Vagueness, Logic and Ontology (Ashgate New Critical Thinking in Philosophy)

VAGUENESS, LOGIC AND ONTOLOGY The topic of vagueness re-emerged in the twentieth century from relative obscurity. It deals with the phenomenon in natural language that manifests itself in apparent semantic indeterminacy – the indeterminacy, for example, that arises when asked to draw the line between the tall and non-tall, or the drunk and the sober. An associated paradox emphasises the challenging nature of the phenomenon, presenting one of the most resilient paradoxes of logic. The apparent threat posed for orthodox theories of the semantics and logic of natural language has become the focus of intense philosophical scrutiny amongst philosophers and non-philosophers alike. Vagueness, Logic and Ontology explores various responses to the philosophical problems generated by vagueness and its associated paradox – the sorites paradox. Hyde argues that the theoretical space in which vagueness is sometimes ontologically grounded and modelled by a truth-functional logic affords a coherent response to the problems posed by vagueness. Showing how the concept of vagueness can be applied to the world, Hyde’s ontological account proposes a substantial revision of orthodox semantics, metaphysics and logic. This book will be of particular interest to readers in philosophy, linguistics, cognitive science and geographic information systems.

ASHGATE NEW CRITICAL THINGING IN PHILOSOPHY

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Series Editorial Board: David Cooper, Durham University, UK Sean Sayers, University of Kent, UK Simon Critchley, New School for Social Research, USA; University of Essex, UK Simon Glendinning, London School of Economics, UK Paul Helm, Regent College, Canada David Lamb, University of Birmingham, UK Peter Lipton, University of Cambridge, UK Tim Williamson, University of Oxford, UK Martin Davies, Australian National University, Australia Stephen Mulhall, University of Oxford, UK John Post, Vanderbilt University, UK Alan Goldman, College of William and Mary, USA Simon Blackburn, University of Cambridge, UK Michael Friedman, Stanford University, USA Nicholas White, University of California at Irvine, USA Michael Walzer, Princeton University, USA Joseph Friggieri, University of Malta, Malta Graham Priest, University of Melbourne, Australia; University of St Andrews, UK Genevieve Lloyd, University of New South Wales, Australia Alan Musgrave, University of Otago, New Zealand Moira Gatens, University of Sydney, Australia

Vagueness, Logic and Ontology

DOMINIC HYDE University of Queensland, Australia

© Dominic Hyde 2008 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the publisher. Dominic Hyde has asserted his right under the Copyright, Designs and Patents Act, 1988, to be identi¿ed as the author of this work. Published by Ashgate Publishing Limited Gower house Croft Road Aldershot Hampshire GU11 3HR England

Ashgate Publishing Company Suite 420 101 Cherry Street Burlington, VT 05401-4405 USA

Ashgate website: http://www.ashgate.com British Library Cataloguing in Publication Data Hyde, Dominic Vagueness, logic and ontology. - (Ashgate new critical thinking in philosophy) I. Vagueness (Philosophy) I. Title 160 Library of Congress Cataloging-in-Publication Data Hyde, Dominic. Vagueness, logic, and ontology / Dominic Hyde. p. cm. -- (Ashgate new critical thinking in philosophy) Includes bibliographical references and index. ISBN 978-0-7546-1532-3 1. Vagueness (Philosophy) 2. Ontology. 3. Logic. 4. Sorites paradox. 5. Semantics (Philosophy) I. Title. B105.V33H93 2007 110--DC22 2007005502 ISBN-13: 978-0-7546-1532-3

Contents Preface

ix

1

Vagueness 1.1 Vagueness Introduced 1.2 The Sorites Paradox 1.3 Kinds of Vagueness 1.4 Extending the Concept 1.5 Higher-order Vagueness

1 1 8 16 19 27

2

Russell’s Representational Theory 2.1 Incompleteness, Lack of Speci¿city and the Source of Vagueness 2.2 Russell’s De¿nition 2.3 Vagueness, Logic and the World

33 35 37 48

3

Descriptive Representationalism 3.1 Against Elimination 3.2 Against Reduction 3.3 Against Supervenience 3.4 The Precision of Scienti¿c Language 3.5 Summary

51 52 59 60 66 70

4

Going Non-classical: Gaps and Gluts 4.1 Supervaluationism 4.2 Subvaluationism 4.3 Summary

73 73 93 103

5

Ontological Vagueness 5.1 Vague Identity 5.2 The Evans Argument 5.3 Vague Objects 5.4 The Vague Identity Thesis 5.5 Vague Properties and States of Affairs 5.6 Vagueness ‘in the World’ or ‘of the World’?

105 109 114 127 141 147 150

6

Vague Individuation and Counting 6.1 Vagueness and Individuation 6.2 Vagueness of Count 6.3 Formalizing Count-phrases 6.4 De¿nite Descriptions

153 153 160 168 172

Vagueness, Logic and Ontology

vi

6.5 6.6 7

The Pinillos Argument Summary

The Logic of Vagueness 7.1 First-order Vagueness 7.2 Objections to Truth-functionality 7.3 Truth and the T-schema 7.4 Higher-order Vagueness

Bibliography Index

173 178 179 179 186 193 197 211 219

Dedicated to those for whom staring keenly into vague spaces is a pleasure in itself.

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Preface This book is an exploration of the philosophical concept of vagueness as it arises in natural language, and the logical and metaphysical issues that it gives rise to. As has recently been remarked in the press, ‘Vagueness is huge!’, and there is an enormous amount of literature on the topic. For this reason this exploration is selective. We shall be concerned with vagueness understood as a semantic phenomenon. There may sometimes simply be no fact of the matter whether a vague term like “bald” applies to a person or not. Their hair-cover may be such that we are simply unable to say whether they are bald or not, and this is a fact about the concept itself, as opposed to a fact about us and our ignorance of its applicability conditions – as an epistemic theorist would have us believe – or a fact about the way we use language – as a pragmatic theorist would have us believe. These epistemic and pragmatic accounts of the phenomenon of natural-language vagueness have been the subject of intense debate in the last decade but I remain unconvinced of their adequacy. Vagueness remains, in my view, a semantic matter, and so I am concerned to explore the phenomenon from this perspective. Even thus restricted, there are issues yet remaining that the book does not cover. It is already too long and so I have passed over some matters others may think deserve attention, or more attention than I have given them. I can only hope that there is content enough for readers to consider it worthwhile nonetheless. The topic of vagueness has a long and venerable philosophical history, dating back nearly two and a half thousand years, and has re-emerged as an area of intense interest in the last three decades, attracting enormous attention. As one of the hottest topics in contemporary analytic philosophy, it has generated a very large amount of literature, and the puzzles and problems it generates are fascinating and complex, none more so than the sorites paradox. This paradox remains one of the most perplexing logical puzzles in philosophy. The response to the paradox, and logical puzzlement about vagueness more generally, defended in this book is an approach that recommends a truth-functional logic. While non-truth-functional approaches have dominated semantic accounts, early work during the revival of interest in the topic in the 1970s also focused on truth-functional approaches, most notably fuzzy logic and associated many-valued approaches. Of late, book-length treatments of the topic have appeared defending the popular non-truth-functional approach that is supervaluationism, as well as epistemic and pragmatic approaches. These scholarly treatments have put on record a range of arguments aimed at undermining any truth-functional response. I hope that this book will go some way to reinvigorating interest in such a response and will bolster those trying to articulate such a position. The book also defends an attitude towards vagueness that sees it as at least sometimes ontologically grounded. Not only are our words and concepts vague, but

x

Vagueness, Logic and Ontology

so too the things that our words and concepts pick out. The view that there might be vague objects, properties, or states of affairs was long thought madness by many. As is the way with good philosophy, this attitude has been increasingly clearly articulated as a thesis to be argued for, and increasingly argued against by at least some. This book adds argument to the growing literature in defence of such a view. Chapter 1 begins by trying to get clear on the phenomenon under scrutiny, vagueness, and the paradox that emphasizes its puzzling nature, the sorites paradox. The varieties and extent of vagueness within natural language are then investigated. Vagueness paradigmatically arises in respect of the application of a predicate (e.g. “is bald”) along some dimension (e.g. number of hairs), but terms can be vague in other ways and the phenomenon also extends well beyond predicates. An account and discussion of the so-called ‘problem of higher-order vagueness’ concludes this introduction to vagueness. Chapter 2 investigates Bertrand Russell’s work on vagueness, especially his 1923 paper “Vagueness”. Working with a semantic conception of vagueness, Russell proposes that the phenomenon is merely semantic, i.e. it is a feature of representations but not that which is represented thereby. Language is vague (as, indeed, is any representational system) in so far as it misrepresents that which it seeks to describe. Vagueness is thus of no metaphysical import. In detailing exactly how, on this view, vague language misrepresents, Russell articulates the very essence of an account of vagueness as merely representational. Given that such a view of the general nature of vagueness, what I term a ‘representationalist’ view, dominates current thinking on the matter, his neglected analysis provides important insight into this popular position. Despite his view the necessity of vagueness in language, Russell thought that the phenomenon and associated paradox were of no consequence to logic. We are not enmeshed in paradox despite the ubiquity and necessity of vagueness in natural language, since logic is not applicable to natural language. More recent representationalists have thought otherwise. In Chapter 3 we turn to consider those representationalists who take logic to be applicable but seek to defuse any supposed challenge posed by vagueness by arguing that it is in some sense super¿cial. Quine, for example, takes it to be eliminable. Quine, Carnap and more recent fellowtravellers would then be able to show that vagueness is of no consequence to either logic or metaphysics, but the logically conservative representationalism being proposed is found wanting. A less conservative approach is described and evaluated in Chapter 4. While typically representationalist, and thus unÀinching on the metaphysical irrelevance of vagueness, the popular supervaluationist approach treats vagueness as a semantic phenomenon that logic must take seriously. It is to be accommodated by truth-value gaps, yet despite the abandonment of bivalence the resulting paracomplete logic of vagueness is generally considered to require minimal revision. In its classical form, it is said that much of classical logic remains intact. An earlier paraconsistent variation invoking truth-value gluts, subvaluationism, is also considered. It is argued that both of these logical responses to vagueness are, despite appearances and arguments to the contrary, equally conservative, but arguments defending such logical conservatism are unconvincing. Many-valued rivals cannot be ruled out so easily.

Preface

xi

One particular argument in defence of super- and subvaluationism appeals to an underlying commitment to representationalism. The metaphysically conservative nature of the representationalist view of vagueness might be thought to justify the logical conservatism of these logics. In Chapters 5 and 6 we return to consider ontological vagueness and the related issues of vague objects; vague identity; and counting and individuation of vague objects. The view that there might be vague objects, and that the world might more generally be vague, is found to have much to recommend it. Therewith, the metaphysical defence of the logics considered earlier collapses. With the truth-value glut response to vagueness set aside in Chapter 4, Chapter 7 goes on to describe and defend a truth-functional, truth-value-gap logic able to accommodate vagueness. The logical system is described initially as a three-valued model of ¿rst-order vagueness and defended against arguments that the commitment to truth-functionality and the demands of a non-bivalent approach present insuperable dif¿culties. Finally, the vexing phenomenon of higher-order vagueness is analszed and the logical system is weakened to accommodate it. The semantic phenomenon of vagueness is accounted for by truth-value gaps, as many others have suggested, but just as the logic of ¿rst-order vagueness was characterized by Łukasiewicz’s three-valued logic L3, the paradox-free logic of vagueness per se (accommodating higher orders of vagueness) is characterized by L∞. Very many people have assisted with helping get this book and the thoughts contained in it to publication and I want to thank them all. Special thanks go to Richard Sylvan, Graham Priest, Bertil Rolf, Tim Williamson, Brian Garrett, Mark Colyvan, Lloyd Humberstone, Philosophy staff and students at The Australian National University and Queensland University, and the many audiences whose questions and incredulity helped me see problems and possibilities that helped form my views. I would also like to thank publishers for allowing me to reproduce material from my: “Why Higher-Order Vagueness is a Pseudo-Problem”, Mind 103 (1994) and “Higher Orders of Vagueness Reinstated”, Mind 112 (2003) in Chapter 1; “Rehabilitating Russell”, Logique et Analyse 137 (1992) in Chapter 2; “Vagueness, Ontology and Supervenience”, The Monist 81 (1998) in Chapter 3; and “From Heaps and Gaps to Heaps of Gluts”, Mind 106 (1997) in Chapter 4. Thanks also go to the Australian Research Council for support for the project with Discovery Grant DP0209051.

xii

Vagueness, Logic and Ontology — “If you hit a rock hard enough and often enough with an iron hammer, some mollycules of the rock will go into the hammer and contrariwise likewise.” — “That is well known”, he agreed. — “The gross and net result of it is that people who spend most of their natural lives riding iron bicycles over the rocky roadsteads of the parish get their personalities mixed up with the personalities of their bicycles as a result of the interchanging of the mollycules of each of them, and you would be surprised at the number of people in country parts who are nearly half people and half bicycles.” Mick made a little gasp of astonishment … — “Good Lord, I suppose you’re right.” — “And you would be unutterably Àibbergasted if you knew the number of stout bicycles that partake serenely of humanity.” Here the sergeant produced his pipe … — “Are you sure about the humanity of bicycles?” Mick enquired of him. “Does it not go against the doctrine of original sin? Or is the Molecule Theory as dangerous as you say?” Flann O’Brien, The Dalkey Archive

Chapter 1

Vagueness The eastern gateway to Australia’s arid centre is marked by ephemeral rivers and wetland areas that skirt the central deserts. A keen diarist seeking to note the exact moment at which they enter one of these wetland areas will be hard pressed. The dif
culty is not simply a matter of their potentially limited grasp of the continent’s wetland ecology, but arises because it seems that there simply is no exact moment at which one enters into the area. There would seem to be no sharp line delimiting the particular wetland from the surrounding country; there are regions that would seem to be part of the wetland and regions not part of the wetland, yet intuitively there is no sharp line that separates them from each other. It seems obvious that no fence can mark the exact boundary of the wetland area. Analogously, there would seem to be no sharp line delimiting situations where the term “wetland” itself applies from situations where it does not. The wetland in which the diarist
nds themselves was not always a wetland area, yet it seems impossible to identify any precise moment in its evolutionary history as that moment at which the term was
rst applicable. We are confronted with vagueness. What exactly is vagueness and why is its analysis of interest? This chapter starts with a discussion of vagueness in its most common setting, natural language, and proceeds initially by discussing the paradigmatic concept of vagueness as applied to predicates. Then, having described a sufficiently thick notion of predicatevagueness, we are in a position to consider that ancient conundrum which presents vagueness as a problem – the sorites paradox. Various kinds of predicate-vagueness are subsequently distinguished and the concept is extended beyond predicates to other parts of language, necessitating, in due course, some further quali
cation of the notion of predicate-vagueness. A general account of what it means to speak of vagueness in language is offered. With this taxonomy of vagueness to hand we can pursue our investigations into its cause and logic with a broad understanding as to the phenomenon being addressed and the nature of the challenge it presents. 1.1 Vagueness Introduced “Vague” is an ambiguous term yet this book is concerned with vagueness in only one speci
c sense. In this
rst section, therefore, let us try to delimit that sense of vagueness around which the subsequent discussion revolves.

Vagueness, Logic and Ontology

2

1.1.1 What is vagueness? Apparent lack of sharp boundaries is prevalent in our use of natural language. Consider our above-mentioned diarist camping in a wetland area fringing Australia’s arid heart. At what moment did they enter? If no moment can be identi
ed as that which marks their entry, then how can they have entered at all? Likewise we may ask, seemingly to no avail, at what instant did the autumn leaves turn brown or did that person become rich, famous, bald, tall or an adult. These predicates – “(is) in a wetland”, “(is) brown”, “(is) rich”, and so on – are all examples of predicates whose limits of application seem essentially inde
nite or indeterminate, and they are typical examples of what are termed vague predicates. Consider the predicate “tall”, for example. We might line up a crowd of people starting with the shortest and progressing monotonically to the tallest. The crowd seems not to be clearly partitioned into two mutually exclusive and exhaustive sets of those to whom the predicate applies and those to whom it fails to apply. The transition from one set to the other would seem not to be precise and one might ask rhetorically, as Diogenes Laërtius is reputed to have done, ‘Where do you draw the line?’. The most common instances of vague predicates are those for which the applicability of the predicate just seems to fade off, as in the above examples, and it consequently appears that no sharp boundary could conceivably be drawn separating the predicate’s positive extension from its negative extension. The behaviour of vague predicates is thus contrasted with such precise predicates as “greater than two” de
ned on, say, the natural numbers. We can divide the domain of natural numbers, N, into two sharp, mutually exclusive and exhaustive sets: P – = {0,1,2} and P + = {3,4,5,…}; the set P – comprising those natural numbers determinately failing to satisfy the predicate “greater than two” and the set P + comprising those natural numbers that determinately satisfy it. The sense of vagueness we shall be working with, then, can already be distinguished from another sense in which language is often said to be vague – vague in the sense of inexact, unspeci
c or general. Consider, for example, the claim that there are between two hundred and one thousand species of Eucalyptus trees. It might be responded that this claim is ‘vague’ and one could be a lot more ‘precise’. However, it is easy to see that vagueness in this sense is quite different from vagueness as I have described it above. A thoroughgoing discussion is to be found in Chapter 2, but the distinction can be made with enough intuitive force for present purposes as follows. Being between 1.01 and 3.24 metres is an inexact description of someone’s height but it is not vague in the sense of there being indeterminate limits to its application – it will be true if their height lies between these two
gures and false otherwise. I can make a much more exact estimation of their height which nonetheless is more vague, e.g. approximately 2 metres. Increasing exactness is consistent with a decrease in precision whilst a decrease in exactness is consistent with an increase in precision. It might be thought that vagueness always involves some inexactness (as Russell seems to have thought) but the terms are nonetheless distinct. When speaking of vagueness henceforth we will be concerned with the sense outlined above, reserving the more explicit terms like “inexactness” to describe this other sense.1 1

Sorensen (1989) has found the prevalence of equivocation here common enough to

Vagueness

3

The symptom of vagueness alluded to above, our inability to draw a sharp line between those things in the predicate’s positive extension and those in its negative extension, is tantamount to there being borderline or penumbral cases for the predicate in question – cases which jointly constitute the borderline region or penumbra for the vague predicate. Intuitively, such cases are those where there are objects to which the predicate meaningfully applies (i.e. objects in the predicate’s domain of signi
cance) yet for which it appears essentially indeterminate whether the predicate or its negation truthfully applies. That is to say, there are situations where a language user, having carried out all the empirical and conceptual research possible concerning the case at hand, will nonetheless still be unable either to apply the predicate determinately to an object to which the predicate may be said to apply meaningfully or to apply its negation determinately. This apparent indeterminacy or inde
niteness, taken as the sine qua non of this, and most other, discussions of vagueness, is not due to the lack of knowledge of facts or of meanings that one could in principle come to know – hence the use of “essential” above. Notice that the characterization of borderline cases is given in terms of an agent’s ability to apply predicates, rather than in terms of the semantic properties of the predicate. In this way we avoid the charge, properly levelled at many discussions of vagueness in the twentieth century, of invoking a theoretically laden de
nition of vagueness which, from the outset, foreclosed on that response to vagueness which claims it to be an epistemic phenomenon (though, as we shall see, it is not an option that will be pursued in what follows). Notice also that though we commonly speak of a predicate’s vagueness in terms of there (actually) being borderline cases for the predicate, there is a weaker sense of vagueness that does not depend on mere contingencies regarding what actually exists – namely, the very possibility of there being borderline cases. In this weak sense a predicate does not cease to be vague simply because its borderline cases cease to exist; the logical possibility of their existence is enough to guarantee the conceivability of borderline cases and it is this, rather than the actual existence of borderline cases, that is of logical interest. This weak sense we shall describe as intensional vagueness2 and depends only on the conceptual possibility of borderline cases as opposed to the stronger extensional vagueness which requires the actual existence of a borderline case. Analogously, though extensional precision – the actual absence of borderline cases – may sometimes be of interest (e.g. in assessing claims to actual truth), we shall generally be more interested in the logically necessary absence of borderline cases, necessary precision. With this in mind we shall often speak of vagueness and precision simpliciter, where these are to be understood respectively as intensional vagueness and necessary precision. In subsequent chapters we shall enquire further into the cause or source of the essential indeterminacy underpinning vagueness. With the ensuing discussion already too long, I shall set aside an epistemic analysis of the indeterminacy involved, and consider vagueness as a semantic feature of language. We will thus be warrant a discussion explaining the distinction, urging that philosophers revise their language and put aside the inexactness sense once and for all, thereby making ‘the streets of speculation just a little bit safer for the philosophers of tomorrow’. This equivocation is further discussed in Chapter 2, §1 and Chapter 3, §4. 2 Following Fine (1975: 266).

4

Vagueness, Logic and Ontology

concerned to properly analyse this semantic indeterminacy and enquire into the possibility as to whether this vagueness could somehow be ‘in the world’. The exact causes of these borderline cases – the reason for the failure of any enquiry to resolve such cases one way or the other – is unquestionably the major watershed amongst analytic philosophers writing on the topic of vagueness; it is therefore not something that can be settled at this preliminary stage. For the moment we need to focus more clearly on its content. Cashing out our intuitions regarding borderline cases even further, we must distinguish borderline cases from cases where one is at a loss as to what to say due to meaninglessness, ambiguity or context-sensitivity. In the
rst instance, vagueness is to be contrasted with meaninglessness. My inability to decide whether the creature now before me is one of Lewis Carroll’s ‘snarks’ derives from the meaninglessness of the term “snark”. To be sure, it is, in a sense, indeterminate whether the creature I see is a snark, but this indeterminacy is not to be confused with that arising from vagueness. Meaningful terms can be vague, as evidenced by the examples given earlier. Nor should vagueness be confused with ambiguity. Consider the ambiguous word “bank” – meaning (amongst other things) a
nancial institution or a river’s edge. It may be essentially indeterminate whether or not to agree to the claim “Jo went to the bank yesterday” since it may be used to assert more than one proposition. An audience may be unable to determinately respond one way or the other since they may
rstly need to determine what proposition is being asserted. This contrasts with borderline cases in the relevant sense. We may be unable to say whether or not Jo, having a moderate income and substantial assets, is wealthy, but this is not due to our being unable to determine what is meant by “wealthy” – that is presumably understood. The term “wealthy” may be taken to have an unambiguous, vague meaning as opposed to having two or more distinct meanings. Our inability to say whether or not Jo is wealthy arises as a result of her being a borderline case of “wealthy”. Borderline cases can arise in our use of unambiguous words and ambiguous words need have no borderline cases. Context-sensitivity is similarly distinct in so far as context-sensitive words might be vague, even if a context is
xed, and precise words can be context-sensitive. Consider the word “tall”, for example. What counts as tall can vary from context to context; tall pygmies are usually not tall basketball players. Yet even when this relativity to context is
xed, e.g. tall relative to basketball players, there might still be borderline cases – is he a tall basketball player or not? There may also be situations where one is unable to say whether or not the word applies simply because the correct ascription depends on the context and this phenomenon occurs regardless of the vagueness of “tall”; that is, this inability does not constitute a borderline case. Suppose “tall” were to mean “above average in height”, which intuitively makes it a precise term; nonetheless there may be situations where I am uncertain whether to apply the term or not since it depends on whether the average is taken over pygmies or basketball players. To be sure, there are those (e.g. Diana Raffman) who present strong arguments for the role of context in any adequate analysis of vagueness; nonetheless, though said to be related, the two notions are distinct. Intuitively, then, vagueness is to be distinguished from meaninglessness, ambiguity and context-sensitivity. Thus vagueness is introduced as a phenomenon affecting predicates and, as such, is characterized by the presence of borderline cases as they have been described above. This initial focus on predicate-vagueness reects

Vagueness

5

one aspect of what we might describe as the paradigmatic concept of vagueness: vagueness as applied to predicates and characterized by the presence of borderline cases. This is the concept of vagueness encountered in the work of early twentiethcentury theorists such as Peirce (1902), Russell (1923), Black (1960), Church (1960), Quine (1960) and Alston (1964). It is not surprising that discussions of vagueness have tended to focus on the concept of vagueness as applied to predicates since this is where issues surrounding vagueness originate historically. Problems with vagueness arose in antiquity in the context of the sorites puzzles, puzzles understood as exploiting the apparent lack of sharp boundaries for certain predicates. In subsequent sections we shall consider the sorites puzzles in detail (§2), look at various types of predicate-vagueness (§3), and see how one might extend the concept of vagueness beyond predicates to other semantic categories (§4), but for the moment we need to consider more carefully that other aspect of the paradigmatic concept of vagueness – the relationship between vague predicates and their borderline cases. A predicate having no borderline cases is precise; borderline cases are therefore necessary for vagueness, as already remarked. However, are they suf
cient? Is a predicate’s vagueness really characterized or de¿ned by its having borderline cases? According to a naive paradigmatic concept of vagueness we might suppose it is – a predicate is vague if and only if it has borderline cases. On this understanding of vagueness, the presence of borderline cases is both necessary and suf¿cient. However, even having restricted the notion of a borderline case so as to exclude indeterminacy due to meaninglessness, ambiguity, and context-speci
city, meeting the criteria for having such borderline cases still might seem insuf
cient for vagueness. Acknowledgement of what has become known as the phenomenon of higher-order vagueness results in further quali
cation of the paradigmatic concept of vagueness, expanding on the notion of a borderline case. We can illustrate the problem using a familiar example from Sainsbury (1991) – the predicate “child*”. It is to count as true of all those people who have not yet reached their sixteenth birthday, false of all those who have reached their eighteenth birthday, and neither true nor false of all other people. Now, for a seventeen-yearold it is neither determinately true that they are a child* nor is it determinately false that they are a child* – they do not determinately satisfy the predicate nor do they determinately satisfy its negation. So it would seem that a seventeen-year-old counts as a borderline case for the predicate “child*” thereby making it vague, even though, intuitively, the predicate is perfectly precise. It is indeed the case that the predicate fails to draw a single sharp boundary between its positive and negative extensions; thus there are cases of indeterminacy. Nonetheless, the fact that it draws two sharp boundaries, one between its positive and borderline cases and one between its borderline and negative cases, seems to disqualify it as a candidate for vagueness proper; it is rather an incomplete predicate.3 3 Wright (1987: 244ff) argues that the vague predicate “red” does effect a tripartite division of its range of signi
cance; it has a penumbra of borderline cases but this penumbra is itself precise. Sainsbury (1991: 176ff) convincingly rebuts Wright’s argument by denying Wright’s criterion for the correct use of “red”. Even were Wright’s argument successful, there is no reason to think that it would generalize to all vague terms.

6

Vagueness, Logic and Ontology

We seem to have confused vagueness with incompleteness. Incomplete predicates effect a sharp tripartite division of their range of signi
cance, whereas vague predicates appear to draw no sharp boundaries. Of course, there is no reason why predicates cannot be both vague and incomplete, but now their vagueness will be evidenced by borderline cases for some of the three categories. What is being gestured at or sought when we dismiss cases like those above cannot yet be fully explained by means of the simple notion of borderline cases between the positive and negative extensions of a predicate; something more needs to be said. Early in the contemporary debate Russell (1923: 87) acknowledged an extra ingredient. The fact is that all words are attributable without doubt over a certain area, but become questionable within a penumbra, outside of which they are again certainly not attributable. Someone might seek to obtain precision in the use of words by saying that no word is to be applied in the penumbra, but fortunately the penumbra itself is not accurately [precisely] de
nable, and all the vaguenesses [sic] which apply to the primary use of words apply also when we try to
x a limit to their indubitable applicability.

In the same vein, Alston (1967: 218) more recently echoed this when, after having endorsed the borderline case conception, he continued: … “middle-aged” is vague, for it is not clear whether a person aged 40 or a person aged 59 is middle-aged. Of course there are uncontroversial areas of application and nonapplication. At age 5 or 80 one is clearly not middle-aged, and at age 45 one clearly is. But on either side of the area of clear application there are inde
nitely bounded areas of uncertainty.

The now common response to the vagueness of the penumbra itself is simply to say that the penumbra has borderline cases. Thus, in an attempt to get at this extra ingredient by means of the notion of a ‘borderline case’, that is, from within the paradigmatic concept of vagueness, talk moves to a hierarchy of borderline cases and the paradigmatic concept is iterated. In the last three decades this phenomenon of higher-order vagueness has come to the fore in discussions of vagueness. Higher-order vagueness arises, as we have seen, because vague predicates typically fail to draw any apparent sharp boundaries within their range of signi
cance. The paradigmatic concept we have been discussing initially attempts to accommodate the intuition that there is no apparent sharp boundary between the positive and negative extension of a predicate by describing the presence of a penumbra or borderline region. So, for example, with the predicate “red” the absence of any apparent sharp boundary between the red and the non-red is initially described by reference to borderline cases. The requirement that there be no apparent sharp boundary is thus satis
ed by eliminating the sharp boundary and replacing it with a borderline region. Yet the region itself might paradoxically appear sharply bounded – unless it too has borderline cases. There is no more an apparent sharp boundary between the positive extension and the borderline region than there was between the positive extension and the negative extension. So, if the presence of a penumbral region is taken as de
nitive of vagueness, then it is not itself characterized merely by the existence of borderline

Vagueness

7

cases; there must be borderline cases of borderline cases. But why stop here? There appears to be no more reason to suppose that there is a sharp boundary between the determinately determinately red and the vaguely determinately red than there was to suppose a sharp boundary between the determinately red and vaguely red. ‘At no point does it seem natural to call a halt to the increasing orders of vagueness’, as Fine (1975: 297) expresses it, so the iteration seems endless; borderline cases echo throughout the categories in the sense that any categorization using a vague predicate (e.g. categorizing things according to whether they are determinately red or not) admits of borderline cases. Exactly what this amounts to will be discussed further in §1.5 and a more rigorous account of it must wait until Chapter 7. For now, the lesson of higher-order vagueness is that vague predicates draw no apparent sharp boundaries, not merely that they apparently fail to draw a sharp boundary at the
rst level, or the
rst and second levels, etc. 1.1.2

How pervasive is vagueness?

Now that we have an initial grasp of what predicate-vagueness amounts to, where does it arise – just how pervasive is it? Once we start looking we see vagueness everywhere. The language of the social sciences, for example, seems ineradicably vague. Terms like “neighbourhood”, “community” and “state of recession” all exhibit an apparent lack of sharp boundaries of application thus giving rise to borderline cases. Menges and Skala (1974: 55) describe the situation thus: In the
eld of social sciences one must … pull down the natural barriers between things and thrust the ‘real’ into ‘arti
cial’ new units. … Social science concepts, due to the speci
c formation process, are in principle vaguer than natural science concepts. To that extent the problem of vagueness is particularly important in the social sciences and it is necessary to draw upon suitable formalized theories.

This is sometimes given as a reason for supposing the social sciences to be among the ‘second-grade’ disciplines. Vagueness, seen as a aw to be eliminated, is a suspect and second-grade feature of language and thus any discipline relying on such language is itself second-grade. The behavioural sciences also employ terms that are vague like “depressed”, or “schizophrenic”, and as such they too are often described as second-grade. So too with the biological sciences with their use of terms like “species”, “ecological community”, etc. This supposes, of course, that the physical sciences, in so far as they are a paradigm of a ‘
rst-grade’ discipline, are free of vagueness – as, indeed, they are commonly supposed to be. In the language of the physical sciences vagueness is often seen as a cancer to be eradicated; precision is often seen as one of the aims of a language for such science, though it is questionable whether perfect precision is achievable.4 Beyond the sciences, the languages of ethics and law exhibit the phenomenon of vagueness. In ethics, terms such as “voluntary euthanasia” are vague and their use has been criticized on the grounds that the ensuing indeterminacy of application will (or could) lead down ‘slippery slopes’, from cases of legitimate application in situations where the term is applied to cases where it clearly ought not to apply; the slippery ethical slope is then the slide from the ethical legitimization of the former 4

This will be discussed in more detail in Chapter 3, esp. §4.

Vagueness, Logic and Ontology

8

cases to the legitimization of the latter cases. Again, the underlying thought is that vagueness must be eliminated, in this case from ethical theories, and in this way we will be able to draw a sharp line demarcating legitimate from illegitimate ethical practices. Vagueness infuses the theory with an instability that leads to untoward consequences and thus one must reject the theory.5 In law, courts try to precisify language in an attempt to provide sharp distinctions between the legal and the illegal, borderline cases being seen as problematic for the instigation of legal action. Yet lengthy debates before a judge as to the possibility of, say, counting an act as one of ‘legally justi
able defence’ (and thus permissible by law) are at least sometimes due to the vagueness of the legal terms involved; if we are to draw a line somewhere and thus resolve the vagueness of the offending term for future legal use, the argument concerns where in the borderline area it will be drawn.6 Of course, the fact that language is vague does not mean that we cannot resolve borderline cases by invention or stipulation. In the case of arti
cial languages we, as stipulators, may attempt to eradicate vagueness by using only precise predicates. However, whilst we are free to precisify wherever and whenever we choose, the transformed predicate is more than a mere re
nement in sense of the original vague term; it is a rede
nition. Rede
ned legal concepts such as “drunk” or “alive” and scienti
c concepts such as “blue” (interpreted as “reecting 445–500 nanometres wavelength of light”) may diverge considerably from many natural-language users’ applications of them. Abortion debates and IVF-programme controversies are cases in point. The question of their admissibility as surrogates for the original vague predicate will be discussed in future chapters. The simple fact is that vagueness is ubiquitous. 1.2

The Sorites Paradox

We remarked earlier that the fact that the paradigmatic concept of vagueness centres on predicates is of little wonder given the historical roots of the problem. The reason for this is that vagueness comes to us through history as more than a mere curio. Our attempts to arrive at an understanding of the concept of vagueness arise quite naturally from a series of puzzles – puzzles that are generally understood as depending crucially on the lack of sharp boundaries for the predicates involved. In modern philosophy, these puzzles are usually presented as a class of paradoxes known as the sorites paradox. In coming to appreciate the puzzlement that these paradoxes engender, we will see why the widespread presence of vagueness has been seen as such a problem. 1.2.1

From puzzle to paradox

Diogenes Laërtius (1925: ii 108) attributed seven puzzles to the logician Eubulides of Miletus, a contemporary of Aristotle. These include, amongst others: The Liar: A man says that he is lying. Is what he says true or false?; The Hooded Man: You say 5

A recent book devoted to this problem is Walton (1992). Hart (1955, esp. §V) discusses this problem; also Hart (1961: 120ff). More recently, see Endicott (2000). 6

Vagueness

9

that you know your brother. Yet that man who just came in with his head covered is your brother and you did not know him; and The Bald Man or The Heap: Would you describe a man with one hair as bald? Yes. Would you describe a man with two hairs as bald? Yes. Would you describe … You must refrain from describing a man with a million hairs as bald, so where do you draw the line? Alternatively: Would you describe a single grain of wheat as a heap? No. Would you describe two grains of wheat as a heap? No. Would you … You must admit the presence of a heap sooner or later, so where do you draw the line? This last puzzle, when presented as a series of questions about the application of the predicate “bald”, was known as the falakros puzzle and, when presented as a series of questions about the application of the predicate “heap”, was known as the sorites puzzle (from the Greek word soros meaning a heap). Later, the whole class of puzzles of this type became known as sorites puzzles. It is not known whether Eubulides actually invented the sorites puzzle. Some scholars have attempted to trace its origins back to Zeno of Elea, claiming his paradox of the Millet Seed as a sorites puzzle. However, the evidence seems to point to Eubulides as the
rst to employ the puzzle. Nor is it known just what he had in mind when he formulated it. Many targets have been suggested, but he is said to have been exclusively interested in logic and the general consensus is that it was for its delightful puzzlement alone that he proffered such a conundrum.7 It was, however, employed by later Greek philosophers to attack various positions, most notably by the Sceptics against the Stoics’ claims to knowledge. Describing this early history is a fascinating scholarly exercise in itself, but there is little point in repeating here what has already been done in this area. With the important early texts translated in Long and Sedley (1987), and discussions in Burnyeat (1982), Barnes (1982), Mignucci (1993) and Williamson (1994) covering the history of the sorites from antiquity to the twentieth century, there is little reason to add to the already suf
cient writings. These puzzles of antiquity are now more usually described as paradoxes, that is, as apparently valid arguments with apparently true premises and an apparently false conclusion. Though the conundrum can be presented informally as a series of questions whose puzzling nature gives it dialectical force, it can be, and was, presented as having logical structure as well. The following argument form of the sorites was common:8 7

For more on this see Barnes (1982). Some Stoic presentations of the argument also used a form which replaced all the conditionals, “if A then B”, with “not(A and not-B)” to stress that the conditional should not be thought of as being a strong conditional, but rather the weak Philonian conditional (the modern material conditional) according to which “if A then B” was equivalent to “not (A and not-B)”. Such emphasis was deemed necessary since there was a great deal of debate in Stoic logic regarding the correct analysis for the conditional. In thus judging that a connective as weak as the Philonian conditional underpinned this form of the paradox they were forestalling resolutions of the paradox that denied the truth of the conditionals based on a strong reading of them. However, in judging that a connective as strong as the Philonian conditional really was employed in the sorites arguments, since the argument thus interpreted was valid, they reduced their range of possible responses to the paradox to denying either the apparent truth of the premises or the apparent truth of the conclusion. They opted for the former. 8

Vagueness, Logic and Ontology

10

A man with 1 hair on his head is bald. If a man with 1 hair on his head is bald then a man with 2 is. If a man with 2 hairs on his head is bald then a man with 3 is. : . If a man with 9,999 hairs on his head is bald then a man with 10,000 is. ∴

A man with 10,000 hairs on his head is bald.

It is paradoxical since, though it appears valid, its premises appear true whilst the conclusion seems false. (If a man with 10,000 hairs does still seem bald, then, since it is arbitrarily chosen, pick any number k – say, a million – for which he does not and extend the argument until we can infer, contrary to appearances, that a man with k hairs is bald.) Innumerable sorites paradoxes can be expressed in this form. For example, one can present the original paradox which gave its name to the whole family. Since one grain of sand is not describable as a heap and if one is not then two is not, so two grains of sand do not constitute a heap. Again, if two is not then three is not, so three grains do not constitute a heap, … etc. So, for any number of grains of sand k, k grains of sand do not constitute a heap yet we rightly feel that there are piles of sand describable as heaps. Similarly, if one is prepared to admit that there are piles of sand describable as heaps, then one could prove that one grain of sand counts as a heap since the removal of one grain at any stage cannot make the relevant difference! The falakros paradox leads, as we have just seen, to the conclusion that any number of hairs still makes for baldness if one hair does, but can also be run using a variant negative form – so if anyone is not bald then everyone is not bald! Thus the argument can take both a positive and negative form. The predicates “is bald”, “is a heap”, as well as “is small”, “is a child”, etc. are all soritical – i.e. can
gure in sorites, or soritical arguments, also known as ‘little-by-little’ arguments. This standard form of the sorites can be schematically represented as follows: Standard Sorites Fa1 Fa1 > Fa2 Fa2 >Fa3 :. Fak–1 > Fak ∴

Fak

(for any number k)

where “>” represents the connective “if … then …”; F represents the soritical predicate, e.g. “is bald”; and the series 〈a1, …, ak 〉 represents the sequence of subjects with regard to which F is soritical, e.g. 〈a man with 1 hair on his head,…, a man with k hairs on his head〉.

Vagueness

11

Having described this standard form, we can state conditions under which any argument of this form is soritical.9 Firstly, the series 〈a1 … ak 〉 must be ordered; for example, scalps ordered according to number of hairs, heaps ordered according to number of grains of wheat, and so on. Secondly, the predicate F must satisfy the following three constraints: (i) it must appear (determinately) true of a1, the
rst item in the series; (ii) it must appear (determinately) false of ak, the last item in the series; (iii) each adjacent pair in the series, an and an+1, must be suf
ciently similar as to appear indiscriminable in respect of F – that is, both an and an+1 appear to satisfy F or neither does. Under these conditions F is soritical relative to the series 〈a1, …, ak〉 and any argument of the above form using F and 〈a1, …, ak〉 will be soritical.10 It is easy to see now that sorites arguments come in pairs, each of the pair identi
able either by the predicate involved (thus we might speak of positive and negative versions) or the series involved (thus we might speak, as the ancients did, of sorites by adding or subtracting). It is easy to verify that F will be soritical relative to 〈a1, …, ak〉 if and only if not-F is soritical relative to 〈ak, …, a1〉; i.e. for every positive sorites by adding (e.g. using “is bald”) there corresponds a negative sorites by subtracting (e.g. using “is not-bald”). And F will be soritical relative to 〈ak, …, a1〉 if and only if notF is soritical relative to 〈a1, …, ak〉; i.e. for every positive sorites by subtracting (e.g. using “is a heap”) there corresponds a negative sorites by adding (e.g. using “is not a heap”). That key feature of soritical predicates which drives the paradoxes, constraint (iii), is what Crispin Wright describes as tolerance.11 Predicates such as “is bald” or “is a heap” appear tolerant of suf
ciently small changes in the relevant respects – namely number of hairs or number of grains. The degree of change between adjacent members of that series relative to which F is soritical would seem too small to make any difference to the application of the predicate F.12 Yet large changes in the relevant respects will make a difference, even though large changes are the accumulation of small ones which do not seem to make a difference. This is the very heart of the conundrum which has delighted and perplexed so many for so long. How might we respond then to the paradoxical nature of a standard sorites argument? The options are clear. One can: 9

Conditions described by Barnes (1982: 30–32). Some logic texts describe multi-premise syllogisms, polysyllogisms, as sorites arguments; e.g. Copi (1972: ch. 7, §5); Luce (1958: ch. 8). Polysyllogistic arguments are similar to sorites as we have de
ned them in so far as they are chain-arguments; however, polysyllogisms need not be paradoxical and sorites as we have de
ned them need not be syllogistic in form. The usage we have adopted is the more usual these days. 11 Wright (1975: 333–4). 12 Naturally, if the degree of change between adjacent members of some other series 〈b , 1 …, bj〉 exceeds the limits of tolerance of the predicate F, then condition (iii) above is not met, and though F may be soritical relative to the series 〈a1, …, ak 〉 it will not be soritical relative to the series 〈b1, …, bj〉. For example, the predicate “is small” is soritical relative to the series 〈1,2,3, … 10,000〉 since it appears true of 1, false of 10,000, and seems tolerant of a difference of 1; however, it is (arguably) not soritical relative to the series 〈1,100,200,300, … 10,000〉. The predicate is (arguably) not tolerant of such large changes. 10

12

(i) (ii)

(iii) (iv)

Vagueness, Logic and Ontology

Deny that the problem can legitimately be set up in the
rst place; that is, logic does not apply to soritical expressions. Assume logic does apply to soritical expressions but deny that the argument is valid. Iterated modus ponens is not valid for the conditional “>”. Since the argument is valid by the canons of classical logic, this response amounts to a refutation of classical logic. Assume logic does apply to soritical expressions and that the argument is valid, but deny one of the premises. Assume logic applies, and that the argument is both valid and has true premises and so accept its conclusion.

Option (i) is what Haack (1974: ch. 6, §4) describes as the ‘no-item’ strategy; soritical expressions are beyond the scope of logic. Well-known advocates of this approach include Frege and Russell. Frege (1903: §56) thought that predicates with fuzzy boundaries of application, vague predicates, lack sense and hence cannot
gure in sentences having truth conditions. Russell (1923: 88–9), whose views will be considered in more detail in Chapter 2, claimed that ‘all traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life’ where vague language abounds. Since soritical expressions are vague (a point we shall return to shortly), then logic does not apply to them. This response to the sorites paradoxes owed much to ideal-language doctrines popular earlier last century and was associated with the demand for logically perfect languages. Ordinary language, in so far as it fell short of perfection, was deemed un
t for serious consideration, and vagueness, like so many other phenomena in natural language, was seen as a defect to be eliminated. An obvious problem with this approach is that logic is relegated to a ‘celestial realm’ (as Russell puts it), and the fact that we do logically evaluate everyday discourse speaks against such an approach. The fading of ideal-language doctrines and respect for ordinary ways of talking have meant that this approach is no longer viewed as tenable. Options (ii) and (iii) both presuppose natural language to be in order as it is and attempt to describe how it is that a logic of vagueness, in particular a logic of soritical predicates, shows us a way out of the paradoxes that beset such language. Having accepted that the conclusion of soritical arguments is false, opponents of the paradox are then divided into what Barnes has described as the radical opponents – those who endorse option (ii) and claim that the argument is invalid, and the conservative opponents – those who endorse option (iii) and claim that the argument, though valid, has some non-true premise.13 The incredulity with which option (ii) is generally met is a measure of the popularity of option (iii), given that most theorists these days are concerned to avoid the more hard-nosed options presented by (i) and (iv). The conservative opponent typically picks the conditional premises as the place to attack the argument. Though 13 Of course these options are by no means exclusive; it may be that the argument is both invalid and has false premises; thus the sorites paradox is doubly dissolved. No one has, to my knowledge, pursued such a course and the reasons are obvious. It is dif
cult enough to resolve the paradox by either route alone; since either will itself be enough to resolve the matter there is no perceived need to engage in the doubly dif
cult task of convincing an audience that our intuitions are wrong on two counts.

Vagueness

13

it appears that adjacent subjects in the sorites series are suf
ciently similar in respect of F to treat them alike in this regard, appearances are deceiving. There are a number of accounts of vagueness which take up this option. One is the epistemic view of vagueness, championed by Williamson and Sorensen, according to which classical logic remains completely intact, even in the context of vagueness. Another is the popular supervaluationist account which modi
es classical logic in a conservative way, preserving iterated modus ponens and (in one sense at least) all classical theorems while abandoning bivalence. Another less conservative, non-bivalent approach is offered by many-valued logics, as we shall see as the discussion unfolds. It is this option that I shall be defending in this book. Yet another alternative pursuing option (iii), conservative at least to the extent that classical logic remains completely intact, takes issue with the categorical premise, at least the categorical premise of positive sorites arguments. According to Unger (1979b) and Wheeler (1979), all such positive, soritical predicates like “is bald”, “is a heap”, “is a stone”, etc. apply to nothing. Negative sorites arguments using predicates like “is not bald”, “is not a heap”, “is not a stone”, on the other hand, are deemed sound, thus conforming with option (iv). The response is therefore a mixed response, with different diagnoses offered for different paradoxes. Indeed, the very soundness of the negative sorites serves to emphasize the falsity of the categorical premise of the corresponding positive version. Each of a positive and negative pair of arguments cannot consistently be counted as sound since the soundness of one counts against the soundness of the other. The validity of sorites reasoning in conjunction with acceptance of the conditional premises serves to show that the predicate in question is either all-encompassing or empty and applying to nothing at all. It applies everywhere in the sorites series if anywhere, and applies nowhere if not everywhere. Unger and Wheeler propose a view in which positive soritical predicates apply nowhere. Given a choice of predicate-extension that is all or nothing, they opt for nothing. The explanation of the non-applicability of such terms lies in the fact of their incoherence.14 Dummett (1975), on the other hand, pursues option (iv) pure and simple. Considering options (i)–(iii) as failed responses to the paradox, Dummett feels forced to accept the view according to which sorites paradoxes are sound, exhibiting valid reasoning from true premises to a conclusion which is nonetheless false. Sorites-prone terms are radically incoherent. Such a view has also been advocated in Quine (1981) and Rolf (1984). Since it is advocated as an option of last resort, if option (iii) can be shown to succeed, as I shall argue, then the motivation for adopting such a response is blocked. It would indeed be surprising if, contrary to all appearances, soritical predicates were so radically defective; they certainly don’t appear so. Moreover, as we shall see later in this chapter, further facts about the nature of vagueness will render this response untenable by its own lights. Matters are further complicated by the fact that soritical arguments can take other forms. So, though we might initially focus on the standard form, what is required is a response to the sorites paradox in all its forms. Two other forms can be discerned. One variant replaces the set of conditional premises with a universally quanti
ed 14

Like the Barber Paradox, then, the positive sorites paradox is taken to establish the falsity of a key existence assumption – that there is a heap to begin with, for example.

Vagueness, Logic and Ontology

14

premise and the sorites paradox is seen as proceeding by the inference pattern known as mathematical induction. With n a variable ranging over the natural numbers, the paradox is now represented as follows: Mathematical Induction Sorites Fa1 ∀n(Fan > Fan+1) ∴

∀nFan

So, for example, it is argued that since a man with one hair on his head is bald and since the addition of one hair cannot make the difference between being bald and not bald (for any number n, if a man with n hairs is bald then so is a man with n+1 hairs), then a man with n hairs on his head is bald regardless of what number n one chooses. Yet another form is a variant of this inductive form.15 Assume a man with k hairs on his head is not bald. Then by the least-number principle (equivalent to the principle of mathematical induction) there must be a least such number, say i + 1, such that it is not the case that a man with i + 1 hairs on his head is bald. Since a man with one hair on his head is bald, it follows that i +1 must be greater than 1. So, there must be some number n (that is, i) such that a man with n hairs counts as bald whilst a man with n + 1 does not. Thus it is argued that though a1 is bald, not every number n is such that an is bald, so there must be some point at which baldness ceases. This variant can be represented as follows: Line-drawing Sorites Fa1 not-Fak ∴

∃n(Fan & not-Fan+1)

Any argument having the form of any of the above sorites arguments will be soritical just if conditions similar to those set down for the standard sorites are met. Now obviously, given that sorites arguments have been presented in these forms, ‘the sorites paradox’ will not be solved by merely claiming, say, iterated modus ponens or universal instantiation to be invalid; all forms must be addressed one way or another. One would hope to solve it, if at all, by revealing some general underlying fault common to all forms of the paradox. No such general solution could depend on the diagnosis of a fault peculiar to any one form. On the other hand, if no general solution seems available, then ‘the sorites paradox’ will only be solved when each of its forms separately have been rendered toothless. In either case, to single 15

See Cargile (1969: 193); Rolf (1984: 220).

