Vagueness in Context

VAGUENESS IN CONTEXT

This page intentionally left blank

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.

Preface

vii

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

viii

Preface

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

Preface

ix

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

x

Preface

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’’

This page intentionally left blank

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

1

2. Interlude: The Place and Role of Model Theory

45

3. A Start on Model Theory

60

4. Connectives, Quantifiers, Logic

88

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

125

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

165

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

190

APPENDIX: Waismann on Open-Texture and Analyticity

210

References Index

216 221

This page intentionally left blank

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.

2

The Nature of Vagueness

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

The Nature of Vagueness

3

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.

4

The Nature of Vagueness

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,

The Nature of Vagueness

5

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

6

The Nature of Vagueness

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’’.

The Nature of Vagueness

7

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.

8

The Nature of Vagueness

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

The Nature of Vagueness

9

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.’’

10

The Nature of Vagueness

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.

The Nature of Vagueness

11

[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.

12

The Nature of Vagueness

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

The Nature of Vagueness

13

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.

14

The Nature of Vagueness

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

The Nature of Vagueness

15

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

The Nature of Vagueness

16

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.

The Nature of Vagueness

17

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

18

The Nature of Vagueness

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.

The Nature of Vagueness

19

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.

20

The Nature of Vagueness

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

The Nature of Vagueness

21

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’’.

22

The Nature of Vagueness

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.

The Nature of Vagueness

23

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.

24

The Nature of Vagueness

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].

The Nature of Vagueness

25

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.

26

The Nature of Vagueness

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.

The Nature of Vagueness

27

(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

28

The Nature of Vagueness

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’’.

The Nature of Vagueness

29

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).

30

The Nature of Vagueness

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.

The Nature of Vagueness

31

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.

32

The Nature of Vagueness

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

The Nature of Vagueness

33

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’’.

34

The Nature of Vagueness

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’’

The Nature of Vagueness

35

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

36

The Nature of Vagueness

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’’).

The Nature of Vagueness

37

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

38

The Nature of Vagueness

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.

The Nature of Vagueness

39

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.

40

The Nature of Vagueness

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.

The Nature of Vagueness

41

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.

42

The Nature of Vagueness

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.

The Nature of Vagueness

43

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

The Nature of Vagueness

44

‘‘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

46

Place and Role of Model Theory

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

Place and Role of Model Theory

47

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.

48

Place and Role of Model Theory

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.

Place and Role of Model Theory

49

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

50

Place and Role of Model Theory

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

Place and Role of Model Theory

51

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.

52

Place and Role of Model Theory

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

Place and Role of Model Theory

53

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

54

Place and Role of Model Theory

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.

Place and Role of Model Theory

55

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

56

Place and Role of Model Theory

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].

Place and Role of Model Theory

57

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

58

Place and Role of Model Theory

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

Place and Role of Model Theory

59

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.

A Start on Model Theory

61

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

62

A Start on Model Theory

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.

A Start on Model Theory

63

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

A Start on Model Theory

64

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,

A Start on Model Theory

65

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.

66

A Start on Model Theory

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

A Start on Model Theory

67

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).

68

A Start on Model Theory

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.

A Start on Model Theory

69

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,

A Start on Model Theory

70

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’’.

A Start on Model Theory

71

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.

72

A Start on Model Theory

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.

A Start on Model Theory

73

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.

74

A Start on Model Theory

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.

A Start on Model Theory

75

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,

76

A Start on Model Theory

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.

A Start on Model Theory

77

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.

78

A Start on Model Theory

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

A Start on Model Theory

79

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.

80

A Start on Model Theory

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

A Start on Model Theory

81

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.

82

A Start on Model Theory

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.

A Start on Model Theory

83

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

A Start on Model Theory

84

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

A Start on Model Theory

85

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

86

A Start on Model Theory

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.

A Start on Model Theory

87

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

Connectives, Quantifiers, Logic

89

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

90

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.

Connectives, Quantifiers, Logic

91

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. On Sunday, the weather was borderline nice, and they did not do anything fun. The father might maintain that he never intended for the material conditionals Ca ! B and :Ca ! F to be forced at the base, rejecting the interpretation imposed in the previous chapter. What he had in mind instead was that (Ca ) B) and (:Ca ) F ) should hold at the base. He notes, with sinister satisfaction, that this is consistent with the status of the weather remaining unsettled and the family doing nothing. In the present philosophical/logical treatment, the issue comes down to whether the father’s promises allow him (and the family) to remain unsettled (indeterminate) about the weather, or whether the promises ‘‘force’’ him to make a decision about it.

92

Connectives, Quantifiers, Logic

hold at the base. This would require that if we sharpen to put #j in the extension of B, and if #i has less hair, then Bmi is forced at that sharpening. No matter what other choices the conversationalists make, and no matter how other indeterminacies come out, there is always a further sharpening in which #i is bald. To me, this does not seem strong enough for the penumbral connection. When the conversationalists explicitly declare that #i is bald, they thereby implicitly declare that any man with less hair is also bald. This relation is expressed with the stronger conditional. However, not much turns on the distinction between ‘‘ ) ’’ and ‘‘ )  ’’ here. The weaker requirement is close enough. Notice that it is too weak to require only that the material conditionals, (Bmj&Smimj) ! Bmi, and (:Bmi&Smimj) ! :Bmj be weakly forced at the base. This would allow the possibility that there is a sharpening in which, say, #902 is bald, but there is no sharpening of that in which #877 is bald (assuming that this is another borderline case). Conversely, it would be too strong to require that the material conditionals (Bmj&Smimj) ! Bmi, and (:Bmi&Smimj) ! :Bmj be forced at the base. This would entail that no matter what decisions are made, and no matter how the future contingents turn out, there is always a further sharpening in which the material conditionals are true. This requires that, for each such conditional, there is always a further sharpening in which the antecedent is false or the consequent is true. So in any possible scenario, one or the other of these must eventually get a truth-value. However, as we saw, there are situations in which the baldness-state of a given man in the series cannot get a truth-value without causing a jump, which amounts to an unsharpening. Formally, there are sharpenings in the relevant frame which themselves have no sharpening in which a given man is in either the extension or the anti-extension of B. Such is tolerance (of which more momentarily). Thus, I submit that something like our intuitionistic conditionals are needed to express the penumbral connection. Let us briefly consider another sort of penumbral connection. Suppose we have a series of patches running from clearly red to clearly orange. In the base of the corresponding frame, the items in the middle are not in the extension of ‘‘red’’ (R) nor in the extension of ‘‘orange’’ (O). In various sharpenings of the base, such patches may fall either in the extension of R or the extension of O. However, it would offend against the meaning of the words (or other necessary truths) if the same patch were both red and orange. So we require that if m is the name of one of the patches, both (Rm ) :Om) and (Om ) :Rm) hold at the base. The present resources give us another way to formulate the tolerance of vague predicates, and the system sheds light on what is intuitively correct about the crucial inductive premise(s) of sorites arguments, and what is incorrect about

Connectives, Quantifiers, Logic

93

them. Let us return once more to our series of 2,000 men. In the model-theoretic framework the principle of tolerance is (or entails): Suppose M satisfies a formula representing a statement that two objects a, a 0 in d differ only marginally with respect to P. Then if Pa is true in M, then it is not the case that :Pa 0 is true in M; and if :Pa is true in M, then it is not the case that Pa 0 is true in M. As usual, we assume that ‘‘bald’’ is tolerant, and that adjacent men in our series differ only marginally in their state of baldness. Let Rxy express the statement that x differs only marginally from y in the amount and arrangement of their hair. So for each i, Rmimiþ1 would be true at the base of the frame. The foregoing account of vagueness, as modeled by the model theory, would not sanction the following, as an articulation (or consequence) of tolerance: Bmi ) Bmiþ1. This would say that if we sharpen to put a man mi in the extension of B, so that man mi is bald in that context, we thereby put miþ1 in the extension of B in the same sharpening. This leads to inconsistency (assuming Bm0 and :Bm2,000). As we noted above, several times, it is consistent with tolerance for a man to be judged bald and for the baldness state of a marginally different man to be left unjudged. Formally, it is consistent with tolerance for mi to be in the extension of B and for miþ1 to fail to be in the extension of B. The foregoing account would also reject the following, weaker conditional as an articulation of tolerance: Bmi )  Bmiþ1. This would say that if we sharpen to put a man #i in the extension of B, then we are committed to put #(i þ 1) in the extension of B. That is, every sharpening of the situation itself has a sharpening in which Bmiþ1 is true. It is an easy exercise to show that this also leads to inconsistency (again, assuming Bm0 and :Bm2,000). But, as we saw, it is possible that a group competently judge mi to be bald, and then jump once they consider miþ1. In this case, when they declare miþ1 to be not bald, they implicitly retract their judgment concerning mi. With the retraction, the new situation is not a sharpening of the previous one. Again, jumps amount to unsharpening. The tolerance of baldness is that one cannot judge marginally different pairs differently. In the model theory, this is a requirement that if mi is in the extension of B, then miþ1 is not in the anti-extension of B. What this means is that for each i, the conditional Bmi ! Bmiþ1 is weakly forced at the base of the frame, as is the quantified version Vn(Bmn ! Bmnþ1). This entails that the frame contains no partial interpretation in which B is sharp (violating Fine’s [1975] completability requirement), which is as it should be in a proper model of the sorites series, when tolerance remains in force. As noted in Ch. 1, one insight, or intuition, that underlies the inductive premise is that if two adjacent items in a sorites series are judged pairwise, or

94

Connectives, Quantifiers, Logic

at the same time, they will be judged alike. This is enough to ensure that there is no acceptable sharpening in which Bmi is true and Bmiþ1 is false. In the model theory, this just is the requirement that the conditional (Bmi ! Bmiþ1) is weakly forced. If the system were bivalent, then the weak forcing of a conditional would entail its truth, and this would violate tolerance. But our system is not bivalent. There is no danger of paradox, however, since modus ponens (or ! -elimination) fails for weak forcing. To see this, consider a frame with three partial interpretations: M, M1, and M2. In the base M, both A and B are indeterminate; in M1, A and B are true; and in M2, A is indeterminate and B is false: M1 : A; B

M2 : :B

j

j

M In this frame, A and (A ! B) are both weakly forced at M (since they are never false in the frame). But B is not weakly forced at the base. Weak forcing is indeed a weak property. At this point, we formally, and officially, add the conditional ‘‘ ) ’’ to the object language. The formation rule is straightforward, and I have already given its truth condition: Let F ¼ hW,M i be a frame, and let N [ W and s be a variable-assignment. Then (F )C) is true 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. Notice that if (F )C) is true at N under s, and if N  N 0 (in the same frame), then (F )C) is true at N 0 under s. So monotonicity is maintained. We are not quite finished with the definition. In the non-bivalent framework, we also have to give a falsity condition for (F )C). One possibility would be to declare that (F )C) is false at N under s if it is not the case that (F )C) is true. In this case, however, monotonicity would be lost. To see this, consider a frame with three partial interpretations: M, M1, M2. In the base M, A is true and B is indeterminate; in M1, B is false; and in M2, B is true. Then (F )C) is not true at the base (in light of M1) and so :(F )C) would be true there (under the proposed falsity condition). But (F )C) is true at M1, and so :(F )C) fails to be true. A natural falsity condition for ‘‘ ) ’’ is its stable failure: Let F ¼ hW,M i be a frame, and let N [ W and s be a variable-assignment. Then (F )C) is false at N under s in F if for any N 0 in W such that N  N 0 , it is not the case that (F )C) is true at N 0 under s. That is, (F )C) is false at N if (F )C) is not true in any sharpening of N in the frame. It is immediate that the system is monotonic: if a sentence F is true in a

Connectives, Quantifiers, Logic

95

partial interpretation in a frame, then F is true in any sharpening of that partial interpretation in that frame. I suppose we could complicate things even more by adding the other conditional ‘‘ ) ’’ to the object language, but the gain in expressive power would be marginal. Notice that F )C is forced at a partial interpretation in a frame if and only if F ) C is forced there (assuming a classical meta-theory).3 Notice that with the addition of the ) connective, it is no longer true that there is no sentence that is forced at every partial interpretation in every frame. It is immediate that any formula in the form F )F is true and thus forced at every partial interpretation in every frame. In addition, we now have the wherewithal to express the usual penumbral connections in the object language, as above. This removes a minor defect of the system so far. Notice, however, that we still have no direct way to express the principle of tolerance in the object language. We get to that in the next section. 2. NEGATION In typical intuitionistic systems, negation is a defined operator: :F iff (F ! ?), where ‘‘?’’ is a sentence that cannot be true (e.g. ‘‘0 ¼ 1’’). We could follow that here, by adding a constant ‘‘?’’ to the object language, with the condition that ? is false in every partial interpretation. Instead, we add an intuitionistic-style negation to the object language, mirroring the truth conditions of the defined operator in Kripke structures. The idea is that a formula F is true if there is no sharpening in which F is true: Let F ¼ hW,M i be a frame, and let N [ W and s be a variable-assignment. Then F is true at N under s in F if for any N 0 in W such that N  N 0 , it is not the case that F is true at N 0 under s. It is straightforward that F is true at N under s in F if and only if F is forced at N under s in F. Clearly, if F is true at N under s, and if N  N 0 (in the same frame), then F is true at N 0 under s. So far, so good: the system remains monotonic. Recall that :F is false in a partial interpretation if and only if F is true in that partial interpretation. So F is true at N if and only if :F is weakly forced at N. Of course, our two negations are quite different conceptually. A sentence in the form Pa is true in a partial interpretation in a frame if the object denoted by a is never in the extension of P in any sharpening of the given partial interpretation in the frame. A sentence in the form :Pa is true in a partial interpretation if the 3 If the conditional ‘‘ )  ’’ were added, it would have a nice property. A formula (F ) C) is forced at a partial interpretation in a frame under an assignment if and only if (F ) C) is true at that interpretation in that frame under that assignment. So the truth of (F ) C) amounts to the forcing of (F )C).

96

Connectives, Quantifiers, Logic

object denoted by a is in the anti-extension of P. The latter is a much stronger statement, amounting to a positive judgment that Pa is false. The former is a weaker statement that Pa is never judged to be true (in any sharpening). Since we are enforcing consistency in the model theory, :Pa entails Pa, but the latter can hold without the former in the non-bivalent framework.4 As with the conditional, we need a falsity condition for ‘‘  ’’, preferably one that preserves monotonicity. We mimic the strategy for ‘‘ ) ’’, and get an interesting result: Let F ¼ hW,M i be a frame, and let N [ W and s be a variable-assignment. Then F is false at N under s in F if for any N 0 in W such that N  N 0 , it is not the case that F is true at N 0 under s. In other words, F is false at N if for each N 0 in W such that N  N 0 , there is a partial interpretation N 00 in F such that N 0  N 00 and F is true at N 00 . That is, F is false at N if and only if F is forced at N. We now have a handy way to express forcing in the object language: Theorem 7. Let N be a partial interpretation in a frame F and let s be a variable-assignment. Then a formula F is forced at N under s in F if and only if :F is true at N under s in F. The following is also straightforward: Theorem 8. A formula :F is true in a partial interpretation in a frame under an assignment if and only if F is true at that partial interpretation in that frame under that assignment. So a formula F is forced at a partial interpretation in a frame if and only if F is true there. We can also express the notion of weak forcing with the new connective. Recall that a formula F is weakly forced at partial interpretation N in a frame F (under an assignment), if there is no sharpening of N in F in which F is false (under the assignment). This happens just in case :F is true (or forced). With the new negation, we can now express the principle of tolerance in the object language. As above, let Bx say that x is bald, and let Rxy express the statement that x differs only marginally from y in the amount and arrangement of their hair. The proper statement of tolerance is (T1) 9x9y(Rxy & Bx & :By). 4 Let us make one final visit to the family invoked in x5 at the end of the previous chapter. Recall that Ca says that the weather on Sunday is nice and F says that the family goes to a movie. When the father promised that if the weather is not nice on Sunday, they will go to a movie, perhaps he meant (or committed himself to) Ca ! F (or maybe Ca ) F). In that case, by refusing to declare that the weather is nice, he is automatically committed to go to the movie. The children would like this interpretation.

Connectives, Quantifiers, Logic

97

This formula should be true at the base of any frame that represents a situation in which tolerance is in force. We can reformulate this in the rough form of the inductive premise of a sorites argument: (T2) VxVy(Rxy ) (Bx&:By)). This, too, can be made true at the base of a frame which represents a situation in which tolerance is in force. What we do not have, and do not want, is: VxVy(Rxy ) (Bx ) By)). This says that if x is marginally different from y, and if we sharpen to make Bx true, then we thereby (or are forced to) make By true. Under (T1) or (T2), if Rxy, and if we sharpen to make Bx true, all we can conclude is :By, that By is weakly forced in that context. But this has no untoward consequences.

3 . Q U AN T I F IE R S It is often pointed out that in ordinary supervaluation semantics, an existentially quantified sentence 9x F can be super-true (in a partial interpretation) without any of its instances being super-true (in that interpretation). Consider, for example, the standard supervaluationist treatment of a sorites series for ‘‘tall’’ which embraces completely sharp sharpenings, following Fine’s [1975] completability requirement. As noted in the previous chapter, the sentence There is a height x such that people of height x are tall while people 0.01 inches shorter are not tall. comes out super-true, since it comes out true in every completely sharp sharpening (see Keefe 2000: 183, who agrees that this is an unattractive feature of the system). However, there is no height i such that ‘‘people of height i are tall while people 0.01 inches shorter are not tall’’ is super-true. Burgess and Humberstone [1987: 226] put the complaint well: It is disconcerting to be told that while it is true that something is F, there is nothing of which it is true that that thing is F . . . The usual reaction to this is to say that if we are told that from some n, [F(n)], then we are entitled to ask ‘‘Which n?’’. The reply, ‘‘Oh, for no particular n’’ appears sophistical.

And Williamson [1994: 153]: ‘‘According to supervaluationism . . . ‘Something is F ’ is sometimes true when no answer to the question ‘Which thing is F ?’ is true. In this sense, super-truth is elusive.’’ When it comes to truth in a partial interpretation, the present system has no such problem. A sentence in the form 9x F is true at a partial interpretation under an assignment if and only if some instance of F is satisfied at the partial interpretation under the assignment. But when it comes to forcing, the supervaluationist bug

98

Connectives, Quantifiers, Logic

(or feature) emerges. A sentence in the form 9x F can be forced in a partial interpretation in a frame even though no instance of F is forced in that partial interpretation in that frame. Consider, for example, a frame whose domain of discourse has only two objects, a,b. At the base M, neither a nor b are in the extension of P and neither a nor b are in the anti-extension of P. So (allowing a and b to name themselves) Pa and Pb are both indeterminate at the base. There are two sharpenings M1, M2 in the frame. In M1, a is in the extension of P and b is in the anti-extension. In M2, a is in the anti-extension of P and b is in the extension: M1 : Pa; :Pb

M2 : :Pa; Pb

j

j

M Then 9xPx is forced at the base, since it is true at both M1 and M2. But neither Pa nor Pb is forced at the base. Indeed, neither Pa nor Pb is weakly forced at the base. There are two common responses to the situation from the supervaluationist camp. First, one can admit that it is indeed counterintuitive for an existentially quantified sentence to be true (i.e. super-true) without it having any true instances. But the theorist quickly adds that the supervaluationist framework is to be preferred due to its overwhelming compensating advantages (following Keefe 2000). The other response is to argue that there is no conceptual block to (super-) true existentially quantified sentences without true instances, and properly tutored intuitions will show us that this is how it should be. Analogs of either response are available to a defender (i.e. me) of the present system. The second sort of response is easily made. The statement that 9xF is forced in a partial interpretation in a frame does not require that it have a forced instance. All that the forcing of 9x F requires is that the existentially quantified sentence itself be eventually true (in the frame). To illustrate this with future contingents, suppose that in a boarding school, there is a requirement that each pupil eat a vegetable with dinner, and that they provide a choice of peas, carrots, and spinach in the dining hall. Let Harry be a student at the school. Harry’s situation is modeled with a frame in which, at the base, nothing (yet) is true about which vegetables he eats. There are three ‘‘sharpenings’’ which fulfil the directive, one in which he eats peas, one in which he eats carrots, and one in which he eats spinach (plus some others where he eats two or three vegetables). So it is (literally and figuratively) forced at the base that there is a vegetable x such that Harry eats x. But there is no vegetable that Harry is forced to eat. He is not forced to eat peas, nor is he forced to eat carrots, nor is he forced to eats spinach. I submit that there is nothing counterintuitive about this. But perhaps this does not sit well with the claim that forcing, rather than truth in a partial interpretation, is the main semantic notion behind reasoning with vague predicates. To satisfy as many potential opponents as possible, I propose to add

Connectives, Quantifiers, Logic

99

another quantifier to express the stronger existential condition.5 Let N be a partial interpretation in a frame F, and let s be a variable-assignment. Say that Ex F holds at N under s in F if there is an instance of F that is forced at N in F. Formally, Ex F is true at N under s in F if there is a variable-assignment s 0 that agrees with s at every variable except possibly x such that F is forced at N under s 0 in F. By definition, then, the truth of Ex F at a partial interpretation in a frame requires that it have a forced instance (compare Burgess and Humberstone 1987: x7). The following is straightforward: Theorem 9. A formula Ex F is forced at N under s in F if and only if 9x F is forced at N under s. If 9x F is true at N in F, then Ex F is true at N under s. If Ex F is true at N under s, then 9x F is forced at N under s. The first clause is that the two quantifiers have the same ‘‘forcing conditions’’. They differ only in their truth conditions. Neither converse of the latter two conditionals holds in general. Once again, we need a falsity condition (that preserves monotonicity). We follow the lead of the connectives introduced above: Ex F is false at N under s in F if for every partial interpretation N 0 in F such that N  N 0 it is not the case that Ex F is true at N 0 under s in F. This is equivalent to: Ex F is false at N under s in F if there is no partial interpretation N 0 in F such that N  N 0 and 9x F is true at N 0 under s in F. In other words, Ex F is false at N in F if and only if :9x F is weakly forced at N. There is a similar, but interestingly different situation with the universal quantifier. Let F be a formula that contains a variable x free. It is possible for each instance of F to be forced at a partial interpretation in a frame without VxF being forced there. To see this, let G be a frame with infinitely many partial interpretations, M0, M1, . . . The domain consists of the natural numbers, and let us assume that the language contains the arabic numerals, with their usual denotations. At each Mi, the extension of the predicate P is the set {0, . . . , i} and the anti-extension of P is empty. So if i j then Mi  Mj. Notice that for each i, Pi is forced at the base M0, but there is no partial interpretation in G in which VxPx is true. Thus, VxPx is not forced at any partial interpretation in the frame. For better or worse, the ordinary supervaluational framework does not have this feature, thanks to completability. Let F be a formula with x free, and suppose that each instance of F is super-true in a given partial interpretation M. 5 In supervaluationist systems, the disjunction operator has a similar feature (or bug) to that of the existential quantifier. A sentence in the form (F _ C) can be super-true without either F being super-true or C being super-true. And in the present framework, (F _ C) can be forced at a partial interpretation at a frame without either F or C being forced there. It is straightforward to add a stronger disjunction to the system that does not have this feature (see Burgess and Humberstone 1987).

100

Connectives, Quantifiers, Logic

Let N be an acceptable, completely sharp sharpening of M. Then each instance of F is true in N, and so Vx F is true in N. A fortiori, Vx F is super-true in N. I propose to follow the above lead, and introduce another universal quantifier. This one is potentially weaker than the standard quantifier. Let N be a partial interpretation in a frame F, and let s be a variable-assignment. Say that Ax F holds at N under s in F if every instance of F is forced at N in F. And say that Ax F is false at N if for every partial interpretation N 0 in F such that N  N 0 , there is an instance of F that is not forced at N 0 . Formally, Ax F is true at N under s in F if for every variable-assignment s 0 that agrees with s at every variable except possibly x, F is forced at N under s 0 in F; Ax F is false at N under s in F if for every partial interpretation N 0 in F such that N  N 0 , it is not the case that Ax F is true at N 0 under s in F. It is straightforward that Ax F is true at N if and only if Ax F is forced at N. We have that if Vx F is forced at N then Ax F is true at N. As above, the converse fails. However, we do have that if N is a partial interpretation in a frame F whose domain of discourse is finite and if s is a variable-assignment, then a formula Vx F is forced at N in F under s if and only if Ax F is true at N in F under s. Indeed, suppose that the domain consists of n objects, which are named by a1, . . . , an, and suppose that Ax F(x) is true at N in a frame F. Let N  N 0 in F. Then we have to find a partial interpretation N 00 such that N 0  N 00 and Vx F(x) is true at N 00 . Since Ax F(x) is true at N, F(a1) is forced at N. So there is a partial interpretation N1 such that N 0  N1 and F(a1) is true at N1. We also have that F(a2) is forced at N. So there is a partial interpretation N2 such that N1  N2 and F(a2) is true at N2. By monotonicity, F(a1) is also true at N2. Continuing, there is a partial interpretation N3 such that N2  N3 and F(a3), as well as F(a1) and F(a2), are true at N3. Eventually, we reach a partial interpretation N 00 ¼ Nn where all of F(a1), . . . , F(an) are true. By construction, N 0  N 00 , and Vx F(x) is true at N 00 . There is no analog of this for ‘‘E’’ and ‘‘9’’. Recall that the example we used to distinguish the forcing of 9x F from the truth of Ex F had only two elements in its domain of discourse. Notice also that the truth and falsity conditions for ‘‘)’’, ‘‘’’, ‘‘A’’, and ‘‘E’’ agree with those for ‘‘ ! ’’, ‘‘:’’, ‘‘V’’, and ‘‘9’’, respectively, in completely sharp partial interpretations, since those are bivalent and have no proper sharpenings. 4. PREVIEW OF COMING ATTRACTIONS: A SECOND AT TEM PT AT DET ERMI N AC Y Depending on how things go with (what passes for) higher-order vagueness, a determinacy operator in the object language might represent a limit of the value of the present system as a model of vagueness. To use the terminology of

Connectives, Quantifiers, Logic

101

Ch. 2: x1, we may encounter a glaring artifact of the system. We turn to this in detail in the next chapter. Here we introduce and explore a crude determinacy operator, perhaps as a first approximation. Recall that in Ch. 1: x7 above, we defined a sentence to be (e-)determinately true if our thoughts and practices in using the language have established truth conditions for the sentence, and externally determined contextual factors (like the comparison class) and the non-linguistic facts have determined that these conditions are met. The base of a frame is supposed to represent the relevant thoughts, practices, external contextual factors, and non-linguistic facts (among other things, perhaps), and a formula is forced at the base if the frame itself guarantees that its truth is inevitable: every sharpening itself has a sharpening in which the formula is true. This suggests truth and falsity conditions for the new operator: Let F ¼ hW,Mi be a frame and N [ W. Let F be a formula and s a variableassignment. Say that DET(F) is true at N under s in F if F is forced at the base M under s in F. And DET(F) is false at N under s if it is not the case that F is forced at the base M under s in F. Notice that for any formula F, DET(F) is fully bivalent in the sense that DET(F) is either true or false (and so never indeterminate) at any partial interpretation in any frame under any variable-assignment. Theorem 10. (i) DET(F) _ :DET(F) is true, and thus forced, at every partial interpretation in every frame. (ii) (DET(F) ! F) and (DET(F) ) F) are both forced in every partial interpretation in every frame. Proof: (i) is immediate (assuming a classical meta-theory). For (ii), if the antecedent DET(F) is true at a (sharpening of a) partial interpretation N in a frame F, then F is forced at the base of F and thus at N (and any sharpening of N). It follows that both conditionals are also forced at N. I presume that item (ii), at least, is a welcome result. Let P be a predicate and a an object in the field of P. A common way to express the fact that a is a borderline case is to say that it is not the case that definitely Pa and is it not the case that definitely not-Pa: ( )

:DET(Pa) & :DET(:Pa).

This is true in a partial interpretation of a given frame if and only if the denotation of a is neither in the extension nor the anti-extension of P at the base of the frame. This, of course, is what is wanted, at least on the current conception of ‘‘borderline’’. Suppose that an atomic sentence Pa is indeterminate at the base of a frame F but is true at a partial interpretation N in F. Then both Pa and :DET(Pa) are true at N. In other words, Pa is true at N, but is not determinately true there.

102

Connectives, Quantifiers, Logic

The DET operator is evaluated at the base, and the sentence Pa is indeterminate (i) there, and remains indeterminate throughout the frame. This is in line with the present contention that truth is not the same as e-determinate truth (see Ch. 1 x7). Roughly, a sentence involving a borderline case of a vague predicate is true in a conversation if it is competently judged to be true (in that conversation). Such a sentence need not be determinately true. Notice that DET(F) has the same truth-value in every partial interpretation in each frame (if the variable-assignment is held fixed). So monotonicity is maintained, vacuously. We thus have: Theorem 11. DET(F) ! DET(DET(F)) and :DET(F) ! DET(:DET(F)) are true at every partial interpretation in every frame. This may not be a welcome result. In x1 of the previous chapter, I argued that a weak negation introduces unwanted precision into the object language. In the present framework, the DET operator brings some of that same precision, and it may be unwanted here as well.6 If so, it is due to a shortcoming, or an artifact of the present framework. In particular, the system does not deal with so-called ‘‘higher-order’’ vagueness. Once again, the base of a frame represents (or models) what is fixed by truth conditions, external contextual factors, and (non-linguistic) facts. The various partial interpretations in a frame represent various ways that competent speakers can go, consistent with the meaning of the terms and the non-linguistic facts (and how other indeterminacies turn out). However, each partial interpretation, including the base, is a set-theoretic construction. As such, the extension (and the anti-extension) of each predicate is a (sharp) set. The system, as developed thus far, does not allow any indeterminacy between what is determinately true and what fails to be determinately true. Let P be a predicate that corresponds to a vague predicate in English. As noted all along, a frame can model the lack of a sharp border between the determinate Ps and the determinate non-Ps, but it leaves us with two sharp borders, one between the determinate Ps and the indeterminate cases, and one between the indeterminate cases and the determinate non-Ps. Intuitively, one would like to say that a given object a is a second-order borderline case of a predicate P if it is a borderline case of the predicate ‘‘borderline P ’’. That is, a is a second-order borderline case of P if it is not determinately the case that Pa is determinately true and it is not determinately the case that Pa is not determinately true: ( )

:DET(DET(Pa)) & :DET(:DET(Pa)).

6 One cannot define a weak, external negation in terms of the present determinacy operator. A formula in the form :DET(F) is not, in any sense, a negation of F. Indeed, :DET(F) is compatible with the truth of F outside the base.

Connectives, Quantifiers, Logic

103

Notice that ( ) is obtained from the above ( ) by substituting DET(Pa) for Pa. However, Theorem 12. Let F be a frame, N a partial interpretation in F and s a variableassignment. Then if F is any formula, [:DET(DET(F))&:DET(:DET(F))] is false at N under s in F. Proof: As noted above, DET(F) is either true or false at N under s in F. From Theorem 11, DET(F) ! DET(DET(F)) and :DET(F) ! DET(:DET(F)) are both true at N. So if DET(F) is true at N, then so is DET(DET(F), and if DET(F) is false at N, then :DET(F) is true at N and so is DET(:DET(F)). Either way, [:DET(DET(F))&:DET(:DET(F))] is false at N. If ( ) does express the borderline-borderline status of Pa, then Theorem 12 is that there is no higher-order vagueness registered in the system.7 This may be an artifact of the model theory. With real natural languages such as English, there may well be vagueness concerning what ‘‘the thoughts and practices in using the language have established’’ (if not concerning what the non-linguistic facts are). Consider our standard sorites series, consisting of 2,000 men in various stages of baldness. The first few are determinately bald and the last few are determinately not bald. Somewhere in the middle, there are indeterminate cases, where competent speakers can go either way. But is there a last e-determinately bald man and thus a first borderline bald man in the series? Is there a number n, such that man n is e-determinately bald, so that competent speakers cannot go either way and must declare #n to be bald, but they can go either way with man n þ 1? This strains intuitions. To repeat some apposite quotes, 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’’ (Raffman 1994: 41 n. 1); ‘‘no sharp distinction may be drawn between cases where it is definitely correct to apply [a vague] predicate and cases of any other sort’’ (Wright 1976: x1). However, a frame designed to model this sorites series will have just such a man (as well as a last borderline bald fellow whose neighbor is e-determinately non-bald). In calling the two sharp boundaries in each partial interpretation, and thus the apparent lack of higher-order vagueness, a (possible) artifact of the system, I acknowledge that it is (or may be) a blemish. It would be a fatal flaw if the model theory were intended as some sort of analysis of vagueness, giving the whole truth about the phenomenon. How can we claim to reveal the truth about the lack of a sharp border between the Ps and the non-Ps by introducing two unnatural sharp borders (see Sainsbury’s [1990: x4] critique of the supervaluation approach)? But the present approach makes no such claim. The system developed here is designed 7 Keefe [2000: 210–11] argues against using a determinateness operator to express higher-order vagueness.

104

Connectives, Quantifiers, Logic

to model what happens with what may be called ‘‘first-order vagueness’’ (see Ch. 2: x1). It illustrates how the gradual transition from clearly or determinately P to clearly or determinately non-P is handled in competent language use. The system includes sharp borders between the determinate and the indeterminate because we are not (yet) interested in that transition. We have to walk before we can run. Of course, as critics of the modeling approach are quick to point out (see Keefe 2000: ch. 2 x3), it will not do to just leave it like this. To say the least, my overall account of vagueness will be seriously compromised if I cannot say anything about the vagueness of determinacy. Indeed, it will be compromised if the account of the vagueness of determinacy does not follow the same philosophical lines as the account of so-called first-order vagueness. The promised account of higher-order vagueness will be provided (or at least started) in the next chapter. But for now, I insist that there is some value in having a model of so-called firstorder vagueness even if it does not model higher-order vagueness very well (or at all). Having identified this potential artifact, we must be careful not to draw any philosophical conclusions about the phenomenon of vagueness that turn on it. In particular, results that turn on the sharpness of the DET operator, especially when it is iterated, need not reflect anything about the phenomenon of vagueness in natural language.

5. TR UT H A ND VALIDIT Y As noted in the previous chapter, the supervaluationist defines validity as the necessary preservation of super-truth, following the slogan that truth is supertruth: 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. In the present framework, sentences forced at the base of a frame correspond to those that are (e-)determinately true: sentences for which our thoughts and practices in using the language have established truth conditions and the external contextual factors and non-linguistic facts have determined that these conditions are met. This is perhaps the closest counterpart to ‘‘super-truth’’. So one might adapt the supervaluationist slogan that truth is super-truth to ‘‘truth is forcing at the base of the relevant frame’’. This would suggest the following definition: Let G be a set of sentences and F a sentence. Then G e F if for every frame F, if every member of G is forced at the base of F, then F is forced at the base of F. Let us call this external validity.

Connectives, Quantifiers, Logic

105

Recall Fine’s [1975] completability requirement that every acceptable partial interpretation has at least one acceptable, completely sharp sharpening. Theorem 2 of the previous chapter is that it follows from the completability requirement that if the DET operator does not occur in any sentence in G or in F, then G S F if and only if F is a classical consequence of G. Something similar would hold here, if we restrict ourselves to frames that satisfy the completability requirement. In x2 of the previous chapter, however, I argue that the completability requirement is not warranted in the present model of vagueness, or at least there is reason to try to get by without it. Since completely sharp sharpenings violate tolerance, they are inconsistent with the normal use of vague predicates. The supervaluationist perhaps agrees (e.g. Keefe 2000: 183 n. 14, 190 n. 18; see Ch. 3 x2 above), but claims that such sharpenings are nevertheless part of the truth conditions for the indicated sentences, and thus part of the definition of validity. Completely sharp sharpenings are merely technical devices used in the definition of truth. So, in a sense, they are an artifact of the supervaluationist system, and do not correspond to anything directly invoked in reasoning with vague predicates. In contrast, the present sharpenings represent extensions and anti-extensions for the vague predicates (and other things like possible outcomes for future contingents) consistent with the meaning of the terms, external contextual factors, and non-linguistic facts. Only these partial interpretations play a direct philosophical role in the justification for the overall logical framework. Since the present system does not have the completability requirement, it does not enjoy the direct and automatic connection to classical logic of Theorem 2. But, as argued in Ch. 2 above, we should not presuppose classical logic anyway. We should first determine how vague predicates function in reasoning, and figure out the logic from that basis. The notion of e-determinacy is a relatively global matter, holding or failing independent of the shifting contexts of various conversations. A more local notion of ‘‘true in a context’’ corresponds to ‘‘forced at a partial interpretation in a frame’’. There is a corresponding internal, or local version of validity:8 Let F be a frame, G a set of sentences, and F a sentence. Then G i F in F if for every partial interpretation N in F, if every member of G is forced at N in F, then F is forced at N in F. We say that G i F simpliciter if for every frame F, G i F in F. The following is immediate: Theorem 13: Let G be any set of sentences and F any sentence. If G i F then G e F. 8 Timothy Williamson [1994: 147–8] defines a version of local validity in the supervaluationist framework, and points out that the notion of super-truth ‘‘plays no role in the definition’’, but he argues that it is not the proper notion of validity for that view, in light of the slogan that truth is super-truth. Rosanna Keefe [2000: 174 n. 10] agrees. Michael Dummett [1975], however, suggests local validity.

106

Connectives, Quantifiers, Logic

For sentences without the DET operator, the converse of Theorem 13 holds as well. First, a lemma: Lemma 14: Let F1 ¼ hW1,M1i and F2 ¼ hW2,M2i be frames, and let N be a partial interpretation in both frames such that {N 0 [ W1 j N  N 0 } ¼ {N 0 [ W2 j N  N 0 }. If a formula C does not contain the DET operator, then for every variable-assignment s, C is forced at N under s in F1 if and only if C is forced at N under s in F2. The proof is straightforward. Notice that with the exception of the DET operator, the various clauses in the definition of truth and forcing make reference only to the sharpenings of the given partial interpretation in the given frame. So if the DET operator does not occur in the formula, then all that matters concerning its forcing at a partial interpretation in a frame are the sharpenings of that partial interpretation in that frame. Theorem 15: Suppose that the DET operator does not occur in any sentence in G or in F. If G e F then G i F. Proof: Suppose that the DET operator does not occur in any sentence in G or in F. Assume that G e F. Let N be a partial interpretation in a frame F ¼ hW,M i, and suppose that every member of G is forced at N in F. Let F 0 ¼ hW 0 ,N i be the frame in which W 0 ¼ {M 0 [ W j N  M 0 }. That is, the base of F 0 is the partial interpretation N and the partial interpretations in F 0 are the partial interpretations of F that are sharpenings of N. By Lemma 14, since every member of G is forced at N in F, then every member of G is forced at the base N of F 0 . Recall that G e F. So F is forced at the base of F 0 . Applying Lemma 14 again, F is forced at N in F. So G i F. The two notions of validity do come apart when the DET operator is involved. If a sentence F is forced at the base of a frame, then DET(F) is true, and so is forced, at every partial interpretation in the frame. So we have that {F} e DET(F). However, it is not the case that {F} i DET(F). A sentence can be forced at a partial interpretation in a frame without being (e-)determinately true in that partial interpretation. One can define external validity in terms of internal validity: Theorem 16. If G is a set of sentences, let DET(G) be {DET(C) j C [ G}. G e F if and only if DET(G) i F. Even if the DET operator is not involved, the local, internal notion of validity makes better sense of reasoning with vague predicates. As in x6 of the previous chapter, one way to see this is to think about the status of deductions, and, in particular, the status of lines in a natural deduction system. When one writes a sentence in a deduction, what is the content of the line? Suppose, for example, that one makes an assumption. What, exactly, is being assumed? When one reaches a conclusion based on some assumptions, what, exactly, is being concluded?

