately, the last one is redundant, in light of monotonicity: Theorem 5. Let F ¼ hW,Mi be a frame, N [ W, and s a variable-assignment. Then (F ) C) holds at N under s in F if and only if (F ) C) holds at N under s in F. 1 Burgess and Humberstone [1987: 216] define a connective much like ‘‘ ) ’’. To complete logical space, we might define a fourth conditional: (F ) C) holds at N under s in F if for any N 0

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Connectives, Quantifiers, Logic Proof: (1) First, suppose that (F ) C) holds at N under s in F. Let N 0 be an interpretation in F such that N  N 0 and F is true at N 0 under s. Then F is forced at N 0 under s. So C is forced at N 0 under s. So (F ) C) holds at N under s in F. (2) Now suppose that (F ) C) holds at N under s in F. Let N 0 be a partial interpretation in F such that N  N 0 and F is forced at N 0 under s in F. We have to show that C is forced at N 0 under s in F. Suppose that N0 is an interpretation in the frame F such that N 0  N0. We have to show that there is a sharpening N 00 of N0 in F such that C is true at N 00 under s. Since F is forced at N 0 in F under s, there is a sharpening N1 of N0 in F such that F is true at N1 in F under s. Since (F ) C) holds at N under s in F, we have that C is forced at N1 under s in F. So F has a sharpening N 00 of N1 (and thus of N0) such that C is true at N 00 under s. Theorem 6. Let F ¼ hW,M i be a frame, N [ W, and s a variable-assignment. (1) If (F )C) holds at N under s in F, then (F ) C) holds at N under s in F. (2) If (F !C) is forced at N under s in F, then (F ) C) holds at N under s in F. Proof: (1) is immediate. For (2), let F ¼ hW,Mi be a frame, N [ W, and s a variable-assignment. Assume that (F !C) is forced at N under s in F, and let N 0 [ W be a sharpening of N such that F is true in N 0 under s. We have to show that C is forced at N 0 in F under s. So let N0 [ W be a sharpening of N 0 . We must find a sharpening N 00 of N0 in the frame F such that C is true at N 00 under s. Since (F ! C) is forced at N and N0 is a sharpening of N in the frame F, there is a sharpening N 00 of N0 such that (F !C) is true at N 0 under s. By monotonicity, F is true at N 00 under s. So C is true at N 00 under s.

The converses of (1) and (2) do not hold in general. To see this, consider a frame G which consists of four interpretations: O, M1, M2, M3. The base O is the completely indeterminate partial interpretation, in which every predicate has both an empty extension and an empty anti-extension. In M1 the denotation of a constant a is in the extension of a predicate P, and the denotation of another constant b is neither in the extension nor the anti-extension of P. The interpretation M2 is a sharpening of M1 in which both the denotation of a and the denotation of b are in the extension of P. In M3, the denotation of b is in the anti-extension of P. So neither M1 nor M2 is a sharpening of M3; and M3 is not a sharpening of M1 nor is M3 a sharpening of M2. A diagram illustrates this frame: M2 : Pa; Pb j M1 : Pa M3 : :Pb j O j

in W such that N  N 0 , if F is forced in N 0 under s, then C is true at N 0 under s. However, I do not see much use for this conditional.

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Notice that both Pa and Pb are forced at M1 and M2, but not at O or M3. So Pa )  Pb holds at O in G. But (Pa ! Pb) is not forced at O in G, since the material conditional is not true at M3, and M3 has no proper sharpenings. Also, (Pa ) Pb) does not hold at O, since Pa is true at M1, but Pb is not true at M1. There is no straightforward logical relationship between the forcing of (F !C) and the holding of (F )C) in a partial interpretation in a given frame. In the aforementioned frame G, (Pa ! Pb) is forced at M1, but (Pa ) Pb) does not hold there. And (Pa ) Pb) holds (vacuously) at M3, but (Pa ! Pb) is not forced there. The various conditionals can be used to explicate different sorts of relations among sentences with vague terms.2 Consider, first, the ‘‘penumbral connections’’ introduced in Fine [1975]. Recall our standard sorites series consisting of 2,000 men in various stages of baldness. Man 1, Yul Brynner, has no hair whatsoever; man 2,000, Jerry Garcia, has a full head of hair; and for each i < 2000, man i has only slightly less hair than man i þ 1, arranged in the same way. Suppose someone proposed a sharpening in which man 922 is bald (so #922 is in the extension of B), but man 915 fails to be bald (either by being in the anti-extension of B or in neither the extension nor the anti-extension). This conflicts with the meaning of ‘‘bald’’. It is axiomatic that if a man is bald, then so is any man with less hair (if it is arranged the same way). Suppose that we add a binary predicate Sxy for ‘‘x has less hair than y, arranged in the same way’’. We stipulate that if i < j 2000, then Smimj is true at the base. Then the penumbral connection is that for each i,j < 2000, (Bmj&Smimj) ) Bmi, and (:Bmi&Smimj) ) :Bmj hold at the base. This says exactly what we want to say: if we sharpen to put #j in the extension of B, so that man #j is bald, and if #i has even less hair (in which case h#i,#ji is in the extension of S), then #i is in the extension of B in that sharpening as well. Instead of this, we might require that the weaker, (Bmj&Smimj) ) Bmi, and (:Bmi&Smimj) ) :Bmj 2 Let us briefly return to the family invoked in x5 at the end of the previous chapter. The father promised that they would go to a ball game if the weather was nice on Sunday, and he promised that they would go to a movie if the weather was not nice. O