Vagueness

15

out something peculiar to one of the forms will not, by itself, solve the problem. Of course, some solution peculiar to one of the forms might lie at the heart of the matter in so far as this feature is implicitly involved in all the other forms, but then we will only have a general solution once we have pointed out how this feature underlies all forms of the sorites. 1.2.2

Soriticality and vagueness

Having thus set out and commented upon the nature of sorites puzzles and paradoxes, it is easy to see intuitively that soritical predicates are vague; soritical predicates appear tolerant so it would seem that there cannot be any sharp boundaries within the predicate’s range of signi
cance. Are all vague predicates soritical, though? To be sure, the phenomenon of vagueness is typi
ed by soriticality. The reasons for this presumably are to be found in the fact that vagueness is historically rooted in sorites puzzles and because this is where the challenge presented by vagueness seems most forcefully presented. Yet, one might think, the lack of apparent sharp boundaries or presence of borderline cases can arise for predicates that are not soritical. If one takes vague predicates to be predicates which fail to sharply partition, in any way, their range of signi
cance, then further questions of ordering the range and doing so in such a way as to make adjacent members apparently indiscriminable in respects relevant to the application of the predicate are simply left unanswered. As such, there is no reason to think all vague predicates soritical. One could of course de¿ne vagueness in such a way that all and only soritical predicates are vague. However, there may indeed be broader issues concerning the apparent deviant semantic behaviour of vague predicates, famously exempli
ed by Frege’s worries about concepts that have ‘fuzzy boundaries’, that motivate our treating some nonsoritical predicates on a par with soritical ones – that is, for our treating some nonsoritical predicates as vague. Rolf (1981: 98), for example, cites the possibility of a mathematical predicate having borderline cases without there necessarily being any ordered series of objects in the predicate’s range of signi
cance with regard to which it could be said to be soritical. Shapiro (2006: 4) also doubts that soriticality is necessary for vagueness.16 It may be that an ordering is unobtainable or, given some ordering, it may be that adjacent members of any such series are suf
ciently dissimilar to falsify any claims to tolerance on behalf of the predicate. The issue, I think, remains open. Comments to the effect that vague predicates are “typically” soritical, though apparently true, do not settle the matter.17 16 Greenough (2003: 270) presents argument to show that the tolerance underlying soriticality follows necessarily from vagueness understood as typi
ed by borderline cases. However, his proof (see assumption 2) seems to rely on a further assumption that vague terms are associated with some dimension of comparison so that the assumed “intolerance” is a matter of intolerance along some presumed dimension of comparison. The step from borderline cases to soriticality is made all the easier for the presumption, as Greenough ingeniously shows, however the presumption requires defence. 17 See, for example, Keefe (2000: 7).

Vagueness, Logic and Ontology

16

1.3

Kinds of Vagueness

Alston (1967: 219) draws a distinction between what he calls degree-vagueness and combinatory vagueness. Degree-vagueness consists of those cases in which the vagueness stems from the lack of precise boundaries between application and nonapplication – or at least their apparent lack – along some dimension. This is the standard kind of vagueness alluded to in most discussions, furnishing standard examples of soritical terms. “Bald” fails to draw any sharp boundaries along the dimension of hair quantity; “heap” fails to draw any sharp boundaries along the dimension of grain quantity; etc. This kind of vagueness is contrasted with combinatory vagueness. Another, more complex, source of indeterminacy of application is to be found in the way in which a word may have a number of logically independent conditions of application. A signi
cant example is the word “religion.” If we consider clear cases of religions, such as Roman Catholicism and Orthodox Judaism, we
nd that they exhibit certain striking features, each of which seems to have something to do with making them religious. These include: (1) Beliefs in supernatural beings (gods). (2) The demarcation of certain objects as sacred. (3) Ritual acts focused around sacred objects. (4) A moral code believed to be sanctioned by the gods. (5) Characteristic feeling, such as awe and a sense of mystery, which tend to be aroused in the presence of sacred objects and which are associated with the gods. (6) Prayer and other forms of communication with the gods. (7) A world view, that is, a general picture of the world as a whole, including a speci
cation of its over-all signi
cance, and a picture of the place of the individual in the world. (8) The individual’s more or less total organisation of his life based on the world view. (9) A social organisation bound together by the preceding characteristics. The existence of a plurality of distinguishable conditions of application does not in itself render a term vague. We can distinguish two conditions of application of the word “square”: being a rectangle and having all sides equal. Here there is a de
nite answer to the question of what combination of these conditions is necessary and what combination is suf
cient for the application of the term, the answer being that each of the conditions is necessary but not suf
cient and that their combination is suf
cient. With “religion” it seems clear that the combination of all the features listed above would be suf
cient to guarantee application of the term. But what feature, or combination thereof, is necessary? And is any subset of the features suf
cient? There do not seem to be de
nite answers to these questions.

This indeterminacy is further emphasized by noting that, though some people are inclined to take beliefs in supernatural beings as necessary for religion, the term is nonetheless applied to systems, such as humanism, Hinayana Buddhism and communism, where belief in supernatural beings is lacking. Ritual too seems unnecessary, considering that it has disappeared from the Quaker movement though the movement is still sometimes described as a religion. Similarly controversial are all claims to the effect that some subset of conditions (1)–(9) is suf
cient to make something a religion. Communism for example could be said to have sacred objects in the form of Lenin’s body and the writings of Karl Marx, a de
nite world view, and a way of life based on it, yet it is not clear whether this is suf
cient for calling it a religion.

Vagueness

17

In such cases we have a variety of conditions, all of which have something to do with the application of the term, yet are not able to make any sharp discriminations between those combinations which are, and those which are not, suf
cient and/or necessary for application. There will be certain combinations, like the one exempli
ed by the case of Communism, in which we get uncertainties and disagreements among uent speakers of the language as to whether the word is applicable. We may call this ‘combinatory vagueness.’ (Ibid.)

Another distinction due to Burks (1946: 481–2) is that between linear and multidimensional vagueness. The former is described as follows: Linear vagueness is best illustrated by means of qualities which form a continuum, such as the colors of the color spectrum. One may de
ne blue ostensively by presenting objects of various shades of blue, and counter-instances of various shades of purple and green. Since it is impossible to detect color differences beyond a certain point, borderline cases, such as blue-greens, will arise.

As Rolf (1981) points out, there is a mistaken (though common and usually harmless enough) assumption here that colours vary along one dimension, whereas colours may vary according to hue, brightness and saturation. Colours can vary in a multidimensional way. The idea of linear vagueness is simple enough, though. Consider “small number” applied to the ordinals; here there seems only one dimension of variation – up and down the number series. Multi-dimensional or non-linear vagueness, on the other hand, is said to arise in connection with qualities which are such that their instances cannot be ordered linearly. The concept of chair is an example … How much of a back does a chair need in order to be a chair rather than a stool? At what point does an article of furniture cease being a chair and become a chaise longue? Various features are involved in being a chair and a piece of furniture may be a borderline case because it lacks one or another of these, or because it has them all but not to a suf
cient extent. Since one cannot give these various features weighting factors (it would be arbitrary to say that having a back is one-third as important as having a certain length of seat, for instance), it is clearly impossible to arrange chairs and non-chairs in a linear sequence. Burks (1946: 482)

One can of course arrange some subset of them in such a sequence. Take that subset that varies along only one dimension, e.g. the height of the back. A soritical argument could be run using the term “chair” in this way, so “chair” is soritical; however, its multi-dimensional vagueness means that it is not soritical relative to its whole range of signi
cance, but only to some linear subset. Linear predicates totally order their range of signi
cance whereas non-linear ones, multi-dimensional ones, only partially order their range, providing a plurality of linear sub-orderings or chains. This is why multi-dimensionally vague predicates could only ever be soritical relative to a subset of their range – soriticality is relative to some chain or linear, totally ordered sequence. One might well wonder now whether the distinction between degree-vagueness and combinatory vagueness coincides with Burks’s distinction between linear and

18

Vagueness, Logic and Ontology

multi-dimensional vagueness. The answer is that it does not. Terms that exhibit linear vagueness are degree-vague since their vagueness stems from lack of sharp boundaries along some dimension. And combinatory vagueness is obviously multidimensional. However, it would seem that multi-dimensionally vague terms can be degree-vague (so multi-dimensionally vague terms need not be combinatory vague and degree-vague terms need not be linearly vague). Generally we can say that, given vague predicates of degree G and H (say), their conjunction, G & H, will be a multi-dimensionally vague predicate of degree. “Knowledge” provides a good example of this non-linear degree-vagueness in so far as it is de
ned as justi
ed, true belief; beliefs can come in degrees, as can justi
cation. Thus, though p may be true and justi
ed, if it counts as a borderline case for “belief” it will be a borderline case for knowledge. On the other hand, we could
x on some true belief q that is a borderline case for “justi
ed” and thereby generate another borderline case for knowledge. (The Sceptics concentrated on this justi
catory aspect to show the soriticality of “knowledge”, thereby presenting the Stoics with a problem in so far as soritical concepts were seen as problematic.) The fact that p and q are both in the signi
cance range of “knowledge” though neither can be said to be any more or less a case of knowledge than the other, shows “knowledge” to be multi-dimensionally vague. Yet these borderline cases for “knowledge” are clearly not due to combinatory vagueness; they do not arise as a result of any indeterminacy regarding what to count as necessary and suf
cient conditions for knowledge. Alston (1967: 219) draws yet another distinction that bisects that already drawn between combinatory and degree-vagueness – vagueness of application versus vagueness of individuation. We began this chapter by describing the “no apparent sharp boundaries” phenomenon typical of vague predicates and then proceeded to focus more carefully on this signature of vagueness via cases where there appeared to be no sharp boundary to the application of the predicate. Such cases present us with vagueness in the predicate’s application conditions, i.e. vagueness of application. The term “wetland” for example has vagueness of application in so far as there are regions for which it is indeterminate whether or not the term applies. Yet as we noted at the outset, in addition to the predicate itself seeming indeterminate in extension, almost any particular wetland region would seem indeterminate in spatiotemporal extent. So “wetland” also exhibits vagueness in regard to how much land to include in something that is determinately described as a wetland. Quine (1960: 126) describes the two types of vagueness thus: Commonly a general term true of physical objects will be vague in two ways: as to the several boundaries of all its objects [what Alston has dubbed vagueness of individuation] and as to the inclusion or exclusion of marginal objects [what Alston has dubbed vagueness of application]. Thus take the general term “mountain”: it is vague on the score of how much terrain to reckon into each of the indisputable mountains, and it is vague on the score of what lesser eminences to count as mountains at all. To a lesser degree “organism” has both sorts of vagueness. Thus under the
rst heading there is the question at what stage of ingestion or digestion to count food a part of the organism … Under the second heading there is the question whether to count
lterable vira as organic at all.

Vagueness

19

“Mountain” exhibits vagueness of application in so far as there are topographical features which are borderline cases for the term “mountain”. It may seem essentially indeterminate whether or not some particular topographic feature is a mountain. And “mountain” exhibits vagueness of individuation in so far as there may be spatiotemporal regions which are borderline cases for being part of that mountain. As Alston explains it, the term exhibits vagueness of individuation since there is an indeterminacy as to what we are to count as one mountain or two. It may seem essentially indeterminate whether or not some particular topographic feature, which is a mountain, is a mountain as opposed to many mountains (e.g. two). So predicates may exhibit vagueness of individuation as well as vagueness of application. Moreover, according to Quine (ibid.), singular terms like “Mount Rainier” exhibit vagueness of individuation. Insofar as it is left unsettled how far from the summit of Mount Rainier one can be and still count as on Mount Rainier, “Mount Rainier” is vague. Thus vagueness affects not only general terms but singular terms as well. A singular term naming a[n] … object can be vague in point of the boundaries of that object …

In other words, the vagueness of a singular term N consists in its being vague what is denoted by N in the sense that the boundaries of the object denoted by N are vague. There are spatial points which count as borderline cases for being part of the denotation of N. For instance, it may be vague whether “Mount Rainier” denotes the spatial region including that point on the lower ridge to the north-east (pointing). Many singular terms exhibit this kind of vagueness. Russell (1923) also points to there being vague singular terms, focusing instead on temporal considerations. Russell considers, for example, the gradual processes of a person’s birth and death which give rise to temporal instants or intervals which count as borderline cases for being part of temporal history of the denotation of the name “Ebenezer Wilkes Smith”. Vagueness in the spatial or temporal boundaries of an object denoted by N will suf
ce for the vagueness of N. So too will vagueness in its mereological or modal boundaries. 1.4

Extending the Concept

The extension of the concept of vagueness to include names as well as predicates quite naturally leads one to enquire after vagueness in other grammatical categories and to wonder about the relationships between the vagueness of various categories. Can the logical connectives be vague? Is the vagueness of a singular term suf
cient for the vagueness of any sentence in which it
gures? It is common to explicitly describe the vagueness of a sentence – for example, its being vague whether the sun is a hot star – in terms of its appearing to be neither determinately true nor determinately false. The idea is simple. The sentence “The sun is a hot star” describes a state of affairs which appears to neither determinately obtain nor determinately not obtain. That state of affairs which actually does obtain in our world counts as a borderline case for the denotation of the sentence “The sun is a hot star”. The vague sentence is a borderline case for the truth-predicate,

20

Vagueness, Logic and Ontology

appearing to be neither determinately true nor determinately false. (Since the borderline case is, in this instance, an actual state of affairs the vagueness is extensional. An intensionally vague sentence is one for which there could conceivably be a borderline case.) Alternatively, and equivalently, we may say that a sentence S is vague just if its extension – the set of possible worlds in which S obtains – is vague; i.e. there is some world a such that it is indeterminate whether it is in the extension of S. Earlier remarks attempting to distinguish borderline cases from other types of indeterminacy apply here in so far as the notion of borderline cases is employed (just as, of course, they apply when we speak of the vagueness of a name in terms of there being borderline cases for its denotation). So too do remarks concerning higherorder vagueness. Thus the apparent semantic indeterminacy is not due to relativity, ambiguity, meaninglessness, incompleteness, etc. And the apparent truth-value gap is not itself sharp. And again, just as we could only gesture at the notion of a borderline case for a predicate rather than give any strict reductive de
nition, this account of sentence-vagueness only points towards an intuitive understanding of what sentence-vagueness consists in. But it’s enough to get on with. Given the basic recognition of a sentence as vague, it is this sentence-vagueness which leads us to say that the denotation of the subject term counts as a borderline case for the predicate. That is to say, vagueness is primarily (though not paradigmatically) an attribute of sentences, the recognition of which leads us to point out that the predicate involved has a borderline case and subsequently leads us to say that the predicate itself is vague. The vagueness of a sentence is seen as giving rise to the vagueness of the predicate involved. However, the leap from sentence-vagueness to predicate-vagueness can be contested. Whilst sentence-vagueness does generate borderline cases for predicates, it is questionable whether this presence of borderline cases is suf
cient for predicatevagueness. Once we have recognized that singular terms or names can be vague, a sentence can be vague by virtue of our predicating a precise predicate of a vague singular term. Whilst the vagueness of the sentence entails that the predicate has a borderline case, it does not follow that the predicate itself is vague. Thus it appears we may have to amend our account of predicate-vagueness offered in §1. Rolf (1980) invites us to consider the following two sentences: (1) (2)

The sun is a hot star. The sun has a diameter of exactly 1.39 × 109 metres.

Both are assumed to be vague and the sun is therefore to be considered a borderline case for both predicates involved.18 However, whilst the vagueness of sentence (1) seems correctly attributable to indeterminacy surrounding what to count as a hot star – that is, predicate-vagueness – sentence (2) seems vague by virtue of the vagueness of individuation of “the sun”. Intuitively the predicate “has a diameter of exactly 1.39 × 109 metres” is precise, even though the sun constitutes a borderline case for the predicate. 18

The use of ‘exactly’ in sentence (2) should not be treated as synonymous with ‘determinately’, which would of course render sentence (2) false rather than indeterminate.

Vagueness

21

Like Rolf, I take this intuition to be well founded. However, Rolf’s means for making the requisite revision depends on our having to hand some notion of ontological vagueness or fuzziness, and it is hotly contested whether this notion is even coherent. (Though I think it is, it will not be until Chapter 5 that such an opinion will be argued for.) A much less question-begging, and thus more widely acceptable, account is to be preferred and can be described, replacing talk of the fuzziness of an object with the vagueness of a singular term. Sentence (1) is vague by virtue of the vagueness of the predicate; any vagueness in the subject term is passive. To see this, assume all apparent indeterminacies in delimiting that thing denoted by “the sun” to be resolved determinately one way or the other. So, for example, (pointing) that outer solar are is to be considered either a determinate part or a determinate non-part of the object denoted by “the sun”, and so on. The various resolutions provide us with a range of admissible delimitations of the sun. The temperature variation across the various admissible delimitations is too small to make any difference to the application or non-application of the predicate “is a hot star”. Any admissible delimitation of the sun is of such a temperature to count as a borderline case for being a hot star. So, even were “the sun” considered precise, denoting any of the admissible delimitations, its referent would be a borderline case for the predicate “is a hot star”. The vagueness of the singular term is thus irrelevant to the vagueness of the sentence; it is passive vagueness. The source of the vagueness in sentence (1) is to be found in the vagueness of the predicate. Sentence (2), on the other hand, is vague speci
cally because it is vague just what counts as the sun. The vagueness of the subject term is, in this case, active. There are ways of resolving its vagueness, taking “the sun” to clearly denote some admissible delimitation, so that the sentence appears determinately true, and ways of so doing that makes it appears determinately false. On some admissible delimitations it is clear that the object denoted determinately has the relevant diameter; on others it is clear that it determinately does not. The sun is, therefore, only a borderline case for the precise predicate “has a diameter of exactly 1.39 × 109 metres” because we are unable to determine precisely what counts as the sun. This is the source of the sentence-vagueness. The general picture can be put as follows. Given the above accounts of sentenceand name-vagueness, we shall say that the vagueness of an atomic sentence Fa is due to the vagueness of the subject term a if and only if the vagueness in a is active in the context of F, that is, if and only if there are distinct admissible delimitations of a some of which are clear satis
ers of the predicate and others which are clearly not. Of course, if the singular term
guring in the vague sentence is precise, then trivially any vagueness in the singular term is passive – there is none – so in all such cases the vagueness is due to the vagueness of the predicate. Predicates we now prefer to describe as precise have (or could have) borderline cases in the sense of a borderline case we have hitherto been using, as witnessed by sentence (2) above, but we can now see our way clear of such deviant cases. A stricter notion of predicate-vagueness, which takes account of cases like these, while preserving the relation between predicate-vagueness and the (possible) presence of borderline cases, can be described by further constraining what counts as a ‘borderline case’.

22

Vagueness, Logic and Ontology

Let us characterize something as a resilient borderline case for a predicate F just if: (i) it is a borderline case for the predicate; and (ii) no two admissible delimitations of it (i.e. no two admissible resolutions of the apparent indeterminacy regarding what to count as that thing) are such that one is a determinate satis
er of F whilst the other is a determinate non-satis
er of F. Then, though the sun constitutes a borderline case for the predicate “has a diameter of exactly 1.39 × 109 metres”, it is not a resilient borderline case – there are ways of determinately individuating the sun as having exactly that diameter and ways of determinately individuating the sun as not. On the other hand, the sun is a resilient borderline case for the predicate “is a hot star” – it counts as a borderline case no matter how it is individuated. Similarly deviant cases of indeterminacy surrounding what to count into the extension of a name can be conceived of. However, having carefully chosen an account of name-vagueness, they do not give rise to deviant cases of namevagueness as de
ned. The deviancy that arose with regard to predicates – something’s being a borderline case only because we were unable to determine precisely what to count as that thing – also underlies cases where a certain spatiotemporal region is a borderline case for being part of the denotation of some name N only because it appears indeterminate precisely what to count as that spatiotemporal region. For example, suppose “Quine National Park” denotes a precisely delimited spatio-temporal region one of whose spatial boundaries runs along Russell River. It could appear indeterminate whether “Quine National Park” denotes that spatial region including Russell River, but the borderline case, Russell River, will only give rise to such apparent indeterminacy by virtue of the apparent indeterminacy surrounding what to count as Russell River – it will not be a resilient borderline case and therefore will not threaten the precision of the term “Quine National Park”. Given the earlier de
nition of the vagueness of a singular term (a singular term N is vague if and only if there are spatio-temporal points which count as borderline cases for being part of the denotation of the N), despite Russell River being a borderline case for being part of the denotation of “Quine National Park”, since it is not a point but, rather, a possibly indeterminate collection of points, it will not make for the vagueness of “Quine National Park”. On the other hand, the existence of any region as a borderline case for being part of the denotation of a term N will make for the vagueness of N just if it is a resilient borderline case (in which case the indeterminacy must be due to the vagueness of N) and spatio-temporal points are necessarily resilient borderline cases if they are borderline cases at all. With the basic conceptions of sentence-vagueness and name-vagueness we are thus able to amend our account of predicate-vagueness via the notion of ‘resilience’, thereby excluding deviant cases arising from the vagueness of names. Rather than carrying the quali
ed notion of ‘resilient borderline cases’ along with us in subsequent chapters, though, for convenience I shall simply speak of borderline cases as constitutive of vagueness. Borderline cases are generally resilient and, where they are not and this is of import, we can invoke the distinction just described. We have now to hand conceptions of predicate-, name- and sentence-vagueness which serve to underpin the intuitions described earlier regarding how the categories of vagueness interact. Assume that some atomic sentence, Fa, is vague; it appears essentially indeterminate whether the denotation of a satis
es F. So the denotation

Vagueness

23

of a is a borderline case for F. Moreover, if a is precise the denotation of a is a resilient borderline case for F, so F is vague. On the other hand, if F is precise then the denotation of a is not a resilient borderline case for F so, since it is non-resilient though a borderline case nonetheless, the vagueness in predicating F of the denotation of a is attributable to the vagueness concerning the denotation of a. In other words, given a vague sentence, if the predicate is precise then the name is vague and if the name is precise then the predicate must be vague. The vagueness of the whole must be attributed to the vagueness of some part; there must be some vague part which is active in the sense that it causes the vagueness of the whole sentence. Of course, just because the vagueness of the sentence is blamed on the predicate, it does not follow that the name must be precise; witness sentence (1). In such a case what we can say is that any vagueness in the subject term is passive. This is a restricted theory of vagueness in the sense that vagueness is only de
ned for predicates, names and sentences. Any phrase in one of these categories can be said to be vague and when quizzed on what this means, we can explain the vagueness via the notion of a borderline case. Ultimately, of course, worries attend this explanation (higher-order vagueness seems a problem for borderline-case accounts of vagueness in general) but we are pointing in the right direction. Intuitively, the vagueness of phrases in these three grammatical categories consists in apparent essential indeterminacies regarding what they denote or represent – vagueness of extension. Does “hot star” denote a property including the sun as an exemplar? Does “Mount Rainier” denote something that includes that lower point on the marginal ridge? Does “The sun has a diameter of exactly 1.39 × 109 metres” denote that state of affairs which obtains in our world? What fact does it represent? Such rhetorical questions, however, cannot be posed for phrases in categories which do not purport to represent anything. It makes no sense to speak of borderline cases for logical notions like “all”, “some”, “many”, “few”, “and”, “not”, or for modi
ers like “roughly”, “rapidly”, “heavy” and “large”. In generalizing our theory of vagueness to encompass these categories as well, we shall take the vagueness of the three denoting categories as basic and outline a more general recursive account of vagueness, going beyond phrases which purport to represent, beyond sentence-, name- and predicate-vagueness. The idea is that a non-denoting phrase will count as vague if and only if it causes the vagueness of a phrase of which it is a part; that is, just if it occurs properly within a vague phrase all of whose other components are precise. We must locate the source of the phrase’s vagueness and, if all other components are precise, there is only one candidate.19 Thus we shall simply expand upon the notion of vagueness already arrived at whereby the vagueness of a sentence, if not attributable to the subject term, must be blamed on the remaining predicate. The underlying principle being followed is that precision is inherited. It is already encapsulated in our discussion of the denoting phrases; if a precise predicate is predicated of a precise subject term, then the resulting sentence will be precise. Precision is inherited to this extent. The idea driving the generalization amounts to a generalization of this principle to all phrases, denoting and non-denoting. The general principle is that: if all but one of the constituent sub-phrases of a complex phrase are precise, then, if the complex phrase 19

What follows draws on Rolf (1980: §3–4) and Rolf (1981: ch. 1).

24

Vagueness, Logic and Ontology

is vague, so is that one remaining constituent sub-phrase. In other words, if all but one of the constituent sub-phrases of a complex phrase are precise, then, if the one remaining constituent sub-phrase is precise, so must the complex phrase be precise. This principle then can be used to recursively de
ne the vagueness of a phrase, of an arbitrary non-denoting grammatical category: Def. (1) A non-denoting phrase P is extensionally vague if and only if there is an extensionally vague complex phrase containing P as a sub-phrase all of whose other constituents are extensionally precise. As usual: Def. (2) A phrase P is (intensionally) vague iff it is logically possible that P is extensionally vague. (As always, when we speak of something’s being vague simpliciter we shall interpret this as the weaker claim that it is intensionally vague, unless speci
ed otherwise. The relations that hold between intensional vagueness, extensional vagueness, extensional precision and necessary precision described in §1.1 can now be said to hold for any phrase P in general.) Of course, the concept of vagueness thus generalized is derivative upon the vagueness of denoting phrases in the sense that when we say of a phrase that it is vague, ultimately this, is to understood in terms of its making for or causing the vagueness of some denoting phrase (sentence, name, or predicate). Hence vagueness, when analysed, is ultimately to be analysed in terms of this vagueness of extension and any explanation as to the source of vagueness rests with an explanation of this vagueness (of denoting phrases). Since sentence-, name- and predicate-vagueness underpin the recursive generalization of vagueness to all grammatical categories, it is their vagueness which does the explanatory work and begs for further analysis. As an example using the above de
nition, consider the vague sentence “Many photons passed through the slide S between t1, and t2”. Assume that the predicate “passed through the slide S between t1 and t2” is precise. Then the quanti
er phrase “many photons” is vague. Suppose further that the predicate “is a photon” is precise. It follows then that the quanti
er “many” is vague. In this way modi
ers can also be shown to be vague. The predicate “is a large rectangle” is vague. Yet “rectangle” seems precise, at least if applied to ideal bodies. So it must be the modi
er “large” which is vague. We can also show that many of the logical constants are precise. Even if the constants include vague sentences in their argument range, it is generally agreed that the classical propositional connectives retain their classical functional roles when only precise sentences are considered as arguments; i.e., given only the possibility of determinate truth and falsity as inputs, the outputs of the truth-functions for conjunction, disjunction, implication and negation are always either determinate truth or falsity. So, every truth-functional compound of precise sentences is precise, and if a truthfunctional compound of sentences is vague then one of the constituent sentences is vague. Vagueness comes in, ultimately, via the atomic sentences, not via any of

Vagueness

25

these constants. There are no surprises here. This view of the constants simply reects the view that vagueness is only ever a problem in the propositional calculus if the propositional parameters are permitted to range over vague (as well as precise) sentences – that is, if the formal language is taken to include vague sentences. The universal and existential quanti
ers can also be shown to be precise. The precision of these constants contrasts with the vagueness of constants peculiar to epistemic logics like “It is known that”, which is vague.20 Consider a perfectly precise true sentence A about which I have only vague beliefs. It might be vague whether I know that A. The counterfactual conditional “if A were the case B would be the case” – i.e. “A ■→ B” – is also vague. Note
rstly that the vagueness of counterfactual sentences is explicitly acknowledged in the preferred Lewis account.21 The fact that counterfactuals may, on this theory, sometimes appear essentially indeterminate is taken to reect the fact that they would ordinarily be thought of as vague anyway. Where, according to our theory of the invasion of vagueness, is vagueness in the sentence “A ■→ B” to be located? It can be ‘blamed upon’ the logical connective. The connective can be shown to be vague since it may be vague whether B would be the case were A the case even though A and B are themselves precise. Lewis’s analysis also treats the constant itself as vague, analysing it in terms of the vague notion of ‘similarity’. To this extent this analysis is in exactly the right direction – employing vague terms to analyse a vague logical constant. As Rolf (1981) notes, since precision is inherited it follows that one cannot de
ne a vague phrase or concept in purely precise terms. This claim, following from our theory of the invasion of vagueness, is important in discussing higher-order vagueness and also underpins the view that we cannot hope to de
ne vague natural language terms by means of a purely precise ideal language. The latter point will be taken up again in Chapter 3; we shall return to the former point shortly. Before following up the consequences of the inheritance of precision, however, let us consider that other aspect of the relation between the vagueness or precision of parts and the vagueness or precision of wholes – how the vagueness of the parts affects the vagueness of the whole. According to the above theory of the invasion of vagueness, precision is inherited. The Inheritance of Precision: If all the constituent phrases of a complex phrase are precise, then the complex phrase is precise. What of the converse claim, though? If a complex phrase is precise, then are all of its constituent phrases precise? In short, the answer is no. This negative response conicts with what Rolf has dubbed ‘the infection theory of vagueness’. A typical expression of the infectious or active nature of vagueness is that of Rosenberg (1975: 309): ‘If the meaning of one term is connected to that of another, then the vagueness of either will infect the other.’ Russell (1923) relied heavily on this idea of infectious 20 This vagueness once prompted Dummett (1973: 285–8) to reject epistemic logics. He thought that the vagueness of these notions rendered them un
t for logical treatment. 21 See Lewis (1973: §1.2).

Vagueness, Logic and Ontology

26

vagueness, claiming that all language is vague. His argument to this effect can be seen to depend on two crucial premises:22 (I)

The Weak Infection Theory of Vagueness: If all the constituent phrases of a complex phrase P are vague, then so is P;

and (II) The Strong Infection Theory of Vagueness: If P is a phrase taking Q as an argument, then, if Q is vague so is P. (II) can be seen to be false by means of the fact that the classical conjunction connective “&” is precise though it takes vague sentences as arguments. (It is easy to see that (II) is false even if the vagueness of Q is extensional.) We also saw earlier that there could be precise predicates taking vague names as arguments. So (II) is false. As a consequence, contra Russell (1923: 88), it does not follow from the fact that the truth predicate takes vague sentences as arguments that the truth predicate is itself vague. (I) can also be counterexempli
ed. Consider the sentence “A red hill is red”. Both the subject term and the predicate are vague; nonetheless the sentence is true and necessarily so. Thus the sentence is precise though all its constituent phrases are vague. The vagueness of each of the constituent phrases, when combined to form a complex phrase, logically cancel each other out, as it were. This cancelling effect has an analogue in cases where extensionally vague phrases combine to form an extensionally precise complex phrase. For instance, the sentence “Caesar attained much power” is actually true and thus is extensionally precise in spite of the fact that both “Caesar” and “attained much power” are extensionally vague. This example points to the falsity of an extensional variant of (I), namely: if all the constituent phrases of a complex phrase P are extensionally vague, then so is P. Appreciating the falsity of this variant involves an appreciation of the fact that we can truthfully describe aspects of our world in purely vague terms; embracing vagueness does not mean that we have to wallow in it to our detriment. In so far as the foregoing theory of the spread of vagueness and precision is right then vagueness is not unquali
edly infectious. One consequence is that, though Russell’s account of the nature of vagueness is satisfactory from at least one point of view (as we shall see in Chapter 2), his account of the invasion and spread of vagueness is not. Though precision is inherited and vague phrases cannot be de
ned in purely precise terms, precise phrases can be de
ned in purely vague terms. So we now have to hand a characterization of the vagueness of language, applicable to any grammatical category, and an account of the interaction of vague and precise language in terms of which passive vagueness can be distinguished from active vagueness. 22

See Rolf (1981: 5–12).

Vagueness

1.5

27

Higher-order Vagueness

Let us now, as promised, return to reconsider the phenomenon of higher-order vagueness. As already noted, the phenomenon presents real dif
culties for any attempt to provide a characteristic-sentence approach to de
ning vagueness. A logical de
nition, providing necessary and suf
cient conditions expressible in a
nite sentence, seems hard to come by. Further complications that higher orders of vagueness seem to pose for attempts to model the phenomenon of vagueness have led some, e.g. Wright (1987, 1992) and Tye (1990, 1994), to question whether or not there really is vagueness at higher orders. When all is said and done, however, I think we are forced to acknowledge its presence in at least some cases and, having done so, we then have little reason to avoid the conclusion that things are as they appear to be, namely that high-order vagueness is an essential aspect of what it is to be vague. As a
rst step towards establishing this we need to recognize that “vague” is itself vague – it is a homological or autological term. It may sometimes appear indeterminate whether to count something as a borderline case for a predicate or not; thus it might be vague whether the predicate has any borderline cases or not. In such circumstances the predicate would constitute a borderline case for vagueness. An argument to this effect is provided by Sorensen (1985). Consider the sequence of predicates “1-small”, “2-small”, “3-small”, and so on, de
ned on the natural numbers. The nth predicate in the list is de
ned in such a way as to apply to only those integers that are either small or less than n. Using these disjunctive predicates we are able to construct a sorites paradox for the predicate “vague” itself. Namely: “1-small is vague. If “n-small” is vague, then “n + 1-small” is vague. ∴

“106-small” is vague.

The predicate “1-small” is as vague as “small” since both predicates clearly apply to 0 and both apply in exactly the same way to all other integers. The same holds for “2-small” and “3-small”. Each of these two predicates apply to the integers in exactly the same way as “small” does; “2-small” has 0 and 1 as clear instances whilst “3-small” has 0, 1 and 2 as clear instances, and since 0, 1 and 2 are all clearly small, it follows that “2-small” and “3-small” are as vague as “small” itself. However, we eventually reach predicates where the “less than n” clause has the effect of making some integers clear instances of the predicate “n-small” whereas they were borderline cases for the predicate “small”. Some borderline cases are eliminated. Still further down the series of predicates – at “q-small”, say – we
nd that all borderline cases for “small” have been eliminated from the borderline region of “qsmall” and thus the predicate “q-small” has no borderline cases at all and is therefore precise. For example, it is clear cut whether or not to apply the predicate “106-small” to any integer; if the integer is less than 106, then the predicate clearly applies and if the integer is 106 or greater, then, since it is clearly neither small nor less than 106, the predicate clearly does not apply. Yet, as Sorensen points out, it is a vague matter

28

Vagueness, Logic and Ontology

where along the sequence those predicates with borderline cases end and those without borderline cases begin. Consequently “vague” is vague. A number of points of interest follow from the homological nature of vagueness, two of which are worth noting at this point. The
rst, noted by Sorensen (1985), is that this property of vagueness puts paid to option (iv) as a viable response to the sorites. Any suggestion to the effect that vague predicates are incoherent enmeshes its advocate in a thesis which, by its own lights, is incoherent. Vague predicates are incoherent and one such predicate is “vague” itself. Assuming its advocates put it forward as a coherent response to vagueness, such a view then is self-refuting and must therefore be rejected. So too with the nihilistic approach to vagueness advocated by Unger and Wheeler, discussed earlier, according to which atomic vague predicates (e.g. “is red” or “is a table”, as opposed to their negations) have empty extension and apply to nothing. If true, then such an approach is committed to there being no vague predicates since “vague” is itself vague and would therefore lack extension. Any claim to have some vague predicate about which we can speculate is on a par with claims to have some object, a chair, say, about which we can speculate. We
nd in the latter case, that there can be no such thing, and so too therefore in the former case, there is no such thing to speculate about. The cost of such a theory then is the denial of there being anything about which to theorize. The second point speaks directly to the current concern regarding the existence of higher-order vagueness. Since “vague” is vague, it cannot be de
ned in purely precise terms. This follows from the more general fact that vagueness is sui generis – since precision is inherited, no vague phrase can be de
ned in purely precise terms. Somewhere in our characterization of the vague concept of vagueness we are forced to employ vague terms. Hence, implicitly, when characterizing predicate-vagueness in terms of borderline cases we must suppose the notion of a borderline case (and associated notions like penumbra and borderline region) to be vague itself, because there is no other candidate for vagueness in the analysans. Predicate-vagueness is characterized, at least in part, by ‘there being borderline cases’, with any additional conditions imposed to distinguish vagueness from mere incompleteness only importing further talk of borderline cases (e.g. borderline borderline cases), yet the existential quanti
er is precise, so the remaining notion of a borderline case must be vague.23 Thus there must be at least some borderline borderline cases. This undermines any attempt to avoid the supposed dif
culties associated with higherorder vagueness by either denying there to be any higher orders of vagueness (as Wright has tried to suggest), or by claiming it to be vague whether there are higher orders of vagueness (as Tye has suggested). The phenomenon of higher-order vagueness is de
nitely real and any complications that it embroils us in as a consequence are simply unavoidable. This conclusion was initially proposed in Hyde (1994); Tye (1994) subsequently tried to resist it, proposing instead that it is vague whether there are any higher orders of vagueness. Agreeing that the vagueness of “vague” entails that there is higherorder vagueness somewhere, he seeks to avoid what he earlier described as ‘the 23 The imposition of additional conditions might invoke conjunction, but it too is precise, like the existential quanti
er.

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29

gratuitous complications’ (1990: 535) that ensue by committing to neither the existence of higher-order vagueness nor its non-existence. There is, he suggests, simply no fact of the matter whether there are higher orders of vagueness – “vague” is not vague; it is merely vaguely vague. Such a view, however, proves ultimately to be untenable. He begins by noting that if “vague” is vague, then it follows that there are predicates which are vaguely vague (1994: 44). Supposing Sorensen’s argument for the vagueness of “vague” were sound, then it would follow that there are vaguely vague predicates. This in turn then presents us with an alternative explanation of the soritical nature of “vague” evidenced by Sorensen’s argument – it is vaguely vague and thus not vague. So, he claims, Sorensen’s argument is not sound after all. Thus we may claim unconditionally that Sorensen’s argument is unsound. The alternative explanation, though, is not coherent. We can show that “vague” cannot, on pain of inconsistency, be vaguely vague. To see this, suppose that the predicate “vague” is vaguely vague. Thus it is indeterminate whether “vague” is vague. Yet this is just equivalent to claiming that “vague” is a borderline case for the predicate “vague”. Thus “vague” has a borderline case – namely itself. So “vague” is vague. Alternatively, we can argue as follows. Supposing that the predicate “vague” is vaguely vague, we can infer that it is therefore vague whether “vague” is vague and subsequently enquire after the source of this vagueness. If “vague” were not vague, then neither the subject term nor predicate of the sentence “‘vague’ is vague” would be vague; each would be precise, and so, given the Inheritance of Precision, it would follow that the sentence was not vague, contrary to our assumption. Consequently it must be the case that “vague” is vague. In summary, then, if “vague” is vaguely vague, then “vague” is vague. This is just a particular instance of the more general fact that if any predicate is vaguely vague, then “vague” is vague, which can be established by generalizing on the foregoing argument, and which is the converse of the principle which Tye employs in his reasoning outlined above. Now the inconsistency in Tye’s position is apparent since if “vague” is vague, then “vague” is not vaguely vague. Not only is Sorensen’s account of “vague” not refuted by Tye’s argument, of the two apparently rival accounts only Sorensen’s seems immune to charges of inconsistency. “Vague” is a homological term: it is vague. Not only is “vague” vague, however. In light of the foregoing reasoning we can also see that some predicate is vaguely vague if and only if “vague” is vague, and therefore recognize the existence of predicates that are vaguely vague. Such predicates present us with examples of borderline cases of borderline cases, as Tye acknowledges. Higher-order vagueness is a real phenomenon. We can neither claim that it determinately does not exist, nor that it is vague whether it exists. It determinately exists.24 There are then borderline cases of borderline cases. What does this show in respect of higher-order vagueness as a general signature of vagueness? Must being 24

For further criticism and subsequent defence of this argument for higher-order vagueness see Deas (1989), Varzi (2003) and Hyde (2003).

30

Vagueness, Logic and Ontology

a borderline case for a predicate itself always admit of borderline cases in order that the predicate in question count as vague? Is higher-order vagueness and the absence of sharp boundaries to a predicate’s borderline cases an essential aspect of what it is for a predicate to be vague, as opposed to merely incomplete, say? There is some room for debate here, but ultimately I think that we should endorse that common conception of vagueness according to which vague terms lack sharp boundaries of any kind and therefore possess borderline cases and borderlines cases of borderline cases and so on ad in¿nitum. Tye objects that one can only conclude from the foregoing discussion that some vague predicates are higher-order vague. ‘It could be, for example, that the only vague predicates admitting border border cases are very peculiar and atypical. So, … it could be that no everyday vague predicates are also higher-order vague’ (1994: 44). And indeed it could be, but such a possibility seems remote indeed. Most theorists from Russell onwards endorse a conception of vagueness which recognizes higher-order vagueness as part of what it is to be vague, and Tye offers no reason why this common conception ought to be revised other than to suggest that it involves us in ‘gratuitous complications’ (1990: 535). However, having been compelled to admit the existence of some higher-order vagueness, any complications can hardly be considered gratuitous. And if we are compelled to deal with the complications that follow from the acceptance of some higher-order vagueness, then it seems that the oodgates are open and we are left with no compelling reason to reject the orthodox view according to which every vague predicate is higher-order vague.25 Once we have characterized the concept of vagueness, the next question to emerge centres on the cause or source of such vagueness in natural language. Remembering that the notions basic to our generalized concept of vagueness were the vagueness of sentences, names and predicates, this question then amounts to our enquiring after the cause of our inability to describe sharp boundaries to the extension of sentences, names and predicates. Where does the associated apparent indeterminacy have its source? Is it perhaps pragmatic, reecting a fact about our use of language? Perhaps, as Lewis (1975: 34–5) suggests and Burns (1991) elaborates on more fully, we have simply failed to choose between one of a number of precise languages. Vagueness arises as a result of the way we as language-users equivocate over which in a cluster of precise languages is the language in use in the community. As Keefe puts it, ‘[a]ny talk about our vague, natural language should then be reducible to sentences about our (vague) use of precise languages’ (2000: 142). Such a view is claimed to have the virtue of casting vagueness as harmless so far as logic, semantics and metaphysics are concerned. Each precise language is governed by classical logic and semantics, and vagueness as a pragmatic feature is of no metaphysical import. Or is it perhaps epistemic, due to our inability to know, for example, the precise boundaries of a predicate’s application? Some modern theorists like Williamson 25 In requiring that the indeterminacy characteristic of vagueness is itself vague, one might be tempted to think that a worrisome circularity is lurking in the background, but the worry is misplaced. Were we to characterize “vague” using the term “vague”, then a vicious circularity would indeed arise, but we are not. Rather, we are characterizing “vague” using vague terms and this is no more a problem than characterizing “meaningful” in meaningful terms.

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31

(1994) and Sorensen (1988, 2001) argue that it is. It has also been suggested that this was the response adopted by the Stoics.26 On this view the apparent indeterminacy regarding the application of, say, “red” to some object is merely apparent. Our lack of clarity concerning the boundaries to the application of a predicate is merely a manifestation of our own ineliminable ignorance. In every case where the predicate may be said to be meaningfully applicable there is a determinate fact of the matter as to whether the predicate applies or not. It is simply that in certain situations we are in principle barred from coming to know which is the case – namely, where one is close to the sharp semantic boundary – and so it appears that there is no determinate fact of the matter. For example, it appears that there is no sharp boundary to the extension of the predicate “red”, yet there is one, though we cannot know where it lies. This analysis of vagueness clears the way for a solution to the sorites paradox which centres on the rejection of one of the conditional premises in the standard sorites – option (iii). According to this conservative solution the sorites is valid though unsound. Other theorists eschew the extreme semantic realism inherent in such an admittedly counterintuitive account, maintaining rather that our inability to know where the boundary lies is a reection of there actually being no determinate fact of the matter. It is true that there is no knowable sharp boundary but this is because there is, as a matter of fact, no sharp boundary. Vagueness is properly a semantic phenomenon. Theorists taking this line, by far the majority, can be further distinguished from one another. Most, it would seem, take vagueness to be merely semantic. That is, though they accept that there is no determinate fact of the matter as to whether “red” applies to some borderline case, there is no sense in which this can be said to be grounded in the world. The vagueness is not, in any sense, ontological; it is purely semantic. Various responses to the sorites paradox attend this analysis of vagueness. Some theorists, for example Russell, in accepting that vagueness is really semantic, go on to conclude that vague language is beyond the scope of logic, thus solving the paradox by denying the problem legitimacy in the
rst place – option (i). Others, while accepting that logic applies to vague language, deny that the paradoxical argument is sound – option (ii) or (iii). Like the epistemic theorist they might opt for a conservative solution though admitting that vagueness necessitates some revision of classical logic and semantics – a conservative extension of classical logic is proffered. Alternatively, they might prefer a radical solution and argue that the semantic indeterminacy that results from vagueness undermines classically valid inferences invoked in sorites reasoning. The purely semantic approach contrasts with a theory of vagueness which admits of ontological indeterminacy, analogous to semantic indeterminacy which it is sometimes taken to underpin. The vagueness of the predicate “red” (for instance) might, on this ontological account, reect the vagueness of the property denoted by “red”, namely redness. More controversially, the vagueness of terms like “Mount Rainier” might, on this account, reect the vagueness of the object referred to by the name, namely Mount Rainier itself. Theorists taking this approach might conceivably opt for either a conservative solution to the sorites or a radical solution, 26

See Mignucci (1993) and Williamson (1994).

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32

denying the validity of sorites arguments. In arguing for this approach, I shall pursue a conservative solution according to which the standard sorites is unsound by virtue of some conditional premises not being true. Of course, mixed accounts of vagueness are also conceivable. Some borderline cases may be epistemic, some semantic and perhaps in some cases ontologically grounded. Or we may get all three kinds of ‘vagueness’ at once: epistemic, semantic and ontological indeterminacies might all contribute simultaneously to make for the vagueness of a word or phrase. Care is required, though. As one might suspect, arguments for epistemic or semantic analyses of vagueness may depend on the perceived virtues of their conservatism, requiring minimal revision to one’s web of belief.27 In such cases, to forego such conservatism anywhere would seem to open the oodgates to more radical analyses, encouraging their adoption where previously it was denied. Typically, theorists offer one account of vagueness. Since each approach seems to carry with it some degree of incredulity, our collective intuitions concerning vagueness seem unsatis
able simultaneously, and to advocate mixed analyses would seem to incur a burden over and above that incurred by each ingredient theory individually. Strong reason would then need to be offered to justify this additional cost. I know of no such reason and therefore of no bene
t that would offset this additional cost. In line with this approach, advocates of the epistemic account commonly offer an epistemological analysis of vagueness as universal and subsequently claim it as a virtue of their theory that the phenomenon of vagueness requires no revision of classical semantics, logic or metaphysics. Advocates of the purely semantic account, while being able to help themselves to the epistemic account at no extra cost to semantics, logic or metaphysics, also typically universalize to avoid the perceived epistemological excesses commonly attributed to the epistemic account. In thus denying the ontological account any room in the theoretical landscape they are also relieved of the need to revise their metaphysics. The rest of this book will be taken up with the description and evaluation of those accounts of vagueness which view it as (at least) a semantic phenomenon, whether or not its source or cause is taken to lie solely in the representational system that is natural language or, as ontological theorist, would have it, in that which such systems seek to represent. My claim will be that vagueness is at least sometimes ontologically grounded and that its presence in natural language necessitates a revision of logic, semantics and metaphysics. This being said, it is to vagueness as a semantic phenomenon that we must now turn.

27

See especially Williamson (1994).

Chapter 2

Russell’s Representational Theory It is a commonplace view that instances of apparent semantic indeterminacy are, contra epistemic and pragmatic theorists, real. Of course, whether this semantic indeterminacy is analysable as a purely semantic phenomenon, or is perhaps seen as a reÀection of some underlying ontological indeterminacy, is an independent issue, constituting a further, major watershed in approaches to vagueness that will be taken up later. Regardless, vagueness is admitted as a semantic phenomenon; vagueness is an essential semantic feature of speci¿c terms in natural language. Like so many philosophical positions, the semantic view of vagueness has dominated twentieth-century philosophy (so much so that alternative views were, until recently, often precluded by de¿nition). Bertrand Russell (1923: 85–6) speaks for many when, by way of an initial explanation of vagueness, he asks us to consider the various ways in which common words are vague, and let us begin with such a word as “red.” It is perfectly obvious, since colours form a continuum, that there are shades of colour concerning which we shall be in doubt whether to call them red or not, not because we are ignorant of the meaning of the word “red,” but because it is a word the extent of whose application is essentially doubtful. This, of course, is the answer to the old puzzle about the old man who went bald. It is supposed that at ¿rst he was not bald, that he lost his hairs one by one, and that in the end he was bald; therefore, it is argued, there must have been one hair the loss of which converted him into a bald man. This, of course, is absurd. Baldness is a vague conception; some men are certainly bald, some are certainly not bald, while between them there are men of whom it is not true to say they must either be bald or not bald.