Connectives, Quantifiers, Logic

107

In an ordinary natural deduction system, typical of introductory logic texts, the content of a line C is that C is true. So if one makes an assumption, one is assuming that the sentence on the line is true (or that a sentence in that form is true). So, when it comes to vague predicates, the issue comes down to what notion of truth is to be in play when reasoning. If a deduction is meant to track external validity (e), then it would be natural to hold that the content of a line C is that C is forced at the base of a fixed, but unspecified frame. That is, the content is that the line is that C is (e-)determinately true. This is consonant with the aforementioned intuition, not shared here, that ‘‘there is no apparent way whereby a statement could be true without being definitely so’’ (Wright 1987: x5). As we saw, this intuition supports, or at least suggests external validity. Consider, once again, the rule of _-elimination, sometimes called ‘‘proof by cases’’: If G  F _ C, G,F  w and G,C  w, then G  w. It is tedious but straightforward to verify that if the DET operator is not involved in the premises or conclusion, then the rule is valid for external validity. It is worth going through the reason for this, to see exactly how it turns on features of formulas that lack the DET operator. Suppose that G e F _ C, G,F e w, and G,C e w, and that the DET operator does not occur in any F, C, w, or in any sentence in G. Let F be a frame in which every member of G is forced at the base. Then F _ C is forced at the base of F. We have to show that w is also forced at the base of F. Let N be a partial interpretation in F. We have to show that there is a partial interpretation N 0 such that N  N 0 and w is true at N 0 . Since F _ C is forced at the base of F, there is a partial interpretation N1 in F such that N  N1 and F _ C is true at N1. So either F is true (and thus forced) at N1 or C is true (and thus forced) at N1. Also, by monotonicity, every member of G is forced at N1. Now consider the frame F 0 consisting of the partial interpretations in F that are sharpenings of N1 (whose base is thus N1). Then by Lemma 14, every member of G is forced at the base of F 0 , and so is either F or C. So by hypothesis, w is forced at the base of F 0 . So there is a partial interpretation N 0 in F 0 (and so in F) such that N1  N 0 and w is true at N 0 . We have that N  N 0 . In a natural deduction system, the rule of _-elimination is employed as follows: (i) We have a line F _ C (resting on some assumptions perhaps). (ii) We assume F and deduce w. (iii) We then assume C and deduce w. (iv) Then we discharge the assumptions and assert w outright, on the assumptions behind the disjunction. On the present interpretation, focusing on external validity, this reads as follows: (i) We have established (from some assumptions perhaps) that a sentence F _ C is forced at the base of a frame F. (ii) We assume that F is forced at the

108

Connectives, Quantifiers, Logic

base of the frame F, and deduce that w is forced at the base of F. (iii) We assume that C is forced at the base of F and deduce that w is forced at the base of F. (iv) We discharge the assumptions and assert that w is forced at the base of F, on the assumptions behind the disjunction. But, as we saw in x6 of the previous chapter, the reasoning does not seem valid. It does not follow from the fact that F _ C is forced at the base of F that either F is forced at the base of F or that C is forced at the base of F. So why think that by considering those two cases, we have exhausted the possibilities, and can conclude w on the basis of the disjunction? In a sense, it is a lucky accident that _-elimination is valid for external validity, when the DET operator is not involved. The suspicion concerning _-elimination for external validity is confirmed when we note that the rule is not valid for external validity in general. As noted above, we have {F} e DET(F) and so {F} e DET(F) _ DET(C). Similarly, {C} e DET(F) _ DET(C). But we do not have {F _ C} e DET(F) _ DET(C). Indeed, a disjunction can be forced at the base of a frame even if neither disjunct is forced there. Let us turn to internal validity (i), and repeat our above questions in this context. When one writes a sentence in a deduction, what is the content of the line? Suppose that one makes an assumption. What, exactly, is being assumed? If we have conclusion based on some assumptions, what, exactly, is being concluded? If the deduction is tracking internal validity, the content of a line in a deduction is that the line is forced at a given partial interpretation (not necessarily the base) in a given frame. The content of ‘‘assume C’’ is something like ‘‘assume that we have sharpened in a manner that forces C’’. That is, making assumptions can indicate a further sharpening. Discharging an assumption takes us back to the partial interpretation before the sharpening took place. From this perspective, an _-elimination is interpreted as follows: (i) We have established (from some assumptions perhaps) that a sentence F _ C is forced at a partial interpretation N in a frame F. (ii) We assume that F is forced at a sharpening N 0 of N in the frame F (where perhaps N 6¼ N 0 ), and deduce that w is forced at N 0 . (iii) We then assume that C is forced at a sharpening N 00 of N in the frame F and deduce that w is forced at N 00 . (iv) We discharge the assumptions and assert that w is forced at the original partial interpretation N of F, on the assumptions behind the disjunction. This reasoning is valid, and does not turn on the presence or absence of the DET operator. If a disjunction F _ C is forced at a partial interpretation N in a frame, then every sharpening of N in the frame has a sharpening in the frame in which F _ C is true. In the latter, either F is true (and thus forced) or C is true (and thus forced). So the assumptions that F is forced at a sharpening of N in the frame (at (ii)) and that C is forced at a sharpening of N in the frame (at (iii)) do

Connectives, Quantifiers, Logic

109

exhaust the relevant cases. If w is forced at all such sharpenings in the frame, then w is indeed forced at the given partial interpretation N in the frame (given that the disjunction F _ C is forced there).

6. NATURAL DEDUCTION I propose now to go over the usual inference rules for natural deduction systems, to see which are valid for internal validity. We have a number of versions of the connectives and quantifiers to consider. We have just dealt with the elimination rule for disjunction, and the introduction rule is immediate. To summarize, Theorem 17. ( _ ) (i) If G i F _ C, and G,F i w, and G,C i w, then G i w. (ii) if G i F, then G i F _ C and G i C _ F. We now consider the other connectives and quantifiers: Theorem 18. (&) (i) if G i F&C then G i F and G i C. (ii) if G i F and G i C then G i F&C. Proof: It is immediate that if F&C is forced at a partial interpretation in a frame, then so are F and C. This entails (i). Suppose that F and C are forced at a partial interpretation N in a frame F. Suppose that N 0 is a sharpening of N in F. Then there is a sharpening N1 of N 0 in F in which F is true, and there is a sharpening N 00 of N1 in which C is true. Since both F and C are true at N 00 , (F&C) is true there. So (F&C) is forced at N. This entails (ii). Theorem 19. (:, ) (i) if G i F and G i :F then G i C; and if G i F and G i F then G i C. (ii) if G,F i C and if G,F i C, then G i F. Notice that the usual introduction rule for ‘‘:’’ fails. We have, for example, that Pa&:Pa i Pa and Pa&:Pa i :Pa, but not i:(Pa&:Pa). Indeed, suppose that the object denoted by a is neither in the extension nor the anti-extension of P in any partial interpretation in a given frame F. Then :(Pa&:Pa) is not true in any partial interpretation of F, and so :(Pa&:Pa) is not forced at the base of F. To be honest, I do not know of a natural introduction rule for the ordinary negation ‘‘:’’ for our model theory, much less a rule that makes for a sound and complete deductive system. There is a complication due to the play with antiextensions in the semantics for this negation. To force an atomic formula in the form :C, we have to make sure that the anti-extensions of the embedded relation has certain members in relevant places in the frame. In general, the conclusion of the introduction rule for : would, of course, be a sentence in the form :F, for an arbitrary F. Suppose that a formula C has no occurrences of : or the new connectives, quantifiers, and operator (, ), E, A, DET). Then C is not false in any partial interpretation, under any assignment, in which every anti-extension is

110

Connectives, Quantifiers, Logic

empty. So :C is not true in such a partial interpretation under any assignment. It follows that :C is not forced in any frame unless that frame contains at least one partial interpretation with at least one non-empty anti-extension. So no such sentence is logically true (i.e. a consequence of the empty set). A natural deductive system for our language would thus have to make sure that no sentence of this form can occur in a deduction with no undischarged premises. I do not see how to accomplish this. Perhaps the introduction rule would require different clauses for :F, depending on the logical make-up of the sentence F. Getting back to good news, both versions of the double-negation rule hold: Theorem 20. {::F} i F, and {F} i F. Proof: A sentence (or formula) in the form : :F is true at a partial interpretation N (under an assignment) if and only if :F is false at that partial interpretation (under that assignment) if and only if F is true at that partial interpretation (under that assignment). This establishes the first entailment. Now suppose that F is forced at a partial interpretation N in a frame F. Let N 0 be a sharpening of N in F. If there is no sharpening of N 0 in F in which F is true, then F is true at N 0 (and at every sharpening thereof) in F, which contradicts the forcing of  F at N. So there is a sharpening of N 0 in F in which F is true. Thus F is forced at N (cf. Theorem 8 above). Of course, this reasoning uses classical logic in the meta-theory. For what it is worth (not much), this observation does allow a generic introduction rule, of sorts, for the ordinary negation ‘‘:’’ operator (or anything else, for that matter). If formulas C and C follow from a set G of sentences and :F, then :F follows from G. Yes, I realize that this is a cheat. Moving on, Theorem 21. ( ! , )) (i) If G i F ! C and G i F, then G i C; if G i F ) C and G i F, then G i C. (ii) if G,F i C then G i F ) C. Proof: Suppose that G i F ! C and G i F. Suppose that every member of G is forced at a partial interpretation N in a frame F. We have to show that C is forced at N in F. Let N 0 be a sharpening of N in F. Then there is a sharpening N1 of N 0 in F such that F ! C is true at N1, and there is a sharpening N 00 of N1 such that F is true at N 00 . Since both F and F ! C are true at N 00 , C is true at N 00 . The other clauses are immediate from the definition of ‘‘ ) ’’. Notice that the introduction rule fails for the material conditional. We have {Pa} i Pa, but recall that (Pa ! Pa) is not true in the completely indeterminate partial interpretation O, in which every predicate has both an empty extension and an empty anti-extension. So (Pa ! Pa) is not forced at the base of the frame whose only partial interpretation is O, and thus i (Pa ! Pa) fails. I know of no natural introduction rule for the material conditional. Indeed, a formula in the form F ! C is equivalent to :F _ C, and we have no introduction rule for :, as above. Of course, we can cheat here too, exploiting

Connectives, Quantifiers, Logic

111

double-negation elimination. Let’s get on to the quantifiers: Theorem 22. (V,A) Let F(a) be the result of substituting the constant a for every free occurrence of the variable x in F. (i) If G i Vx F then G i F(a), and if G i Ax F then G i F(a). (ii) If G i F(a) then G i Ax F, provided that a does not occur in any member of G or in F. Proof: (i) is straightforward. For (ii), suppose that G i F(a) and that a does not occur in any member of G or in F. Let N be a partial interpretation in a frame F such that every member of G is forced at N in F. We have to show that every instance of F is forced at N. Let s 0 be a variable-assignment. Let F 0 be a frame that is just like F except that the constant a denotes s 0 (x) in every partial interpretation in F. Let N 0 be the counterpart of N in F 0 . Then every member of G is forced at N 0 in F 0 (recalling that a does not occur in any member of G). So F(a) is forced at N 0 . Since a does not occur in F, this entails that F is forced at N under s 0 in F. Thus Ax F is forced at N in F. The introduction rule for the ordinary universal quantifier (V) fails. Indeed, we have (by (i)) that {AxPx} i Pa. Thus, if the introduction rule for V held, we would have {AxPx} i VxPx. But in the previous section, we saw an example of a frame that forces AxPx at its base, but fails to force VxPx. Here, again, I know of no natural introduction rule. We will provide a diagnosis of this difference between our quantifiers in x8 below. In light of the last clause of Theorem 9 above, our two existential quantifiers (9,E) behave the same in the semantics (since forcing is the main component of validity). It is straightforward that the usual introduction and elimination rules hold for both quantifiers: Theorem 23. (9,E) Let F(a) be the result of substituting the constant a for every free occurrence of the variable x in F. (i) If G i Ex F, and G,F(a) i C, then G i C, provided that a does not occur in F, C, or any member of G. And if G i 9x F, and G,F(a) i C, then G i C, provided that a does not occur in F, C, or any member of G. (ii) If G i F(a) then G i 9x F, and G i Ex F. We turn, finally, to the DET operator. Define a formula to purely determinate if it is built up from formulas in the form DET(F) using the formation rules for the other connectives and quantifiers. More formally, for any formula F, DET(F) is purely determinate; and if C and w are purely determinate, then so are :C,  C, C&w, C _ w, C ! w, C ) w, VxC, AxC, 9xC, and ExC. In x4 above, we noted that a formula in the form DET(F) is fully bivalent in that for any frame F, any partial interpretation N in F and any variableassignment s, DET(F) is either true or false at N in F under s. A straightforward induction shows that the same goes for any purely determinate formula. Consequently, if F is purely determinate, then it has the same truth-value at

112

Connectives, Quantifiers, Logic

every partial interpretation N in a frame F that it has at the base of F. The following is thus straightforward. Theorem 24. (DET) (i) If G i DET(F) then G i F. (ii) If G i F, and if every member of G is purely determinate, then G i DET(F). These are the introduction and elimination rules for the modal operator for S5 (see Corcoran and Weaver 1969).9 We should not make too much of this, since the key features of the DET operator may be due to artifacts of the system. In particular, the bivalence of purely determinate formulas follows from the fact that each formula has a sharp extension and a sharp anti-extension at each partial interpretation, and this turns on the fact that our system does not model higher-order vagueness. It follows from the foregoing results that the Barcan formula and its converse both hold for the new universal quantifier: {AxDET(F)} i DET(Ax F)

and

{DET(Ax F)} i AxDET(F)

Consider the first of these. The premise AxDET(F) is that for every object a in the domain of the frame, it is forced at the base that F holds of a. Given the definition of the quantifier, it follows that Ax F is true, and thus forced, at the base. We also have DET(Vx F) i VxDET(F). However, the converse, the Barcan formula for the ordinary universal quantifier, fails. In x3 above, we defined a frame in which, for each a in the domain, Pa is forced at the base, but that VxPx is not true at any partial interpretation of the frame. In that frame, VxDET(Px) is true, and is thus forced at the base, but DET(VxPx) is false at the base and thus at every partial interpretation in the frame.10 Concerning the existential quantifier, we have both converses to the Barcan formula: 9xDET(F) i DET(9x F)

and

ExDET(F) i DET(Ex F).

However, neither Barcan formula holds. In x3, we defined a frame whose domain consisted of only two objects, a, b. We saw that 9xPx is forced at the base of this frame (since it is true at both proper sharpenings of the base in the frame). But neither Pa nor Pb is forced at the base. It follows that both DET(9xPx) and DET(ExPx) are forced at the base, but neither 9xDET(Px) nor ExDET(Px) are forced at the base. We do have the Barcan formula in one sense, however. By definition, if Ex F is true (and not just forced) at the base, then ExDET(F) holds at the base. 9 The connection between determinateness and S5 is noted in Williamson [1994: 149], in the context of supervaluationism. 10 In modal logic, typical situations in which the converse to this Barcan formula fails involve variable domains. That is, the possible worlds accessible from a given world may not have the same objects. By stipulation, this does not happen here. The inference fails for other reasons. Something analogous holds for the existential quantifier.

Connectives, Quantifiers, Logic

113

7. CLASSICAL VALIDITY AND CLASSICAL LOGIC If F is a formula in the present language, then let F 0 be the result of replacing each occurrence of ‘‘ ) ’’ with ‘‘!’’, replacing each occurrence of ‘‘’’ with ‘‘:’’, replacing each occurrence of ‘‘A’’ with ‘‘V’’, and replacing each occurrence of ‘‘E’’ with ‘‘9’’. If G is a set of formulas, then let G 0 be {C 0 j C [ G}. Theorem 25. Let G be a set of sentences and F a sentence in our language, none of which contain the DET operator. If G i F, then F 0 is a classical consequence of G 0 . Proof: Suppose that F 0 is not a classical consequence of G 0 . Then there is an interpretation that satisfies every member of G 0 but does not satisfy F 0 . Let M be the corresponding completely sharp partial interpretation, and let FM be the frame whose only partial interpretation is M. Then a formula C is forced at M in FM under an assignment s if and only if C is satisfied at M under s. Finally, note that the truth and falsity conditions for ‘‘)’’, ‘‘’’, ‘‘A’’, and ‘‘E’’ agree with those for ‘‘!’’, ‘‘:’’, ‘‘V’’, and ‘‘9’’, respectively, in completely sharp partial interpretations, since those are bivalent and have no proper sharpenings. So every member of G is forced at the base of FM, but F is not. As noted above (several times), the converse of Theorem 25 fails. For example, there are frames in which Pa _ :Pa and even Pa ! Pa are not forced. There is, however, a partial converse to Theorem 25. The various theorems of the previous section indicate that the standard intuitionistic elimination rules are sound for all of the connectives, and the introduction rules are sound for everything but ‘‘!’’, ‘‘:’’, and ‘‘V’’. Moreover, the rule of double negation elimination is sound for both negations (Theorem 21). Thus, Theorem 26. Let G be a set of sentences and F a sentence in our language, none of which contain ‘‘DET’’, ‘‘!’’, ‘‘:’’, or ‘‘V’’. If F 0 is a classical consequence of G 0 , then G i F. Proof: If F 0 is a classical consequence of G 0 , then by the completeness of firstorder logic, F 0 can be deduced from members of G 0 in a standard classical natural deduction system. But as we just saw, the corresponding rules are sound for internal validity (i) for the indicated connectives and quantifiers. So we see that our semantics does sanction classical logic after all, if we restrict ourselves to certain connectives and quantifiers. The introduction and elimination rules for classical logic are sound for those connectives, even for formulas that contain the DET operator (which, in turn, is governed by the rules for S5). Perhaps these are not unreasonable connectives to use when reasoning with vague predicates. The main drawback of this language is that it does not allow one to express the proper, strong negation for vague predicates (:). Indeed, neither anti-extensions nor falsehood play a direct role in the semantics of the indicated operators.

114

Connectives, Quantifiers, Logic

There is a straightforward link between the present model theory and standard supervaluationist treatments, which shows the latter to be a restriction of the former. In the previous chapter, we saw the supervaluationist’s claim that classical logic is preserved turns on the completability requirement and the slogan that truth is super-truth (see Theorem 2). That carries over. Define a frame F to be closed if for each partial interpretation N in F, there is a completely sharp N 0 in F such that N  N 0 . Closed frames are those that respect Fine’s [1975] completability requirement. To put it metaphorically, in a closed frame, every indeterminacy is eventually resolved. I argued above, early and often, that closed frames do not respect the ordinary use of some vague terms. Nevertheless, we can characterize the supervaluationist’s notion of validity in the present system, using our local, internal notion of validity: Let G be a set of sentences and F a sentence. Say that G s F if for each closed frame F, G i F in F. That is, G s F if for every partial interpretation N in every closed frame F, if every member of G is forced at N in F, then F is forced at N in F. Recall that the examples we used to show that ‘‘)’’, ‘‘’’, and ‘‘A’’ differ from their respective counterparts ‘‘!’’, ‘‘:’’, and ‘‘V’’ consisted of non-closed frames. In each case, there is at least one partial interpretation that has no completely sharp sharpening in the frame. This turns out to be essential: Lemma 27. {F ) C} s F ! C; {F} s :F; {Ax F} s Vx F. Proof: Assume that F ) C is forced at a partial interpretation N in a closed frame F. Let N 0 be a sharpening of N in F. Let N 00 be a completely sharp sharpening of N 0 in F. Then F ) C is forced at N 0 . We have that either F is true at N 00 or F is false at N 00 . If F is true at N 00 , then C is true at N 00 (since F ) C is forced and N 00 is the only sharpening of itself in the frame). If F is false at N 00 , then F ! C is true at N 00 . So either way, F ! C is true at N 00 . Thus, F ! C is forced at N 0 . The other cases are similar. The converses are immediate. If we restrict ourselves to closed frames, there is no logical difference between the corresponding connectives. Theorem 25, Theorem 26, and Lemma 27 together give us a direct counterpart to Theorem 2 of the previous chapter: Theorem 28. Let G be a set of sentences and F a sentence in our language, none of which contain the DET operator. G s F if and only if F 0 is a classical consequence of G 0 . We now provide a criterion for a frame to be closed: Theorem 29. Let F ¼ hW, M i be a frame. Then F is closed if and only if for each n-place atomic predicate P, the sentence Vx1 . . . Vxn(Px1 . . . xn _ :Px1 . . . xn) is forced at the base M of F.

Connectives, Quantifiers, Logic

115

Proof: The left to right direction is a consequence of Theorem 28. For the converse, suppose that Vx1 . . . Vxn(Px1 . . . xn _ :Px1 . . . xn) is not forced at the base of a frame F. Then there is a partial interpretation N such that Vx1 . . . Vxn(Px1 . . . xn _ :Px1 . . . xn) fails to be true at any sharpening of N in F. Since Vx1 . . . Vxn(Px1 . . . xn _ :Px1 . . . xn) can never be false, it follows that Vx1 . . . Vxn(Px1 . . . xn _ :Px1 . . . xn) is indeterminate (i) in every sharpening of N in F. So there is no completely sharp sharpening of N in F. So F is not closed. The use of an instance of excluded middle in Theorem 29 nicely illustrates what is going on, but just about any classical logical truth would do. For example, a frame F is closed if and only if for each n-place atomic predicate P, the sentence Vx1 . . . Vxn(Px1 . . . xn ! Px1 . . . xn) is forced at the base M of F.

8. KRIPKE STRUCTURES AND INTUITIONISTIC LOGIC Frames were deliberately modeled after Kripke structures for intuitionistic logic. It is worthwhile to explore the similarities and differences. Let me briefly review the theory.11 A Kripke structure T for a formal language is a triple hA, R, H i, where A is a set of interpretations of the language, H [ A, and R is a partial ordering on A. It is required that if K [ A, then RHK (so H is the analog of the base of a frame). Each member K of A is itself an ordinary (non-partial) interpretation of the given language. That is, each K is a pair hd, I i, where d is a non-empty set, the domain of K, and I assigns an appropriate denotation and extension (but not an antiextension) to each non-logical item in the language in the usual way. The following is also required: Let K1 ¼ hd1,I1i and K2 ¼ hd2,I2i be members of A. If RK1K2, then d1  d2 and I2 extends I1, in the sense that if P is an n-place predicate and ha1 . . . ani is in the extension of P in N1, then ha1 . . . ani is in the extension of P in N2. Roughly, the idea is that the members of A represent states of possible knowledge, and RK1K2, indicates that what is knowable at K2 is an extension of what is knowable at K1. That is, anything knowable at K1 is still knowable at K2. Let T ¼ hA, R, Hi be a Kripke structure, let K ¼ hd, I i be a member of A, and let s be an assignment of members of d to the variables. A relation of satisfaction, between formulas, Kripke structures, members of A, and variable assignments is 11 See Kripke [1965] and Dummett [2000: 134–5, 145–50]. Dummett calls Kripke structures, ‘‘Kripke trees’’.

116

Connectives, Quantifiers, Logic

defined recursively, as follows:12 If P is an n-place predicate and t1, . . . , tn are terms, then K satisfies Pt1 . . . tn in T under s if ha1, . . . , ani is in the extension of P in K, where, for each i, ai is the denotation of ti in K. K satisfies F&C in T under s, if K satisfies F in T under s and K satisfies C in T under s. K satisfies F _ C in T under s, if K satisfies F in T under s or K satisfies C in T under s. K satisfies F ! C in T under s, if, for every K 0 in A such that RKK 0 , if K 0 satisfies F in T under s, then K 0 satisfies C in T under s. K satisfies 9x F(x) in T under s, if there is an assignment s 0 that agrees with s on every variable except possibly x such that K satisfies F(a) in T under s 0 . K satisfies Vx F(x) in T if, for every K 0 ¼ hd 0 ,I 0 i in A such that RKK 0 , and every assignment s 0 on K 0 that agrees with s at every variable except possibly x, K 0 satisfies F(a) in T under s 0 . The satisfaction condition for the existential quantifier requires that the embedded formula hold for an instance in the interpretation in question. The condition for the universal quantifier requires that the embedded formula hold in all ‘‘future domains’’, even if more objects are added. It is not enough for the formula to hold for the objects in the given interpretation. Thus both the existential quantifier and the universal quantifier are strong. As noted above, it is customary to define negation as follows: :F if and only if (F ! ?), where ‘‘?’’ is an absurd sentence. Instead, I just give the resulting satisfaction condition: K satisfies :F in T under s, if, for every K 0 in A such that RKK 0 , K 0 does not satisfy F in T under s. Let G be a set of sentences and F a sentence. We say that G K F if for any Kripke structure T and any interpretation K in T, if K satisfies every member of G in T, then K satisfies F in T. It is well known that the Kripke semantics is sound and complete for the intuitionistic predicate calculus: G K F if and only if F can be deduced from members of G in the intuitionistic predicate calculus. One important difference between Kripke structures and the present framework of frames is that in each frame, every partial interpretation has the same domain. Following Dummett [2000: 147], say that a Kripke structure T has a fixed domain if every interpretation in it has the same domain of discourse. We define yet another notion of validity: G Kf F if for any fixed-domain Kripke structure T and any interpretation K in T, if K satisfies every member of G in T, then K satisfies F in T. 12 With an abuse of language, I make no distinction here between ‘‘satisfaction’’ and ‘‘truth’’. I use the former term here to help distinguish the defined notion from that of truth-in-a-partialinterpretation (in a frame) under an assignment, as defined in the previous chapter.

Connectives, Quantifiers, Logic

117

If the variable x does not occur free in a sentence F, then {Vx(F _ C)} Kf (F _ VxC), even though the latter is not an intuitionistic consequence of the former.13 Dummett states that this inference characterizes Kripke structures with fixed domain: G Kf F if and only if F is deducible from members of G in the deductive system obtained from intuitionistic logic by adding the rule: (FX) If G ‘ Vx(F _ C) and x does not occur free in F, then G ‘ F _ VxC. Notice that in the present semantics for vague languages, the truth conditions for the ‘‘new’’ connectives ‘‘)’’, ‘‘’’ are the same as the satisfaction conditions for the intuitionistic ‘‘!’’, ‘‘:’’, respectively, in Kripke structures (see x2 above). This, of course, is no accident. Moreover, the given truth conditions for ‘‘&’’ and ‘‘_’’ in the present semantics are exactly the same as the satisfaction conditions for those connectives in Kripke structures; and the truth conditions for ‘‘9’’ and ‘‘V’’ are equivalent to the corresponding satisfaction conditions in fixed-domain Kripke structures. The main differences concern the truth conditions for sentences in the form F ! C and :F in the present semantics, since those make reference to the falsity conditions for the embedded sentences. In particular, the truth conditions for sentences in the form Pa ! Qb and :Qa make reference to the anti-extensions of P and Q. As in the previous section, if F is a formula in the present language, then let F 0 be the result of replacing each occurrence of ‘‘)’’ with ‘‘!’’, replacing each occurrence of ‘‘’’ with ‘‘:’’, replacing each occurrence of ‘‘A’’ with ‘‘V’’, and replacing each occurrence of ‘‘E’’ with ‘‘9’’. If G is a set of formulas, then let G 0 be {C 0 j C [ G}. Theorems 25 and 26 are that if none of the sentences contain DET, ‘‘!’’, ‘‘:’’, or ‘‘V’’, then Gi F if and only if F 0 is a classical consequence of G 0 . In light of the foregoing connections between frames and (fixed-domain) Kripke structures, one might have expected that the connection would be with something in the neighborhood of intuitionistic logic, augmented with the rule (FX). We can diagnose the situation. The present notion of validity (i) is defined in terms of forcing, while validity for Kripke structures (K or Kf) is based on satisfaction, and there is a crucial disanalogy between those. As might be expected, the key item is double negation elimination for the new connective ‘‘’’, as registered in Theorem 25 above, {F} i F. In words, if F is forced (or true) at a partial interpretation N in a frame, then F is forced at N in that frame.14 We did not (and could not) show that if F is 13 Consider a Kripke structure with two interpretations K , K , with RK K . The domain of K 1 2 1 2 1 is {0} and the domain of K2 is {0,1}; and the extension of a monadic predicate P in both K1 and K2 is {0}. Then Vx(9y:Py _ Px) is satisfied by K1 in this Kripke structure, but 9y:Py _ VxPx is not. 14 The proof of Theorem 25 uses classical logic in the meta-theory. In particular, we assumed that if N is any partial interpretation in a frame, then either there is a sharpening of N in which F is true or there is no such sharpening.

118

Connectives, Quantifiers, Logic

true at N in a frame, then F is true at N in that frame. Indeed, consider a frame that consists of two partial interpretations, M, N, with M  N. At the base M, the object denoted by a is not in the extension of P, but in N, a is in the extension of P: N : Pa j M Then Pa is true (and thus forced) at M, but Pa is indeterminate (i), and thus not true, at M. In the foregoing philosophical and model-theoretic treatment, I suggested that it is appropriate to define validity in terms of forcing, if one is to model correct reasoning with vague predicates. Someone might agree with the bulk of the present philosophical/semantic treatment of vagueness, and accept the overall semantic framework, but hold that the central notion should be that of truth at a sharpening in a frame, instead of forcing. After all, some critics of the supervaluation approach have argued that when it comes to the truth/satisfaction conditions for atomic formulas, at least, what happens in various sharpenings is irrelevant (see e.g. Tye 1989). The fact that some atomic sentence does or does not come out true under various sharpenings does not tell us anything about its truth-value when we use the predicates in the real world and do not (further) sharpen them. One cannot make a similar complaint for formulas containing the new connectives ‘‘’’, ‘‘)’’, since those are explicitly defined in terms of sharpenings. Of course, our critic might wonder if these connectives correspond to anything concerning our actual reasoning with vague predicates, but set that aside. The partial critic might define yet another notion of validity: G ii F if, for any partial interpretation in any frame F, if every member of G is true at N, then F is true at N. The subscript ‘‘ii’’ stands for ‘‘internal, intuitionistic’’, or else the lowercase Roman numeral for the number two. This notion of validity is intuitionistic. If none of the sentences contain DET, ‘‘!’’, ‘‘:’’, or ‘‘V’’, then G ii F if and only if F 0 can be deduced from members of G 0 in the intuitionistic predicate calculus, augmented with the rule of inference (FX). To further tighten the connection between frames and Kripke structures, it is straightforward to import the present notion of forcing into the Kripke framework. Say that a sentence F is Kripke-forced at an interpretation K [ A in a Kripke structure T ¼ hA,R,Hi if, for every K 0 [ A such that RKK 0 , there is an interpretation K00 [ A such that RK 0 K00 and F is satisfied by K 0 . In effect, F is Kripke-forced at K in T if and only if ::F is satisfied at K in T. That is, the Kripke-forcing of a formula amounts to the satisfaction of its double-negation. If we assume classical logic in the meta-theory, there is an analog to Theorem 21: a sentence in the form ::F is Kripke-forced at an interpretation in a given

Connectives, Quantifiers, Logic

119

Kripke structure if and only if F is Kripke-forced at that interpretation in that Kripke structure.15 Say that G Kc F if, for every interpretation K in every Kripke structure T, if every member of G is Kripke-forced at K in T, then F is Kripke-forced at K in T. It follows that G Kc F if and only if F is a classical consequence of G. This is the last notion of validity to be defined in this book. I promise. The connection with Kripke structures and intuitionistic logic lends some perspective to the difference between our two universal quantifiers. Let F be a formula with x free. Recall, from x3 above, that it is possible for each instance of F to be a forced at a partial interpretation in a frame without the universal generalization Vx F being forced there. So we defined a new quantifier thus: Ax F is true at N under s in F if for every variable-assignment s 0 that agrees with s at every variable except possibly x, F is forced at N under s 0 in F. We have that Vx F entails Ax F (i.e. Vx F i Ax F), but not conversely. The forcing of an instance F(a) of F amounts to, or is analogous to, the satisfaction of ::F(a) in a Kripke structure for intuitionistic logic. Since we are considering fixed-domain Kripke structures, the satisfaction of each ::F(a) at a given interpretation amounts to the satisfaction of Vx::F(x). So, in effect, a sentence in the form Ax F corresponds to the intuitionistic Vx::F. In contrast, the forcing of the quantified sentence Vx F amounts to the satisfaction of ::Vx F. So Vx F is stronger than Ax F in the same way that, in intuitionistic logic, ::Vx F is stronger than Vx ::F. It is straightforward to transfer the frame used to motivate the new universal quantifier (in x3) into a Kripke structure that satisfies Vx ::Px, but not ::VxPx. All you have to do is ignore the (empty) anti-extensions.

9. ADDENDUM ON PO TENTIAL INF INITY AND CHOICE SEQUENCES The issues underlying our two quantifiers are in the neighborhood of the traditional Aristotelian distinction between actual and potential infinity. Roughly, the meaning of Vx::Px is that for each x, it is eventually true that Px; while the meaning of ::VxPx is that it is eventually true, all at once so to speak, that for every x, Px. While we are on the subject of intuitionism, let me briefly show how the present system of frames can further model this distinction between potential and actual infinity. I focus on another aspect of intuitionistic mathematics, choice sequences. 15 The restriction to fixed-domain Kripke structures can be relaxed here, since ::(Vx(F _ C)) does entail ::(F _ VxC) in ordinary intuitionistic logic.

120

Connectives, Quantifiers, Logic

Like the classical mathematician, the intuitionist can identify real numbers with Cauchy sequences of rational numbers.16 The classical mathematician, however, regards each sequence as a complete, actual infinity. As L. E. J. Brouwer [1948] put it, the classical mathematician holds that ‘‘from the beginning the nth element is fixed for each n’’. The intuitionist disagrees, insisting that the sequences are only potentially infinite. The entire sequence is never before us, as it were. Instead, we only have the ability to continue it as far as desired: there is no bound on how many elements can be obtained. Early in his career, Brouwer held that (in effect) every Cauchy sequence must be given by a rule: Let us consider the concept: ‘‘real number between 0 and 1’’ . . . For the intuitionist [this concept] means ‘‘law for the construction of an elementary series of digits after the decimal point, built up by means of a finite series of operations’’. (Brouwer 1912: 85)

Later, Brouwer supplemented rule-governed sequences with what are sometimes called ‘‘free-choice sequences’’. He envisioned a ‘‘creative subject’’ with the power to freely produce further members of an evolving sequence. The analogy to be exploited here is between the activity of the creative subject continuing a Cauchy series and the members of a conversation deciding borderline cases of a vague predicate. The analogy holds surprisingly well—up to a point. A creative subject (or a rule) giving better and better approximations to a real number corresponds to conversationalists sharpening a vague predicate again and again. In both cases, the ‘‘work’’ is never finished: the sequence is never completed and the predicate is never completely sharpened (so long as tolerance remains in force). Consider a language that has a monadic predicate B, in addition to the usual non-logical terminology for the rational numbers. Consider a frame F ¼ hW,Mi for this language. The domain of each member of W is the set of rational numbers, and each partial interpretation N [ W represents an approximation to a real number (or numbers). In particular, the extension of B consists of lowerbounds of the real number being approximated, and the anti-extension of B consists of upper bounds of the number. We stipulate that the following hold: (1) at any partial interpretation N [ W, if a rational number x is in the extension of B and if y x, then y is in the extension of B. That is, Vx((Bx&y x) ) By) holds at the base M of F. 16 A sequence a , a , . . . of rational numbers is Cauchy if for each rational number e > 0 there is a 1 2 natural number N such that for all natural numbers m, n, if m > N and n > N then j am  an j < e. Two Cauchy sequences a1, a2, . . . and b1, b2, . . . converge to the same real number if, for each rational number e > 0, there is a natural number N such that for all natural numbers m > N, j am  bm j < e. A Cauchy sequence a1, a2, . . . is given by a rule if (i) there is an effective procedure for calculating the members an, and (ii) there is an effective procedure for calculating the bound N, given e.

Connectives, Quantifiers, Logic

121

(2) at any partial interpretation N [ W, if a rational number x is in the antiextension of B and if y x, then y is in the anti-extension of B. That is, Vx((:Bx&y x) ) :By) holds at the base M of F. (3) for each partial interpretation N [ W, there are two distinct rational numbers a < b, such that neither a nor b are in the extension of B at N and neither a nor b are in the anti-extension of B at N. In other words, in every partial interpretation, the extension of B and the anti-extension of B are separated by an interval. In symbols, 9r > 0VxVy((Bx&:By) ! j x  y j r) is weakly forced at the base (i.e. it is never false). (4) for each N [ W and each rational number r, there is a sharpening N 0 of N in W and two rational numbers x,y such that x is in the extension of B in N 0 , y is in the anti-extension of B in N 0 , and (y  x) < r. That is, Ar9x9y(Bx&:By&(y  x) < r) is forced at the base M (i.e. for each r, the formula is eventually true). Items (1) and (2) are like penumbral connections for (some) vague predicates. If John is bald and Harry has less hair than John (arranged the same way), then Harry is bald. Similarly, if x is a lower bound of a real number and y < x, then y is a lower bound of that real number. Item (3) is a rough analog of the principle of tolerance. It guarantees that each approximation is only an approximation. Indeed, if 9x > 0VyVz((By&:Bz) ! j y  z j x) were false at a partial interpretation N, then the extension and the anti-extension of B would constitute a Dedekind cut, defining a unique real number. Such a partial interpretation would be a completely sharp sharpening, an actual infinity. So clause (3) says that the creative subject (or the rule) never ‘‘finish’’ and produce a real number. This is the ‘‘potential’’ part of potential infinity. Item (4) guarantees that arbitrarily close approximations are available. For each rational number r, and each partial interpretation N in the frame F, there is a sharpening N 0 of N in F such that the approximation represented by N 0 is within r. This is the ‘‘infinity’’ part of potential infinity. With item (4), the analogy with ordinary vague predicates breaks down. I argued above, several times, that completely sharp sharpenings are not appropriate in the semantics of vague predicates. If the items in the domain are dense in the borderline area, I would think that tolerance would also rule out what we may call arbitrarily close sharpenings. The analog of (3) for a vague predicate should thus be stronger, stating that there is a fixed rational number r such that the extension and the anti-extension are always separated by at least r (in every

122

Connectives, Quantifiers, Logic

acceptable sharpening). In other words, the sentence VxVy((Bx&:By) ) j x  y j r) should be forced at the base. This is inconsistent with (4). Say that a frame F ¼ hW,M i for the present language is linear if, for any partial interpretations N, N 0 in F, either N  N 0 or N 0  N. A frame that corresponds to a rule-governed Cauchy sequence would be linear, since for each approximation given, the rule would supply a unique next approximation. Classically, each linear frame meeting the foregoing conditions defines a unique real number. However, a frame that corresponds to a free-choice sequence need not be linear. It is part of the spirit of free creation that each approximation is consistent with different (perhaps incompatible) ways of ‘‘going on’’. There is no determinate fact of the matter what choices the creative subject will make. The intuitionist argues that the law of excluded middle is not valid for real analysis. We saw above that if the connectives are chosen carefully, and interpreted in terms of forcing, then classical logic holds in the present framework (Theorems 25 and 28 above). Nevertheless, we can still sanction some of the crucial intuitionistic insights. In the framework developed above, one can state that the real number (or every real number) determined by a partial interpretation N in a frame F is rational. This happens if there is a rational number r such that every rational number x < r is eventually in the extension of B and every rational number y > r is eventually in the anti-extension of B. In this case, the frame ‘‘converges’’ to r. In the present language, the number determined by N in F is rational if (RAT) 9rVxVy((x < r )Bx)&(r < y ):By)) is forced at N. The formula (RAT) says that each ‘‘branch’’ from N converges to a rational number, but it allows that different branches may converge to different rational numbers. Similarly, the real number(s) determined by N in F is (are) not rational if for every rational number r, either there is a rational number x > r that is eventually in the extension of B or there is a rational number y < r that is eventually in the anti-extension of B: (IRR) Ar9x((r < x&Bx) _ (x < r&:Bx)) Consider the (linear) frame consisting of the following partial interpretations. In the base N1 the extension of B is {x j x 3} and the anti-extension of B is {x j x 4}. At N2 the extension of B is {x j x 3.3} and the anti-extension of B is {x j x 3.4}. At N3 the extension of B is {x j x 3.33} and the anti-extension of B is {x j x 3.34}. At N4 the extension of B is {x j x 3.333} and the anti-extension of B is {x j x 3.334}, etc.

Connectives, Quantifiers, Logic

123

... j N4 : B3:333, :B3:334 j N3 : B3:33, :B3:34 j N2 : B3:3, :B3:4 j N1 : B3, :B4 Then (RAT) is forced at the base N1, with r ¼ 1/3. The real number determined by this frame is 1/3. Now consider the frame consisting of the following partial interpretations. In the base M1 the extension of B is {x j x < 3} and the anti-extension of B is {x j x > 4}. At M2 the extension of B is {x j x < 3.1} and the anti-extension of B is {x j x > 3.2}. At M3 the extension of B is {x j x < 3.14} and the anti-extension of B is {x j x > 3.15}. At M4 the extension of B is {x j x < 3.141} and the anti-extension of B is {x j x > 3.142}, etc., following the decimal expansion of p. ... j M4 : B3:141, :B3:142 j M3 : B3:14, :B3:15 j M2 : B3:1, :B3:2 j M1 : B3, :B4 Then (IRR) is forced at the base M1. This frame determines the real number p. Using classical reasoning in the meta-language, one can show that for every linear frame F that satisfies (1)–(4), either (RAT) is forced at the base of F or (IRR) is forced at the base of F. Indeed, let c be the least upper bound of the unions of the extensions of B in the partial interpretations in F. Either c is rational or c is irrational. In the former case, (RAT) is forced at the base of F and in the latter case, (IRR) is forced at the base of F. But for the intuitionist, this argument just begs the question. In taking the least upper bound of the union of the frames of F and then applying excluded middle, the intuitionist would charge us with treating the frame F as a completed, actual

124

Connectives, Quantifiers, Logic

infinity. I do not apologize (here) for using classical logic in the meta-theory. In so doing, however, we do not seem to have a nice model of intuitionistic analysis, or the potential infinity, so long as we restrict ourselves to linear (or rule-governed) frames. However, once we allow non-linear frames, via free choice sequences, we do not have that for every frame F satisfying (1)–(4), that either (RAT) or (IRR) is forced at the base of F. Indeed, consider the most generic frame G possible. For each pair r,s of rational numbers with r < s, G has a partial interpretation Nrs in which the extension of B is {x j x < r} and the anti-extension of B is {x j x > s}. At the base of G, the extension of B and the anti-extension B are both empty. It is straightforward to see that there is no partial interpretation N in G such that (RAT) is forced at N, nor is there a partial interpretation N in G such that (IRR) is forced at N. Indeed, each approximation in G is consistent with a further series of sharpenings that converges to a rational number and each approximation in G is consistent with a different further series of sharpenings that converges to an irrational. This, I suggest, is the key insight behind the rejection of the validity of excluded middle, once free choice sequences are in the picture. Theorem 28 still holds. We do have that (RAT) _ (RAT) is forced at every partial interpretation in every frame. But in a given partial interpretation in a frame, (RAT) just says that (RAT) is never forced. It does not follow from this that (IRR) is forced at that partial interpretation. As an exercise, the reader is invited to reconstruct Brouwer’s theorem that all functions on the real numbers are continuous in the present framework. This is enough for the present sidetrack away from vagueness. We still have to deal with (what passes for) higher-order vagueness and with the possibility of vague objects.

5 Refinements and Extensions I: So-Called ‘‘Higher-Order Vagueness’’ I do not suggest that this simple observation puts an end to the lure of sorites reasoning, which, like a virus, will tend to evolve a resistant strain. Must there not be an outer limit to the things to which it is mandatory to apply ‘‘red’’, and a first member of the series with respect to which we have licence to withhold? The answer is ‘‘No: ‘mandatory’, too, is boundaryless’’ . . . Sainsbury [1990: x6] it may be misleading to think of higher-order vagueness in
as a species of vagueness in
. Higher-order vagueness in
is first-order vagueness in certain sentences containing
. Williamson [1999: 140] Our intuitions seem to run out after the second or third order of vagueness. Perhaps this is because our understanding of vague language is, to some extent, confused. One sees blurred boundaries, not clear boundaries to boundaries. Fine [1975: x5]

1 . W H I T H ER HI G H E R - O R D E R V A G U E N E S S ? Higher-order vagueness is vagueness concerning borderline cases of vague predicates or vagueness concerning determinacy. Consider our standby sorites series consisting of 2,000 men. The first, Yul Brynner, has no hair whatsoever, and the last, Jerry Garcia in his prime, has as full a head of hair as one could desire. After Mr Brynner, each man in the series has slightly more hair than his predecessor, arranged in roughly the same way. Intuitively, there is no (sharp) border between the men that are bald at the beginning and those that are not bald at the end. The fellows in the middle are the borderline cases, as analyzed and modeled in the previous chapters. According to the present thesis of open-texture, competent speakers can sometimes go either way with borderline cases.