This conception not only typi¿es the view of vagueness as semantic but exempli¿es what Bertil Rolf has described as the traditional concept of vagueness: as well as describing vagueneness as a semantic phenomenon of natural language, this concept endorses the paradigmatic conception of vagueness as applied to predicates and characterized by the presence of borderline cases – identi¿ed in Chapter 1.1 Russell’s conception also typifies the common view according to which vagueness, though a semantic phenomenon, in no way reÀects any underlying ontological indeterminacy. There is a certain tendency in those who have realized that words are vague to infer that things also are vague. … This seems to me precisely a case of the fallacy of verbalism – the fallacy that consists in mistaking the properties of words for the properties of things. 1

The naivety of this as a characterization has already been noted in Chapter 1, §4. For the moment we can ignore this complication though it will become important later.

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34

Vagueness and precision alike are characteristics which can only belong to a representation, of which language is an example. They have to do with the relation between a representation and that which it represents. Apart from representation, whether cognitive or mechanical, there can be no such thing as vagueness or precision; things are what they are, and there is an end of it. Nothing is more or less what it is, or to a certain extent possessed of the properties which it possesses. Russell2 (1923: 84–5)

To be sure, Russell might be understood as agnostic in relation to metaphysical matters, claiming merely that it is meaningless to ascribe vagueness to the world rather than false, and leaving open the possibility that the indeterminacy of representations is in some sense a reÀection of indeterminacy in the represented, though not meaningfully describable as vagueness therein. On this view, though that being represented can be talked about, “vague” and its opposite “precise” are relational attributes that cannot be applied to it as opposed to the representations themselves. It is, therefore, akin to a category mistake to ascribe vagueness or precision to that which language represents. They are features that can only be applied in the realm of representations. However, we might still meaningfully ask whether the indeterminacy manifested in representations is evidence of or has its source in indeterminacy in that which is represented. What we shall see is that, in the course of charitably interpreting Russell’s de¿nition of vagueness, he must be understood as claiming not. The world is ‘sharp’, as we shall say, and exhibits no indeterminacy or ‘fuzziness’ in any sense relevant to an explanation of the vagueness of representations. Vagueness is merely semantic and is to be seen as a feature of linguistic representations, and representations more broadly, that is not attributable in any way to that which they are taken to represent. It is what I shall term a representational theory – the phenomenon manifests itself in representations, linguistic and otherwise, and occurs speci¿cally as a result of indeterminacies that arise in the relation of representation itself. The reasons for the widespread popularity of such a view are, I think, not hard to ¿nd. It recognizes that feature of natural language that initiated the problem in the ¿rst place – vagueness – as semantic, while denying that this has any import at all for metaphysics and promises to be of no more than super¿cial import for logical theory. Of course, some account is required of the relation between vague natural language and the world, but this will be constrained by the fact that the world is sharp. In this chapter I want to consider this representational theory of Russell’s. It will be instructive in part because it presents us with the most conservative approach to 2

One might well wonder whom Russell had in mind when describing this fallacy. Over ¿fty years later Margalit similarly protested that ‘things are what they are. … The ascription of vagueness to objects may yield the quantities-turn-qualities kind of “logic” (“dialectical” or otherwise), which commits the fallacy of verbalism.’ Margalit’s criticism (see Margalit 1976) seems directed at the dialectical approach to vagueness popularly endorsed by orthodox Marxists. As we shall see in Chapter 4, the turn-of-the-century Marxist scholar Plekhanov (1937 [1908]) used just such an analysis of sorites situations and had no qualms in admitting the dialectical nature of the world. Perhaps this was Russell’s target too.

Russell’s Representational Theory

35

semantic vagueness. For Russell, as with some later representational theorists like Quine, the metaphysical conservatism inherent in the representational approach offers a basis for logical conservatism as well, that is for the retention of classical semantics and logic without extension or revision. The ‘deep structure’ or ‘canonical regimentation’ of natural language, Ideal Language, is said to mirror the world whose structure remains completely classical. Logic is parasitic on this Ideal and thus the logic of Principia Mathematica remains intact (at least so far as vagueness is concerned). The ‘super¿cial’ structure of natural language should not misguide us as to the true logical structure of language, just as the ‘super¿cial’ structure of the world (Appearance) should not misguide us as to its true structure (Reality). The classical logical theory can no longer be viewed as a normative theory governing natural-language argumentation since natural language has been shown to exhibit features that that theory forbids. Its adequacy is defended by means of language of a suitable ideal type. In the subsequent chapter we shall look at the Quinean variant on this theme, before turning to consider accounts that propose logics of vague natural language. For now though, let us concentrate on Russell. The Russellian position on vagueness is described most notably in his 1923 paper “Vagueness”, with further remarks to be found scattered through his other writings. These other remarks, found for example in The Analysis of Mind and An Inquiry into Meaning and Truth, do not represent a departure from the 1923 treatise. His view – vagueness is a semantic feature of natural language having its source in the representational nature of language and not in that which is represented, the world, which is sharp – remained unchanged. For this reason we can concentrate on the 1923 work, deferring to other remarks if and when required. 2.1

Incompleteness, Lack of Speci¿city and the Source of Vagueness

Before presenting his de¿nition of vagueness, it will help to look at some examples Russell presents as being examples of vagueness to see just where he takes the source of vagueness to be. We have already noted above that he sees vagueness as applicable to representations in general, be they ‘cognitive or mechanical’, so linguistic and perceptual representations are both items which can exhibit the phenomenon. Moreover his account of the source is primarily concerned with examples from cases of perception. What is clear is that the knowledge that we can obtain through our sensations is not as ¿negrained as the stimuli to those sensations. We cannot see with the naked eye the difference between two glasses of water of which one is wholesome while the other is full of typhoid bacilli. In this case a microscope enables us to see the difference, but in the absence of a microscope the difference is only inferred from the differing effects of things which are sensibly indistinguishable. It is this fact that things which our senses do not distinguish produce different effects – as, for example, one glass of water gives you typhoid while the other does not – that has led us to regard the knowledge derived from the senses as vague. And the vagueness of the knowledge derived from the senses infects all words in the de¿nition of which there is a sensible element. (1923: 87)

36

Vagueness, Logic and Ontology

This feature of perception whereby the perceived item (that item having all and only those properties perceived of the thing in question) lacks properties possessed by the thing perceived is more commonly described as the incompleteness or selectivity of perception. Assuming that a representation R is less speci¿c than a representation S just in case S speci¿es more about the world than R does (i.e., the information content of S properly includes that of R), then the more selective or incomplete a perception, the less speci¿c the perceived item or representation is. For example, a photograph of a man from a great distance will be less speci¿c (ceteris paribus) than one taken at short range since there will be features discernible in the latter not discernible in the former. Given Russell’s analysis, lack of speci¿city and vagueness share a common source. Vagueness in our knowledge is, I believe, merely a particular case of a general law of physics, namely, the law that what may be called the appearances of a thing at different places are less and less differentiated as we get further away from the thing. When I speak of ‘appearances’ I am speaking of something purely physical – the sort of thing, in fact, that, if it is visual, can be photographed. From a close-up photograph it is possible to infer a photograph of the same object at a distance, while the contrary inference is much more precarious. That is to say, there is a one–many relation between distant and close-up appearances … I think all vagueness in language and thought is essentially analogous to this vagueness which may exist in a photograph. (1923: 91)

And at times he goes so far as to completely conÀate vagueness with lack of speci¿city. … it is obvious that what you see of a man who is 200 yards away is vague [less speci¿c] compared to what you see of a man who is 2 feet away; that is to say, many men who look quite different when seen close at hand look indistinguishable at a distance, while men who look different at a distance never look indistinguishable when seen close at hand … there is less vagueness [more speci¿city] in the near appearance than in the distant one… (1923: 91)

Russell is quite clearly wrong to conÀate the two separable notions and, as we shall see, it is not something he was consistent about. Nonetheless, his (intermittent) confusion in this regard can be recti¿ed without damage to his overall account of representational vagueness. As will become apparent, despite the fact that incompleteness and representational vagueness are in some respects analogous, they can be separated out and Russell’s sometimes confusing account shows us the way. As a matter of historical note, such confusion can be found in the writings of numerous other early contributors on the topic. Cases in point are Duhem (1954: 178–9), Austin (1962: 125–31), Alston (1964: 84–5) and Rosenberg (1975: passim). This confusion fuels the misconception that, since ‘a theory couched in inexact terms … has a much higher probability of being true than any one of the exact theories with which it is incompatible, … unfalsi¿ability … follows in the train of vagueness …’ (Rosenberg 1975: 299–300). Vagueness is unjustly portrayed as an enemy of Popperian scienti¿c theorizing, echoing Russell’s (1923: 91) claim that

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37

a vague belief has a much better chance of being true than a precise one … If I believe soand-so is tall, I am much more likely to be right than if I believe that his height is between 6 ft 2 in and 6 ft 3 in … Science is perpetually trying to substitute more precise beliefs for vague ones; this makes it harder for scienti¿c propositions to be true than for the vague beliefs of uneducated persons to be true, but makes scienti¿c truth better worth having if it can be obtained.

The scienti¿c ‘second-gradeness’ of vague beliefs or propositions – which has, in the past, rendered them a target for suspicion, a defect for elimination – is based upon this confusion. Though it is true that I am much more likely to be right if I believe so-and-so is tall (a vague belief) than if I believe that his height is between 6 ft 2 in and 6 ft 3 in (a more precise belief), it is not true in general that vaguer beliefs or propositions have a better chance of being true than more precise ones, as evidenced by my being much more likely to be right if I believe so-and-so to be between 1 ft and 8 ft in height (a relatively precise belief) than if I believe that he is tall (a vaguer belief). It is not that more precise propositions are less likely to be true than vague ones, but that more speci¿c propositions are less likely to be true than less speci¿c ones. Given propositions A and B, A is more precise than B if A’s borderline cases are also B’s but not conversely and whenever B is clearly (not) applicable to a situation A is (not). Precisi¿cations thus preserve determinate truth and falsity of application of propositions to situations and reduce the number of borderline cases. As a consequence A need not entail B since A might determinately apply to a situation where B is only doubtfully applicable (as the above counterexample shows) and so may, in fact, be more likely to be true than the vaguer B. What we can say in general is that a more speci¿c belief or proposition has a much lower chance of being true than a less speci¿c one. If A is more speci¿c than B, then A entails B, and this explains why more speci¿c propositions are less likely to be true than less speci¿c ones. Having noted this common confusion in early discussions of vagueness, also apparent in some of Russell’s thinking on the matter and against which we have been so pointedly warned by Sorensen (1989), let us move on to Russell’s de¿nition and see how it might serve to capture the notion of representational vagueness. 2.2

Russell’s De¿nition

Russell (1923: 89–90) presents the following ‘de¿nition’ of vagueness. [A] representation is vague when the relation of the representing system to the represented system is not one–one, but one–many. For example, a photograph which is so smudged that it might equally represent Brown or Jones or Robinson is vague. A small-scale map is usually vaguer than a large-scale map, because it does not show all the turns and twists of the roads, rivers etc., so that various slightly different courses are compatible with the representation that it gives … Passing from representation in general to the kinds of representation that are specially interesting to the logician, the representing system will consist of words, perceptions, thoughts, or something of the kind, and the would-be one–one relation between the representing system and the represented system will be

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38

meaning. In an accurate [precise] language, meaning [denotation] would be a one–one relation; no word would have two meanings … In actual languages, as we have seen, meaning is one–many.

The most thoroughgoing treatment of this account is that presented in Rolf (1982). Therein Rolf presents two interpretations of Russell’s de¿nition: R1.

The representation r is vague if and only if there are two different entities, x and y, such that it is logically possible that r represents x and it is logically possible that r represents y.

Or R2.

The representation r is vague if and only if it is logically possible that there are two different entities, x and y, such that it is logically possible that r represents x and it is logically possible that r represents y.

The distinction between the two is simply that between intensional and extensional vagueness. In the context of the current discussion this distinction is of no import and I shall speak of ‘an interpretation’. This interpretation may be thought inadequate as regards its use of mere ‘logical possibility’. Representations that are intuitively precise are such that it is logically possible that they might represent a whole range of things; mere logical possibility is too weak. In fact (and this seems to be what Rolf has in mind anyway) the logical possibilities are constrained. As Rolf (1982: 70) explains it, the de¿niens of R1 and R2 are true just when a representation r has certain ‘essential properties’ but these do not determine whether or not r represents x or y – for two, perhaps non-actual but logically possible, entities x and y. Both alternatives are possible inasmuch as x and y only differ on some property that is not determined either way by the properties essential to the representation. That is, both alternatives are logically possible to the extent that they are consistent with the essential properties of the representation. What then is this notion of an ‘essential property’? Let us say that an essential property of a representation r is any property constitutive of r itself – qua representation – that is, a property that could, in principle, be directly given to or known by an agent to be a property of r on the basis of the representation r itself (which might be a painting or photo, knowledge, beliefs, propositions, or other linguistic items depending on whether the representation is mechanical, cognitive or linguistic). For example, a colour photo, p, showing a blue ball, has as one of its essential properties that what is represented is represented as being blue and, though it may, as it happens, in fact be a photo of a green ball this is simply an extrinsic property of p. As another example, consider a predicate. Properties essential to a predicate (qua representation) are just the semantic properties of the predicate; for example, that the predicate F denotes the property α, or would apply to objects possessing this property. In focussing on the essential properties of a representation we are interested merely in how that which is actually represented is represented as being – something

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intrinsic to the representation itself. In the above example of the photograph, the ball that is actually photographed is green yet represented as being blue. In this case the represented – the ball – is represented as being a colour which it in fact is not and to this extent the photograph is a misrepresentation; however, this kind of misrepresentation has no bearing on whether or not the representation is vague – extrinsic considerations ¿gure in determining that the representation actually misrepresents. There is another way, though, that representations can misrepresent. Misrepresentation can also occur due to the fact that a representation of x presents it as being something that it cannot be by virtue of features intrinsic to the representation itself – the representation’s essential properties. Such is the case with vagueness. Consider Russell’s smudged photo. Its essential properties do not have the distinctness which would entail either that the photo does not represent Brown or that it does not represent Robinson. It is therefore possible relative to the essential properties of the photo that it is a photo of Brown (the essential properties do not rule this out) or of Robinson (the essential properties do not rule this out either). The photo underdetermines its referent and it is this, according to Russell, which constitutes its being vague. So x and y are not just logically possible referents of the representation r. We can be more speci¿c. They are logical possibilities constrained by the fact that they must be consistent with what is already represented as being the case or logical possibilities relative to the essential properties of r. Letting Σr stand for the set of essential properties of the representation r, and abbreviating this notion of (relative) possibility as “◊Σr”, we can then de¿ne ■ΣrB =df ~◊Σr~B and ∇ΣrB =df ◊ΣrB & ~■ΣrB. Returning then to the Russellian de¿nition of vagueness, x and y are logically possible referents of r subject to the restriction that they be consistent with what is already represented as being the case. Thus we may now interpret Russell’s de¿nition more adequately as follows: R3. The representation r is (extensionally) vague if and only if there are two different entities, x and y, such that it is logically possible relative to Σr that r represents x and it is logically possible relative to Σr that r represents y. Or R4. The representation r is (intensionally) vague if and only if it is logically possible that there are two different entities, x and y, such that it is logically possible relative to Σr that r represents x and it is logically possible relative to Σr that r represents y. Less formally, we can say that a representation is vague just if there are, or could be, various different referents compatible with the representation given. On the basis of the representation itself one cannot determine its referent – various possibilities remain open, the representation lacking the distinctness which would entail any particular referent as being represented; i.e. the representation lacks the distinctness required to determinately represent anything.

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As will become clear, in the above de¿nition we have the very essence of an account of representational vagueness. Russell can be seen as the paradigmatic representationalist. Representations are (representationally) vague when the relation of the representing system to the represented – which in the case where the representing system is linguistic will be the relation of meaning or denotation – is one–many. 2.2.1

Russell’s de¿nition and borderline-case vagueness

Echoing concerns raised by Black (1949: 29) and Kohl (1969: 37ff), Rolf (1982: 71) contends that the foregoing de¿nition is unable to differentiate vagueness from generality. If r is a general word like “man”, it can represent – denote – many different men and would therefore seem to count as vague on Russell’s de¿nition. But a general word can, at least prima facie, be precise. So ‘Russell’s de¿nition of vagueness cannot by itself distinguish generality from vagueness’. As we shall see, this criticism is misguided, but in Rolf’s defence one might be excused for thinking “man” is vague according to Russell since Russell himself sometimes seemed confused as to whether general terms were vague. In Russell (1921: 182), for instance, he does say “‘I met a man” is vague since any man would verify it’, but quickly goes on to contradict himself by claiming (p. 184) that ‘[a] vague word is not to be identi¿ed with a general word, though in practice the distinction may often be blurred.’ And as we have seen, there is some confusion in Russell (1923) concerning the distinction between vagueness and lack of speci¿city. Nevertheless, preceding his de¿nition of vagueness he makes the following declaration (1923: 88): when we speak of a proposition having a certain degree of vagueness, there is not one de¿nite fact necessary and suf¿cient for its truth, but a certain region of possible facts, any one of which would make it true. And this region is itself ill-de¿ned: we cannot assign to it a de¿nite boundary. This is the difference between vagueness and generality. A proposition involving a general concept – e.g. “This is a man” – will be veri¿ed by a number of facts, such as “This” being Brown or Jones or Robinson. But if “man” were a precise idea, the set of possible facts that would verify “this is a man” would be quite de¿nite. Since, however, the conception “man” is more or less vague, it is possible to discover prehistoric specimens concerning which there is not, even in theory, a de¿nite answer to the question, “Is this a man?” As applied to a such specimens, the proposition “this is a man” is neither de¿nitely true nor de¿nitely false. [My italics]

So Russell does informally distinguish vagueness from generality. Lack of determinate or de¿nite boundaries and reference to borderline cases are invoked to do this, as the italicized claims show. The success of the de¿nition follows from its ability to formally incorporate these insights regarding the distinction, as we shall see. The error underlying the criticism of Russell’s de¿nition seems to arise as a consequence of confusing a term’s application with its denotation. A term is general (not to be confused with its being ambiguous3) when it is applicable to a number of 3

Ambiguous terms can be taken to have (at least) two semantically distinct uses individuated by two distinct sets of semantic properties with distinct (though perhaps indeterminate) extensions.

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different objects in virtue of some common property; however, it does not denote or represent each of these things to which it applies. To the extent that it is acceptable to speak of a general term denoting or representing at all, it may be said to denote the class comprising those things rather than the things themselves. So, though a general term like “man” does indeed apply to a number of different persons, it cannot be said to denote or represent each or any of them individually (though it may be taken to denote the class having those persons as members, mankind). By attending to this distinction between a term’s application and its denotation we see that Russell’s de¿nition invoking the one–manyness of representation is not thereby committed to counting general terms as vague. Establishing the adequacy of Russell’s de¿nition will also serve to answer a second criticism levelled at Russell’s de¿nition in Rolf (1982): the de¿nition makes no use of the notion of a borderline case. Russell certainly invokes the notion informally in order that a requisite distinction concerning vagueness and generality can be made. Furthermore he explicitly says in his initial consideration of just what one means when one speaks of vagueness (1923: 85) that common words such as “red” are vague since ‘there are shades of a colour concerning which we shall be in doubt whether to call them red or not, not because we are ignorant of the meaning of the word “red,” but because it is a word the extent of whose application is essentially doubtful [indeterminate]’. So Russell certainly invokes the notion of a borderline case as a criterion of vagueness and it will become evident that it also ¿gures, if only implicitly, in his de¿nition. This traditional criterion of vagueness – the presence of borderline cases – is implicit in Russell’s de¿nition and enables his de¿nition to distinguish vagueness from generality.4 We have already seen that the one–manyness of the representation relation in Russell’s de¿nition arises, not as a result of the one vague representation r actually representing many referents, but as a result of each of a plurality of distinct referents being possible relative to the essential (semantic) properties of r – Σr. What close inspection reveals is that this modal relation – “what r might represent given Σr” – is one–many just when reference varies depending on whether or not the representation’s borderline case(s) is (are) to be included in or excluded from the range of the representation function. In other words, that characterizing feature of vague representations on Russell’s account – one–manyness of denotation – has its source in exactly that which characterizes representations as vague on the traditional account – namely, borderline cases. The Russellian de¿nition differs from the more common account in terms of borderline cases only in that the former incorporates the concept of borderline cases into the speci¿c (representational) theory of vagueness; subsumed within the more speci¿c account, it no longer features explicitly.5 How then are borderline cases responsible for the one–manyness of the representation relation and vice versa? This question is best answered, I think, by 4 Rolf (1982) points to the possible equivalence between Russell’s de¿ntion and the borderline-case criterion. In what follows this equivalence is established. 5 Borderline borderline cases (i.e cases for which it is indeterminate whether or not it is indeterminate) similarly give rise to one–manyness, though now it is manifested in the one–manyness of the denotation of “is a borderline case for F ”, or, equivalently, in the plurality of candidates for the class of possible denotations for F.

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contrasting the traditional conception of vagueness, borderline-case (semantic) vagueness, with another notion employed by Russell – ‘accuracy’. In the preamble to his de¿nition of vagueness he says the following (1923: 89): One system of terms related in various ways is an accurate representation of another system of terms related in various other ways if there is a one–one relation of the terms of the one to the terms of the other, and likewise a one–one relation of the relations of the one to the relations of the other, such that, when two or more terms in the one system have a relation belonging to that system, the corresponding terms of the other system have the corresponding relation belonging to the other system. Maps, charts, photographs, catalogues, etc., all come within this de¿nition in so far as they are accurate.

That is to say, one structured system is an accurate representation of another if there is a one–one relation correlating elements of one to elements of the other by means of which the former can be interpreted as a model of the latter. When such a structure-preserving correlation obtains between any systems X and Y, X is said to be isomorphic to Y. Consider, for example, a map which consists of various marks on paper related in various ways, a legend saying what each type of mark represents, a scale, contours, and so on. If this map is accurate or isomorphic to that which it purports to represent, then each mark correlates one–one with (i.e. denotes, represents) something in the region mapped, and the relations that obtain on the map, when interpreted and scaled, correspond to relations that obtain in that region. If, for example, a dot designated “Mt Glorious” is 30 cm from another dot designated “UQ” in a direction designated “north-west” on a 1:100,000 map of Brisbane, then, using the intended or canonical one–one correlation, the map represents the elevation Mt Glorious as 30 km north-west of the University of Queensland in the city of Brisbane. Moreover, the map is then accurate in so far as this is the case. If our geographical world is not as described, then the correlation, though one–one, is not structure-preserving and we may suppose the map to be inaccurate in this regard. The map is a factually inaccurate representation of our world because it misrepresents our world; nonetheless there is some world it can be taken to accurately represent – namely a world which, were it to exist, would be as the map describes. Though factually inaccurate, the map is isomorphic to some possible world. Thus, factually inaccurate representations are contrasted with accurate ones. But there is another contrast we can draw. When we come to consider vague naturallanguage descriptions, we are, on the representationalist view, forced to concede that such descriptions could never be isomorphic to the world. There is a different type of inaccuracy inherent in vague language, logical inaccuracy – a different type of misrepresentation, logical misrepresentation. Vague representations are thus more analogous to the impossible pictures of M.C. Escher. Assuming the consistency of the world, these pictures represent things as being a way they cannot be – they logically misrepresent the world; the inaccuracy is a logical one. So too with vague representations. Were vague language ever isomorphic to the world, then the world would have to exhibit that feature analogous to vagueness, yet this, according to representationalists

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(and, I shall argue, Russell in particular), is impossible. As a matter of necessity, that represented by language is never vague or fuzzy so any possible candidate referent for a vague term will fail to share that property analogous to vagueness – fuzziness. Any candidate for reference, anything in the range of the denotation function, is sharp. So vague descriptions can never be considered accurate representations and hence cannot stand in a one–one relation to that represented. Russell’s smudged-photo example is an apt illustration of this inaccuracy and the paradigm for vagueness, in particular for vagueness in language. ‘[A]ll vagueness in language and thought is essentially analogous to this vagueness’ (1923: 91). Being smudged is, like vagueness on Russell’s view, a property a photograph possesses by virtue of the nature of its relation to that photographed. The indistinctness in the photograph amounts to the necessary inaccuracy of the representation rather than the (impossible) indeterminacy of the represented. Just as the represented is never smudged in itself, so too the represented can never be vague, nor anything analogous, in itself. Smudged or vague representations are inaccurate in a way that mere factually inaccurate representations are not. Using that example of vagueness where our intuitions seem strongest – the case of a vague predicate F – we can show, more speci¿cally and ¿rstly, that representational vagueness results in one–manyness. One–manyness is a necessary condition for representational vagueness. To say that the predicate is vague as traditionally understood is to say that there is some borderline case for F, a, say. Suppose further that this is the predicate’s sole borderline case, with d being such that F determinately does not apply, and objects b and c being such that the predicate determinately does apply. So what property might F denote? Since its denotation must be sharp, there are two candidates consistent with its semantic properties, that is, consistent with the fact that b and c must determinately possess the property denoted by F whilst d must determinately not possess the property denoted by F. Either the borderline case a determinately possesses the property denoted by F or it determinately does not. The predicate’s borderline case must be resolved at the ontological level, one way or the other.6 The vague predicate’s denotation might consistently be said to be the property α1 with extension {b, c} and anti-extension {a, d} or the property α2 with extension {a, b, c} and anti-extension {d}. Moreover, this one–manyness regarding what the predicate might represent is one–manyness about whether or not to include the borderline case in the class of items possessing the property denoted. Since the vagueness in the representation has no analogue in what is represented, the vagueness can (and must) be resolved in one of the two ways described, each being consistent with the essential properties of the representation – in this case, the semantic properties of F – though neither being necessary. 6 Thus, for the sake of the argument, we are simply assuming that property-sharpness is the ontological analogue of predicate-precision – there are no borderline cases for the property. Again, I am setting aside the distinction between resilient and non-resilient borderline cases for the moment (see n. 1, this chapter). So, having already admitted earlier that our current concerns with vagueness are really concerns with the traditional, naive conception of vagueness, we now note that talk of fuzziness, in so far as it is simply an analogue of vagueness, is really talk of a naive conception of fuzziness. More sophistication will be added in due course, but to do so now would simply confuse things.

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In the case of predicate-vagueness, then, we can see that not only does the presence of some borderline case give rise to some one–manyness – thereby contrasting vagueness with accuracy. The very borderline case itself is the source of the referential indeterminacy. Secondly, we can do more than merely contrast vagueness with accuracy – merely establishing that borderline cases give rise to one–manyness. The contrasting notion instantiated, one–manyness, is instantiated solely by terms that are borderline-case vague. That is to say, we can do more than merely show that vagueness is contrasted with accuracy by showing that one–manyness is a necessary condition for vagueness – we can also show that one–manyness is suf¿cient for vagueness. The simplest way to show this is to establish that a precise predicate must stand in a one–one correlation with a sharp property, so precision is suf¿cient for one–oneness. A one–one correlation (that will also be factually accurate) between a precise predicate F and a sharp property can be obtained by simply taking the property, α, to be that which all objects describable as F have in common. Hence the set of objects having the property α is the set of objects satisfying F. So, by de¿nition, the property α is correlated one–one with the predicate and α is sharp since the fact that everything either determinately satis¿es the predicate or determinately doesn’t entails that all objects instantiate α determinately or determinately not – there are no borderline cases for α. Concerns might be raised as to whether the foregoing suf¿ces in the presence of ambiguity. More exactly, predicates that are precise though ambiguous may appear not to stand in a one–one correlation with a sharp property but exhibit one–manyness due to the plurality of denotations derived from the various disambiguations of the predicate. The Russellian account of vagueneness might therefore seem to conÀate vagueness and ambiguity. To be sure, there are those who have pointed to a close connection between the two phenomena; some supervaluationists for example have been tempted by the seeming likeness between the two. Nonetheless it would be a mistake to equate the two and any theory which did so would be the worse for it.7 But Russell’s account can distinguish them. An ambiguous predicate exhibits indeterminacy of sense and is one–many only in so far as it admits of a plurality of senses each of which may exhibit determinate reference. Russellian vagueness, on the other hand, amounts to a predicate with ¿xed sense exhibiting indeterminacy of reference resulting in many possible referents. Russellian one–manyness is therefore suf¿cient for vagueness given that it is understood that the one–many relation obtains between predicates with ¿xed sense and properties. Seen in this way, terms that are vague on Russell’s account are more akin to indexicals for which indeterminacy of reference may arise despite their having a ¿xed sense. In contrast to both indexicality and ambiguity, though, Russellian one–manyness is not resolvable by appeal to context. Though more like the former than the latter, it stands apart from both by virtue of the enquiry-resistant nature of the indeterminacy involved. So, conjoining the two claims arrived at earlier, one–manyness of denotation is both necessary and suf¿cient for traditional predicate-vagueness on the assumption that the represented is sharp. Hence Russell’s de¿nition is a de¿nition of traditional predicate-vagueness if the represented is sharp. 7

See Sorensen (1998) for grounds for distinguishing the two.

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On Russell’s account the above-mentioned one–one correlation is not possible with vague predicates because they could only ever be so correlated with a fuzzy property and there are none. However, if there could be fuzzy properties, then the same one–one correlation could then be employed. In other words, if the represented could be fuzzy, then not all vagueness need result in one–manyness; a vague predicate G could accurately represent or denote some fuzzy property β, say. So we can further add that Russell’s de¿nition is a de¿nition of traditional predicatevagueness only if the represented is sharp.8 Summing up the foregoing, we may say that if the represented is sharp, then all traditional vagueness of denoting phrases results in one–manyness and vice versa, and if all traditional vagueness of denoting phrases results in one–manyness and vice versa, then the represented is sharp. Conjoining these two claims, it follows that Russell’s definition of predicate-vagueness and the traditional definition of predicate-vagueness are equivalent if and only if the represented (property) is sharp. The truth of this conjecture serves to show that Russell has successfully de¿ned traditional vagueness for all and only those predicates where vagueness is semantic but not ontological. That is, it con¿rms that, despite criticism, Russell has indeed given a de¿nition of traditional representational vagueness for predicates. Subsequent amendments to Russell’s de¿nition to avoid the naivety of the traditional aspect of vagueness (discussed in Chapter 1), to then establish its adequacy as an account of the representational vagueness of any denoting phrase and, ¿nally, to extend its application beyond denoting phrases will establish that the de¿nition provides the foundation for any adequate representational account of vagueness. Russell has got the semantic analysis of representational vagueness exactly right. Thus, charitably interpreted, we should take Russell’s remarks concerning the impossibility of vagueness in the world to mean that things are neither vague nor fuzzy. The agnosticism contemplated at the outset of this chapter is best rejected, and Russell is best understood as a representationalist about vagueness. 2.2.2

Extending Russell’s account

How then are we to generalize on this result to establish that Russell’s de¿nition can be extended to provide a de¿nition of representational vagueness in general? The ¿rst point we need to address, prior to establishing its adequacy beyond the category of predicates to all denoting phrases, is whether Russell’s de¿nition can be modi¿ed to provide necessary and suf¿cient conditions for predicate-vagueness simpliciter as opposed to traditional predicate-vagueness. The point of difference, recall, is that traditional vagueness is evidenced by the presence of borderline cases rather than the presence of resilient borderline cases required for vagueness proper. Nonetheless, the foregoing arguments can be reworked using resilient borderline cases as opposed 8 The de¿nition can be seen to distinguish vagueness from ambiguity since an ambiguous predicate need be one–many only in the sense that it admits of many interpretations each of which may uniquely refer, whereas Russellian one–manyness amounts to a uniquely interpreted predicate having many possible referents.

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to the simple borderline cases. The obvious amendment amounts to our supposing, when we cite something as a borderline case in argument (a, say), that it be resilient (as described in Chapter 1, §4). This caveat is then carried on through the argument and proofs. In this way, though the precise predicate “has a diameter of exactly 1.39 × 109 metres” exhibits one–manyness of denotation – (it being indeterminate whether it denotes a property instantiated by the sun or another distinct property not so instantiated) and might therefore appear to be vague on the Russellian account, since the indeterminacy centres on the inclusion or exclusion of a non-resilient borderline case it is not suf¿cient for Russellian vagueness. A suitably amended account invokes one–manyness where the many differ on resilient borderline cases – resilient difference, as we might term it. Assuming this required caveat to be in place, it remains to be seen whether this characterization of predicate-vagueness can be generalized to any grammatical category. The obvious way to proceed is to attempt to employ the generalization described in Chapter 1 from predicate-vagueness to the vagueness of all phrases capable of exhibiting the phenomenon. The initial generalization is from predicate-vagueness to vagueness of denoting phrases more generally – i.e. predicates, singular terms and sentences. The unifying aspect of all these categories of vagueness is vagueness in extension, which Russell’s conception of vagueness is readily able to capture. Though the discussion so far has centred on predicate-vagueness, Russell’s account deals more generally with the vagueness of representations, among them denoting phrases, and it is not hard to see how the equivalence between the presence of (resilient) borderline cases and the indeterminacy of extension might be seen more generally as applicable to denoting phrases. We have already seen how a borderline case for the predicate F gives rise to sharp properties α1 and α2 which can be said to be borderline cases for the denotation of the relevant predicate. Assuming the necessary sharpness of properties, the presence of borderline cases for the denotation of F could be shown to be equivalent to the underdetermination of any particular property as the referent by the semantic properties of the predicate. So, too, we can see that a borderline case for the denotation of a singular term N is equivalent to the underdetermination of any particular sharp object by the semantic properties of N. The argument of the previous section serves as the key to this insight. By redescribing a predicate’s extension as the extension of the relevant vague singular term N, we see that the fact that some spatio-temporal point a is neither determinately in the extension of N nor determinately not in the extension of N means that there are two possible sharp candidates for the denotation of N consistent with its determinate semantic properties (that is, consistent with a point b’s being a determinate part of the extension of N, point d’s being a determinate nonpart; and so on). One determinately includes a as a part, the other determinately excludes it. Moreover, if the presence of borderline points in the denotation of N is tantamount to there being a plurality of possible referents, then objects must be sharp since were fuzzy objects logically possible then it would be logically possible that a vague singular term N determinately denotes a fuzzy object having as an indeterminate part that spatio-temporal point that was neither determinately nor determinately not in the extension of N. Indeterminacy in extension makes for a plurality of possible

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sharp referents, and a plurality of sharp referents entails some indeterminacy of extension. A similar argument can be given for sentences. One can redescribe the extension of F, speaking instead of the extension of the sentence S consisting of the set of possible worlds where S is true. If S is vague, then there could be worlds where it is neither determinately true nor determinately false. That is, there could be some possible world, w, such that it is indeterminate whether or not it is in the extension of S. The extension of S has a possible borderline case, w. Yet if states of affairs are sharp, then, given the equivalence between a state of affairs and the set of worlds wherein it obtains, any set of possible worlds that is a candidate for the extension of S is sharp. That is, every world, including w, is either a determinate member of the set or a determinate non-member. Hence there are two possible candidates for the extension of S – the set determinately including world w or the set determinately excluding w. Moreover, if the presence of borderline cases for the denotation of S is tantamount to there being a plurality of possible extensions (sets of worlds), then states of affairs must be sharp since were fuzzy states logically possible then it would be logically possible that a vague sentence S determinately denotes a fuzzy state of affairs, having as its extension an indeterminate set of worlds that neither determinately included world w nor determinately excluded it. Thus Russell’s account of the semantic vagueness of denoting phrases exactly captures what it sets out to describe (given obvious amendments to take account of resilience) if and only if properties, objects and states of affairs – that is, the world – are sharp. De¿nitions R3 and R4 adequately characterize extensional and intensional representational vagueness of denoting phrases. That is, Russell’s de¿nition of vagueness adequately characterizes the (semantic) vagueness of denoting phrases if and only if the world is sharp. Given the claimed success of Russell’s characterization of the representational vagueness of denoting phrases, it remains only to show how it can be further extended to characterize the representational vagueness of any grammatical category – that is, how it can be extended to characterize the representational vagueness of language in general. Russell himself relied on an infection theory of vagueness to extend to talk of vagueness for non-denoting phrases like the logical constants. In thus extending via a theory which we have already seen to be fallacious (see Chapter 1, §4), his generalized account, implicit in Russell (1923), is misguided. However, supposing that we are correct in claiming the adequacy of his account as applied to the representational vagueness of denoting phrases, then to generalize we need only make use of the recursive extension described in Chapter 1, §4 and encapsulated in Def (1) and Def (2). If Russell has got things right for the basic cases (of denotational vagueness), then, since the recursive extension will preserve this adequacy, extending by means of Def (1) and Def (2) will result in a characterization of vagueness in language appropriate for all and only those cases of representational vagueness.9 9 We have discussed Russell’s theory of vagueness only for linguistic items, but I see no reason to think that we could not extend his account as analysed to include all forms of

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Let us pause to consider what has been said. It has been suggested by a number of philosophers that Russell’s de¿nition of vagueness is defective for a variety of reasons. I believe they are mistaken to the extent that, though Russell himself may perhaps have had something else in mind, the reconstruction above is consistent with the spirit of his proposal for denoting phrases and preserves the essential ingredient of ‘one–manyness’. Moreover, far from being an inadequate de¿nition of vagueness, his de¿nition (or, if you prefer, the foregoing interpretation of his de¿nition) is both suf¿cient and necessary to the task of characterizing representational vagueness when conjoined with an appropriate theory extending the concept of vagueness from denoting phrases to any grammatical category in general. Thus the frequent charge that Russell’s de¿nition of vagueness in terms of the one–manyness of denotation cannot account for ordinary usage of the word “vague” is seen to be false. Russell has accounted for the most popular use of “vague”, the representationalist’s use, if not the most general and theory-free use. 2.3

Vagueness, Logic and the World

It is not hard to see how Russell’s analysis of vagueness might provide a foundation for the development of a formal logical analysis of vagueness. Consider his analysis as applied to a vague predicate F. It is representationally vague if and only if its denotation relation is one–many in the sense that there are at least two possible (sharp) properties the predicate could denote consistent with the predicate’s own semantic properties. By correlating each of the possible (sharp) properties with linguistic items, namely precise predicates denoting these properties, we can shift to intra-linguistic talk of vagueness as a relation between vague and precise predicates. The vagueness of F will now be said to amount to there being at least two possible ways of making the predicate precise consistent with its semantic properties. That is, vagueness amounts to a plurality of ‘admissible precisi¿cations’. More particularly, though, to say that it is vague whether Fa is to say that it is possible relative to the semantic properties of F that a instantiates the property denoted thereby and it is similarly possible that it not instantiate the property (the indeterminacy is a matter of underdetermination in this sense). It is thus contingent relative to the semantic properties of the predicate whether a instantiates the property denoted by F. Vagueness in respect of F is thus a species of F-relativized contingency, ∇ΣF. The logic of vagueness per se is then given by the logic of relativized contingency. Russell, however, never began down this path – a path pointing in the general direction of what is now a common logical response to vagueness, a supervaluation account, or some allied logic. The operator “∇Σr” was not, in his view, to be treated as a logical operator. Instead he defended classical logic in much the same way Quine does, as we shall see. Vagueness is admitted as an essential semantic feature representation, linguistic or otherwise. Indeed, he himself presented his account in full generality. In so doing we would ¿nally arrive at a complete explanation of his theory of the vagueness of representations in general. Our concern, however, is with language and we shall restrict ourselves to its study.

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of natural language, yet the supposedly sharp world is describable in a precise ideal language to which the scope of logic is limited. This ideal language plays the role that natural language plays for epistemic theorists of vagueness. Such theorists claim natural language is precise and are sceptical about our ability to know the precise set of linguistic rules governing such language. Russell (1923: 85) denies this, as we have seen. Nonetheless he holds a view very close to theirs with regard to his ideal language, which is precise though unknowably so and which is able to support classical logic. Were we able to occupy the epistemic high-ground (indeed heaven, the epistemically highest), all would be clear and classical. All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life, but only to an imagined celestial existence. Where, however, this celestial existence would differ from ours, so far as logic is concerned, would be not in the nature of what is known, but only in the accuracy [precision] of our knowledge. (1923: 89)

In the imagined logical heaven precise symbols are employed since we are there able to represent the sharp world to ourselves accurately, with precision, and in such a situation classical logic is therefore not threatened by vagueness. However, arguing in this way for the retention of classical logic while satisfying the intuition that natural language is vague, Russell has distanced logic so much as to make it virtually useless. It is no longer appropriate for the assessment of the validity of many arguments in natural language – in particular, the sorites paradox whose soritical expressions Russell considers beyond the scope of logic. This, of course, is Russell’s solution to the paradox. Nearly a century on, philosophers are suspicious of theories that make essential reference to such ideals. What we should concern ourselves with is natural language as it is. After all, if logic is a theory of valid argumentation, then surely a logic of those terms or concepts used in argument is what is of primary interest to us. The obvious step for the representationalist is simply to take the logic of “∇Σr” seriously. Moreover, the orthodox metaphysical position underlying the representational analysis of vagueness is not adequately defended by Russell. Claims encountered at the beginning of this chapter to the effect that ‘the world is what it is’ do not constitute the required defence since one may simply ask: What is it that it is, sharp or fuzzy? Taking a closer look, one is left with the feeling that Russell and Margalit have succumbed to just the kind of deep-seated metaphysical prejudice pointed to by Dummett (1981: 440). To say that ‘things are what they are’ is certainly true, but tautologous. I may claim something to be what it is even though what it is is vague. Assertions of self-identity need not (perhaps even cannot) be indeterminate or false, and if this is all Margalit means when he says that ‘they are not what they are in degrees’, one might agree and yet claim that a defence of the metaphysical view underlying representationalism is still lacking. Things are what they are, but what counts as that, beyond the tautological reply already proffered? To have any consequences for the advocate of a fuzzy world, the above statements must be read not as denying the indeterminacy of being-self-identical, but as denying the indeterminacy of being-φ, for some property φ.

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Russell’s claim that nothing is to a certain extent possessed of the properties it possesses surely only holds for those properties a thing possesses determinately, of which self-identity may be thought to be one. Is it not possible that something may be to a certain extent possessed of a property if it possesses that property to a certain extent? Given that things are what they are, they are only precise if they are precise. (Hardly controversial metaphysics.) From this, the thesis that they are precise can only ¿gure as the conclusion given the premise that they are precise. Thus, as yet, the metaphysical issue in question remains either question-beggingly assumed or simply asserted. As we shall see, more recent advocates of the representational account have attempted to provide arguments for this claim, thus bolstering a representational view. In addition, even if the metaphysical basis for the representational approach can be secured and one accepts Russell’s analysis of the semantics of vague terms, his means for defending classical logic – grounding logic in some celestial realm – are generally unacceptable these days. The implications for logic – in other words, what the representationalist goes on to say about logic – have changed somewhat with the development of modal logics. These are the issues to which we now turn.

Chapter 3

Descriptive Representationalism Russell’s account of vagueness as merely representational coupled with his view of the celestial nature of logic would defuse any tension between vagueness and classical logic. We noted two key elements in this defence that emerged from the previous chapter: an a priori acceptance of the precision of the world or ideal worlddescription, and a view of logic that relegates it to the realm of the celestial, mirroring the structure of some ideal language or corresponding reality behind appearances. Both of these elements have, since Russell, been rejected by various authors in giving representational accounts of natural-language vagueness, and in the next two chapters I want to take up these variations on the representationalist’s theme with a view to their implications for logic. Beginning with the former point, let us now turn our attention to attempts to defend classical logic and metaphysics by means of a variant account of vagueness that eschews a priori arguments in favour of arguments for the actual existence of a descriptively complete, precise ideal language. If sound, these arguments establish the precision of the world, thereby characterizing it as a representational account. I shall refer to this type of representationalism as descriptive representationalism since it is from our supposed ability to actually describe the world precisely that we are entitled to infer its precision. The appearance/reality or representation/represented distinction coupled with the view that vagueness is an inevitable feature of appearances or representations forced Russell to maintain that though all representations are vague, the represented – the world – can nonetheless be known to be precise by means of a priori reasoning; he advocated what we might term a priori representationalism. But what if vagueness is not thought to be an inevitable feature of representations? Then it is conceptually possible that the world be precisely describable. Arguments might then be sought to establish whether or not the world is, as a matter of fact, completely describable in some precise ideal language and thereby precise. Arguments establishing such an ideal-language description would provide a case for a representationalist account of vagueness, though distinct from Russell’s, and provide a defence of classical logic. Let us be clear on how this project is to proceed. The preceding Russellian account dealt with vague terms in natural language considered as in themselves ineliminably vague (vagueness was taken to be a semantic feature); moreover (and more importantly in the present context), the class of vague terms was itself considered an essential part of any complete description of the world. What is now being suggested is that, although vague terms are irremediably vague – there being no disagreement as to the semantic nature of vagueness – these terms can, as a class, be dispensed with in providing such a description; that is, language as a system of representation can be purged of its vague elements – while of course preserving the language’s descriptive power, thereby ensuring the non-triviality of the claim.

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What shall we count as ‘a complete description of the world’, though? Obviously it depends in part on what we take to be ‘the world’. If we are prepared to wield Ockham’s Razor with gusto and accept a minimal ontology, then a complete description is perhaps more easily had. It also depends on how rigidly one construes the completeness constraint. Is close enough good enough? This issue is the source of a distinction between: (i) those descriptive representationalists who maintain that the class of vague terms can as a whole be dispensed with, and this without any loss of descriptive completeness – I take Carnap as making this claim; and (ii) those pragmatically minded descriptive representationalists, such as Haack and Quine, who maintain that the class of vague terms can as a whole be dispensed with at perhaps some cost to descriptive completeness, but a cost that is nonetheless worth paying for the simplicity it affords – namely, our thereby being able to defend classical logic with what Quine (1981: 91) describes as its ‘sweet simplicity’. So the representationalist might agree that natural language is vague in its description of a precise world but claim that this vague language makes no essential contribution to the completeness of the description. It is in some sense eliminable (§1), redundant (§2), or super
cial (§3). 3.1

Against Elimination

A common move is to advocate the elimination of vague terms. The suggestion is that one could, in principle, reconstruct an ideal or regimented language containing only precise terms by means of which a precise world could be described. Quine (1977), for instance, suggests that though the common person’s ontology is vague and untidy, it really does not matter so long as we can think of ourselves as approximating a scienti
c language free of vagueness. Carnap (1950: ch. 1; 1966: part II) and Haack (1974: ch. 6; 1978: ch. 9) have also endorsed this ideal-language approach. It should be noted, though, that Carnap and Haack have not explicitly declared any interest in the ontological issue that presently concerns us in our attempt to arrive at a satisfactory analysis of vagueness. They are concerned to address the logical problems posed by vagueness. However, they effectively defuse the ontological issue at the outset by claiming the class of vague terms as a whole to be eliminable. For the eliminativist the ontological question is effectively trivial or irrelevant. This eliminativist response to the problem of vagueness is described by Putnam (1983: 274–6) as the ‘“what me worry” response’. On this view, the eliminativist, though committed to a precise world that is not precisely describable in natural language, can answer the question “Why suppose the world to be precise if it is not completely describable as such?” by suggesting that it is indeed, as a matter of fact, completely describable in an ideal language or that our interests are best served by assuming this to be the case. The burden of proof confronting the representationalist with regard to the assumption of a precise world vaguely described by natural language would thus be relieved. Equally so, the burden of proof regarding the appropriateness of classical logic; the precision of the world or language used to justify classical logic is seen as a matter of fact to obtain or at least be affordable.

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The claim that the world is completely describable in an ideal language free of vagueness is thus crucial in shifting the burden of proof back onto advocates of ontological accounts of vagueness and non-classical logics of vagueness. It is my contention that such a claim does not withstand close scrutiny. An obvious way in which this programme might initially be problematic is that the replacement of vague terms by precise surrogates requires that there be some such surrogates. It has, as we have seen, been suggested by Russell that no such terms are available: all language is vague. However, his arguments for this claim were found wanting (in Chapter 1) and so we may grant that there may be a nonempty class of precise terms available as possible replacements for vague terms. So how then are vague terms to be eliminated without loss? 3.1.1

Comparative descriptions

It has sometimes been suggested that such elimination can be achieved by replacing attributive expressions with precise comparative ones. Briey, the thought is this: where vagueness arises, for example with predicates like “red”, one might attempt to replace the vague attributive expressions with corresponding comparative expressions, for example “is redder than”. If comparatives are free of vagueness and all vague terms are replaceable by comparatives without loss of descriptive power, then it would seem that the elimination programme is feasible. Both Carnap (1950; 1966) and Peacocke (1981: 133–5) have made suggestions along these lines. Such a manoeuvre is doomed to failure, however. In the
rst instance, vague singular terms cannot be eliminated in this way. Suppose the claim “Mt Glorious is high” to be vague and suppose that the vagueness of the predicate “is high” can be eliminated in favour of the comparative “is higher than” – “Mt Glorious is higher than Mt Nebo” (say). Nonetheless, this latter claim might be vague by virtue of the vagueness of the singular terms involved – it might depend on how one delimits Mt Glorious and Mt Nebo, and that will be a vague matter. Even if such replacement seemed promising in the case of predicative terms, it does not in the case of vague singular terms. Secondly, this kind of systematic replacement applied to vague predicates requires that the predicates being replaced be predicates of degree, where the degrees are totally ordered. Following Wright (1975: 348), let us characterize a predicate F as a predicate of degree just in case the comparative “is less/more F than” makes sense and iteration of one of these relations may transform something satisfying F into something satisfying not-F or vice versa. (Iteration is required to ensure the non-triviality of the notion of being a predicate of degree – every predicate F admits of at least two degrees, F and not-F.) The requirement for a total ordering of degrees is met just if the relation “is less F, more F or F to the same degree” always applies to any two objects in F’s range of signi
cance. The need for such requirements is obvious. If the vagueness of a predicate F is to be avoided by the use of comparatives, then obviously the comparative locution must make sense. Furthermore, if the only way in which the comparative “being less/more F than” makes sense is in there being two degrees – not-F (being less F than something that is F) or F (being more F than something that is not-F) – then the use of comparatives will not avoid problems of vagueness since this use of comparatives is merely

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elliptical for describing something as F or not-F and the problem of vagueness will simply re-arise for the comparative locution. So F must be a predicate of degree in the above sense. The demand that the ordering be total, noted by Peacocke, is required to avoid the possibility of its being vague whether one thing is as F as, or more F than, or less F than some other thing. It must always be the case, for any pair 〈a,b〉 in the range of signi
cance of the relevant predicate F, that one of the three comparative relations determinately holds. Of course, many vague predicates do admit of degrees – indeed, Wright takes this as an essential feature of soritical predicates and Engel (1989: 24) claims that every predicate, or ‘virtually’ every predicate, admits of differences of degree. Yet many vague predicates, such as colour-predicates or predicates like “is a chair”, exhibit non-linear or multi-dimensional vagueness. Their degrees are not totally ordered. Vagueness is therefore not eliminated by the substitution of comparative relations. Comparatives might be indeterminate as well. Thirdly, even if subject terms were eliminable in this way and demands for total ordering were met, what justi
es this restriction in our description of the world to comparative locutions? The substitution of a vague and categorical description for a description of how things are relative to each other leaves out much of our ordinary talk about the world. Many properties we take to be properties of things in the world are left out of such a description. Descriptive completeness is not preserved by such a revision of ordinary talk. 3.1.2

Science and the ideal language

Perhaps the most popular means of resolving the problem of identifying a precise, ideal language is by way of that class of terms that many consider undoubtedly precise and thus perfect candidates – scienti
c terms. By their use, it is claimed, one can either precisify previously vague terms, devising precise criteria for their application using quantitative terms, or simply remove vague terms in favour of precise scienti
c ones. A complete description of the world can be given in the precise language of an ideal science.1 Consider the term “hot”, for example. This vague qualitative notion may either be precisi
ed by de
ning a new term “hot1” to be “more than 300° Kelvin” (say) or eliminated altogether in favour of quanti
ed descriptions of the entropy of a system, e.g. “temperature of x° Kelvin”. Such was the programme of Carnap in his 1966 Philosophical Foundations of Physics, for instance, more recently endorsed by Haack and Quine. We can ‘tidy up’ (Haack) or ‘regiment’ (Quine) ordinary language in this way. To be sure, it is practically cumbersome to have to defer to protracted tests, etc. to ensure the proper use of such regimented language. Aware of this, Haack speaks of ‘the advantages that vague ways of speaking undoubtedly possess’. However, this practical dif
culty is not a dif
culty in principle. Competent speakers endowed with complex measuring devices could in principle speak such a language containing precise substituends for their vague counterparts. Temporarily setting aside the question of whether or not the language of science 1 Copeland refers to this as ‘the crispness postulate’ (1995: 91), which he takes to be quite probably false.

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is precise, it remains to be shown that whatever can be said in the vague language of ordinary discourse can, in principle, either be eliminated in favour of talk in the (purportedly) precise language of science or eliminated altogether without loss. Having identi
ed a prima facie candidate for the ideal language, the language of science, the question remains – is it suf
cient to the task of providing a complete description of the world? Can we say all we need to say in precise terms? Physical-quality terms already considered like “hot”, “heap” or “bald”, which are absent from the ideal description, are said by Quine to depend on mere verbal conventions, not matters of fact, so regimentation – their elimination in favour of precise substituends – is not problematic. Descriptive completeness is not threatened. ‘Where to draw the line … is not determined by the distribution of microphysical states, known or unknown; it remains an open option’ (Quine 1981: 94). In accordance with Quine’s physicalist outlook we are therefore free to resolve indeterminacies any way we please so long as the proposed resolution does not conict with the distribution of microphysical states. Similarly for terms describing intentional states, moral and aesthetic terms such as “good” or “beautiful”, and descriptors like “noble” or “slovenly”. After all, language describing the contents of beliefs, desires, intentions, indeed any psychological state, might be essentially vague by virtue of the vagueness of the content of the relevant state, and the terms “belief”, “desire”, etc. themselves are essentially vague. (Of course intensional language is also problematic for reasons other than its vagueness, and the responses are varied.) Physicalists like Quine claim that we are free to regiment such language, resolving any indeterminacies any way we like. Though there are costs to common sense with such regimentation, these costs are not ones we need worry about. For anyone who takes common sense to constitute a partial constraint on philosophical theories, the cost of such a programme may seem excessively high. No complete ideal-language description of the world will include anything’s being hot or tall or red. Nor, it would appear, can it include anything’s being beautiful, good, noble, believed or desired, and so on. Perhaps more surprisingly, though, descriptions of things as tables, chairs, persons, stones or indeed as any ordinary thing are all eliminated – all the respective terms are vague and as a consequence are absent from the ideal language. These consequences are most clearly brought to light by considering the sceptical arguments of Peter Unger and Samuel Wheeler whereby the extent of the linguistic revision entailed by the regimentation programme is made obvious.2 Unger and 2 See especially Unger (1979a; 1979b; 1979c; and 1980a) and Wheeler (1979). Grim (1982) has argued that the consequences of accepting the line of argument advocated by Unger and Wheeler are far more extensive than even they recognize. The debate is then joined by Abbott (1983) and continued in Grim (1983). If Grim’s line of argument is accepted as sound, then the point I go on to make is strengthened. However, I think Grim’s argument is awed. Quine (1981: 92ff) mentions Unger’s argument but goes on to discuss it in the context of the Problem of the One and the Many, on which Unger has also written – see Unger (1980b). While I agree that issues surrounding vagueness can be employed to motivate this latter problem (see Chapter 5), Quine’s digression here is simply confused.

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Wheeler argue that all soritical terms, since incoherent, must be eliminated, with corresponding ontological consequences. The general impact of these arguments comes from their showing just how much of language is soritical and consequently just how radical the consequences for common-sense or folk ontology might be. Their speci
c bearing on our present concern with the regimentation programme is as follows. Those terms that must be eliminated according to Unger and Wheeler due to their soritical nature must be eliminated from Quine’s regimented language since any term is vague if soritical, and absent from the precise regimented language if vague. Unger and Wheeler, then, in showing just how much of everyday language is soritical, show just how far-reaching the regimentation programme is. (It should be noted that Unger and Wheeler, far from seeing the extent of the revision as a possibly troubling consequence, whole-heartedly embrace the extensive scepticism which ensues. Of course, their claim that all soritical terms are incoherent is incoherent by its own lights since the argument for the vagueness of “vague” discussed in Chapter 1 is easily adapted to establish the soriticality of ‘soritical’. See the end of this section for more remarks in this vein.) Let us diverge then briey to consider their position. Their sceptical arguments proceed by considering the sorites of Eubulides (Wheeler), or some variation thereof (Unger), e.g. the sorites-of-decomposition, the sorites-of-slicing-and-grinding and the sorites-of-cutting-and-separating. By way of example, consider the sorites-ofdecomposition which purports to establish the non-existence of stones. Unger (1979b: 120ff) invites us to entertain the following three propositions: (1) (2) (3)

There is at least one stone. For anything there may be, if it is a stone, then it consists of many atoms but a
nite number. For anything there may be, if it is a stone (which consists of many atoms but a
nite number), then the net removal of one atom, or only a few, in a way which is most innocuous and favourable, will not mean the difference as to whether there is a stone in the situation.

It is then argued that they form an inconsistent set. Consider a stone, composed of a certain
nite number of atoms, n, say. (That there is such a thing is guaranteed by (1) and (2) above.) If we or some physical process should remove one atom (any piece of matter just below the threshold of perceptibility would do), without replacement, then there are left n–1 atoms, presumably still constituting a stone (by (3) above). Now, after another atom is removed, there are n–2 atoms; and so on. After n atoms have been removed in similar stepwise fashion there are no atoms left at all, yet we must still be supposing that there is a stone there (since again, by (3) above, no one removal could make any difference, so there is no difference at any stage). But this contradicts (2). There is then, claims Unger, a blatant inconsistency in our thought. ‘However discomforting it may be, I suggest any adequate response to this contradiction must include a denial of the
rst proposition, that is, the denial of the existence of even a single stone.’ The term “stone” is an inconsistent or incoherent notion by virtue of its vagueness rendering it susceptible to soritescontradictions, and so there are no stones.

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Thus we have, according to Unger, an argument for the claim that there are no stones – an argument which can be generalized to refute the existence of all ordinary things.3 In showing just how many terms are susceptible to sorites-contradictions, the arguments of Unger and Wheeler highlight the extent of the regimentation programme. For both of them, this sceptical conclusion is expressible in both the material and formal mode. That is, it is at one and the same time a claim about language and about the world, applying equally well to the terms denoting ordinary things (e.g. “stone”) and to the things thereby denoted (e.g. a stone); ‘our results concern words and things alike’ (Unger 1979b: 147). And so too for Quine; the soritical nature of terms describing ordinary things, which renders them incoherent according to the above argument, precludes them from
guring as terms in the ideal regimented language and so, since ontological accounting only makes sense relative to this regimentation, there are no ordinary things. Such consequences of regimenting natural language have long been known to obtain. It is reported by Galen in his On Medical Experience that the Empirical Doctors, in attempting to defend their use of the vague notion of “suf
ciently many” in their account of inductive inference, replied to the attack of the Dogmatic Doctors as follows: according to what is demanded by the analogy [by parity of reasoning], there must not be such a thing in the world as a heap of grain, a mass or satiety, neither a mountain nor strong love, nor a row, nor strong wind, nor city, nor anything else which is known from its name and idea to have a measure of extent or multitude, such as the wave, the open sea, a ock of sheep and herd of cattle, the nation and the crowd.4

The consequences of banishing such vagueness seem unacceptable. As Quine (1981: 94–5) himself acknowledges: ‘At this point, if not before, the creative element in theory building may be felt to be getting out of hand, and second thoughts … may arise.’ Unger (1979b: 124–5) is not unaware of such concerns. In an exaggerated characterization of the concerns of many he acknowledges the Mooreian gambit of clutching onto common sense at the expense of anything else, most especially any philosophical reasoning. According to this way of thinking it is always most appropriate to reply to philosophical challenges as follows. We are more certain that there are tables than of anything in the contrary philosophical reasoning. Hence, while we may never be able to tell what is wrong with the reasoning, at least one thing must be wrong with it. 3 Some physical objects remain, impervious to Unger’s argument – ‘certain particulars which are prominent in the physical sciences, … [e.g.] electrons, hydrogen atoms, and water molecules’ (Unger 1979b: 122). Presumably mereologically simple objects, those objects that are not composed of any
ner objects, are also immune; ‘certain sub-atomic particles may provide an example’ (Unger 1979a: 241). In addition, Unger leaves open the possible existence of physical (though not ordinary) objects – even middle-sized ones (Unger 1979b: 150–51), in contrast to Wheeler, who bolsters his argument against ordinary things by arguing against all middle-sized objects (Wheeler 1979: 164). 4 Galen, On Medical Experience, XVI 1, p. 114 W; cited in Barnes et al. (1982: 34). See also Long and Sedley (1987: 222–3).

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But of course, nothing as strong as ‘the Mooreian gambit’ is required to motivate second thoughts concerning the theory being built by Quine, Unger and others here. Second thoughts arise here because such consequences conict so markedly with common sense. Though it may not always be appropriate to clutch onto common sense at the expense of anything else, including philosophical reasoning, it might sometimes be appropriate and this might well be one of those times. We might reasonably claim that we are more certain that there are ordinary things (or that terms describing ordinary things are coherent) than that the foregoing philosophical reasoning, including its commitment to classical logic, is correct. After ‘second thoughts’ on the matter, though, Quine nonetheless opts for a defence of this strategy. Regimentation via elimination is costly, though costeffective – this is the Quinean defence. Though it seems such regimentation is not possible without some descriptive incompleteness, nonetheless the resulting loss in utility is offset by gains in simplicity afforded by the retention of classical logical theory. All things considered, our adherence to classical logic optimizes the drive for evidence and the drive for system – that is, utility and simplicity.5 The adequacy of this defence depends, amongst other things, upon how one evaluates the cost–benefit equation. Is the loss in descriptive completeness adequately offset by the gain in simplicity afforded by the retention of classical logic? Well, that depends
rstly on how little the loss in descriptive completeness is deemed to be, and secondly on how great the gain in simplicity afforded by classical logic is deemed to be. In regard to the
rst issue we have already seen that the costs appear to be high indeed. The second issue surely cannot be addressed until competing logical theories are available for evaluation as regards their complexity.6 Moreover, it is the simplicity of the overall philosophical theory in which the logic is embedded which is of paramount importance, rather than the theory itself. Recall Poincaré’s insistence on retaining Euclidean geometry in the face of pressure to abandon it in favour of a non-Euclidean alternative with the development of Relativity Theory early last century. Whilst Euclidean geometry is simpler than its non-Euclidean rival, the new physical theory in which it was embedded was not and Poincaré’s proposal was ultimately rejected. Quine’s judgement is thus doubly premature and this renders his defence inadequate. It is simply not clear whether the gains in simplicity outweigh the loss engendered by the programme of regimentation. How could it be, even for Quine, who is able to
x the debit column (the cost of descriptive loss) at a level favourable to his case (that is, low)? Balancing the books with unknown credit can only be done by making presumptions – which here amount to the supposition that classical logic will provide a simpler philosophical theory of vagueness than any rival that may appear. 5 Note that this is not the position advocated by Unger and Wheeler. They simply assume the correctness of classical logic and embrace what they take to be the consequences thereof. Since they assume what is at issue in the current context (the adequacy of classical logic in the face of vagueness), their position cannot be invoked to defend Quine’s regimentation programme; it merely illustrates the costs associated therewith. 6 Margalit (1976: 216–17), whilst endorsing the Quinean programme, admits that this latter concern prohibits evaluation of the cost–bene
t equation.

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Let’s take stock of our position, then. It has been suggested that classical metaphysics is not threatened by vagueness since the class of vague terms as a whole can be eliminated from any description of the world. A suitably regimented or ideal language purged of its vague elements, for example the language of science, is suf
cient to the task. However, vagueness is such a pervasive feature of natural language that this programme of regimentation seems to threaten descriptive completeness for all but those with the sparsest of ontologies. Thus it seems likely that regimentation is not possible without some loss. Can this loss be offset by gains in the simplicity of that logical theory deemed adequate after regimentation, classical logic? Quine’s inclination is to answer in the af
rmative ‘for the simplicity of theory it affords’, but this cost–bene
t analysis is premature. Of course, the foregoing does not constitute a knock-down argument against eliminativism but it does highlight the common problem associated with such a strategy. Moreover, further and more serious problems attend the very statement of the elimination thesis. As remarks by Putnam (1983: 277), Rolf (1981: 135), Sorensen (1985: 135–6) and Williamson (1994: 169) all suggest, there appears to be no way to state the doctrine without undermining it. Since, as we have seen in Chapter 1, “vague” is itself vague – indeed, soritical – it follows that it will not be part of the vocabulary of the regimented ideal language but is merely one of many defective vague terms of unregimented natural language. Yet an explication and defence of the Quinean position requires that we make use of the term. The very statement of the position requires its use. Thus the theory being advocated must itself employ language which, by the theory’s own lights, is to be eliminated. The very fact that any discussion of vagueness itself already involves us in the use of vague terms shows the untenability of any elimination programme. 3.2

Against Reduction

One might be tempted to respond by claiming that, though a complete description of the world must include the aforesaid language since such terms are, as a class, ineliminable, their apparent resistance to regimentation is illusory in the following sense – they are reducible to talk in some precise ideal language. Reduction is commonly understood as a relation between theories – in the case at hand, between a vague natural-language theory of the world, TV, and a precise ideallanguage theory typically said to be provided by the language of science, TP. Beyond this, conceptions vary. A strong reduction claim is that of de
nitional reduction. To say that TV can be de
nitionally reduced to TP is to say that vague terms are analytically de
nable in precise terms, and one can use those de
nitions to derive within TP all the statements of TV as suitably rewritten statements of TP. Such a strategy would provide a defence of classical metaphysics since a purely precise ontology, the ontology of the reducing theory, can be retained. Similarly, one might think, the logical challenge posed by vagueness can be met – vagueness is no threat to classical logic since vague talk is translatable into talk in some precise language without loss of descriptive power. We cannot make do without the descriptive ability made possible by vague language, but this ability can be underwritten by precise language via translations generated by the de
nitions.

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This prima facie plausible position does not withstand closer scrutiny, however. As Rolf (1980: 321) has pointed out, no vague phrase can be de
ned solely by means of precise ones. This is a straightforward consequence of the fact that precision is inherited. As we saw in Chapter 1, §4, if all the constituent parts of a complex phrase Q are precise, then Q is also precise. So it follows that if we have a class of precise phrases, we cannot de
ne any vague phrase by means of them (though there is nothing in principle to stop us from de
ning precise phrases by means of vague ones, so reduction in the opposite direction appears possible). So, for example, if all the primitive terms of physics are precise (as is often supposed), predicates such as “is red” cannot, by virtue of their vagueness, be de
ned in terms of the language of physics. De
nitional reduction cannot succeed. Even weaker reduction relations are also ruled out. Consider Nagel’s (1961) comparatively broad concept of derivational reduction. Such reduction requires the availability of bridge laws which connect terms of the theory to be reduced, for example TV, with those of the theory to which it is reduced, for example TP. The problem follows from the fact that these bridge laws are subject to the following requirement: Each primitive predicate P of the theory being reduced, TV, is connected with a co-extensive predicate Q of the reducer, TP, in a biconditional law of the form: For all x, Px iff Qx. And similarly for all relational predicates. If the requirement on the bridge laws is not met, then derivational reduction fails, and it seems it cannot be met. Where P is some vague predicate (e.g. “is red”) and Q is a predicate of the precise language (e.g. “is in microstate m”), there are serious doubts as to whether the requisite biconditional law can obtain. A vague term cannot be determinately co-extensive with a precise term since a term’s being vague entails its having borderline cases, which is exactly what fails to be the case for precise terms. Of course, there may be conditionals which nonetheless warrant the truth of the biconditional in question. However, in contexts where indeterminacy might arise, presumably the biconditional law must obtain determinately, thus requiring determinate coextension (at least), and this is not to be had. It seems therefore that vague talk cannot be reduced to talk in some precise language – be it the language of science, or anything else.

3.3

Against Supervenience

Having beaten a retreat from eliminativist and reductionist strategies, the representationalist might attempt a
nal stand resting their hopes on supervenience claims. The use of such claims to justify a parsimonious ontology is neither new nor uncommon. The paradigm example of supervenience is that relation said to obtain between the mental and the physical. Quine (1977: 186–8), for instance, alludes to the existence of this relation when he says:

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If a man were twice in the same physical state, then, the physicalist holds, he would believe the same things both times, he would have the same thoughts, and he would have all the same unactualized dispositions to thought and action. … It is not a reductionist doctrine of the sort sometimes imagined … what it does say about the life of the mind is that there is no mental difference without a physical difference.

Having made what amounts to a supervenience claim, he then continues – arguing for ontological parsimony on the basis of this relation – in favour of a purely physical ontology: It is a way of saying that the fundamental objects are the physical objects. It accords physics its rightful place as the basic natural science without venturing any dubious hopes of reduction of other disciplines. … If there is no mental difference without a physical difference, then there is a pointless ontological extravagance in admitting minds as entities over and above bodies … Thus it is that the physicalist comes out with an ontology of just physical objects, together with the sets or other abstract objects of mathematics; no minds as additional entities.

Adapting this justi
cation of the physicalist’s ontology to current circumstances, then, the thought is this: if vague natural-language descriptions could be shown to supervene on descriptions in some precise language, then it would be ontologically superuous to admit vague entities over and above the precise. It would thus be pointless to admit the world to be vague. This route to a precise ontology has been suggested by Peacocke (1981: 132–3). Suppose we have a language L containing vague expressions. Then the suggestion that the world itself is not vague is the suggestion that there will be some conceivable language L1 which contains no vague expressions and which has the following property: it is a priori that if two situations agree in all respects describable using the language L1, then they agree in all respects describable using the language L. This is a form of supervenience.

Given this form of supervenience, Peacocke goes on to de
ne what it is for vagueness to be ‘super
cial’. I shall say that the vagueness of a vague expression E is super¿cial if for any language L whose sole vague expression is E, there is some language L1 containing only sharp [precise] expressions, and such that the descriptions of L supervene on those of L1 in the sense just explained. … On this construal, then, the thesis that the world itself is vague would be the thesis that not all (possible) vague expressions have merely super
cial vagueness.

In other words, the world is precise if and only if all vagueness is ‘super
cial’ – that is, the world is precise if and only if all possible natural-language descriptions supervene, in the above sense, on precise descriptions. Quite clearly, such a thesis, coupled with the view that all vagueness is ‘super
cial’, would justify a purely semantic (i.e. representational) account of vagueness. Though natural language is vague, the world is not. The plausibility of this line of argument depends, though, on our preparedness
rstly to accept the ‘super
ciality’ of vagueness in the sense described above and then to accept, as a

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consequence, its super
ciality in the everyday sense of a surface feature of language without metaphysical import, a mere surface feature. Consider,
rstly, the supposed ‘super
ciality’ of vagueness. It amounts to the claim that any vague language supervenes on some precise language. That is, for every language L containing a vague expression there is some conceivable language L1 which contains no vague expressions and which stands in the following supervenience relation to L: it is a priori that if two situations agree in all respects describable using the language L1, then they agree in all respects describable using the language L. Just what this precise language will be is left open. To convince the sceptic that there will be some conceivable language which contains no vague expressions and which can stand as a relatum of the supervenience relation, advocates of such a claim must either argue for this as a matter of conceptual necessity, that is, that to assume otherwise leads to incoherence, or argue for it by providing some particular instance. It is hard to imagine an argument as strong as the former which is not circular or question-begging in some way. An argument of the latter type is the argument most usually endorsed, if only implicitly. The almost universally agreed candidate to instance the existential claim is the language of science. If this is not how the claim is supported, then we are owed some other reason for accepting it as even plausible. Suppose then that the existential claim is instanced by the language of science and suppose further that it is precise. The question then is whether it is a priori that if two situations agree in all physically describable respects then they agree in all respects describable in vague language. The answer is that it clearly is not. Consider the vague predicate “is blue”, and assume that we have two objects a1 and a2 that agree in all respects describable in physical terms (e.g. both a1 and a2 reect light of 480 nanometres wavelength, etc.). Assume also that a1 is blue. The question we need to address is whether or not it is a priori that, given these assumptions, a2 is blue. The answer is obviously that it is not. That any two objects alike in physical terms are alike as regards their colour is something we have learned from scienti
c enquiry – it is an a posteriori matter. Peacocke offered “many” as an example of a vague expression whose vagueness is ‘super
cial’ and we can agree that it certainly does
t his form of supervenience. The concept of number or cardinality is caught up in the analytic explication of or the understanding of “many” and it is a priori that the covariance relation obtains. So we can agree with Peacocke that the vagueness of some expressions supervenes, in his sense, on precise ones. That is, that the vagueness of some expressions is, as Peacocke puts it, ‘super
cial’. However, not all vague expressions are like this. Colour concepts, as we have just seen, are not. So, even supposing one is prepared to accept the language of science as the subvening precise language, vagueness is not in general ‘superficial’. An appropriately modi
ed supervenience relation is, however, already implicit in Peacocke’s (1981: 133) remarks concerning the ‘super
cial’ vagueness of “many”. The truth values of sentences containing “many” will supervene on those sentences not containing “many” but containing cardinality quanti
ers: there cannot, for example, be two situations with respect to one of which some sentence of the form “Many Fs are G”

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is true and with respect to the other is false, if the two situations have the same number of Fs being G.

Supervenience is to be explicated in modal rather than epistemological terms. More speci
cally, the form of supervenience required is that there cannot be two situations differing in vaguely describable respects which are alike in precisely describable respects. In other words, no two possible situations alike in precisely describable respects differ in vaguely describable respects. Thus the relation is one of strong supervenience.7 (Of course, in light of the a posteriori nature of the relation between colour concepts and physical ones, the vague descriptions can be said to strongly supervene on precise descriptions only if one is prepared to accept the existence of necessary a posteriori truths. To the extent that one doubts the existence of such a class one will doubt the possibility of there being an a posteriori relation which holds of necessity and thus doubt the existence of any relation of strong supervenience.) In light of the point made in §2 that the vague cannot be reduced to the precise, any such supervenience claim would be refuted if it could be shown that the suggested supervenience relation entailed reducibility – echoing concerns surrounding the psychophysical case where a reductive supervenience relation is often viewed with suspicion. In the context of that debate many complain that nonreductive strong supervenience is not to be had. In the case under consideration, though, the usual argument for ensuing reducibility exhibits the fallacy of composition. Suppose we admit that for any supervenient predicate P there exists a predicate with which it is co-extensive in every world. This predicate is constructed by noticing that: for supervenient property P there is a set of properties, Q1, Q2, … in the subvenient set such that each Qi is necessarily suf
cient for P. Assume that this list contains all the subvenient properties each of which is suf
cient for P. Consider then their disjunction: Q1 or Q2 or … (or ∪Qi for short). This disjunction may be in
nite; however, it is a well de
ned disjunction … It is easy to see that this disjunction is necessarily coextensive with P.8

What is required though is that ∪Qi correspond to a precise predicate. Whilst each element of the set ∪Qi is ex hypothesi precise, the set of all properties each of which is suf
cient for P is a vague set. There will be precise properties for which it is indeterminate whether they are suf
cient for P. For example, there are heights for which it is indeterminate whether being such a height is suf
cient for being tall. Consequently, it will be indeterminate whether the relevant property is a member of the set of properties suf
cient for tallness. Given that ∪Qi is vague, the corresponding predicate will also be vague. It does not follow from the precision of each Qi that their composition ∪Qi is precise. The strong supervenience relation said to obtain between the vague and the precise is, for all that has been said so far, non-reductive. 7 This is the supervenience relation employed by Williamson (1994: 202–3) in the context of vagueness. 8 Kim (1990: 19–20).

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Let us grant then that vague descriptions may be said without contradiction to strongly supervene on the precise physical descriptions. The question remains whether we ought accept that the world is precise, with vagueness a mere surface feature of natural language. One might, of course, stipulate that this is simply what one means by ontological precision.9 However, advocates of ontological vagueness will rightly complain that this simply fails to address their claims. What is needed is some explanation as to why one should think supervenience sufficient for ontological precision. A tempting approach, I think, is the idea that we can explain why supervenience is suf
cient by means of descriptive completeness. For example, psychophysical supervenience is often taken as grounds for claiming the completeness on physical descriptions of the world. A similar claim might be made regarding the supervenience of the vague on the precise, i.e. a complete description of all the facts can be had in precise terms. This, coupled with the view that such a description is incompatible with ontological vagueness, might serve to explain why supervenience suf
ces to rule out ontological vagueness. More exactly, the explanation might be put as follows. The strong supervenience of the vague on the precise warrants the claim that a complete description of the world can be given in terms of the precise subvening descriptions and, moreover, the fact that a complete description can avoid vague language altogether means that the world is precise. Let us then consider each of these claims in turn, beginning with the claim for the completeness of the precise description. Turning again to the psychophysical case, physicalists like Lewis and Jackson certainly claim that a complete description can be given in subvening (physical) terms. This follows from the fact that ‘completeness’ itself is to be spelt out in terms of supervenience. Thus says Jackson (1994: 103) ‘the place to look … for illumination regarding the sense in which materialism [i.e. physicalism] claims to be complete … is at various supervenience theses.’10 Similarly, then, we might attempt to explain how the strong supervenience of the vague on the precise guarantees the completeness of the precise description by actually explicating the notion of ‘completeness’ in terms of strong supervenience. To say that vague descriptions of a situation strongly supervene on precise ones is just to say that any vague description of a situation is necessitated by the subvening precise description. From this, entailments are said to quickly follow by adapting ‘a straightforward and familiar argument’ from the psychophysical case.11 Suppose Π is a true statement giving the precise description of a situation and Λ is 9 This is just what Peacocke was advocating and, given the absence of any published accounts of ontological vagueness at the time of his writing, such a stipulation was not unreasonable. But the debate has moved on since then with the publication of Rolf (1980), Burgess (1990), Tye (1990, 2000), van Inwagen (1988, 1990), Zemach (1991), Akiba (1999), Dummett (2000), Parsons (2000) and Morreau (2002). 10 A similar route from supervenience to completeness is implicit in Lewis (1983), especially pp. 361–4. 11 Jackson (1994: 105) is arguing from an even weaker global supervenience claim to entailment.

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the true statement describing that situation in vague terms. Given strong supervenience, every situation of which Π is true is a situation satisfying the vague description Λ. More generally, every world in which Π is true is a world in which Λ is true. So, it is argued, Π entails Λ. This simply mimics the view given in the psychophysical case where Kim (1990: 12) claims that the strong supervenience of the psychological on the physical ‘supports in a straightforward way the assertion that the psychological character of a thing is entailed … by its physical nature’. If the vague description of a situation is entailed by the precise description, then the precise description can be said to be complete in the sense that it explicitly mentions all there is to say in describing the situation or entails all that there is to say. As we have already seen, however, any purported entailment is not a priori, the entailments are not conceptual and one cannot deduce the vague description from the precise one,12 and this brings us to the second claim mentioned above – that the availability of a complete precise description of any situation, or the world in general, is suf
cient for claiming the world to be precise. Is this plausible? In one sense of completeness one might think that it is. Were the vague description of a situation analytically entailed by the precise description on which it supervened, then there is a clear sense in which the description at the precise subvening level would be complete. And on the supposition that vague descriptions, in following by virtue of facts about meaning alone, carry no ontological commitment over and above that implicit in the subvening precise descriptions from which they follow, one is left with whatever ontology one accepts on the basis of purely precise descriptions of the world. This ontology, it might be argued, is purely precise. So, if the subvening description could be shown to be complete in this strong sense, then the phenomenon of semantic vagueness really would seem to be super
cial and of no consequence for metaphysics.13 However, weak completeness, identi
ed above, does not justify ‘dubious denials of existence’ – as Lewis (1983: 358) puts it. There is a sense then in which the strong supervenience of the vague on the precise justi
es the claim that a complete description of the world can be given in precise terms and, according to the foregoing argument, a sense in which the availability of a complete description in precise terms warrants our treating the phenomenon of semantic vagueness as irrelevant to questions of ontology. Nonetheless, the supervenience claim thereby supports ontological parsimony in favour of the precise only if there is no equivocation here in what one means when one speaks of ‘completeness’. And there obviously is. A description in precise terms is complete, at best, only in the weak sense of its
xing or entailing a posteriori the relevant vague description, yet it is only by means of completeness in the strong sense – the existence of the relevant analytic entailments – that the foregoing argument assured ontological parsimony. 12 Of course, one might seriously doubt that one can legitimately speak of ‘entailment’ at all under such conditions. 13 One further problem not addressed here is the fact that if the entailments were taken to uniquely privilege the ontology implicit in the subvening descriptions, then precise descriptions of a situation cannot strongly supervene on vague ones. This asymmetry is necessary to ensure that there be no parallel privileging of the ontology implicit in the vague descriptions. The arguments for asymmetry in the psychophysical case are discussed in Kim (1990: 13–17).

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Worse still, the assurance of ontological parsimony by means of strong completeness is questionable. The existence of analytic entailments from the precise description of a situation to the supervening vague description might indeed show that vague descriptions carry no ontological commitment over and above that implicit in precise descriptions; however, the existence of such entailments might merely serve to underscore the fact that precise descriptions carry an implicit commitment to vague entities from the outset. The view that the analytic consequences of a statement (or class of statements) introduce no new ontological commitment merely suggests that commitments owing from the statement (or class thereof) remain unchanged when we consider its logical closure under analytic entailment. This alone says nothing about the speci
c content of the commitment. Even if vague descriptions were analytic consequences of precise descriptions (thus warranting strong completeness claims), vagueness would have yet to be shown to be metaphysically inconsequential. For all that has been said so far, then, the strong supervenience of the vague on the precise is compatible with ontological vagueness. Whilst it is undoubtedly true, as Williamson (1994: 249) observes, that ‘the idea of vagueness in things themselves … repels [some], because it promises to forbid a complete description of all the facts in precise scienti
c words’, we can now see that the promise of a complete description in a suitably strong sense is empty anyway, and where redeemable in a weakened sense it is not incompatible with ontological vagueness. 3.4

The Precision of Scienti¿c Language

Underwriting much of the foregoing is an assumption that there exists an ideal language containing only precise terms by means of which the world is completely describable, and the paradigm of such a language is the language of science. This is an assumption that has reverberated throughout the preceding discussion. As in other elimination, reduction or supervenience debates where the language of science is frequently presumed to exhibit desired features (for example, it is non-moral, nonmental, extensional discourse), so too here. To the list of ‘virtues’ is added the further supposed virtue of precision. But is the language of science precise, and if not, could it ever be completely precise? An initial problem with the assumption of precision arises from Russell’s claim that all language is vague. This strong claim would deliver a negative verdict on the above question but, as we have seen, this claim cannot be sustained. Further doubts concerning the precision of scienti
c discourse might arise from the supposed vagueness of empirical terms, a view originally expressed by Benjamin (1939) and Burks (1946). They argue that all empirical language is necessarily vague, so, since science must have an empirical vocabulary, the language of science is vague. Must empirical vocabulary be vague and must it be considered part of the language of science? The claim that empirical vocabulary must be vague is premised on the idea that empirical vocabulary is ostensively de
ned and, since ostensively de
ned terms are necessarily vague, empirical terms are necessarily vague. Ostensively de
ned terms are necessarily vague since they are de
ned by reference to
nite positive and

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negative samples of objects which are taken respectively to instance the correct application of the term and to counter-instance its application, yet between the instances and counter-instances there are or could be cases which will count as borderline cases. For example, “red”, it is claimed, is taught ostensively by our being presented with some determinate instances and counter-instances as paradigms and our then applying the term to future cases by means of their resemblance to these paradigms. In this way borderline cases will arise since it may be indeterminate whether or not some particular case resembles any of the paradigm cases. Körner (1960) and Quine (1960) also argue for the vagueness of ostensively de
ned terms in this way. The indeterminacy of future reference makes for vagueness. But this view is no more sustainable than Russell’s, considered earlier. As Rolf (1981: 96) makes clear, this process of ostensive de
nition does not necessarily lead to vagueness.14 Consider a number-theoretic predicate “twiddle” ostensively de
ned by offering 1, 4, 9, 16, 25, 36 and 49 as twiddles (to which we could add the numbers 2, 5, 10, 15 and 82 as non-twiddles). ‘If the student is intelligent enough, he will guess that 64, 81 and 100 are also twiddles. … The guess may be right or wrong. What is important is that he guesses “at” a precise content which he may be unable to express.’ Guesses as to the correct application of the ostensively de
ned term to future cases go beyond the cases presented yet the guesses might ‘aim at perfectly precise contents’. Though many ostensively de
ned terms are vague this is not necessarily the case. This counterexample also puts paid to attempts by Wright to establish the vagueness of ostensively de
ned terms via the notion of “observationality”. Wright’s general argument is that ostensively de
ned terms are observational, observational terms are tolerant and terms that are tolerant are vague; thus ostensively de
ned terms are vague. (Recall that ‘tolerance’, perhaps the most important notion underlying the sorites paradox, is explained as follows. A term is tolerant if there is some degree of change in that to which the term applies, which is too small in respects relevant to the application of the term to make any difference to the application of the term. For example, the predicate “bald” is tolerant with respect to the loss of a single hair, “red” is tolerant with respect to a change of 1 Å in the light-reective property, and so on.15) Rolf concedes that “observational”, though notoriously dif
cult to pin down, can be given a reasonable interpretation so that any observational term is vague; however, it is then patently false that ostensive terms are observational – the above example of de
ning “twiddle” ostensively will be a counterexample. So not all ostensively de
ned vocabulary is vague. However, since some is, the way seems clear for Benjamin and Burks to argue that science must be infected with vagueness, since either it must make use of empirical terms some of which are vague or (using the Wright variation) it must make use of observational terms all of which are vague. Haack (1974: ch. 6), though not explicitly considering this move, seems to advocate a retreat to the quantitative, theoretical terms of mature science. Such a retreat seems necessary if a precise language of science is to be had since it seems unavoidable that the practice of science makes use of at least some vague 14 15

Haack (1974: ch. 6) argues for the same negative claim, but less convincingly. Wright (1975: 337ff).

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language in its experimental phase and during phases of conceptual innovation, whether this be innovation in mathematical concepts – whose language is considered as part of the language of science – or innovation in physical concepts. Thus, says Rolf (1981: 101–2): Platonistically speaking, conceptual innovation involves the invention of new predicates and names for newly discovered properties and entities. In this process, it is more important to classify the clear cut cases than to delimit the predicates of the classi
cational system so that they have no borderline cases. It would be a happy coincidence if such a process gave rise to precise predicates [and names]. Therefore, as long as discovery and conceptual innovation are among the practices of a science and as long as there is more for it to discover, precisi
cation will always lag behind.

This echoes the views of Benjamin (1939: 430) expressed more than six decades ago. ‘If the place of vague ideas in science is denied, the mystery of scienti
c discovery becomes more than ever a mystery. For discovery occurs only as a result of the
rm conviction that no scienti
c advancement is possible without an unwavering faith in the cognitive value of hunches, vague insights and intuitions.’ Thus we see an obvious route to attempt a defence of the claim that the world can be completely described in the precise language of science – exclude those parts of scienti
c discourse that are vague, while maintaining descriptive completeness. This narrowing of what we mean by “the language of science” presumably is not thought to threaten us with scepticism (a position one can always adopt to that degree required to maintain descriptive completeness – though obviously at some cost; see §1) since what is jettisoned is merely the preamble or ‘rough draft’ required for a full-blown scienti
c description of what is. However, this suggests that what is being offered in defence of the claim that there is a precise complete scienti
c description of the world is a promissory note to the effect that when science is complete there will be a precise complete scienti
c description of the world. It is hard to see any grounds on which this defence might rest other than sheer (questionbegging) faith in precision. A further line of attack on the claim that the language of science provides the means for completely describing the world in precise terms derives from quantum mechanics. Such an attack threatens even the weakest of defences that completed science will eventually provide a precise description of the world. The threat need not come from showing that claims to precision are false; it can be far more modest than that. It might merely aim to raise sceptical doubts concerning the existence of a precise scienti
c description from within a contested part of science itself – doubts, though, which advocates of the precision-claim must quash in order to substantiate their claims to precision. It can, in other words, merely amount to a shifting of the burden of proof back onto those advocating precision. The sceptical attack is, as I said, motivated by considerations in perhaps the most disputed current scienti
c theory – quantum mechanics (QM). The fact that the theory is so disputed legitimately plays into the hands of those sceptics challenging the claim that the language of science is precise. If any of the disputes surrounding QM involve the questioning of the assumption of precision, then the sceptics’ case would seem sound. Quine and other like-minded theorists could not then assume the

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language of science to be precise without begging the question against those positions which question this assumption. Some discussions of QM certainly include claims that suggest that the theory requires vague concepts in order to resolve perceived problems to do with the descriptive incompleteness or predictive incompleteness of the theory. For example, Teller (1979: 349–50) claims that in the case of continuous quantum-mechanical quantities quantum mechanics provides only descriptions with spread, not point-valued descriptions. … If quantum-mechanical systems have exact point values, the dispersed quantummechanical descriptions seriously misrepresent the nature of these systems. … Physicists sometimes deal with this situation by tacitly giving up the point-value theory and acquiescing in the orthodox theorist’s talk of imprecise or ‘imperfectly de
ned’ positions and momenta.

Lockwood (1989: 339) also says that ‘according to quantum mechanics, particles do not possess precise positions or momenta’. Similarly, Redhead (1987: 45) suggests that one response to problems attending measurement of observable quantities is that the quantity ‘has an unsharp or “fuzzy” value’. However, further reading through these discussions reveals just the sort of confusion between precision and exactness discussed in Chapter 1, §1.1. Teller equates the abandonment of ‘exact point values’ with the acceptance of ‘imprecise’ values, which themselves are associated with ‘descriptions with spread’. Redhead goes on to say that the terminology of unsharp or “fuzzy” values is prevalent in some elementary textbooks on QM. But what is really intended is that the observable quantity does not possess a value at all. What the QM system does in reality possess is a propensity or potentiality to produce various possible results on measurement [from some range of values], in respect of the observable quantity.

It is the dispersion or inexactness of values that makes for the absence of precise values. What is being discussed then is the idea that particles have position etc. that can at best be described to lie within some interval – that is, they take on inexact values with the potential to take on some exact value on measurement – though, of course, this interval itself is precisely describable. Notably, Quine himself feels no threat from QM, noting merely that ‘presumably regions are always wanted rather than single points – sometimes because of indeterminacy at the quantum level and sometimes for more obvious reasons’ (1977: 191). These disputes over quantum-mechanical indeterminacy do not have to do with indeterminacy in the sense relevant to discussions of vagueness.16 The frequent confusion between vagueness and inexactness has again surfaced. 16 Other discussions where “vague” is used to mean inexact or unspeci
c in interpreting QM indeterminacy include Glymour (1971), especially p. 746, and Krips (1987), especially ch. 1, §6. The latter is especially interesting. It explicitly recognizes and invokes standard discussions of vagueness in the philosophical literature yet still confuses the various types of indeterminacy involved.

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Nonetheless QM does appear to provide legitimate grounds for doubting the precision of scienti
c language. Firstly, the theory makes essential reference to an ‘observer’. Since this term is vague, the language of the theory cannot make do without some vagueness. Secondly, recent claims by Lowe also provide a convincing case for rejection of the supposed virtue of precision in the language of science. Lowe points to cases where a free electron a is captured by an atom to form a negative ion which later reverts to a neutral state, releasing electron b. ‘According to currently accepted quantum-mechanical principles there may simply be no objective fact of the matter as to whether or not a is identical to b’ (1994: 110). The pair 〈a,b〉 is, quite simply, a borderline case for identity and thus electron a’s spatiotemporal extension is vague. Following the distinction drawn by Alston between vagueness of application and vagueness of individuation, we can characterize this vagueness of the term “electron” by saying it exhibits vagueness of individuation – it may be vague on the score of exactly what parts of space-time to include in a speci
c electron’s extension. In particular, any speci
c electron entering into a superposed or entangled state would seem indeterminate in spatio-temporal extent. So “electron” exhibits vagueness of individuation. Of course, currently accepted theories might subsequently be abandoned; however, the foregoing is suf
cient to establish the sceptic’s point – there is reason for thinking that completed science may well be required to make use of vague language. Further evidence comes from the biological sciences. The foregoing discussion has concentrated on vagueness in the so-called ‘hard science’ of physics, but many important concepts in biology (e.g. “species”, “organism”, etc.), ecology (e.g. “biosphere”, “community”, etc.) and other so-called ‘softer’ sciences are vague. Of course, one might simply persist with the view that such sciences are second-grade sciences that supervene on physics, and insist that one’s commitment was only ever to the precision of the language of physics (not science per se). But we have already seen the problems that attend even this more exact thesis. The burden of proof is thus effectively pushed back onto advocates of the precision of completed science. In addition to the failure of supervenience to guarantee ontological precision, the obtaining of the supervenience relation itself is threatened for want of an identi
able subvening base. 3.5

Summary

In this chapter we have considered that response to the problem of vagueness according to which vagueness, though a semantic phenomenon, does not entail any revision of the classical metaphysical view that the world is not vague, nor does it necessitate any revision of classical logic or semantics. This view was supported by arguments that purported to show that the world was in fact completely describable in a precise ideal language or, at least, that our interests were best served by supposing this to be the case. The precision of the world then followed from the availability of a complete precise description thereof, and the superuous nature of vague language as a whole was taken to relieve any tension between the vagueness of natural language and classical logic. The retention of classical metaphysics, logic and semantics thus depended on the

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claim that there exists a complete precise ideal-language description of the world and, as we have seen, such a claim lacks adequate supporting evidence. To justify such a claim by eliminating vague terms is too costly to ordinary ways of talking. Any attempt to reduce vague terms is impossible. The supervenience of vague descriptions on precise ones would make available a complete description in precise terms but no subvening base language has been clearly identi
ed. And even if it were, the resulting supervenience claim would not seem to warrant completeness in a sense strong enough for the required ontological parsimony. When all is said and done, such an ideal language appears not to exist. It remains no closer than the celestial realm to which Russell banished it. Let us look then to natural language.

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Chapter 4

Going Non-classical: Gaps and Gluts In this chapter we turn to two post-Russellian responses to the problem of vagueness which eschew ideal-language approaches and instead propose logics of natural language encompassing vague terms. The ¿rst we shall consider is that account known as supervaluationism. This logic, what we might term a paracomplete logic, admits truth-value gaps to accommodate vagueness and has, as we shall see, an obvious paraconsistent analogue admitting truth-value gluts. The former is perhaps the most popular current response to problems posed by vagueness. The latter, developed in detail as a response to vagueness some two or three decades earlier, has much deeper roots. As we have seen, Russell thought that vagueness demonstrated the inapplicability of logic to natural language. Indeed, in the absence of nihilistic accounts like Unger’s, vagueness (considered as a semantic phenomenon, as we are supposing it to be) demonstrates the inapplicability of classical two-valued logic to natural language but Russell, rather than revise classical two-valued logic or natural language, simply restricted the scope of logic to an ideal precise language that was unobtainable in principle and located in some ‘celestial realm’. Russell’s view of logic as utterly divorced from natural-language analysis has, as already noted, been viewed with suspicion in more recent times and two distinct responses have predominated: (i) retain classical two-valued logic, both classical logic and its semantics, and regiment natural language to obtain an ideal language that is constructible in principle – that is, revise the data; or (ii) accept natural language as being in order as it is and modify classical two-valued logic to enable the evaluation of arguments involving vague language – that is, revise classical semantics and/or classical logic. The ¿rst of these responses has just been described in Chapter 3. It is to the second more enlightened response that we now turn. 4.1

Supervaluationism

Let us turn ¿rstly to the explication and analysis of that revision of classical twovalued logic advocated by supervaluationists in response to vagueness, the logic SpV. An informal account appears to have ¿rst been proposed in Mehlberg (1958). Mehlberg was a former student of the Lov-Warsaw School of philosophy, itself a well-known centre of logical innovation and origin of a number of non-classical logics – one of them, as we shall see in §2, being an alternative logic of vagueness proposed a decade earlier by Jaśkowski. Despite its early advocacy by Mehlberg, supervaluationism as applied to the phenomenon of vagueness is generally considered a reinterpretation of the ‘presuppositional languages’ of van Fraassen, formally described in his (1966).