126

Refinements and Extensions I

Intuitively, it also seems that there is no sharp boundary between the (determinately) bald men at the start and the borderline bald men in the middle, nor is there a sharp boundary between the borderline bald men in the middle and the (determinately) non-bald men at the end. By definition, a second-order borderline case of ‘‘bald’’ is a borderline case of ‘‘borderline bald’’. Equivalently, a second-order borderline case of ‘‘bald’’ is a borderline case of either ‘‘determinately bald’’ or ‘‘determinately non-bald’’. Call such a person borderlineborderline bald. Say that a man is determinately-determinately bald if he is determinately bald and (determinately) not borderline-borderline bald. Presumably, the first few men in our list are determinately-determinately bald. Say that a man is determinately-determinately not bald if he is determinately not bald and (determinately) not borderline-borderline bald. Jerry Garcia was determinatelydeterminately not bald if anybody ever was. Higher-order vagueness need not stop at this second level. Is there a sharp boundary between the determinately-determinately bald men and the borderlineborderline men? One would think not. To paraphrase Mayor Giuliani, why start now (with precise borders)? Say that a man is borderline-borderline-borderline bald if he is either a borderline case of ‘‘determinately-determinately bald’’ with ‘‘borderline-borderline bald’’, or if he is a borderline case of borderline-borderline bald with ‘‘determinately-determinately non-bald’’. This is third-order vagueness. And we define determinately-determinately-determinately bald and determinatelydeterminately-determinately non-bald accordingly. We will not inquire about the border between the men that are determinately borderline bald and those that are (merely) borderline-borderline bald. Enough is enough. But is there a sharp border between the men that are determinatelydeterminately-determinately bald and those that are borderline-borderlineborderline bald? Presumably not. So we have fourth-order vagueness—and so on through fifth-order, sixth-order, etc. For each natural number n, say that a man is n-determinately bald if he is determinately-determinately- . . . -determinately bald, with n iterations. Define n-determinately non-bald and n-borderline similarly. If we stick with our series of 2,000 men we will eventually run out of distinctions to make, provided we want at least one man in each category. A finite series can support only finitely many distinctions, no matter how vague they all seem to be. Let me be more precise (or determinate) about this. For each natural number i, let pi be the number of a man in our list who is (determinately) (i þ 1)borderline bald (provided that there is such a man). That is #pi lies between i-determinately bald and i-borderline bald. So man pi is not a determinately i-determinately bald man. That is, he is not (i þ 1)-determinately bald. Nor is man pi (determinately) i-borderline. Similarly, if man #piþ1 exists, then he is not determinately (i þ 1)-borderline. But by definition, #pi is determinately (i þ 1)-borderline bald. So pi 6¼ piþ1.

Refinements and Extensions I

127

A fortiori, pi > piþ1. And, of course, i is arbitrary. Since the series is finite, there is a number n such that there is no pn. That is, there is no man in our series who is on the borderline between n-determinately bald and n-borderline.1 It follows that either no man in our list—and Mr Brynner in particular—is n-determinately bald, or else the last n-determinately bald man in our list is followed immediately by an n-borderline man. The cut-off lies there. Of course, it does not follow that the borderline between n-determinately bald and n-borderline bald is actually sharp. If our series had only two men, Yul Brynner and Jerry Garcia, it would have a ‘‘sharp’’ cutoff between bald and not bald. The problem, perhaps, is that our series is not sufficiently dense. Nevertheless, if we assume that someone is (or could be) n-determinately bald, then the borderline area between n-determinately bald and n-borderline lies entirely between two of the men in our series (assuming that Yul Brynner is n-determinately bald). Recall that each man in our series has only marginally more hair than his predecessor— sufficiently small for us to cram 2,000 distinct heads between our extremes. So the area between n-determinately bald and n-borderline fits within a very narrow range, a few hairs at most. Moreover, the point about higher-order vagueness is general. Suppose that we had started with a series of 2,000,000,000 men instead of 2,000. The difference between each man and the next would amount to no more than a few hair molecules, much less than a single hair. Nevertheless, there is still a number m such that there is no pm in the expanded series. If someone is, or can be, m-determinately bald, then the entire borderline area between m-determinately bald and m-borderline bald lies within a few molecules on a single hair. In general, as i increases, the borderline area between i-determinately bald and i-borderline approaches zero.2 If there is vagueness at arbitrarily high levels, the borderline areas become arbitrarily small. There is, of course, a clear mathematical difference between a vague borderline area a few angstroms long and a sharp border, but that difference cannot amount to much in our philosophical analysis of vagueness, especially if we are to maintain the thesis that vagueness turns on response-dependence or judgmentdependence. Humans can only discriminate so much. So, for all practical purposes, most of these higher-order borders are sharp. Moreover, if we consider uncountable sorites-like sets, we can even squeeze out the tiny borderline areas, forcing absolute sharpness. Consider a set S of (continuum-many) potential NBA basketball players, ordered by height. The tallest member of S is 9 feet 6 inches tall—exactly (whatever that might mean). The shortest one is exactly 4 feet tall. And for each real number r between 9.5 and 4, S contains a player that is exactly r feet tall. For what it is worth, we now Graff [2003] makes essentially the same point, against the very notion of higher-order vagueness. It will not help much here to assert that there is a number m such that for all n > m, there are no n-determinately bald men. We could look at the mixed borderlines in the middle, say between determinately borderline-borderline bald and borderline-borderline-borderline bald. 1 2

128

Refinements and Extensions I

have the mathematical resources to support infinitely many non-empty orders of vagueness. For each n, there might be a player who is n-borderline-tall (for an NBA player). Unlike our finite series of bald men, we do not run out of players. Not yet. The run to higher orders need not stop with the finite levels. Say that a player is o-determinately tall if for each natural number n, he or she is n-determinately tall. If we assume that there can be o-determinately tall players, is there a sharp border between those and the rest (i.e. those that are shorter than someone n-borderline tall, for some natural number n)? If not, then we have o-order vagueness. Define a player to be o-borderline tall if he or she lies on the borderline between o-determinately tall and the rest—he or she is neither determinately o-determinately tall nor determinately not tall, nor, for any n, (determinately) n-borderline tall. We continue up through the ordinals: let a be an ordinal, and assume that we have defined ‘‘b-determinately tall’’ and ‘‘b-borderline tall’’, for each b < a. If a is a successor ordinal, we ask if there is a sharp boundary between the (a  1)determinately tall players and the (a  1)-borderline players. If not, we define ‘‘a-determinately’’ and ‘‘a-borderline’’, as above. If a is a limit ordinal, then define a player to be a-determinately-tall if he or she is b-determinately tall for each b < a. If there is no sharp border between those players and the rest, we define b-borderline accordingly. It is possible that there is an ordinal a, no larger than the continuum, such that our first player, 9 feet 6 inches tall, is not a-determinately tall. If there is no such ordinal a, then we must encounter a sharp border somewhere in the transfinite recursion. Otherwise, we’ll end up with a proper class of degrees of vagueness, and there aren’t enough real numbers for that. One can get a similar result without (explicitly) running through the ordinals. Say that a man (whether he appears in one of our lists or not) is absolutely determinately bald if he is bald and his baldness is not tainted by any vagueness at all (see Sainsbury 1990). In other words, someone fails to be absolutely determinately bald if either he is not bald, or if he exemplifies some iteration of borderline bald. We are forced to conclude that absolutely determinate baldness is a sharp predicate (perhaps empty). Indeed, suppose that Karl is a borderline case of absolute-determinate baldness. Then he is indeed tainted by some degree of vagueness concerning ‘‘bald’’, and so Karl is a (determinate) case of not absolutely determinate baldness. Karl is not a borderline case of that predicate after all. Consider a forced-march sorites series using our 2,000 men, or perhaps our longer one of 2,000,000,000 men. This time, we will use the predicate ‘‘absolutely determinately bald’’. Presumably, Yul Brynner, the first fellow in both lists, is absolutely determinately bald (if anybody is). He has no hair at all, and so there

Refinements and Extensions I

129

is no foothold for some level of borderline baldness. I do not know about the second man. But how can a few hairs—or a few molecules of a single hair— remove any baldness at any level? If #2 is not tainted with any level of borderline baldness (or non-baldness), then what about #3? Sooner or later, probably sooner, we will come to the first man in the list who has some level of borderline baldness or non-baldness. It seems that a few hairs, or a few molecules, can make a difference concerning absolutely determinate baldness. There is an interesting analogy between absolute or iterated higher-order vagueness and Michael Dummett’s [1991: 315–16] notion of the indefinite extensibility of, say, the cardinal numbers. He writes: to someone who has long been used to finite cardinals, and only to them, it seems obvious that there can only be finite cardinals. A cardinal number, for him, is arrived at by counting; and the very definition of an infinite totality is that it is impossible to count it . . . [This] prejudice is one that can be overcome: the beginner can be persuaded that it makes sense, after all, to speak of the number of natural numbers. Once his initial prejudice has been overcome, the next stage is to convince the beginner that there are distinct cardinal numbers: not all infinite totalities have as many members as each other. When he has become accustomed to this idea, he is extremely likely to ask, ‘‘How many transfinite cardinals are there?’’ How should he be answered? . . . If it was, after all, all right to ask, ‘‘How many numbers are there?’’, in the sense in which ‘‘number’’ meant ‘‘finite cardinal’’, how can it be wrong to ask the same question when ‘‘number’’ means ‘‘finite or transfinite cardinal’’? . . . And merely to say, ‘‘If you persist in talking about the number of all cardinal numbers, you will run into contradiction’’, is to wield the big stick, but not to offer an explanation.

In the present case, we start with someone who, initially, encounters a threepart division of the universe of (possible) men, those that are bald, those that are non-bald, and those that are borderline bald. Then she realizes, or conjectures, that these are not sharp predicates, and so she postulates a new kind of vagueness, which involves borderline-borderline cases, and then borderline-borderlineborderline cases, etc. As we saw, if our beginner is particularly determined, or warped, she may carry her process of higher-orders vagueness into the transfinite. But sooner or later, she will wonder about the predicate ‘‘absolutelydeterminately bald’’. This is analogous to Dummett’s beginner wondering about how many transfinite cardinal numbers there are. As far as I can determine, there is no analog of Cantor’s paradox, and so unlike Dummett’s beginner, ours does not run into contradiction. But she does run into precision, which may be almost as bad. By definition, there cannot be any borderline cases of absolutely determinate baldness. As noted in Ch. 1, an epistemicist about vagueness (e.g. Williamson 1994, Sorenson 2001) argues that every legitimate predicate has a sharp, but unknowable boundary. A single molecule of hair makes the difference between bald and not-bald, a single thousandth of a nanosecond marks the boundary of

130

Refinements and Extensions I

‘‘child’’, etc. The foregoing reasoning seems to suggest that we need some sharp boundary—between absolutely determinately bald and its complement. And, as far as I can tell, it seems that we have no way of determining exactly where this boundary is (if it exists). So our conclusion seems to support epistemicism, at least for the predicate ‘‘absolutely determinately bald’’. The epistemicist would urge one who accepts this to postulate sharp, unknowable boundaries everywhere. Hung for a sheep; hung for a lamb. One who wishes to reject epistemicism generally, but is convinced by the foregoing reasoning, has to motivate the fact that ordinary, or what may be called ‘‘first-order’’ predicates, such as ‘‘bald’’ and ‘‘red’’, do not have sharp boundaries, but higher-order analogs of these predicates do. One of the motivations for running up the ladder of higher-order vagueness is an intuition that there should be no relevant sharp borders anywhere in the series. The idea is that there is no sharp line separating the bald cases from those of any property that is incompatible with baldness and has something to do with the amount and arrangement of hair on the head. To repeat our oft-repeated paraphrase of Raffman [1994: 41 n. 1], there is tolerance between baldness ‘‘and any other category—even a ‘borderline’ category’’. And 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’’. Let us call this the ‘‘NSB thesis’’, for ‘‘no sharp boundaries’’. Sainsbury [1990] is an attempt to sustain this intuition—eschewing the notion of a borderline case (at any level). If we allow that there are borderline cases, the NSB thesis would tell us that they are not sharply separated from the determinate cases, and so we are off and running up the hierarchy of higher-order vagueness. Or else we can eschew the very notion of a borderline case. The forgoing reasoning shows that it is indeed difficult to maintain the NSB thesis. The ultimate forced march and the puzzles with higher-order vagueness are tough nuts to crack. Indeed, it is not easy rigorously to formulate a consistent and interesting NSB thesis. First, as we saw above, several times, we cannot just take a negation operator for granted. In previous chapters, we had occasion to discuss three different operators: a strong internal negation (:), a weak external negation (), and an intuitionistic-style negation (). The intuitive NSB thesis is itself formulated with operators in the neighborhood of negation. The thesis is that there is no sharp boundary between the Ps and anything else. Let’s try. Since we have run out of negation signs, I’ll use the English word. Consider a sorites series a0, a1, . . . , a1,000,000,000 for a predicate P. The NSB thesis would say that there is no sharp boundary between the Ps and anything else. That is, there is no x such that P holds of ax and something incompatible with P holds of Paxþ1. This suggests that the following should be true: ( ) not-9x(Pax¬-Paxþ1).

Refinements and Extensions I

131

Whatever the ‘‘not’’ in ( ) means, it is reasonable to assume that the intuitionistic introduction rule is good for it: If G, F ‘ C and G, F ‘ not-C, then G ‘ not-F. Then, given the usual (intuitionistic) rules for the existential quantifier and conjunction, we can derive a contradiction from ( ), Pa0, not-Pa1,000,000,000, and some trivial truths about the first 1,000,000,000 natural numbers.3 As we saw in the previous chapter, the introduction rule fails for the strong, internal negation : of our model theory. If tolerance is in force, then it is never true (and so never forced) that Pan&:Panþ1. So far, so good. Given tolerance, however, some instances of this are bound to be indeterminate, and so :9x(Pax&:Paxþ1) is always indeterminate (i), never true (and so never forced). There is a NSB thesis formulated with two of our negations. The formula ( ) 9x(Pax&:Paxþ1) is true, and thus forced, at every partial interpretation in every frame that respects tolerance. Indeed, ( ) just is an expression of tolerance in the object language, as in x2 of the previous chapter. Nevertheless, since the embedded negation : is strong, ( ) itself is too weak to fully express the NSB thesis. The idea is that there should be no boundaries between any incompatible predicates. We might try to replace the embedded :Panþ1 with something like Panþ1-is-indeterminate (or perhaps borderline). On the present view, however, Panþ1-is-indeterminate is compatible with Panþ1 itself being true in a sharpening (other than the base). A fortiori, Panþ1-isindeterminate is compatible with Pan. The appropriate contrast with Panþ1-isindeterminate is Pan-is-determinate. So the NSB thesis should entail: not-9x(Pax-is-determinate¬-Paxþ1-is-determinate). Once again, we are off and running up the hierarchy. For what it is worth, I’d like to maintain the NSB thesis. It does have intuitive appeal. The trick is to figure out a way coherently to formulate the thesis, and then figure out a way to stop the run up the hierarchy of higher-order vagueness without introducing or postulating sharp boundaries anywhere. For now, I will map out an alternative way to look at the issues. The next task is to analyze the phenomena of higher-order vagueness from the present perspective. We encounter Humpty Dumpty again.  3 See Wright [1987: x5]. The premises are (1) Pa , (2) not-Pa 0 1,000,000,000, and (3) ( ). It is manifest that not-Pa0 contradicts Pa0, so not-not-Pa0 follows from the first premise. Now assume (i) not-not-Pan. Assume (for reductio) (ii) not-Panþ1. Assume (also for reductio) (iii) Pan. Then Pan¬-Panþ1, and so 9x(Pax¬-Paxþ1). This contradicts ( ). So, discharging (iii), not-Pan. But this contradicts (i). So, discharging (ii), not-not-Panþ1. So by induction, or else, by repeating the above reasoning 1,000,000,000 times, once for each instance, we conclude not-notPa1,000,000,000 (discharging (ii) ). This contradicts (2).

132

Refinements and Extensions I

2. OPEN-TEXTURE, BORDERLINES, AND BORDERLINES TO BORDERLINES Let F be a predicate in a natural language such as English. Recall the definition of ‘‘determinateness’’, from Ch. 1 (drawn from McGee and McLaughlin 1994: x2): ‘‘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’’. We defined an object a to be a borderline case of F if Fa is ‘‘unsettled’’, if it is neither the case that a is determinately an F nor that a is determinately a non-F. According to the present ‘‘open-texture’’ account, in some situations, competent speakers can go either way with borderline cases. There is thus an important difference—a difference in kind—between predicates such as ‘‘bald’’ and predicates such as ‘‘borderline bald’’, as the latter is understood here. As noted in x8 of Ch. 1, the meaning of the word ‘‘bald’’ makes no reference to meaning or the judgments of competent speakers. For example, one of the entries for ‘‘bald’’ in Random House Webster’s Unabridged Dictionary reads, in part, ‘‘having little or no hair on the scalp’’. This definition, which I assume is correct, makes no reference, even implicitly, to competent users of the language. It concerns only the amount of hair on a person’s head. To be sure, what counts as ‘‘having little or no hair’’ can vary with the comparison class, paradigm cases, and/or contrasting cases. A person can have little or no hair vis-a`-vis Jerry Garcia but not vis-a`-vis me (alas). According to the present account, what counts as ‘‘having little or no hair’’ can also vary with the shifting calls in a conversation. But this contextual relativity is not a component of the meaning of the word ‘‘bald’’. On the present view, context figures in the determination of the extension of the predicate, not its meaning. In contrast, the meaning of ‘‘borderline bald’’ does contain an explicit reference to ‘‘the thoughts and practices of speakers of the language’’. By definition, a man is borderline bald if the application of the predicate ‘‘bald’’ is unsettled—if the thoughts and practices of competent English speakers, together with the nonlinguistic facts, have not fixed the truth-value of the sentence. In other words, the present notion of ‘‘borderline bald’’ has an explicit meta-linguistic component. To take another example, consider a sorites series consisting of colored patches. The leftmost patch is clearly red and the rightmost is clearly orange, and there are some borderline cases in the middle. The present point is that ‘‘borderline red’’ is not a color, on a par with ‘‘red’’ and ‘‘orange’’. Of course, color concepts are part of the meaning of ‘‘borderline red’’, but there are linguistic components of the meaning as well. So I agree with the passage from Timothy Williamson [1999: 140] at the start of this chapter: ‘‘it may be misleading to think of higher-order vagueness in a as a species of vagueness in a. Higher-order vagueness in a is first-order vagueness in certain sentences containing a.’’

Refinements and Extensions I

133

To be sure, there is nothing to prevent someone from introducing intermediate predicates that are on par with ‘‘bald’’ and ‘‘red’’. Define a man to be sort-of-bald if he has about as much hair as #864 in our standard baldness sorites series, and that this hair is arranged in roughly the same way. So defined, the meaning of ‘‘sort-of-bald’’ is purged of any reference to the thoughts and practices of competent speakers of the language. Like ‘‘bald’’, ‘‘sort-of-bald’’ concerns only the amount and arrangement of hair. And like any vague predicate, the extension of ‘‘sort-of-bald’’ will vary with external and internal context. As above, this does not say anything about its meaning. That meaning is given here, by explicit definition. Similarly, in any of those large boxes of Crayola crayons, you will find one that is about midway between red and orange, leaning toward the orange side. Such a crayon probably has an interesting name printed on it, but let us just call the color ‘‘reddish-orange’’. To repeat, ‘‘reddish-orange’’ is a color, and as such, the predicate differs in meaning from the present ‘‘borderline red’’, or ‘‘borderline between red and orange’’. With all due respect to Humpty Dumpty, we are the masters of language use, and can define new terms as we please. One might even call the new predicates ‘‘borderline bald’’ and ‘‘borderline red’’ (or ‘‘borderline red-orange’’). That would introduce an ambiguity with the foregoing notion. In any case, I do not wish to make any claims concerning the meaning of the English word ‘‘borderline’’ or the English phrases ‘‘borderline bald’’ and ‘‘borderline red’’, even as those terms are used in philosophical discourse. In the philosophy of language, the word ‘‘borderline’’ is becoming a term of art. The main thesis of this work is that vagueness is due to open-texture regarding borderline cases—in the present sense of ‘‘borderline’’. If English gives another sense to ‘‘borderline’’, then so be it. I also have no stake in the use of the term ‘‘higher-order’’ (except in its rather different use in logic). I thus have no quarrel with some authors who claim that there is no such thing as higher-order vagueness. I think I agree with some of them. The present plan is to characterize and model certain phenomena that might go under that name, hoping to shed light on the underlying philosophical issues. Predicates such as ‘‘sort-of-bald’’ and ‘‘reddish-orange’’ do not give rise to special problems in the area of higher-order vagueness.4 The new predicates are of course vague. Indeed, they were defined to be vague, using words such as ‘‘roughly’’. And one can construct sorites series invoking the new predicates. Consider, for example, the men in original series from Yul Brynner to #864, our defined paradigm of ‘‘sort-of-bald’’. We can then do a forced march on this sorites, instructing our conversationalists to judge each man, in turn, as to whether he is bald or just sort-of-bald. The result would have the same overall pattern as the original forced march. The conversationalists would agree that the 4 Here I am especially indebted to conversations with Diana Raffman, and to the treatment in Graff [2000: x3].

134

Refinements and Extensions I

first few men are bald and that the last few are sort-of-bald. If tolerance remains in force, there would be range in the middle, where they would go back and forth, exhibiting open-texture and backward spread (see Ch. 1). These men would be the borderline cases (in the present sense) between ‘‘bald’’ and ‘‘sortof-bald’’. Similarly, a sorites series, constructed with Crayola crayons, would identify some borderline cases between the red and the reddish-orange patches. The extension of ‘‘bald’’ is not the same in the original series and in its subset. Some men who are determinately bald in the first series lie in the borderline area in the sub-series. There is at least one man i, such that competent speakers always judge man i as bald when he is considered vis-a`-vis Yul Brynner and Jerry Garcia in his prime, and yet this same man i is sometimes (competently) judged ‘‘sortof-bald’’ (or even ‘‘not bald’’) vis-a`-vis Yul Brynner and #864, our paradigm for ‘‘sort-of-bald’’. This is to be expected. As noted above, many times, it is a commonplace that the extensions of vague predicates vary with (external) context. One crucial contextual component is the contrasting cases. A family might have a paltry income with respect to oil sheiks, but not so with respect to a philosophy department. It is the same commonplace that a man can be bald with respect to Jerry Garcia and not so with respect to our paradigm for sort-of-bald. But, once more, this is not higher-order vagueness. It is just the vagueness of ‘‘bald’’ and of ‘‘sort-of-bald’’, understood in their respective (external) contexts. But, again, I do not wish to quibble over terminology. The point is that the present account handles this phenomenon in stride. It is no different than the ordinary, so-called ‘‘first-order vagueness’’. We can iterate the procedure of defining intermediate predicates. To get to the next level, pick someone who lies in the borderline area between ‘‘bald’’ and ‘‘sort-of-bald’’, say man #714, and define him to be a paradigm of a new predicate, ‘‘more-or-less-sort-of-bald’’. And there will be a new sorites series, from Yul Brynner to #714. Similarly, a large box of Crayolas has paradigms for all sorts of colors intermediate between just about any two standard colors, and provides fanciful names for them. But the same analysis applies. As we keep shifting one or both of the ‘‘endpoints’’ of a sorites series, we create ever new (external) contexts, and ever new borderline areas. It is not seriously in dispute whether the extensions of vague predicates vary with such contexts. It should also be clear that as we iterate this process, and keep defining ever more intermediate predicates, the (vague) borderline areas between them will shrink. In the limit, we would have a unique predicate that applies to each man in the series—2,000 different baldness states—and we will have a unique predicate for each crayon, just as the folks at Crayola contend. Of course, these predicates are all sharp in the attenuated sense that the given series contains no borderline cases (in the present sense of ‘‘borderline’’) of those predicates. If we want to illustrate the vagueness of the new refined predicates, we will have to insert fresh men into the gaps and purchase a larger box of Crayolas.

Refinements and Extensions I

135

3 . W H A T P A S S E S F O R H IGHER-ORDER VAGUENESS Things are not as straightforward with the predicate ‘‘borderline bald’’, as that notion is defined here in terms of determinacy and open-texture. Again, a man is borderline bald if it is neither the case that he is determinately bald nor that he is determinately not bald. Combining this with open-texture, a man is borderline bald if speakers can sometimes go either way in their judgments without undermining their competence. Truth be told, I do not know whether ‘‘borderline bald’’ is vague, and so I will explore some options. Notice that a borderline case of ‘‘borderline bald’’ is also a borderline case of ‘‘determinately bald’’ or a borderline case of ‘‘determinately non-bald’’. Those predicates lie on the other side of the relevant ‘‘borders’’, so to speak. Let me repeat the definition once more. To say that a man is determinately bald means that the thoughts and practices of speakers of the language determine conditions of application for ‘‘bald’’, and the non-linguistic facts determine that these conditions are met. So our man is borderline bald if the thoughts and practices of competent language-users, together with the non-linguistic facts, do not determine whether the man is bald or whether he is not bald. I assume here that there is no vagueness in the relevant non-linguistic facts (such as the number and arrangement of hairs on our man’s head). I also assume, just for the sake of simplicity, that there is no relevant vagueness in what counts as a ‘‘thought’’ and a ‘‘practice’’. There is only one more place to look. If there is vagueness in ‘‘borderline bald’’, it turns on the vagueness of ‘‘speaker of English’’. More specifically, this vagueness must turn on what it is for the thoughts and practices of competent speakers to determine a verdict (or not). So what passes for ‘‘higher-order vagueness’’ of, say, ‘‘bald’’, is (first-order) vagueness in another predicate, again following Williamson (perhaps not in a way he would approve). A man is a borderline case of ‘‘borderline bald’’ if it is not determinate that he is determinately bald nor is it determinate that he is borderline bald.5 If the foregoing analysis of ‘‘determinate’’ is any good, it applies here. And the foregoing analysis is good. So a man is a borderline case of ‘‘borderline bald’’ if the thoughts and practices of speakers of the language determine conditions of application for ‘‘determinately bald’’ and for ‘‘borderline bald’’, and the nonlinguistic facts do not determine that either of these conditions are met. To spell out part of the former condition, the thoughts and practices of speakers of the language determine conditions of application for ‘‘the thoughts and practices of speakers of the language determine conditions for the application of ‘bald’ and the non-linguistic facts determine that those conditions are met.’’ 5 Another borderline case of ‘‘borderline bald’’ would be a man for whom it is not determinate that he is determinately not bald nor is it determinate that he is borderline bald. This case is exactly analogous to the one considered.

136

Refinements and Extensions I

So what is it for the thoughts and practices of speakers of the language to determine conditions for the application of a predicate such as ‘‘bald’’? Of course, this is a big question, concerning how language functions, and, in particular, how words obtain their meanings. One cannot provide a fully adequate answer to this question without settling just about every open and controversial issue in linguistics and the philosophy of language. Since I have no special insights to bring on these global matters, I must be more speculative and perhaps naive. Hopefully, enough can be said to shed light on our present topic concerning what passes for higher-order vagueness. I argued, in x8 of Ch. 1, that the vagueness of such terms as ‘‘red’’ and ‘‘bald’’ turns on the response-dependence or, better, judgment-dependence of those predicates (following Raffman 1994, 1996, and Wright 1987). In other words, the judgments of competent speakers of the language (under normal conditions) itself determines the extension of the predicates, at least in part.6 In present terms, the judgments of competent speakers are an essential component of the ‘‘conditions of application’’ of such vague predicates as ‘‘bald’’ (even if they are not part of the meaning). Recall our Thesis (B), from Raffman [1994]: [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. Although this thesis is substantially correct for vague predicates, it leaves open the question of which is the chicken and which is the egg. Following Raffman [1994: 70], ‘‘B is true with respect to borderline cases because our competent judgments of borderline cases determine their category memberships . . . ’’ In the borderline area, then, there is no consensus among competent speakers (under normal conditions), even after the external context (consisting of the comparison class, the paradigm or contrasting cases, etc.) is fixed. Open-texture comes into play. Moving to the next level, a borderline case of ‘‘borderline case of ‘bald’ ’’ would be a case in which it is not determinate whether there is consensus among competent speakers concerning the use of the word ‘‘bald’’. One option, perhaps, would be to introduce a parameter for statistical significance among the judgments of competent speakers. Then we would formulate (what passes for) higher-order vagueness with strong, scientific-sounding precision: so and so is a borderline case of ‘‘borderline bald’’ to the extent that 94 per cent of competent speakers agree that he is bald in normal contexts. For the sake of simplicity, I propose to interpret ‘‘consensus’’ to involve unanimity. If at least one competent speaker of the language would judge a man 6 There is room for vagueness concerning what counts as the ‘‘normal conditions’’ under which competent speakers exercise their competence. For simplicity, I propose to ignore that here. Assume that we have specified some conditions, which we hold fixed throughout the discussion. The alternative would be to apply the present account of vagueness to ‘‘normal conditions’’. I see no conceptual difficulties in this, but it does promise to complicate matters considerably.

Refinements and Extensions I

137

to be bald (under some fixed normal conditions—see n. 6—in some conversational context) and at least one competent speaker would judge the same man to be not bald (under the same normal conditions but a different conversational context), then the man is borderline. But even if we assume that counterfactuals such as these are determinate, this definition does not eliminate the possibility of (what passes for) higher-order vagueness. As above, with the simplifying assumptions in place, any vagueness of ‘‘borderline bald’’ must turn on vagueness in ‘‘competent speaker’’. Who are the competent speakers? In the present case, concerning baldness, a competent speaker is someone that understands the language and is employing normal perceptual mechanisms under the fixed, favorable conditions. I presume that most of us—readers (and the writer) of this book, for example—are clear cases of competent speakers, or at least competent users of the word ‘‘bald’’. If a person (not one of us) judges Jerry Garcia to be bald, or judges Yul Brynner to be not bald under those conditions, then she is clearly incompetent. She either does not understand the word, has misperceived, or has succumbed to some other failure, such as a mental lapse. But we have learned from Plato that identifying paradigm cases and paradigm noncases does not always shed light on the relevant philosophical issues. What is it to be borderline competent?

4. OPTIONS I can see three ways to proceed from this theoretical juncture. One is to hold that ‘‘competent speaker of English’’ or ‘‘competent user of the word ‘bald’ ’’ is not vague at all. For each person, it is determinate whether or not he or she is competent, or competent in the use of that word. This would entail that there is no analog to higher-order vagueness in our account of vagueness: the predicate ‘‘borderline bald’’ would be sharp. On this option, in our original series, there is a single number, i, such that man i is determinately bald and man i þ 1 is borderline. In other words, competent speakers all judge man i to be bald, and someone who does not is incompetent, and yet at least some competent speakers judge man i þ 1 to be not bald in at least some normal circumstances. This strains intuitions. How can a few hairs here and there, or even a few molecules on a single hair, make a difference between whether a judgment of baldness is competent or not? As epistemicists remind us, however, there is no definitive account of how words get their meanings or, indeed, on how communication is achieved in the use of language. So we cannot be sure about whether statements about competent speakers are sharp or vague. And all I claim to do here is to explore one option. As just noted, in Ch. 1, I argued that vagueness is tied to judgmentdependence. All vague terms are judgment-dependent, at least in their borderline

138

Refinements and Extensions I

regions.7 The contrapositive of this is that if a term is not judgment-dependent, then it is not vague. So if ‘‘competent user’’ is not judgment-dependent, even in part, then it is not vague. In other words, if the correct story of language acquisition and language use allows for no judgment-dependent component in whether a given person is a competent user of that language, then the indicated predicates will not display any vagueness. There will be no borderline cases of ‘‘borderline bald’’. If there is some indeterminacy there, it is not due to vagueness. As noted, this line is (vaguely) reminiscent of what epistemicists say about vagueness generally, without the disclaimer concerning other sources of indeterminacy (e.g. Williamson 1994, Sorenson 2001). These philosophers argue that the legitimate predicates of natural language, including ‘‘bald’’ and ‘‘red’’, all have sharp boundaries. They predict that the correct account of how words acquire meanings supports this. Williamson argues that we do not, and in some cases cannot, know the location of some of these sharp boundaries, because we are unaware of all the factors that go into fixing the boundaries. Language is a public affair, involving the thoughts and practices of a large community. No one user is aware of the thoughts and practices of every member of this community. An epistemicist might argue that if one is going to take a line like this for the phrase ‘‘competent user of the word ‘bald’ ’’, he or she should adopt the same account for ‘‘bald’’. What is the relevant difference? Simplicity demands that if we are going to wax epistemicist at the second level (so to speak), we should be epistemicist from the start. Again, hung for a sheep, or hung for a lamb. I maintain, however, that there is a difference. I do not plead (complete) ignorance concerning how vague words such as ‘‘red’’ and ‘‘bald’’ get their extensions. As elaborated in the previous chapters of this work, the indicated words are judgment-dependent, at least in part. Perhaps they are learned through paradigm instances. The judgment-dependence leads to a range of cases over which there is no consensus among competent speakers, and the open-texture thesis provides an account of how the indicated words are deployed in conversational contexts, showing how extensions are fixed and how communication is achieved. An advocate of the option under consideration concedes (or speculates) that the phrase ‘‘competent user’’ is not judgment-dependent, and (so) is not vague. She pleads ignorance on how that phrase acquires its meaning and its extension. Admittedly, it does strain intuitions to hold that ‘‘competent speaker’’ and ‘‘borderline bald’’ are sharp (or at least not vague). Although intuitions are fallible, perhaps we should explore other options before settling on this account. A second possibility is to introduce a (further) relativity. The idea is that, strictly 7 Perhaps this is too strong. A more modest claim is that the present account only applies to terms whose vagueness turns on judgment-dependence. The present discussion requires the stronger claim, which is put forward tentatively. Notice, however, that there may be other sources of indeterminacy besides vagueness (see Ch. 1, x8).

Refinements and Extensions I

139

speaking, it does not make sense to say that a man is borderline bald simpliciter, but only that he is borderline bald relative to a fixed class S of speakers of the language. In particular, a man is borderline bald relative to S if at least one member of S would declare him to be bald under some normal conditions (in some conversational context) and a member of S would declare him to be not bald under normal conditions (in a different context). To maintain this option, the reader should replace most of the previous talk of ‘‘competent speaker’’ with ‘‘member of S ’’. I presume that for a given set S of speakers, the predicate ‘‘borderline bald relative to S ’’ is sharp. It is defined in terms of the responses the members of S would make in various situations. This assumes that the extent of the relevant situations is fixed and sharp (see n. 6), and that the counterfactuals concerning how they would respond are determinate. Of course, some choices for the set of speakers have bizarre outcomes. Suppose, for example, that a set T contains a person who, for some strange reason, keeps seeing (what she thinks is) lots of hair when she looks at Yul Brynner’s head in normal light (or else our subject is confused about the meaning of ‘‘bald’’). Then, assuming that T contains at least one ordinary speaker, Yul Brynner comes out borderline bald relative to T. This is despite that fact that, on the present account, Yul Brynner is (or was) determinately bald. The thoughts and practices of speakers of language have fixed conditions for the application of ‘‘bald’’ (i.e. having little or no hair on the scalp), and the non-linguistic facts determine that those conditions are met in this case. He has no hair whatsoever. The problem might be attenuated if it could be specified that we are interested only in sets of speakers whose responses are consistent with the determinate cases (and determinate non-cases). That is, we are only interested in speakers that always get the determinate cases right. But then we would need a criterion for figuring out just which speakers those are, the very problem at issue. An advocate of this, second option, despairs of giving a sharp criterion for distinguishing the good (i.e. competent) speakers from the bizarre ones. So she bites this bullet, via the relativity. Of course, typical discussions of baldness and borderline baldness, both in philosophy and in ordinary life, do not make explicit reference to sets of speakers. For example, there was none in this work, before the present, perhaps desperate, option was raised. An advocate of this approach might hold that ordinary discussion nevertheless presupposes a fixed, but unspecified set of speakers. In most cases, it does not matter which class is picked, so long as it is reasonable—so long as the responses of members of the class substantially agree with language users generally. Or else an advocate of this option might propose a regimentation of the language in which the parameter is introduced as a theoretical response to the difficulties in saying who is competent. However the second option is interpreted, an advocate of it maintains that once a class of speakers is picked, the notion of ‘‘borderline bald’’ is sharp. The

Refinements and Extensions I

140

sorts of cases that might lead one to think that ‘‘borderline bald’’ is vague are precisely those where the specification of the class S is relevant. The cases which lead one to think of higher-order vagueness are those in which there are reasonable choices S, S 0 , such that the case is borderline bald relative to S but not relative to S 0 . Our advocate insists that this is not a case of vagueness, higher-order or otherwise. Rather, it is a situation in which a suppressed parameter matters. Our third theoretical option here is to hold that ‘‘competent speaker’’ or ‘‘competent user of the word ‘bald’ ’’ is indeed vague. This is the closest we will come to higher-order vagueness. The plan is to apply the present account of vagueness, in terms of open-texture and judgment-dependence, to the phrases in question. This option threatens to iterate in a vicious manner. For now, let us focus on the phrase ‘‘competent user of the word ‘bald’ ’’. Later (x5) we will explore a second track, focusing on the general predicate ‘‘competent speaker of English’’. The vagueness of ‘‘competent user of the word ‘bald’ ’’ can be illustrated via a forced march sorites on our fallback series consisting of 2,000 men, ranging from Yul Brynner to Jerry Garcia. Assume that we have assembled a group of (competent) subjects who are discussing the men in the series. This time, we ask them questions in the form: Would someone who judged man #i as not bald be competent? Start with i ¼ 1, and proceed in order. We only allow two answers, ‘‘yes’’ and ‘‘no’’, to the questions. As in Ch. 1, this is only for convenience, and not essential to the situation. The result would be the same if we allowed such answers as ‘‘no fact of the matter’’, or ‘‘indeterminate’’, or even silence. The important feature is that the conversationalists must answer each question by consensus. If they want to give an intermediate verdict, or a non-verdict, they must all agree to it. Unlike the conversationalists in the original sorites (perhaps), the members of this group must be convinced of the truth of the present account of vagueness, involving lack of consensus, determinacy, e-determinacy, competence, and opentexture.8 That is, they should answer in the framework of the present account of vagueness. Otherwise, the results might not come out correctly. On some views concerning vagueness, for example, a subject would give the same answer to the present question as she would to ‘‘Is man #i not bald?’’ The first question put to our conversationalists is whether someone who calls #1, Yul Brynner, not bald is competent. Presumably, they would answer ‘‘no’’, since, by hypothesis, they are all competent speakers, and they know that any competent speaker must call a man with no hair at all ‘‘bald’’. The conversationalists would give the same answer to the second question, concerning someone who calls #2 not bald. They would answer ‘‘no’’ to questions 3, 4, etc. Eventually, they will enter the borderline area for the predicate ‘‘may be called 8

See Ch. 1, x7 for a definition of ‘‘e-determinate’’.