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Describing the supervaluation theory

So what is the theory of supervaluations and how does it address the problems that arise in the context of vagueness? As a theory of vagueness in particular, it was developed in Przelecki (1969), Lewis (1970), Dummett (1975), Fine (1975) and Kamp (1975). Notable recent advocates include Keefe (2000), and Varzi (2001). The theory begins from the assumption that a typical natural-language predicate F has objects that are clearly or determinately in its extension (the set of objects of which F is determinately true) and objects that are clearly or determinately in its anti-extension (the set of objects of which F is determinately false). These extensions are exclusive, reÀecting the supposition that no natural-language sentence is both determinately true and determinately false. Moreover, every precise predicate is such that every object to which it could signi¿cantly be said to apply is either determinately in the extension or determinately in the anti-extension of the predicate; that is, every precise predicate is such that its determinate extension and anti-extension are exhaustive. Hence every precise sentence in natural language is considered either determinately true or determinately false. So, if we are concerned only with that fragment of natural language which is precise, then determinate predicate extensions, and determinate truth and falsity, are exclusive and exhaustive. However, vague predicates are such that their determinate extension and antiextension are not exhaustive; there are objects which are in neither. And thus, assuming a full complement of names in the language, there are sentences for which the possibility arises of their being neither determinately true nor determinately false. The supervaluationist then equates determinate truth with truth simpliciter, or ‘supertruth’, thus de¿ning a concept of truth for which bivalence fails and in terms of which we may describe borderline cases as giving rise to truth-value gaps. On this view, where a is a borderline case of F, the indeterminacy of Fa amounts to its being neither true nor false.1 Note that on this account of truth the semantics for the precise fragment of natural language is decidedly classical – truth and falsity are considered exclusive and exhaustive. This assumption is not essential. A supervaluational model structure could equally well be built upon an underlying semantics that was non-classical, e.g. intuitionist, relevant, etc. In this sense a supervaluationist approach merely aims to provide a non-bivalent semantic superstructure sensitive to vagueness which collapses to one’s preferred underlying semantics where vagueness does not arise. However, since it is traditionally a development of a non-classical semantics from a classical base, and this tradition has circumscribed the ensuing issues, problems and debate, supervaluationism as it is discussed and debated is now synonymous with this classically oriented theory – classical supervaluationism. We shall continue in this vein and take classical supervaluationism as our object of focus, referring to it simply as supervaluationism. 1 Problems associated with its denial are discussed by Williamson (1994: ch. 7, §2). Reminiscent of Geach’s argument against multi-valued logics (1972: 80–81) and similar to Haack’s defence of bivalence (1974: 66–8), it is argued that bivalence must hold for any truthpredicate satisfying the T-schema, on pain of contradiction. Fine acknowledges the tension, and rejects the T-schema for the supervaluationist conception of truth (1975: 296ff). Keefe follows suit and defends the stance at some length (2000: ch. 8, §3).

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The supervaluationist semantics for natural language including vagueness then deviates initially from classical semantics to the extent that some sentences are neither true nor false. The non-bivalent logic SpV is thus an example of what I shall term an incomplete logic. Let us say that a logic is complete if and only if, for any valuation or model of any contradictory pair of sentences A and ~A, one or the other must be true in the valuation or model. I.e. = A, ~A.2 Classical logic is a paradigm of a complete logic. A logic will count as incomplete just in case it is not complete – i.e. for some sentence A, neither it nor its negation need be evaluated as true. I.e. ≠ A, ~A. Non-bivalent approaches to the problem of vagueness like SpV that postulate truth-value gaps exemplify logics which are incomplete since, given the shared assumption that a sentence is false if and only if its negation is true, to admit sentences which are neither true nor false is to admit sentences which fail to be true while their negations also fail to be true. Thus: ≠SpV

A, ~A.

Now there is obviously a trivial sense in which a logic might be incomplete – namely, if whenever a sentence and its negation fail to be true every sentence fails to be true. This is not the sense of incompleteness to which truth-value-gap approaches in general, and SpV in particular, are committed. Such approaches countenance quarantined gaps by rejecting the spread-principle – B = A, ~A – according to which if there are truth-value gaps anywhere, then they are everywhere. That is, they accept that some sentence B can be true whilst not every sentence or its negation is. Gaps do not implode everywhere – the logic is non-implosive. So in addition to incompleteness there is also a commitment to the non-triviality of the incompleteness. I.e. B ≠ A, ~A. Let us say that a logic which is non-trivially incomplete is paracomplete. The cornerstone of mainstream responses to the logical and semantic problems posed by vagueness, and SpV in particular, amounts to the view that vagueness necessitates a paracomplete response at worst. Thus: B ≠SpV A, ~A. Given this non-classical constraint on the semantics of vague expressions, we may wonder to what extent classical logic remains intact. For instance, if a sentence A and its negation ~A are indeterminate, what of their conjunction and disjunction? Are they likewise indeterminate? Are the classical laws of excluded middle and noncontradiction still theorems? Do inferences like modus ponens or proof by cases remain valid in such semantics? What supervaluationists aim to do in developing a logic of vagueness is to admit truth-value gaps while respecting what they describe as ‘penumbral connections’. To paraphrase Fine (1975: 269ff), suppose that a certain blob is a borderline case of “red” and let S be the sentence “the blob is red”. Though we may agree that S is 2 “=” represents the generalized, multiple-conclusion consequence relation. Given a set of sentences Σ (the multiple-premise set) and a set Γ (the multiple-conclusion set), we shall say that Σ = Γ if and only if whenever all the members of Σ are true then some member of Γ is true. (For more on multiple-conclusion consequence see Shoesmith and Smiley 1978.)

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indeterminate, as is its negation, ~S , nonetheless their conjunction should count as false since they are contradictories. The boundary of the one shifts, as it were, with the boundary of the other. (The sentence S & S, on the other hand, will count as indeterminate since it is taken to be equivalent to S, so since a conjunction with indeterminate conjuncts is sometimes indeterminate and sometimes not, “&” will not be truth-functional on this account.) Similarly, since S and ~S are complementary over the given colour range, their disjunction, S V ~S, is true (so “V ” will not be truth-functional either). Penumbral connection is the possibility that just such logical relations hold among indeterminate sentences. The supervaluationist’s claim then is that penumbral truths must be respected (and, as a consequence, some non-truth-functional approach must be sought). Supervaluationists insist then that classical theorems, in so far as they reÀect these supposed penumbral connections, must be respected despite the failure of bivalence in accommodating vagueness. It is this which makes classical supervaluationism – supervaluationism which extends on classical semantics – especially interesting. It sets itself the goal of developing a semantics which, despite being non-bivalent, nonetheless retains classical laws which might appear threatened by failures of bivalence, e.g. the laws of excluded middle and non-contradiction. This conservative goal is especially challenging in a way that it would not be if the underlying laws being retained were not threatened by non-bivalence. Supervaluationists thus require a paracomplete logic of vagueness, yet aim to retain the laws of classical logic, among them the law of excluded middle (LEM). Consequently, the paracomplete logic that is sought differs crucially from other paracomplete logics that have been proposed to deal with vagueness, for example Łukasiewicz’s three-valued logic or Kleene’s popular strong three-valued logic, where LEM fails. The logic that is sought is what, following Arruda (1989), we may describe as a weakly paracomplete logic. That is to say, though it admits of nontrivial incomplete valuations which do not make true either A or ~A (i.e. B ≠ A, ~A), nonetheless all such non-trivial valuations make true A V ~A. (By contrast, strongly paracomplete logics do not distinguish between the non-truth of A V ~A and the non-truth of both disjuncts, and consequently admit of non-trivial valuations that do not make A V ~A true despite making something true.) Thus: B ≠SpV A V ~A. Given these constraints, then, the challenge is to construct a semantics for vague statements which would generate such a weakly paracomplete logic. The supervaluationist model of vagueness attempts to deliver just such a semantics. Interestingly, the Russellian conception of vagueness as ‘one–manyness of denotation’ offers a way into just such a model (though historically Russell’s theory seemed to play no part in the development of supervaluationism). As we have seen, Russell de¿ned a denoting phrase to be vague if and only if its denotation relation was not one–one but one-many in the sense that there are at least two possible candidates for denotation consistent with the truths involving that linguistic item. Any referent capable of serving as the denotation of the linguistic item in question must be sharp, yet the presence of borderline cases makes it essentially indeterminate what this referent is. Take the predicate “red”, for example, and

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suppose there to be only three possible objects a1, a2 and a3, where a1 is determinately red, a3 is determinately not red and a2 is the predicate’s sole resilient borderline case. Preservation of the truths involving a1 and a3 requires that any (sharp) property in the denotation range of the predicate must have a (sharp) extension that includes a1, excludes a3, and might include a2 or might exclude a2; it must either include a2 or exclude a2 (since the world is sharp, a2 is either in the extension of the property denoted by the predicate or it isn’t), yet it is indeterminate which. There are therefore two possible candidates for denotation – the set {a1, a2} or the set {a1} – consistent with a1’s being red and a3’s being not red. Any possible referent for the predicate is sharp, but the presence of a resilient borderline case makes for two distinct possibilities. By associating each of the two respective possible denotations with linguistic items – precise predicates “red1” (with satis¿er a1) and “red2” (with satis¿ers a1 and a2) – we can shift talk of vagueness in terms of denotation relations to intra-linguistic talk of a relation between vague and precise predicates. We can then say that the vagueness of the predicate “red” amounts to there being two ways of making the predicate precise, consistent with a1’s being determinately red and a3’s being determinately not red. Let us call this ‘making precise subject to the consistency constraints’ admissible precisifying. Then an admissible precisi¿cation of some vague predicate F is a precise predicate Fi such that any determinate satis¿er for F is a satis¿er for Fi and any determinate satis¿er for not-F is a satis¿er for not-Fi. The vagueness of the predicate “red” now amounts to there being two admissible precisi¿cations, “red1” and “red2”. Moreover, the determinateness of a1’s being red now corresponds to its being both red1 and red2, the determinateness of a3’s being not red corresponds to its being both not red1 and not red2, whilst a1’s being a resilient borderline case corresponds to its being red2 but not red1. Generalizing to any predicate, an object a is determinately (not) in the extension of predicate F if and only if a is in the (anti-)extension of every admissible precisi¿cation of F; an object a is a resilient borderline case for a predicate F if and only if a is in the extension of some admissible precisi¿cation and in the anti-extension of some admissible precisi¿cation. So a predicate F is vague with resilient borderline case a if and only if there are (at least) two admissible precisi¿cations F1 and F2 such that a satis¿es one but not the other. Similar accounts can be given for the vagueness of n-place relations generally and names. Analogously, an admissible precisification of an atomic sentence S is a constrained way of making all the constituent expressions of S precise; it is another atomic sentence which is precise (is either true or false), subject to the constraint that it is true (false) in all those circumstances in which S is true (false). An atomic sentence then is true (false) if and only if it is true (false) for all admissible precisi¿cations; it is indeterminate in truth-value if and only if it has (at least) two admissible precisi¿cations, one of which is true and one of which is false. The semantic value of an atomic sentence is thus given by ‘supervaluating’ over all admissible valuations which, in turn, are given by valuations on admissible precisi¿cations. For example, the sentence “The blob is red” may be vague and so neither true nor false because there are two admissible precisi¿cations of the sentence “The blob is red1” and “The blob is red2” (assuming the subject term to be precise), one of which

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is true and the other false. We could draw a sharp line between the red and non-red in various ways, thereby making the sentence precise in ways that make for a difference in truth-value. For many atomic sentences, of course, how we draw the line will have no effect on the truth-value. For example, any determinately red blob will, by virtue of the constraint on what counts as an admissible precisi¿cation, count as red no matter how one precisi¿es. Various admissible precisi¿cations of an atomic sentence will only differ on truth-value ascriptions if the sentence ascribes a predicate to its borderline case. Thus we can rewrite Russell’s account of vagueness in terms of a relation between a linguistic item and its admissible precisi¿cations. Yet nothing we have said so far suf¿ces to justify or explain the preservation of classical laws and classically valid inferences. Extending the account of vagueness of atomic sentences to complex ones involving logical constants in the usual truth-functional manner will threaten classical laws. However, there is another alternative suggested by the above account of vagueness for atomic sentences which would permit the retention of classical laws. Rather than evaluating atomic sentences by means of their admissible precisi¿cations and then extending to all sentences of the language via recursive clauses for the logical constants, one might evaluate all sentences in the same way, regardless of their logical complexity. So any sentence of the language is true (false) if and only if it is true (false) for all admissible precisi¿cations and indeterminate if and only if there are (at least) two admissible precisi¿cations thereof, one of which is true and the other false. This is the crucial ‘insight’ of supervaluationist semantics. The truth-value of a sentence, simple or complex, is given by supervaluating over all admissible (classical) valuations. Consider a particular instance of the law of excluded middle – say, “The blob is red or the blob is not red”. Regardless of how one precisi¿es the constituent expressions, the precisi¿cation of “The blob is red” comes out true or false, so it or its negation is true. So, assuming the truth-conditions of complexes in an admissible precisi¿cation are just the classical truth-conditions (that is, where no vagueness is present, classical logic applies by default), then the disjunction of “The blob is red” with its negation must come out true in each and every admissible precisi¿cation. In this way, contra the above reservations, all instances of the law turn out to be (determinately) true. If we now de¿ne SpV consequence in the obvious way, we shall say Σ =SpV Γ if and only if whenever all the members of Σ are (determinately) true then some member of Γ is (determinately) true. Such an account of consequence will indeed establish all classical (CL) theorems as theorems of SpV, as desired, since in the special case where Σ = ∅ and Γ is a singleton set, it is easily shown that: (I)

=SpV

A if and only if =CL A.3

This con¿rms, for example, that though the principle of bivalence is rejected (i.e. despite the failure of the logical theory to be complete), the law of excluded middle remains valid. Thus: 3

See Williamson (1994: ch. 5, §3).

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LEM: =SpV A V ~A.4 Where a restriction is placed on the consequence relation to the effect that Γ be a singleton set, the multiple-conclusion consequence relation narrows to the more commonly studied single-conclusion consequence relation as de¿ned in Fine (1975). This relation is co-extensive with classical consequence and admits as valid all and only those inferences that are classically valid.5 Thus: (II) Σ =SpV A if and only if Σ =CL A. It is not hard to show then that rules like modus ponens, contraposition, conditional proof, proof by cases and reductio are all SpV-valid. Notably, as we shall see, all these principles fail in a language extended to include a determinacy operator, “D”. But even in the unextended language currently under consideration, cracks are already evident in the avowedly conservative logical veneer of SpV. Multipleconclusion consequence already deviates from its classical counterpart. As a result of being paracomplete but only weakly so, SpV fails subjunction: A V B ≠SpV A, B.6 This non-classical feature of SpV manifests the non-truth-functional account of disjunction required to underwrite a weakly paracomplete theory of vagueness and, as we shall see shortly, is a major source of concern when evaluating the system’s adequacy. How can it make sense to deny the truth of two unacceptable claims considered separately while accepting them jointly by accepting their disjunction? The anomaly is evidence of the fact that SpV’s preservation of classical singleconclusion consequence incurs a correlative cost at the level of classical multipleconclusion consequence more generally. Classical multiple-conclusion consequence is preserved in SpV only in the following restricted sense: (III) Σ =SpV A1 V A2 V … V An 4

if and only if Σ =CL A1, A2, …, An.

Note that Russell does say that the ‘law of excluded middle’ fails in the presence of vagueness (1923: 85–6); however, this is best understood as a rejection of the Principle of Bivalence. Russell frequently blurred the distinction between them. 5 This de¿nition characterizes what Williamson identi¿es as the property of global validity. (See Williamson 1994: 147–8.) It is the account of supervaluation-validity endorsed by Fine. (See Fine 1975: 283–4.) An alternative property, local validity (according to which an argument is SpV-valid just in case, in all admissible precisi¿cations, whenever the premises are true the conclusion is true) is endorsed by Dummett. (See Dummett 1975: 311.) Problems attend the local account (see Williamson 1994) and I shall assume the global account throughout what follows. As it happens, though the local account of validity is strictly stronger than the latter, both accounts validate all and only those single-conclusion inferences validated by the classical account of consequence. (See Williamson, 1994 and Keefe 2000: 174ff.) 6 Since, were subjunction to hold, substituting ~A for B, LEM would mandate completeness (i.e. one of A and ~A must be true).

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Thus we see how Russell’s account of vagueness at the simple or atomic level seems able to be generalized. The resulting weakly paracomplete logic SpV allows for truth-value gaps while respecting what supervaluationists claim are ‘penumbral connections’.7 4.1.2

Defending supervaluationism

The conservatism of the supervaluationist approach expressed through (I) and (II) above is frequently cited as a major virtue. Retaining classical logic to that extent while accommodating vagueness would be a signi¿cant achievement if, indeed, it is achievable.8 There are, however, problems with such an approach, problems which have been addressed in two quite distinct ways. One popular defence proceeds by recourse to representationalism; i.e. it appeals to a substantive philosophical thesis about the source of vagueness according to which it is merely semantic and in no way ontologically grounded. Supervaluationism is frequently associated with such a representationalist view of vagueness, and the defence makes this explicit. Remarks by Dummett seem to speak in favour of this understanding of the SpV account. ‘[I]f we suppose that all vagueness has its source in the vagueness of certain primitive predicates, relational expressions and quanti¿ers, we may stipulate that a statement, atomic or complex, will be de¿nitely true just in case it is true under every sharpening of the vague expressions of these kinds which it contains’ (1975: 311). And so, with the retention of an account of validity in terms of truth-preservation, classical laws and (singleconclusion) inferences are preserved. As we shall see shortly when we come to evaluate speci¿c consequences of the SpV account, Fine (1975) and Varzi (2001) seem similarly motivated. An alternative defence, articulated in Keefe (2000), dissociates the logical account of vagueness offered by supervaluationism from metaphysical issues and opts instead for an explicitly pragmatic defence. Irrespective of how the theory is defended, two sources of tension need to be defused. Tension arises on the one hand between the general phenomenon of vagueness and classical laws, and, on the other hand, between the sorites paradox and classical (single-conclusion) inference. Doesn’t the presence of vague language within the scope of logic threaten the validity of some classical laws, e.g. the law of excluded middle? Moreover, doesn’t the sorites paradox cast doubt over the validity of some classical inferences involved?9 7 Fine’s conditions of stability, ¿delity, completability and resolution (1975: 268–79), and further elaborated on in Keefe (2000: ch. 7, §2), are all implicitly assumed in the foregoing discussion. 8 More generally, retaining one’s preferred underlying logic – classical or otherwise – to that extent in the face of vagueness would be a virtue if achievable. (Recall that we are considering what I earlier termed classical supervaluationism, thus the default logic is assumed to be classical.) 9 Of course, as will become clear, the two sources can be traced to an (untenable) conservatism with respect to the central notion of logical consequence.

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In defence of sorites-validity What of the latter question: can a logic of vagueness reasonably be supposed to preserve the inferences employed in sorites reasoning? If so, how is the paradox resolved? By way of example, consider the following version: A man with 1 hair on his head is bald. For any n, if a man with n hairs on his head is bald, then a man with n + 1 hairs on his head is bald. ∴

A man with a million hairs on his head is bald.

The conclusion is deemed unacceptable yet the reasoning is valid by classical lights. By (II) above then the SpV theorist accepts the argument as valid, but deems it unsound. Despite the prima facie truth of the premises, supervaluationists deny the truth of the second, universally quanti¿ed conditional premise. Of course, the mere non-truth of a premise does not entail its falsity given the failure of bivalence; nonetheless for the SpV theorist the paradox in this form does indeed have a false premise. The universally quanti¿ed conditional premise is false. As a consequence it is true (simpliciter) that there is some n for which a man with n hairs on his head is bald whilst a man with n + 1 hairs on his head is not. This is so since for every admissible precisi¿cation of the vague predicate “bald” it is true that there is some such falsifying n, it is therefore determinately true that there is some such n and so true (simpliciter).10 The truth of there being some such falsifying n however is not to be confused with another, very similar claim which in SpV is nonetheless quite distinct: namely, that there is some n of which it is true that it is the falsifying n; i.e. there is some n for which it is true that a man with n hairs on his head is bald and a man with n + 1 hairs on his head is not bald. The truth of there being a hair-splitting n no more entails there being an n of which it is true that it is hair-splitting than the truth of A V ~A entails the truth of A or the truth of ~A. There is no n which is the hair-splitter in every precisi¿cation (though, as we have just seen, in every precisi¿cation there is a hair-splitting n). The distinguished n shifts relative to distinct admissible precisi¿cations – a phenomenon referred to by Fine as ‘the truth-value shift’. In response, there are two points worth taking up here. Firstly, in claiming the major premise is false, SpV might seem committed to the precision of the predicate “bald”. Were this charge sustainable, classical consequence would be retained in the face of the sorites by denying the existence of the very phenomenon being modelled; classical consequence would be retained only by ignoring that which was thought to be a threat. In claiming the premise is false the SpV account accepts the existence of a hair-splitting n, and this seems to imply the semantic precision of “bald”. Dummett, while advocating the SpV approach to vagueness, speaks for many when he acknowledges that ‘[the SpV] solution may, for the time being, allay our anxiety over identifying the source of the paradox. It is, however, gained at the cost of not really taking vague predicates seriously, as if they were vague only because we had 10 SpV must, of course, be presumed to generalize in some way to a solution to all the sorites forms to count as adequate to the task of modelling vagueness.

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not troubled to make them precise’ (1975: 312). It is simply that the semantic decision as to which is the correct precisi¿cation remains unmade. Fine’s view (echoed subsequently by Keefe) is that one need not, indeed should not, take vagueness ‘seriously’ in the sense of denying the existence of a hairsplitting n, but that one should deny the implication from its existence to the precision of the predicate “bald”.11 Fine does not elaborate other than to suggest that due attention to the ‘truth-value shift’ falsi¿es any such implication. What he appears to have in mind and what Keefe explicitly suggests is that precision of the predicate would only follow if one could be in a position to say which n is the hairsplitter, yet SpV, while asserting that there is some hair-splitting n, nonetheless refrains from asserting of any n that it is the hair-splitting case. Thus the admittedly untoward implication is avoided by assuming that precision follows from the existence of a cut-off point only if the existence claim is one for which an instance could be given, yet the sense of “there is” employed by the SpV theorist when they claim that there is a cut-off point is not the sense in which it is reasonable to ask what the cut-off point is. On this analysis, objectors baulk at claiming the quanti¿ed conditional premise false since they mistakenly take it to imply the precision of the predicate in question. SpV, in blocking the implication, undermines the force of the objection. But even if the diagnosis to this point were accepted, rejecting the implication by appealing to the requisite sense of “there is” incurs considerable additional cost. The use of “there is” in this sense has led many to claim that it is not used in its usual sense in SpV.12 This is the second point of dispute over the SpV solution to the sorites. The described defence bears closer scrutiny. Consider the two similar claims whose difference nonetheless underwrites the SpV response: (1) It is true that there is some n such that a man with n hairs on his head is bald whilst a man with n + 1 hairs on his head is not bald. i.e. T“∃n(B(n) & ~B(n + 1))”. (2) For some n, it is true that a man with n hairs on his head is bald whilst a man with n + 1 hairs on his head is not bald. i.e. ∃nT“(B(n) & ~B(n + 1))”. In an attempt to chart a course between the acceptance of paradox and the rejection of vagueness, it is recommended that one accept (1) while rejecting (2). Acceptance of (1) is considered reasonable (vagueness is not so ‘serious’ as to warrant its rejection) and does not preclude the vagueness of the relevant predicate, whereas acceptance of (2) would do so for it would amount to a rejection of the truth-value shift, yet one should reject (2). 11

Keefe cites this as ‘one of the least appealing aspects of the theory’ (2000: 183–6). See, for example, Rolf (1984: 232), Sainsbury (1995a), Williamson (1994: 153–4). Note that the unusual behaviour of the existential quanti¿er is mimicked by the universal quanti¿er. In SpV the falsity of a universally quanti¿ed statement (e.g. the major premise of a sorites paradox of the form currently under discussion) does not entail the existence of a false instance. 12

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In response to this counterintuitive recommendation, Varzi (2001) makes explicit one line of defence that remarks of Fine’s gesture at, namely a justi¿cation by appeal to an associated metaphysical thesis taken to underlie the formal theory of SpV. Vagueness is a real enough phenomenon to warrant logical reform but is not serious enough to be ontologically grounded – it is purely representational. Bringing to mind Russell’s account of vagueness as representational, the suggestion appears to be that the vague predicate “bald” vaguely refers to some precise property baldness, though exactly which property this is is an indeterminate matter, there being a plurality of admissible candidates. On this view it is true to say of baldness that it has some cut-off point, and this is all that (1) asserts. A commitment to the precision of the predicate “bald”, on the other hand, would involve a commitment to there being a unique point of which we can truly say that it is the cut-off point to baldness. Since the predicate is not precise, no such cut-off point from the range of candidates admitted by the vague predicate can be truly identi¿ed, and since (2) claims otherwise it is therefore to be rejected. On this understanding of SpV, Fine’s ‘truth-value shift’ is simply a reÀection of the boundary shift among the various admissible denotations of the imprecise designator “bald”. So supervaluationists advocating SpV as a logic of representational vagueness might, as Varzi does, claim to justi¿ably accept (1) on the grounds of ontological precision, but add that this does not imply that the predicate “bald” is precise unless one accepts the implication from (1) to (2) – that is, unless one thinks that claiming the existence of a cut-off point implies one can specify precisely what this cut-off point is. SpV understood in this way, it is said, correctly rejects the implication. Surprising though it may seem, truth does not distribute over “∃”. Yet, as we have seen above, objectors will then query the account of existential quanti¿cation provided by SpV – this was the second problem pointed to earlier in reference to the SpV solution to the sorites paradox. Since SpV now interpreted as a logic of representational vagueness claims that the consequences of such a philosophical approach to vagueness mandate the acceptance of (1) despite the rejection of (2), the underlying philosophical theory will now incur the ensuing burden of the non-standard analysis of existential quanti¿cation. If the defence given by Varzi is plausible, then any representationalist should indeed accept (1) while rejecting (2), but must also consequently admit to the non-standard behaviour of existential quanti¿cation. Does this speak in favour of accepting the counterintuitive analysis of existential quanti¿cation? Committed representationalists will presumably think so, but others might consider the cost more than the underlying theory can bear. The cost–bene¿t analysis here is dif¿cult, if not impossible, to adjudicate. Note however that the defence (even assuming it plausible) depends crucially on a commitment to representationalism. What arguments are offered in favour of this central commitment, and are they convincing? This is an issue we will return to in the next chapter, where the central argument, the Evans Argument, will be scrutinized and found wanting – representationalism should be rejected. Until then we need but note that SpV as defended on the above grounds depends on the plausibility of a substantive theory of vagueness. Keefe opts for a quite different response. Rather than any metaphysical defence, the choice of theory is made on pragmatic grounds. ‘I advocate the indirect argument

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that we should accept the phenomenon because of its role in an altogether successful theory of vagueness’ (2000: 182). Agreeing that (2) precludes the vagueness of the predicate involved, it is indeed to be rejected. Nonetheless (1) is true, thus undermining the soundness of the sorites paradox above. This acceptance of (1) is admittedly counterintuitive; however, the costs are said to be offset by the overall bene¿ts of the SpV theory – ‘any costs [that accrue by virtue of the acceptance of (1)] are easily worth paying given the advantages of the theory’ (2000: 183). Attempts to mitigate the costs, though, are seriously Àawed. Firstly, claims that the major premise of the paradox would not always be assented to, even if plausible, hardly serve to show the plausibility of the truth of its negation, i.e. the truth of (1). Of course, they would, if successful, serve to show that those who object to (1) on the grounds that the major premise is true are on shaky ground; the truth of the premise is not beyond question. But in the absence of the Principle of Bivalence one might agree with Keefe on the non-truth of the major premise and nonetheless object to its being counted false, as (1) demands. Arguably, the most compelling (and frequently cited) reason for objecting to (1) is that already discussed earlier – namely, its apparently untoward consequence in the form of (2). On this point Keefe seeks to redress the balance in favour of (1) by pointing to the fact that in SpV the untoward consequence simply doesn’t follow. ‘The rest of my defence turns on the fact that supervaluationism can distinguish between [(1) and (2)]’ (2000: 184). But this is not itself a non-question-begging defence unless the distinction, whose maintenance depends on the failure of the inference from (1) to (2), is itself independently plausible. And it is not, at least not without further argument, for as we have seen it requires an interpretation of existential quanti¿cation at odds with our ordinary understanding. The costs of accepting (1) are not mitigated, but rather, the attempted mitigation simply shifts the costs of accepting (1) onto the semantic analysis of quanti¿cation. Not surprisingly, then, the apparent anomaly surrounding quanti¿cation is one which supervaluationists are keen to dispel. They sometimes appeal to facts about the behaviour of the quanti¿ers in other contexts and Keefe is no different in this respect. The change of scope involved in the inference from (1) to (2) is compared to syntactically similar scope changes involving the existential quanti¿er. The generally agreed unacceptability of the latter is then offered as evidence for the (required) unacceptability of the former. For example, most, if not all, people would agree that its being true that someone ought to perform action X does not entail there being someone of whom it is true that they ought to perform action X. Similarly, it seems plain that its being true that some seat has been promised to me on my booked Àight to destination Y does not entail there being any seat of which it is true that it has been promised to me. In each case, the change in scope of the quanti¿er is deemed illicit. Obligation and promising do not distribute over “there is”. Similarly, claims the supervaluationist, it is to be expected that truth does not distribute over “there is” and objections to the theory based on an assumption to the contrary are misguided. The claims to similarity, however, do not obtain in the relevant sense. To be sure, there is a syntactic similarity to all three cases considered. But this is irrelevant. The claim underlying the objection is not that truth ought to distribute over the quanti¿er, since any operator ought to do so. Clearly some operators do not; examples to do

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with obligation and promising are cases in point. The claim is that truth ought to distribute by virtue of what “there is” means. Given our ordinary understanding of the meaning of existential quanti¿cation, an existential claim is true if and only if it has some true instance. Truth and the existential quanti¿er interact in this way given a proper semantic understanding of the quanti¿er. There is no analogous reason for thinking that operators representing obligation and promising should behave similarly; moreover their not doing so reveals nothing salient concerning the meaning of “there is” and so nothing salient concerning the illegitimacy of the change of scope involved in the inference from (1) to (2). We are left with no reason for rejecting the strong inclination that the inference is valid and consequently SpV is strongly counterintuitive in this regard. In defence of classical laws Analogous problems confront the SpV account of disjunction. This brings us back to the former source of tension identi¿ed earlier – the tension between vagueness and classical laws. Isn’t it the case, as many seem to suppose, that the presence of vague language within the scope of logic threatens the validity of some classical laws, e.g. the law of excluded middle? Not according to SpV, and nor should it according to supervaluationists. When the objector argues that, for vague A, A V ~A fails because neither A nor ~A – e.g. “Tim is tall or Tim is not tall” fails because Tim is neither tall nor not tall – the SpV theorist responds by admitting that neither A nor its negation are (super)true, and so were “V ” truth-functional, A V ~A would fail to be (super)true but, just as with “∃”, truth does not distribute over “V ”, i.e. the connective is not truth-functional and, as noted earlier, subjunction fails. Though the SpV theorist accepts failures of bivalence, this would only entail the failure of LEM were disjunction to be truthfunctional, which it is not. On this view the objection, like that which arose in response to the SpV solution to the sorites above, is based on the mistaken supposition that acceptance of LEM commits one to semantic precision (if everything’s red or it isn’t then “red” is precise); however, one should not confuse LEM with the Principle of Bivalence. One should not confuse the claim that A V ~ A is (super)true with the claim that A is (super)true or ~A is (super)true. SpV accepts the former while denying the latter. That is to say, for vague A, the SpV theorist accepts: (3) T“A V ~A” yet denies: (4)

T“A” or T“~A”.

The latter is denied by virtue of the vagueness of A (just as (2) was denied by virtue of the vagueness of the predicate B) yet (3) is accepted (just as (1) was).13 As with (1), it is the acceptance of (3) that gives rise to misgivings and leads to doubts concerning the adequacy of SpV. Unlike (1), though, where the 13

Just as earlier concerns centring on the existential quanti¿er extend to the universal quanti¿er, so too here claims centring on disjunction extend to conjunction in the obvious way. Despite the non-falsehood of both A and ~A, their conjunction is nonetheless always false, thus guaranteeing the validity of the law of non-contradiction.

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supervaluationist simply bit the bullet and sought to minimize the damage done by undermining the supposedly untoward consequences, (3) has been defended not as a cost worth paying but, rather, as a claim that all theories should embrace. (3) is straightforwardly true and follows from the logical truth of excluded-middle claims which themselves can be seen to be mandatory despite some people’s intuition to the contrary. In the case of (3), objectors are quite simply wrong. Costs attend its rejection, not its retention. This is because (3) reÀects what supervaluationists refer to as ‘penumbral connections’. Of course, to defer to talk of ‘penumbral connections’ as a means of defending a commitment to (3) is only successful to the extent that the existence of penumbral connections is independently defensible. Obviously, then, to defend claims for the existence of penumbral connections it would simply beg the question to cite the need to retain classical laws. That is the very issue in question here, so argument independent of classical laws must be presented. And indeed it is. Keefe cites Edgington’s arguments for the non-truth-functionality of disjunction as evidence of such ‘penumbral connections’ (though Edgington is intent on developing an alternative, non-truth-functional logic to SpV). Edgington presents a number of arguments for non-truth-functional disjunction. Initially, “Sibling” means the same as “brother or sister”. There are sex changes; and they are not instantaneous. Therefore, at times, while someone is de¿nitely a sibling, it is indeterminate whether they are a brother, or a sister. Therefore, someone can be de¿nitely a brother or a sister, without being de¿nitely a brother, or de¿nitely a sister: a disjunction can be de¿nitely true without either disjunct being de¿nitely true. (1997: 310)

Problems attend the ¿rst assertion, though. “Sibling” does not in fact mean the same as “brother or sister”. In the absence of borderline cases the terms are extensionally equivalent however, “sibling” simply means “having (at least one of) one’s parents in common” and thus does not name a simple disjunctive category, but, rather, spans the categories of “brother” or “sister”, or anything in between. Edgington offers a further argument for non-truth-functional disjunction, one which would generalize to a defence of LEM in the context of vagueness, and which seeks to explicitly force a disjunctive reading of the key term. A library book can be such that it is not clear whether it should be classi¿ed as Philosophy of Language or Philosophy of Logic; but if we have a joint category for books of either kind, it clearly belongs there. It is not unusual for a term in one language to require a disjunctive translation in another. Suppose a language trivially different from English which has one word “bleen”, for “blue or green”. Something can be de¿nitely bleen, but neither de¿nitely blue nor de¿nitely green. Therefore, something can be de¿nitely blue or green, while neither de¿nitely blue, nor de¿nitely green. (1997: 310) Curiously, many who object to the non-truth-functionality of disjunction and the retention of LEM rest content with the non-truth-functionality of conjunction and the retention of the law of non-contradictory (LNC). A notable example is Burgess and Humberstone (1987), where SpV is modi¿ed to allow for failures of LEM yet LNC is retained. This asymmetric treatment of the dual notions of disjunction and conjunction and the closely related laws LEM and LNC seems dif¿cult, if not impossible, to justify. We will return to the issue of vagueness and LNC later.

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The argument is similarly unsuccessful. If one means by “a joint category for books of either kind” a category that includes all those in the category Philosophy of Language and all those in the category Philosophy of Logic then the book in question is not clearly in this simple disjunctive category. (Just imagine the librarian moving books onto the new shelves purchased to house books in the new category. Any book from either of the older categories is placed there. Is it clear that the contested book should be placed there? Surely not.) Of course, if the “joint category” is one which spans books from Philosophy of Language, Philosophy of Logic, and all in between, then the book will clearly belong in this category but this is no longer the simple disjunctive category required for the argument to establish the existence of a de¿nitely true disjunction with admittedly indeterminate disjuncts. More generally, terms with ‘disjunctive translations’ like “bleen” do de¿nitely apply to objects which admittedly do not de¿nitely satisfy either of the disjuncts if the ‘disjunctive translation’ names a span that covers each of the two disjunct categories and all in between. But where the term’s ‘disjunctive translation’ is a mere disjunction and names a simple disjunctive category, as required for the argument to succeed, it is unclear whether the term de¿nitely applies in cases where neither of the disjuncts de¿nitely apply. Only by equivocating on exactly what one means by a disjunctive translation can the argument succeed. It is indeed true, as Edgington says, that ‘[i]f these examples are acceptable, there cannot be any general objection to “A or not A” being de¿nitely true when neither disjunct is de¿nitely true’, but the examples are not acceptable. Both Edgington and Keefe have failed to establish (3) and have failed, more generally, to establish the acceptability of a non-truth-functional analysis of disjunction or the acceptability of penumbral connections. Opting for a weakly paracomplete response as opposed to a strongly paracomplete one that abandons LEM, restores subjunction and endorses a truth-functional analysis of disjunction remains a costly option to pursue from a purely pragmatic point of view. As with concerns above arising from the retention of classical consequence, the retention of the classical laws themselves, especially LEM, in conjunction with the resulting non-truth-functional account of disjunction might again be defended by claiming that such a logic follows from a representational account of vagueness. Fine seems to respond in this way, offering representationalism as a defence. Since we cannot precisely describe the precise world, we cannot in general say precisely how it is, though we can say that A V ~A is the case if we can say that the world is precise. LEM is indeed counterintuitive in the context of vagueness, but the merely semantic nature of vagueness does not impugn LEM. ‘Suppose I press my hand against my eyes and “see stars”. Then LEM should hold for the sentence S = “I see many stars”, if it is taken as a vague description of a precise experience’ (1975: 285). If vagueness is merely semantic, as the representationalist takes it to be, then LEM is prescribed and so defensible. Fine seems to go further though. ‘There is however, a good ontological reason for disputing LEM. … LEM should fail for S if it is taken as a precise description of an intrinsically vague experience’ (ibid.). On this approach, a representationalist account of vagueness is considered not only suf¿cient for the retention of LEM but also necessary. The failure of bivalence in SpV ensures the ability to encompass semantic vagueness within the scope of logic, whilst, if Fine is correct, the retention

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of LEM (amongst other things) stands or falls with the view that vagueness is not ontological. Laws like LEM are hostage to representationalism. Of course, it is only the suf¿ciency-claim that is relevant to the issue currently under discussion – the legitimacy of retaining LEM. LEM is defensible in so far as vagueness is not ontological, so it is claimed, and so, since the representationalist assumes vagueness not to be ontological, LEM is defensible tout court. SpV is again defended by appeal to a substantive theory of vagueness – a theory we shall consider and reject in the next chapter. 4.1.3

Extending supervaluationism: SpV+

The foregoing defence of SpV was couched in metatheoretic terms. Rolf (1984) objects that the theory is stuck with the paradoxical nature of the sorites no matter how much explanation is offered at the level of metatheory. The problem about accepting the major premise of the foregoing sorites as false is simply that it runs counter to our conviction that a man can cease being bald by the addition of one hair, but this conviction can be expressed in the object-language, so why should the elaborate metalinguistic theory be relevant here? If the paradox can be framed in the object-language, then it should be met in the object-language. This point is made even more emphatic by Rolf’s suggestion that one need have no metatheory; if someone formulates a sorites in the object-language, using only those logical principles of the object-language as the SpV theorist accepts, it appears completely beside the point to accuse that person of some sort of error in their metatheory. Similarly one might object to the SpV defence of LEM on the grounds that it too demands ascent into the metalanguage. In each case it is an explanation of (super)truth and its interaction with the logical constants that supposedly defuses any tension thought to be implicit in the SpV account. Yet, if the explanatory role played by (super)truth could be ¿lled by an object-language expression, then this objection could be met. No ascent into metatheory would be required. One obvious way in which (super)truth could be mirrored in the object-language is by denying Tarski’s prohibition on semantic closure and admit (super)truth into the object-language itself. A less controversial way of capturing the role of (super)truth within the object-language is by means of an expression which is not itself a truth-predicate. Many accounts of the supervaluationist response to vagueness (e.g. Fine 1975 and Keefe 2000) include just such an expression: “It is determinately (or de¿nitely) the case that …”. One might think that such an expression (or, if semantic closure is deemed acceptable, the (super)truth-predicate) is independently necessary on the grounds of expressive completeness. Presumably, the fact that it might be vague whether some colour counts as red or whether so-and-so is tall is a fact that ought to be expressible in the object-language. To this end, the SpV object-language is extended to include just such an expression, subject to the following constraint: Determinately A if and only if “A” is (super)true. “Determinately” can then be given a semantics analogous to the possible-worlds semantics for a language including the expression “necessarily” (and reminiscent of

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Russell’s analysis of determinacy/indeterminacy of Chapter 2, §2). Now, by way of analogy to the de¿nition of contingency in modal logic, one can de¿ne what it is for something to be indeterminate: Indeterminately A =df not determinately A and not determinately not-A. In the now-extended language, SpV +, one can express the idea of its being vague whether Tim is tall by claiming that it is indeterminate whether Tim is tall. Representing these expressions by operators “D” and “I ”, we can now answer Rolf’s criticism. In the absence of any metatheoretic notions one can now express the supervaluationist resolution of the sorites paradox in the object-language. When it is objected that a man cannot cease being bald by the addition of one hair (there is no cut-off point), the SpV + theorist can respond by saying that it is indeed the case that one hair cannot make the difference between being determinately bald and determinately non-bald (there is no determinate cut-off point). In other words. SpV + denies “∃n(DB(n) & D~B(n + 1))” which, given D(A & B) if and only if DA & DB, is equivalent to the denial of: (2+) ∃nD(B(n) & ~B(n + 1)), the SpV + object-language formulation of SpV’s metalinguistic expression (2). However, this does not impugn the fact that there determinately is some cut-off point; it does not imply the denial of: (1+)

D(∃n(B(n) & ~B(n + 1))).

Just as SpV justi¿ed their resolution of the (standard) sorites by endorsing (1) while denying (2), so too SpV + endorses (1+) while denying (2+). Moreover, in answer to Rolf’s criticism of SpV, the supervaluationist’s position is expressible in the objectlanguage of SpV +. The earlier objection to the supervaluational resolution of the sorites, which was diagnosed as relying upon the fallacious inference from (1) to (2), is now recast as relying on the fallacious inference from (1+) to (2+). Supervaluationists keen to pursue a representationalist defence have emphasized the apparent analogy with the modal inference from ■∃xPx to ∃x■Px (for example, inferring that some number is necessarily the number of the planets from the claim that necessarily some number is the number of the planets). Such inferences are commonly said to be fallacious where the key term “the number of the planets” is a non-rigid designator varying its denotation across worlds. Analogously, supervaluationists like Fine and Varzi can re-express their representationalist defence as claiming all vagueness as de dicto and on that basis point to the illegitimacy of the scope change evident in the shift from (1+) to (2+). Where the modal theorist points to the presence of non-rigid designators as undermining certain inferences, the supervaluationist points to the presence of what have been referred to as ‘imprecise designators’.14 What the objector is effectively 14 See, for example, Thomason (1984). The analogy with modality is stressed in Lewis (1988). And Akiba (2000) is also keen to stress the similarities.

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doing, the SpV + theorist may say, is inferring that something is determinately the cut-off point between baldness and non-baldness from the fact that it is determinate that something is the cut-off point. Yet the designator “the cut-off point” varies its denotation across precisi¿cations and, as such, the inference is invalid. This is just to rephrase Fine’s “truth-value shift” explanation of the fallacy of inferring (2) from (1). Similarly, the SpV defence of the retention of classical laws can be re-expressed in SpV + without recourse to metatheoretic notions like (super)truth. When it is objected that LEM fails since, for example, Tim may be neither tall nor not tall, the SpV + theorist can respond by admitting that it is indeed the case that Tim may be neither determinately tall nor determinately not tall but that this does not impugn the fact that he must determinately be one or the other. That is to say, for vague A, the SpV + theorist accepts: (3+) D(A V ~A), yet denies: (4+) DA V D~A. Just as SpV sought to justify their retention of classical laws by appealing to ‘penumbral connections’ and denying the implication from (3) to (4), SpV + will similarly appeal to penumbral connections and deny the implication from (3+) to (4+). As is now well known, with the introduction of “D”, despite the retention of all classical theorems, the classically valid rules of contraposition, conditional proof, proof by cases and reductio are rendered invalid.15 This divergence between classical consequence and SpV + consequence is an undesirable aspect of the extended logic of supervaluationism to the extent that the theory’s inherent conservatism is considered a virtue, as indeed it is by many supervaluationists. Williamson too notes the fact, inviting speculation on the extent to which this undermines supervaluationism’s right to claim to be a theory of least mutilation.16 But the much-vaunted conservatism was already overblown. SpV already deviates signi¿cantly from classical consequence. To see this we must ¿rst consider assertion and denial in the context of a supervaluationist response to vagueness. 15 See Williamson (1994: 151ff), and Keefe (2000: 176ff). Interestingly, the failure of these principles is due to the fact that the de¿nition of validity adopted earlier, global validity (see n. 5, this chapter), validates the inference “A therefore DA”. The failure of the consequence relation so de¿ned to satisfy the Deduction Theorem (as evidenced by its failing conditional proof) might cast doubt on the plausibility of the de¿nition. Moreover, second thoughts may arise concerning the seeming implausibility of Dummett’s local validity given its failing of the above problematic inference and its subsequent success in satisfying the Deduction Theorem and validating the contested rules cited. 16 Keefe (2000: 180–81) responds to Williamson’s probing, not by showing how insigni¿cant the extent of divergence is, but by showing that, however signi¿cant, it is properly traceable to vagueness. This seems to miss the point.

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Assertion, denial and logical consequence

Supervaluationism is a theory which explicitly recognizes truth-value gaps, abandoning the Principle of Bivalence in the face of incompleteness as evidenced by applications of predicates to their borderline cases. There will be a sentence A which is not true and so too its negation. Given the plausible view that we should deny what is not true, it follows that supervaluationists should deny both A and ~A. Moreover, given that one ought not assert what one also denies, and thus ought assert only truths, supervaluationists should refrain from asserting either of A and ~A. It follows then that the denial of A must be separated from the assertion of its negation, unless supervaluationists be required both to assert ~A (by way of denying A) and not assert ~A (since it is not true).17 Restall (2005) presents additional arguments for their separation independent of considerations of truth-value gaps, but the weaker claim that supervaluationists, at least, to ought separate the two notions is all that is relevant for current purposes. The distinctness of denial and assertion means that we cannot characterize constraints on denial by simply appealing to constraints on assertion. Constraints on denial must be independently characterized along with constraints on assertion. Such constraints are captured by an adequate theory of logical consequence. As Restall (2005: 191–3) argues, [i]t is common ground that logical consequence – whatever it amounts to – has some kind of grip on assertion and denial, [the speech-acts associated with] acceptance and rejection. … Logical notions are nothing if they have no applicability to regulate cognitive states of agents like us, and the content of such states. … If an agent’s cognitive state, in part, is measured in terms of those things she accepts and those she rejects, then valid arguments constrain those combinations of acceptance and rejection. … [A valid] one-premise, oneconclusion argument from A to B constrains acceptance/rejection by ruling out accepting A and rejecting B.

More generally, a multi-premise entailment such as Σ = A constrains acceptance and rejection in the obvious way, ruling out accepting all of Σ while rejecting A. As Restall goes on to point out, this understanding of the role of logical consequence 17 To be sure, as Keefe points out (2000: 155, n. 1), denial of A may sometimes be expressed by a locution ‘which is hard to distinguish in practice from the assertion of the negation’ – “not A” with an emphasis on “not” – nonetheless they are distinct. Keefe is ambivalent on the matter, suspecting they may amount to one and the same thing and viewing any resulting incoherence on what she sees as possible conÀict arising from the absence of a clear-cut rule for assertion. However, she fails to heed the full force of the reductio argument just presented, treating denial of non-truths and assertion of only truths as distinct and competing rules for assertion which may conÀict rather than seeing the latter rule as directly following from the former. Parsons (2000: 20), on the other hand, goes on to treat denial as assertion of alternative ‘exclusion negation’, ¬, but this will lead to an in¿nite regress. Given higher-order vagueness, there will be borderline cases between A and ¬A where denial of both is appropriate, and so we should refrain from asserting either. Denial of A then will need to be distinguished from the assertion of ¬A and attempts to cast denial as the assertion of a negation will require yet another species of negation. And so on.

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‘has the advantage of symmetry’. It does not privilege acceptance over rejection, assertion over denial. Recognition of the entailment equally mandates that those accepting all of Σ cannot, on pain of cognitive incoherence, go on to reject A and that those rejecting A cannot, on pain of incoherence, go on to accept all of Σ. This view of consequence and its role in regulating patterns of acceptance and rejection does, however, have repercussions for our understanding of logical consequence. Suppose an agent, a supervaluationist, say, rejects both A and B. Can such an agent coherently accept their disjunction, A V B? That, of course, depends on whether or not the multiple-conclusion consequence relation A V B= A, B holds – i.e. it depends on whether subjunction is accepted. To think that it is renders incoherent the acceptance of the disjunction while rejecting each of the disjuncts. Since supervaluationists, as we have seen, do accept some disjunctions (e.g. A V ~A) while rejecting each disjunct, they must, as we have also seen, fail subjunction. The generalized consequence relation describes the logical constraint that supervaluationists are forced to reject. Now, were denial the assertion of negation and rejection the acceptance of negation, then rejection of each of A and B would be equivalent to acceptance of ~A and ~B, and the coherence of accepting A V B while rejecting both disjuncts would simply be equivalent to the coherence of accepting A V B, ~A and ~B. And that is representable without recourse to a multipleconclusion consequence relation as A V B, ~A, ~B= ∅. Given the non-equivalence of denial and asserted negation, though, there is no avoiding the generalized multipleconclusion consequence relation as the means for representing the point at issue. Whether or not inference as ordinarily understood only ever takes one from premises to a single conclusion, as some seem to think (e.g. Keefe 2000: 198), it would be foolish to think that the logical notion associated with its evaluation – a single-conclusion consequence relation – was of paramount importance. Inference matters precisely because good inference generates constraints on acceptance and rejection, and this underlying value shows the importance of single- and multipleconclusion entailments equally. Just as multiple-premise, single-conclusion consequence establishes the incoherence of rejecting the conclusion while accepting all premises, single-premise, multiple-conclusion consequence establishes the incoherence of accepting the premise while rejecting all conclusions. The ‘symmetry’ associated with the cognitive-constraint account of logical consequence coupled with the relative independence of denial and negation thus points to another symmetry between premise-sets and conclusion-sets associated with logical consequence: each may be empty, a singleton, or many-membered. This having been said, those who, like Keefe, accept the failure of subjunction in SpV as an inevitable result of the acceptance of LEM have, at least implicitly, manifested a preference for one logical principle (LEM) over another (subjunction). Where it is explicitly acknowledged, the preference is justi¿ed by playing down the signi¿cance of the failure of subjunction on the grounds that it is not a principle employed in ‘ordinary life’. Keefe claims, for example that ‘[subjunction] fails according to supervaluationism … But we do not use multiple-conclusion arguments in ordinary life and it is reasoning in vague natural language that is in question’ (2000: 198, n. 24). Even supposing that ordinary arguments are not multipleconclusion arguments (a doubtful supposition), we are now in a position to see that this focus on an asymmetric, single-conclusion consequence relation is not justi¿ed.

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There is a clear sense in which we do and must use multiple-conclusion consequence in ordinary life. The presumed relative irrelevance of subjunction as compared with principles capable of being represented by a single-conclusion consequence relation is illusory. Summing up, then, a paracomplete response to vagueness must choose between LEM and subjunction. Supervaluationists opt for a weakly paracomplete account, retaining LEM. But we have seen their arguments for that theorem unconvincing or, in the case of the representationalist defence, resting on an as yet untried assumption as to the merely semantic nature of vagueness. The supposed virtue of recognizing ‘penumbral connections’ is not established. Now we can also see that the cost of their recognition is signi¿cant indeed and the failure of subjunction would already represent a considerable departure from classical, regulative, logical principles. Unless a representationalist theory of vagueness can be defended (and even then, contra Fine and Varzi, we might doubt it suf¿cient to defend SpV), a weakly paracomplete approach, in general – and supervaluationism, in particular – looks hard, if not impossible, to defend. 4.2

Subvaluationism

Closely allied to supervaluationism is another non-classical approach to vagueness – that revision of classical two-valued logic known as subvaluationism, SbV. The formal system was ¿rst proposed as an account of vagueness in Jaśkowski (1948). Jaśkowski, a student of the Lov-Warsaw School of philosophy, published his account a decade before Mehlberg, a former student of the same School, proposed the now popular supervaluationist account. Subvaluationism is the paraconsistent dual of supervaluationism and admits truthvalue gluts where the paracomplete SpV admits truth-value gaps. A logic is said to be consistent if and only if, for any valuation or model of any contradictory pair of sentences A and ~A, they cannot both be true in the valuation or model. I.e. A, ~A = ∅. Classical logic is a paradigm of a consistent logic. A logic will then count as inconsistent just in case it is not consistent – i.e. for some sentence A, both it and its negation can be true together. I.e. A, ~A ≠ ∅. Assuming a sentence to be false if and only if its negation is true, approaches to the problem of vagueness like SbV that, as we shall see, postulate truth-value gluts, exemplify logics which are inconsistent. A, ~A≠ SbV ∅. As with incompleteness, however, logics might only admit inconsistency in a trivial sense, so that whenever a sentence and its negation are both true in a theory, every sentence and its negation is true. This is not the sense of inconsistency to which truth-value gluts in general, and SbV in particular, are committed. Such approaches countenance quarantined gluts by rejecting the spread-principle – A, ~A= B – according to which if there are truth-value gluts anywhere then they are everywhere. That is, they accept that a sentence and its negation might be true in a theory without every sentence and its negation being true. Gluts do not explode everywhere – the logic is non-explosive.

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So in addition to the admission of inconsistency there is also a commitment to the non-triviality of the inconsistency, i.e. A, ~A≠ B. As is now standard, we shall say that a logic which admits non-trivial inconsistent theories is paraconsistent. The foundation of SbV as an approach to vagueness is that vagueness necessitates a paraconsistent response. Thus: A, ~A≠ SbV B. Subvaluationism pursues a paraconsistent approach for reasons similar to those presumed to ground supervaluationism and other paracomplete responses. Just as supervaluationists et al. cite that not uncommon response to borderline cases where speakers display an unwillingness to assert either A or ~A as evidence of truth-value gaps, subvaluationists heed that not uncommon response where speakers display a willingness to assert both. A preparedness to assert two contradictory claims is taken as prima facie evidence of the truth of each. A paraconsistent response has been pointed to by a range of theorists in the past. In Marxist philosophy key examples of dialectical situations are provided by focusing on what to say about the application of vague predicates to borderline cases. A seedling in a process of becoming a tree is said to be both a seedling (by virtue of what it was) and not a seedling (by virtue of what it will become); a man growing a beard is at some stage both bearded and not bearded. Terms such as “seedling” and “bearded” are vague predicates in the modern sense, yet for classical Marxists they are characteristically dialectical, issuing in contradictions. The Marxist theoretician Plekhanov (1937 [1908]: 114) proposes a ‘logic of contradiction’ as the means to a solution to ‘the riddle of the bald man’ – i.e. the sorites paradox. This solution was no curiosity in the far reaches of Marxist scholarship but a commonly and popularly cited virtue of Marxist dialectics and evidence of the inadequacy of classical logic. (See Milosz 1980: 50.) Of course this approach, as it stands, is far from worked out but it does strongly suggest a paraconsistent approach. In a more illuminating and recent discussion, McGill and Parry explicitly advocate vagueness as grounds for a paraconsistent dialectical logic. ‘In any concrete continuum there is a stretch where something is both A and ~A. … There is a sense in which the ranges of application of red and non-red [in so far as “red” is vague] overlap, and the law of non-contradiction does not hold’ (1948: 428). In agreeing with McGill and Parry that vagueness constitutes a case for paraconsistency, da Costa and Wolf (1980: 194) suggest that one requirement of a paraconsistent dialectical logic ‘is that the proposed logic be interpretable as a logic of vagueness’. Da Costa’s view that vagueness be treated paraconsistently can be traced to an earlier suggestion in Jaśkowski (1969), a reprint of Jaśkowski (1948). Therein Jaśkowski, a student of Łukasiewicz, described a paraconsistent ‘discussive logic’, one of whose main applications was to serve as a logic of vague concepts – concepts which he saw as giving rise to inconsistency. Within the pioneering Brazilian tradition of research into paraconsistent logics, this work was picked up and subsequently elaborated on in Arruda and Alves (1979), and da Costa and Doria (1995). They persisted with Jaśkowski’s claim that discussive logic be looked on as a logic of vagueness. Some idea of the extent to which Jaśkowski’s work, and more

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particularly his view on vagueness, has influenced the development of paraconsistent logic in Brazil can be gleaned from Arruda (1989). This explicit interest in vagueness from a paraconsistent perspective is not restricted to the Brazilian school. A paraconsistent approach to vagueness has been pursued by other non-classical logicians and philosophers. (See Peña 1989; Priest and Routley 1989a. Lewis 1982 considers such an approach on the strength of the questionable analogy between vagueness and ambiguity, but ultimately endorses a paracomplete approach.) The main problem with many of these suggestions that vagueness warrants a paraconsistent analysis is that while they point in a paraconsistent direction, they do not explain in any detail how vagueness is to be analysed from either a formal or philosophical point of view. Jaśkowski’s discussive logic reinterpreted as a dualization of supervaluationism presents us with both a formal analysis of vagueness and a philosophical interpretation as informative as its paracomplete rival.18 4.2.1

Describing the subvaluation theory

So what is the theory of subvaluation and how does it address the problems that arise in the context of vagueness? Subvaluational semantics begins from the agreed view that when, for example, a vague predicate “heap” has some borderline case a, say, it is indeterminate whether a is a heap since a is neither determinately a heap nor determinately not a heap. There are admissible ways of precisifying “heap” so that a is a clear counter-instance and there are other admissible ways of precisifying “heap” so that a is a clear instance. The difference is that subvaluational semantics treats borderline cases for a vague predicate like “heap”, as cases to which the predicate both applies and does not apply. That is, if a is a borderline case for “heap” then “a is a heap” is true and “a is not a heap”, is true (i.e. “a is a heap” is false). Where supervaluational semantics de¿ned truth simpliciter (or supertruth) as applying to a sentence just in case that sentence was true no matter how one admissibly precisi¿ed any vague constituents of the sentence, i.e. just in case the sentence was true for all admissible precisi¿cations, subvaluational semantics de¿nes truth simpliciter (or subtruth) as applying to a sentence just in case that sentence is true for some admissible precisi¿cation. Whilst determinate truths are still those sentences which remain true for all admissible precisifications, determinate falsehoods are still those which are false for all admissible precisi¿cations, and indeterminate (vague) sentences still those which are true on some but not all admissible precisi¿cations, this third class now consists of those sentences that are both true simpliciter and false simpliciter (as opposed to neither true nor false simpliciter). Indeterminate sentences take on both truth-values. Unlike paracomplete responses to vagueness where indeterminacy is analysed as underdetermination, paraconsistent responses – and SbV in particular – analyse indeterminacy as overdetermination. Determinate truth is now considered a matter of truth only and determinate falsity a matter of falsity only. Since truth and falsity 18

The formal duality between SpV and SbV was originally examined in detail in Varzi (1994 and 1995). The suggestion that they can be seen as dual philosophical accounts of vagueness was presented in Hyde (1997).

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are taken to be exhaustive (i.e. completeness is assumed), indeterminate sentences are now considered neither true only nor false only but, rather, both true and false. It is easy to show that for such sentences, e.g. “a is a heap”, both it and its negation are true. Moreover, an evaluation which ascribes both truth-values to such a sentence might nonetheless ascribe just the value “false” to another sentence “b is a heap”. The logic is clearly paraconsistent. Given this non-classical constraint on the semantics of vague expressions, we may wonder to what extent classical logic remains intact. For instance, if a sentence A and its negation ~A are indeterminate and thus both true and false, then what of their conjunction and disjunction? Are they likewise indeterminate? Are the classical laws of excluded middle and non-contradiction still theorems? Do inferences such as modus ponens and proof by cases remain valid in such semantics? Like supervaluationism, subvaluationism seeks to minimize logical revision necessary to accommodate vagueness and so, against a classical background, will seek to preserve all classical tautologies. Subvaluationists thus require a paraconsistent logic of vagueness yet aim, in particular, to retain the law of noncontradiction (LNC). More particularly still, not only should contradictions therefore always be false – i.e. = ~(A & ~A) – but they should also never be true – i.e. A & ~A = ∅; the distinction is, of course, non-trivial in the current paraconsistent context. Consequently, the paraconsistent logic that is sought differs crucially from others that have been proposed to deal with the liar paradox, for example. There the most plausible candidate is the logic of Priest (1979), LP, where contradictions are sometimes true. The logic that is sought here, however, is what, following Arruda (1989), we may describe as a weakly paraconsistent logic; though it admits of nontrivial inconsistent theories which contain both A and ~A – i.e. A, ~A≠ B – nonetheless no such non-trivial theory includes A & ~A. Thus: A & ~A= SbV B. (By contrast, strongly paraconsistent logics like LP do not distinguish between the truth of A & ~A and the truth of both conjuncts, and consequently they admit of nontrivial valuations making A & ~A true, i.e. A & ~A≠ B.) De¿ning SbV consequence in terms of preservation of truth simpliciter, i.e. subtruth, satis¿es the foregoing constraints. Thus Σ =SbV Γ if and only if whenever all the members of Σ are true then some member of Γ is true (i.e. whenever all the members of Σ are true in some admissible precisi¿cation then some member of Γ is true in some admissible precisi¿cation). Or, equivalently for current purposes, it is impossible for all the members of Σ to be true and all the members of Γ to be not true.19 Such an account of logical consequence will obviously establish all classical (CL) theorems as theorems of SbV: (I′)

=SbV A if and only if ≠CL A.

19 As with SpV consequence, a rival account of consequence could be given analogous to SpV-local-validity: Σ =SbVΓ iff for some admissible precisi¿cation, whenever all the members of Σ are true, some member of Γ is true. Again, however, I shall retain the global account described.

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So, for example, though the principle governing the exclusivity of truth-values is rejected, the law of non-contradiction is preserved in both of the following senses: LNC: =SbV ~(A & ~A), and LNC′: A & ∼A =SbV ∅. Where a restriction is placed on the consequence relation to the effect that the premise-set Σ be a singleton set, the multiple-conclusion consequence relation is coextensive with classical consequence and admits as valid all and only those inferences that are classically valid. Thus: (II′)

A=SbV Γ

if and only if A=CL Γ.20

But multiple-conclusion SbV consequence more generally considered deviates from its classical counterpart. As a result of being paraconsistent but only weakly so, SbV fails adjunction: A, B≠SbV A & B.21 This non-classical feature of SbV manifests the non-truth-functional account of conjunction required to underwrite a weakly paraconsistent theory of vagueness and is a major source of concern when evaluating the system’s adequacy. How can it make sense to accept two claims considered separately while rejecting them considered jointly? Classical multiple-conclusion consequence is preserved in SbV only in the following quali¿ed sense: (III′) A1 & A2 & … & An =SbV Γ

if and only if A1, A2, …, An =CL Γ.22

How then might such a paraconsistent approach to vagueness resolve the sorites paradox? To answer this question, consider the standard (i.e. many-conditionals) form of the paradox. A man with 1 hair on his head is bald. If a man with 1 hair on his head is bald, then a man with 2 is. If a man with 2 hairs on his head is bald, then a man with 3 is. :. If a man with 9,999 hairs on his head is bald, then a man with 10,000 is. ∴

A man with 10,000 hairs on his head is bald. 20

Proof follows from a simple generalization of the proof of (II) in Hyde (1997: 648). Since, were adjunction to hold, substituting ~A for B, LNC′ would mandate consistency (i.e. not both of A and ~A could be true). 22 For a proof see Hyde (1997: 655). 21

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Since SbV does not preserve classical consequence unrestrictedly, the possibility arises of the paradox being discounted by virtue of its invalidity. One diagnosis available to the SbV theorist is exactly that – the premises, including conditional premises, are all true but modus ponens is not unrestrictedly valid. Consider the sentence “A pile of n grains of sand is a heap” where a pile of n grains counts as a borderline case for “heap”. The sentence is true and false, so it is true. Since it is also false, then the material conditional “If a pile of n grains of sand is a heap, then a pile of n – 1 grains is a heap” is true by virtue of the falsity of its antecedent. Nonetheless, a pile of n – 1 grains of sand might be determinately not a heap, thus making the sentence “A pile of n – 1 grains of sand is a heap” false.23 So: Heap(n), Heap(n) → Heap(n – 1) ≠SbV Heap (n – 1), for some n. As is familiar from other paraconsistent logics, then, modus ponens for material implication, i.e. disjunctive syllogism, is not valid in SbV. (Notice that what is claimed is that modus ponens is here denied for material implication – the implication relation typically assumed in modelling the sorites conditional. Beall and Colyvan (2001) point out that this analysis of the sorites may hide the problem rather than solve it, since it assumes what many consider a very weak reading of conditionality. Stronger conditionals are de¿nable and alternative responses are then available to the SbV theorist. Entailment is a much stronger conditional for which modus ponens clearly holds, but it does not provide an interpretation of the sorites premises which renders them true. A mid-strength connective, “ →”, just strong enough to satisfy modus ponens, can be explicitly de¿ned in SbV but it is easily shown that this does not provide an interpretation of the sorites-conditionals which renders all the premises true.24 There is, of course, an analogous problem lurking here for the supervaluationist, who can equally be charged with assuming an overly strong reading of the conditional which is suf¿ciently strong to validate modus ponens but not weak enough to provide an interpretation of the sorites premises which renders them true.25) The foregoing simple paraconsistent reinterpretation of supervaluational semantics reproduces exactly the ¿rst formal system of paraconsistent logic – discussive logic – developed by Jaśkowski over ¿fty years ago, which already at the time was claimed to be applicable to vagueness. Jaśkowski (1969 [1948]: 149) suggested that: 

23

Higher-order vagueness may complicate matters here, but even the simpli¿ed approach will subsequently be found untenable. The simpli¿cation is thus harmless. Notice that the conditional premise will be counted both true and false in SbV under the conditions described, whereas SpV would count it neither true nor false. 24 De¿ne “A → B” as follows: A → B is true simpliciter iff either A is false in every admissible precisi¿cation or B. The aforementioned counterexample to modus ponens now renders the corresponding (ponendable) conditional “Heap(n) → Heap(n – 1)” false. The newly de¿ned connective is, in fact, just Jaśkowski’s discussive implication, “→D”, suitably reinterpreted, which he explicitly introduced to recapture a conditional satisfying modus ponens. For further discussion of discussive implication see Priest and Routley (1989b: 158f). 25 See Hyde (2001).

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logical researchers so far have been taking into consideration such deductive systems which are symbolic interpretations of consistent theories, so that theses in each such systems are theorems in a theory formulated in a single symbolic language free from terms whose meanings are vague. But suppose that theses which do not satisfy those conditions are included into a deductive system. It suf¿ces, for example, to deduce consequences from several hypotheses that are inconsistent with one another in order to change the nature of the theses, which thus no longer reÀect a uniform opinion. The same happens if the theses advanced by several participants in a discussion are combined into a single system, or if one person’s opinions are so pooled into one system although that person is not sure whether the terms occurring in his various theses are not slightly differentiated in their meanings. Let such a system which cannot be said to include theses that express opinions in agreement with one another, be termed a discursive [or discussive] system. To bring out the nature of the theses of such a system it would be proper to precede each thesis by the reservation: “in accordance with the opinion of one of the participants in the discourse [discussion]” or “for a certain admissible meaning of the terms used.” Hence, the joining of a thesis to a discussive system has a different intuitive meaning than has the assertion in an ordinary system. A discursive assertion includes an implicit reservation of the kind speci¿ed above, which … has its equivalent in possibility Pos [or ◊]. Accordingly, if a thesis α is recorded in a discussive system, its intuitive meaning ought to be interpreted so as if it were preceded by the symbol Pos [◊], that is, the sense: “it is possible that α ”. That is how an impartial arbiter might understand the theses of the various participants in the discussion.

The subvaluation semantics outlined above clearly describes a discussive system in Jaśkowski’s sense. Instead of some thesis being counted true simpliciter just if it is true ‘in accordance with the opinion of one of the participants in the discourse’ or true ‘for a certain admissible meaning of the terms used’, we have de¿ned a thesis as true simpliciter if true for a certain admissible precisi¿cation of the terms used. Jaśkowski’s long-standing proposal to treat vagueness from a paraconsistent perspective by means of a discussive logic is simply reinterpreted so as to be the dual of the dominant paracomplete supervaluationist approach. More exactly, where Σ and Γ are sets of sentences of the shared language of SpV and SbV and ~Σ=df {~A: for all A ∈ Σ}: Σ =SpV Γ 4.2.2

if and only if

~Γ=SbV ~Σ.26

Defending subvaluationism

Given the duality between supervaluationism and subvaluationism, it is unsurprising that subvaluationism faces objections that are the exact dual of those pressing against a supervaluationist account. Problems arise for SbV as regards both its resolution of the sorites paradox and its retention of classical laws. Before turning to these matters, though, let us deal immediately with the very general objection that SbV must be inadequate merely by virtue of its very paraconsistency. Keefe claims that ‘many philosophers would soon discount the paraconsistent option (almost) regardless of how successfully it treats vagueness, on the grounds of the unappealing commitments and features of the logical framework 26

For a proof see Hyde (1997: 656).

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as a whole, in particular the absurdity of p and ~p both being true for many instances of p’ (2000: 197). As a sociological observation, this is quite possibly true. Many discount paraconsistency as an option on the grounds that it simply must be wrong. But the assumed absurdity does not obtain merely by virtue of its being presumed to obtain, and the arguments offered are not conclusive either.27 Paraconsistency per se has not been shown to be absurd any more than paracompleteness has. Both options are available for considered application. Fine suggests that a paraconsistent approach might rest on a fundamental confusion or might masquerade as rival when it is, in fact, a mere verbal variant of its paracomplete dual. Some have thought that a vague sentence is both true and false and that a vague predicate is both true and false of some [borderline] object. However, this is a part of the general confusion of under- and over-determinacy. A vague sentence can be made more precise; and this operation should preserve truth-value. But a vague sentence can be made to be either true or false, and therefore the original sentence can be neither. (1975: 266–7)

The argument, however, is not compelling. All parties to the debate can agree that precisi¿cation should resolve any vagueness surrounding the sentence in question, and that this will resolve borderline cases consistent with what is already determinate. Clear cases should remain unaffected by attempts to resolve indeterminacies involving borderline cases. As Fine’s (1975: 268) Stability Condition subsequently makes clear, precisi¿cations therefore should undeniably preserve any determinate truths (i.e. cases of truth only) or determinate falsehoods (i.e. cases of falsehood only) involving the sentence irrespective of its vagueness. But to also require that they preserve truth-value per se begs the question against the paraconsistentist. By their lights there are no truth-value gaps and every sentence is truth-valued. Precisi¿cation cannot and should not preserve truth-value, but should instead preserve determinate truth-value. A vague sentence can be made to be either determinately true or determinately false, and therefore the original sentence can be neither. But whether the original indeterminate sentence is neither determinately true (true only) nor determinately false (false only) by being neither true nor false or, contra Fine, by being both true and false remains open, neither option a confusion. Under- and over-determination can provide coherent analyses of indeterminacy. Perhaps, then, as Fine (1975: 267) speculates, ‘this battle of gluts and gaps may be innocuous, purely verbal’. There is, one might think, no real choice to be made here since when advocates of the rival approaches disagree as to whether or not some inference is truth-preserving, they each have in mind a different conception of truth and the disagreement is, in fact, purely nominal. But there is a substantive issue here about the nature of truth in natural language. The question of which, if any, of the two ensuing accounts of logical consequence is correct is far from innocuous. The validity of modus ponens, for example, depends upon it. In describing a logic of vagueness one is attempting to provide an account of non-fallacious reasoning in natural language. Does the natural-language conditional satisfy modus ponens? Is 27 Those interested may wish to look at the debate in Priest (2006 [1987]), Sainsbury (1995a), and Beall et al. (2004).

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natural-language conjunction adjunctive? Is natural-language disjunction subjunctive? These are substantial questions on which the two approaches disagree. Whilst the disagreement arises by virtue of differing conceptions of truth, the debate is non-verbal since each party is to be understood as providing an unequivocal account of the central semantic notion of truth. Fine (1975: 267) does not think the matter is merely verbal either, deciding in favour of a paracomplete approach. ‘[i]t is the gap-inducing notion [of truth] that is important for philosophy. It is the one that directly ties in with the usual notions of assertion, veri¿cation and consequence. The glut-inducing notion has a split sense; for it allows truth to rest upon either correspondence with fact or absence of meaning.’ But it is easy to see how the glut-inducing notion too ties in with assertion and veri¿cation, and its interaction with logical consequence has already been made clear. Subvaluationism admits of a sentence A which is true and so too its negation. Were neither assertible, we could agree with Fine that something is amiss since surely their status as recognizable truths makes them available for assertion. But there is no non-question-begging reason to suppose that each of the contradictory pair of sentences is not assertible. Subvaluationists start from the view that each is assertible. It is thus consistent with the view that all truths should be able to be asserted. Similar remarks apply to veri¿cation. As for the charge of being committed to a ‘split sense’ of truth, this presumes an analysis of vagueness as absence of meaning or what Fine earlier describes as ‘under-determination of meaning’. Subvaluationists can agree that cases of vagueness are cases where there is no determinate fact of the matter, i.e. meaning is indeterminate but, again, the question at issue is begged if it is supposed, as Fine does, that this species of indeterminacy is to be analysed as underdetermination as opposed to overdetermination. The glut-inducing notion of truth can rightfully claim to be philosophically important. Conceding the coherence and distinctness of a paraconsistent approach considered generally, more particular concerns centre on subvaluationism itself. Like SpV, the non-truth-functionality of the subvaluationist response weighs heavily against it. More particularly, the failure of adjunction is a major concern. The feature is well known in discussive logic and its non-adjunctive nature has often been remarked upon and is frequently considered a major obstacle to the plausibility of this paraconsistent approach. (See, for example, Priest and Routley 1989b: 158.) The implausibility transfers immediately to SbV. The failure of conjunction to satisfy this most basic of rules counts against its interpretation as a natural-language conjunction. False conjunctions with no false conjunct are counterintuitive indeed. The most obvious SbV counterexample to adjunction is that which establishes it as a weakly paraconsistent system: A, ~A≠SbV A & ~A. Where A is vague, counterexamples to adjunction readily accrue. Keefe (2000: 198) takes this particular departure from classically acceptable reasoning to be a major obstacle to the acceptance of SbV, as indeed it is. Such a weakly paraconsistent account of logical consequence seems dif¿cult, if not impossible, to defend. But we should be careful here, as elsewhere, to draw the

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appropriate lessons from such an anomaly. It is, in Keefe’s view, testimony to the inadequacy of a paraconsistent response, a weakly paracomplete one (supervaluationism) being acceptable. SbV must reject an instance of an acceptable multipremise, single-conclusion consequence relation (adjunction), whereas SpV is only required to reject an instance of an acceptable single-premise, multi-conclusion consequence relation (subjunction). The former involves the rejection of an acceptable ‘ordinary’ inference properly counted as part of the provenance of ‘traditional’ logic, whereas the latter involves the rejection of a principle not part of ‘ordinary’ inference and beyond the scope of ‘traditional’ logic. Evaluating the two logical theories then as regards their ability to account for ‘ordinary’ inference and ‘traditional’ logic, SpV is supposedly superior. But we have already seen that claims for the relative irrelevance of subjunction are misplaced given a proper understanding of assertion and denial in the context of gaps and gluts. The proper lesson to be drawn from the objectionable failure of adjunction is that a weakly paraconsistent account of vagueness should be abandoned by virtue of its being weakly paraconsistent, just as the proper lesson to draw from the objectionable failure of subjunction is that a weakly paracomplete account of vagueness should be abandoned by virtue of its being weakly paracomplete. There is no relative difference here between the two approaches. Pressing the failure of adjunction more strongly, Keefe (2000: 200) also points to the related fact that SbV is forced to differentiate between seemingly equivalent forms of paradox. Although the standard sorites consisting of a categorical premise and many conditional premises is declared invalid, the closely related form where all premises are conjoined is valid but has a now false premise, and so too for the mathematical induction form of the paradox. This ‘unappealing lack of uniformity in locating blame results in denying most intuitions associated with the sorites argument: it is not valid, at least in some forms, one of the premises is not true, in other forms, and different ways of stating what is apparently the same argument are actually stating crucially different arguments’. Of course, the same is true of Keefe’s preferred SpV responses to the paradox in its many forms and no comparative disadvantage is manifested by the observation in relation to SbV. Supervaluationists similarly deny ‘most’ of our intuitions associated with the sorites argument: it is sound, at least in some forms (e.g. the line-drawing form); one of the premises is not true in other forms (e.g. the standard sorites); and different ways of stating what is apparently the same argument are actually stating crucially different arguments (e.g. the standard sorites with no false premise and the seemingly equivalent form with conjoined premises which does indeed have a false premise, the resulting conjunction). What the objection properly points to in relation to both a weakly paracomplete approach and a weakly paraconsistent approach is the fact that they are equally counterintuitive. Semantic anomalies already discussed in relation to supervaluationism ¿nd their analogue in subvaluational semantics. Naturally, defences can be mounted by analogy with supervaluationism. To wit, vagueness demands a modi¿cation of classical semantics, namely the admission of the truth of contradictory pairs of sentences, yet, it might be contended, ‘penumbral connections’ must nonetheless be respected by the logic and thus contradictions themselves must always be false. However, the arguments for penumbral connection

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are as unconvincing in present circumstances as they were previously when considering the supervaluationist response to vagueness. Just as no compelling reason has been given for thinking LEM should hold in the face of truth-value gaps, so too arguments for LNC and, more particularly, LNC′ are unlikely to succeed in the face of truth-value gluts. Opting for a weakly paraconsistent response as opposed to a strongly paraconsistent one that both abandons LNC′, restores subjunction and endorses a truth-functional analysis of disjunction, remains a costly option to pursue from a purely pragmatic point of view. As with supervaluationism, though, LNC′ might again be defended by claiming that it follows from a representational account of vagueness. Adapting Fine’s response to the retention of LEM in SpV, a representational view of vagueness might be appealed to by way of defence. Since we cannot precisely describe the precise world, we cannot in general say precisely how it is, though we can say that A & ~A is not the case if we can say that the world is precise. To be sure, LNC′ is counterintuitive in the context of a paraconsistent approach to vagueness, yet the merely semantic nature of vagueness does not impugn LNC′. Again, as we shall see in the next chapter, this defence, even were it thought plausible assuming representationalism about vagueness, fails to the extent that arguments for representationalism fail. In addition to its being paraconsistent and non-adjunctive, a further concern might be expressed about the resolution of the sorites paradox described above. The failure of modus ponens might be objected to. But given that the failure is a direct result of the failure of disjunctive syllogism in paraconsistent circumstances, the objection amounts simply to scepticism concerning a paraconsistent approach and is no additional objection. Moreover, the subvaluationist can point to the fact that some intuition has to give way in resolving the problems attending the phenomenon of vagueness, and the failure of “→” to satisfy modus ponens (i.e. the failure of disjunctive syllogism) is no more objectionable than the failure of “∃” to satisfy standard semantic clauses (as required by SpV), or the failure of conditional proof (as required by SpV+, or the rejection of the law of excluded middle (as required by strongly paracomplete approaches). Logical innovation appears inevitable. That modus ponens should be excluded from possible revision seems unprincipled. 4.3

Summary

I have argued that super- and subvaluationism offer distinct but equally inadequate responses to the phenomenon of vagueness. While non-classical semantics and an associated logic are undoubtedly required to deal with the phenomenon, the responses considered, in seeking to retain classical theoremhood, are thereby committed to weak paracomplete and paraconsistent responses respectively. The costs are high, as we have seen. Moreover, the costs seem equally weighted against each and do not speak against paracompleteness or paraconsistency per se. We should look to strongly paracomplete and strongly paraconsistent systems for a more acceptable logic of vagueness.

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Having said this, the task of pursuing either as a response to vagueness is already work enough and, while the foregoing considerations point equally to paracompleteness and paraconsistency as avenues of exploration, I shall limit the ensuing discussion to a consideration of a paracomplete response, a strongly paracomplete response. Paraconsistency and its defence as an account of vagueness and the sorites paradox would embroil us in that other notable puzzle bequeathed to us by Eubulides of Miletus, the liar paradox, and that is best left for another day. Thus interest turns to a strongly paracomplete response to vagueness. Before moving on to evaluate such a response, though, we must turn to consider the possibility of ontological vagueness. In the course of evaluating the weak logics presented, a partial defence by appeal to a representationalist view of vagueness was rejected on the grounds that the necessary supposition to the effect that the world is not vague was unsubstantiated. We must now move to redeem the promissory note undermining such a defence.

Chapter 5

Ontological Vagueness And so we come to the question of whether the world might in some sense or other be vague. The idea that vagueness in language, and representations more generally, might at least sometimes have its source in that which is represented, including and especially the world, has not been widely accepted. I think this reluctance is ultimately misplaced: objects, properties and relations can all be vague, or so I shall argue. However, one does not have to look far to get an idea of just how dimly such a proposal is viewed by some. As we have seen, Russell saw the proposal as symptomatic of a particular fallacy. The tendency, having realized that words are vague, to infer that things might also be vague Russell took to be a case of the fallacy of verbalism – ‘the fallacy that consists in mistaking the properties of words for the properties of things’. Vagueness and precision alike, though characteristics of language, cannot be attributed to things: ‘things are what they are, and there is an end of it. Nothing is more or less what it is, or to a certain extent possessed of the properties which it possesses.’ More famously, perhaps, Dummett once claimed that ‘the notion that things might actually be vague, as well as vaguely described, is not properly intelligible’ (1975: 314). He has since recanted by suggesting that this view, according to which reality cannot be vague, is nothing more than deep-seated metaphysical prejudice (1981: 440), most recently proposing an account of reality which does indeed admit of vagueness (see Dummett 2000). Much as prejudice ruled out consideration of an epistemic approach to vagueness for much of the twentieth century, this metaphysical prejudice in favour of a precise ontology was, for a long time, enough to rule out general consideration of an ontological account of vagueness. To be fair, until very recently an ontological account has not had vociferous articulation, with few early proposals providing suf¿cient detail as to what such an account might look like to counter the deep misgivings many have towards such supposedly ‘dark metaphysics’. But proposals have been made nonetheless. The Marxists’ view expressed by Plekhanov (1937 [1908]) invites a construal of the dialectical nature of vagueness as having an ontological basis. Burks (1946) proposed an explanation of the vagueness of predicates by suggesting that the relevant universals they represent are themselves vague, and objects may have them only to some degree or other. More recently, Rolf (1980), Burgess (1990), Tye (1990, 2000), van Inwagen (1988, 1990), Zemach (1991), Akiba (2000, 2004), Dummett (2000), Parsons (2000), Morreau (2002) and Rosen and Smith (2004) have all advocated ontological vagueness of one kind or another. The most common sense in which it is said that the world might actually be vague is that according to which there might be objects that are vague. Others, like Burks, have suggested the existence of vague properties, while some, like Parsons, see such attributions as too ¿ne-grained, opting instead for a view according to

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which there are vague states of affairs below the level of which attributions of ontological vagueness are to be avoided. Reasons in favour of ontological vagueness are not hard to ¿nd. Firstly, and most obviously, common sense seems to speak in its favour. Tables, chairs, clouds, mountains, landscapes, persons, cats and dogs all seem to be vague – in many cases we cannot determinately specify exact spatial or temporal boundaries for such things, and the most obvious explanation for this is that they simply lack sharp boundaries. As with most material objects, there will sometimes be no fact of the matter as to what something’s exact spatial parts are and no fact of the matter as to exactly when it came into and went out of existence. Of course, there is a popular way in which philosophers try to resist such an ontological attitude. Lewis’s oft-quoted response captures the essence of what many of those who resist think. ‘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 has been fool enough to try to enforce a choice of one of them as the of¿cial referent of the word “outback”’ (1986: 212). So too presumably with clouds, cats, dogs and mountains, etc., but Tye speaks for many when he retorts that ‘[i]ntuitively, there is no vagueness or indeterminacy in which entity, “Everest”, for example, denotes. It denotes a single mountain in the Himalayas, a bloody great mountain, indeed the highest mountain in the world’ (2000: 196). Denotation itself is typically determinate. When speaking using “Felix”, “Spot” and “Everest”, there is typically no indeterminacy in which entity is denoted. Looked at more broadly, Lewis’s response echoes Russell’s more general representationalist approach to vagueness, and the counter-intuitive nature of Lewis’s response simply reÀects what we might call ‘the Problem of Multiple Denotation’. Given the essential one–manyness of denotation required by a representational or semantic account of vagueness, denotation, reference and representation more generally are typically not determinate. The ubiquity of semantic indeterminacy therefore means that exactly what is represented by a given representation is typically left open. There are many candidates, no one of which is determined. As McGee and McLaughlin recognize, we learn ‘a disturbing philosophical lesson’ that the semantic approach to vagueness forces us to accept ‘the inscrutability of reference’ (2000: 130).1 Such an approach thus invites the view, expressed clearly by Russell, that vague representations are defective since they fail to represent in the obvious way that their precise cousins do – i.e. determinately, directly. Consequently, as Williamson notes, the opposing idea of vagueness in the world is initially attractive. It acknowledges a direct relation between our vague ordinary words and the facts we use them to describe, for example, between an utterance of “Everest is high” and the fact that the mountain Everest has the property of being high (1994: 249). Whilst it is, no doubt, sometimes the case that the vagueness of our representations exceeds the vagueness of that which is represented and the 1 McGee and McLaughlin, in embracing the semantic theory, point to the inscrutability of reference as a fact about language to be accepted rather than an untoward consequence undermining the theory.

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representing relation is itself somewhat vague in such cases (just consider a blurry photo, for example), in typical cases, where terms are vague, it might seem simplest, all things considered, or even mandatory to suppose that the relation that obtains between words and what they describe is as direct as it would be were our languageprecise and we might reasonably suppose representation to be one–one, as opposed to one–many. To put the point another way, simplicity might suggest that, wherever possible, we retain a precise relation of representation, whatever our theory of representation prior to considerations of vagueness. In this way, we avoid unnecessarily complicating aspects of representation, reference and denotation incurred by those who adopt a representational approach to vagueness. Secondly, there are related reasons arising from the Problem of the Many (Unger 1980b, Geach 1980). Consider Tibbles the cat. As is often the case with cats, suppose there to be a number of hairs, h1 … h100, in the process of coming loose. For each hi it is indeterminate whether it is a part of Tibbles. Which aggregate of matter then counts as Tibbles? There are numerous precise candidates, each having their parts determinately and lacking any borderline parts, and if vague cats are to be ruled out, then there are only these candidates. Consider then this plurality of distinct candidates corresponding to the various ways of resolving the borderline parts – the object that determinately includes all but h100, the object that determinately includes all but h99, the object that determinately includes all but h100 and h99, etc. Differing in their parts, each is distinct from the others, yet all have equal claim (if any) to being Tibbles. If only precise candidates are admitted, then it seems we must admit all if we admit any. Since there is a cat, Tibbles, there must in fact be many cats. The standard approaches to the problem are well known. We might, following Unger (1979b), deny that there is any cat at all and, since there was nothing particular about cats in this respect, we might go on to deny the existence of material things more generally. Problems with this approach have already been canvassed in Chapter 3, §1. Alternatively, then, we might admit all of the numerous precise candidates, admit that there are many cats, and seek ways to make this conclusion palatable. For example, Lewis (1993) argues that strictly speaking there are indeed many but, since each differs only marginally from the others, the many cats are almost one and in everyday circumstances being almost-identical is enough for identity. Thus, though there are many cats, strictly speaking, their being almost one is suf¿cient for our counting them intuitively as one. Problems also attend this approach (see Hudson 2001). We might, instead, seek ways to defend common sense and claim that there is only one cat. Following Geach (1980), for example, we might invoke a notion of relative identity and claim that the precise candidates are different objects but the same cat. Or we might invoke the formal semantics afforded by supervaluationism and claim that, despite its not being true of any candidate that it is the cat, it is nonetheless true that some one candidate is the cat. Finally, we might suggest that it is a simple brute fact that some one candidate is the cat and that the others are not. Given the dif¿culties that attend these responses, we might wonder if, all things considered, the heart of the problem rests with the refusal to countenance vague

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objects.2 In many ways, the most obvious intuitive response to the problem of the many is to simply acknowledge the existence of a single spatially vague cat Tibbles, having borderline parts h1 … h100. If we can make sense of spatially vague objects – objects with fuzzy spatial boundaries – then we can respond to questions concerning how to individuate the unique object that is Tibbles in the same way that we do when called upon to individuate any precise material object, and the Problem of the Many simply dissolves. Its paradoxical nature is a direct consequence of the refusal to countenance vague objects. Consider what we would say in cases where we are apparently confronted by Nibbles, say, the precisely bounded cat. Nibbles has no borderline parts and all of h1 … h100 are determinately part of Nibbles. Now consider the many distinct aggregates of cat-stuff each determinately lacking some or more of these hairs. Aren’t each of these cats?, we might ask. Aren’t there then many cats, paradoxically, just as with Tibbles above? In such a case it seems easy to answer “no”. Each of the aggregates is a part of a cat (part of Nibbles, in fact) and therefore not a cat itself. The cat is, or is constituted by, ‘the maximal lump of cat-stuff’, as Keefe and Smith put it (1997: 51), that (determinately) includes all of the cat-stuff included in each of the aggregates and (determinately) excludes anything not included in any of the aggregates. There simply is no problem or paradox analogous to that arising from the case of Tibbles. It is the supposed absence of just such a unique maximal lump in the case of the vaguely bounded Tibbles that requires of us that we therefore seek the cat amongst the range of seemingly equally deserving precise candidates. But an acceptance of the existence of vague objects offers the prospect of recognizing that there is no unique precise maximal lump while maintaining that a unique but vague maximal lump nonetheless exists. (See Lowe 1982, Lewis 1993.) Nibbles and Tibbles are each but one cat: the former is a precise object and the latter is what it at ¿rst blush appears to be – a vague object with imprecise boundaries. A ¿nal set of considerations in favour of the view that the world is or might be vague arises from rumination on reasons pushing in precisely the opposite direction, against the view that the world might be vague. As we saw in Chapter 3, there is a signi¿cant school of thought according to which the world is completely describable in precise scienti¿c terms. From this perspective, the idea of vagueness in the world repels. ‘The idea repels, because it promises to forbid a complete description of all the facts in precise scienti¿c terms.’3 As already noted, however, the desire for such a description and its underlying ‘metaphysical proclivities’ (as Williamson describes them) is misplaced. A complete description of ‘all the facts’ seems unachievable in precise scienti¿c terms. The language of science appears to be descriptively incomplete if precise, and vague if descriptively complete. Far from a well-motivated desire excluding the idea of ontological vagueness, the failed prospects for a complete scienti¿c description of the world strongly suggest advocacy of the idea. For, if a complete scienti¿c description of the world cannot avoid vagueness, then we have good reason for supposing the world to be vague – 2 Morreau (2002) emphasizes the connection. Lewis (1993) mentions ontological vagueness as a way out of the problem but baulks at the suggestion. Keefe (2000) follows Lewis. 3 Williamson (1994: 249).

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our current best scienti¿c theories describe it as such. As Colyvan (2001) has pointed out, anyone with broadly naturalist tendencies ought to feel the tug of such reasoning. More explicitly, one might simply take the view that the best explanation of the ineradicability of vagueness in our scienti¿c theorizing is that the world it describes is itself vague. ‘To maintain that good science cannot be rid of vague representations is to say that the world constrains us to be vague even where we would most like to be precise’ (Copeland 1995: 91). In addition to these considerations in favour of ontological vagueness, countervailing considerations have also arisen in a number of quarters. Central among these has been an argument due to Gareth Evans which focuses on the connection between the possibility of ontological vagueness and de re vague identity, and claims that the impossibility of the latter entails the impossibility of the former. Much of the subsequent debate has taken its lead in one way or another from the extensive literature spawned by Evans’s argument and so we turn to that debate to learn its many lessons prior to an explicit characterization of ontological vagueness and the issues it gives rise to. 5.1

Vague Identity

Consider a pile of rubbish by the roadside.4 After having been swept into the gutter by a council worker, it is then left for the day. Some of the rubbish blows away. More rubbish accrues in the adjacent area and is later swept into the remaining pile, which, again, is left to stand for a time. After suf¿ciently many episodes of partial dispersal and addition we are confronted by a pile of rubbish which is neither determinately the same as the original pile nor determinately distinct from it. There will be a time when there is simply no determinate fact of the matter as to whether the original and later pile are one and the same pile of rubbish. We are confronted with a case of vagueness of identity. Yet another common example of vague diachronic identity involves a case of socalled ‘disrupted persons’. Suppose that a person, Alpha, enters a room and is subsequently subjected to some disruption that alters them psychologically and/or physically (involving, perhaps, some discontinuity in relevant respects). Depending 4 This example derives from Parsons (1987 and 2000: 120–21). Broome (1984) discusses an analogous case. Parsons’s detailed studies explicitly avoid using the term “vague”, focusing instead on what he describes as ‘indeterminacy’ because of what he sees as unwanted associations with the term “vague”. In particular, the indeterminacy in question is not vagueness in the sense of being semantic in nature. Nor should the indeterminacy be understood as vagueness in the sense of degree vagueness (as described in Chapter 1); the indeterminacy is not a matter of degree and does not lead to ‘degrees of truth-values’. Parsons also distances his account of indeterminacy from any commitment to vague objects and is theorizing merely about what I have termed ¿rst-order vagueness. Despite all these quali¿cations, however, Parsons points to cases of indeterminacy of identity of relevance to the current discussion. That he postulates non-degree, first-order, ontological indeterminacy of identity, independent of any commitment to vague objects, does not undermine the value of such cases to the more general debate now at issue.

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on one’s view of personal identity, the details of the disruption can be ¿lled in in such a way as to leave it vague whether the person who later leaves the room, Omega, is the same as the one who entered. It is vague whether Alpha is Omega.5 Cases of vague synchronic identity also appear easy to generate. Adapting an example from Quine, suppose that a linguistic community under investigation has been found to have named a local mountain “Aphla” and a local mountain “Ateb”. They are unaware of the fact that the visible peak they refer to as “Aphla” and the visible peak they refer to as “Ateb” are separated by a saddle whose dimensions in relation to the peaks is such that there is no fact of the matter whether the saddle separates two mountains or is the mid-region of a single, topologically complex mountain extended over two peaks joined by the saddle. It would seem then that it is vague whether Aphla is Ateb.6 Similarly, where it is vague whether an isthmus is part of an island, it is presumably vague whether the island is one and the same thing as the island plus isthmus. Geographers may debate the point, but short of settling whether or not the isthmus is a part of the island, there is simply no fact of the matter as to whether the island and the composite island plus isthmus are one. Geach’s puzzle about 1,001 cats can be developed in a similar way. Cats frequently shed hairs and other bits of matter, and for this reason it is sometimes unclear exactly what to count as parts of a cat. It may be unclear, for example, whether to count a loose hair as part of the cat the hair is in the process of falling from or not. There is an object that has that hair as a determinate part and another which determinately lacks that hair as a part. Call each such a ‘p-cat’. Many other p-cats presumably exist, each corresponding to a way of settling all indeterminacies concerning what to count as parts of the cat. There are, therefore, many p-cats. There is, however, but one cat. The unclarity we face concerning its individuation is a matter of there being no determinate fact of the matter as to whether the cat is identical to any given p-cat.7 Again, the lack of an answer is taken as a reÀection of the absence of any determinate fact of the matter. Temporal, spatial and mereological considerations also have obvious modal analogues generating vague transworld identities. For example, if we were to consider the possibility of the original rubbish pile changed in the ways described earlier, then the newly considered object would be such that there is no determinate fact of the matter as to whether it and the actual pile are one and the same pile of 5

See Par¿t (1984: 238–41), Van Inwagen (1988) and Parsons (2000: 8). For van Inwagen (1988) the case for vague objects, resting as he assumes it to do on vagueness of identity, depends crucially on such a case of diachronic identity. Deemed more compelling than any purported example of vague synchronic identity, it nonetheless is said to require a commitment to three-dimensionalism to be justi¿ably counted a case of vague identity. Fourdimensionalists are thus taken to be in a position to resist the argument from vague identity to vague objects. I think van Inwagen is wrong to suggest the four-dimensionalist can avoid cases of vague diachronic identity and vague objects. We return to this below. 6 The example is from Burgess (1990). Cook (1986) presents a similar variant. 7 Lewis (1993), of course, agrees that the sentence “the cat = p-cat” for any p-cat is vague, but denies that it is interpretable de re. There is no precise thing that is the cat (and so no pcat that is the cat) because “the cat” is a vague name lacking unique denotation. It exempli¿es ‘the inscrutability of reference’ embraced by semantic theorists like McGee and McLaughlin (2000).

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rubbish. Would the original pile of rubbish survive such changes? What about persons: what possible changes could I survive and remain the same person? Surely sometimes there is simply no fact of the matter. In each case, we seem to encounter de re vagueness of identity.8 As we shall see later, such vagueness is a result of the vagueness of the boundaries – temporal, spatial, mereological or modal – of the objects concerned. 5.1.1

Split indeterminate identity

The idea that there are cases of de re vague identity seems further evidenced by the apparently innocuous possibility of there being a series of small alterations that can be made to some initial object x leading eventually to some other object z where, moreover, we deny that any one alteration transformed x into z. There must therefore be some object y which is neither determinately the same as nor determinately different from either of x or z. This presents a case of so-called split indeterminate identity (SII).9 Such cases are characterized by there being objects x, y and z where there is no fact of the matter as to whether x = y, nor whether y = z, yet it is determinately the case that x ≠ z. For example, a series of disruptions to a person a might eventually result in a person c such that there is a clear difference between a and c, yet no single disruption may be thought suf¿ciently drastic as to change a into c. Rather, it may be contended that the change proceeds via someone, b, who has resulted from suf¿ciently many disruptions to what was originally a so that it is vague whether a is b, and who is suf¿ciently many disruptions short of what will eventually be c so that it is vague whether b is c. We shall return to consider split indeterminate identity more closely in the next chapter. For the moment note that an immediate objection against SII from transitivity will fail. Suppose for the sake of argument that indeterminate identity is a relation (a contentious supposition, as we shall see later when considering the Evans Argument). Supposing it then to be the case that it is indeterminate whether x = y, and indeterminate whether y = z; it does not follow that it is indeterminate whether x = z. (Were it to logically follow, cases of SII would, of course, be incoherent, as our imaginary objector contends.) Vague or indeterminate identity is not transitive. For example, if it is vague whether a = b, then by symmetry it is vague whether b = a yet it is not vague whether a = a. So it does not follow from its being vague whether x = y and vague whether y = z, that it is vague whether x = z; x may be determinately the same as z and, according to SII, may be determinately distinct from z. As we shall see, other arguments have been offered against SII, and indeterminate (or vague) identity more generally, but they too will be found to be fallacious. A special and even more contentious case of indeterminate identity, a particular species of SII, is what might be termed synchronous split indeterminate identity (SSII), where objects x and z are considered cotemporaneous. Parsons (2000) offers such a case of indeterminate identity as a means for resolving the puzzle of 8 9

For more examples see Kripke (1980: 51, n. 18); Wiggins (1986: 172–3). See Pinillos (2003).