Refinements and Extensions I

141

not bald by a competent speaker’’. As with the original series, they will probably maintain the consensus on the negative answer for a while, even in the borderline area. Eventually, however, a sufficient number of conversationalists will want to demur from this answer and the consensus will not be maintained. Given the ‘‘forced march’’ instructions, there will be a first member in the series, say #788, on which they agree that someone who calls that man not bald is competent. This happens even though they had just agreed (reluctantly for many of them) that someone who calls #787 not bald is not competent. As in Ch. 1, when this jump occurs, then some items, such as the statement that someone who calls #787 not bald is not competent, are removed from the conversational score. The context has shifted. This could be verified by reversing the order, repeating the question for #787, #786, etc. Assuming that tolerance remains in force, the conversationalists would exhibit backward spread, as they move backward and forward through the borderline area. In each case, they are exercising their options via open-texture—so long as they remain in the borderline area of the phrase ‘‘competent response concerning ‘bald’ ’’. Details of the phenomena can be found in Ch. 1, and need not be repeated here. In sum, on the present option, what passes for second-order vagueness is vagueness in the phrase ‘‘competent user of the word ‘bald’ ’’. There are folks who are determinately competent users of the word ‘‘bald’’; there are folks who are determinately incompetent; and there are still others who are borderline competent. Open-texture applies to the latter cases. 5 . H O L D O N T I GH T , W E ’ R E G O I N G U P So far, so good (perhaps). Recall that our first two options for what passes for second-order vagueness yield sharp boundaries, either absolutely or relative to a class of speakers. Unlike those, the third option can be iterated. Indeed, it calls out for iteration. What happens when we move to (what passes for) thirdorder vagueness? Are ‘‘determinately competent user of the word ‘bald’ ’’ and ‘‘borderline competent user of the word ‘bald’ ’’ themselves vague notions? The foregoing analysis applies here, almost word for word. To reiterate one conclusion, the issue of third-order vagueness comes down to the status of ‘‘competent user of ‘competent user of the word ‘‘bald’’ ’ ’’. Let us go over the same three options concerning this predicate that we had with the embedded ‘‘competent user of the word ‘bald’ ’’. First, one might hold that ‘‘competent user of ‘competent user of the word ‘‘bald’’ ’ ’’ is not vague, perhaps because it is not judgment-dependent. The move looks ad hoc, if not outright contradictory. How can we hold that ‘‘competent user of the word ‘bald’ ’’ is judgment-dependent and vague, and yet ‘‘competent user of ‘competent user of the word ‘‘bald’’ ’ ’’ is not? Surely, this strains intuitions if anything does. The more complex predicate has the same components as the simpler one;

142

Refinements and Extensions I

we just use three of the words twice. How can that mark a change from a vague, judgment-dependent predicate to a sharp (or otherwise non-vague) one? Second, we can introduce a relativity. That is, we maintain that, strictly speaking, it does not make sense to say that a speaker is a borderline competent user of ‘‘competent user of the word ‘bald’ ’’ simpliciter, but only that she is borderline relative to a fixed class S of speakers of the language. As above, for a fixed set S, the predicate ‘‘borderline competent user of ‘competent user of the word ‘‘bald’’ ’ relative to S ’’ is arguably sharp. But if we go relativist at this stage, why did we not do so at the previous one? Again, we are talking about the same components. The third option is that ‘‘competent user of ‘competent user of the word ‘‘bald’’ ’ ’’ is itself vague, because it is judgment-dependent (at least in part). The vagueness could be illustrated with a forced march sorites series. In the spirit of conservation, let’s reuse our series of 2,000 men in various stages of baldness, and reassemble our group of competent conversationalists who are convinced of the correctness of the present account of vagueness. This time, we put questions like the following to them: Suppose that a person p judges the following: A person q who judges man #i to be not bald is not competent. Is that judgment by person p competent? I will refrain from reformulating the proposed results involving backward spread and open-texture. The present plan—the third option applied to the third option—again calls for iteration, giving us counterparts to fourth-order vagueness, fifth-order vagueness, etc. Presumably, we would then have the specter of the borderline areas getting narrower as we move up through the (counterparts of ) orders. The result would be predicates that, for all practical purposes, are sharp (see x1 above). I suppose that someone might even iterate the iteration into the transfinite. The theorist would give an algorithm for producing the questions put to the conversationalists at the second, third, fourth, etc. level. Then the oth level would consist of a series of questions in the following form: Suppose someone answers ‘‘no’’ to all of these questions for person i. Is that a competent judgment? Then we could go to level o þ 1, and onward up through the ordinals. As above, this meta-process seems to force us to recognize a sharp predicate. The situation is not as bleak as all this would indicate, but I must get even more speculative and explore even more options. The air is getting rarefied, and I will be brief. Notice, first, that even at finite levels, there is no reason to expect the various borderline regions to get smaller and smaller. Indeed, if we work with actual human conversationalists, and not idealized versions thereof, the results might

Refinements and Extensions I

143

not even be consistent. The borderlines at the third level might overlap with, or include, those of the fourth level. For example, some subjects might rule someone who answers a certain way competent at the third level and rule that last answer incompetent when it comes up the next level, or vice versa. In effect, that person would judge herself to be incompetent (not in those words, of course). I would think that competence in language use does not require that answers to these recherche´ questions line up in a certain way. Humans are not perfect. This is another place where the conversational record might require a paraconsistent logic. A contradiction with a penumbral connection between the levels need not commit the conversationalists to every sentence of the language. And it need not interfere with general communication. Also, if we use actual human conversationalists, the experiment will break down at the fourth or fifth level, if not before, for the simple reason that humans cannot parse questions that long. The above question for the third level is complicated enough. A fortiori, we need not worry about the transfinite levels. So it seems that in order to sustain the argument that the higher levels lead to sharp predicates, we have to both enforce consistency on the questions and answers, and idealize on the abilities of humans to parse and reflect on sentences. But it is not clear that we can coherently do so. As argued in x6 of Ch. 1, a subject’s actual response in the borderline area of a vague predicate (at any level) may be caused by extraneous features, such as her attention span or state of fatigue. This is the very thing we are trying to idealize away. In sum, even sticking to the third level, I am not sure that there is always a determinate truth-value to such counterfactuals as: A suitably idealized version of person r would respond negatively to the following: Suppose that an idealized person p judges that a person q who judges man #722 to be not bald is not competent. Is the judgment by person p competent? Moreover, this indeterminacy is not due to judgment-dependence, and so does not give rise to vagueness. Or at least the present account of vagueness does not apply to it. Suppose, on the other hand, that the counterfactuals are, in fact determinate. There is a fact of the matter as to how a highly idealized version of each subject would respond to the various queries. Then, if the responses are consistent with the various penumbral connections, we do converge on ever sharper boundaries as we go up the surrogate orders, as shown in x1 above. But I see nothing counterintuitive in this. The sharpness comes from the assumed penumbral connections and the assumed idealization. Recall that on the present program, the issue corresponding to second-order vagueness concerns the status of ‘‘competent user of the word ‘bald’ ’’, and we supposed that this notion is vague. The issue of third-order vagueness comes

144

Refinements and Extensions I

down to the status of ‘‘competent user of ‘competent user of the word ‘‘bald’’ ’ ’’. And we assumed that this notion is also vague. The issue of fourth-order vagueness comes down to the status of ‘‘competent user of ‘competent user of ‘‘competent user of the word ‘bald’ ’’ ’ ’’.9 Is it reasonable to assume that these predicates are all different? Is there really a difference between, for example, a competent user of the word ‘‘bald’’, and a competent user of ‘‘competent user of the word ‘bald’ ’’? If there is a difference here, is there one between the latter and a competent user of ‘‘competent user of ‘competent user of the word ‘‘bald’’ ’ ’’? It makes sense to distinguish competence with some words and phrases from competence with others. For example, I take myself to be a competent user of the vocabulary of baseball, but not the vocabulary of cricket. However, it does not seem to make sense for someone to be a competent user of the word ‘‘strike’’ (in baseball) but not the word ‘‘out’’. Competence concerns chunks of language, not individual words and phrases. It is arguable that after the first level or two (if not before), the various iterated competencies concerning the word ‘‘bald’’ are in the same chunk. At some point, perhaps, it just comes down to being a competent user of the language, full stop, or at least a competent user of the various competencies associated with the word ‘‘bald’’. As noted above, after the second level, we keep reusing the same words and nesting quotations. So what is it to be a competent speaker of the language, or a competent judge when it comes to iterated judgments of competence? Once again, this broaches large questions concerning language acquisition and use, and I have no special insights to bring on that score. Notice, however, that when it comes to judgmentdependent terms such as ‘‘bald’’, there seems to be a deep circularity in the notion of ‘‘competent’’. Intuitively, someone is competent only if her responses conform to everyone else’s. But not everyone’s responses are relevant. By definition, a competent speaker’s judgments will not, or at least need not, conform to those of an incompetent speaker. So someone is competent only if her responses conform to those of the competent users. Friedrich Waismann [1951b: 122] points to essentially the same problem: For one thing, to speak of the ordinary use of language is . . . questionable, implying as it does, that there is such a thing, and a unique one, and that one can find out what it is. But how ought one to determine what this ordinary use is, e.g., in a case of doubt? What ought one to do—to ask people? Any people? Or only the competent ones? And who is to decide who is ‘‘competent’’—the leading circles of society, the experts of language, the writers just in vogue? And supposing there are people generally considered competent— what if they disagree?

In the end, it may be that there is no non-circular way to specify who the competent users are. If this is correct, then even if we assume that competence is vague and (thus) judgment-dependent, we cannot assemble a group of competent 9 I never thought I would have occasion to embed quote marks this deeply in a sentence. For some interesting advice on this, see Boolos [1995].

Refinements and Extensions I

145

subjects to help us decide who is competent, without begging the question. We would have no criteria to decide who to include as the proper judges. This suggests that at the second or perhaps third level, we have to fall back on our second option, the one that produces a relativity. We have to specify, or postulate, a group S of speakers, who we assume to be the competent ones, and all judgments are relative to this group S. The idea is that, strictly speaking, it does not make sense to say that a speaker is competent, or borderline competent, or incompetent simpliciter, but only that she has this status relative to S. As above, once the class of speakers is fixed, the relevant predicates may be sharp, depending on the status of the counterfactuals concerning how the members of S would respond. In any case, the iteration is blocked. A different choice of the background class of speakers would result in different extensions for ‘‘competent’’, but that is to be expected. Given the circularity in the intuitive notion of ‘‘competent speaker’’, this may be the best route to take. The reason why we need not actually specify the class of speakers is that in the vast majority of cases, it does not matter. This ends the speculations concerning what passes for higher-order vagueness. I now turn to the model theory developed in the previous two chapters, showing how it needs to be modified to account for the phenomena discussed above. 6. REPRESENTORS AND ARTIFACTS Recall that in the Kripke-structure-supervaluationist-style model theory developed in Ch. 3, a partial interpretation consists of a domain of discourse and a function that provides interpretations for each non-logical item in the language. What makes the interpretation ‘‘partial’’ is that each predicate is assigned a disjoint pair of sets, an extension and an anti-extension. The predicate is ‘‘sharp’’ in the partial interpretation if its extension and anti-extension exhaust the domain. A frame F is a structure hW,M i such that W is a collection of partial interpretations, M [ W, and every partial interpretation N in W is a sharpening of the base M. All of the partial interpretations in a given frame have the same domain. In each partial interpretation, the extensions and anti-extensions are ordinary sets—the sort of thing studied in standard set theory. So each partial interpretation provides what are, in effect, two sharp boundaries for each relation symbol: one between instances which are in the extension of the relation (in the interpretation) and instances which are not in the extension, and another between instances that are in the anti-extension and instances that are not in the anti-extension. The indeterminate cases also constitute a sharp set, the complement of the union of the extension and the anti-extension. It follows that in each partial interpretation N, each sentence is true, false, or indeterminate. Similarly, for each sentence F, each frame F, and each partial interpretation M in F, F is (determinately) forced at M in F, :F is forced at M in F, or neither is. Those are the only options.

146

Refinements and Extensions I

This sharpness does not disqualify the system as a model theory for natural language. As noted in Ch. 2, the plan is to use a precise meta-language to model a vague object language. The model theory does not have to match up perfectly with the logical and semantic phenomena being modeled in order to illuminate those phenomena. With models generally, we sometimes need to pay attention to artifacts, features of the model that do not correspond to the reality being modeled. In particular, we must make sure that we do not conclude anything about the language being modeled on the basis of artifacts in the modeling system. The crucial question here, then, concerns the extent to which the sharp borders of the extensions and anti-extensions are representors or artifacts of the system. That is, do these sharp borders represent something in the semantics of vague predicates of natural language? Recall that a given atomic sentence Pa can be true in a given partial interpretation and another sentence Pa 0 indeterminate, even if P is vague and the denotation of a can barely be distinguished from the denotation of a 0 in the relevant respect. Since extensions in the partial interpretations are (sharp) sets, this is bound to happen when we model sorites series. All we need is for denotation of a to be in the extension of P and for the denotation of a 0 not to be in that extension. Similarly, Pa can be forced at an interpretation in a frame and Pa 0 not be forced at the same interpretation in the same frame. This much is unproblematic for partial interpretations other than the base of a given frame. In those partial interpretations, the sharpness of the extensions and anti-extensions is a representor, not an artifact. Recall that a partial interpretation in a frame represents a possible conversation that is consistent with the meanings of the terms, the non-linguistic facts, etc. In other words, the truths in a given partial interpretation represent a range of judgments that competent speakers of the language might make if they understand the language and are perceiving correctly. On the present open-texture thesis, if an object a lies in the borderline area of a vague predicate P—if it is not e-determinate that Pa, nor is it e-determinate that :Pa—then in some circumstances competent speakers can go either way. In such a context, they may assert Pa without undermining their competence, and they may assert :Pa without undermining their competence. To be sure, they cannot assert both Pa and :Pa in the same context without sinning against logic (see Ch. 1 for details). Now suppose that a and a 0 both lie in the borderline area of P. Then if a and a 0 are sufficiently close, then a speaker cannot assert Pa and deny Pa 0 at the same time without sinning against tolerance. But there is nothing untoward about a competent speaker, or a group of competent speakers in a conversation, holding Pa to be true while not saying anything yet about Pa 0 . She or they can affirm one and leave the other unjudged.10 Indeed, 10 As noted in Ch. 1, it would violate tolerance for a speaker to assert Pa and consciously decide to leave Pa 0 unjudged (in the same context), or to assert Pa and assert that Pa 0 is unjudged (in the same context). The present claim is only that it does not violate tolerance for a speaker to assert Pa and not say anything about Pa 0 .

Refinements and Extensions I

147

in a typical sorites situation, this must happen. It is not possible to judge every member of a sorites series and remain true to tolerance (and logic). In general, the extensions of vague predicates vary with the flow of conversation. This is registered in the various partial interpretations in a given frame. However, the sentences that are true at the base of a frame represent e-determinate truths, sentences whose truth is fixed by the meanings of the terms, the nonlinguistic facts, and the external context—prior to any conversational decisions concerning borderline cases. The important issue concerns whether the two sharp boundaries at the base of a given frame are representors or artifacts. Suppose, then, that the language has a monadic predicate B designed to represent baldness. According to the model developed in the previous chapters, the extension of ‘‘e-determinately bald’’ in a given frame F consists of the extension of B at the base. And, again, this is a set with sharp boundaries. The sharpness of the determinateness relation in the model is reflected in the fact that for any sentence F, DET(F) is either true or false, never indeterminate, at every partial interpretation in every frame. Is this an artifact of the model, or does it represent something about the logical or semantic behavior of vague predicates? Here is where the analogs of higher-order vagueness come to the fore. If there is no vagueness concerning e-determinacy, competence and the like, then the sharpness of the extensions and anti-extensions at the base of a frame, and the bivalence of the DET operator, are representors. The DET operator works correctly; it is a good model of e-determinacy. On the other hand, if there is something in the neighborhood of (what passes for) higher-order vagueness, then the sharp boundaries at the base of each frame are artifacts. In this case, we must be careful not to draw any general conclusions about vagueness from the sharpness and bivalence of the DET operator. The acceptability of this sharpness is due to the fact that in previous chapters, we were only interested in what happens as we go from ‘‘bald’’ to ‘‘not bald’’, and the like. We were not (yet) interested in what happens near the boundary between ‘‘e-determinately bald’’ and ‘‘unsettled’’. We had to walk before we could run. Now we are interested in these other boundaries. I argued in x2 above that the phenomenon that passes for higher-order vagueness—if anything does—turns on the vagueness in the notion of competency. The so-called higher-order vagueness concerning a word such as ‘‘bald’’ depends on the status of the expression ‘‘competent user of the word ‘bald’ ’’. Once again, I do not know whether being a competent speaker of English, or a competent user of the word ‘‘bald’’, are vague notions. Instead of adjudicating this matter, I noted three options. The first is that competence is a sharp notion. To be more precise, if there is some indeterminacy involved in the notion of competence, it is not due to any response-dependence or judgment-dependence, and so any such indeterminacy does not fall within the purview of the present account of vagueness. So, for simplicity, we treat the relevant predicates as sharp. The second option is to take competence as relative to a fixed class S of speakers. To say that a person p is competent relative to S, or that p is a

148

Refinements and Extensions I

competent user relative to S of a particular word, is for p’s responses to agree with those of at least one member of S. On this second option, once the class S is fixed, we can think of ‘‘competence-relative-to-S ’’ as sharp. Counterfactual conditionals fix its boundary. On both these options, there is no analog of higher-order vagueness, and so for simplicity, we assume that the relevant notions are sharp. This is reflected in our model theory, as it stands. Recall that a frame F ¼ hW,M i consists of a set of partial interpretations, one of which is designated as the base. On the first option, a frame would represent a model of the language, or logical possibility concerning it, reflecting the range of possible extensions of the predicates. On the second, a frame would represent a model of the language once a class S of speakers is determined, to fix the extension of ‘‘competence’’. That is, each frame would represent the possible extensions of the predicate relative to a single class of speakers. The sharpness of the determinacy operator reflects the fact that competence is sharp, either by itself or once a class of speakers is fixed. In general, a sentence F is indeterminate in a frame F if :DET(F)& :DET(:F) holds at the base of F (which happens if and only if it holds at every partial interpretation of F ). In this case, the sentence F is true at some partial interpretations in F, but only indeterminately so, since :F holds at other partial interpretations in the same frame. Let a be an object and P a predicate. It is straightforward that a is a borderline case in the frame F if Pa is indeterminate, i.e. that :DET(Pa)&:DET(:Pa) holds at the base of F. Similarly, the object a is a second-order borderline case of the predicate P if DET(Pa) is indeterminate. That is, a is a second-order borderline case of P if :DET(DET(Pa) )&:DET(:DET(Pa) ) is true. As we saw in the previous chapter, however, this never happens in the present model theory. Indeed, we have that the DET operator satisfies the S5 axioms (Theorem 24 of Ch. 4 x6): DET(F) ! DET(DET(F) ) and :DET(F) ! DET(:DET(F) ) are both true at every partial interpretation in every frame (Theorem 11). It follows that if F is any formula, then [:DET(DET(F) )& :DET(:DET(F) )] is false at N under s in F (Theorem 12). On the first two options, this is as it should be. Once again, on those options, there is no analog of higher-order vagueness, and so the sharpness of the DET operator is a representor in our model theory, representing the sharpness of e-determinacy. 7. IMPROVING THE MODEL THEORY (IF WE NEED TO), OR GETTING UP TO A SLOW JOG Now we turn to the more interesting third option, where competence and so e-determinacy are themselves vague. Here is where we get analogs of

Refinements and Extensions I

149

higher-order vagueness. On this option, then, the sharpness of the DET operator in the model theory is a glaring artifact. To get a more accurate model of vagueness, one that accommodates the analog of second-order vagueness, we must complicate the system.

7.1. A Loan from our Supervaluationist Friends As a first approximation, we adapt the treatment of higher-order vagueness in the supervaluationist framework of Kit Fine’s classic [1975: x5]. In the model theory developed in the previous chapters, the unit—the analog of a model in model theory—is a frame. Each partial interpretation in a given frame represents a set of judgments that are consistent with the meaning of the terms and the nonlinguistic facts. To accommodate second-order vagueness, we take the unit of model theory to be a set Y of frames, all with the same language, and all involving the same universe of discourse. Let F ¼ hW,M i be a frame in Y. The base M of F represents one acceptable way to fix the extensions of predicates like ‘‘e-determinately bald’’ or ‘‘competent use of the term ‘bald’ ’’, and the other partial interpretations in F represent possible extensions of phrases such as ‘‘bald’’ within the parameters set by the base M. The sum total of frames in the set Y represents the various ways that the extensions of e-determinacy or competency can be fixed, consistent with the meaning of the terms and the non-linguistic facts. In the present framework, the analog of ‘‘super-truth’’ is ‘‘forcing at the base’’. Recall from x4 of the previous chapter that DET(F) holds at a frame N if F is forced at the base M of F. And :DET(F) holds at N if it is not the case that F is forced at the base. So any sentence in the form DET(F) is fully bivalent. It is never indeterminate. In effect, the DET operator represents what e-determinacy comes to once we settle on a particular (acceptable) way to fix the extensions of predicates such as ‘‘e-determinately bald’’. So let us rename the operator DET1. To handle second-order vagueness, we introduce another operator DET2, for second-order determinacy, with the following clause: DET2(F) holds at a partial interpretation N in a frame F in a set Y (under an assignment) if F is forced at the base of every frame in Y (under the assignment). Notice that DET2(F) holds at a given partial interpretation if and only if DET2(DET1(F) ) holds in that partial interpretation. So suppose that B is a predicate for ‘‘bald’’ and a is a singular term naming a man. Then DET2(Ba) holds if, for every acceptable way of delimiting the acceptable sharpenings, a is in the extension of ‘‘bald’’ in every one of the acceptable sharpenings. I presume that Yul Brynner is in this category. As above, a man b is a borderline case of ‘‘bald’’ in a given frame F if :DET1(Bb)& :DET1(:Bb) holds at the partial interpretations in F. Similarly, b is a secondorder borderline case of ‘‘bald’’ if :DET2(DET1(Bb) )&:DET2(:DET1(Bb) ) holds at the frames in Y. This happens if and only if there is a frame F in Y such that Bb (and so DET1(Bb) ) is forced at the base of F and there is another frame F 0

150

Refinements and Extensions I

in Y such that Bb is not forced at the base of F 0 . That is, on one acceptable way of delimiting the acceptable sharpenings, b is bald on all acceptable sharpenings (according to that way), but there is another acceptable way to delimit acceptable sharpenings, and on one of those acceptable sharpenings, b is not bald. To accommodate third-order vagueness from this perspective, we would take the unit to be a set of sets of frames. Each set of frames in the unit would represent one acceptable range of ways of indicating what the range of acceptable sharpenings is. We would define an operator DET3, and third-order borderline cases accordingly. The procedure generalizes in a straightforward manner.11 We can carry it out as far as we want—to the level where either sharpness, effective sharpness, or boredom results (see x1 above). At that level, there is no point to further iteration. For present purposes, this system is not completely satisfactory. It uses the present model theory for vagueness at the first level, but switches to a straight supervaluationist account for the higher levels. Moreover, starting with the second level, all that the system considers are (the analogs) of complete sharpenings of determinacy. The conceptually rich notion of open-texture is not invoked beyond the first level. Also, in a sense, the various levels are all fixed separately. The vagueness of, say, baldness, is addressed only after the vagueness of e-determinately bald is eliminated by choosing a (complete) sharpening of e-determinacy. The vagueness of baldness is addressed in a frame, and each such frame is a member of a set of frames that delimits the next level of vagueness.

7.2. A Better Model On our third theoretical option above, the notions of competency, e-determinacy, and borderline are all vague. Consistency demands—and simplicity suggests—that we apply the same account of vagueness to those notions that we applied to such predicates as ‘‘bald’’ and ‘‘red’’. As we saw, the present philosophical account of vagueness applies to, and illuminates, the vagueness of the notions that give us our analog of higher-order vagueness. Or so I argue. The technical problem is to slot this into the formal framework. The first thing to do is remove the artifactual DET operator from the system. We will replace it with something that better models e-determinacy, in light of what passes for higher-order vagueness. Recall that in the present model theory, a partial interpretation in a given frame represents a collection of judgments that are consistent with the meaning of the terms and the non-linguistic facts. As we saw above, with our third option, the analog of second-order vagueness turns on 11 Fine [1975: x5] briefly presents an ingenious system that accommodates all the (finite) iterations at once. For Fine, a ‘‘borderline’’ is a sequence c0, c1, c2 . . . , where c0 is a complete sharpening, c1 is a set of complete sharpenings, c2 is a set of sets of complete sharpenings, etc. He then defines a single determinacy operator that automatically focuses on the right item in the sequence for each of its iterated embeddings.

Refinements and Extensions I

151

the vagueness of ‘‘competent user of English’’, or of more specific predicates such as ‘‘competent user of the word ‘bald’ ’’. By hypothesis, the notions are judgmentdependent (at least in part) and open-textured. So to accommodate the phenomena, we need a mechanism to track possible judgments concerning those notions. To provide a realistic model of actual linguistic use, judgments about these ‘‘higher-order’’ matters can be made simultaneous with ‘‘first-order’’ judgments, like those of baldness, color, wealth, and height. Of course, there are penumbral connections that relate various judgments to others. So as with the basic system, the unit is still a single frame in the form hW,M i. We add a new sentential operator CP to the language, with the formation rule: If F is a formula, then so is CP(F); CP(F) has the same free variables as F. The intuitive idea is that CP(F) holds if a judgment that F is competent. In the cases that interest us here, CP(F) holds if someone can judge that F without sinning against the meaning of the terms in F and the non-linguistic facts. Notice that, on the present view, competence is not closed under conjunction. If h is a borderline case of B, then CP(Bh) and CP(:Bh) are both true, but so is :CP(Bh&:Bh). That is, even if one can competently assert Bh and one can competently assert its negation, one cannot competently contradict oneself (dialetheism notwithstanding). In the semantics, a formula in the form CP(F) is treated as atomic, even if F is syntactically complex. In general, its semantic value (true, false, forced, etc.) at a sharpening in a frame cannot be read off the value of the embedded formula F anywhere in the frame. Of course, interesting and sometimes subtle penumbral connections control the relationships between the values of F and CP(F). Stay tuned. In some cases, what counts as a competent judgment on a given matter depends on previous judgments, both about what are allowable competent judgments and about other things, such as who is and who is not bald. In conversations, these matters are recorded on the score. In the model theory, the extension of predicates like CP(Bx) vary from context to context—from sharpening to sharpening. In light of the foregoing analysis, we can define an operator that corresponds to e-determinacy as follows: DET(F)  def F&:CP(:F). In words, F is e-determinately true if F is true and a judgment that :F is incompetent: one has no option to say that F is false. The first conjunct in the definition of DET is perhaps redundant when it comes to vague or judgment-dependent notions. If, in fact, it is incompetent (in the present sense) to deny that a man m is bald, then m is bald. As will be seen below, however, the first conjunct is needed for non-judgment-dependent notions.

152

Refinements and Extensions I

As competence is understood here, its extension need not be altered as (firstorder) borderline cases are called in a given conversation. Suppose, for example, that a given man a lies in the borderline area for ‘‘bald’’. Then in some context C, a competent speaker may call him bald. So Ba is true in C. But, by hypothesis, a is, and remains, a borderline case of B. So CP(:Ba) remains true in C. This is a counterexample to F ! DET(F) and to the widely held inference from F to DET(F). So long as F is true, but not e-determinately so, it remains that one can competently deny F in some contexts.12 To be sure, if F is true in context C, then one cannot deny F in the context C (on pain of inconsistency). However, one can deny F in other contexts that are consistent with meaning, nonlinguistic fact, and externally determined contextual features. A speaker initially in context C who denies F shifts to another context C 0 , a context incompatible with C. In the terminology of Ch. 1, borrowed from Lewis [1979], when the shift to the next context occurs, items (such as F) are removed from the conversational score. In the model theory, C and C 0 represent sharpenings on different branches: the partial interpretations corresponding to C and C 0 are not sharpenings of each other. The present point is that if CP(:F) holds in C, then a move from C to a context like C 0 is allowed. To be sure, the present interpretation of CP, and the concomitant notion of e-determinacy are tightly connected to the present account of vagueness, involving judgment-dependence and open-texture. Advocates of other accounts of vagueness disagree on the use of the word ‘‘competent’’. On some views, if someone competently asserts that a man h is bald, then she thereby holds that anyone who says otherwise has made a mistake—he has either misperceived the amount or arrangement of h’s hair or does not understand the meaning of the words. Say that a partial interpretation N 0 is a competence-sharpening of a partial interpretation of N if they have the same domain of discourse and, for every formula F, the extension of CP(F) in N is a subset of the extension of CP(F) in N 0 and the anti-extension of CP(F) in N is a subset of the anti-extension of CP(F) in N 0 . In other words, N 0 is a competence-sharpening of N if N 0 is a sharpening of N for the ‘‘language’’ consisting of formulas in the form CP(F). Say that a frame hW,M i is acceptable if, for every partial interpretation N [ W, any formula F, and any variable assignment s, if CP(F) is true at N under s, then there is a competence-sharpening N 0 of N in W such that F is true at N 0 under s. In other words, a frame is acceptable if, whenever it has a partial interpretation N on which it is competent to assert F, then there is a competence sharpening of N in which F is true. Every competent judgment is realized somewhere in the frame. In acceptable frames, CP acts like a kind of ‘‘possibility’’ operator in modal logic, with ‘‘competence-sharpening’’ as the accessibility relation. 12 As noted above, I may be using the notion of ‘‘determinacy’’ and thus ‘‘competence’’ in somewhat non-standard ways. According to most authors who use the notion, there is no way for a sentence to be true without being determinately true. Wright [1987: x5], for example, writes that ‘‘there is no apparent way whereby a statement could be true without being definitely so’’.

Refinements and Extensions I

153

Notice, however, that so long as we think of CP as vague (and vagueness is understood in terms of open-texture), then there are acceptable frames in which a sentence F is true at a partial interpretation N 00 even though :CP(F) holds at another partial interpretation N. In this case, N 00 represents a context which is incompatible with N concerning CP. Suppose that in the course of a conversation, a person p1 declares that a certain man h is bald: she says Bh. Another person p2 disagrees, and says :Bh. Moreover, the two conversationalists agree on the ‘‘external’’ contextual factors, like the comparison class and/or the paradigm or contrasting cases. Suppose that neither p1 nor p2 retracts her view on the baldness status of Mr h in light of the other’s assertion.13 On the present account of vagueness, there are two different attitudes that p1 can take to p2 (and there are two attitudes that p2 can take to p1). First, p1 might admit that p2 is within her rights in denying that h is bald. That is, p1 might hold that h lies in the borderline area of baldness. Assuming that p1 adopts something in the neighborhood of the present open-texture account of vagueness, she holds that competent speakers can go either way in the borderline area. She goes one way and her interlocutor p2 goes another. This would be for p1 to hold (or admit) CP(:Bh). Our speakers p1 and p2 are in different legitimate contexts, and p1 recognizes this. On the other hand, p1 might hold that p2 is not within her rights as a speaker: she is at least locally incompetent. As far as p1 is concerned, p2 has either misperceived h’s head, or she misunderstands the meaning of the word ‘‘bald’’, or she is mistaken about the comparison class, or has some other sort of cognitive shortcoming. In this case, p1 maintains that h is not a borderline case of the predicate ‘‘bald’’: she holds :CP(:Bh). Since she also holds Bh, then by the above definition, p1 thus holds DET(Bh): it is e-determinate that h is bald. Further queries can reveal what sort of attitude p1 has toward p2. The present resolution of (what passes for) higher-order vagueness has it that statements in the form CP(F) and CP(:F) are themselves vague and judgmentdependent. Their own truth-values can vary from context to context—from sharpening to sharpening. In the terminology of Lewis [1979], the score in a conversational context C not only contains information about which borderline cases have been decided (in C ), it also contains information about allowable shifts to contexts incompatible with C. That is, the allowed ‘‘jumps’’ are themselves contextually determined. Thus, on the present account of higher-order vagueness, there need not be a sharp distinction between the two perspectives sketched just above. That is, it

13 See also Gaifman [2005]. In the scenarios presented in the previous chapters (and below), we insisted that the members of a conversation achieve consensus on any verdicts they assert. That assumption is not in effect here. Moreover, here we do not assume that the participants in the conversation are all competent speakers of the language, nor that each agrees that the other is competent. There is no need to figure out if the two of them are in the same context or not.

154

Refinements and Extensions I

may be indeterminate whether CP(:Bh), and thus indeterminate whether p2’s assertion is a competent one.

7.3. Penumbral Connections with Competence For the time being we will only deal with formulas in the form CP(F) in which F does not contain any instances of the ‘‘CP’’ operator. This amounts to a restriction to what passes for second-order vagueness. On the present view, the vagueness of a predicate B is due to its responsedependence or, better, judgment-dependence. So it follows that if Ba holds, then it is possible for someone to competently judge that Ba. Let F ¼ hW,M i be a frame, let N [ W, and let s be a variable assignment. One might think that (Ba ! CP(Ba) ) should be true, or at least forced, at the base of every acceptable frame. However, this is not quite right. We might have that CP(Ba) is itself indeterminate. That is, Ba might be a borderline case of CP. If Ba is also indeterminate in the partial interpretation in question, then (Ba ! CP(Ba)) is indeterminate. Moreover, if there is a ‘‘branch’’ on which CP(Ba) and Ba remain indeterminate, then (Ba ! CP(Ba) ) will not be forced. At the very least, however, we should require that if B is vague, then (Ba ! CP(Ba) ) is weakly forced at every interpretation. It is never false. In other words, it should never be the case that Ba is true and CP(Ba) false. We can do better than this in formulating the proper penumbral connection in the object language. We should have that in each partial interpretation, the extension of Bx is a subset of the extension of CP(Bx), and the anti-extension of Bx is a subset of the anti-extension of CP(:Bx). Recall, from Ch. 4, x1 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. Our penumbral connection is that if B is vague, then Vx(Bx ) CP(Bx) ) should hold at the base of each acceptable frame F. One can include an object b in the extension of B only if it is competent to judge that Bb. Sounds reasonable. We assume throughout that if B is vague, then so is its (strong) negation. So if B is vague, then Vx(:Bx ) CP(:Bx) ) must also hold at the base of each acceptable frame. Notice that the scheme (F ) CP(F) ) need not hold in general. In particular, the scheme can fail for predicates that are not judgment-dependent. Let P be a precise predicate, perhaps a natural-kind term or a mathematical predicate of natural numbers. Global anti-realism aside, there might very well be objects t such that Pt holds, but that we have no even mildly reliable means to ascertain that Pt. In such a case, it would be epistemically irresponsible to assert that Pt. One who knows the meaning of the terms and all available non-linguistic information is still not in position to assert Pt. In other words, :CP(Pt). For similar reasons, an epistemicist would reject our two penumbral connections, Vx(Bx ) CP(Bx) ) and Vx(:Bx ) CP(:Bx) ). Let T be a predicate for ‘‘tall’’, and suppose that a man h is in fact tall but is tenuously close to the border

Refinements and Extensions I

155

between tall and not-tall, say a few angstroms away. Then Th. But a judgment that Th would violate Williamson’s [1994] margin for error principles that govern vague predicates. When our knowledge is inexact, one has no business making judgments that are so close to being wrong (see Williamson 1994: x8.3). So for the epistemicist, :CP(Th). Roy Sorenson (2001: e.g. Ch. 3, see also 1988) holds that there are some false propositions that anyone who understands the language must hold. One example has the form: if a person of height m meters is short, then someone of height m þ .000001 meters is short. For most values of m, this is true, but it is false for a few values. Still, we must hold all of them. For such ‘‘forced analytical errors’’ or ‘‘blindspots’’, we have :F&:CP(:F). I digress. The present account is not epistemicist. Let B be a vague predicate. Then, as above, we have (Ba ) CP(Ba) ) and (:Ba ) CP(:Ba) ) as penumbral connections. The contrapositive of the latter involving the ‘‘intuitionistic’’ negation defined in x2 of Ch. 4 is CP(:Ba) ) :Ba. This entails :CP(:Ba) )  :Ba. In words, if it is true in a partial interpretation N that it is incompetent to deny that Ba, then there is no sharpening of N in the frame in which a occurs in the anti-extension of B. This seems utterly correct. We might say more. A stronger penumbral connection is: :CP(:Ba) ) Ba. That is, if it is incompetent to deny that Ba, then a is in the extension of B. On the present usage of the terms, this seems correct as well, and so we add another penumbral connection:14 Vx(:CP(:Bx) ) Bx). As above, it is not the case that (:CP(:F) ) F) holds (or should hold) in general. If F is precise, then it very well may be the case that it is incompetent to deny F even when F is false. Just above, we defined the determinacy operator: DET(F)  F&:CP(:F) ). Given our three-valued semantics, we do not have that DET(F) ! F always holds. If both CP(:F) and F are indeterminate, then DET(F) will be 14 This penumbral connection does not entail the previous one. The contrapositive of :CP(:Ba) ) Ba is Ba ) :CP(:Ba). From this, we can infer :Ba ) :CP(:Ba). That is, if a is in the anti-extension of B, then there is no sharpening in which it is incompetent to deny Ba. It does not follow from the consequent that it is competent to deny Ba. The incomparability of our penumbral connections is related to the fact that our system has, in effect, two negations. A sentence in the form :Ba says that a lies in the anti-extension of B, while Ba is the weaker statement that a is not in the extension of B in any sharpening. We saw in the previous chapter that standard, classical inferences hold for the weaker negation.

156

Refinements and Extensions I

indeterminate, as will the material implication in question. However, it is immediate that DET(F) ! F is weakly forced and that DET(F) ) F holds at every partial interpretation at every frame. In x4 of the previous chapter, we defined DET(F) to hold at a partial interpretation in a frame if F is forced at the base of the frame. The result was that DET(F) is sharp—either true or false at every partial interpretation in every frame—and that DET(F) ! F and thus DET(F) ) F hold everywhere. So in that system, the connection between DET(F) and F is an immediate consequence of the overall structural features of the model theory. Here, however, we ‘‘float’’ both F and CP(F) (if both are vague), allowing their extensions (and anti-extensions) to vary with the judgments of competent speakers. So the connections between them have to be built in as explicit penumbral connections. Many of the ordinary penumbral connections delimited in previous chapters can be reformulated with the CP operator. Suppose, for example, we have a predicate R for ‘‘red’’ and a predicate O for ‘‘orange’’, and are discussing a series of patches running from red to orange. The patches in the middle are in the borderline area. Each of them may be judged red and each may be judged orange, but none can be judged red and orange. In symbols, 9x(CP(Rx)&CP(Ox))&Vx(:CP(Rx&Ox) ). In light of the above penumbral connections, the second conjunct prevents any partial interpretations in which the extensions of R and O overlap. This is another case in which competent judgment is not closed under conjunction. Recall, once more, 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 not to have P. Assume that P is tolerant, and let Rxy say that x and y differ only marginally (allowing that marginal difference may itself be vague). The principle of tolerance has a direct formulation: VxVy(Rxy ) (:CP(Px&:Py) ) ). This sentence should be forced at the base of those partial interpretations that model contexts in which tolerance is in force.15 Assume that a frame F represents a sorites series for a vague predicate P. That is there is a sequence of objects a0, . . . , an in the domain, such that Pa0, :Pan, and for each i < n, Raiaiþ1 all hold at the base of F. Suppose that the principle of tolerance, as formulated here, also holds at the base, as do the above penumbral connections concerning CP. Then there is no partial interpretation in F in which 15 If we assume VxVy(Rxy ) Ryx), as seems reasonable, then it follows that VxVy(Rxy ) (:CP(Py&:Px) ) ).

Refinements and Extensions I

157

P is sharp. In other words, the present ‘‘object language’’ formulation of tolerance, together with other penumbral connections, entail the systematic formulations of those connections in the previous chapter. This is perhaps another advantage of the present program of subjecting competence to the present treatment of vagueness. While on the subject of penumbral connections with the CP operator, let us illustrate the present framework and model theory with a forced march style sorites exercise, in which a vague predicate and statements about competent judgments interact. Consider one last time—I promise—our series of 2,000 men, ranging from Yul Brynner who, you will recall, has no hair whatsoever, to Jerry Garcia in his prime. For each natural number n < 2,000, let mn be the nth man in the series. So ‘‘man 297 is bald’’ is symbolized Bm297. Reassemble our competent group of speakers to discuss various matters related to the men in our series. We will ask the conversationalists a series of questions and, following the lead of Ch. 1, we insist that they answer by consensus. For convenience, we allow them only two answers to each question, ‘‘yes’’ and ‘‘no’’. Unlike the simple forced march from Ch. 1, here we assume that the conversationalists at least implicitly accept the present open-texture account of vagueness.16 As above, this is to make sure that their responses on questions concerning what is a competent response can differ from their responses on the ‘‘first-order’’ questions. The plan is to put the following questions to the conversationalists, coming from both ends of the series: (A) Is man i bald? Bmi? (B1) Is someone who judges man i to be bald thereby incompetent? :CP(Bmi)? (B2) Is someone who judges man i to be not bald thereby incompetent? :CP(:Bmi)? We begin by asking questions from list (A) in order, starting with the first man, Yul Brynner: Is man 1 bald? Is man 2 bald? . . . Call this the (Aþ) series. After a while we stop this and start asking questions from list (A) coming from the other end: Is man 2,000 ( Jerry Garcia) bald? Is man 1,999 bald? . . . This is the (A) series. After some of these, we switch to questions from the (B1) list, starting at the beginning of the list (B1þ), and then starting from the end (B1). After a bit of that, we switch to questions from the (B2) list, yielding the (B2 þ ) and (B2) 16 I could subsume this under the assumption that the conversationalists are themselves competent. Indeed, if they are competent speakers of the language, then they should know how vague terms are deployed, and the present account is the correct account (in case you have not noticed this already). However, I am not so pompous as that. More seriously, competent speakers need not be aware, even implicitly, of the correct account of vagueness, just as they need not be aware of the correct grammar. Otherwise, there would be preciously few competent speakers of English, maybe none at all (or at most one). Moreover, the present notion of CP is a technical part of the present account; it need not match the English word ‘‘competent’’. We assume that the conversationalists deploy the CP notion as it is presented here.

158

Refinements and Extensions I

series. Then we go back and continue the (Aþ) series and then the (A) series from where we left off, etc. As noted in Ch. 1, to develop the account of conversational score in full generality, we will need a paraconsistent logic to handle situations when contradictory items are placed on the conversational record unnoticed. In the forced march sorites from Ch. 1, it was easy for the conversationalists to maintain consistency, tolerance, and any other penumbral connections in play. The situation was arranged for just this purpose. In the present case, it will not be quite as easy to maintain consistency, but the situation is still artificial enough that we can manage. Assume that tolerance applies to the three predicates invoked in these lists, Bx, CP(Bx) and CP(:Bx). A few hairs cannot make a difference between bald and not bald, nor can a few hairs make a difference concerning whether a certain judgment about baldness or non-baldness is competent or incompetent (on the present option concerning so-called higher-order vagueness). The answers to questions from the (Aþ) series will, of course, be ‘‘yes’’ for a while at the beginning of the exercise. By assumption, the speakers are all competent and the first men in the series are e-determinately bald. Similarly, the answers to the questions from the (A) list will be ‘‘no’’ for a while. Questions from the (B1þ) list will be answered ‘‘no’’ at first, and answers from the (B1) list will start ‘‘yes’’, with the opposite responses from the corresponding (B2) lists. These uniform answers will continue as they enter the borderline areas for the three properties. On the present account of vagueness, this is to be expected. Borderline cases are those for which competent speakers can go either way (in some circumstances), and as they first enter the various borderline areas, they go one way rather than the other for a while, prompted by their previous responses to similar questions. As in Ch. 1, they need not be aware that they can go either way as they make the various judgments on those particular cases. This puts propositions such as Bm917, :Bm1,266, CP(Bm1,012), :CP(:Bm1,070) on the conversational record (assuming that those are indeed borderline cases of the indicated properties). Let us briefly review the outcome of the simpler forced march described in Ch. 1. There the scenario continued until the participants were ‘‘forced’’ to violate tolerance, after a fashion. As competent speakers, they cannot and so will not go through the entire series, and call Jerry Garcia bald. So at some point, they call a man not bald even though they had just judged his predecessor to be bald. Say that this ‘‘jump’’ happens with man 982. I claimed that at that point, some items come off the conversational score, namely Bm981, Bm980, and perhaps a few more. Tolerance is maintained in the conversation by retracting certain of the previous judgments. So long as the retracted cases are all in the borderline area, all is well concerning meaning and non-linguistic fact (and externally determined contextual factors). The participants remain competent. In sum, what we may call the ‘‘normal’’ march through the series adds items to the score

Refinements and Extensions I

159

as they make judgments (once they enter the borderline area). The ‘‘extraordinary’’ jumps remove items. In the present scenario, there are six forced march sorites dovetailed together. Apparent violations of tolerance can occur in any of them. In part, which series has the first jump depends on how far up (or down) they go on each before switching to another series. Suppose that the first jump occurs in the (B1þ) series. In particular, suppose that at one point, the conversationalists say that a judgment that man 994 is bald is competent (and so CP(Bm994) is on the record), but a minute later they say that a judgment that man 995 is bald is incompetent: :CP(Bm995). This pair of judgments violates tolerance. So analogous with the situation from Ch. 1, the earlier one is removed from the score, as are a few of its predecessors (CP(Bm993), CP(Bm992), etc.). At that point, we could query the group about those cases. Indeed, it would be natural to pick up the B1 series at that point, since it is a penumbral connection that if :CP(Bm995) then :CP(Bmn) for any n > 995. As in x5 of Ch. 1 above, I predict that the conversationalists would go backward and forward through (part of ) the borderline area, displaying what Raffman [1994], [1996] calls backward spread. The B1þ jump that occurs with CP(Bm995) might result in the removal of some items put on the conversational record from the other series. Suppose, for example, that prior to this jump in the B1þ series, the group had judged man 990 to be bald, so that Bm990 is on the record (presumably from the Aþ series). When they realize that they have just committed (for the time being at least) to :CP(Bm995), they may retract the judgment that Bm990, because they find that man 990 is too close to one that cannot competently be judged bald, i.e. man 995. If it mattered to us, we could explicitly put the question of Bm990 to them again. That is, we would remind them that they have just judged :CP(Bm995) and then ask them about Bm990, picking up the A series at that point.17 It is manifest that one cannot simultaneously and self-consciously maintain :CP(Bmn) and Bmp if n p. This is not exactly a contradiction, but clearly some norms of meaning and assertion are violated. A person who says this has declared his own judgment incompetent. The pair :CP(Bmnþ1), Bmn would violate tolerance (since anyone who holds Bmn is committed to CP(Bmn) ). It is more or less an empirical matter to determine which other pairs :CP(Bmn), Bmp can be competently maintained. In the above, highly artificial situation, tolerance will come into play before anything else does, but, in general, tolerance is not the only penumbral connection that gives rise to situations in which items are removed from the conversational record. First, the participants in a conversation might contradict themselves. They might agree to Bm870 when :Bm870 is already on the conversational score. In other 17 As noted in Ch. 1, whether a given item is on the conversational score is itself a vague matter. Usually, the borderline cases of this notion do not interfere with a conversation. If it ever mattered whether a given proposition is presupposed, the participants in the conversation could just ask about it, and make its status explicit.