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Theseus’s Ship. It is supposed that a ship sets sail and at sea a process takes place whereby each plank of the original ship is replaced with a new plank so that eventually all parts of the original ship have been replaced with new parts. Is the resultant ship simply the original ship, now rejuvenated? Considerations arising from other everyday examples of repair and rejuvenation would suggest an af¿rmative answer. But matters are complicated by the further supposition that the discarded parts of the original ship are assembled into a ship materially identical to the original. Is the resultant ship simply the original ship, now reassembled? In the absence of the ship with new parts, considerations arising from everyday examples of disassembly and reassembly would suggest an af¿rmative answer. When confronted by the ship with new parts and the ship materially identical to the original, we appear to have equally strong reasons for counting each identical to the original ship which set sail, yet by Leibniz’s Law the resultant ships are each apparently distinct from one another and thus, in light of the transitivity of identity, they cannot both be identical to the original lest they be counted identical to one another. Parsons’s solution invokes indeterminate identity. There is, he contends, simply no fact of the matter as to whether either resultant ship is identical to the original ship: it is indeterminate whether the ship with new parts is the original rejuvenated and indeterminate whether the ship with old parts is the original reassembled. We are faced with an example of split indeterminate identity where each of those objects said to be vaguely identical to the original are considered cotemporaneous.10 Such a response contrasts with others that might be considered. Adapting a solution offered in Prior (1968) to the analogous problem of ¿ssion, one might contend that, in fact, each of the resultant ships is identical to the original, identity is transitive, and consequently the resultant ships are one and the same thing; Leibniz’s Law is to be rejected. Alternatively, one might reject the implied symmetry concerning strength of reasons in favour of each resultant ship being counted identical to the original and claim that one but not the other is the original. (Parsons considers such responses but ¿nds them unconvincing, claiming that the ship-example can be adjusted to undermine intuitions that one clearly wins out over the other and that general criteria of identity that might be used to settle such cases – e.g. sameness of parts, etc. – lack general appeal.) Finally, one might accept the symmetry concerning the strength of reasons in favour of each ship being identical to the original and accept Leibniz’s Law and the transitivity of identity, and claim that neither of the resulting ships is the original.11 In respect of each question, “Is the resulting ship identical to the original?”, considerations arising solely from the resulting ship in question and the original ship, i.e. ignoring the alternative resulting ship, suggest an af¿rmative answer. To this extent a Prior-type response seems correct. Yet the apparent non-identity of the two resulting ships and apparent symmetry of the situation suggest a negative answer in each case. Our temptation to individuate the original in such a way as to 10

Parsons (2000: 1–8). The Prior approach has it that there is but one ship in the scenario described. Alternatively, one might claim there are two ships, or that there are three. Parsons claims that there is no fact of the matter as to whether there are two or three. 11

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make it identical to either of the resulting ships is undermined by the existence of some other thing. Perhaps, then, as Parsons suggests, there simply is no fact of the matter. Now it might be objected that whether an object considered at one time is identical to an object considered at a later time is an intrinsic feature of the object in question and is independent of extrinsic facts like the existence or non-existence of some other thing. From this perspective a Prior-type response is admittedly problematic, yet all other responses canvassed, including Parsons’s, violate this supposed fact that identity is intrinsic. Admittedly, if one takes this view, then, regardless of what one makes of the other responses to the puzzle of Theseus’s Ship, Parsons’s response – exemplifying SSII – will be thought contentious even if one admits cases of indeterminate identity, even split indeterminate identity. In denying the (determinate) identity of the original and resulting ships Parsons’s response will fall foul of the objection that extrinsic considerations are relevant to an object’s identity. Consequently, it might be contested whether or not the puzzle of Theseus’s Ship is best described as a case of synchronous split indeterminate identity. I agree with Parsons’s description of the case – identity is not intrinsic.12 In any event, there are other less obvious cases of SSII, cases involving vague objects and their precisi¿cations, as we shall see in §4. But irrespective of where one stands in relation to the above concerns regarding identity and intrinsicness, cases of SSII are not objectionable by virtue of their commitment to de re vague identity.13 As we shall see, de re vague identity does not succumb to the arguments levelled against it. In light of the examples discussed above, one might plausibly require that the identity relation sometimes admit of indeterminacy. That is, one might take the view adopted by Parsons and others that any reasonable account of identity must admit of the possibility of there being no determinate fact of the matter as to whether x is y, for some x and y – identity can sometimes be ontologically indeterminate, or as it is commonly put, there can be de re vague identity statements. This view has come to occupy centre-stage in debates about the very possibility of ontological vagueness. Since Gareth Evans’s landmark 1978 paper “Can There Be Vague Objects?”, debates over vague identity and vague objects have often been closely linked.14 By apparently linking the two, Evans’s brief note was seen as offering a substantive argument where previously there had been only prejudice. ‘Where previously those of us sympathetic to the view that there might be vague objects had nothing to engage with except the unreasoned intonement of a mantra … Evans was now presenting, perhaps for the ¿rst time, what appeared to be a cogent argument against that view.’15 It appeared to be cogent, but debate has raged since as to just 12

See Weatherson (2003:§7.1) for argument. Gallois (1998) wrongly rejects Parsons’s solution to the puzzle of Theseus’s Ship on just such grounds. 14 Salmon (1981: 243–4) subsequently presented a similar argument. So too Wiggins (1986: 173–6). 15 Burgess (1989: 113). 13

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how much substance can be elicited from what has become known simply as ‘the Evans Argument’. 5.2

The Evans Argument

In the light of the Àurry of papers and discussion on the matter since Evans’s paper appeared, it is difficult to uncontroversially state the Evans Argument and simultaneously show how it was supposed to offer a reductio ad absurdum of the view that there could be vague or fuzzy objects. The argument has been interpreted in a variety of ways and one might sometimes be excused for thinking it has been all things to all people. Nonetheless, after stating the argument we can, I think, get a fairly clear idea as to how it at least appeared to resolve the debate. The representationalist certainly seemed to many to be in possession of a powerful argument in favour of their view. So, heeding the words of Lewis (1988) that the Evans Argument, while being well known, is frequently misunderstood, we should determine just what the argument is and how it aims to achieve its end. Evans (1978) begs us to consider the possibility that the world might itself be vague. Rather than vagueness being a de¿ciency in our mode of describing the world, it would then be a necessary feature of any true description of it. It is also said that amongst the statements which may not have a determinate truth-value as a result of their vagueness are identity statements. Combining these two views we would arrive at the idea that the world might contain certain objects about which it is a fact that they have fuzzy boundaries. But is this idea coherent?

The question is subjected to the ensuing logical proof – the Evans Proof – which he presents as follows. Assume “a” and “b” are singular terms and that the sentence “a = b” has indeterminate truth-value. Let the operator “∇” be that sentential operator expressing the notion of indeterminacy, i.e. “∇A” is to be understood as expressing the view that it is indeterminate or vague whether A in the sense that A is indeterminate in truth-value. Then: (1) ∇(a = b). (1) tells us that we may ascribe the property ˆx [∇(x = a)] to b:16 (2) ˆx [∇(x = a)]b. And yet: (3) ~∇(a = a) 16 Note that Evans has implicitly appealed to the symmetry of identity. Abstraction on the ¿rst term will yield the abstract mentioned only if one ¿rstly appeals to symmetry to infer ∇(b = a) from ∇(a = b).

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thus: (4) ~xˆ [∇(x = a)]a. From (2), (4) and Leibniz’s Law: (5) ~(a = b) which Evans claims contradicts the initial assumption that the identity statement “a = b” has indeterminate truth-value. If we are not yet convinced of this, he invites us to strengthen the conclusion, (5), as follows. Let “∆” (“determinacy”) be the dual of “∇”. On the assumption that “∆” and “∇” generate a modal logic as strong as S5 we can pre¿x (1)–(4) and Leibniz’s Law with the determinacy operator and so strengthen the conclusion to: (5′) ∆~ (a = b). This, according to Evans, ‘is straightforwardly inconsistent with (1)’. What then is the moral concerning the possibility of vague objects? How might this ¿nal remark of the Evans Proof contribute to an argument bearing upon the ‘coherence’ of the notion of ‘objects having fuzzy boundaries’? The argument is of reductio ad absurdum form, having as its implicit conclusion some statement involving ontological vagueness, fuzziness. In order that a reductio proof might entail anything at all about fuzziness, it must engage the notion at some point in the proof before going on to show that the assumption involving fuzziness entails absurdity. Evans himself explicitly takes the abstract in (2) to describe a property, and thus ontological issues would appear to be engaged at that point. The assumption of (1) and subsequent preparedness to infer (2) would involve a commitment, Evans seems to think, to ‘a fact about b’, namely that it has the property of being such that it is vague whether it is identical to a. Coupled with his opening remarks on objects having fuzzy boundaries as a conjectured source of vagueness in identity statements, it seems that the Evans Argument proceeds by: (i) (ii)

inviting us to consider a necessary condition for the very possibility of there being vague objects with their ‘fuzzy boundaries’; and purporting to show that this necessary condition is logically unsatis¿able, issuing in logical contradiction.

Evans’s question as to the possibility of vague objects is answered in the negative by implication. But how does the Argument proceed in detail, and is it sound? Intense interest has naturally centred on part (ii) of the Argument, the Evans Proof itself, and its ¿rst contestable move is the abstraction inference from (1) to (2). This inference involves quanti¿cation within the scope of the indeterminacy operator, “∇”, which has sometimes been criticized as fallacious. Its status is not so easily determined, however, and is said to depend on the sense in which the indeterminacy is thought to apply, de dicto or de re, i.e. depending on whether the vagueness is attributable to

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the semantic vagueness of the terms involved and thus attributable to representational vagueness, or must be thought of as being due to the vagueness of that to which the expression refers and thus attributable to ontological vagueness. According to this line of reasoning, and I think it is correct, if the indeterminacy apparent in (1) were thought to be de dicto, mere semantic indeterminacy arising from our ¿xing the reference of “a” and/or “b” by vague descriptive means, then the attribution of a property to b, say (being such that it is vague whether it is identical to a), is fallacious. The indeterminacy is merely semantic and not attributable to any object. Lewis (1988: 128–9), assuming as Evans does that abstracts produced by quanti¿cation within the scope of “∇” denote properties, puts the point as follows: If vagueness is semantic indeterminacy, then wherever we have vague statements, we have several alternative precisi¿cations of the vague language involved, all with equal claim to being ‘intended’. These alternative precisi¿cations play a role analogous to alternative worlds in modal logic. The operator “it is vague whether …” is analogous to an operator of contingency, and means “it is true on some but not all of the precisi¿cations that …”. A term like “Princeton” that denotes different things on different precisi¿cations is, analogically speaking, non-rigid. When “a” is non-rigid, the … [change of scope] … is fallacious. It is analogous to the fallacious modal equivalence between “It is contingent whether the number of planets is nine” (true) and “The number of planets is such that it is contingent whether it is nine” (false) … .17

As Sainsbury (1995b: 65f) points out, we need to be careful regarding the strength of the suggested analogy between necessity and vagueness, and to be clear as to what is being claimed here. To be sure, a designator like “Princeton” will, analogically speaking, count as ‘non-rigid’ since (given it is a vague name) it has no unique precisi¿cation. Being ‘non-rigid’ in this sense is the essence of what it is to be vague. The analogy extends thus far. However, unlike the modal case (at least as conventionally understood), a designator’s being ‘non-rigid’ (i.e. vague) does not, in general, render the change of scope fallacious. Such a fallacy obtains when a change of scope occurs with respect to a term which is vague and, moreover, whose vagueness is representational – or, as Lewis puts it, ‘semantic’. If indeed the vagueness of “Princeton” is of this kind, then the designator is not only ‘non-rigid’, having a range of alternative precisi¿cations, but is also such that its non-rigidity is a consequence of its referent having been only imprecisely ¿xed with no candidate being thereby determined (i.e. a consequence of the vagueness of its reference relation) as opposed to its being a consequence of its referent having been precisely ¿xed and a candidate thereby determined, that though one is indeterminate in itself (i.e. a consequence of the vagueness of its referent). Following Thomason (1984), let us then de¿ne a designator to be a precise designator (to be distinguished from being a precise term) if and only if there is something determinately denoted thereby and so it is not vague what the designator picks out – i.e. there is no vagueness or indeterminacy in ¿xing the reference of the designator. Recalling Russell’s analysis of vagueness from Chapter 2, such designators exhibit one–oneness of denotation. A designator is an imprecise 17

Thomason makes essentially the same point in reference to the move in the Evans Argument from (3) to (4); see Thomason (1984: 331).

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designator if and only if it is not precise, that is, if and only if there is no thing determinately denoted thereby and its reference is ¿xed in such a way that it remains vague which from amongst the range of admissible candidates its referent actually is. Thus we might gesture vaguely at a portion of the map intending to pick out some precisely demarcated area, naming ‘the area pointed at’ “Princeton” when in fact no such candidate is determinately individuated thereby. The gesturing is indeterminate between a number of distinct precisely demarcated areas, it being vague of each whether it is the referent of the term “Princeton”. Such designators exhibit onemanyness of denotation. We shall sometimes say of a precise designator that it precisely designates the thing determinately denoted thereby, and of an imprecise designator that it imprecisely or vaguely designates any of the many admissible candidate referents. (Of course, if all vagueness is representational, then a precise designator is also ‘rigid’, having a unique precisi¿cation. Or, as Russell would say, if all vagueness is representational, then designators exhibiting one–oneness of denotation are precise, i.e. all vagueness is then one–manyness, a matter of vague designation. If all vagueness is representational, then precise designators present cases where it is not vague what the designator picks out nor can that which is determinately picked out be in any sense vague; thus determinate reference to a determinate object is secured and there is no scope for vagueness to enter. If vagueness can in some way be attributed to the referent of a designator, then cases of precise, though ‘non-rigid’, designators may arise; i.e. in Russellian terms, in such cases we may have vagueness without one–manyness. In such cases we would be confronted with a term which is precise qua designator, though vague qua name.) Now the point being made above by Lewis can be stated as follows. When a designator is imprecise, then the change of scope as exempli¿ed in inferences like that from (1) to (2) is fallacious. To secure its validity we must suppose the designator in question to be a precise designator. With the foregoing distinctions to hand we see that (1) is to be understood as involving vagueness de dicto just in case the terms “a” or “b” are imprecise designators and de re otherwise, i.e. just in case the terms therein are precise designators.18 So, if the vagueness invoked in (1) is de dicto with either or both of the singular terms “a” or “b” imprecise designators which in fact exhibit one–manyness of denotation, then a scope fallacy obtains in either the inference from (1) to (2) or from (3) to (4). We cannot suppose that in each case the identity claim involves predicating something of an object a or b since with respect to imprecise designators no object is determinately denoted by the relevant singular term. The move alluded to by Lewis, our being able to quantify into the scope of “∇”, is sanctioned in each case only if the singular terms are precise designators with some object determinately denoted thereby. So, if the vague identity involved is understood as indeterminacy de dicto, then the proof is fallacious, as Lewis suggests. 18 We are assuming that the vagueness of identity claims is not attributable to the identity predicate itself. This follows from the fact that though the predicate has borderline cases, it does not possess the required resilient borderline cases to count as vague. The only candidates for the vagueness of identity claims are thus the singular terms therein. The nature of their vagueness, de dicto or de re, determines the nature of the vagueness of (1) overall.

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This result is hardly surprising. Whether Evans’s intended reductio proof is ultimately sound or not, it does not preclude the possibility of there being semantically indeterminate identity statements – any supposed proof to the effect that there can be no semantically indeterminate identity statements is blocked.19 However, if the singular terms are indeed taken to be precise designators and the vagueness is assumed to be de re and ultimately attributable to ontological indeterminacy, no scope fallacy is said to obtain. From the assumption of (1) the inference to (2) is sanctioned: de re vague identity involving precise designator “b” is taken to entail b’s having the property denoted by the abstract, legitimating the view that in such cases one is committed to the attribution of the relevant property to the relevant object. This, continuing Evans’s apparent line of reasoning, is something to which the believer in vague objects is committed (and which leads to inconsistency).20 As we shall see, this is where the Evans Proof goes subtly awry, but more on this diagnosis later. For the moment let us continue with our search for a plausible interpretation of the sometimes-cryptic reasoning. 19

Note that Sainsbury (1995b: 66–8) suggests that the proof is not blocked by the supposed invalidity of the abstraction-inference from (1) to (2) or (3) to (4). Abstraction is valid even given a de dicto reading of the vague identity statement. Endorsing the rest of the Evans Proof, the consequence according to Sainsbury is that all vague identity claims are incoherent. Nonetheless, just as de re vague identity is considered by Evans a necessary condition for their being vague of objects, so too according to Sainsbury is de dicto vague identity a necessary condition for their being vague names. As a consequence, he argues, there can be no vague names and he subsequently sets out to explain how we might have been misled into thinking otherwise. However, to the extent that such a semantic reconstrual of the abstraction-inference is possible it cannot simultaneously be supposed that the generated abstracts are legitimate substituends in the subsequent application of Leibniz’s Law. Permissive abstraction in conjunction with a view that all such abstracts denote properties possessed by objects denoted by the remaindered singular term (and so are relevant to the individuation of objects by means of Leibniz’s Law) quickly leads to absurd results. Moreover, as we shall see in §3, abstraction in the presence of de dicto vagueness is indeed illegitimate. 20 There are other versions of the argument, that due to Wiggins for example, which do not make use of abstraction but simply employ a substitution principle like the following: a = b → (∇(c = a) → ∇(c = b)), for any a, b, and c. Now it might be wondered, as Rasmussen (1986: 82–3) does, how any such argument can reduce to absurdity the supposition that there can be vague identities due to ontological vagueness since that Evans inference which guarantees that the vagueness of the identity claim is not semantic, the abstraction-inference, is not employed in such proofs. However, the substitution principle itself is commonly restricted, being said to be valid only if the names “a” and “b” are precise designators. Just as the inference from a wide to a narrow scope reading of (1) is said, by analogy with modal contexts, to be legitimate only if the names in (1) precisely designate, it is argued, by analogy with modal contexts, that the principle of substitution is only valid if the names “a” and “b” precisely designate. If they did not then the substitution made would be analogous to the following in a modal context: “(the number of the planets = 9) → (it is contingent that the number of the planets = 9 → it is contingent that 9 = 9)” which is false. If the substitution principle is taken to guarantee substitutivity within the scope of “∇”, then “a” and “b” are precise designators. Either proof of the incoherence of vague identity is taken to go through (if at all) only on the assumption that “a” and “b” are precise designators – that is, on the assumption that any indeterminacy in identity is ontological.

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Given the argument as interpreted so far, the Evans Proof supposes that a de re reading is both suf¿cient and necessary for the validity of the inferences from (1) to (2) and (3) to (4). All and only those prepared to entertain the possibility of vague identity statements involving precise designators are committed to (2) and (4) (and thus fall foul of the incoherence entailed by their joint assumption). The negative conclusion that Evans goes on to draw in respect of such statements is then neutral on the more general issue of whether or not there are indeterminate identity statements. What is denied by Evans is the possibility of there being indeterminate identity statements that are interpretable de re. Note that one need not further attribute to Evans the view that there are indeterminate identity claims. His proof is silent on this issue. His claim is simply that if there are, they must (on pain of incoherence) be read de dicto. In this sense, then Evans is not committed to the vagueness of identity statements.21 Nonetheless, such a view seems easily justi¿ed by merely concocting or citing an example using a singular term that is vague by virtue of having its reference ¿xed by vague descriptive means. ‘Example: “Princeton = Princeton Borough”. (It is unsettled whether the name “Princeton” denotes just the Borough, the Borough plus the surrounding Township, or one of the countless somewhat larger regions.)’22 The acceptability by most of the belief in the existence of vague identity statements is further evidenced by the response Evans’ argument was often said to elicit: namely that he had endorsed a fallacious proof of the absurd conclusion that there can be no vague identity.23 Though the underlying belief in the existence of vague identity statements is well-founded, the response is misguided as we have seen. Evans is not committed to the impossibility of vague identity statements. Nor, on the other hand, is he committed to their existence, though such a commitment seems hardly controversial. Having now given an interpretation of what is going on in the initial part of the Evans Proof, we see that the validity of the inference from (1) to (2) is said to be justi¿ed by the assumption that (1) be interpreted as a de re vague identity relation obtaining between a and b. Thus interpreted, (1) might then be taken to constitute the postulated necessary condition for the possibility of there being vague objects appealed to in part (i) of the Argument. It is certainly an inviting construal of the Argument. Evans was apparently trying to prove a de re reading of (1) to be inconsistent, thereby proving the possibility that there might be vague objects to be incoherent. This suggests that he took de re indeterminate identity to be a necessary condition for their being possible. It is relatively uncontroversial that it is a suf¿cient condition but, of course, it is only the identity clause as a necessary condition for fuzziness that is crucial to the Argument. Many have certainly taken this line or assented to claims that could plausibly lead to this view. 21

See also Garrett (1988: 131). Lewis (1988: 128). Garrett (1991: 341) cites another example. See also Williamson (1994: 253–4) and Heck (1998: 280) for alternative means of generating vague identity statements. 23 See Lewis’s comment (1988: 129), ¿rst paragraph; Rolf (1980: 72–3); and Noonan (1982: 3–6). 22

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Returning again then to the conjectured interpretation of the Evans Argument described by (i) and (ii) above, we can now state more explicitly what appears to be the intended interpretation as follows. Evans is, it seems, inviting us to consider the possibility of de re vague identity as a necessary condition for the possibility of there being vague objects with their ‘fuzzy boundaries’, and subsequently invoking what we have called the Evans Proof, which purports to show that this necessary condition is internally inconsistent. Now that we have a plausible account of where object-vagueness was considered to engage with the Evans Argument, we can look to see what the reductio proof purports to show regarding this vagueness. 5.2.1

The Evans Proof

Assume there to be some object, b, possessing the property ˆx [∇(x = a)]. By the law of self-identity, a = a. Moreover, ‘even if a were a vague object, we still ought to be able to obtain a (so to speak) perfect case of identity, provided we were careful to mate a with exactly the right object. And surely a is exactly the right object to mate with a. There is a complete correspondence. All their vagueness matches exactly.’24 So we may reasonably suppose it to be determinately the case that a = a. Given the determinacy of self-identity, it follows that it is not the case that it is vague whether a is identical to a, i.e. ~∇(a = a). Thus (given the validity of abstraction) a may be said not to possess the property ˆx [∇(x = a)], i.e. ~xˆ [∇(x = a)]a. Now, in accordance with identity criteria speci¿ed by the contrapositive of Leibniz’s law, the law of de¿nite difference: (DD) ∃F(Fx & ~Fy) → ~(x = y), for objects x and y, and property F, it is said that, since a and b differ on some property, they are not identical, ~(a = b). This, according to Evans, ‘contradict[s] the assumption, with which we began, that the identity statement “a = b” is of indeterminate truth-value’. However, given the lack of outright inconsistency between the two claims, the contradiction is not yet apparent. Alternative variations on the Evans Proof sometimes ¿nd themselves in a similar situation. For example, the contraposed variant proposed by Wiggins and Sainsbury, having purportedly established that if a = b, then determinately a = b, similarly suggests that indeterminate identity is as a consequence incoherent. Sainsbury (1988: 47), for instance, claims that showing that identity is not a vague relation is equivalent to showing ‘that questions of the form “Is this thing … the same as that thing …?” have de¿nite [determinate] answers. The suggestion is that, quite generally: if β is α, then β is de¿nitely [determinately] α.’25 However, it is no more obvious how a proof of the latter suggestion would provide de¿nite answers to the relevant questions concerning identity relations, thus eliminating the possibility of such relations obtaining vaguely, than it is obvious how the Evans Proof to this point 24

Wiggins (1986: 175). See also Akiba (2004: 417). Note that in Sainsbury (1995a) the erroneous claim is corrected. 25

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serves to show the incoherence of ‘the assumption with which [it] began’. To say that questions of identity have determinate answers presumably means that answers like “determinately yes” or “determinately no” are the only ones possible, but the suggestion is only to the effect that if the answer is af¿rmative, then the answer “determinately yes” is required. What about answers in the negative? If it is not the case that one thing is the same as another, is it determinately different? The lack of an established answer is as much a problem for the Wiggins/Sainsbury variant as is the lack of a derived contradiction in the Evans Proof. No contradiction appears immediately derivable, as yet. Of course, the argument to this point would, if valid, immediately serve to rule out one particular position with respect to vague identity. De re vague identity would be shown to be inconsistent with the obtaining of identity simpliciter. It is perhaps worth noting in passing then that if, as has been suggested,26 Evans interpreted “∇A” to mean “A is true but indeterminately so”, then a contradiction would now be derivable given the proof of ~(a = b) from ∇(a = b). Such a reading of “∇” would make the Evans Proof strictly analogous to versions of arguments against contingent identity. From the assumption that an identity statement involving only rigid designators is true but contingently so we can, by the obvious and usual reasoning, supposedly prove that a ≠ b. De re contingent identity is thus said to be incoherent. Analogously, on this reading of “∇”, de re vague identity is purportedly incoherent. The position being ruled out, however, is not one generally thought a contender anyway. Few, if any, are prepared to interpret “∇” in this way – most preferring an interpretation according to which “∇A” means “It is indeterminate whether A” and is made true by either the vagueness of A or the vagueness of not-A – and even if one does, the subsequent diagnosis I shall offer of the Àaw in the reasoning of the Proof to this point will invalidate such an argument thus interpreted anyway. Though an interesting aside, it should not delay us longer. So is it possible coherently to maintain that, though a is not identical to b, it is nonetheless indeterminate or vague whether a is identical to b? If such a position is not already inconsistent, as Evans seems to think it is, it is ultimately unstable, he further suggests, because on the assumption that “∇” and ‘its dual’ “∆” generate a modal logic as strong as S5, the assumptions of the Proof may be taken to be determinate, pre¿xed with “∆”, and so we may infer the stronger conclusion ∆~(a = b). This latter claim is then taken to straightfowardly contradict our initial assumption that ∇(a = b). The ¿rst thing to note with this strengthening of the argument is the introduction of “∆” as the dual of “∇”. That is: (Dual) ∆ A ↔ ~∇~A. Given that it is indeterminate whether A if and only if it is indeterminate whether not A, it follows that the operators represent contradictory notions: (C) ∆ A ↔ ~∇A. 26

See Parsons (2000: 45).

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Thus, pursuing Lewis’s analogy between “∇” and contingency, the dual notion “∆” is then analogous to non-contingency. To say A is determinate is thus to say that it is determinate whether A, a claim which by the above principles is equivalent to its being determinate whether not-A. Either the determinate truth of A or the determinate falsity of A is suf¿cient for A’s being determinate. This reading of “∇” and “∆” as duals contrasts with a rival reading according to which “∆” is to be read as “it is determinate that …” and which supports the principle: (T) ∆A → A. As Heck (1998) points out, this reading of “∆” – now analogous to “■” – undermines the duality principle. Accepting (Dual) would require a reading of “∇” – now analogous to “◊” – according to which premise (3) of the Evans Proof is equivalent to the claim that ∆~ (a = a), which, given (T), entails ~(a = a), i.e. a rejection of self-identity.27 Acceptance of (T) would thus require abandoning duality and treating “∇” and “∆” as analogous to contingency and necessity respectively. (C) must also then be rejected in favour of a principle expressing the fact that the operators represent not contradictory notions but contrary ones: ∆A → ~∇A. Acceptance of (Dual) on the other hand demands that we abandon (T) and treat the Evans operators as analogous to contingency and non-contingency respectively. On either interpretation, Evans’s strengthened conclusion is inconsistent with the assumption of de re vague identity as expressed by means of “∇”. Heck (1998) pursues the strengthened Proof with an interpretation of “∆” satisfying (Dual) as Evans suggested, with the Proof then said to depend for its validity on the key S5like principle: (5∆) ∇A →∆∇A. Strengthened in this way, it is easily shown that, assuming the validity of the unstrengthened Proof, the strengthened Proof is valid and thus de re vague identity is reduced to absurdity. As Heck points out, though, Evans’s own intentions aside, the interpretation of determinacy most commonly invoked in discussions about vagueness and modelled previously by logics like supervaluationism is that interpretation satisfying (T) above. With “∆” interpreted in this way as analogous to “It is necessary that”, in conjunction with an S5-like principle syntactically equivalent to (5∆) and a further plausible principle, an analogue of the modal rule of necessitation: (Det) if = A then = ∆A, 27 Williamson’s acceptance of the analogy between the pairs “∇” and “∆” and “◊” and “■” with the associated commitment to (D) and (T) leads him to reject the reading of “∇A” as “It is indeterminate whether A” and propose instead “It is not determinate that not A”. (See Williamson 1994: 304, n. 11.) Given the problems that then attend the reading of (3), such a reading should be rejected.

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we can similarly reduce the assumption of de re vague identity to absurdity assuming the unstrengthened Proof is already valid. To see this, notice ¿rstly that (5∆) would justify the claim that premise (1) of the proof is determinate: (1′) ∆∇(a = b). Moreover, given (Det), premise (3) and the Liebnizian principle DD can both also be said to be determinate: (3′) ∆~∇(a = a) and (DD′) ∆(∃F(Fa & ~Fb) → ~(a = b)). Thus, on the assumption that the unstrengthened Proof is valid – i.e. from (1), (3) and DD we can infer (5) – and that each of (1), (3) and DD can be said to be determinate in the manner just described, we might well agree with Evans that the original conclusion (5) can be strengthened to (5′). If (5) is a valid consequence of (1), (3) and DD and each of these assumptions holds determinately, then, given the distribution of “∆” over “→”, it is determinate that (5). That is to say, given (1), (3) and DD, it follows by the principles described that ∆~(a = b). And this does indeed contradict the assumption with which we began, namely the assumption that there can be de re vague identities. Having said all this, we can avoid getting bogged down in details relating to the strengthened Proof and its evaluation.28 The appeal to (5∆) on either interpretation of “∆” is widely thought to be unjusti¿ed, the frequent (and, as we shall see in Chapter 7, entirely justi¿ed) criticism being that such a principle excludes higherorder vagueness. Moreover, regardless of the legitimacy of the strengthening, the unstrengthened argument is frequently considered already suf¿cient to establish an inconsistency if valid – as Evans suggested – yet, as I shall argue, it is already invalid anyway – despite Evans’s and others’ view to the contrary. The view that the unstrengthened Evans Proof, if valid, already succeeds in reducing (1) to contradiction is supported by the claim that though no explicit contradiction has been formally derived, a semantic inconsistency is seemingly already to hand – the truth of (1) is already inconsistent with the truth of (5). Of course, in order to see this, some semantic framework for “∇” must be employed and in the absence of any speci¿ed semantics one must be imported. Various obvious choices present themselves. Restricting ourselves to paracomplete logics, as discussed earlier (Chapter 4,§3), notice that the weak paracomplete logic already considered, SpV, validates such a position, as do strongly paracomplete systems like Kleene’s three-valued logic, K3, Łukasiewicz’s L3, and paracomplete many-valued logics.29 On such semantics it is true that ∇(a = b), i.e. “a = b” is indeterminate in 28

For further discussion of the issues see Heck (1998) and Williamson (1994: 304, n. 11). All such systems have been advocated as semantics for “∇”. Whether they offer plausible semantics for vagueness understood as an ontological phenomenon, as required for the Proof to properly engage with its intended target, is another matter. Nonetheless, since at this point we are trying to interpret Evans’s position charitably, being generous in this regard is appropriate. 29

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truth-value, just in case it is neither true nor false that (a = b). Assuming the Proof valid, the conclusion would then follow to the effect that “~(a = b)” is true. This contradicts the claim of vague identity on the further, plausible assumption that the truth of “~(a = b)” entails the falsity of “(a = b)”.30 Inconsistency obtains. Nonetheless, the Evans Proof (and, as a consequence, the associated Evans Argument) is far from compelling anyway. Its validity is questionable. In particular, the supposition that from the assumed de re reading of the vague identity claim (1), “∇(a = b)”, we are entitled to infer that b possesses the property denoted by the abstract “xˆ [∇(x = a)]” lacks justi¿cation. De re vague identity involving precise designator “b” is unjusti¿ably taken to entail b’s having the property denoted by the abstract, falsely legitimating the view that in such cases one is committed to the attribution of the relevant property to b – a property not shared by a and which thus counts as evidence of (5), the non-identity of a and b. To be sure, the Evans Proof shows that one cannot consistently claim that an object b (for example) possesses the property supposedly denoted by “xˆ [∇(x = a)]” (since, as we’ve seen, the property itself is one which can subsequently be used to determinately differentiate a from b, thus contradicting b’s possession of the supposed property in the ¿rst place). Yet, it remains to be seen whether there is any inconsistency involved in (1) interpreted de re. As Parsons (2000: 48f) has argued, to suppose that the identity in question is de re vague does not itself constitute grounds for claiming that b possesses the property denoted by the abstract “xˆ [∇(x = a)]”. We should, I think, concede that the process of abstraction itself is legitimate to the extent that the vagueness in (1) is de re. Abstraction thus viewed is simply a way of dispelling scope ambiguities that arise when conÀating de re and de dicto attributions of indeterminacy.31 The change of scope is then justi¿ed since “b” counts as a precise designator (and the same holds for abstraction with regard to “a” since it too counts as a precise designator given the de re attribution of vagueness). The question then is how could we understand (1) so constrained if not as the attribution of the property to b? That is, doesn’t (2) simply describe b as having the property denoted by the abstract, as Evans claims? In short, ‘No’. Abstraction reÀects the fact that the relevant indeterminacy concerns b itself rather than the (precise) designator “b” but it is a further claim to then go on to suggest that this indeterminacy amounts to b’s possessing the property of being vaguely identical to a. Intensional contexts already undermine our general preparedness to accept that every abstract denotes a property, so construing “∇” intensionally would be suf¿cient to thwart such a claim. However, an intensional reading is not necessary for the claim’s rejection (contra Burgess 1990). Abstracts and the question of what, if anything, they denote may be problematic for reasons other than intensionality. Firstly, impredicativity and accompanying paradox looms as a concern. As Parsons points out, from the point of view of one advocating the possibility of de re 30 The assumption is, in effect, that “~” represents a strong, internal negation whose truth is equivalent to the falsity of the negand as opposed to a weak, external negation whose truth requires only the non-truth of the negand. That the negation is indeed strong, validating the foregoing reading, follows from the negation attaching to Evans’s claim (3) being strong and the relevant contraposed version of LL, DD, using strong negation throughout. 31 For discussion, see Burgess (1990: 266ff).

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vague identity there is a striking similarity between the above Evans abstract and its role in the Evans Proof and the well-known Russell set-abstract and its role in Russell’s paradox. As already noted above, the Evans Proof shows that one cannot consistently claim that an object possesses a property denoted by “xˆ [∇(x = a)]”. From the point of view of the de re vague-identity theorist the supposition that the abstract in question denotes a property leads immediately to inconsistency (i.e. that the property both does and does not apply to b, say). Moreover, this inconsistency is a direct consequence of the impredicative nature of the property de¿ned by means of the relevant abstract. Given the usual de¿nition of identity in terms of coincidence of properties and the permissibility of intersubstitutivity of de¿nitional equivalents within the scope of “∇”, the claim that it is indeterminate whether a is b, i.e. ∇(a = b), amounts to the claim that it is indeterminate whether a and b share all properties, i.e. ∇∀F(Fa ↔ Fb). Thus the abstract “xˆ [∇(x = a)]” can be rewritten as “xˆ [∇∀F(Fx ↔ Fa)]” and the supposition that the former abstract denotes a property now amounts to the claim that the latter abstract denotes a property – a property which inherits the impredicativity associated with the de¿nition of identity which gave rise to it. The candidate property is de¿ned by reference to a totality of which it forms a part, namely the totality of all properties. Now, not only is the candidate property de¿ned impredicatively, but it is precisely this impredicativity which generates inconsistency. If the de re identity claim “∇(a = b)” were to be analysed as b’s instantiation of a property denoted by “xˆ [∇(x = a)]”, then the very property postulated, when added to the totality of properties in terms of which it is de¿ned, is thereby precluded from being instantiated by b and contradiction immediately ensues. This phenomenon whereby impredicativity generates paradox is already familiar from the case of Russell’s paradox. In that case the supposition that the abstract “the set of all sets that are not members of themselves” denotes a set leads immediately to inconsistency. Moreover this inconsistency is a direct consequence of the impredicative nature of the set de¿ned by means of the abstract since contradiction arises when the supposedly denoted set is added to the totality of sets in terms of which it is de¿ned. Parsons is quite right to point to the similarity. Even more striking is the similarity between the Evans Proof and Grelling’s paradox. This paradox deals explicitly with properties and forcefully exhibits the consequences of always supposing that abstracts denote properties. Consider the impredicative abstract “the property of all properties that are not applicable to themselves”, i.e. “xˆ [x is heterological]”. If this abstract denotes a property (that property of properties that is heterologicality), then, given the impredicativity of the abstract involved, we can ask whether the property supposedly denoted thereby is one of those to which the property applies. If it is, then it is heterological (since it satis¿es “xˆ [x is heterological]”) and yet, since it applies to itself and so is by de¿nition homological, it also follows that it is not heterological. The supposition that the impredicative abstract denotes a property leads to inconsistency by virtue of the impredicativity involved. This being the case, two possibilities present themselves. One might accept the supposition and permit unrestricted property-abstraction, embrace the ensuing paradox and consequently pursue a non-classical, paraconsistent response. This approach to property-abstraction mirrors the paraconsistent, naive set theorist’s

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attitude to set-abstraction. Such a view, with its liberal account of properties, requires abandonment of reductio ad absurdum as a valid rule of inference and thus the Evans Proof fails to establish the incoherence of de re vague identity any more than Russell’s paradox establishes the incoherence of the Russell set. Alternatively, one might restrict property-abstraction and reject the supposition that any abstract denotes a property. In the case at hand, the latter option is to be preferred since there is a restriction which is not ad hoc which justi¿es the view that the abstract involved in the Evans Proof does not denote a property but, rather, describes the absence of fact concerning property possession. Keefe (1995: 187–8) makes the point nicely. … the key issue for the assessment of [the Evans Proof] is whether delta-predicates [i.e. abstracts involving “∇”] denote properties … I maintain that they do not. The indeterminacy operator plays the role of indicating that it is indeterminate whether something has a given property. Expression of this should not be taken to be the (determinate) ascription of another property. If it is indeterminate whether a is F, there is no fact of the matter about whether it is F – the facts do not thereby determine that a has a property accounting for this indeterminacy. Not every statement about a can be construed as specifying that a has a property. Some statements might describe a mode in which it has a property, whilst … statements containing the indeterminacy operator express that it is indeterminate whether it has a given property. If we allow that it can be genuinely indeterminate whether something has a particular property, then we must deny that “it is indeterminate whether …” denotes a further property. Assuming that it does … begs the question against this possibility.

Its being ‘a fact about b’, as Evans puts it, that b is vaguely identical to a is thus to be explained by saying that there is no fact of the matter as to whether a is b. (2) thus reÀects ‘a fact about b’ in the sense that b is such that there simply is no fact as to whether it is identical to a. It expresses there being no fact of the matter. But this absence of fact is not itself to be explained by pointing to a property that b possesses, i.e. to the instantiation of a further property, but rather by pointing to the absence of fact in respect of instantiation. In short, then, when all is said and done, Evans’s claim that the de re vague identity claim “∇(a = b)” ‘reports a fact about b which we may express by ascribing to it the property “xˆ [∇(x = a)]”’ is to be rejected. De re ascriptions of vagueness are not to be explained as the possession of a property involving “∇” (i.e. possession of a property denoted by a delta-predicate) by an object named in the ascription. If one insists (e.g. by stipulation) that abstracts are to be understood as denoting properties, then the inference from (1) to (2) in the Proof will be fallacious since a de re reading of (1) does not entail the legitimacy of abstraction understood in this metaphysically loaded way. Alternatively, if one understands abstraction as a device which merely makes explicit scope distinctions suf¿cient for differentiating de re from de dicto ascriptions of indeterminacy, as I have suggested (following Burgess), then the inference from (1) to (2) is naturally valid, but the interpretation that Evans places on (2), subsequently legitimating it for use in DD, is not warranted and the inference from (2) and (4) to (5) is invalid. By dilemma, then, the Proof is invalid.

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The Proof’s invalidity therefore undermines the soundness of the Evans Argument, relying as it does on the presumed incoherence of de re vague identity for which the Proof is evidence. Even supposing, what remains to be shown, that de re vague identity is necessary for object-vagueness, no adequate reason has yet been given for thinking that the representationalist’s case is established and vagueness of objects is incoherent. 5.3

Vague Objects

How might we then proceed to a positive characterization of object-vagueness? It is to this we now turn. When this is completed we will be in a position to scrutinize that other assumption of the Evans Argument, the claim that object-vagueness entails the existence of de re vague identities. 5.3.1

Characterizing vague objects

If indeed we accept that Tibbles is the vaguely bounded cat she appears to be, then what account are we to give of such vagueness? The answer is becoming increasingly familiar. Cases of such spatial indeterminacy are described in Sainsbury (1989) and defended in Burgess (1990), Tye (1990, 2000), Morreau (2002) and Rosen and Smith (2004), and all articulate a similar criterion for what it is for an object to be vague or ‘fuzzy’ – a form of vagueness of composition. There may sometimes be no fact of the matter whether a particular collection composes to constitute Tibbles since at a particular point in time there may be no fact of the matter as to whether the spatially extended object Tibbles includes all that the collection includes. Tibbles’s spatial boundaries are vague in the sense that exactly what composes the object at the time in question is a vague matter. This much having been said, there are two obvious ways of proceeding depending on just how we further elaborate the collection in question. Are we speaking of collections of matter that constitute the object or collections of parts that constitute the object? Taking the collection to be a mere quantity of matter, we might then suppose that the spatial indeterminacy exhibited by Tibbles concerns indeterminacy of material composition. On the other hand, taking the collection to be a collection of cat-parts (e.g. whiskers, tail, head, paws, etc.), we might suppose that the spatial indeterminacy concerns mereological composition. The distinction, emphasized by Morreau, can be seen in the difference between our speaking of a certain portion of matter as being simply ‘part of’ Tibbles, and our speaking of a certain thing (e.g. a whisker) being ‘a part of’ Tibbles. With this distinction in mind, then, we might proceed to elaborate the spatial indeterminacy exhibited by Tibbles by speaking of indeterminacy concerning the matter that composes or constitutes a, or by speaking of indeterminacy concerning the parts that compose a. Sainsbury (1995b) explicitly considers vagueness of material composition (which he subsequently rejects as justifying a belief in the existence of vague objects for reasons independent of current concerns and to which we will return below). Burgess and Morreau, on the other hand, focus on vagueness of mereological composition. Tye, despite defending ontological vagueness against

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Sainsbury, speaks not of Sainsbury’s concern with material composition but rather of mereological composition. Tibbles’s vague spatial boundaries can be characterized in terms of vague composition of either kind. Since material objects can be said to have parts (even if only improper parts) yet not all objects with parts are material, it would seem that the mereological conception of vague objects affords a more general approach enabling not just the characterization of vague concrete objects like Tibbles but also more abstract objects like species, nations and families. Species evolution might throw up individuals for which it is indeterminate whether they are part of an existing species or not, nations may be vague in respect of their territorial parts, and a family may be such that it is indeterminate whether someone is a part of it or not. Tibbles is not alone in having borderline parts. Let us say initially then that an object a counts as mereologically vague if it has some borderline part, i.e. if there is some y such that it is vague whether y is a part of a: (P) ∃y∇(y is a part of a). Most material objects will, at some time or other, count as vague in this sense. Things are forever coming apart from the surface of material objects, and doing so gradually so as to produce occasions when there is no fact of the matter whether they are or are not a part. Abstract objects, in so far as they persist through change, can similarly be vague. When, for example, did Marie’s new beau become ‘part of the family’? Such things generally take time and gradually come to be, and so at some time the family will be a vague object. So too with species (just think of the gradual process of evolution) and nations (just think of the gradual process of nationbuilding). Two familiar problems immediately come to mind in relation to (P). Firstly, we will need to appeal to resilient borderline parts to avoid faulty attributions of vagueness, and secondly, we will need to insist on a de re understanding of (P) to avoid trivializing the content of the ontological claim of object-vagueness.32 (The problem of higher-order vagueness is also lurking in the background throughout. Some will object that we are only dealing here with ¿rst-order vagueness, properly speaking. We will continue to theorize in this way, returning to consider higherorder vagueness in Chapter 7, §4.) The ¿rst problem is the ontological analogue of a problem that beset the characterization of predicate-vagueness in Chapter 1. Just as having a borderline case was deemed insuf¿cient for a predicate’s being vague and something stronger, namely a resilient borderline case, was required, so too here. Having borderline parts will not be suf¿cient for an object’s being vague since intuitively there can be precise objects having borderline parts – that is to say, there can be precise objects that are such that it can sometimes be indeterminate whether a thing is a part of it. Recall the discussion of Chapter 1, §4 concerning Quine National Park and Russell 32 Both problems are touched on by Sainsbury (1989) and Tye (2000), though Tye subsequently denies that the ¿rst is a problem – no precise object can have borderline parts – and the second is subsequently dealt with in a spirit similar to that pursued below.

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River. Analogous to problems there concerning the vagueness of names and borderline cases, with the current ontological turn we might now suppose the Park’s southern boundary to be a sharp boundary drawn so as to run parallel with and include most if not all of Russell River. It may then be vague whether the River is a part of the Park since, despite the sharpness of the Park’s boundary, vagueness attends the drawing of the River’s boundaries. The fact that some part is such that it is vague whether it is part of a might similarly be explicable by pointing to the vagueness of the part.33 Thus we ought to require that an object a count as vague only if it has some resilient borderline part, i.e. something for which it is vague whether it is a part of a and, moreover, which does not present us with a case of a determinate part and a case of a determinate non-part when the indeterminacy as to how to delimit the part is variously eliminated. In this way we can safely say that the indeterminacy being claimed as evidence of vagueness of a does not arise by virtue of the vagueness of that which counts as its borderline part – any vagueness associated with the part itself is inert. Henceforth, unless otherwise stated, we shall suppose that when talking of borderline parts we mean resilient borderline parts. The second problem emphasizes the need to properly capture the ontological basis for (P). As with the case of vague identity, for the necessary ontological bite such a claim must be understood de re. One way to do this, as we saw in the identity case, is to suppose it to mean that object a is such that, in respect of some y, it possesses the property of having y as a borderline part. Another way, consistent with the preferred analysis arising from the Evans Argument, is to suppose it to mean that object a is such that, in respect of some y, it is vague whether it possesses the property of having y as a part. For reasons already outlined in respect of vague identity I take the latter to be a preferably less loaded interpretation of what it means to make de re ascriptions of part–whole indeterminacy. Using the machinery of abstraction already discussed, we can make explicit the de re interpretation by recasting the mereological vagueness of a as expressed in (P) as: (MV) ˆx [∃y∇(y is a part of x)]a.34 For example, Tibbles counts as a mereologically vague object since there is something, namely the borderline whisker, in respect of which Tibbles is such that it is vague whether it has that thing as a part. An obvious concern with a characterization of mereological object-vagueness by means of (MV) emerges from the rejection of a much more permissive account of vague objects. Zemach (1991) suggests such an account whereby an object counts 33 Tye’s argument to the contrary (2000: 201) is question-begging, assuming as it does that if an object a has sharp boundaries, then it does not have vague composition: ‘there is no object which is such that it is indeterminate whether it is a part of a’. This just restates the thesis to be defended. 34 This follows Sainsbury (1989). Sainsbury (1995b) offers the variation: ∃yx ˆ [∇(y is a part of x)]a. Neither (MV) nor Sainsbury’s later variation makes explicit the requirement that y must be a resilient borderline part. To do so would overly complicate things. We shall leave it as an implicit requirement.