160

Refinements and Extensions I

words, the Aþ and A series may overlap before they notice. Similarly, they may assert :CP(Bm1,022) when CP(Bm1,022) is on the score. These declarations alone do not undermine their competence so long as man 870 is a borderline case of baldness and the sentence :Bm1,022 is a borderline case of CP. In such cases, the prior item is removed from the score, along with any other sentences on the score that would violate tolerance in light of the new pronouncement. This follows the principles of accommodation in Lewis [1979] (see x3 of Ch. 1 above). Other things equal, what is said will be construed as correct if possible, updating the score as needed. In these cases, the score is updated by removing items from it. Second, they may assert Bm987 when :CP(Bm987) is on the score, or vice versa. In this case, the Aþ and B1 lists come into conflict with the penumbral connection that for a simple predication F involving a vague predicate, F entails CP(F). Here again, the principles of accommodation require that the earlier item be removed from the conversational score, along with any other items that conflict with the tolerance of the recent pronouncement. In general, when any of the penumbral connections is violated, the principles of accommodation require that the earlier, conflicting pronouncements be removed from the record. Consistency, tolerance, and penumbral connections are all enforced in this way. So long as the removed items are in the borderline area—and they will be so long as our conversationalists remain competent—all is well with the meaning of the vague terms and the non-linguistic facts. Such is open-texture. Of course, the original forced march from Ch. 1 and the present dovetailed forced march are highly artificial situations. Some thinkers may think that even if I have correctly described the behaviors in these bizarre situations, we should not conclude much about how vague predicates function in real life. I do not know how to respond, other than to put the overall view on the table and let its merits speak for themselves. I submit that the general principles invoked in the present forced marches apply in general. During a conversation, items are placed on the score when borderline cases of vague predicates are decided, one way or the other, and items are removed from the score when things said in the conversation conflict with what is on the record. The conflict can come from logic or from a penumbral connection which, presumably, always applies, or conflict can come from tolerance if it is in force. In the model theory, the partial interpretations in a frame represent the possible combinations of atomic formulas and statements of competence that can be simultaneously true, given whatever penumbral connections are in force. The sentences true in a given partial interpretation of a frame include e-determinate truths and the items temporarily on the conversational score. In the semantics, these combinations determine which formulas are forced at each partial interpretation of the frame, including its base. And this, in turn, fixes the consequence relation, as in the previous chapter. This relation, I submit, is a model of correct reasoning with vague predicates.

Refinements and Extensions I

161

7.4. Up We Go, Again The main difference between the present model theory and the more simplified one developed in the previous chapter concerns the formal, object-language analog of e-determinacy. Let F be a formula that does not contain the CP operator. In the previous chapter, we defined the formula DET(F) to be true at a partial interpretation in a frame F under a variable assignment if and only if F is forced at the base of F under the variable assignment. So defined, DET(F) is always sharp, in the sense that under a fixed variable assignment, each formula in that form is either true or false, never indeterminate, and it has the same truthvalue in every partial interpretation in the frame. In the present model theory, DET(F) is defined as F&:CP(:F). If F itself is vague, then CP(F) is treated as a vague atomic predicate, subject to open-texture. In the model theory, the truthvalues of such formulas can vary from partial interpretation to partial interpretation: sometimes true, sometimes false, sometimes indeterminate. This represents the thesis that competent pronouncements concerning competence (and thus e-determinacy) can vary from conversation to conversation, depending on the calls of competent speakers. So much for what passes for second-order vagueness. I conclude this long section with a brief account of the model theory for what passes for third- and even higher-order vagueness. Again, the phenomena in the area of second-order vagueness concern the status of predicates such as ‘‘competent user of the word ‘bald’ ’’ (see x4 above). We explored three options. On two of these, the predicate in question is sharp (or at least not vague) and on the third it is vague. The analog of third-order vagueness similarly turns on the status of predicates such as ‘‘competent user of the phrase ‘competent user of the word ‘‘bald’’ ’ ’’. And, in theory, we have the same options as we did at second-order. At the analog of third-order vagueness, the first two options each require two different notions of (e-)determinacy, a vague one corresponding to the competency with expressions such as ‘‘bald’’ (at second-order, so to speak), and a sharp (or at least non-vague) one corresponding to propositions that concern the competency of users of the word ‘‘competent’’ (third-order). The latter concerns the status of formulas that themselves contain the CP operator. It is straightforward to represent these two options in the model theory. First, we maintain the restriction of the CP operator to formulas that do not themselves contain any instances of the CP operator. On the options in question, competence concerning what is competent is not vague, and so we do not need to invoke our model theory to account for it. For clarity of exposition, we rename the determinacy operator DET1. That is, DET1(F) is an abbreviation for F&:CP(:F). Let B be a vague predicate, say for baldness. In a frame F, the extension and anti-extension of CP(Bx) at the base represent the e-determinate cases of ‘‘competent user of the predicate B’’. The extension consists of those cases that

162

Refinements and Extensions I

a competent speaker must apply the predicate to, and the anti-extension consists of those that a competent speaker is not allowed to apply the predicate to. The rest of the cases are the borderline cases of ‘‘competent user of B’’. In various circumstances, competent users can go either way on those cases, via open-texture. We use an operator like the one defined in the previous chapter for determinacy concerning competence, here assumed sharp (or at least not vague). Let F be a formula, N a partial interpretation in frame F, and s a variable assignment. Say that DET2(F) is true at N under s if F is forced at the base of F. As in the previous chapter, each sentence in the form DET2(F) is true or false, never indeterminate, at each partial interpretation, and it has the same truth-value at each. The third option for the analog of third-order vagueness is to maintain that predicates such as ‘‘competent user of ‘competent user of the word ‘‘bald’’ ’ ’’ are genuinely vague. And, of course, the vagueness of the predicates in question is to be understood in terms of open-texture, since that is what vagueness comes to. It is a simple matter to handle this formally. We drop the new DET2 operator, and allow formulas in the form CP(F) in which the formula F contains instances of the CP operator. If we want to limit things to (what passes for) the third-order vagueness, we would insist that no occurrence of CP that has an occurrence of CP in its scope should itself be in the scope of a CP operator in F. The first section of this chapter discusses the prima facie inevitability of having the run to higher orders end with sharp (or at least non-vague) predicates. Suppose that the first sharp ‘‘order’’ is the nth. The formal analog of this would be a restriction to formulas in which the CP operator is nested at most n1 deep. After that, we would use a DET operator like that of the previous chapter. If i < n, then DETi(F) is :CP(:F); DETn(F) would be ‘‘F is forced at the base’’. The arguments in the first section of this chapter notwithstanding, there is something pleasing about the possibility of not cutting off the run to these analogs of higher- and higher-order vagueness. Perhaps all the predicates in question are vague. We can (begin) to model this in the present framework by removing all restrictions on the CP operator. It can be nested as deep as one desires. We would thus have only one notion of (e-)determinacy, for all formulas: DET(F) if and only if F&:CP(:F). On this plan, determinacy is always tied to the CP predicate. Thus, the base of each frame plays no special role. For simplicity, we can just stipulate that at the base, the extension and anti-extension of every vague predicate, including those in the form CP(F), is empty. In that partial interpretation, every object is a borderline case of every vague predicate, and it is borderline competent to assert any sentence. Of course, this degenerate partial interpretation does not represent a competent use of the predicates. A model-theoretic structure such as this is nicely consonant with the foregoing thesis that vagueness is due to response-dependence. On an extreme version of this view, if there are no responses, then there are no truth-values. The base of

Refinements and Extensions I

163

a frame would represent such a situation. Notice also that at the base, there are no responses, and thus no truths or falsehoods, concerning what is and what is not a competent use. We might call this extreme response-dependence the ‘‘Copenhagen view of vagueness’’. I won’t pursue it further. 8. CONCLUSION: WHERE HAVE WE GOT TO? Recall the quote from Fine [1975: x5] that serves as an epigraph to this chapter: ‘‘Our intuitions seem to run out after the second or third order of vagueness. Perhaps this is because our understanding of vague language is, to some extent, confused. One sees blurred boundaries, not clear boundaries to boundaries.’’ A survey of the literature indicates that some philosophers’ intuitions do not even make it to the second or third order. Along similar lines, we had several occasions to paraphrase Raffman [1994: 41 n. 1]: there is tolerance between baldness ‘‘and any other category—even a ‘borderline’ category’’. And 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’’. In x1 above, I called this the ‘‘NSB thesis’’, for ‘‘no sharp boundaries’’ (see Sainsbury 1990). For what it is worth, I confess to sharing these intuitions, especially the thought expressed in the last sentence of the Fine passage. But, as noted at the outset, the forced march is a tough nut to crack. And it is not easy to provide a coherent, rigorous, and interesting formulation of the NSB thesis. Strictly speaking, the foregoing account satisfies Fine’s intuitions, and thus a variation on the NSB thesis. The unhelpful reason is my endorsement of the second epigraph to this chapter, from Williamson [1999: 140]: ‘‘it may be misleading to think of higher-order vagueness in a as a species of vagueness in a. Higher-order vagueness in a is first-order vagueness in certain sentences containing a.’’ On the present view, there is no higher-order vagueness, strictly so-called. Nevertheless, the account here does have (or may have) analogs to higherorder vagueness, namely vagueness concerning the competent use of certain predicates and phrases. The higher and higher orders correspond to deeper and deeper embeddings of ‘‘competent user of ‘competent user of ‘‘competent user of . . . . . . ’’ ’ ’’. Perhaps this is what those who talk about higher-order vagueness are on to. The argument of the first section can be adapted to this surrogate notion. Suppose we have before us a finite sorites series. Assume that the various surrogate higher-order predicates are all different from each other and that the penumbral connections canvassed in x7.3 above all hold. In particular, it is not possible for a vague sentence F and :CP(F), the statement that it is not competent to assert F, both to be true in the same context. This would be to declare oneself to be incompetent. Can one competently do that? Given all this, the reasoning from x1 above then holds. There is a limit to how high the

164

Refinements and Extensions I

surrogate orders go—if there is to be an indeterminate instance of each surrogate higher-order predicate. And there will be sharp boundaries somewhere in the series. But, as noted above, these presuppositions can be resisted. It is natural, I submit, to demur from the penumbral connections. It seems that it is not possible to maintain consistent judgments that satisfy all of them, all at once, without violating tolerance. But this is not a big deal. Communication surely does not require that much consistency. We can always enforce consistency on the conversational record, perhaps artificially. If a group of conversationalists say F and, a bit later, say :CP(F), then the former is removed from the conversational score. With an ordinary (first-order) sorites series, it is not possible for competent subjects to judge every member of the series and satisfy tolerance (duh). Similarly, it is not possible for them to make a lot of judgments about what is and what is not a competent judgment, all at once. Alternately, it does not do much damage to the overall philosophical view, or the intuitions that underlie it, to concede that the run up the surrogate orders terminates (somewhere) with a sharp boundary. We can reject tolerance at the higher orders. As we saw above, a few times, to talk of responses to—or judgments on—the horribly complex predicates, we have to idealize on the limitations of ordinary human attention span, and, ultimately, human life span. Real humans simply won’t give responses to most of these questions, since they won’t live long enough to hear the question. If it makes sense to talk about the judgments of highly idealized beings, there is no damage to intuitions if we hold that the responses show no variation and open-texture. That is, there is no intuitive reason for tolerance to hold of such idealized responders. Indeed, if our idealized speakers respect the penumbral connections concerning competent judgments, their responses will have to show a sharp boundary in some surrogate higher-order series (if not the first-order one). Certainly this is so for the surrogate for predicates such as absolutely-determinate bald and their ilk. But this says little about how ordinary predicates, such as ‘‘bald’’, and even ‘‘competent user of the language’’ are deployed by us ordinary, non-idealized speakers of natural languages.

6 Refinements and Extensions II: Objects, Identity, and Abstracts . . . I find it hard to disagree with the proposition that there are [vague] objects. I take the Australian outback to be an example of one. David Lewis denies that ‘‘there’s this one thing, the outback, with imprecise borders’’ [1986: 212]. But this view seems to go against common sense. There is such an object. I have driven across it. And it simply is a fact about the outback that it has a fuzzy boundary. Copeland [1994: 83]

1. WHITHER VAGUE OBJECTS? The present study, so far, focuses exclusively on vague predicates such as ‘‘bald’’, ‘‘tall’’, and, with the last chapter, predicates such as ‘‘determinately bald’’ and ‘‘borderline bald’’. Intuitively, what makes a predicate vague is that there is some indeterminacy over whether it applies to some objects: vague predicates have borderline cases. Some aggregates of sand are borderline heaps, some potential NBA players are borderline tall (for an NBA player), some men are borderline bald; and perhaps some men are borderline-borderline bald. At least prima facie, there also seem to be vague objects. In this chapter, we look at some variations on this theme, and then extend the model theory to handle them.

1.1. Blurry Boundaries Geographic entities such as mountains and seas are paradigm cases of one sort of vague object. A standard example is Mount Everest. Is there a fact of the matter where the mountain begins? There seem to be chunks of real estate that are borderline parts of Mount Everest. Yet, it is one mountain; the locution ‘‘Mount Everest’’ is a singular term that at least purports to denote an object—one object. A cloud is perhaps the clearest instance of a vague object. In what sense is it determinate, of each molecule of water, whether it is part of the cloud, or not?1 1 I have benefited considerably from conversations with Michael Morreau, and from his [2002] illuminating account of vagueness in the boundaries of objects, in pretty much the present sense.

166

Refinements and Extensions II

Arguably, most physical objects seem to exhibit at least some vagueness at their boundaries; perhaps they all do. Consider a piece of dried-up skin that is loosely attached to my foot, but will fall off shortly (if I don’t pick it off first). Assume that it still contains a few live cells. Is that piece of skin part of me? Or part of my body? Or part of my foot? Consider a piece of plastic that is hanging down from, and barely attached to, the cracked headlight of a car. Is that part of the car? Most natural and artificial objects gain and lose parts all the time, sometimes at the microscopic level. Cells die and fall off organisms and are replaced by others, as they ingest their food. Molecules leave and join a picnic table. At a more macroscopic level, people have haircuts and organ transplants. At what point do the various objects become or stop becoming a part of the object in question? This is not to mention quantum effects at the sub-microscopic level. We are told that there is indeterminacy—no fact of the matter—concerning the position and/or the momentum of each subatomic particle. Suppose that we have reasonably precise information about the momentum of a given particle. Then its location is indeterminate. We are told that it is not just a matter of our inability to locate the particle. It simply has no determinate location—to the extent that it has determinate momentum. Since, ultimately, a given macroscopic physical object O is composed entirely of subatomic particles, there is at least potential indeterminacy as to what, exactly, the borders of O are. Since the physics is hard to interpret, and perhaps contentious, we will stick to the more ordinary examples. Sorites is not hard to come by. Consider a series of 1,000,000,000 points on the surface of the earth in a line, each one millimeter from its neighbors. One end of the series is at the summit of Mount Everest and the other is 1,000 kilometers away. Clearly, the first umpteen points are on the mountain and the last umpteen are not on the mountain. Prima facie, a principle of tolerance applies here. If two points are only a millimeter apart, then how can it be the case that one is part of the mountain and one is not?

1.2. Counting Objects A closely related matter concerns the identity, or count, of certain objects. Suppose we look at the sky and see two distinct clouds separated by about 100 meters. Slowly they get blown together, to form a single cloud. At what point is there a single cloud there, and not two distinct clouds? Consider, for example, the point at which the two masses touch each other at several places, overlapping over a small part of space, say 6 per cent of each. Is that one cloud or two? Or consider Scotland’s famous hills. Suppose there is a mass of land with two peaks, separated by a few hundred feet, and with the land in between rather high up. Is there one hill there, with a flowing peak, or are there two distinct hills? This phenomena of ‘‘how many’’ is rather general (although perhaps not quite as general as the phenomena of indeterminate or blurry boundaries). Consider an amoeba in the process of mitosis. At what point are there two, and not one?

Refinements and Extensions II

167

A more important example, perhaps, revolves around the abortion debate. At what point does a sperm-egg pair, or a fetus, or whatever, become a separate person? When are there two organisms—mother and child—or two human beings, or two persons (if those are different questions), and not just one?2 Some say that the bifurcation occurs at conception, some at implantation, some at ensoulment, some at viability, some at birth. Notice that with the possible exception of ensoulment, which depends on some contentious metaphysics, pronouncements like this do not remove all the vagueness. Conception, implantation, viability, and birth are themselves vague events. There does not seem to be a determinate instant at which they occur. It is a continuing theme of this book that vagueness can be, and routinely is, lessened in the course of conversation, but it is rarely, if ever, eliminated altogether. Again, a sorites is not hard to come by. Consider a series of moments, each a single millisecond from its neighbors. At the start we have a sperm and egg separated by a few centimeters. At the end we have a 1-year-old child who developed from that pair. Is there a first instant in the series when the pair becomes a determinate zygote, a first instant at which it is determinately implanted, a first instant at which it is determinately viable, and a first instant at which it is determinately born? Most importantly, at what point do we have two human beings in the scenario, and not just one? Along similar lines, we are told that death is a process, and typically does not occur in an instant. So we can envision a morbid sorites series, consisting of moments of time separated by a millisecond. It begins with a determinately alive, but dying, person and a loved one nearby, so that it is determinately true that there are two living human beings. The series continues until the dying one has determinately passed away, so there is only one living human being left in the scene. We can then ask, of each millisecond in the series, how many live human beings it contains. As with the beginning of life, certain contentious pronouncements propose to lessen the vagueness, citing (permanent) cessation of respiration, (permanent) cessation of brain function (or higher brain function), irreversible coma, etc. And, again, there are serious moral issues involved. Prima facie, it is not immoral to use the heart of a person who has died in a transplant. Since time is of the essence, we need a criterion to determine who, exactly, has died and who hasn’t.

1.3. Vagueness in Abstraction So far, then, we have two possible sources of vagueness concerning objects. One is the ‘‘part of ’’ relation. There are, or at least seems to be, clear and borderline 2 A pro-life advertisement begins with the statement that those who favor abortion on demand say that a woman has a right to control her own body, and then adds, ‘‘Aren’t they forgetting someone?’’ Although this question begs the question, it nicely highlights the present issue. At what point is there another ‘‘someone’’ to take into account?

168

Refinements and Extensions II

cases of one object being a part of another. Is the piece of dried skin part of the foot or not? The other source, involving the clouds, hills, amoebae, and human beings, concerns individuation of the relevant objects. When do we have two and not one object (or organism) of a given type? Of course, the two sources are closely interrelated and may, at root, be identical. Whether the sky contains one cloud or two depends on what the (spatial) boundaries of a cloud are. Whether a given moment in a fetal or morbid sorites contains one person or two depends on the temporal boundary of ‘‘live human being’’. Nevertheless, it will prove useful to keep these sources separate, noting the danger of treating the same issue twice. To labor a truism, the issues surrounding physical boundaries concern physical objects. The count issues also concern physical objects. Another source of vagueness arises with objects introduced via a process of abstraction (or nominalization). An abstraction principle is any proposition in the form: (ABS) VaVb(S(a) ¼ S(b)  E(a,b) ), where a and b are variables of a given type (typically individual objects or properties/sets of objects), S is a higher-order operator, denoting a function from items of the given type to objects, and E is a relation over items of the given type. In what follows, I will usually omit the initial universal quantifiers. In his development of logicism, Gottlob Frege (1884, 1893) employed three abstraction principles. One of them, used for illustration, comes from geometry: The direction of l1 is identical to the direction of l2 if and only if l1 is parallel (or identical) to l2. Presumably, Frege intended the ‘‘direction of ’’ operator to apply primarily to Euclidean (or otherwise abstract) lines. Otherwise issues of vagueness would arise. Is there a determinate and sharp ‘‘parallel’’ relation between physically drawn line tokens? Stay tuned. A second Fregean abstraction principle was dubbed N ¼ by Crispin Wright [1983] and is now called Hume’s principle: (#Fx ¼ #Gx )  (F  G ), where ‘‘#’’ is the cardinal-number operator and ‘‘F  G ’’ is an abbreviation of the second-order statement that there is a one-to-one relation mapping the Fs onto the Gs. Hume’s principle states that the number of a property F is identical to the number of a property G if and only if the Fs are equinumerous with the Gs. The third instance is Frege’s ill-fated Basic Law V: (EF ¼ EG )  Vx(Fx  Gx), stating that for all properties F, G, the extension of F is identical to the extension of G, if and only if every F is a G and every G is an F. Hume’s principle is consistent, but Basic Law V is not (for the former, see e.g. Boolos [1987]; for the latter, see any treatment of Russell’s paradox).

Refinements and Extensions II

169

In the philosophy of mathematics, there is some dispute concerning the status of those abstraction principles that are true (or true over a restricted range). The neo-logicist holds that acceptable abstraction principles are akin to implicit definitions. If certain conditions are met, then a given abstraction principle is true by stipulation (see e.g. Hale and Wright 2001). Others hold that true acceptable abstraction principles are theorems derived from explicit definitions. Whatever their status, abstraction principles are common fare in mathematics. Much of abstract algebra revolves around them. Notice that it follows from (ABS) that the embedded relation E is an equivalence: reflexive, symmetric, and transitive. For example, S(a) ¼ S(a) is a truth of logic, an axiom of identity. So it follows from (ABS) that E(a,a). So wherever the abstraction operator S applies, the relation E is reflexive. The symmetry and transitivity of E similarly follow from the symmetry and transitivity of identity, respectively. Most abstraction principles are foundationally unproblematic. The mathematician can simply identify the abstracts with the equivalence classes. There is a potential problem only when the equivalence classes are not sets. The serenity and clarity of pure mathematics is compromised when the messy physical universe gets involved, as we try to apply the abstraction principles to the physical world (or ‘‘the real world’’, as some might say). Inevitably, vagueness rears its head. Consider a room that contains only Yul Brynner and a borderline case of bald man, and let T be the property of being a bald man in that room.3 What is the number T ? Is it 1, or 2, or is it some other thing, a vague number indeterminate between 1 and 2? What is the mathematics for these vague numbers? In ordinary, non-mathematical discourse, concerning ordinary physical objects, we sometimes invoke what look like abstraction principles (in the form (ABS) ). For example, we speak of the heights of individuals, indicating whether those are the same or different. Harry has the same height as Sarah, but not as Joe. This seems to involve the following: The height of a is the height of b if and only if a and b are of the same height, where a, b are variables ranging over people. Call this the ‘‘Height Principle’’. If the linguistic practice is coherent, then heights are abstract objects, but they are abstracts of physical objects. Let us call such objects quasi-abstract.4 Is the practice of introducing quasi-abstract objects coherent? When we say that the height of a is identical to the height of b, we might mean that they are of exactly the same height. In this sense, the identity is false if a is, say .00001 angstroms taller than b—assuming that differences this small make physical sense. Although we are not usually this strict when talking about heights I am indebted to George Schumm here. According to Charles Parsons [1990], an abstract object is ‘‘quasi-concrete’’ if it has concrete instances. A typical example is the type shared by all tokens of a 12-point Times Roman letter ‘‘e’’. It seems that the present quasi-abstract objects are also quasi-concrete. 3 4

170

Refinements and Extensions II

(perhaps never), this reading gives us the best chance for the relation on the right of the Height Principle—being of the same height—to be an equivalence. Even here, however, things are problematic, due to the aforementioned vagueness concerning the boundaries of physical objects such as human bodies. What are we to make of possible differences so small that quantum indeterminacy is a factor? Well, I promised to waive that. Suppose that two people a, b are identical (molecule for molecule), except that a has a small scab of dead and dying skin cells loosely attached to the top of his (completely bald) head, about to fall off. Prima facie, it is indeterminate whether the scab is part of a or not (see Morreau 2002). If the scab is not part of a, then he is the same height as b, but otherwise he is not, since the scab makes him a bit taller. In any case, when speaking of heights, ordinary speakers of ordinary language are not so strict. When one says that the height of a person a is the same as the height of a person b, she means that they are roughly the same height, depending on the standards that are in effect in the given conversation. Such is the case with most ordinary-language abstractions. Consider the following principles, all in the form (ABS): The height of a is identical to the height of b if and only if one cannot tell them apart when they stand back to back. The weight of a is identical to the weight of b if and only if they balance each other in a pan scale. The income group of a is identical to the income group of b, if and only if a and b are roughly equi-incomed. The color of a patch a is identical to the color of a patch b if and only if they cannot be distinguished by observation. These should all fail as abstractions, since the relations on the right are not equivalence relations. In particular, being indistinguishable by sight, balancing a pan scale, and the like, are not transitive. In each case, one can easily construct a very short sorites series, in which the first item is clearly different (in weight, height, etc.) from the last, but each item in the series bears the relevant relation to its neighbors.5 Since the relations on the right are not transitive, it should not be possible to speak of objects on the left. Identity is transitive, if anything is. Nevertheless, we do speak of heights, weights, income groups, and colors with ease, all the time, and we seem to know what we are doing, if not what we mean. Michael Dummett [1975] argues that such talk is incoherent, due to sorites. It would be best to avoid such conclusions, if possible. The issue here is reminiscent of Peter Unger’s [1975: 65–8] argument that nothing (or hardly anything) is flat. The analogous conclusion here would be that no two distinct people have the same height or the same weight, or that no two people are in the same income group unless their incomes match to the penny. 5 Delia Graff [2001] challenges the non-transitivity in the case of colors. If she is right, then we can drop this example. There are plenty of others.

Refinements and Extensions II

171

As we saw in x3 of Ch. 1, David Lewis [1979] concedes that, strictly speaking, Unger is right about flatness, but there is a looser sense in which such statements as ‘‘Kansas is flat’’ are true enough. Presumably, the same goes for statements concerning the identity of heights, weights, and income groups. It is a strength of the present view of vagueness that it can handle this (supposed) loose sense of truth, and, as we shall soon see, it can handle the notion of a vague quasi-abstract object. At least some instances of the above issue concerning ‘‘how many’’ can be brought under the quasi-abstractionist fold. Define two points in space to be ‘‘co-clouded’’ at a given moment if there is a sizeable and sufficiently continuous and connected mass of cloud-stuff that encloses both at the moment. Consider the following: The cloud of a is identical to the cloud of b if and only if a is co-clouded with b. Of course, this is not to say that clouds are themselves (quasi-)abstract objects. They are concrete, since composed of concrete water molecules. But a resolution to the issue of vague, quasi-abstract objects might work here as well. 2. ENTER OPEN-TEXTURE Back at the beginning of Ch. 1, we had trouble giving a characterization vagueness that is neutral between various theories of vagueness. It is an ironic situation. The main contenders—supervaluationism, fuzzy logic, epistemicism, contextualism—claim to be competing accounts of one and the same phenomenon. Yet they have trouble agreeing on a description of what it is that they are competing accounts of. I will not try to be neutral here, especially since there is controversy over the very issue of whether there are vague objects, or vague identity, and over what it is for an object to be vague. The present view, of course, is that vagueness is tied to open-texture, the thesis that in the borderline region of a vague predicate, competent speakers can (sometimes) go either way. The extensions of vague predicates vary from conversation to conversation, even after the relevant external contextual parameters are fixed. The purpose of this section is to accommodate the foregoing phenomena of vague objects and/or (what passes for) vague identity. The next section indicates the changes in the model theory needed to incorporate the phenomena. To recapitulate, we found three overlapping situations which involve something in the neighborhood of vague objects. The first is where there is, or may be indeterminacy, concerning whether one item is a part of another. The typical case is that of vagueness in the boundaries of physical objects. The second situation involves the individuation or count of various items, such as clouds, hills,

172

Refinements and Extensions II

amoebae, and people. Are there one or two in a given scenario? The third issue concerns quasi-abstract objects, the results of an (attempted) abstraction, where the embedded relation is not an equivalence due to the failure of transitivity.

2.1. The Easy Cases: Fuzzy Boundaries In the foregoing account of vagueness, developed in previous chapters, we focused on monadic predicates, such as ‘‘bald’’ and ‘‘red’’. But this was just a matter of convenience. The philosophical/semantic/model-theoretic account applies directly to vague relations, such as that of being brighter, balder, or smarter. A borderline case of, say, ‘‘smarter’’ would be a pair of people a, b, for whom it is not determinate, in the present sense, whether one is smarter than the other. To paraphrase our definition, borrowed from McGee and McLaughlin [1994: x2], the thoughts and practices of speakers of the language determine conditions of application for the statement ‘‘a is smarter than b’’, and for the statement ‘‘a is not smarter than b’’, and the (the non-linguistic facts) about a and b are such that neither condition is met. From our thesis of open-texture, competent speakers of the language can sometimes go either way in this case without compromising their competence. In practice, the conversational score tracks which borderline cases of this, or any other vague predicate or relation, have been called. Statements are removed from the record when required by consistency, penumbral connection, or tolerance. With formal languages, it is often convenient to think of an n-place relation as a (monadic) property of n-tuples of items on the domain. So construed, the binary relation of ‘‘smarter’’ would correspond to (or be identified with) a property of ordered pairs of people. The foregoing account of vagueness would then be applied directly to this property. Returning to the matter at hand, the relation of ‘‘part of ’’ invoked in the examples involving clouds, seas, people, and, indeed, just about every physical object, is a binary relation. The foregoing account applies without modification to this relation, or, if you wish, to the corresponding property of ordered pairs of objects. Some objects are determinate parts of others, some are determinate non-parts, and some are borderline parts. For example, let x be an arbitrary chunk of land of about a square meter somewhere on planet Earth. Then the thoughts and practices of speakers of our language have determined conditions of application for the statement that ‘‘x is part of Mount Everest’’, and for the statement ‘‘x is not a part of Mount Everest’’. In some cases, the facts concerning the location of x determine that one of these conditions is met, in which case, x is determinately part of, or determinately not part of, the mountain. In other cases, neither condition has been met, and x is a borderline part of the mountain. The same goes for the aforementioned hunk of skin on my foot, about to fall off. If s denotes the hunk and f denotes my foot, then at some point between the

Refinements and Extensions II

173

time when s started to separate and now, when it is about to fall off, s was a borderline part of f. At such a point, a competent speaker would not err if she said it was part of my foot, nor would she err if she denied this. In general, the ‘‘part of ’’ relation forms the central item in mereology, although there are important differences between physical parts and mereological parts. What is suggested here is a vague mereology, or a vague part–whole relationship. Morreau [2002] has shown that there is nothing incoherent about this. In sum, the first type of indeterminacy, concerning whether one item is a part of another, is handled by the present account in stride. No extension or addendum or nuances are needed to handle it. The reason why there is no real problem here (or at least none that is not already shared with the overall account of vagueness) is that the situation of parts and physical boundaries does not require us to modify ontology or the identity relation. We can assume that the class of objects stays fixed throughout the discussion. In a given context, the fixed domain might consist of rivers, feet, scabs, hunks of skin, clouds, chunks of land, molecules, and the like. It is just that some of these objects (if not all of them) have fuzzy boundaries, so to speak, and the ‘‘part of ’’ relation among them is vague. I submit that this vagueness is of a piece with that discussed throughout this book. In short, our first type of vague object does not require us to say anything substantial about the identity relation, or what it is that individuates each object.

2.2. The Hard Cases: Quasi-Abstract Objects Let us now turn to quasi-abstract objects, introduced by principles in the form (ABS): VaVb(S(a) ¼ S(b)  E(a,b) ), The interesting cases are those where a potential sorites series demonstrates that the embedded relation E is not transitive. Let us focus on income groups. For each person x, let INCx be his or her income group. Suppressing the initial quantifiers, the abstraction principle is: INCa ¼ INCb if and only if a and b are roughly equi-incomed. Call this the ‘‘Income Principle’’. The embedded relation, of being roughly equiincomed, is vague. Imagine a sorites series consisting of 8,000,001 people. The first has an annual income of exactly $20,000, and for each n in the series, save the last, the annual income of person n þ 1 is greater than that of person n by one penny. So the last person enjoys an annual income of $100,000. Fix the external context in such a way that the first person is not much above the poverty line and the last is comfortably middle-class. Clearly, the first person is not roughly equi-incomed with the last, nor are they are in the same income group.

174

Refinements and Extensions II

Now assemble a bunch of conversationalists who start discussing the income status of members of the series. They ask themselves whether person 1 is roughly equi-incomed with person 2. Surely yes. Then they ask if person 1 is roughly equi-incomed with person 3. Surely yes. Those incomes differ by only 2¢. They continue in order, asking if person 1 is roughly equi-incomed with person n. And for a time, they will answer ‘‘yes’’. Eventually, they will enter the borderline area of this binary relation or, equivalently, the borderline area of the monadic property of being roughly equi-incomed with person 1. As they do, certain propositions, such as ‘‘person 782,000 is roughly equi-incomed with person 1’’ go on the conversational record. Of course, the conversationalists will not go through the entire series, and end up saying that the last person is roughly equi-incomed with the first. In the external context of the conversation, that would reveal incompetence. At some point, they will jump and explicitly deny that a certain person in the line-up, say #797,000, is roughly equi-incomed with person 1. Assuming that tolerance remains in force, at that point some items get removed from the conversational record, such as the statement that #796,999 is roughly equi-incomed with person 1. We then take the conversationalists back down the series, asking, first, if #796,999 is roughly equi-incomed with #797,000, and then, after an affirmative answer, asking if #796,998 is roughly equi-incomed with #797,000. Eventually, they will jump again, and for the second time, items will be removed from the score. The principle of induction for the sorites would be something like this: If two people are roughly equi-incomed, and a third person’s income differs from one of them by at most one penny, then that person is roughly equi-incomed with the each of the first two. As usual, this leads to paradox. To follow the theme of this book, the correct principle of tolerance is this: If someone judges two people to be roughly equi-incomed, and a third person’s income differs from one of them by at most one penny, then she cannot judge that the third person as anything other than roughly equiincomed with each of the first two. If the judge is forced to rule on the third person, she can satisfy tolerance by either declaring the third person to be roughly equi-incomed with each of the first two, or retracting her original statement that the first two are roughly equi-incomed. So far, nothing is new here, but we have not yet talked about quasi-abstract objects, nor about vague identity. We have not yet invoked the Income Principle. So let us introduce income groups into the picture. Suppose that the pair h#1,#734,000i is in the borderline area of the relation of being roughly equiincomed. Then the statement INC#1 ¼ INC#734,000 is not determinately true, nor is it determinately false. When a group of conversationalists say that, #1 and #734,000 are roughly equi-incomed, they implicitly (or perhaps explicitly) assign

Refinements and Extensions II

175

them to the same income group. So once the judgment is made, it is true that INC#1 ¼ INC#734,000 (in that context). In another context, it might be true that INC#1 6¼ INC#734,000. In other words, identity statements are themselves subject to indeterminacy and open-texture. So now we have to deal with ontological issues. What, exactly, are income groups, heights, weights, etc.? Is it coherent that identity is a vague relation? In a much-discussed one-page note, Gareth Evans [1978] argues that the notion of vague identity is incoherent. We can adapt the reasoning to the present case. Suppose, as above, that it is indeterminate whether INC#1 ¼ INC#734,000. Then INC#734,000 has the property of ‘‘being neither determinately identical to INC#1 nor determinately distinct from INC#1’’. Call this property y. Clearly, INC#1 is determinately identical to INC#1 if anything is, and so INC#1 does not have the property y. So INC#734,000 has a property that is not shared by INC#1, namely y. So, by (the contrapositive of ) Leibniz’s law on the indiscernibility of identicals, we conclude that INC#734,000 6¼ INC#1. The income groups are (determinately) distinct after all. We have a few options for interpreting the Income Principle, and the other principles that deliver quasi-abstract objects. Only some of these run up against the Evans argument. I take it as a desideratum that our ordinary talk of such things as income groups, heights, colors, and weights is coherent, if not literally true, and I will explore only alternatives that sanction this, Unger [1975] and Dummett [1975] notwithstanding. One possibility is to think of an income group as a dynamic set, like the Supreme Court or the Boston Red Sox. Such things gain and lose members over time. Earl Warren used to be a member of the Supreme Court, but he is no longer; Nomar Garciaparra used to be a member of the Red Sox, but he is no longer, alas.6 Under this interpretation, income groups gain and lose members as borderline cases of the relation on the right-hand side of the Income Principle are called or retracted in the course of a conversation. So, for example, in the above scenario, when it is declared that #796,999 has roughly the same income as #1, then #796,999 goes into the income group of #1 (and #1 goes into the income group of #796,999). When that statement is implicitly retracted a minute later, #796,999 is removed from the income group of #1 (and vice versa). In a sense, income groups are quasi-equivalence classes. On this reading, the statement on the left-hand side of the Income Principle is not actually a statement of identity between objects. That is, INCa ¼ INCb does not say that the income group of a and the income group of b are the very same income group. Rather, it says that the groups have the same members at the contextually indicated time, or perhaps it only says that a is in the income group of b and b is in the income group of a. 6

See Uzquiano [2004] for an illuminating account of dynamic sets.

176

Refinements and Extensions II

Given the nature of dynamic groups, this is not a terrible abuse of language. Suppose, for example, that at a certain point in time, the members of a certain basketball team, the Wizards, are the nine justices of the Supreme Court. Then it would be true, in a sense, that the Supreme Court is the Wizards, even though these are different dynamic sets. Their members were different in the past, and are likely to be different in the future. But, right now, they are ‘‘identical’’, not in the sense that they are the same dynamic group, but that they have the same members. Although it is not usual to think of other quasi-abstract objects, such as heights, weights, and colors, as dynamic sets, with temporary members, one might conveniently interpret them that way. In any given context, the height of a given person is the set of people that are (competently) judged to be of about the same height in that context. The same goes for weights, colors, and the like. Let a be a person. As the context changes, the members of the height of a change, but it remains the same height throughout, just as the one and only Supreme Court changes its members over time. Returning to incomes, notice that there must be at least two people whose incomes differ by a single penny and that there is a possible context in which one of them is not in the income group of the other. Otherwise, we would be stuck with sorites, having to conclude that #1 and #8,000,001 are in the same income group in every context. Admittedly, it is counterintuitive that two people whose incomes differ by a single penny are not in the same income group in every context. How much difference can a single penny make? Nevertheless, this feature of the view may not strain intuitions that much. It is the familiar principle of tolerance, treated in previous chapters at exhausting length. As we keep reminding ourselves, when it comes to vagueness, something has to give with intuitions. The principle of tolerance does entail that if the incomes of two persons differ by one penny, then they are never in different income groups. If they both get classified at the same time, they must be classified together. This is similar to our conclusion from Ch. 1 that it is not possible to classify every member of a sorites series as either bald or non-bald at the same time, on pain of violating tolerance, but it does not follow that we must classify close cases differently. To sum up this option, the Income Principle is false if we read its left-hand side as an identity between income groups. Indeed, the income group of a person a is not the very same income group as that of b unless they have exactly the same income. A charitable reading of the Income Principle is to take INCa ¼ INCb as a statement that the income groups have the same members in a given context. So construed, the principle says that if a and b are roughly equi-incomed in a given context, then their respective income groups have the same members in that context. As noted above, it is clearly coherent to note that two different dynamic groups have the same members at one point in time. For example, it might happen that over a period of a few hours, the executive board of a given

Refinements and Extensions II

177

organization is the starting infield on a given softball team. In saying this, we don’t mean to say that the two groups, qua groups, are the same. What we mean is that for a time, the two groups have the same members. Since, on this reading of the Income Principle, we do not have a genuine identity on the left-hand side, we do not encounter indeterminate identity statements, much less indeterminate identities. Consequently, we do not come up against the conclusion of the Evans argument. In a sense, the situation with quasi-abstract objects is of a piece with the above treatment of objects with vague parts. To say that a person a is in a given income group is to say that a is a ‘‘part of ’’ this income group. Just as the small hunk of skin is a part of my foot in some conversational contexts and not in others, one person is in, or part of, an income group in some contexts and not in others. So we do not have an uncomfortable view on ontology to deal with. A more radical way to interpret the Income Principle is take its left-hand side literally, as invoking a genuine identity between the income groups themselves. Given the vagueness of the relation on the right, it follows that for some pairs of people, a, b, it is not determinate that the income group of a is identical to the income group of b, nor is it determinate that the income groups are distinct. We have vague identity statements, and perhaps even vague identities. It is part of the foregoing account that each vague predicate is responsedependent or, better, judgment-dependent, at least in its borderline area (see especially Ch. 1 x8). The extension of a vague predicate in a given context is, in part, a function of the judgments of competent speakers concerning this extension in that context. For at least some objects, an item falls in the extension (or anti-extension) of a vague predicate only if it is competently judged to be such under appropriate circumstances. If there is no judgment (one way or the other), then there is no fact of the matter as to where the object falls. This holds, at least, for objects in the borderline area of the predicate. An extreme version of the view, dubbed the ‘‘Copenhagen view of vagueness’’ in Ch. 5, is that the conditional holds for every object in the field of the predicate. But perhaps we need not go to such extremes. The present reading of the Income Principle extends the judgmentdependence to ontology and/or identity. The thesis is that what identities hold in a given context is, at least in part, a function of the judgments that have been made, implicitly or explicitly, in that context. Suppose, for example, that in the course of a conversation, someone says that a man whose income is $80,000 is roughly equi-incomed with a woman who makes $92,000. This establishes an income group containing these two people, and including everyone whose income is in between them. This information goes on the conversational score. Suppose that, a few minutes later, someone else in the conversation notes that a person with an income of $99,000 is roughly equi-incomed with the woman. This explicitly extends the membership in the given group. It now stretches (at least) from $80,000 to (at least) $99,000. But now suppose that later on, the

178

Refinements and Extensions II

conversationalists identify the latter income with someone who makes $106,000, and they agree that the latter is not roughly equi-incomed with the first man, who makes a paltry $80,000. In other words, at this point, the conversationalists identify the $80,000 and $92,000 incomes, but distinguish those two from both $99,000 and $106,000. The conversational record is thereby updated to include this information, retracting the statement that $92,000 is roughly equi-incomed with $99,000. This retraction is analogous to a jump in the forced march sorites treated above, ad nauseam. There, the jump results in changes to the extensions and anti-extensions of predicates. Here, the jump results in a change in identity and in ontology. Before the jump the conversationalists had a single income group in their sights, after the jump they have two. As with the forced march sorites, conversationalists cannot assign every income to an income group without violating tolerance. On the present view, as they make and change various assignments, the ontology of quasi-abstract objects changes to fit their pronouncements, with the conversational record keeping track of which identifications have been made. What are we to say about incomes that do not belong to the groups under discussion at any given moment? For example, suppose that person a makes $75,000, person b makes $79,999.99, and person c makes $120,000. What is the status of the income group of a, the income group of b, and the income group of c, during the above conversation? There are two theoretical options. One is to hold that the relevant income groups do not exist (at the time). On this view, throughout the above-discussed conversation, there is no such thing as the income group of a, the income group of b, or the income group of c. Of course, the conversationalists could go on to discuss the incomes of one or more of those folks, in which case the requisite income groups would be created (and, if necessary, adjustments made to other items on the conversational score). But they do not discuss those incomes, and so the income groups do not exist during that conversation. This option represents a further concession to Humpty Dumpty. In the foregoing account of vagueness, competent speakers are masters of the extensions of vague predicates, at least in their borderline areas. On the present option of present view, competent speakers are also in control of ontology, at least for quasiabstract objects introduced by vague abstraction principles. In other words, on the present interpretive option, the judgments of competent speakers determine which objects exist. This interpretive option is not challenged by the Evans argument. On the present perspective, there are no indeterminate identities, or at least none between existing quasi-abstract objects. As a conversation proceeds, income groups come into existence and go out of existence, depending on the decisions of the participants. But whenever income groups exist at a given point in a conversation, any pair of them is (determinately) identical or distinct at that point. In some contexts, a given statement of identity between income groups may lack a

Refinements and Extensions II

179

truth value, but that is because at least one of the groups does not exist in that context. I presume that the indiscernibility of identicals does not apply to objects that do not exist. If the income group of a does not exist at a given moment, it is not the case that this very income group has the property of not existing at the time, nor does it have the property of being indeterminately identical to the income group of b, for any b. There simply is no ‘‘it’’ to have or lack properties. A second interpretive option is to hold that all the possible income groups exist, all the time. We do not create or destroy income groups just by talking. However, some identities and inequalities between these income groups lack truth values. For example, throughout the above conversation, it is neither true nor false that the income group of a is identical to the income group of b. Indeed, throughout the above conversation, it is neither true nor false that the income group of a is identical to that of someone who makes a single penny more or a single penny less. So long as tolerance is in force, at least some identities between income groups must lack truth values. Notice, however, that on this option it is nevertheless true that the income group of a is identical to the income group of a. Also, at the end of the conversation, it is true that the income group of a is distinct from that of c, since a makes less than $80,000, and this income has been explicitly distinguished from $106,000, which is less than the income of c. This option represents a different but related concession to Humpty Dumpty. Here we do not hold that competent speakers are masters of which quasi-abstract objects such as income groups exist, but speakers are in charge of what identities hold between and among quasi-abstract objects. So speakers sometimes competently determine how many income groups are under discussion. Suppose we are talking about the incomes of three different people. How many income groups are there? In some contexts, there may be one, in others two, and in still others three. In yet other contexts, it is indeterminate how many income groups there are. The second interpretive option thus allows that there are indefinitely many income groups—exactly how many is a function of the context. The groups themselves do not come into or go out of existence, but identities (and inequalities) among them vary in the course of a conversation. In particular, we do have genuinely indeterminate identities. So this option runs afoul of the conclusion of the Evans argument. Let me at least briefly sketch how a resistance to this argument might be motivated for our present theorist. A full defense would take us too far afield, both from the scope of this book and from my own competence and interests. I presume that it is not an option simply to reject the indiscernibility of identicals. At the very least, this principle borders on being analytic of the notion of identity. The key move, I suggest, is to deny that such predicates as ‘‘is determinately identical to the income group of a’’ refer to properties, or at least to properties

180

Refinements and Extensions II

that are relevant to the Leibniz principle.7 After all, it is clear that not every predicate counts toward the indiscernibility of identicals. For example, Karl may know that Cicero was Roman and not know that Tully was Roman. So Cicero has, and Tully lacks, the ‘‘property’’ of being known by Karl to be Roman. This does not entail that Cicero is not identical to Tully. Presumably the inference fails because ‘‘Karl knows that x is Roman’’ is not the right sort of predicate. It does not represent a property. An advocate of the strict reading of the Income Principle would argue that ‘‘being determinately identical to’’ is similarly disqualified. One question-begging way to proceed on the present issue is via a modus tolens on the Evans argument. Since, on this view, there are indeterminate identities, it follows that the predicates in question are not appropriate to the Leibniz principle. The motivation for this view, I presume, is the strength of the overall theory of quasiabstract objects. By itself, this move has the flavor of Lakatosian monster-barring. We can do better. The philosophical literature contains many reasons why intensional predicates do not count toward the indiscernibility of identicals. What we need here is an (independently) motivated reason to exclude the determinacy predicates used in the Evans argument. I submit that the present account of vagueness supplies such a reason: the predicates in question are intensional. Chapter 5 extended the present account of vagueness to include the phenomenon that passes for higher-order vagueness. On one of the theoretical options presented there, the key thought is that determinacy is itself a judgmentdependent matter. In the present context, statements in the form ‘‘a is determinately identical to b’’ might themselves be judgment-dependent. As such, they are intensional. Suppose, for example, that someone makes a competent judgment in the form F(a), and suppose that a is (determinately) identical to b. It does not follow that the person could, or would, judge that F(b), unless, of course, she knows that a ¼ b. Since, on the option under consideration, statements of determinacy are intensional, it is reasonable to exclude such predicates from the indiscernibility of identicals. We briefly return to the Evans argument in the next section, dealing with the model theory, and again in the next chapter.

2.3. Counting Objects Our remaining exemplar of vague objects concerns the identity or distinctness of objects with physical boundaries. When are there two clouds, or just one? When are there two hills, or just one? When are there two amoebae, or just one? Here we can be brief, since the resolution uses the resources invoked with quasi-abstract 7 Terence Parsons and Peter Woodruff [1995] also defend the coherence of indeterminate identities. They point out that the Evans argument invokes the contrapositive of the Leibniz principle of the indiscernibility of identicals, and that in a three-valued system, the contrapositive of a valid inference need not be valid. Second, they suggest that predicates that invoke indeterminacy need not express properties, although they do not suggest that determinacy is intensional. Parsons and Woodruff also provide a nice model, in a crisp, bivalent meta-language, to illustrate the coherence of vague identity.

Refinements and Extensions II

181

objects. We turn to Humpty Dumpty once more, noting that ontology and/or identity varies from context to context. Suppose that we have two distinct groups of people lying on their backs watching the sky at the same time, dreamily. We assume that all the people are competent speakers of English, and they exercise normal powers of observation throughout. When they start, there are two distinct clouds above them, by anyone’s lights. Over the course of four hours the clouds slowly move around and eventually merge into one. We ask each group how many clouds there are, and we repeat the question every twenty seconds. At the start, of course, both groups answer ‘‘two’’. And they will continue with this answer for a while, at least until the clouds barely touch each other, and for a bit after that. At some point, of course, each group will change its answer to ‘‘one’’, but probably not at the same time. After the switch, a group may continue to answer ‘‘one’’ until the end of the time period, or they may backtrack, especially if one of the clouds splits off again temporarily or a new cloud starts to form. At the end, both groups agree that there is only one cloud. Suppose that the first group first answers ‘‘one cloud’’ at 1:45:40, changes back to ‘‘two clouds’’ at 2:15:00, back to ‘‘one cloud’’ at 2:20:20, and sticks with that answer until the end. The second group first answers ‘‘one cloud’’ at 1:55:40, and does not change again after that. At various times, the two groups had different ontologies in their respective conversations—different objects in their respective domains-of-discourse. During the ten-minute interval from 1:45:40 until 1:55:40, the domain of the first group contained a single cloud while the domain of the second contained two, and from 2:15:00 until 2:20:00 it was the reverse. This is so even though, in some sense, both groups were talking about the same stuff all along—the same sky with the same water molecules, etc. On the view in question, the individuation of vague objects such as clouds is a judgment-dependent matter, and the two groups made different judgments, both competently, along the way. The some goes for issues of continuity, or individuation over time. We might ask the groups about the two clouds that started. Call them a and b. Does one of them, say a, survive and become the larger cloud, with cloud b thus destroyed (or absorbed Bork-like into the first)? Or is the single cloud at the end a new object, numerically distinct from each of the two original clouds? These, too, are judgment-dependent matters, handled in the foregoing manner. 3. BACK TO MODEL THEORY, ONE LAST TIME This chapter concludes with the changes to the model theory required to accommodate the various phenomena of vague objects, under the various interpretive options. Recall that a frame is a structure hW,M i such that W is a collection of partial interpretations and M [ W. The designated partial interpretation

182

Refinements and Extensions II

M is the base of the frame, and every partial interpretation N in W is a sharpening of M, written M  N. By the definition of the sharpening relation, all the partial interpretations in the frame have the same domain. I argued above (x2.1) that the foregoing philosophical and semantic account easily handles vagueness in the boundaries of physical objects and, in particular, vagueness over whether one object is a part of another. The vagueness of the ‘‘part of ’’ relation between physical objects is modeled on exactly the same lines as those of any vague predicate, such as our standard exemplar ‘‘bald’’. Our paradigm case of this sort of vagueness concerns the status of a hunk of skin loosely attached to my foot, about ready to fall off. Let a denote my foot, and let b denote the hunk. And, by way of contrast, let c denote the big toe on that foot and let d denote a nail clipping that has just been (completely) removed from the foot. Let P be the binary ‘‘part of ’’ relation. Intuitively, the toe is determinately part of the foot, the nail is determinately not part of the foot, and the status of the hunk of skin is ‘‘unsettled’’. The situation is modeled with a frame in which the pair hc,ai is in the extension of the relation P at the base, and thus in the extension of P in every partial interpretation in the frame. So Pca is determinately true in the frame. Similarly, the pair hd,ai is in the anti-extension of P at the base, and so Pda is determinately false throughout the frame. And the pair hb,ai is neither in the extension nor the anti-extension of P at the base. Moreover, there is at least one sharpening of the base in the frame in which hb,ai is in the extension of P, and there is at least one sharpening of the base in which hb,ai is in the anti-extension of P. At the first of these partial interpretations, Pba is true (but not determinately true), while at the second :Pba is true (but not determinately true). Let us turn to quasi-abstract objects, characterized by abstraction principles in the form (ABS): VaVb(S(a) ¼ S(b)  E(a,b) ), where a and b are variables of a given type (typically individual objects or properties/sets of objects), S is a higher-order operator, denoting a function from items of the given type to objects (in the range of the first-order variables), and E is a relation over items of the given type. In the cases of interest, E is reflexive and symmetric, but not transitive, due to tolerance. The paradigm case is that of income groups, but the same treatment applies to heights, weights, colors, etc. The two interpretive options for quasi-abstract objects broached above are modeled differently in the semantics. One of them, the first, is quite straightforward. On that reading, recall, income groups are dynamic sets that gain and lose members over time. The present system of frames is ideally suited to model the shifting extensions (and anti-extensions) of such ‘‘sets’’, whether they are vague or not. We add two binary predicate symbols: Rap: person a is in income group p. pq: the group p has the same members as the group q.

Refinements and Extensions II

183

Suppressing the initial quantifiers, the Income Principle should thus read: INCa  INCb if and only if E(a,b), where E is the (vague) relation of being roughly equi-incomed. Suppose first that we interpret the Income Principle as invoking the usual material bi-conditional. Let E(m,n) be indeterminate. In this case, the ‘‘identity’’ on the left would also be indeterminate, and so the whole bi-conditional is indeterminate, according to the adopted truth table for the material conditional (see Ch. 3 x1). So in the cases of interest, there will be some acceptable partial interpretations in which the Income Principle fails to be true. So interpreted, then, the best we can say about this abstraction is that in acceptable frames, it is never false. In the technical jargon from Chs. 3–4 above, the Income Principle is weakly forced. We can do better by invoking the intuitionistic-style conditionals developed in x1 of Ch. 4. The idea behind the Income Principle is that in any sharpening in which E(a,b) is true, INCa  INCb should also be true, and vice versa. So the Income Principle is: (INCa  INCb ) E(a,b) ) & (E(a,b) ) INCa  INCb). This should hold at the base of each acceptable frame. In effect, we treat the Income Principle as a penumbral connection between the relation of roughly equi-incomed and the  relation between income groups. The Income Principle thus says that if, in any sharpening of the base in a frame, a and b are equiincomed at that sharpening, then their respective income groups have the same members at that sharpening, and conversely. Some further penumbral connections are: INCa  INCa (INCa  INCb) ) (INCb  INCa) ( (INCa  INCb)&(INCb  INCc) ) ) (INCa  INCc) INCa  INCb ) Vx(RxINCa ) RxINCb) The first three of these say that, at each partial interpretation, the income groups are reflexive, symmetric, and transitive (respectively) with respect to the ‘‘’’ relation, and the last says that at each partial interpretation—in each conversational context—if two groups have the same members (in that partial interpretation), then any member of one is a member of the other. With the Income Principle, this requires that in each partial interpretation of the frame, the extension of E is an equivalence. This, I presume, is a reasonable requirement, and not problematic. Since E is vague, there will be pairs that are neither in its extension nor in its anti-extension. So, in a sense, E itself does not represent an equivalence relation.8 8 There may be an issue at the base of the frame, where the extension of E consists of those pairs of folk who are determinately roughly equi-incomed. For simplicity, we prescind from such issues here. They broach (so-called) higher-order vagueness, which would be handled along the lines of the previous chapter.

184

Refinements and Extensions II

So much for the easier interpretation of quasi-abstraction. With this one, we do not really get vague objects. Instead, we get dynamic sets whose members shift around with the extension and anti-extension of a vague relation. What is vague is the relation of ‘‘having the same members’’ among dynamic sets. On the second interpretation of the Income Principle, and kindred quasiabstractions, we did end up with vague statements of identity. And we had two readings of this, one of which (the second) is at loggerheads with the conclusion of the Evans argument against vague identity. Let us deal with that second reading first. Recall that, so interpreted, all defined quasi-abstract objects exist, but the identity relation between them is vague. For example, if a and b are people with different incomes, then the income group of a exists and the income group of b exists, but it may be indeterminate whether these income groups are identical or distinct—whether they are one or two. Recall that the meta-theory for the model theory is ordinary set theory, which (I presume) is not vague. As noted throughout the above treatment, we use a precise meta-theory to model the phenomena of vagueness in an object language. Consequently, there will inevitably be gaps between model and modeled. In particular, the meta-language does not have singular terms denoting objects with a vague identity relation (or with vague boundaries, etc.). But the meta-theory can model such objects. The key is to introduce a binary relation symbol ‘‘  ’’ to stand for the vague identity relation. It is treated as non-logical, and subject to the by now familiar treatment of vague predicates. Using the intuitionistic-style implication, the Income Principle should read: (INCa  INCb ) E(a,b) ) & (E(a,b) ) INCa  INCb). This should hold at the base of each acceptable frame. For each natural number i 15,000,000, let ai be a person whose annual income is exactly $.01i. So a0 is completely destitute, a1 is all but destitute, while a15,000,000 enjoys an income of $150,000. At the base of a frame F, let the extension of E consist only of the pairs hai,aii, and let the extension of ‘‘  ’’ consist of only the pairs hINCai,INCaii. That is, the only incomes that are determinately identical are those that are exactly identical, to the penny. We noted above that one cannot determinately identify any (distinct) incomes, no matter how close they are. Let the anti-extension of E at the base consist of those pairs hai,aji in which j i  j j 1,000,000. And let anti-extension of ‘‘  ’’ at the base consist of those pairs hINCai,INCaji in which j i  j j

1,000,000. That is, we determinately distinguish two incomes if they differ by at least ten grand. To enforce tolerance in this frame, we further insist that if, say, j i  j j < 20,000, then INCai  INCaj and E(ai,aj) are weakly forced. In other words, if j i  j j < 20,000, then  :(INCai  INCaj) and  :E(ai,aj)

Refinements and Extensions II

185

both hold at the base. This means that if the incomes of two people differ by less than $200, then we never put them in distinct income groups. So, as noted above, people whose incomes are close are not always identified, but they are never distinguished in any partial interpretation in the frame. As with the interpretation of income groups as dynamic sets, we should also add penumbral connections requiring the extension of the ‘‘  ’’ relation to be reflexive, symmetric, and transitive. Since it is vague, it is not an equivalence. The various partial interpretations in the frame would represent various ways of identifying and distinguishing income groups consistent with the Income Principle and the tolerance of the embedded relation of being roughly equi-incomed. As noted just above, this option, on which identity itself is vague, runs afoul of the conclusion of the Evans argument. The key premise in this argument, of course, is the Leibniz principle of the indiscernibility of identicals: if a ¼ b, then every property enjoyed by a is enjoyed by b. I’d rather not introduce explicit talk of properties into the model theory. It is complicated enough. Fortunately, we can get by with predicates. In standard treatments of logic, one of the principles of identity is the scheme: (ID) VxVy(x ¼ y ! (F(x) ! F( y) ) ), one instance for each formula F(x) not containing y free. In the present model theory, (ID) does not hold in complete generality, independently of any considerations about vague objects. There are partial interpretations in which a ¼ b is true, but F(a) and F(b) are both indeterminate. It follows that the corresponding instance of (ID) is also indeterminate. As usual, however, we would hope that no instance of (ID) comes out false, so that each instance is at least weakly forced in each frame. Even more, we would like: (ID ) VxVy(x ¼ y ) (F(x) ) F( y) ) ) to hold in each acceptable frame. That is, if an identity a ¼ b holds at any partial interpretation and F(a) holds at a sharpening in the given frame, then F(b) holds at that sharpening as well. We can insist on this, as a penumbral connection concerning identity. With our vague identity relation on board, the principle becomes: (ID ) VxVy(x  y ) (F(x) ) F( y) ) ). Let us introduce this as well. If we stick to the original model theory, developed in Chs. 3 and 4 above, we have a rather hollow victory for the defender of vague identity. The Evans argument cannot be formulated in the given formal language, for the simple reason that the object language cannot express the notion of determinacy (putting aside the artifactual ‘‘sharp’’ determinacy operator introduced in Ch. 4 x4). That is, there are no predicates in the formal language that correspond to ‘‘indeterminately identical to b’’and ‘‘determinately identical to a’’.

186

Refinements and Extensions II

In Ch. 4 x4, and more fully in Ch. 5, the model theory was expanded to allow a determinacy operator DET. The point was to accommodate (what passes for) higher-order vagueness. Formally, at least, the best way to avoid the conclusion of the Evans argument is to insist that (ID ) and (ID ) be restricted to formulas that do not contain the DET operator. As above, this is not (completely) ad hoc. It is well known that if the formal language contains intensional operators such as ‘‘it is believed that’’ or ‘‘Harry knows that’’, then formulas with such operators must be excluded from (ID). The present view is that vagueness is a judgmentdependent matter. Whether a given predicate is vague is a function, at least in part, of the judgments of competent speakers. On at least one of the theoretical options explored in the last chapter, the determinacy operator is itself vague, and thus judgment-dependent. That is, the extension of formulas in the form DET(F) is, in part, a function of the judgments of competent subjects. As noted, such judgments are sensitive to the mode of presentation. So, statements of determinacy are intensional, and so should be excluded from (ID), (ID ), and (ID ). Finally, the other reading of the Income Principle, and other such abstractions, is that the quasi-abstract objects come into and go out of existence in the course of a conversation. Suppose, for example, that it is established in the course of a conversation that a pair of people have roughly the same income. Then the existence of an income group containing those folks is also established. Later, if the identification of the incomes is retracted, that particular income group no longer exists. The natural framework for formalizing this option is free logic. Happily, this is readily accommodated, since we are using (what amounts to) a three-valued system already. However, we now confront another artifact of our set-theoretic metalanguage. Ordinary set theory does not employ a free logic. The theory has no nondenoting singular terms, and it has no room for non-existent objects—whatever those would be. But, as usual, we can model the phenomena in the rich language. To do this, we introduce a monadic predicate ‘‘e’’ for existence. So if a is a person, then eINCa says that the income group of a exists. And, in the spirit of this study, ‘‘e’’ is vague. That is, we allow indeterminate statements of existence. To model the free logic, we have to reinterpret a key aspect of the model theory. Recall that in a given frame F, every partial interpretation has the same domain. The natural interpretation, of course, is that the members of this domain are the objects that we use the frame to talk about. That is, the members of the domain exist, according to the frame. Here, of course, we cannot follow this natural reading, since only some of the members of the domain exist, those in the ‘‘extension’’ of e. How are we to think of the other members of the domain? One can think of them as ‘‘possible objects’’, or as ‘‘potential objects’’, or perhaps as ‘‘virtual objects’’. So that a statement ex says that the virtual x is actual. But this raises some contentious metaphysics. What are virtual objects? How can we talk about them if they don’t exist? A cleaner interpretation is just to

Refinements and Extensions II

187

think of the members of the domain as names, some of which do not denote anything. On this reading, ex would say that x denotes something. Not much turns on the various ways understanding the status of the members of the domain. We have no occasion here to say anything in the object language about nonexistent or merely virtual ‘‘objects’’ (or about non-denoting names), other than their non-existence. So for any atomic relation R, both Rxy and :Rxy entail ex and ey. In particular, the identity relation holds only between existing objects. On this option, the identity relation is never indeterminate. That is, if two objects exist in the same partial interpretation, then they are either identical or distinct at that partial interpretation, and in all sharpenings thereof. So the conclusion of the Evans argument is maintained. A sentence in the form a ¼ b gets the value i (if and) only if either a or b does not exist (or if either a or b fails to denote something). Since we are restricting attention to quasi-abstract objects, we need not deal with singular terms that determinately fail to denote anything. To reformulate this in material mode, I see no point in introducing virtual objects that determinately do not exist, at least not here. A given income group (or a height, or a color) may fail to exist in a given partial interpretation, but never determinately so. So we stipulate that ‘‘e’’ has an empty anti-extension in every partial interpretation in every acceptable frame. Recall that a partial interpretation N 0 is a sharpening of a partial interpretation N, written N  N 0 , if for each atomic formula F, if F is true at N (under an assignment) then F is true at N 0 (under the same assignment), and if F is false at N, then F is false at N 0 . The only difference between the partial interpretations is that some atomic formulas that are indeterminate (i) at N may get truth values (t or f ) at N 0 . The same goes for the new existence predicate ‘‘e’’. If N  N 0 then objects may exist at N 0 that do not exist at N. To be precise, there may be an object o that exists at N 0 (i.e. eo is true at N 0 ), but the existence of o is i at N. In particular, N 0 may contain more income groups, or heights or colors, than N. In free logic, quantifiers range over existing objects, and the quantifier rules must be adjusted accordingly. One consequence of this is that in the present framework, monotonicity is lost. A universally quantified sentence might be true at a partial interpretation N, but false (or i) in a sharpening N 0 of N, since N 0 may contain more objects than N. To remedy this, we reformulate the satisfaction conditions for the universal quantifier in the image of intuitionism (or, to be precise, to the treatment of the universal quantifier in Kripke structures). The idea is that a formula in the form Vx F is true at a given partial interpretation N in a given frame F (under a variable assignment s) if F is true in every sharpening N 0 of N in F (under every assignment that agrees with s except possibly at x). I forgo details. Now what are we to make of abstractions such as the Income Principle? The idea is that if a person a is roughly equi-incomed with a person b, then the

188

Refinements and Extensions II

income group of a exists, the income group of b exists, and they are identical. And vice versa. So the formula, [E(a,b) ) INCa ¼ INCb] & [INCa ¼ INCb) ) E(a,b)] should be true (and thus forced) at the base of every acceptable frame. Let N be a partial interpretation in a frame in which the Income Principle holds. Then if a pair ha,bi is in either the extension or the anti-extension of E in N, then the existence of both income groups is true at N. That is, if either Eab or :Eab is true at N, then both eINCa and eINCb are true at N. And there is a converse. Suppose that eINCa and eINCb are both true in N. Then, since identity is determinate, either INCa ¼ INCb is true at N or INCa 6¼ INCb is true at N. In the former case, the Income Principle entails that E(a,b) is true at N—the two people are roughly equi-incomed. In the latter case, :E(a,b) is forced at N. To wrap up this long journey, our final exemplar of vague objects concerns cases in which it is indeterminate how many of a given object, or type of object, occur in a given situation. Is it one cloud or two? One amoeba or two? One person or two? As noted in the previous section, this phenomena can be understood in terms of vague existence and/or vague identity (depending on which you find the most palatable, or the least unpalatable). In the treatment of quasi-abstract objects, we developed the resources to model these items. No further difficulties remain. Consider, for example, the splitting amoeba. Let a denote the single amoeba at the start and let b and c denote the two amoebae that result at the end. Consider a moment when it is indeterminate how many of them there are. This is modeled with a frame in which ea _ (eb & ec) is forced at the base, but neither a, b, nor c is in the extension of the existence predicate ‘‘e’’ at the base. That is, either a exists or both b and c do, but it is not determinate that any of those exist. There is one sharpening in which a exists and b and c do not, and another sharpening in which a does not exist, but b and c do. Other aspects of the situation are handled as well. For example, presumably b and c are different. This is expressed with the stipulation that b 6¼ c is weakly forced. One might hold that after the split, a survives as b, with c as a newly created cell. This would correspond to a sharpening in which ea, eb, and a ¼ b are all true. Someone else might hold that a survives as c : ea, ec, and a ¼ c. A third person might hold that a does not survive, but is replaced by b and c : :ea, eb, and ec. These would correspond to two other sharpenings of the base. I speculate that the open-texture thesis is correct for this talk of individual amoebae, and other similar situations. That is, any of these foregoing judgments concerning the survival and creation of these interesting cells is competent, and no one judgment is mandated. When it comes to such matters, Humpty

Refinements and Extensions II

189

Dumpty reigns supreme. In the end, however, it is an empirical matter whether the meaning of English words such as ‘‘amoeba’’ and ‘‘cloud’’ allows this much freedom. This, finally, completes the philosophical and formal account of vagueness. We turn now to the issue of metaphysical vagueness, and to the objectivity of vague discourse.

7 Metaphysical Matters: Language, the World, and Objectivity This notion that the world in itself might be vague strikes me as a particularly dark piece of metaphysics. Copeland [1994: 83] . . . I should like to offer some constructive suggestions. Before doing so, I must warn you that I can’t see any ground whatever for renouncing one of the most fundamental rights of man, the right of talking nonsense. And now I suppose I may go on. People are inclined to think there is a world of facts as opposed to a world of words which describe these facts. I am not too happy about that. What rebels in us against such a suggestion is the feeling that the fact is there objectively no matter in which way we render it. I perceive something that exists and put it into words. From this, it seems to follow that something exists independent of, and prior to language; language merely serves the end of communication. What we are liable to overlook here is the way we see a fact—i.e., what we emphasize and what we disregard—is our work. Waismann [1968: 137, 140]

1 . MET A P H YS IC A L VA GU E N E SS One issue that has not been addressed directly in this book so far is the source of the phenomenon of vagueness. Is it a purely linguistic matter, concerned with how we represent the world via language, or is there a sense in which the world itself is vague? The issue is often formulated in terms of vague objects, the topic of the previous chapter. So construed, the question is whether the world itself contains vague objects, or whether it is just a matter of some indeterminacy in how we represent objects in language. To follow the epigraph of the previous chapter (from Copeland 1994), the Australian outback appears to have vague boundaries. Is this a fact about the world, or is it a fact about how we represent the world?

Metaphysical Matters

191

In the previous chapter, we saw that vague boundaries, or vague parts, are not especially problematic. Statements of identity are another matter. Consider a supposed indeterminate statement a ¼ b. There are at least two ways to understand it. One is that there are these (two?) objects, a and b, and it is indeterminate whether those very objects are identical or distinct. On this reading, the singular term ‘‘a’’ and the singular term ‘‘b’’ each fulfils its task of denoting one particular object. The identity statement concerns the object or objects so denoted. This would be what is sometimes called ontic, or metaphysical vagueness.1 The other interpretation of the identity statement is that it is indeterminate which object ‘‘a’’ denotes and/or it is indeterminate which object ‘‘b’’ denotes. The terms might denote the same object, and they might denote (determinately) different objects. As Mark Sainsbury [1994: x5] puts it, in this case there is ‘‘vagueness concerning what, if anything, our words denote’’. Or David Lewis [1986: 212]: The only intelligible account of vagueness locates it in our thought and language. The reason it’s vague where the outback begins is not that there’s this thing, the outback, with imprecise borders; rather, there are many things, with different borders, and nobody’s been fool enough to try to enforce a choice of one of them as the official referent of the word ‘‘outback’’.

On this reading, the blame for the indeterminacy of the statement a ¼ b is placed on the reference relation, and not on the world. This is (mere) semantic or representational vagueness. Most of the warring parties, with the possible exception of epistemicists, agree that there is, or can be, semantic vagueness. The possibility of metaphysical vagueness is more controversial (but see Merricks 2001). Sainsbury [1994: x5] argues that ‘‘identity is sharp, if anything is’’. If he is correct, then there is no metaphysical vagueness, or at least none concerning the identity of objects. Lewis [1988] shows that the short, but influential argument in Evans [1978] (discussed in x2.2 of the previous chapter) was not aimed at semantic vagueness. According to Lewis, the Evans argument does not tell against indeterminate statements of identity, a large sample of which were taken up in the previous chapter. Rather, the Evans argument tells against indeterminate identities. Following Sainsbury [1994: x2], ‘‘those who hold that vagueness is not in the world, but springs rather from . . . language, can accept de dicto vagueness, [but] they cannot accept de re vagueness’’. Although, as noted, the issue of metaphysical vagueness is usually motivated in terms of singular terms and objects, one can raise the same issue for predicates and properties. Let Harry be a borderline case of our standby ‘‘bald’’, so that the sentence ‘‘Harry is bald’’ is indeterminate, or unsettled. What is the basis of this more or less sad state of affairs? The metaphysical possibility is that the English predicate ‘‘bald’’ expresses (or stands for, or refers to) a single property, the 1 I’ll stick to the term ‘‘metaphysical’’ here, to separate the present matter from the issues concerning vague objects addressed in the previous chapter.

192

Metaphysical Matters

property of baldness, and this very property is vague, having borderline cases. An opposing, semantic view is that there are no vague properties. If the universe contains properties at all, they are not vague. Since the predicate ‘‘bald’’ is vague, it does not pick out a single property. Perhaps the word indeterminately picks out a number of properties at once (but don’t ask which ones), or perhaps it fails to pick out any at all. Again, this is mere semantic, or representational vagueness. There is an interesting and infuriating linguistic barrier to treating the underlying issues. From Berkeley on (if not before), idealists, anti-realists, irrealists, non-cognitivists, and the like have insisted that they are entitled to words and phrases such as ‘‘reality’’, ‘‘object’’, ‘‘the world’’, ‘‘fact’’, and ‘‘state of affairs’’. These locutions, they claim, are part of ordinary language, and are used coherently by ordinary speakers without invoking substantial metaphysical principles. The philosophers in question insist on deflationary, idealist, non-cognitivist, etc. readings of the words and phrases. For Berkeley, for example, a material object just is a bundle of ideas. For a projectivist about morality, the proper explanation of such a statement as ‘‘it is a fact that murder for hire is wrong’’ invokes projections of sentiments. But still, it is a fact that murder for hire is wrong—or so says the projectivist. This makes it hard to discuss the metaphysical issues themselves. Whatever the realist wants to say, in order to express his realism, is accepted by his opponent, once it is suitably (re-)interpreted: ‘‘Of course there are objects; of course there are facts; . . . It is just that . . . ’’ It gets tedious to have to keep writing things like ‘‘in a metaphysically robust manner’’ to various pronouncements and hypotheses. There is a literary device, attributed to Hilary Putnam, of writing words in small capitals when one means to, as Terence Horgan [1994b: 99] puts it, ‘‘be talking about denizens of the mindindependent, discourse-independent, world’’. That is, words written in small capitals are not subject to reinterpretation or irrealist explanation. They are metaphysically loaded, referring to the ultimate constituents, the underlying fabric, of reality. This assumes, of course, that irrealists will not insist on suitably reinterpreting the words in small capitals, despite our command not to do so. The change in font has to mean something. The device also assumes that the underlying distinction makes sense—that we can somehow talk about, or at least refer to, the ultimate constituents of the world (or the world) as they are, independently of language, conventions, form of life, etc.—despite the obvious fact that we have to use language to do this talking. If the distinction at hand cannot be made out, then I do not see how the present debate can be conducted. I lose my bearings. In these terms, the issue at hand concerns whether a term such as ‘‘the outback’’ denotes a vague object in the world, namely the outback. And the issue concerns whether a term such as ‘‘bald’’ stands for a vague property, in this case baldness. The proponent of metaphysical vagueness answers ‘‘yes’’ to these questions. Sainsbury [1994] argues that there is no coherent and even prima facie plausible thesis to be had here. He concludes that ‘‘when we make enough

Metaphysical Matters

193

concessive assumptions to have an intelligible thesis of [metaphysical] vagueness, we have a thesis which can be shown to be false by a few short lines of proof ’’, namely the short argument in Evans [1978]. B. Jack Copeland [1994] is even less concessive, arguing that Sainsbury has not ‘‘managed to find a way of giving coherent expression to the thesis of [metaphysical] vagueness’’. I don’t know whether Copeland [1994: 33] is correct that the issue of metaphysical vagueness is a piece of dark metaphysics, but, obviously, metaphysical matters are involved. Hilary Putnam [1981: 49] defines metaphysical realism to be the view that ‘‘the world consists of some fixed reality of mindindependent objects. There is exactly one true and complete description of ‘the way the world is’ ’’. For the metaphysical realist, the issue of metaphysical vs. semantic vagueness is fairly clear (or should I say ‘‘determinate’’). It depends on whether there are any vague specimens among the fixed reality of mindindependent objects and, presumably, mind-independent properties. On the second clause in Putnam’s passage, the issue comes down to the nature of the complete description of the way the world is. The world is vague if, and only if, the presumed complete description of it is vague.2 Suppose that the complete description contains a vague predicate or a vague singular term. Then, given that this description is complete, one would think that the predicate stands for something, a vague property, or the singular term refers to something, a vague object. But what of those philosophers, such as Putnam himself, who reject metaphysical realism? What are they to make of the issue? Evans [1978] begins: ‘‘It is sometimes said that the world might itself be vague. Rather than vagueness being a deficiency in our mode of describing the world, it would then be a necessary feature of any true description of it.’’ I take the Peano postulates to be a true description of (part of ) the world, and yet there is no vagueness in the Peano postulates, or so I assume. The thesis of metaphysical vagueness is surely that vagueness is a feature of any true and sufficiently extensive description of the world. That is, even if the metaphysical realist is mistaken, and there is no complete description of the world, it may be that every adequate description of it is vague, or that our best descriptions of it are vague. If so, then perhaps there is some sense to metaphysical vagueness. Or perhaps not. Copeland [1994: x4] writes that there is a perfectly intelligible thesis about representations that a supporter of [metaphysical] vagueness might regard as capturing part of what it is they want to say. The fundamental 2 The second clause in Putnam’s account of metaphysical realism might not be completely apt for posing the issue of metaphysical vagueness. For one thing, it invokes semantic notions such as ‘‘descriptions’’. What does the possibility of various kinds of description have to do with the metaphysical nature of the universe? Second, if vagueness involves indeterminacy, there may be some difficulty in the possibility of a complete, true and vague description of the universe. Perhaps the view should be that there is a unique, best description of the world, in the sense that anything more precise would be false or at least untrue. Since metaphysical realism does not figure prominently here, there is no need to adjudicate this matter. Thanks to Julian Cole and Katherine Hawley.

194

Metaphysical Matters

thought of the friend of [metaphysical] vagueness seems to be that it is not (or might not be) up to us whether to be vague or not. That is, a believer in [metaphysical] vagueness denies what I will call the crispness postulate: Beyond a certain point of sophistication the physical sciences will use only non-vague, or crisp representations of the world. To maintain that good science cannot be rid of vague representations is to say that the world constrains us to be vague even where we would most like to be precise.

Let us assume, for the sake of argument, that the crispness postulate is false.3 So, as Copeland puts it, it is not up to us whether to be vague. Unfortunately, this does not locate the source of the vagueness, and does not settle the present issue. We do not yet have metaphysical vagueness. Crispin Wright [1976, 1987] speculates that vagueness is due to the human need for predicates that can be applied on the basis of (casual) observation. Color predicates will do as an example. It would defeat much of the purpose of color-talk if we had to haul out a digital meter every time we need to establish or communicate the color of something. And of course, our powers of observation are limited. Our eyes, ears, noses, taste buds, and hands can discriminate only so much. Indeed, even the discriminatory abilities of digital meters are limited. As noted in Ch. 1 above, this suggests a principle of tolerance for certain predicates and makes them prone to sorites arguments. Moreover, science requires predicates that are applied on the basis of observation, or on the basis of meter readings, and so are subject to at least some vagueness for the same reason. Although I find Wright’s account of the source of (at least some) vagueness eminently plausible, it may not be correct in all its details, and the present point does not turn on his account. All we need here is that a story along these lines is conceptually possible. It very well might be that every sufficiently rich, true description of the world that we are capable of making is infested with vagueness. If so, then it is indeed not up to us whether to be vague. We have to be. Yet the responsibility for this fact lies with us, and our limited powers of observation and detection, and not with the world (except to the extent that we observers and masters of language are part of the world). Following Copeland, the negation of the crispness postulate does not entail anything in the neighborhood of metaphysical vagueness, at least not by itself. Of course, someone who denies the crispness postulate in favor of metaphysical vagueness, or someone who accepts Evans’s statement about true descriptions, might not be limiting the quantifier over descriptions to those that we finite human beings are capable of making. If it is the world that is vague, then no being, human or otherwise, can completely describe it without invoking 3 If she rejects a certain naturalistic assumption, a defender of metaphysical vagueness might agree to the crispness postulate. Our metaphysician might concede that science, or future science, or ideal science, invokes (or will invoke) only precise vocabulary, but she will hasten to add that there is an aspect of reality that science does not deal with, namely its vagueness. Thanks to Stephen Schiffer for this point.

Metaphysical Matters

195

vague language, no matter what their powers of observation and discrimination. Even God has to use a vague language. If this is what the debate over metaphysical vagueness is all about, then I politely decline to participate, due to lack of interest. It is a frustrating situation. Since Kant, if not before, philosophers and scientists have been anxious to tell us something about the ultimate constituents of the world as it is, independent of how we perceive it, and how we describe it. Through science, we have made progress on this, showing, for example, how perception of color works, and which features of reality give rise to various perceptions (assuming, of course, that the relevant ‘‘features of reality’’ are objective). But we see that sometimes it is virtually impossible to completely separate out the part of our (true) representations that is due to the way the world is and the part that we, the human describers, contribute. If we try to say something about the world, as it is independent of our language, form of life, etc., we have to use language. How else can we describe things? And this language will almost invariably contain mind-dependent features.4 To paraphrase Friedrich Waismann, the language was developed to serve human interests. That’s life, or at least our life. Sainsbury [1994: x7] concludes that the very question of whether our language is vague because the world is vague, or whether the world is vague because language is vague suggests that we have a choice of picture. According to one, our world, before we find it, is an undifferentiated sludge. In finding it, we divide it up. If we divide with vague tools, we see the world as containing vague objects . . . but the explanation will lie with us and our tools and not with the world. According to the alternate picture, the world is a certain way before we find it. Our job is to fashion concepts to mirror it. Because it contains vague objects, we find vague objects, and fashion vague concept[s] to match . . . I doubt whether either picture is intelligible, and I therefore doubt whether this route will lead to an intelligible thesis of [metaphysical] vagueness. We cannot think of our world except through our concepts . . .

I will not take sides on the intelligibility of either of Sainsbury’s pictures. I see no reason to challenge naive common sense that the world ‘‘is a certain way before we find it’’—idealism aside. For example, before we found the world, it contained stars, planets, trees, and animals. And before I found the world, it contained my parents and grandparents. But, as Waismann [1968: 140] asks, ‘‘What, then, is the objective reality supposed to be described by language?’’ He points out that language ‘‘contributes to the formation and participates in the constitution of a fact; which, of course, does not mean that it produces the fact’’ (p. 141). Clearly, the concepts that we apply to the world are vague, and, arguably, these concepts have to be vague, given the sorts of beings we are. 4 I am indebted to extensive conversations with Louise Antony and Julian Cole, and to the participants in a seminar on objectivity at Ohio State in the Autumn of 2004.

196

Metaphysical Matters

It seems to me that the business of semantic vs. metaphysical vagueness is of a piece with the analytic-synthetic distinction. Indeed, the coherence of the present debate presupposes something in the neighborhood of analyticity. Consider any sentence S. Quineans agree that it is a patent truism that the truth or falsehood of S is due both to what the words in S mean and to the way the world is. They argue, however, that there is no clean way to separate those factors from each other. I would think that what we say about truth-value here should apply to vagueness as well. Let S be the statement that Harry is bald, where Harry is in fact borderline bald—whatever this comes to. The status of S as borderline is due in part to the meaning of the words ‘‘Harry’’, ‘‘bald’’, and, not to forget Bill Clinton’s example, ‘‘is’’. And the status of S is due in part to the way the world is, to the totality and arrangement of Harry’s hair, for example. But, on the philosophical perspective under discussion, there is no clean way to separate out those factors. Vagueness is due to language and to the way the world is. There is no sense to figuring out which of these is the chicken and which is the egg. Of course, not everyone accepts the Quinean perspective. Moreover, it may very well be that there is a compelling account of language and linguistic function, and a concomitant metaphysics, that makes room for metaphysical vagueness, vagueness that is due to the way the world is, independent of our linguistic practices.5 Following a theme of Robert Kraut [1992], it would come down to whether such an account was explanatorily illuminating—whether it sheds some light on some puzzling feature or aspect of reality, language, or whatever. Let a thousand flowers bloom, if they can. But to bloom here, the flower of metaphysical vagueness should show us something. The account of vagueness developed in this book turns on open-texture, the thesis that linguistic practices—what we say, think, and do—do not fix precise and exhaustive extensions of most predicates. Competent speakers can sometimes go either way with borderline cases of vague predicates. I do not know, and, for now, do not much care whether this makes vagueness a metaphysical or (merely) a semantic and representational matter. The world, and perhaps the world, is a certain way, and we are destined to represent it, or it, using vague language. Douglas Adams’s delightful novel Mostly Harmless, the fifth (and, alas, final) member of the Hitchhiker’s trilogy, begins thus: ‘‘The history of the galaxy has got a little muddled, for a number of reasons: partly because those who are trying to keep track of it have got a little muddled, but also because some muddling things have been happening anyway.’’ In the novel, the muddling things involve time travel. It seems to me that the issue of metaphysical vagueness has become muddled. I won’t say that the philosophers trying to track this feature have themselves become muddled. It is the phenomenon itself that is muddled. 5 For some recent, interesting proposals for metaphysical vagueness see Rosen and Smith [2004] and Akiba [2004]. The former turns on the metaphysics of properties and objects, and the latter proposes a ‘‘precisification’’ dimension to reality, along the lines of temporal dimensions and possible worlds. I briefly revisit the notion of analyticity in the Appendix.

Metaphysical Matters

197

As I see things, vagueness is a linguistic phenomenon, tied to the kinds of languages that humans have evolved, and are continuing to evolve. But it is just as true to say that vagueness is due to the world we find ourselves in and try to negotiate and make sense of. It is also due, just as much, to the kinds of beings we are. I see no need to sort out these factors, and pin the phenomenon of vagueness on any one of them. 2. OBJECTIVITY Despite the aforementioned Kantian dilemma, there does seem to be a distinction between subjective judgments, like ‘‘broccoli is disgusting’’, and objective judgments like ‘‘water molecules contain hydrogen’’ and ‘‘the Yankees are evil’’ (just kidding). Objectivity might be vague, of course, and perhaps there is no pure or complete objectivity. Nevertheless, the distinction does play an important role in both everyday and intellectual discourse, and I presume that it is not an illusion. One item of present interest is the objectivity of vague discourse. If vagueness is (merely) a result of our representational apparatus, and not part of the fundamental fabric of the universe, then one might think that vague statements are not objective. Maybe this is what is meant by vagueness being purely linguistic, and not metaphysical—not part of the ultimate fabric of reality. The terrain here is interesting. It depends on what sort of statement or judgment that we are talking about, and it depends on what we mean by ‘‘objective’’. Let B be a predicate, and let a be the name of an arbitrary object in the field of B. Let b be the name of a borderline case for B, and let c be the name of a determinate case for B (so that Bc is determinately true). We can inquire about the objectivity of a number of different statements concerning this predicate and these objects. Among them are the following: (1) (2) (3) (4)

B is a vague predicate. a is a borderline case of B. Bb. Bc.

According to the view developed in this book, a is a borderline case of B if the sentence Ba is unsettled: language users have established truth conditions for the sentences Ba and :Ba, and the (presumably) non-linguistic facts have not determined that either of these conditions is met. The predicate B is vague if it has, or can have, borderline cases. So on the present account of vagueness, statements of types (1) and (2) concern truth conditions, as established by competent speakers. They are explicitly about linguistic matters. Of course, non-linguistic matters are typically involved as well, especially in type (2) statements. The status of a man as borderline bald depends at least in part on the amount and arrangement of the hair

198

Metaphysical Matters

on his head, and the status of an object as borderline red depends, in part, on the wavelengths of the light it reflects in normal conditions. Still, if we think that objectivity requires independence from human languages, then statements of type (1) and type (2) are not objective. But this is too hasty. Statements of type (1) or (2) are explicitly about language. I would think that, in at least some sense of the term, the fact that the word ‘‘bald’’ is an English word with exactly four letters is objective. It is not a matter of someone judging it to be so, at least not anymore. So far, I see no reason that type (1) and type (2) statements should not be objective in that same sense. On the present thesis of open-texture, competent speakers can sometimes go either way with borderline cases of vague predicates. It follows that vague predicates are at least partially judgment-dependent, at least in their borderline regions (see Ch. 1 x8 with apologies for the multiple hedging). Intuitively, judgment-dependence is a hallmark of the failure of objectivity. After all, what can it mean to be objective, if not ‘‘independent of human judgments’’? So one would think that type (3) statements are not objective. Yet one would think that at least some type (4) statements are objective. Intuitively, it is as objective as anything is that Yul Brynner is bald. He has no hair whatsoever, and that much is not a matter of anyone’s judgments. It is strange that two different cases of the same predicate can differ in their status as objective—or is it? Even these preliminary remarks should make it clear that it is not all that clear just what it is to be objective. There is no consensus on this philosophical matter. Crispin Wright [1992] provides one of the most sustained and detailed treatments of objectivity in recent years. I propose to apply Wright’s criteria to statements of the above types (1)–(4), assuming the correctness of the present account of vagueness. Philosophy is a holistic enterprise. The plan here is to test certain statements against Wright’s criteria for objectivity, and to test Wright’s criteria against intuitions concerning the objectivity of the foregoing statements. The end result, I hope, is that we will learn something more about open-texture, something about objectivity, and something about the ways in which language manages to latch onto a world that, presumably, is not of our making. According to Wright, objectivity is not a univocal concept. There are different notions or axes of objectivity, and a given chunk of discourse can exhibit some of these and not others. This might help to explain why intuitions come into conflict so quickly in the present area. Statements are objective in some senses of the notion, and not so on others. The first item in Wright’s arsenal is the centerpiece of Michael Dummett’s semantic anti-realism, the principle of epistemic constraint (EC) (see, e.g. Dummett 1973). Let P be a variable ranging over declarative sentences in the discourse. The principle of epistemic constraint has two formulations: ‘‘If P is true, then evidence is available that it is so’’ (p. 41) and ‘‘P $ P-may-be-known’’ (p. 75). These formulations seem to be different, depending on what ‘‘evidence is

Metaphysical Matters

199

available’’ and ‘‘may be known’’ come to. But we will let that matter pass, and go with the latter formulation. According to Wright, if epistemic constraint fails for a given area of discourse— if there are (or might be) true propositions in that area whose truth cannot become known—then that discourse can only have an objective interpretation. He writes: To conceive that our understanding of statements in a certain discourse is fixed . . . by assigning them conditions of potentially evidence-transcendent truth is to grant that, if the world co-operates, the truth or falsity of any such statement may be settled beyond our ken. So . . . we are forced to recognise a distinction between the kind of state of affairs which makes such a statement acceptable, in light of whatever standards inform our practice of the discourse to which it belongs, and what makes it actually true. The truth of such a statement is bestowed on it independently of any standard we do or can apply . . . Realism in Dummett’s sense is thus one way of laying the essential groundwork for the idea that our thought aspires to reflect a reality whose character is entirely independent of us and our cognitive operations. (Wright 1992: 4)

In other words, epistemic constraint is necessary for any sort of non-objective account of a discourse. According to Wright, the converse of this fails: epistemic constraint is not sufficient for the failure of objectivity. Even if a given discourse is epistemically constrained, it is consistent to ‘‘retain the idea that [the] discourse . . . answers to states of affairs which, on at least some proper understandings of the term, are independent of us’’. It is only when epistemic constraint holds that Wright’s other criteria—width of cosmological role, the Euthyphro contrast, cognitive command—come into play. Here lies the multi-dimensionality of the notion of objectivity. So we start here with epistemic constraint. Perhaps the most central tenet of Timothy Williamson’s [1994] and Roy Sorenson’s [2001] epistemicism is that every type (3) statement is epistemically unconstrained. For the epistemicist, bivalence holds for vague predicates: every instance is either true or false. In the borderline area, it is simply not known, or knowable, whether the predicate holds. So, for example, a man can be bald without it being knowable that he is bald.6 Williamson holds, in addition, that there are determinate (i.e. knowable) cases of vague predicates which cannot be known to be determinate (i.e. knowable). In other words, the KK-thesis fails. Similarly, there are borderline (and thus unknowable) cases of a vague predicate that cannot be known to be borderline cases of that predicate. So epistemic constraint fails for some type (2) statements. It is part of Williamson’s overall philosophy of language that in the case of vague predicates, we cannot know their truth conditions (although we can, of 6 For the epistemicist, type (4) statements may or may not be epistemically constrained, depending on the predicate in question. If a type (4) statement is true but not knowable, then this unknowability is not due to vagueness.

200

Metaphysical Matters

course, know the T-sentences). We cannot know, for example, the exact cut-off for ‘‘tall’’, even though the cut-off is part of the truth conditions for the predicate. As Williamson sees things, truth conditions are determined in a complex way by the behavior of (competent) speakers of the language, and no speaker is aware of all of this behavior. If truth conditions cannot themselves become known, then I would think that it is sometimes unknowable whether a given predicate is vague. If so, then type (1) statements are epistemically unconstrained as well. Of course, our concern here is with the present account of vagueness, and not with epistemicism, and I propose to avoid, as much as possible, the underlying delicate and complex issues in the philosophy of language. However, since the present account turns directly on truth conditions, we must at least tentatively enter the quagmire. If Dummett’s global semantic anti-realism is correct, then every coherent discourse is epistemically constrained, and so there is no special issue concerning vagueness. Let us suppose, however, that semantic anti-realism is not true in general, and that at least some statements in some discourse are epistemically unconstrained. To make things simple, suppose, further, that some necessary truths are epistemically unconstrained. In particular, let F be an unknowable, necessary truth, and assume that it is known that F is either necessarily true or necessarily false (i.e. that F is not contingent). Let B be a predicate, such as ‘‘bald’’, which is known to be vague. Consider the complex predicate Nx defined as (:F&Bx). For any object b in the range of applicability of B, the thoughts and actions of speakers of the language have established truth conditions for Nb. Moreover, since :F is necessarily false, the facts determine that those truth conditions are never met. That is, for every b, Nb is determinately false, so N cannot have any borderline cases. Thus N is not vague.7 Now consider the complex predicate Vx defined as (F&Bx). By reasoning similar to the above, we see that any borderline case of B is a borderline case of V. So V is vague. It follows that at least some statements of type (1), to the effect that a given predicate is vague, are not epistemically constrained. We can know that V is vague, and that N is not vague, only if we can know that F is not necessarily false. And, by hypothesis, we cannot know this. Under the foregoing assumptions, some statements of type (2), that a given object is a borderline case of a given predicate, are also epistemically unconstrained. Let h be a borderline case of B. Then h is also a borderline case of V, but 7 One might think that the predicate N is vague since it has a vague component (namely B). I would prefer to stick to the official definition of this monograph, which disqualifies N since it can have no borderline cases (in the present sense of ‘‘borderline’’). But I realize that intuitions may differ here. Someone else might complain that predicates such as N do not count since they are mixed. If we were talking about humor, for example, it would not be fair to consider a conjunction of a joke with a statement of physics. There is no need to be definitive here. Perhaps ‘‘vague’’ is, in the end, a term of art, or perhaps N is a borderline case of ‘‘vague’’.

Metaphysical Matters

201

we cannot know that h is a borderline case of V, since by hypothesis, we cannot know that F is not necessarily false. Suppose, again, that h is a borderline case of B, and consider a conversational context in which Bh is true. To make things simple, we can just assume that Bh is explicitly on the conversational record. Then, in that context, Vh (i.e. F&Bh) is true. But no one can know this, since no one can know that F is true. So there are unknowable truths of type (3). Similarly, if p is a determinate case of B, then Vp is also an unknowable truth. So some type (4) statements are also epistemically unconstrained. As noted, these conclusions can be resisted by a global semantic anti-realist, who holds that there are no unknowable truths, and so no propositions like F. The conclusions might also be resisted if one holds that all unknowable truths are contingent, but the details of this would be tricky and would take us too far afield (if we have not strayed too far already). The point here is that truth conditions are sometimes complex. In some cases, it may not be known, or even knowable, if a part of some truth conditions trumps another part of those same truth conditions in a given situation. This leaves open the possibility that it may be unknowable whether a given predicate is vague—even if we know its truth conditions. And it leaves open the possibility that it may not be knowable whether a given object is borderline, or, if it is, whether the predicate holds in a given case. More generally, the truth conditions for a given complex statement may have objective and non-objective parts, and it may not be knowable in a given case which parts actually figure in the actual truth-value of the statement. So it may not be knowable whether the statement is objective. For the rest of this treatment, let us stick to relatively simple predicates, such as our paradigm cases of vagueness: ‘‘bald’’, ‘‘red’’, or ‘‘looks red’’. I take it that such predicates are known to be vague, and so there is no issue of the status of type (1) statements with respect to epistemic constraint. Statements of type (2), that a given object is a borderline case, concern the truth-conditions of the predicate, among other things. Such statements are thus in the neighborhood of what passes for higher-order vagueness in Ch. 5. One of the theoretical options delimited there is that it may not be knowable where the boundary between determinate cases and borderline cases of a vague predicate lies. Something in the neighborhood of epistemicism may hold in this arena. This would make statements of type (2) epistemically unconstrained. Another theoretical option proposed in Ch. 5 is that statements of type (2) are themselves judgment-dependent, and so, presumably epistemic constraint holds for them. This option leads to a hierarchy of ‘‘orders’’. On the present view of vagueness, if b is a borderline case of B, then Bb is true in a given conversational context if that statement is on the conversational record in that context, or is implied by something on the record. So statements of type (3) are epistemically constrained if it is knowable, in general, what is on the

202

Metaphysical Matters

record in a given conversation, and what is implied by what is on the record. This last seems plausible enough (if the logic is effective). If there is a doubt concerning a given statement, the participants can always just ask about it, and make it an explicit part of the conversation, one way or the other. There is some residual uncertainty concerning whether this would change the context. Moreover, whether a given item is on the record is itself a vague matter. I have no stake on how these matters turn out. Turning to type (4) statements, suppose that b is a determinate case of a paradigm vague predicate B. Then, putting nuances aside, no one can competently hold that :Bb, in any conversational context. Does it follow that Bb is knowable? This depends on the epistemic status of matters concerning competence, the truth conditions of the statements, etc. So, once again, we broach issues in the neighborhood of what passes for higher-order vagueness, and other matters that go beyond the scope of the present treatment of objectivity. Let us move on to Wright’s other axes of objectivity. The Euthyphro contrast concerns response-dependence or, better, judgment-dependence, which is, of course, a main theme of the present account of vagueness. I will indulge in a brief summary of the basic view. Recall Thesis (B) from x8 of Ch. 1 (taken, a bit out of context, from Raffman 1994: 69–70): [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. In effect, Thesis (B) is a formulation of epistemic constraint, stating that for the chunk of discourse in question, truth coincides with best opinion. Recall that, even in the best of cases, Thesis (B) has exceptions. We are dealing with typical competent subjects. Whenever Thesis (B) holds, even partially, there is a further question of which is the chicken and which the egg. A judgment-dependent, or Euthyphro, reading of Thesis (B) is that the judgments of competent subjects constitute membership in the category. In other words, what it is for a proposition to be true is for competent subjects to make the appropriate judgments under appropriate conditions. Arguably, humor is judgment-dependent: what makes something funny is that, typically, people tend to find it to be funny. In contrast, a Socratic reading of Thesis (B) is that competent judges accurately track an independent reality; their judgments do not constitute the very truth at issue. As far as I can tell, there is no reason why this distinction needs to be a sharp one. Perhaps the Socratic-Euthyphronic contrast is itself vague. It seems plausible, at least to me, that there are no purely Socratic predicates, and probably very few, if any, purely Euthyphronic exemplars. Stay tuned. A major thesis of the present book, developed explicitly in x8 of Ch. 1, is that vague predicates are judgment-dependent, at least in part, in their borderline area. So, according to the present account, type (3) statements are judgmentdependent. What of type (4) statements, which concern non-borderline cases? Arguably, some vague predicates, such as ‘‘looks red’’ or ‘‘looks bald’’, are

Metaphysical Matters

203

judgment-dependent throughout their range of application—for clear and borderline cases alike. Other vague predicates, such as ‘‘bald’’ or ‘‘rich’’, do not seem to be judgment-dependent for their determinate cases. Prima facie, it does not seem to be a matter of judgment that Yul Brynner is bald or that Bill Gates is rich. We encounter a theme broached just above. In a given context, the predicate gets its extension through both Euthyphronic and Socratic elements, and it is a question of which dominates in a given case. With the determinate cases, the Socratic, non-judgment-dependent elements dominate. In the borderline area, the Socratic elements are indecisive, and the judgment-dependent features take over. As we noted in Ch. 1, 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’’. I submit that the first clause here, that vague predicates are judgment-dependent in their borderline regions, holds in general. The second clause, that the determinate cases are Socratic, holds for some vague predicates and not for others. As noted just above, type (1) and type (2) statements, concerning whether a given predicate is vague and whether a given object is a borderline case, invoke the matters that pass for higher-order vagueness, as in Ch. 5. It depends on whether statements about competent speakers and the like can themselves be vague, and thus at least partially judgment-dependent. The present treatment is officially neutral on this. Moving on to another axis of objectivity, Wright defines an area of discourse to have a ‘‘wide cosmological role . . . just in case mention of the states of affairs of which it consists can feature in at least some kinds of explanation of contingencies which are not of that sort’’ (Wright 1992: 198). The width of cosmological role is ‘‘measured by the extent to which citing the kinds of states of affairs with which [the discourse] deals is potentially contributive to the explanation of things other than, or other than via, our being in attitudinal states which take such states of affairs as object’’. The idea here is that a discourse is apt for an objective construal, on this axis, if statements in that discourse figure in explanations provided within a wide range of other discourses. For example, Wright points out that the wetness of some rocks can explain ‘‘my perceiving, and hence believing, that the rocks are wet’’, ‘‘a small . . . child’s interest in his hands after he has touched the rocks’’, ‘‘my slipping and falling’’, and ‘‘the abundance of lichen growing on them’’ (Wright 1992: 197). So statements about rock wetness figure in explanations concerning perception, belief, the interests of children, the ability to negotiate terrain, and lichen growth. So rock-wetness-discourse has wide cosmological role, and, of course, it is eminently plausible to regard that discourse as objective. In contrast, Wright argues that moral discourse fails to have wide cosmological role: moral ‘‘states of affairs’’ do not figure in explanations of non-moral matters (other than

204

Metaphysical Matters

our moral beliefs, or via our moral beliefs). If this is correct, it tells against the objectivity of moral discourse, at least in part. Type (1) statements and type (2) statements do enjoy some cosmological role. One might cite the vagueness of a predicate, or the borderline status of an object, to explain why a given speaker sometimes hesitates in her verdicts, or why verdicts seem to vary with context. On the other hand, one might think that on the present view of vagueness, this is not much of an explanation. On the present view, it is part of what vagueness is for there to be hesitation and/or variation in the borderline region. So we are just explaining vagueness by citing vagueness. I submit, however, that in some cases at least, the proposed explanation is legitimate and illuminating. By citing vagueness in explanation, we mean to rule out other sources for the hesitancy or variation. We are saying, in effect, that the hesitancy and variation are not due to the speaker being ignorant of the meaning of the term, or ignorant of certain non-linguistic facts, or to her being irrational or having a seizure. The hesitation is due to its vagueness. Even in such cases, however, one might argue instead that the proper explanation of the hesitancy or variation is not the vagueness itself, but the widespread belief that the predicate is vague, or a widespread belief that the instance is borderline. For Wright, the width of cosmological role is compromised if explanations go via propositional attitudes concerning the discourse in question.8 In the present case, it is an empirical matter to determine whether the hesitancy or variation in use is due to any specific beliefs of the speakers. If the predicate in question is vague, and if the instance in question is borderline, then speakers can hesitate, and usage can vary in the borderline area, even if many of the speakers have not formulated an opinion over whether the predicate is vague, or whether the given instance is borderline. Indeed, we can have the hesitation and variation even if the speakers believe (wrongly) that the predicate is not vague, or that the instance is not borderline. They may not have a very good explanation of why they are hesitating. The proper explanation, in some cases at least, is that the predicate is vague, or that the instance is borderline, in the present sense, whether or not the speakers notice this. Type (4) statements, which apply a vague predicate to a determinate case, do enjoy a wide cosmological role. I would think that just about any explanation of any contingent event will involve determinate instances of vague predicates. Wright’s own example, used to illustrate the criterion, concerns the wetness of rocks. The predicate ‘‘wet’’ is surely vague, and, presumably, the rocks in question are determinately wet. Similarly, the baldness of a (determinately bald) man might explain his interest in sun-screen and Rogaine, why he does not carry 8 For example, one can explain why a certain person is shunned in a community by citing the fact that she is morally corrupt. Wright suggests, however, that in this case, the proper explanation is not so much that the person is morally corrupt, but that people in the community believe her to be corrupt.

Metaphysical Matters

205

around a comb, why he does not invest in shampoo, and why he lacks confidence in some social situations. Perhaps these examples go via the propositional attitudes that our man has toward his hair loss. The baldness of a bald man might also explain why he is prone to sunburn on his head. Type (3) statements are the application of vague predicate to a borderline case. Here things are less straightforward. One might think that to some extent, such statements enjoy a cosmological role. Suppose, for example, that some rocks are borderline wet, and the statement that they are wet is on the conversational record in a given context (and so the rocks are wet in that context). The statement can help explain the presence of some lichen growing on the rocks; my perceiving, and hence believing, that the rocks are wet; why there is some slippage when I walk on them, etc. If the rocks are borderline wet, then in another context, the statement that those rocks are not wet may be on the conversational record (and so they will not be wet in that context). And this explains why there is not much lichen growing on them, why the footing on them is not too bad, etc. This does not sound right. If the rocks are borderline wet, then they are wet in a given context only if their wetness follows from certain competent judgments in that context. It is hard to see how the fact that the rocks are competently judged to be wet explains why there is some lichen growing on the rocks, or why there is some slippage when I walk on them. How can a judgment-dependent fact explain something that is not judgment-dependent, especially when the judgment is optional? By hypothesis, the rocks could just as easily be judged to be not wet in a very similar context. That would destroy the explanation. I suggest that in these cases, the proper explanation is that the rocks are wet enough to be competently judged wet, and perhaps also that they are dry enough to be judged as not wet. In effect, the explanation is that the rocks are borderline wet, which is a type (2) statement. The statement that the rocks are borderline wet explains why there is some, but not much lichen growing on them, why there is some, but not much slippage as we try to walk on them, etc. In like manner, the statement that a man is borderline bald might explain why he is interested in Rogaine, but not in sun-screen, or why he combs his hair across the top, to hide the bald spot. Let us now turn to Wright’s final axis of objectivity. Suppose that a given discourse describes mind-independent features of some mind-independent reality. Then if two speakers disagree about something in that area, at least one of them has misrepresented that reality. In typical cases, one of them has a cognitive shortcoming. Suppose, for example, that two people disagree over whether there are at least three elm trees in a given backyard. Then at least one of the people did not look carefully enough, has faulty eyesight or memory, does not know what an elm is, etc. On the other hand, if a discourse does not serve to describe a mindindependent realm, then disagreements in that discourse need not involve cognitive shortcoming on the part of either party. Two people can disagree about whether a certain dish is tasty without either of them having any cognitive

206

Metaphysical Matters

shortcoming. One of them may have a warped sense of taste, or no sense of taste, but there need be nothing wrong with his cognitive faculties. Wright [1992: 92] writes that A discourse exhibits Cognitive Command if and only if it is a priori that differences of opinion arising within it can be satisfactorily explained only in terms of ‘‘divergent input’’, that is, the disputants working on the basis of different information (and hence guilty of ignorance or error . . . ), or ‘‘unsuitable conditions’’ (resulting in inattention or distraction and so in inferential error, or oversight of data, and so on), or ‘‘malfunction’’ (for example, prejudicial assessment of data . . . or dogma, or failings in other categories already listed).

In other words, if cognitive command fails, then (cognitively) blameless disagreement is possible, or at least it cannot be ruled out a priori. This is another place where epistemic constraint is presupposed in Wright’s analysis of objectivity. If an objective discourse is epistemically unconstrained, then it very well might be that speakers can disagree without either of them displaying a cognitive shortcoming. But if epistemic constraint holds for a chunk of discourse, then cognitive command does seem to bear on the objectivity of the discourse. Suppose, then, that two speakers disagree over a type (1) statement or a type (2) statement. One of them says that a given predicate B is vague and the other says that it is not, or one of them says that a is a borderline case of B and the other denies this. On the present account of vagueness, they either disagree about the truth conditions of B, or they disagree on the non-linguistic facts (or on the relationship between the former and the latter). As noted above, the present account of vagueness is neutral on whether some of these matters are epistemically constrained. If there are unknowable truths of this sort, then cognitive command is not relevant. Suppose, then, that statements about truth conditions are epistemically constrained. Then I would think that a disagreement over them does indicate a cognitive shortcoming on the part of at least one of the parties. She has misunderstood the language. Admittedly, intuitions are not all that strong here, and I do not know of any compelling arguments. A disagreement over type (1) or type (2) statements might also turn on a disagreement concerning the non-linguistic facts. This pushes the concern to another arena: it depends on the nature of the non-linguistic facts in question. Cognitive command clearly holds for some type (4) statements (i.e. those that apply vague predicates to determinate cases). If two people disagree over whether Yul Brynner is bald, or over whether Bill Gates is rich, then one of them displays a cognitive shortcoming. He is badly misinformed on some putatively objective matters, such as how much money Bill Gates has or how much hair Yul Brynner sports, or else he has not looked carefully enough, or his perception is at fault, or . . . In light of open-texture, type (3) statements concerning borderline cases are the most interesting ones here. The present view is that if b is a borderline case of B,

Metaphysical Matters

207

then in some contexts, a competent speaker can go either way without sinning against the meaning of the words, or getting the non-linguistic facts wrong. At first blush, then, it would seem that cognitive command fails in the borderline area of a vague predicate. Blameless disagreement is possible. If we stick to the letter of the present view, this is not quite right. Suppose that one speaker competently asserts Bb and another competently asserts :Bb in a different conversation. On the present view, they do not really disagree since the contexts are different. The sentence Bb is true in the first context and false in the second. It is similar to a statement that a certain person is rich, when the comparison class is philosophy professors, and a statement that the same person is not rich, with a different comparison class. Or consider an utterance of ‘‘I am hungry’’ by one speaker and an utterance of ‘‘I am not hungry’’ by another (which, of course, is not to say that vague predicates just are indexicals). No disagreement, and thus no conflict. We seem to have ruled out blameless disagreement in the borderline region by ruling out genuine disagreement. Well, it can’t be this simple either. We might get a genuine disagreement if we hold the context fixed, and assume that our speakers are part of the same conversation at the same time. If b is in fact a borderline case of B (as required for type (3) statements), then our speakers might be in disagreement as to what is already on the conversational record. I won’t venture an opinion on the status of disputes like this. The record is volatile, and will undoubtedly be changed in the course of the dispute. Another possibility is that our speakers are each proposing to update the conversational record in incompatible ways. I see no need to adjudicate the status of this sort of dispute either. Suppose that in a course of a conversation, the statement Bb is unsettled. A participant asserts Bb and someone else challenges this, asserting :Bb, thus preventing the first judgment from going on the conversational score. There is at least a sense in which our two conversationalists disagree. By hypothesis, they are in the same context, and they have a different opinion on the same sentence. In this case, it seems, there is blameless disagreement. Cognitive command seems to fail. Later in the book, Wright [1992: 144] adds some qualifications to the formulation of cognitive command. He writes that a discourse exerts cognitive command if and only if ‘‘It is a priori that differences of opinion formulated within the discourse, unless excusable as a result of vagueness in a disputed statement, or in the standards of acceptability, or variation in personal evidence thresholds, so to speak, will involve something which may properly be described as a cognitive shortcoming.’’ That is, Wright holds that blameless disagreement that turns on vagueness does not undermine cognitive command. Why is there an exception for vagueness? What happened to the original motivation for the criterion of cognitive command, which did not concern vagueness one way or the other? Is this an instance of what Imre Lakatos calls ‘‘monster-barring’’?

Metaphysical Matters

208

Wright [ibid.] goes on to explain the qualification: it will have been obvious to the alert reader . . . that any fully satisfactory version of the Cognitive Command constraint would have somehow to be fashioned to allow for vagueness. Even in the most robustly objective area of enquiry, vagueness—whether in the content of a statement at issue, or in the standards for appraising it, or in what one might style permissible thresholds of evidence—may set up the possibility of disagreements in which nothing worth regarding as a cognitive shortcoming is involved. It’s tempting to say, indeed, 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.

This is dead right, and it is worth taking a moment to put this insight into the present context. I submit that the key item in play here is open-texture. On the present view, once again, borderline cases are ‘‘unsettled’’, in the sense that the relevant objective facts (if there are any) give out, so to speak, and do not determine a verdict. With open-texture, borderline cases of vague predicates are just those where competent speakers can go either way. Once again, in the borderline area, competent judgments determine the extension of the predicate. So, of course, the difference between judgments in this area does not thereby show any cognitive shortcoming on the part of either party. So cognitive command is not relevant to type (3) statements. They are among the explicit, and well-motivated exceptions. With open-texture, we can have blameless difference, and perhaps even blameless, genuine disagreement on such statements, but this alone does not undermine the overall objectivity of the discourse. To conclude, then, if we focus on simple vague predicates, then typical statements of types (1), (2), and (4) easily pass Wright’s various criteria for objectivity. Moreover, there is nothing in the present examples to undermine the intuitions that underlie Wright’s criteria. The statements pass the tests in light of the underlying motivations that lie behind the analysis. So we can at least tentatively conclude that we have something that deserves to be called the objectivity of (some) vague discourse. But this is quite different from the traditional issue of metaphysical vs. (merely) linguistic vagueness. We did not come to the present conclusions through metaphysical inquiry into the underlying fabric of reality. Following Wright, we focused instead on the nature of vague discourses, and the role that such discourses play in our overall intellectual lives. An advocate of metaphysical vagueness might explain the objectivity of vague discourse by citing the existence of mind-independent vague properties and/or vague objects. That is, our metaphysician might claim that the reason why statements of types (1), (2), and (4) are objective is that the language manages to refer to, or denote, or otherwise latch onto vague items that characterize the underlying fabric of reality.9 9

I am indebted to Crispin Wright for helping me to see how to put this point.

Metaphysical Matters

209

The key place where vague discourse fails Wright’s criteria is the judgmentdependence of predicates in their borderline regions. This is, of course, a central feature of the present account of vagueness. I would think that this undermines the explanation of objectivity proposed by the advocate of metaphysical vagueness. Is it possible for the extension of a mind-independent property to be dependent on the judgments of competent speakers of a natural language, even in part? Are human judgments that much in tune with the ultimate constituents of reality? This query is, of course, in line with the impatience with metaphysical vagueness that I expressed in the previous section. In general, the extension of any predicate on a given occasion is a function of its meaning and features from the context of utterance. With vague predicates, judgment-dependent matters sometimes figure in how the extension is fixed. For determinate cases, the judgment-dependent matters in question are trumped by other factors that fix the extension. In the borderline region, the judgmentindependent features give out, and, consequently, the judgment-dependent features of the process dominate. The predicate becomes Euthyphronic. As I see things, every vague predicate has (or can have) such a Euthyphronic region. But this does not undermine the overall objectivity of the predicate, or the discourse in which it figures. If it did, there would be precious little objectivity anywhere.

APPENDIX

Waismann on Open-Texture and Analyticity I close this monograph with a brief account of Friedrich Waismann’s notion of opentexture, its connection to vagueness, and his view on the analytic-synthetic distinction.1 This last is connected with a topic of the final chapter of this book, the objectivity of vague statements. Waismann introduces his notion of open-texture in an attack on crude phenomenalism, the view that one can understand any cognitively significant statement in terms of sense-data.2 The failure of the verificationist program: is not, as has been suggested, due to the poverty of our language which lacks the vocabulary for describing all the minute details of sense experience, nor is it due to the difficulties inherent in producing an infinite combination of sense-datum statements, though all these things may contribute to it. In the main it is due to a factor which, though it is very important and really quite obvious, has to my knowledge never been noticed—to the ‘‘open texture’’ of most of our empirical concepts. (Waismann 1968: 118–19)

To motivate the notion, Waismann employs some thought experiments. Here is one of them: Suppose I have to verify a statement such as ‘‘There is a cat next door’’; suppose I go over to the next room, open the door, look into it and actually see a cat. Is this enough to prove my statement? . . . What . . . should I say when that creature later on grew to a gigantic size? Or if it showed some queer behavior usually not to be found with cats, say, if, under certain conditions it could be revived from death whereas normal cats could not? Shall I, in such a case, say that a new species has come into being? Or that it was a cat with extraordinary properties? . . . The fact that in many cases there is no such thing as a conclusive verification is connected to the fact that most of our empirical concepts are not delimited in all possible directions.

The last observation is the key to Waismann’s notion of open-texture. We language users introduce terms to apply to certain objects or kinds of objects, and, of course, the terms are supposed to fail to apply to certain objects or kinds of objects. As we introduce the terms, and use them in practice, we cannot be sure that every possible situation is covered. This applies even in science, to what are now called ‘‘natural kinds’’: The notion of gold seems to be defined with absolute precision, say by the spectrum of gold with its characteristic lines. Now what would you say if a substance was discovered that looked like gold, satisfied all the chemical tests for gold, whilst it emitted a new sort of radiation? ‘‘But such things do not happen.’’ Quite so; but they might happen, and that is enough to show that we can never exclude altogether the possibility of some unforseen situation arising in which we shall have to modify our definition. Try as we may, no concept is limited in such a way that there is no room for 1 I am indebted to Neil Cooper and Crispin Wright for helping me to put Waismann’s ideas into their proper perspective. 2 Waismann credits William Kneale for suggesting the term as a translation of a German phrase that Waismann had coined: Porosita¨t der Begriffe.

Appendix

211

any doubt. We introduce a concept and limit it in some directions; for instance we define gold in contrast to some other metals such as alloys. This suffices for our present needs, and we do not probe any farther. We tend to overlook the fact that there are always other directions in which the concept has not been defined. . . . we could easily imagine conditions which would necessitate new limitations. In short, it is not possible to define a concept like gold with absolute precision; i.e., in such a way that every nook and cranny is blocked against entry of doubt. That is what is meant by the open texture of a concept. (Waismann 1968: 120)

Immediately after this passage, Waismann relates his notion of open-texture to the topic of the present book: Vagueness should be distinguished from open texture. A word which is actually used in a fluctuating way (such as ‘‘heap’’ or ‘‘pink’’) is said to be vague; a term like ‘‘gold’’, though its actual use may not be vague, is non-exhaustive or of an open texture in that we can never fill up all the possible gaps through which a doubt may seep in. Open texture, then, is something like possibility of vagueness. Vagueness can be remedied by giving more accurate rules, open texture cannot.

This, unfortunately, is the full extent of the treatment of vagueness in Waismann [1968]. His agenda lies elsewhere. Wasimann’s use of the term ‘‘open-texture’’ is not exactly the same as the present one, although they are closely related. His ‘‘open-texture’’ is closer to the present notion of ‘‘indeterminate’’ or ‘‘unsettled’’. As far as I know, Waismann did not explicitly endorse the present idea that competent speakers can go either way in different contexts with a vague predicate, but this seems implicit in much of his work. He does say that vague terms are used ‘‘in a fluctuating way’’. All we need to get to the present thesis of open-texture is to note that at least some of this ‘‘fluctuation’’ is perfectly correct, consistent with the meaning of the terms, the rules for using them, and the non-linguistic facts. I do not know what Waismann means by ‘‘remedying’’ vagueness by giving more accurate rules. It is a commonplace that vagueness can be lessened by ‘‘giving more accurate rules’’ or deciding borderline cases. The modification in the rules for a vague predicate can be made permanent by changing or extending the language—we are masters of that, after all—or the modification can be done on the fly, temporarily, via a conversational record. This, of course, is the main theme of the present work. But to lessen vagueness in this way does not usually eliminate it. Even after sharpening, open-texture—in both Waismann’s and the present sense—remains. Waismann [1968: 122] waxes poetic: ‘‘Every description stretches, as it were, into a horizon of open possibilities: However far I go, I shall always carry this horizon with me.’’ The phrase ‘‘open-texture’’ does not appear in Waismann’s treatment of the analyticsynthetic distinction in a lengthy article (or articles) published serially in Analysis from December 1949 until March 1953.3 Nevertheless, the notion clearly plays a central role in the essay(s). Consider, for example, what he says in the first installment about the notion of ‘‘definition’’: Whether conditions can be specified which determine, without any doubt, what a definition is, I do not know. . . . [I]n view of the indefiniteness with which nearly all the terms of word language are used; and the need for leaving, at least, some freedom for adjusting them to new situations which may crop up—would Aristotle have considered the case of a recursive definition?—I doubt the wisdom of pressing for a hard-and-fast rule . . . It is perhaps better to keep a term like definition 3 Waismann [1949, 1950, 1951a, 1951b, 1952, 1953]. Each of the articles, including the last, ends ‘‘To be continued’’. So I presume that the work was never completed.

212

Appendix

flexible and make a decision, if the need arises, only in individual cases without anticipating the issue. (Waismann 1949: 39)

That is, Waismann proposes that the notion of ‘‘definition’’ be left open, and that we should not seek for a hard and fast characterization of the notion, giving necessary and sufficient conditions for something to be a definition. Flexibility can be a virtue. The broad outlines of Waismann’s philosophy of language are at least relatively clear. Language is an evolving phenomenon. As new situations are encountered, and as new scientific theories develop, the extensions and anti-extensions of various predicates change. The predicates become sharper and, importantly, sometimes the boundaries move. In some cases, at least, there is no need to decide, on hard metaphysical or semantic grounds, whether the application of a given predicate to a new case represents a change in its meaning or an extension or application of its old meaning. And even if we focus on a given period of time, language use is not univocal: Simply . . . to refer to ‘‘the’’ ordinary use is naive. There are uses, differing from one another in many ways, e.g. according to geography, taste, social standing, special purpose to be served and so forth. This has long been recognized by linguists . . . [These] particular ways of using language loosely [revolve] around a—not too clearly defined—central body, the standard speech . . . [O]ne may . . . speak of a prevailing use of language, a use, however, which by degrees shades into less established ones. And what is right, appropriate, in the one may be slightly wrong, wrong, or out of place in others. And this whole picture is in a state of flux. One must indeed be blind not to see that there is something unsettled about language; that it is a living and growing thing, adapting itself to new sorts of situations, groping for new sorts of expression, forever changing. (Waismann 1951b: 122–3)

A similar idea is expressed in the next article in the series (Waismann 1952: 6): ‘‘Language is an instrument that must, as occasion requires, be bent to one’s purpose.’’ And in the final installment: ‘‘What lies at the root of this is something of great significance, the fact, namely, that language is never complete for the expression of all ideas, on the contrary, that it has an essential openness’’ (Waismann 1953: 81–2). The Wittgensteinian themes in the series are often explicit. As Waismann sees things, analyticity is vague, and subject to open-texture, both in his and the present sense of the term: I have defined ‘‘analytic’’ in terms of ‘‘logical truth’’, and further in terms of certain ‘‘operators’’ used in transforming a given sentence into a truth of logic. The accuracy of this definition will thus essentially depend on the clarity and precision of the terms used in the definition. If these were precisely bounded concepts, the same would hold of ‘‘analytic’’; if, on the other hand, they should turn out to be even ever so slightly ambiguous, blurred, or indeterminate, this would affect the concept of analytic with exactly the same degree of inaccuracy . . . I shall try to show that both concepts [‘‘operator’’ and ‘‘logical truth’’] are more or less blurred, and that in consequence of this the conception of analytic, too, cannot be precisely defined . . . [I]t is significant that we do not only ‘‘find out’’ that a given statement is analytic; we more often precisify the use of language, chart the logical force of an expression, by declaring such-and-such a statement to be analytic . . . It is precisely because, in the case of ‘‘analytic’’, the boundary is left open somewhat that, in a special instance, we may, or may not, recognize a statement as analytic. (Waismann 1950: 25)

The appearance of these articles coincides with the initial phases of W. V. O. Quine’s celebrated attack on the notion of analyticity, which was based at least in part on similar

Appendix

213

observations concerning language. The crude phenomenalism that is the target of Waismann [1968] is essentially the second dogma—reductionism—attacked by Quine’s ‘‘Two Dogmas of Empiricism’’ [1951]. And there is more than a family resemblance between the argument alluded to in the first paragraph of the last-quoted passage and that of the opening sections of ‘‘Two Dogmas’’. Quine’s claim, I take it, is not that there are no coherent notions of ‘‘meaning’’, ‘‘necessity’’, and ‘‘analyticity’’ to be had, but that these notions do not have a special role to play in epistemology. So-called analytic truths are known in the same way as any other sentences, via their role in the web of belief. At any given time, there are varied connections between sentences. Some of these connections turn directly on factual matters, some are causal, some are logical, some are mediated by language use, etc. But there is no sense to be had in precisely separating out these factors. To use the now tired slogan, only the web-of-belief, as a whole, faces the tribunal of experience. Quine’s main target, I think, was the special epistemological status assigned to mathematics and other analytic truths by the Vienna Circle, with which Waismann was associated. Given Waismann’s views on the evolving nature of language, and the concomitant notion of open-texture, it would seem that he would agree with Quine that analytic truths are not epistemologically sacrosanct (and thus immune to revision). Indeed, he points out that major advances in science sometimes—indeed usually—demand revisions in the use of common terms, to the extent that what was an analytic truth becomes false: ‘‘breaking away from the norm is sometimes the only way of making oneself understood’’ (1953: 84). Did scientists contradict themselves when they said that atoms have parts? What is it to be an atom? I recall hearing that one of the schoolmen thought that Galileo had gone mad when he declared that the earth moves. Didn’t Galileo understand the meaning of ‘‘motion’’? Was he proposing a new meaning for the word? Waismann [1952] illustrates the general point in some detail with the evolution of the word ‘‘simultaneous’’. The main innovative theses of the theory of relativity violated the previous meaning of that word. Yet Einstein eventually was able to make himself understood. One might think that in cases like these, a new word, with a new meaning, is coined with the same spelling as an old word, or one might think that an old notion has found new applications. Did Galileo and Einstein each discover a hidden and previously unnoticed relativity in certain words? Or did they coin new terms, to replace the old, scientifically misleading terms? According to Waismann, there is often no need, and no reason, to decide what counts as a change in meaning and what counts as the extension of an old meaning to new cases, going on as before. In some cases, we encounter borderline cases (in the present sense) of meaning-change. As Waismann put it, a bit earlier: ‘‘there are no precise rules governing the use of words like ‘time’, ‘pain’, etc., and that consequently to speak of the ‘meaning’ of a word, and to ask whether it has, or has not changed in meaning, is to operate with too blurred an expression’’ (Waismann 1951a: 53). The same goes for the other philosophical notions that come under the Quinean attack: ‘‘The reason for this wavering between opposite poles is that this is a case in which the philosophical antithesis ‘contingent-necessary’ loses its edge. [There is] a wide class of sentences which, so to speak, are on the borderline between necessary and contingent, the a priori and the empirical’’ (ibid. 54). Quine and Waismann thus come to similar conclusions. But they part company on the significance of these conclusions—on where philosophy goes from there. Quine, of course, was out to debunk (or deconstruct) the notion of analyticity showing it to have no

214

Appendix

legitimate scientific or philosophical role to play. The everyday notions of ‘‘meaning’’ and ‘‘same meaning’’ are not scientifically respectable, and should play no part in our attempt to make sense of experience, and thus of the world around us. The same goes for modal notions generally: ‘‘We should be within our rights in holding that no formulation of any part of science is definitive so long as it remains couched in idioms of . . . modality . . . Such good uses as the modalities are ever put to can probably be served in ways that are clearer and already known’’ (Quine 1986: 33–4). He also inherited from the logical positivists a distaste for vagueness, figuring that it should be eliminated from a properly regimented language (see e.g. Quine 1981). Waismann, in contrast, was not out to debunk notions such as analyticity, and he was no enemy of modality and vagueness. For Waismann, these are rich and important notions, essential to any understanding of language. And, for Waismann, understanding language is a key component in coming to understand how we interact with, and come to grasp, the external world. Properly understanding language, and ourselves, requires a grasp of the notion of open-texture. Analyticity is the quintessential open-textured notion. It is far from useless. Of course, there are some sentences that are true solely in virtue of meaning, and thus do not depend on the way the world is. The standard example, ‘‘no bachelor is married’’ is surely one such. So are tautologies. No one in their right mind would spend any time trying to confirm such sentences. With Waismann, however, I submit that the analytic/ synthetic distinction is vague.4 Some of Quine’s favorite examples, such as ‘‘all green things are extended’’, are borderline analytic. Waismann [1951] devotes a lot of space to exploring the status of the sentence ‘‘Time is measurable.’’ Surely this is another case of borderline analyticity. The vagueness of the notion of analyticity, and related notions, is consonant with the present overall account. Indeed, the present view handles this vagueness in stride. Consider, for example, synonymy, which is prima facie a binary relation among linguistic items of the same grammatical type. As is often pointed out, there is a word ‘‘synonymous’’ in English, with an established use. I presume that no one, Quinean or otherwise, would deny that. Speakers apply the word smoothly, without much difficulty (most of the time), and there is wide consensus on a large range of examples (cf. Grice and Strawson 1956). The same goes for other, typical vague terms such as ‘‘bald’’. So, on the present account, whether two expressions are synonymous is sometimes a judgmentdependent matter, at least in part. Calling two expressions synonymous indicates how the expressions are to be treated in the conversation. If the usage becomes established, then we have a communal decision to use words a certain way. And this helps fix how we, the community of language users, approach the world—for the time being. Or expressions might be declared synonymous temporarily, on the conversational score. If synonymy is in fact a vague binary relation, then, in terms of the Ch. 6, meanings are quasi-abstract objects. They can be introduced by an abstraction principle: the meaning of S is identical to the meaning of T if and only if S is synonymous with T. And even the dreaded propositions are legitimate, quasi-abstract objects: a proposition is the meaning of 4 Perhaps there is a sorites series going in short steps from a clearly analytic sentence to a clearly synthetic sentence. If so, then we can do a forced march sorites, with results much like those for baldness. However, on the present view, the possibility of sorites is not a necessary condition of vagueness.

Appendix

215

a declarative sentence. And a sentence is analytic if it can be turned into a logical truth by substitution of synonymous expressions. So much for the digression into Waismann. For what it is worth, I think that the picture of language development and use that he sketches is substantially correct, and it provides some perspective for the present account of vagueness.

References Akiba, Ken [2004], ‘‘Vagueness in the World’’, Nouˆs 38: 407–29. Benacerraf, P., and Putnam, H. [1983], Philosophy of Mathematics, 2nd edn. (Cambridge: Cambridge University Press). Boolos, G. [1987], ‘‘The Consistency of Frege’s Foundations of Arithmetic’’, in Judith Jarvis Thompson (ed.), On Being and Saying: Essays for Richard Cartwright, (Cambridge, Mass.: MIT), 3–20; repr. in Boolos [1998], 183–201. —— [1995], ‘‘Quotational Ambiguity’’, in Paolo Leonardi and Marco Santambrogrio (eds.), On Quine (Cambridge: Cambridge University Press), 283–96; repr. in Boolos [1998], 392–405. —— [1998], Logic, Logic, and Logic (Cambridge, Mass.: Harvard University Press). Brouwer, L. E. J. [1912], Intuitionisme en Formalisme (Gronigen: Noordhoof); trans. as ‘‘Intuitionism and Formalism’’, Benacerraf and Putnam [1983], 77–89. —— [1948], ‘‘Consciousness, Philosophy and Mathematics’’, Philosophy of Mathematics, Benacerraf and Putnam [1983], 90–6. Burgess, J. A., and Humberstone, I. L. [1987], ‘‘Natural Deduction Rules for a Logic of Vagueness’’, Erkenntnis 27: 197–229. Burns, L. C. [1991], Vagueness: An Investigation into Natural Languages and the Sorites Paradox (Dordrecht: Kluwer). Coffa, A. [1991], The Semantic Tradition from Kant to Carnap (Cambridge: Cambridge University Press). Cook, R. T. [2000], ‘‘Logic-as-modeling: A New Perspective on Formalization’’, Ph.D. Dissertation, The Ohio State University. —— [2002], ‘‘Vagueness and Mathematical Precision’’, Mind 111: 225–47. Copeland, B. Jack [1994], ‘‘On Vague Objects, Fuzzy Logic and Fractal Boundaries’’, The Southern Journal of Philosophy 33, suppl.: 83–96. Corcoran, J. [1973], ‘‘Gaps between Logical Theory and Mathematical Practice’’, in M. Bunge (ed.), The Methodological Unity of Science (Dordrecht: D. Reidel), 23–50. —— and Weaver, G. [1969], ‘‘Logical Consequence in Modal Logic I: Natural Deduction in S5’’, Notre Dame Journal of Formal Logic 10: 370–84. Davidson, D. [1984], Inquiries into Truth and Interpretation (Oxford, Clarendon). —— [1986], ‘‘A Nice Derangement of Epitaphs’’, in Richard Grandy and Richard Warner (ed.), Philosophical Grounds of Rationality (Oxford, Oxford University Press), 157–74. —— and Harman, G. (eds.) [1972], Semantics of Natural Language (Dordrecht, D. Reidel). Dummett, M. [1973], ‘‘The Philosophical Basis of Intuitionistic Logic’’, repr. in id., Truth and Other Enigmas (Cambridge, Mass.: Harvard University Press, 1978), 215–47; repr. in Benacerraf and Putnam [1983], 97–129. —— [1975], ‘‘Wang’s Paradox’’, Synthese 30: 301–24; repr. in Keefe and Smith [1997], 99–118. —— [1991], Frege: Philosophy of Mathematics (Cambridge, Mass.: Harvard University Press).

References

217

Dummett, M. [2000], Elements of Intuitionism, 2nd edn. (Oxford: Oxford University Press). Edgington, D. [1992], ‘‘Validity, Uncertainty and Vagueness’’, Analysis 52: 193–204. —— [1997], ‘‘Vagueness by Degrees’’, in Keefe and Smith [1997], 294–316. Evans, Gareth [1978], ‘‘Can there be Vague Objects?’’, Analysis 38: 208; repr. in Keefe and Smith [1997], 317. Fine, K. [1975], ‘‘Vagueness, Truth and Logic’’, Synthese 30: 265–300; repr. in Keefe and Smith [1997], 119–50. Fodor, J., and Lepore, E. [1996], ‘‘What Cannot Be Evaluated Cannot Be Evaluated, and It Cannot Be Supervaluated Either’’, Journal of Philosophy 93: 513–35. Frege, G. [1879], Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (Halle: Louis Nebert), trans. in J. van Heijenoort (ed.), From Frege to Go¨del (Cambridge, Mass.: Harvard University Press, 1967), 1–82. —— [1884], Die Grundlagen der Arithmetik (Breslau: Koebner); The Foundations of Arithmetic, trans. J. Austin, 2nd edn. (New York: Harper, 1960). —— [1893], Grundgesetze der Arithmetik 1 (Hildesheim: Olms). Gaifman, Haim [2005], ‘‘Vagueness, Tolerance and Contextual Logic’’ (forthcoming). Graff, Delia [2000], ‘‘Shifting Sands: An Interest-Relative Theory of Vagueness’’, Philosophical Topics 28: 45–81. —— [2001], ‘‘Phenomenal Continua and the Sorites’’, Mind 110: 905–35. —— [2003], ‘‘Gap Principles, Penumbral Consequence, and Infinitely Higher-Order Vagueness’’, in J. C. Beall (ed.), Liars and Heaps (Oxford: Oxford University Press), 195–221. Grice, P., and Strawson, P. F. [1956], ‘‘In Defense of a Dogma’’, Philosophical Review 65: 141–58. Hacking, I. [1979], ‘‘What Is Logic?’’, The Journal of Philosophy 76: 285–319. Hale, Bob, and Wright Crispin [2001], The Reason’s Proper Study (Oxford: Oxford University Press). Hodes, H. [1984], ‘‘Logicism and the Ontological Commitments of Arithmetic’’, Journal of Philosophy 81: 123–49. Horgan, T. [1994a], ‘‘Robust Vagueness and the Forced-March Sorites Paradox’’, Philosophical Perspectives 8: Logic and Language, 159–88. —— [1994b], ‘‘Transvaluationism: A Dionysian Approach to Vagueness’’, The Southern Journal of Philosophy 33, suppl.: 97–126. Hyde, Dominic [1997], ‘‘From Heaps and Gaps to Heaps of Gluts’’, Mind 106: 641–60. Jeshion, Robin [2001], ‘‘Frege’s Notions of Self-Evidence’’, Mind 110: 937–76. Kamp, Hans [1981], ‘‘The Paradox of the Heap’’, in U. Mo¨nnich (ed.), Aspects of Philosophical Logic (Dordrecht: D. Reidel), 225–77. Keefe, R. [2000], Theories of Vagueness (Cambridge: Cambridge University Press). —— and Smith, P. [1997], Vagueness: A Reader (Cambridge, Mass.: MIT). Kraut, R. [1992], ‘‘The Objectivity of Color and the Color of Objectivity’’, Philosophical Studies 68: 265–87. Kripke, S. [1965], ‘‘Semantical Analysis of Intuitionistic Logic I’’, in J. Crossley and M. Dummett (eds.), Formal Systems and Recursive Functions (Amsterdam: North Holland), 92–130.

218

References

Lewis, D. [1969], Convention (Cambridge, Mass.: Harvard University Press). —— [1970], ‘‘General Semantics’’, Synthese 22: 18–67. —— [1975], ‘‘Languages and Language’’, Minnesota Studies in the Philosophy of Science 7, ed. K. Gunderson (Minneapolis: University of Minnesota Press), 3–35. —— [1979], ‘‘Scorekeeping in a Language Game’’, Journal of Philosophical Logic 8: 339–59. —— [1986], On the Plurality of Worlds (Oxford: Basil Blackwell). —— [1988], ‘‘Vague Identity: Evans Misunderstood’’, Analysis 48: 128–30; repr. in Keefe and Smith [1997], 318–20. Lycan, W. [1984], Logical Form in Natural Language (Cambridge, Mass.: MIT). Machina, K. F. [1976], ‘‘Truth, Belief and Vagueness’’, Journal of Philosophical Logic 5: 47–78; repr. in Keefe and Smith [1997], 174–203. McCarty, C., and Tennant, N. [1987], ‘‘Skolem’s Paradox and Constructivism’’, Journal of Philosophical Logic 16: 165–202. McGee, Vann [1991], Truth, Vagueness, and Paradox (Indianapolis: Hackett). —— and McLaughlin, B. [1994], ‘‘Distinctions Without a Difference’’, Southern Journal of Philosophy 33, suppl.: 203–51. Merricks, Trenton [2001], ‘‘Varieties of Vagueness’’, Philosophical and Phenomenological Research 62: 145–57. Montague, R. [1974], R. Thomason (ed.), Formal Philosophy (New Haven: Yale University Press). Morreau, Michael [2002], ‘‘What Vague Objects Are Like’’, Journal of Philosophy 99: 333–61. Parsons, C. [1990], ‘‘The Structuralist View of Mathematical Objects’’, Synthese 84: 303–46. Parsons, Terrence, and Woodruff, Peter [1995], ‘‘Worldly Indeterminacy of Identity’’, Proceedings of the Aristotelian Society 95: 171–91; repr. in Keefe and Smith [1997], 321–37. Priest, G. [1987], In Contradiction: A Study of the Transconsistent (Dordrecht: Martinus Nijhoff ). Putnam, H. [1981], Reason, Truth and History (Cambridge: Cambridge University Press). Quine, W. V. O. [1951], ‘‘Two Dogmas of Empiricism’’, Philosophical Review 60: 20–43. —— [1960], Word and Object (Cambridge, Mass.: MIT). —— [1981], ‘‘What Price Bivalence’’, Journal of Philosophy 78: 90–5. —— [1986], Philosophy of Logic, 2nd edn. (Englewood Cliffs, NJ: Prentice-Hall). Raffman, D. [1994], ‘‘Vagueness Without Paradox’’, Philosophical Review 103: 41–74. —— [1996], ‘‘Vagueness and Context Relativity’’, Philosophical Studies 81: 175–92. —— [2005], ‘‘Borderline Cases and Bivalence’’, Philosophical Review (forthcoming). Rosen, Gideon, and Smith, Nicholas, J. J. [2004], ‘‘Worldly Indeterminacy: A Rough Guide’’, Australasian Journal of Philosophy 82: 185–98. Sainsbury, R. M. [1988], Paradoxes (Cambridge: Cambridge University Press). —— [1990], ‘‘Concepts Without Boundaries’’, Inaugural Lecture, published by the King’s College London, Department of Philosophy; reprinted in Keefe and Smith [1997], 251–64. —— [1994], ‘‘Why the World Cannot Be Vague’’, The Southern Journal of Philosophy 33, suppl.: 63–81.

References

219

Sa´nchez-Miguel, M. [1993], ‘‘The Grounds of the Model-Theoretic Account of the Logical Properties’’, Notre Dame Journal of Formal Logic 34: 107–31. Sanford, D. H. [1976], ‘‘Competing Semantics for Vagueness: Many Values versus Super-Truth’’, Synthese 33: 195–210. Shapiro, S. [1988], ‘‘The Lindenbaum Construction and Decidability’’, Notre Dame Journal of Formal Logic 29: 208–13. —— [1998], ‘‘Logical Consequence: Models and Modality’’, in M. Schirn (ed.), The Philosophy of Mathematics Today (Oxford: Oxford University Press), 131–56. —— [2001], ‘‘Modeling and Normativity: How Much Revisionism Can We Take?’’, Agora 20: 159–73. Soames, Scott [1999], Understanding Truth (Oxford: Oxford University Press). Sorenson, R. [1988], Blindspots (Oxford: Oxford University Press). —— [2001], Vagueness and Contradiction (Oxford: Oxford University Press). Stalnaker, Robert C. [1999], Context and Content: Essays on Intentionality in Speech and Thought (Oxford: Oxford University Press). Tappenden, J. [1993], ‘‘The Liar and Sorites Paradoxes: Toward a Unified Treatment’’, Journal of Philosophy 90: 551–77. Tennant, N. [1987], Anti-realism and Logic (Oxford: Oxford University Press). Tye, M. [1989], ‘‘Supervaluationism and the Law of Excluded Middle’’, Analysis 49: 141–3. —— [1994], ‘‘Sorites Paradoxes and the Semantics of Vagueness’’, Philosophical Perspectives 8: Logic and Language, 189–206; repr. in Keefe and Smith [1997], 281–93. Unger, P. [1975], Ignorance (Oxford: Oxford University Press). Uzquiano, Gabriel [2004], ‘‘The Supreme Court and the Supreme Court Justices: A Metaphysical Puzzle’’, Nouˆs 38: 135–53. Waismann, Friedrich [1949], ‘‘Analytic-Synthetic I’’, Analysis 10: 25–40. —— [1950], ‘‘Analytic-Synthetic II’’, Analysis 11: 25–38. —— [1951a], ‘‘Analytic-Synthetic III’’, Analysis 11: 49–61. —— [1951b], ‘‘Analytic-Synthetic IV’’, Analysis 11: 115–24. —— [1952], ‘‘Analytic-Synthetic V’’, Analysis 13: 1–14. —— [1953], ‘‘Analytic-Synthetic VI’’, Analysis 13: 73–89. —— [1968], ‘‘Verifiability’’, in Antony Flew (ed.), Logic and Language (Oxford: Basil Blackwell). Williamson, T. [1994], Vagueness (London: Routledge). —— [1999], ‘‘On the Structure of Higher-Order Vagueness’’, Mind 108: 127–43. Wright, C. [1976], ‘‘Language Mastery and the Sorites Paradox’’, in G. Evans and J. McDowell (eds.), Truth and Meaning: Essays in Semantics (Oxford: Oxford University Press), 223–47; repr. in Keefe and Smith [1997], 151–73. —— [1983], Frege’s Conception of Numbers as Objects (Aberdeen: Aberdeen University Press). —— [1987], ‘‘Further Reflections on the Sorites Paradox’’, Philosophical Topics 15: 227–90; repr. in Keefe and Smith [1997], 204–50. —— [1992], Truth and Objectivity (Cambridge, Mass.: Harvard University Press).

This page intentionally left blank

Index _ -elimination 81–2, 107–9, 116, 117 o-order vagueness 128, 142 _(defined) 65 ! -elimination 3, 16, 94, 110 ! -introduction 74, 110–11, 113 a priori knowledge 206, 207, 213 abortion 11, 30, 167 ABS form 168–70, 173, 182 absolute determinacy 128–30, 164 abstraction 167–71, 173–80, 182–8, 214–15 acceptable frame 152–3, 154, 183–5, 187–8 acceptable sharpening 16, 56–7, 66–73, 84–5, 93–4, 105, 147, 149–50, 183 see also penumbral connection accommodation, rules of 14–15, 37, 160 Adams, Douglas 196 Akiba, K. 196 n Alice xi, 31, 43 see also Humpty Dumpty ambiguity 35, 48, 50 analyticity 4, 67, 196, 211–15 anti-realism 38 n, 154, 192, 198, 200–1 see also realism arrow-elimination 3, 16, 94, 110 arrow-introduction 74, 110–11, 113 artifact 50–4, 58, 64, 72, 100–1, 103–5, 145–9, 186 attention span 41, 42, 143, 16 (B), Thesis 37–42, 136, 202–3 backward spread 21, 25, 41, 78, 134, 141, 159 Baez, Joan xi baldness 8, 11, 12, 17–18, 43 Barcan formula 112 Berkeley, George 192 bivalence 18, 23 n, 56, 61, 62, 68, 88, 94, 101, 147, 180 n, 186, 199 blindspot 155 Boolos, George 144 n, 168 borderline case 1–3, 7, 9–10, 11–12, 17, 24 n, 27–31, 39–40, 43–4, 52, 62, 76, 101–3, 125–39, 148, 150 n boundaries, fuzzy 165–6, 168, 170–3, 180–1, 182, 190 boundaryless 9 n, 11, 125 Brouwer, L. E. J. 120 Brynner, Yul 3, 12, 17, 18, 21, 27, 33, 40, 76, 91, 125, 127, 134, 139, 149, 157, 169, 198, 203, 206

Burgess, J. A. 76, 79 n, 80 n, 89 n, 97, 99 Buridan’s ass 28 Carroll, Lewis xi, 5 see also Humpty Dumpty Cauchy sequence 120 choice sequence 120–4 classical logic, see logic, classical closed frame 114–15 Coffa, Alberto 6 n cognitive command 205–8 color 1, 2, 3, 7, 8–9, 12, 19, 22 n, 24, 65, 132, 133, 134, 170, 194, 195 communication 15, 27, 31, 37, 39, 137, 138, 143, 164, 190 comparison class 12, 24, 32–3, 35, 75–6, 132, 136, 153, 207 see also external context competence 5–6, 8–10, 18, 22, 24, 28, 30–1, 37–40, 43–4, 61, 103–4, 132, 135–64, 173–4, 178–80, 196–8, 202–3, 205, 209 competence sharpening 152–3 completability 68, 69, 70–1, 72, 76, 81, 93, 97, 99, 105, 114 completely sharp interpretation, see interpretation, completely sharp compositionality 7, 47 conditional 2, 64, 65, 82–7, 89–95, 110, 183 counterfactual 33–4, 42, 44, 137, 139, 143, 145–8 intuitionistic 65 n, 89–95, 110, 183 consensus 13, 18, 26–7, 39, 44, 136, 140 consequence relation 3, 50–1, 52, 54–5, 59, 80 see also validity consistency 3, 11 n, 17, 20, 39, 96, 142–3, 158, 164 content externalism 6, 31 context 3, 12, 13, 14–15, 20, 21, 25–37, 39, 42, 43, 59, 61, 64, 65 n, 76 n, 77, 89, 105, 132, 152–3, 177, 202–3, 205, 207, 211 external 24, 25, 27, 31–6, 43, 76 n, 88, 101, 134, 147, 152 incoherent 9, 20, 22–3, 70–1 infeasible 9, 20, 22–3, 70–1 internal 24–5, 27 n, 32–5, 43, 76 n contrasting cases 12, 32, 33, 35, 132, 134, 136, 153 see also external context

222

Index

conversation 11–24, 26–7, 28, 29–32, 35–6, 37–40, 58–9, 61, 66, 76–9, 82, 105, 142–3, 147, 151–2, 153, 158–60, 207, 211, 214 conversational record 12–24, 26–7, 32, 35, 58–9, 76–9, 82, 151–2, 158, 207, 211, 214 conversational score, see conversational record Cook, Roy 49, 53 Copeland, B. Jack 165, 190, 193–4 Copenhagen view 163, 177 Corcoran, John 49, 112 cosmological role 203–5 counterfactual conditional 33–4, 42, 44, 137, 139, 143, 145–8 CP operator 151–64 Crayola crayons 133, 134 creative subject 120–2 crispness postulate 194 Davidson, Donald 31, 43, 47 Dedekind cut 121 deduction, natural 51, 80–2, 106–12 deduction theorem 74, 110–11, 113 deflationism 68 n denotation 12, 48, 60, 88, 165, 186–7, 191, 192 DET operator 73–5, 101–3, 104, 106–8, 111–12, 113, 147–8, 149–64, 186 determinacy 5–7, 9, 10–12, 31–6, 40, 54 n, 68 n, 73–5, 79–80, 89, 100–4, 111–12, 125–64, 175, 180, 186–7, 191 absolute 128–30, 164 e-, see e-determinacy determinateness, see determinacy dialetheism 12, 17, 151 dictionary 43, 132 disagreement 153–4, 205–9 disjunction 7, 64, 80–3, 99 n, 107–9 disjunction elimination 81–2, 107–9, 116, 117 double negation 55, 110–11, 117, 118 Dummett, Michael 3, 4, 26, 105 n, 115, 116, 117, 129, 170, 175, 198–200 Dumpty, Humpty, see Humpty Dumpty e-determinacy 33–6, 40, 75–6, 81–2, 102, 105, 147–8, 149–52, 161 EC, principle of 198–202, 206 Edgington, Dorothy 2 EOI principle 19–20 see also tolerance epistemic constraint 198–202, 206 epistemicism 2–3, 7, 26, 28 n, 36, 61, 85–6, 129, 130, 138, 154–5, 191, 199–200, 201 established, see e-determinacy

Euthyphro contrast 38–42, 202–3, 209 see also judgment dependence Euthyphro reading 38–40, 42, 202, 203, 209 see also Euthyphro contrast; judgment dependence Evans, Gareth 175–80, 185–7, 191, 193–4 excluded middle 55–6, 80, 81 n, 115, 122–4 existence 178–9, 186–8 external context 24, 25, 27, 31–6, 43, 76 n, 88, 101, 134, 147, 152 external negation 63–4, 66, 70, 73, 102, 113 external validity 45, 104–9 externalism, content 6, 31 fallacy of four terms 35 Fine, Kit 1 n, 2, 10, 66, 67, 68, 70–1, 76, 105, 125, 149–50, 163 see also completability fixed domain 116–17 Fodor, J. 71–2 four terms, fallacy of 35 forced march 17–31, 43, 76–9, 128–9, 130, 133–4, 140–2, 157–60 forcing 21, 79–80, 82, 86–7, 88–93, 95–101, 104–15, 117–19, 122–4, 146, 149, 156, 161–2 weak 87, 89, 92, 93–4, 95, 96–9, 121, 154, 156, 183, 184, 185, 188 formal language, see language, formal frame 75–80 acceptable 152–3, 154, 183–5, 187–8 closed 114–15 linear 122–4 free logic 186–8 Frege, Gottlob 47–8, 168 future contingents 82–3, 86, 98 fuzzy boundaries, see boundaries, fuzzy fuzzy logic 2, 11, 17, 52–4, 85, 171 FX rule 117–18 Gaifman, Haim 3, 9, 20, 22, 70 n Garcia, Jerry 3, 17, 19, 22, 24, 26, 27, 33, 36, 40, 76, 77, 91, 125, 126, 127, 132, 134, 137, 140, 157, 158 Garciaparra, Nomar 175 Gestalt shift 25, 26, 33, 41 Graff, Delia 3, 12, 34–5, 127 n, 133 n, 170 n Grice, P. 214 group 175–7, 182–4 H principle 73 Hacking, Ian 51 Hale, Bob 169 heap, paradox of the 3, 44, 165, 211 see also sorites

Index Height Principle 169–71 Heraclitus 35 heterologicality 6 higher-order vagueness 6, 31, 40, 45, 54 n, 56, 58, 63, 73, 100, 102–4, 112, 125–64, 180, 183 n, 186, 201, 202, 203 Hodes, Harold 52 homophonic approach 57 Horgan, Terrence 3, 4, 17, 26, 192 Humberstone, L. 76, 79 n, 80 n, 89 n, 97, 99 Hume’s principle 168 humor 38–9, 41, 200 n, 202 Humpty Dumpty xi, 5, 15, 29, 31, 37, 43, 131, 133, 178, 179, 181, 188–9 Hyde, Dominic 11 n ID principles 185–6 idealization 42, 50, 56, 142–3, 164 identity 166–71, 173–81, 183–9, 191 incoherent context 9, 20, 22–3, 70–1 incoherentism 3, 4, 26, 170 income groups 170–1, 173–80, 182–5, 186–8 Income Principle 170, 173–81, 182–8 incompetence, see competence indefinite extensibility 129 indiscernibility of identicals 175, 179–80, 185 inductive premise 3–4, 16, 22–3, 25, 41, 71, 92, 93, 97, 174 see also sorites; tolerance infeasible context 9, 20, 22–3, 70–1 infinity 119–24, 127–9 potential 119–24 intension, see meaning intensionality 48, 54, 180, 186 internal context 24–5, 27 n, 32–5, 43, 76 n internal validity 45, 105–9, 113–15, 117 interpretation 52, 60 completely sharp 62, 65, 67–9, 71–3, 76, 79, 87, 97, 100, 105, 113–15, 120–1 partial 21 n, 61–2 intuitionism 55–6, 65 n, 87, 89, 95, 115–24, 130–1, 183–4, 187 intuitionistic conditional 65 n, 89–95, 110, 183 intuitionistic logic, see logic, intuitionistic intuitionistic negation 95–7, 116–17, 118, 155 IRR principle 122, 123, 124 Jeshion, Robin 48 n judgment dependence 30, 37–43, 44, 127, 136, 137–9, 140, 141–4, 147, 150–4, 162–3, 177, 180, 181, 186, 198, 201, 202–3, 205, 209, 214 jumping 20–31, 66, 78, 92–3, 141, 153, 158–9, 174, 178

223

Kamp, Hans 3, 9, 13, 19–20, 26, 28, 49, 57–9, 65 n, 71, 80, 89 Kant, Immanuel 195 Keefe, Rosanna 2, 10, 16 n, 53, 71–3, 80 n, 83–4, 97, 103 n, 104, 105 KK-thesis 199 Kleene truth table 64–88 Kneale, William 210 n Kraut, Robert 196 Kripke structure 54, 55, 75, 89, 95, 115–19, 187 Lakatos, Imre 180, 207 language, formal 21 n, 46–52, 55, 60, 67, 94, 185–7 Leibniz principle 175, 179–80, 185 Lepore, Ernest 71–2 Lewis, David 12–16, 27, 37, 152, 153, 160, 165, 171, 191 linear frame 122–4 linguistic vagueness 190–7, 207–8 local validity 45, 105–9, 113–15, 117 logic 2, 9, 17, 27 n, 36–7, 43, 45–59, 72, 80–7, 105, 109–19, 160, 185 classical 34–5, 51, 54, 55–9, 61, 68–9, 72, 74, 81, 84, 87, 105, 110, 113–15, 117, 118, 122–4, 155 n free 186–8 fuzzy 2, 11, 17, 52–4, 85, 171 intuitionistic 54, 55–6, 65 n, 87, 89, 92, 95, 113, 115–24, 131, 183, 187; see also excluded middle many-valued 2, 11, 17, 52–4, 85, 171 modal 54, 55, 112, 152–3 as model 46–54, 589; see also artifact paraconsistent 6 n, 12, 17, 143, 158 logical positivists 214 logician’s dilemma 54–9 logicism 168–9 Lycan, William 47 Machina, K. F. 2 McGee, Vann 5–7, 10, 31–3, 36 n, 43, 45, 68 n, 75, 132, 172 McLaughlin, Brian 5–7, 10, 31–3, 36 n, 43, 45, 68 n, 75, 132, 172 many-valued logic 2, 11, 17, 52–4, 85, 171 meaning 3 n, 4, 5–6, 8–10, 13, 27, 29, 31–2, 36–7, 41, 42–3, 66–7, 71–2, 76–7, 79, 88, 102, 132–3, 136, 137–8, 149, 151, 152, 154, 158–60, 196, 204, 209, 212–15 mereology 165–6, 172–3, 177, 191, 213 Merricks, Trenton 47 n, 191 meta-language 21 n, 54–9, 63–4, 67–8, 70, 73, 89, 94–6, 102, 110, 117 n, 118, 123–4, 131, 146, 154, 180 n, 184

224

Index

meta-theory, see meta-language metaphysical realism 192, 193 metaphysical vagueness 190–7, 208–9 mind-independent 193 see also objectivity modal logic 54, 55, 112, 152–3 model, mathematical 17, 43, 49–53, 58, 61, 64–5, 72, 75–6, 80, 86–7, 89, 98, 102–5, 124, 145–8, 150–1, 160, 182, 184, 186–7, 188 see also logic as model; artifact model theory 17, 35, 43, 46, 52, 55, 56, 59, 60–119, 148–63, 181–9 modus ponens 3, 16, 94, 110 monotonicity 66, 68, 73–5, 79, 85, 87, 90, 94, 95, 100, 107, 187 monster-barring 180, 207 Montague, Richard 47 morality 11, 26 n, 30, 167, 192, 203–4 Morreau, Michael 165 n, 170, 173 N ¼ principle 168 natural deduction 51, 80–2, 106–12 naturalism 194 n negation 9, 63–4, 85, 95–7, 109, 110, 113, 116–17, 130–1, 155 n double 55, 110–11, 117, 118 external 63–4, 66, 70, 73, 102, 113 introduction 109–10, 113, 131 intuitionistic 95–7, 116–17, 118, 155 weak 63–4, 66, 70, 73, 102, 113 neo-Fregean program 169 neo-logicism 169 nice number 1 n no sharp boundaries thesis 63, 126, 130–2, 163–4 non-denoting singular terms 48, 186–8 normal conditions 9 n, 39, 136–7, 139 normativity 27–8, 35, 36, 46, 48–9, 159, 213 NSB thesis 63, 126, 130–2, 163–4 object language, see meta-language objectivity 38–9, 47, 190, 195, 197–209, 210 objects, vague 45, 62, 76, 165–89, 190–5, 196, 208 see also boundaries, fuzzy ontic vagueness 190–7, 208–9 open-texture 10–12, 17–18, 32, 37, 40, 44, 61, 63–5, 75–9, 81 n, 82–3, 132–4, 135–7, 140–1, 146, 150, 152, 160, 171–2, 175, 188, 198, 206, 208, 210–15 outback 165, 190, 191, 192 Oxford English Dictionary 43 paraconsistent logic 6 n, 12, 17, 143, 158 Parsons, Charles 169 n

Parsons, Terrence 180 n part–whole relation 165–6, 172–3, 177, 191, 213 partial interpretation 21 n, 61–2 parts 165–6, 167–8, 170, 171, 172–3, 177, 182, 191, 213 penumbral connection 23, 67–8, 69–72, 79, 91–2, 95, 121, 143, 151, 154–60, 164, 183, 185 potential infinity 119–24 pragmatics 12, 14, 16 n, 36–43, 59 precisification, see sharpening Priest, Graham 17 promises 82–7, 91 n, 96 n proof by cases 81–2, 107–9, 116, 117 psychological state 24–8, 32–4, 39–40, 42, 43 see also internal context Putnam, Hilary 192, 193 quantifier 12, 50, 51, 64, 97–100, 111, 112, 116, 119, 187 quantum mechanics 27, 166, 170 quasi-abstract objects 167–71, 173–80, 182–8, 214–15 Quine, W. V. O. 48, 67 n, 196, 212–14 quote marks 144 n Raffman, Diana 3, 18, 22 n, 24–30, 32–5, 37–43, 103, 130, 136, 159, 163, 202 RAT principle 122–4 realism 47–8, 83, 192, 193, 199 metaphysical 192, 193 record, conversational, see conversational record see also anti-realism Red Sox, Boston 175 relativity 12, 14, 24–7, 42, 132, 138–40, 142, 145, 147–8, 213 representor, see artifact response dependence, see judgment dependence Rogaine 204, 205 Rosen, Gideon 196 n S5 112, 113, 148 see also logic, modal Sainsbury, R. M. 9, 10–11, 57, 83–4, 103, 125, 130, 191, 192–3, 195 Sa´nchez-Miguel, M. 49 Sanford, D. H. 71 n satisfaction 52, 60–2, 115–16, 117, 118 see also truth score, conversational, see conversational record second-order vagueness, see higher-order vagueness semantic vagueness 190–7, 208–9

Index semantics 14, 16 n, 19–20, 28, 33–4, 36–43, 45–6, 50–1, 54, 57–8, 59, 61, 62, 64, 75, 113, 117, 146, 182, 191, 193 n see also model theory sequence, Cauchy 120 sequence, choice 120–4 set, dynamic 175–7, 182–4 set theory 13, 21 n, 46, 54, 55, 57–8, 102, 145, 186 shampoo 43, 205 Shapiro, Stewart 49, 56 sharp predicate 10, 57, 62, 68, 70, 71 n, 85, 128, 129, 137, 142–3, 145 sharp relation, see sharp predicate sharpening 10, 16, 45, 49, 56–7, 66–7, 70–1, 75, 76, 77, 93, 105, 108, 118, 120, 149–50, 152, 196 n, 211 acceptable, see acceptable sharpening complete, see interpretation, completely sharp simple truth 14–15, 16–17, 171 sludge 195 Smith, Nicholas 196 n Soames, Scott 3, 9 n, 11, 19, 23, 33 n Socrates reading 38, 39, 41–42, 202, 203 see also Euthyphro contrast; judgment dependence Sorenson, Roy 2, 7, 22 n, 41–2, 85, 129, 138, 155, 199 sorites 3–4, 8, 9, 14, 16, 17–31, 33, 34–5, 36, 40–4, 53–4, 57, 59, 69–72, 76, 91–3, 97, 103, 125–8, 130, 132–4, 142, 146–7, 156–60, 163–4, 166–8, 170, 173–4, 176, 214 n sort-of-bald 133–4 specification point 76 specification space 76 Stalnaker, Robert 13 n Strawson, P. F. 214 sub-valuation 11 n super-truth 10, 11, 14, 16, 36, 45, 56, 68–9, 71–5, 79, 80–5, 97–100, 104–5, 114–15, 149 supervaluation 2, 10, 11, 14, 16, 36, 45, 51, 56–7, 61–9, 71–5, 80–5, 97–100, 103, 104–5, 114–15, 118, 149–50 synonymy 4, 47, 214–15 T principle 70, 71, 72 T-scheme 2 T-sentence 199–200 T1 principle 96, 97 T2 principle 97 tachometer 29–31 Tennant, Neil 4, 51, 56 Theorem 1: 66, 68, 73, 74, 79 Theorem 2: 68–9, 72, 74, 80, 81, 105, 114 Corollary 3: 69, 72, 74

225

Theorem 4: 87 Theorem 5: 89–90 Theorem 6: 90 Theorem 7: 96 Theorem 8: 96, 110 Theorem 9: 99, 111 Theorem 10: 101 Theorem 11: 102, 103, 148 Theorem 12: 103, 148 Theorem 13: 105, 106 Lemma 14: 106, 107 Theorem 15: 106 Theorem 16: 106 Theorem 17: 109 Theorem 18: 109 Theorem 19: 109 Theorem 20: 110 Theorem 21: 110, 113, 118 Theorem 22: 111 Theorem 23: 111 Theorem 24: 112, 148 Theorem 25: 113, 114, 117, 122 Theorem 26: 113, 114, 117 Lemma 27: 114 Theorem 28: 114, 115, 122, 124 Theorem 29: 114–15 Thesis (B): 37–42, 136, 202–3 tolerance 8–9, 12, 18, 19–24, 28 n, 30, 44, 69–72, 77–9, 92–4, 95, 96–7, 103, 105, 121, 130–1, 134, 146–7, 156–60, 163–4, 166, 174, 176, 184 failure of 9, 20, 22–3, 70–1 toupee 43 transfinite levels 128–9, 142, 143 truth 2, 10, 11, 14, 31, 35–6, 38, 45, 52–3, 56, 59, 68, 71, 73, 75, 79, 80–3, 97, 102, 104–8, 114, 116 n, 147, 196, 199, 206–8 enough 14–15, 16–17, 20, 171 simple 14–15, 16–17, 171 super, see super-truth see also determinacy; super-truth truth value 2, 28, 52–3, 62–6, 71 n, 77 n, 80, 88 Tye, Michael 2, 53–4, 71 n, 118 Unger, Peter 15, 170–1 unsettled 7, 9, 10, 12, 18, 43, 44, 132, 197, 207, 211 see also determinacy unsharpening 78, 92, 93 Uzquiano, Gabriel 175 n vagueness 1–215 higher-order, see higher-order vagueness metaphysical 190–7, 208–9 ontic 190–7, 208–9 second-order, see higher-order vagueness

226

Index

validity 10, 17, 35, 44, 45, 47, 52, 68–9, 81, 104–9, 113–15, 116, 117, 118–19 external 45, 104–9 internal 45, 105–9, 113–15, 117 local 45, 105–9, 113–15, 117 see also consequence relation Waismann, Friedrich 10 n, 144, 190, 195, 210–15 weak forcing, see forcing, weak weak negation 63–4, 66, 70, 73, 102, 113 Weatherson, Brian 22 n

Weaver, George 112 Webster’s Unabridged Dictionary 43, 132 Williamson, Timothy 2, 7, 28 n, 36 n, 80 n, 85, 97, 105 n, 112 n, 125, 129, 132, 135, 138, 155, 163, 199–200 Wittgenstein, Ludwig 5–6, 57, 212 Woodruff, Peter 180 n Wright, Crispin 1, 8, 10, 18, 29, 36, 37, 38–9, 49, 63, 103, 107, 130, 131 n, 152 n, 163, 168–9, 194, 198–209 Yankees, New York 197