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as vague just in case it is a borderline case for a vague predicate. But of course this trivializes an ontological approach since it entails that any theorist accepting of borderline cases accepts the existence of vague objects. Representationalists, in particular, would be committed to their existence. Thus, as Sainsbury points out, objects would be vague only in ‘an anodyne way’ (1995b: 64) and such vagueness does nothing to address the concerns which we took to motivate a distinctively ontological approach to vagueness outlined at the beginning of this chapter. We are unwilling to seriously propose the vagueness of an object in any substantive sense simply because it is a borderline case of “red”. To be sure, where “a” is a precise designator we ought to agree that the vagueness of the claim “a is red” is to be explained by saying that the object a is such that it is a borderline case of “red”, but this is due to the vagueness of the predicate involved. That is to say, even assuming that the vagueness of a predication is not attributable to the de dicto vagueness of any designator involved, there remains the possibility of de dicto vagueness in the predicate. In so far as (MV) only presupposes that the vagueness described in (P) is not due to the imprecise nature of the designator “a”, it leaves open the possibility that the vagueness is attributable to the predicate “has y as a part” and is merely de dicto.35 Sainsbury (1995b) objects to (MV) as an adequate characterization of what it is for an object to be vague on just these grounds. Representationalists can equally well accept the existence of cases satisfying (MV) since the mere satisfaction of the appropriate abstract by an object, while admittedly a fact about the object in question and thus ‘ontological’ in that sense, is entirely consistent with the representationalist position. The representationalist can agree that wherever a predicate is vague in its application there is ‘an object with respect to which this vagueness in application arises’ (p. 67). The fact of a’s being a borderline case for “red” is a fact about a, namely that it is an object in respect which vagueness in the application of “red” arises. Similarly, where there is some y such that Tibbles is a borderline case for “has y as a part”, this is indeed a fact about Tibbles; the cat does indeed have vague spatial boundaries in the sense of being a borderline case for the spatial predicate “has y as a part”. But for all that has been said so far, this is explicable by appeal to the vagueness of the spatial predicate, and so is consistent with a representationalist account of vagueness and therefore cannot be an adequate account of a distinctively ontological approach to vagueness. More particularly, Sainsbury assumes that the truth of instances of (MV) are to be explained by the de dicto vagueness of the twoplace predicate “is a part of” rather than by means of the explicitly de re vagueness appealed to in the discussion leading up to the framing of (MV). However, “part of” is vague by de¿nition only if it has a resilient borderline case (in this case, some object-pair 〈x, y〉), yet none has been established. Moreover, no borderline case arising in respect of part–whole predications will ever count as a resilient borderline case. Resilient borderline cases, recall, are those cases that give rise to borderline predications that remain resiliently borderline irrespective of how 35 Of course, there is room here for an ontological account according to which properties are vague and it is this that explains the vagueness of the predicate (i.e. vagueness de re) as opposed to imprecision in what (precise) property the predicate refers to (i.e. vagueness de dicto). More on this below.

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indeterminacies surrounding the individuation of the case in question are resolved (i.e. no two resolutions are such that one produces a clear satis¿er of the predicate and the other produces a clear non-satis¿er). Given the inertness, in this sense, of any vagueness in the borderline case itself, the vagueness of predication was, in such cases, more plausibly said to be due to vagueness in the predicate. Alternatively, where the vagueness of predication is sensitive to ways of resolving indeterminacy in what to count as the borderline case itself, it is assumed that the vagueness of predication arises as a result of the vagueness of the subject term. It is clear then that this account of vagueness will preclude the vagueness of “part of”; any indeterminacy in predication with respect to an object-pair 〈x,y〉 will, of course, be sensitive to ways of resolving the indeterminacy surrounding the object-pair and so no borderline case will be resilient. In fact, the initial characterization of vagueness in Chapter 1 appealing to resilient borderline cases was deemed acceptable exactly because it was thus able to admit to the existence of borderline cases for spatial predicates like “has a diameter of exactly n metres” while recognizing the strong intuition that they are nonetheless precise. Given the various equivalences between spatial predicates and predicates involving part–whole relations, the account is also able to recognize the precision of the associated mereological predicate. Thus we face a stand-off. On the one hand, following the intuition that certain spatial predicates are precise, and by implication associated predicates like “part of”, it follows that the vagueness described in (P) must inhere in the singular terms involved. Given the earlier mentioned further requirement that y be a resilient borderline case for being part of a, any vagueness in y is irrelevant and so the vagueness described in (P) must inhere in “a”. According to the vague-object theorist, this, when conjoined with the importation-inference to yield (MV), entails that the vagueness inheres in that which the singular term (precisely) designates, the object a. So, on the theory of vagueness developing out of our account of vagueness in natural language described in Chapter 1, (MV) is adequate, at least for all that has been said so far. We require good reason to abandon such an account. On the other hand, on the contrary assumption that “part of” is vague, Sainsbury points to a dif¿culty with accepting (MV) as characterizing what it is for an object to be in any sense vague. If “part of” is vague, then (MV) ought to be rejected. But why think it vague when intuitively it is not? Sainsbury does offer reasons to undermine our intuitions here but they are not suf¿ciently compelling. The ¿rst is that the predicate is vague since it ‘meets conventional suf¿cient conditions for vagueness in terms of the possibility of borderline cases’ (1995b: 68). But as we saw in Chapter 1, the conventional suf¿cient conditions were deemed ultimately counterintuitive exactly because they admitted seemingly precise predicates as vague. They therefore provide no defence in current circumstances relating to “part of” in particular, since it is exactly such a contested case satisfying conventional suf¿cient conditions yet seemingly precise that is at issue here. Secondly, Sainsbury points to the fact that the predicate’s application to a material thing ‘is based in part on matters of degree, for example the strength of cohesive forces’ (ibid.). But this just seems to be another way to say that it might sometimes be vague whether a predicate applies to a thing, and this again appears question-begging. Stalemate again.

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The debate so far can be summarized as follows. If vagueness is characterized in accord with our intuitions as described in Chapter 1, (MV) presents an adequate characterization of what it is for an object to be mereologically vague in respect of its spatial or mereological boundaries. If the conventional, paradigmatic, borderlinecase conception of vagueness is accepted, then (MV) will, for the reasons Sainsbury outlines, not be adequate and the only sense in which there are mereologically vague objects will be an ‘anodyne’ one. I have made my case against the conventional conception and offered a more ¿ne-grained conception of vagueness. To the extent that you agree, you can accept (MV). Apart from vagueness in spatial boundaries, as typi¿ed by cases like Tibbles the cat, the mereological account expressed in (MV) also admits vagueness in temporal boundaries on the condition that one treat spatial and temporal parts alike, accept temporal parts theory and the associated four-dimensionalist view of objects and their persistence through time. On such a view, persisting objects like Tibbles are temporally extended as well as being spatially extended, and just as we might confront one spatial part at a point in space but not another (e.g. the tail but not the head), so too we might confront one temporal part at a point in time but not another (e.g. the four-dimensional object Tibbles-now but not Tibbles-as-of-yesterday). Tibbles the persisting object is the mereological composition of its temporal parts (themselves composed of spatial parts unless simples of some kind), and changes in Tibbles over time (e.g. the loss of a hair) are accounted for claiming that the composite Tibbles has distinct temporal parts that vary in their properties (e.g. a temporal part that includes the hair and a later temporal part that excludes the hair). The persisting object is said by the four-dimensionalist to ‘perdure’, as it is frequently put.36 On this approach the spatial vagueness of Tibbles considered at some particular point in time (namely that time at which the tail is a borderline part) amounts to the instantaneous temporal part that is Tibbles at that time, being vague in spatial extent. Tibbles does exist at that time so something composes that temporal part of Tibbles but does so in such a way as to render the temporal part spatially vague. On this view, the seemingly obvious fact that many objects have vague temporal boundaries, coming into and going out of existence gradually, is to be explained by pointing to its sometimes being vague whether a particular object (e.g. an instantaneous object occupying the region of space where there was a cat a second ago) is a temporal part of the vaguely persisting object. Vague perdurance presents another way in which objects might possess borderline parts in the sense expressed by (MV). Those temporal parts that are instantaneous objects (e.g. Tibbles-now) or precisely delimited extended temporal parts (Tibbles-from-1-January-2004-to2-January-2004) are not themselves temporally vague, though they may be spatially vague. However, many temporal parts are, it seems. The improper temporal part of the perduring object Tibbles that is Tibbles itself (i.e. the perduring object itself) is vague. So too the temporal part of Tibbles that is the young Tibbles, for example. The vagueness of temporal boundaries thus presents yet another kind of vagueness of composition captured by (MV), vagueness of temporal composition. 36

Sider (2001) gives a good introduction to four-dimensionalism.

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Three-dimensionalists, on the other hand, offer a different metaphysical account of objects and their persistence through time. According to three-dimensionalists spatial and temporal dimensions are importantly different in that objects may have spatial parts but do not have temporal parts – objects are spatially extended but not temporally extended. They do not persist through time, and do not change by having distinct temporal parts at distinct times they are not the mereological composition of their instantaneous temporal parts. Objects, according to three-dimensionalists, are ‘wholly present’ at each time at which they exist, by which they mean that there is no part of the object (and no temporal part, in particular) not present at each time at which it exists. All of an object’s parts are present whenever the object is. Tibbles persists over time by ‘enduring’, that is by wholly existing at different times, and can be said to change over time by being constituted by distinct things or quantities of matter at distinct times (e.g. being currently constituted by something including the controversial hair yet later constituted by something else not including the hair). A characterization of vagueness in respect of temporal boundaries that is neutral in respect of three- and four-dimensionalism will therefore need to differ from a characterization of vagueness in respect of spatial boundaries and is not captured by (MV). To have vague spatial boundaries at some time as described by (MV) amounts to there being no fact of the matter whether a particular collection P composes to constitute the object in question, a, say, since at that time there is no fact of the matter as to whether the spatially extended object a includes all that the collection P includes.37 So a’s spatial boundaries are vague in the sense that it is vague exactly what composes the object at the time in question. To have vague temporal boundaries on the other hand amounts to there being a time at which there is no fact of the matter whether any collection P composes to constitute the object in question, a; a’s temporal boundaries are vague in the sense that it is vague whether anything composes the object at the time in question since at that point in time there is no fact of the matter as to whether the object a exists.38 Let us say, following Sainsbury (1989), then, that a persisting object a counts as temporally vague if and only if for some time t it is vague whether a exists at t: (P′) ∃t∇(a exists at t). Of course the same restrictions apply as with (P) and it is to be understood de re. Thus, a persisting object a counts as temporally vague if: (TV) ˆx [∃t∇(x exists at t)]a. 37 Whether P can nonetheless still be said to determinately compose something at that time will depend on where one stands with respect to what van Inwagen (1990) calls the ‘Special Composition Question’: under what conditions does a collection compose an object? We shall return to this below when considering criticisms of the view currently being elaborated. 38 Sider (2001), developing ideas from Lewis (1986), claims to prove the absurdity of supposing it ever to be vague whether an object exists, vague existence is absurd. The argument is problematic in the current context since it supposes vagueness to be representational (see e.g. pp. 127–30) and since it appears to be question-begging (see Koslicki 2003). We will return to consider vague existence below.

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So, for example, persons are temporally vague objects since there are times at which it is vague whether they exist, coming into and going out of existence gradually. Whether or not persons (as opposed to bodies) might also be spatially vague will depend on the theory of persons adopted. If, like Parsons (2000: 75), one thinks that persons might not be the kinds of things that can have parts, that it is debatable whether persons are mereological objects individuated by their parts, then they might not be candidates for spatial vagueness. But they seem to present clear examples of temporal vagueness. As conceived so far, then, a commitment to vague objects is equally plausible regardless of one’s commitment to either a three-dimensionalist or fourdimensionalist metaphysical framework. Morreau (2002: 343) claims contrariwise that ‘the hypothesis that ordinary material objects are vague ¿ts naturally within the doctrine that they are three-dimensional continuants’. Considering cases of spatial vagueness as exempli¿ed by Tibbles’s shedding of hair, Morreau contends that de re identity is precise (for the reasons given by Evans) and also that quantities of matter are not the kinds of things that can have borderline parts. Thus if an object is taken to be identical to the matter that constitutes it at a time, as four-dimensionalism suggests, then it must be precise since the identity relation is precise and so too the relatum that is the quantity of matter to which it is identical. But irrespective of one’s views concerning identity, anyone sympathetic to the foregoing account of vague objects ought to allow that quantities of matter can be vague as regards their spatial boundaries. Tibbles is, at a certain time, spatially vague, with some hair as a borderline part. Supposing that the four-dimensionalists will indeed consider the relevant temporal part of Tibbles as identical to some quantity of matter, the obvious thought is to explain the hair being a borderline part by claiming that the matter that makes it up is a borderline part of the quantity of matter that is that temporal part. Why think otherwise? Because, Morreau claims, while persisting material objects can come to have something as a borderline part as a consequence of their undergoing gradual change from (determinately) having that thing as a part to (determinately) not having that thing as a part, the same does not hold true of quantities of matter. ‘Quantities of matter have their parts essentially’, as Morreau puts it, whereas persisting things have temporary parts. As time goes by, some of a quantity of matter may separate from the rest but it remains a part of the quantity nonetheless. Some of the soap I gave you on your birthday may ¿nd its way into your second bathroom but it nonetheless remains part of the soap I gave you. This contrasts with cases where some of the cat’s hairs may come loose and thus cease to be a part of the cat Tibbles, or some of the bar of soap I gave you may come away from the bar and thus cease to be a part of the bar of soap. This difference in the persistence conditions of quantities of matter and material objects shows that something cannot come to be a borderline part of a quantity of matter. But, of course, being a borderline part by coming to be one over time in the transition from being a (determinate) part to being a (determinate) non-part is only one way in which borderline parts can arise, and we should be careful not to become too ¿xated on such cases. Consider Mt Everest at any particular moment. Fourdimensionalism claims that we are confronted by a temporal part of the persisting object. What quantity of matter is that temporal part composed of? There is no

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determinate answer. Not because parts are coming loose from Everest, but because there is no sharp spatial boundary separating Everest from the surrounding country out of which it emerges. The quantity of matter that composes Everest has borderline parts due to vagueness of individuation of the mountain irrespective of whether the persisting object undergoes any change at all over time. Coming to be a borderline part is one way in which composition might be vague; being a borderline part simpliciter is another. Four-dimensionalists are able to recognize the vagueness of ordinary material objects as readily as three-dimensionalists. Compositional, spatial and temporal vagueness aside, indeterminacies might also arise with respect to an object’s modal boundaries. To have vague temporal boundaries amounts, remember, to there being a time at which there is no fact of the matter whether the object in question exists. To have vague modal boundaries amounts to there being a possibility that (i.e. a possible world in which) there is no fact of the matter whether the object in question exists. Let us say, following Sainsbury (1989), then, that a persisting object a counts as modally vague if and only if for some world w it is vague whether a exists at w: of course the same restrictions apply as before and the claim is to be understood de re. Thus a persisting object a counts as modally vague if and only if: (ModV) ˆx [∃w∇(x exists at w)]a. So, for example, persons and nations are modally vague objects since there are possible circumstances in which it is vague whether they exist, various counterfactual situations (e.g. psychological disruption cases for persons, or historical disruption cases for nations) presenting circumstances where there is simply no fact of the matter whether they exist. Again, as with temporal vagueness, vagueness with respect to modal boundaries will be variously explicated depending on whether one endorses modal counterpart theory (and supposes that objects are not modally extended but can only be said to have counterparts in other possible worlds) or whether one thinks that transworld identity makes sense (and objects can be identi¿ed across worlds and are thus modally extended). But on any analysis room must be made for indeterminacy in respect of what ordinarily, pretheoretically, would be described as the range of possible situations (i.e. possible worlds) in which an object can be said to still exist. If this ultimately involves an analysis in terms of counterparts and questions as to whether or not one thing is the counterpart of another or simply involves us in questions as to whether an object in one possible situation is identical to an object in another is ultimately of little consequence. To be sure, if one insists that objects, properly understood, are neither mereologically, spatially, temporally nor modally extended, then they cannot be vague in respect of their boundaries, but the ensuing narrowing of what counts as an object presents a pyrrhic victory. A commitment to vague objects, then, is a commitment to there being objects that are compositionally, spatially, temporally or modally vague. That is, it is a commitment to objects that are vague in respect of their compositional, spatial, temporal or modal boundaries. In addition to issues already canvassed, there are further concerns with vague objects. Famously, as discussed, Evans points to the impossible requirement for de re vague identity. Whether the foregoing account of

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vague objects is so committed is something we will return to shortly. Firstly, though, let us turn to another set of concerns centring on problems attending vague composition and vague existence. 5.3.2

Vague composition and vague existence

According to the foregoing conception of vague objects, for a to have vague spatial boundaries at some time amounts to there being no fact of the matter whether a particular collection P composes to constitute a since at that time there is no fact of the matter as to whether the spatially extended object a includes all that the collection P includes. It might then be wondered whether P composes anything at all at that time. This will depend on where one stands with respect to what van Inwagen (1990) calls the ‘Special Composition Question’: under what conditions does a collection compose an object? Following van Inwagen, restrictive conditions might be proposed to accord with a seemingly common-sense intuition that, in the given situation, a particular region of space-time includes a as an object and nothing else. Then whether or not P composes an object will depend on whether or not it composes a. Consider Tibbles again, a vague object with indeterminate spatial boundaries, various hairs being neither determinately a part of Tibbles nor determinately not a part. Vague parthood goes together with vague composition in the weak sense that there will then be a particular collection of objects P such that it is indeterminate whether P composes to form Tibbles.39 Since, given the restrictive position outlined above, there is no other candidate in the vicinity of Tibbles for P to compose, it must therefore be indeterminate whether P composes anything at all. But if composition can be vague in this much stronger sense, then, it seems, there can be cases where it is indeterminate whether the composition of the elements of P exists. The postulated existence of spatially vague objects thus commits one to the view that there can be cases of vagueness of existence. But, it might be contended, vague existence is absurd and so for that reason spatially vague objects (at least) ought to be rejected. Before turning to the question of whether or not vague existence is indeed absurd, notice that the argument moves from the claim that it is indeterminate whether P composes to form Tibbles (what we might term weak vagueness of composition, characteristic of spatially vague objects) to the claim that it is indeterminate whether P composes anything (what we might term strong vagueness of composition, characteristic of vague answers to van Inwagen’s Special Composition Question) since the only candidate object that P might compose is Tibbles. As Morreau (2002: 336–7) points out, this inference can be resisted. Suppose one thinks, as many do, that composition is completely unrestricted and any collection composes to form some object, however arbitrary and gerrymandered.40 Then, despite the fact that it is 39 There is an obvious temporal analogue where it is vague whether some collection of objects composes to form Tibbles by virtue of the vagueness of Tibbles’s temporal boundaries. 40 Advocates of unrestricted composition include semantic theorists like Keefe, Lewis and Sider, and ontological theorists like Morreau. Whether a commitment to unrestricted composition thereby commits one to four-dimensionalism, as Sider (2001) thinks, remains in my view an open matter. See Miller (2005) for discussion.

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indeterminate whether P composes to form Tibbles, P nonetheless (determinately) composes to form something. It is in fact never indeterminate whether P composes to form something – it always does, on this view. Vague existence would appear to be avoided since composition cannot, on this view, be vague in the strong sense required. Vague existence is not avoided, however. Even if, in accordance with unrestricted composition, we reject strong vagueness of composition and admit the existence of something that P composes, we are still confronted with cases of vague temporal boundaries to objects. In such cases it is sometimes vague whether a (say) exists. There will be some collection P such that it is indeterminate whether P composes a at that time, i.e. there will be weak vagueness of composition, but not because it is indeterminate whether P composes anything, i.e. not because of strong vagueness of composition (it is, on this view, supposed that P always composes something), but because, despite P’s composing something, a’s marginal existence at that time means that it is indeterminate whether the something that P composes counts as a.41 Thus, wherever we have a collection their composition is something, but whether it is a or not may depend on whether a exists and that may be a vague matter, unrestricted composition notwithstanding. So too where objects have vague modal boundaries. In such cases there is now no question that something exists but it is vague whether that something is a. Vague existence in this weaker sense, then, is a consequence of a commitment to temporally or modally vague objects. But should one baulk at such vague existence (and consequently baulk at a commitment to vague objects)? I think not. Note ¿rstly that objections to vague existence that depend upon a rejection of ontological vagueness, notably well-known Lewis–Sider arguments that appeal to the representational theory of vagueness (e.g. Sider 2003), beg the question in the current context and are therefore unpersuasive. So why should the advocate of ontological vagueness baulk at such vague existence? Morreau does, claiming that ‘this idea is genuinely mysterious. How can something neither quite be nor not be there? Must we imagine that the presence of vague objects is somehow a matter of degree, like the intensity of a beam of light?’ (2002: 336). But the sense of mystery is overplayed. Something might be neither determinately there nor determinately not there by being on its way out, as it were, as will be the case with each of us at various points in time. The vagueness of the distinction between being and not being is no more mysterious than the vagueness between being a part and not being a part. Sometimes there may simply be no fact of the matter. Van Inwagen (1990) and Hawley (2002) consider more detailed concerns surrounding vague existence that arise in relation to acceptance of strong vagueness of composition – vague existence in the strong sense ¿rst considered above according to which it is vague whether anything exists.42 Suppose, like van Inwagen, we restrict composition and accept that there can be such cases where it is indeterminate whether the composition of the elements of P exists. Then it might 41 Morreau, accepting the existence of vague objects but rejecting the idea of vague existence, accepts unrestricted composition and is reassured that ‘nothing has any sort of shady presence’ (2002: 337). Morreau focuses exclusively, though, on spatially vague objects. 42 What follows is heavily indebted to both discussions.

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seem that we are committed to there being objects whose existence is shady – what Hawley terms ‘immodest vague existence’. Consider some simples that would typically be described as composing a virus and suppose furthermore that a virus is a borderline case of a life or living thing (van Inwagen 1990: 278). Since, according to van Inwagen’s answer to the Special Composition Question, some simples compose to form an object if and only if they jointly constitute a life, all complex things are living things. Thus it is indeterminate whether the complex collection of simples that we might have ordinarily taken to be a virus compose any kind of thing at all, and it is therefore indeterminate whether the composition of simples exists. Then it might seem that we are committed to there being something, namely ‘the virus’, whose existence is shady. But to thus pick out the object and say of it that it is vague whether it exists requires that one be committed to non-existent objects.43 And that is a commitment that many will be loath to accept. Vague existence, at least of the immodest kind just described, is commonly thought to be problematic. But it does not follow from the modest claim that it is indeterminate whether the composition of P (or indeed anything) exists that the composition of the elements of P (or indeed anything) is such that it is indeterminate whether it exists, any more than it follows from the modest claim that Pegasus does not exist that Pegasus is such that it does not exist. Where denials of existence are analysed in such a way as to obviate any commitment to an object whose existence is denied (e.g. through Russellian means), so too can one analyse denials of determinate existence in such a way as to obviate any commitment to an object whose determinate existence is denied. When we say that it is vague whether some particular thing or other exists, we need not say that there is some thing for which it is vague whether it exists. Consider again, by way of example, van Inwagen’s ‘virus’. It is indeterminate whether the virus (or, more exactly, the composition of simples in the region we would ordinarily say is occupied by the virus) exists. Those, like van Inwagen, unwilling to commit to anything other than the existent will refuse to accept any analysis of vague existence according to which there is something, namely the virus, for which it is indeterminate whether it exists. ‘[To say that existence is vague] cannot mean that there are certain objects that fall into a vague frontier between existence and nonexistence. For suppose χ to be one of those objects. If χ is there to be talked about in the ¿rst place, then χ exists, and in fact de¿nitely exists’ (1990: 240). But vague existence can be accounted for without such a commitment. With existence considered as a second-order property, following Russell for example, to claim something exists is to claim that some property or other itself has the property of being instantiated. Vague existence then amounts to vagueness as to whether that property itself has the property of being instantiated. We can then say, in accord with this commonly employed analysis, that it is vague whether the virus exists since it is vague whether the property of being a virus is instantiated. A second-order 43 It might seem that one could coherently deny that there are any (determinately) nonexistent objects and nonetheless be prepared to accept that in addition to those which (determinately) exist there are those for which there is no fact of the matter. The additional commitment, then, is merely to objects whose existence is vague rather than objects that (determinately) don’t exist. However, the coherence of the former presupposes the coherence of the latter. If something is such that it is vague whether or not it exists, then its (determinate) non-existence must be a possibility for its denial to have substance.

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approach to existence claims is available for use in the analysis of cases of vague existence. As Hawley also shows, a ¿rst-order approach need not fare any worse in respect of an analysis of vague existence, i.e. it need not commit itself to there being something which vaguely exists any more than a second-order approach. To say that it is vague whether the virus exists is now to be analysed as claiming that it is vague whether anything instantiates the property of being an existing virus, or for those shy of such complex properties, it is vague whether anything instantiates both the property of existing and the property of being a virus. This can be the case without there being anything of which it is true that it vaguely instantiates the properties in question. What typi¿es vague existence on either analysis is precisely the fact that though it is vague whether something is F it is not true to say that there is something for which it is vague whether it is F. The vagueness of existence exempli¿ed by the collection of simples ordinarily called a virus presents a case where it is vague whether something is a virus, yet we cannot say of any thing that it is vague whether it is a virus. That is to say, the following holds: ∇∃xFx≠ ∃x∇Fx.44 Similarly, at a particular point in time it might be vague whether there is something that is Tibbles, since it is vague whether Tibbles exists at that point in time. Yet there need be no thing at that point in time of which we can say that it is vague whether it is Tibbles, since given restricted composition the only candidate thing might be Tibbles itself, in which case the cat would, paradoxically, exist at that time at which its existence was indeterminate. Of course, where the existence of the persisting object Tibbles is in question we can say that there is some persisting thing for which, at some time t, it is vague whether it exists (as a consequence of its having vague temporal boundaries). The explicit consideration of a persisting object at a time justi¿es such a claim. However, even here we must be careful, as we have just seen. It remains the case that at the time at which Tibbles’s existence is in doubt we are not committed to the claim that there is some thing which is such that it is vague whether it exists. Moreover, in extreme cases it is not vagueness of a persisting object’s temporal boundaries that generates cases of vague existence but, rather, vagueness as to whether anything can, in a given situation, be said to exist at all given a restrictive account of the conditions under which a collection may be said to compose an object. In such cases those who eschew non-existent objects will not attribute vague existence to an object, but nor need one any more than when one denies existence thereby attribute non-existence to an object. Those who take the view that composition is unrestricted will, of course, not accept cases of vague existence in the strong sense discussed, and so problems arising from such cases will be avoided. Moreover, those who accept non-existent 44 To accommodate this, the formal semantics will admit of domains which have borderline members. (Those of us who accept unrestricted composition can, of course, ignore this complication.)

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entities will have no problem in admitting there being something of which it is vague whether it exists. The problem just discussed, that of strong vagueness of existence or composition, need only be faced by those who reject non-existent entities and restrict composition. And, as we have seen, even then the problem is really no problem at all. For all that has been said, then, we can accept vague existence, and recognize cases of temporally and modally vague objects. So too spatially vague objects, irrespective of whether or not we accept unrestricted composition. Van Inwagen restricts it and accepts the consequences. It is vague whether Tibbles exists at t, and given restricted composition as determined by his restricted answer to the Special Composition Question, it is therefore vague whether anything exists at t. There are, to be sure, more cases of vague existence given restricted composition, but vague existence is acceptable nonetheless. 5.3.3

Simplicity

Another objection that might be raised against an ontological approach to vagueness, and the postulation of vague objects in particular, is that such an approach falls foul of simplicity constraints. When choosing theories we should prefer the simpler to the more complex. Given that no objection has yet been raised to the existence of representational vagueness, it seems that we must acknowledge and account for it; there are cases where, though the represented is not vague, the representation relation is. To go on and further suppose that the represented is also sometimes vague and thus commit to vague objects in addition to precise ones seems to add unnecessary complexity to our theory of vagueness. We should commit only to precise objects and seek to explain all vagueness as representational. A purely representational theory is simpler than one extended to allow ontological vagueness. There is, however, some equivocation on the notion of simplicity. Though adding vague objects to our theory of vagueness means that it is less parsimonious from an ontological point of view in the sense that it admits of an additional kind of object (a vague one) and is in this sense less simple, it is nonetheless simpler in that, with the admission of vague objects, we can better explain cases of vagueness where reference seems to function exactly as in cases where there is no vagueness, i.e. we can account for vagueness while not having to commit to the inscrutability of reference. The more ontologically parsimonious representational theory of vagueness will indeed avoid a commitment to a category of vague objects but at the cost of having to admit to reference being indirect and inscrutable in cases where this seems implausible. The objection is similar to that once raised against the use of non-Euclidean geometries to describe physical space. The non-Euclidean approach invokes a geometry more complex than its Euclidean rival. But this additional complexity in geometry was said to be offset by gains in our overall physical theory through the elimination of anomalies and the subsequent epicycles within the theory to deal with them. So too here. Overall, the ontologically more complex theory enables simpler analysis of many cases of vagueness than that made available by its representational rival. Just as we may have no qualms about hammers per se but object to their use to drive in screws, preferring to enlarge our toolbox to include screwdrivers, so too we

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may have no objection to the notion of representational vagueness but still object to its use to explain all vagueness and enlarge our ontological categories to include vague objects. Just as some things can be done with a hammer but not everything, some vague representations may be vague by virtue of the vagueness of the representation relation itself, but not all representations. To say there are vague objects, then, is to commit to there being objects which are vague in respect of their boundaries – spatial, mereological temporal or modal. Is this suf¿cient to account for vague language in general? Obviously not. There are cases of vague sentences where the vagueness is attributed to the vagueness of the predicate involved – e.g. “The sun is a hot star” – as opposed to those cases where the vagueness is attributable to the singular term and ultimately the object precisely designated thereby – e.g. “The sun has a diameter of exactly 1.39 × 109 metres”. In such cases it seems that an ontological account of the vagueness of the predicate will rest with the predicate’s precisely designating a vague property. Some account is needed therefore of what a vague property is. Before turning to such an account, however, let us deal ¿rst with another matter arising from the Evans Argument, the relation between vague objects and vague identity. 5.4

The Vague Identity Thesis

Given the independent account of vague objects above, is it then true, as Evans seemed to have supposed, that if some such objects exist, then there are cases of de re vague identity? Many contemporary writers seem to have thought so. Sainsbury (1988: 47) endorses the idea by suggesting ‘that if an object were vague, it would be a vague matter what object it is identical with’; obviously then if there are vague objects, then there are de re vague identities, ones involving those vague objects in particular. Wiggins (1986: 173) is also committed, identifying as he does ‘the existence of objects that are indeterminate in respect of identity’ with ‘objects such that it is indeterminate which things they are’. Object vagueness is not just suf¿cient for de re vague identity it is also necessary. In this spirit, Garrett (1988: 130) postulates the following Vague Identity Thesis: ‘The thesis that there can be vague objects is the thesis that there can be identity statements which are indeterminate in truth-value (i.e. neither true nor false) as a result of vagueness … the singular terms of which do not have their references ¿xed by vague descriptive means.’ That is, the thesis that there can be vague objects is the thesis that there can be identity statements involving only precise designators which are indeterminate in truth-value as a result of vagueness, de re vague identities. Of course, the Evans Argument itself does not require that the two mentioned (sub)theses of the Vague Identity Thesis actually be equivalent. It is suf¿cient for the Argument that the thesis postulating the existence of de re vague identities merely be entailed by the thesis postulating the existence of vague objects; the converse is not required. But nor is the converse controversial. The existence of such identities is to be explained by the independently de¿ned vagueness of at least one of the objects precisely designated by the relevant singular terms and thus entails that such objects exist. To suppose in the disrupted-person case that it is vague whether (the

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persisting object that is) person a is identical to (the persisting object that is) person b and admit that the vagueness is not a result of imprecise reference requires an admission that it results from imprecision in at least one of the objects themselves. Supposing then that the controversial aspect of the Thesis is exactly that aspect required for the Evans Argument, what of it? Is de re vague identity a necessary condition for there being vague objects? Frequently it seems, it is supposed that the Evans Proof of the impossibility of such identities is compelling but nonetheless consistent with the thesis that there are vague objects – the necessary condition on their being vague objects captured by the Vague Identity Thesis is to be rejected. Burgess (1990), for example, endorses the existence of vague objects while accepting the impossibility of de re vague identity. And Burgess is not alone here, with Akiba (2004), Morreau (2002), Tye (1990, 2000) and Williamson (1994) also adopting a similar position. Most recently, Noonan (2004) endorses this view as obvious and something ‘everyone knows’. Despite the Evans Proof being sound, there can nonetheless be vague objects since the Evans Argument is unsound by virtue of the falsity of the Vague Identity Thesis. Given that the Evans Proof is unsound, however, we are in a position to recognize the intuitive force of examples of vague identity presented earlier in the chapter and can also come to appreciate the circumstances under which such vagueness arises by virtue of the vagueness of objects named therein. A commitment to vague objects as described earlier will not necessarily involve a commitment to the existence of de re vague identities, but given certain conditions such identities are a consequence of a commitment to vague objects, objects whose existence is no longer threatened by the Evans Argument. Not surprisingly, whether or not vague objects generate de re vague identities will depend on other metaphysical views concerning the nature of objects. Consider, ¿rstly, mereologically vague objects like Tibbles the cat. Suppose that at a particular point in time Tibbles has some single hair h as a borderline part. Suppose further that we accept unrestricted mereological composition, any objects combine to compose an object, and therefore that there is a cat-like object, Tibbles′, that is the sum of Tibbles and h. Assuming the transitivity of the relation of being a part, since Tibbles is determinately a part of Tibbles′, anything determinately a part of Tibbles will determinately be a part of Tibbles′ – there can be nothing determinately part of Tibbles yet determinately not part of Tibbles′. And since the only things that are determinately a part of Tibbles′ are h, the determinate parts of Tibbles, and sums of h and these parts, and these determinate parts of Tibbles′ are not determinately not a part of Tibbles, there can be nothing determinately a part of Tibbles′ yet determinately not a part of Tibbles. So, while Tibbles and Tibbles′ are not in determinate agreement on all parts (there being no determinate agreement on h in particular), neither do they determinately disagree on any part. If identity is a matter of mereological sameness (i.e. x and y are identical if and only if they have the same parts), then they cannot be said to be determinately identical (since they do not determinately agree on all parts), yet nor can they be said to be determinately different (since they do not determinately differ on any part). It is indeterminate whether Tibbles is identical to Tibbles′. So too other mereologically vague objects like clouds, mountains, etc. As Parsons (2000: 73f) makes clear, there are a number of assumptions made. In particular:

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unrestricted mereological composition – i.e. for any x and y there is an object z which is their mereological composition or sum, such that x is a part of z, y is a part of z, and everything that x and y are both parts of, z is a part of too; transitivity – i.e. if x is a part of y and y is a part of z, then x is a part of z; a principle dubbed uniformity of indeterminate parts – if it is indeterminate whether y is a part of x, there is no part of their sum which is determinately not a part of x; and mereological identity – for any objects x and y, x is identical to y if and only if x and y have all and only the same parts. Where these principles can be assumed, mereologically vague objects will give rise to de re vague identities. In fact, where a mereological precisi¿cation of an object x is an object x′ having as determinate parts anything determinately part of x and having as determinate non-parts anything determinately not a part of x, and having any borderline part of x as a determinate part or determinate non-part (so having no borderline parts and being mereologically precise), these principles will ensure that it is de re vague whether x = x′. Being thus committed to vague identities might then be resisted in various ways. While the principles of transitivity and the uniformity of indeterminate parts seem obvious enough, unrestricted composition and mereological identity are more contentious. Firstly, one might question the principle of mereological identity. Mereological objects might share the same parts and yet nonetheless not be identical; they might be differentiated in virtue of some property other than those pertaining to the simple possession of parts. In the spirit of the Evans Proof, Morreau (2002) looks to the property of having-h-as-a-determinate-part. Since Tibbles′ has h as a determinate part but Tibbles does not, they differ in respect of this ‘property’ and so are determinately distinct.45 But for reasons already outlined in respect of the Evans Proof I take it that there are no such properties involving “determinacy”. What then, we might wonder, of the fact that Tibbles is vague whereas Tibbles′ is not? Might this provide the necessary grounds on which to determinately distinguish the two objects? Again, Morreau thinks so, but again it will do so only on the assumption that being vague or being precise constitute properties, and given the implicit use of “determinately” in the analysis of what it is for an object to be vague (or precise), this requires, again, that we admit abstracts involving the notion of determinacy as properties – something already ruled out. The vagueness of Tibbles′ is not a matter of its having a particular property (being such as to have a borderline part h), but, rather, amounts to there being no fact of the matter in respect of the property of having h as a part. There being no fact of the matter whether there is any mereological difference between Tibbles and Tibbles′ is enough to establish that there is no fact of the matter whether they are different, i.e. it is indeterminate whether Tibbles is identical to Tibbles′. Secondly, as already noted, one might object to unrestricted composition. If so, then vague identity might be contested since the existence of the allegedly composite 45 Of course, for Morreau and fellow travellers like Burgess and Tye who accept the existence of vague objects yet reject vague identity by virtue of the Evans Proof, the Evans Argument goes awry in assuming vague identity a necessary condition for the existence of vague objects. For those like myself who accept the existence of vague identities by virtue of the existence of vague objects, the Evans Argument goes awry in assuming the soundness of the Evans Proof.

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object in respect to which the identity question is raised might be contested. As noted when discussing vague existence earlier, if composition is restricted in this way, then existence will be vague. Since it is indeterminate whether h is a part of Tibbles, it is similarly indeterminate whether the sum of h and Tibbles compose to form Tibbles. If composition is restricted in such a way as to admit only Tibbles as existing in a given situation, then it will be indeterminate whether the sum of h and Tibbles, Tibbles′, exists in that situation. Thus we cannot commit to there being a thing such that it is vague whether it is (identical to) Tibbles – i.e. we cannot infer that ∃x∇(x = Tibbles). To be sure, it is vague whether there is something that is Tibbles′ – i.e. ∇∃x(x = Tibbles′) – but this does not issue in vague identity. Spatial vagueness in an object a will only issue in de re vague identity involving a if an appropriate relatum can be found, and restrictions on composition could undermine this. For those like myself who accept unrestricted mereological composition, the objection lacks force. De re vague identities will result from such spatial vagueness. Notice that even when composition is restricted, thus constraining what counts as an object, an appropriately vague identity relation holds between sets whose existence appears not to be in question. Consider the set T consisting of all of the parts of Tibbles and the set Th consisting of all such parts plus h. By Extensionality, sets are identical if and only if they have the same members; and it is indeterminate whether the two sets share all and only the same members – Th has h as a member, while it is indeterminate whether T does, so they do not determinately agree on h, nor is there any member on which they disagree. It is de re vague whether T = Th. Though not a case of de re vague identity involving Tibbles, the spatial vagueness of Tibbles nonetheless is suf¿cient for such a case of vague identity making only minimal assumptions about sets.46 The set-theoretic case aside, there are other cases of spatial vagueness where the existence of the relata seems assured. Consider again the case of Aphla and Ateb. A linguistic community names a local mountain “Aphla” and a local mountain “Ateb”, unaware of the fact that the visible peak they refer to as “Aphla” and the visible peak they refer to as “Ateb” are separated by a saddle whose dimensions in relation to the peaks are such that there is no fact of the matter whether the saddle separates two mountains or is the mid-region of a single, topologically complex mountain extended over two peaks joined by the saddle. There is simply no fact of the matter whether Aphla is part of Ateb (and vice versa); each is mereologically vague. By similar reasoning to the Tibbles case, then, it is vague whether Aphla is Ateb. This seems right to me but, again, others might contest it. As in the Tibbles case one might again object that Evans-style reasoning will establish the de¿nite difference between Aphla and Ateb; they differ as regards having Aphla as a 46 Sainsbury (1990) would object on the grounds that sets simply cannot be vague. I disagree, seeing no reason to exclude abstracta like sets from the category of vague things. (Set-theoretic vagueness will mimic mereological vagueness with talk of parts replaced by talk of members.) Hawley (2002: 133), in accepting the soundness of the Evans Proof, rejects the example of vague identity claiming that Extensionality should be rejected – sets may be determinately distinct though they do not determinately differ on any member. Given the diagnosed flaw in the Proof, we can retain Extensionality and accept the obvious consequence.

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determinate part. Despite each of “Aphla” and “Ateb” being precisely designating singular terms, the objects designated thereby are determinately distinct. But such reasoning is Àawed for the same reason the Evans Proof is. Alternatively, one might accept that such a case gives rise to vague identity, but deny that the terms are precise designators and count the vagueness as merely de dicto. Such is the view of van Inwagen (1990: 243), for example.47 But the argument is not compelling. Van Inwagen contends that the vagueness of the terms is a result of ‘purely linguistic considerations’ and not because of any vagueness in reality. In particular, any vagueness is a result of our ‘rules for naming’. Applying his reasoning to the current example, rules covering cases where mountains are named might, say, stipulate that any material parts contiguous to the named peak count as part of the mountain crowned by that peak unless there is a valley between it and the peak. Whether Aphla is Ateb (i.e. whether the two names name the same mountain) now depends on whether or not the saddle between the peaks counts as a valley or not, and indeterminacy here makes for vagueness in identity. The vagueness in the ‘spread’ of the mountain results from the vague spatial boundary delimited by the rule for naming such things. It is indeterminacy in whether a naming rule is satis¿ed or not that best explains the vagueness, and the vagueness is therefore said to be de dicto. No de re vague identity results. But all that this shows is that the rules for naming do not issue in precision. And, of course, we would not expect them to where a vague object was being named. Our rules for naming stipulate that “Aphla” is vague, not because it vaguely refers but because the mountain that is named thereby is itself vaguely bounded. It has vague boundaries – the valleys that surround it – just as a country occupying a continent might have vague boundaries – the sea that surrounds it (where precisely does the land end and the sea begin?). And just as our rules for naming such countries will issue in vague names reÀecting the vagueness of the country named, so too the names for mountains. To infer from the vagueness of the rules for naming that the vagueness is de dicto is simply to mistake vagueness that is in language for vagueness that is due to language. Of course, sometimes vagueness in our rules for naming results in vague names whose vagueness is indeed de dicto and we would be justi¿ed in resisting any temptation to suppose that any vagueness in identity was de re. For example, suppose that we are looking out across a plain containing two hills side by side. I name one of them “Aphla” where the rules for naming stipulate that the hill named is that one to which I am pointing. Where it is indeterminate whether I am pointing to the hill on the left or to the hill on the right, both being in front of us in the direction in which I am pointing, there is no fact of the matter whether Aphla is the hill we ascended last week (the one on the left) and called “Ateb”. So the identity claim “Aphla is Ateb” is now vague but the vagueness is merely de dicto. “Aphla” is an imprecise designator. Sometimes vagueness in the rules for naming issues in de dicto vagueness, but not always, and we should not suppose otherwise. 47 Van Inwagen discusses a case involving lands “Columbia” and “America”, but it is a simple variant of the current case. Another case involves buildings and is discussed in Cook (1986) and Parsons (2000). Like van Inwagen, Parsons considers the vagueness in such cases to be de dicto.

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We will return to consider such cases in the next chapter, where issues surrounding vague objects and counting are said to pose further problems for de re vague identities over and above the Evans Argument but, for now at least, assuming such identities coherent, we can see how the existence of mereologically vague objects entails the existence of some such identities. Moreover, analogous considerations arise in respect of temporally vague persisting objects. Consider van Inwagen’s case of disrupted persons. Person Alpha enters a room, disruption occurs, and Omega leaves the room. The reader is free to ¿ll in the details of the disruption in whatever way makes Alpha and Omega temporally vague objects with temporal boundaries such that it is indeterminate whether Alpha ceases to exist after the disruption and indeterminate whether Omega exists before the disruption. There is simply no fact of the matter whether the disruption is suf¿ciently great as to make for a change of person. Of course, the three-dimensionalists will describe this as a case where it is vague whether the enduring Alpha is the enduring object Omega, while four-dimensionalists will talk alternatively of its being vague whether the perduring object of which Alpha is a temporal part is the perduring object of which Omega is a part, but in either case we are confronted by de re vagueness of identity. In other cases of temporally vague persisting objects, for example each of us whose temporal extent is indeterminate (either beginning or end or, more commonly, both), the existence of the relata necessary for any consequent de re vague identity will be conditional just as it was with spatially vague objects. Those accepting of unrestricted composition will be confronted by such vague identities. Consider me, for example, and suppose for simplicity that my birth was instantaneous and that my death will be mercifully swift with there being but a single instant t where it is indeterminate whether I exist, alive before t and dead thereafter. Analogous to the case of Tibbles and Tibbles′, there is simply no fact of the matter whether the persisting person that I am is identical to the persisting object determinately like me in every respect except as regards existence at t, with the persisting object determinately existing at that instant. The temporally precise persisting object (which exists given unrestricted composition) and the temporally vague persisting person (me) are vaguely identical, and the vagueness is de re. In fact, where a temporal precisi¿cation of a (persisting) object x is an object x′ exactly like x except having any vague temporal boundary eliminated in favour of a precisi¿cation thereof, it will be de re vague whether x = x′. For those accepting of transworld identity, the obvious analogue will follow for modally vague objects. Counterpart theorists will be confronted with vague identities only given the ability to compose objects and their modal counterparts into composite objects capable of standing in identity relations. So what then of the relation between vague objects and de re vague identity? As expected, it depends on one’s other metaphysical views. The mere postulation of a vague object of one sort or another does not of itself entail the existence of de re vague identities. However, for those like myself accepting of unrestricted mereological composition, the existence of the former entails cases of the latter. Like Evans, I take the existence of vague objects to entail the existence of de re vague identities. Given the plausibility of the converse entailment, I take the Vague Identity Thesis to be true. There is a close connection, as Evans supposed. But, of

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course, as we have already seen, such a connection does not provide the basis for a rejection of vague objects since the required proof of the absurdity of de re vague identity is fallacious for the reasons outlined earlier. 5.5

Vague Properties and States of Affairs

There has been a noticeable asymmetry in the ontological approach developed so far. When speaking of vagueness generally, considerable attention is paid to predicate-vagueness. Yet the foregoing ontological account of vagueness has focused on vague objects with no mention made of ontological vagueness that might arise in respect of vague predicates. If the vagueness of singular terms can sometimes be explained by way of the vagueness of objects referred to thereby as opposed to any vagueness in the reference relation itself, then it seems reasonable to ask whether the vagueness of predicates might also sometimes be explained by the vagueness of properties. Nominalists, and any others sceptical of the ontological status of properties in general, might immediately respond negatively. So too those who question the existence of anything like a property corresponding to a vague predicate, in particular; for example, Carnap and naturalistically minded philosophers have suggested that we need only take ourselves to be metaphysically committed in respect of scienti¿c talk about the world, and such talk is precise. But, in respect of this latter more conservative response, we have already considered attempts to discount the signi¿cance of terms due simply to their vagueness. Chapter 4 concerned itself with just such an approach to vagueness where vagueness was thought eliminable, reducible to the precise, or supervenient on the precise, and the approach was found unsatisfying. Where precise predicates sometimes, at least, may be taken to represent properties, so too with vague ones. Their vagueness is no particular barrier to their representing properties in whatever sense precise predicates might. Of course, a general scepticism towards properties is entirely consistent with this last claim, adding simply that there is no real sense in which precise predicates might represent either. There are no properties, vague or precise. If that is one’s view, then any further deliberation on vague properties is pointless. For realists like myself – or any other theory of properties which is parasitic on realism by, in some way or other, ‘adapting’ and thereby depending upon the realist story – the question remains as to whether or not predicate-vagueness might at least sometimes be associated with the vagueness of the property referred to. A number of writers have certainly thought so and, for reasons similar to those arising in respect of vague singular terms, I think such a view is correct.48 That ‘disturbing philosophical lesson’ forced upon us by a representational approach to vague names, as McGee and McLaughlin describe it, is that reference would, on a such an approach, be ‘inscrutable’ (2000: 130). As was pointed out in respect of names, such an approach thus invites the Russellian view that vague representations are defective since they fail to represent in the obvious way that their precise cousins 48 Defenders of vague properties include Rolf (1980), Sainsbury (1989), Tye (1990), and Rosen and Smith (2004).

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do – i.e. determinately, directly. Without committing ourselves to any particular theory of representation or reference, we seem able to say quite generally that de dicto or semantic vagueness is a defect, or at the very least requires considerable reworking of standard accounts of reference and denotation to accommodate imprecise designators. And the lesson will reapply in the case of vague predicates. Where the predicate might ordinarily be supposed, in some general sense, to represent a property, the vagueness of the predicate would, on a semantic approach, render the representation relation ‘inscrutable’. As Russell would say, representation is one–many. Reference is indirect. Consequently, the opposing idea of vague properties is attractive. It acknowledges a direct relation between our vague ordinary words and the facts we use them to describe, for example, between an utterance of “Blood is red” and the fact that the substance blood has the property of being red (Williamson 1994: 249). Simplicity might again suggest, as in the case of objects, that wherever possible we retain a precise relation of representation, whatever one’s theory of representation prior to considerations of vagueness. In this way, we again avoid unnecessarily complicating aspects of representation, reference and denotation incurred by those who adopt a representational approach to vagueness. So what is a vague property? Following naturally from our account of vague predicates, one might say that a property is vague if and only if it has resilient borderline cases. That is to say, property F is vague if and only if there are (or could be) objects for which it is indeterminate whether they exemplify F and there are no two precisi¿cations of the object such that one determinately exempli¿es F and the other determinately does not. (As with the vagueness of objects, we are setting aside issues to do with higherorder vagueness, returning to the matter in Chapter 7, §4.) Thus we can say that redness is a vague property since there are objects that are not only borderline cases for redness but they are resilient borderline cases. No diminution of any inherent mereological, temporal or modal vagueness in the object would affect its borderline status in respect of redness, and thus any vagueness in the object is irrelevant to the vagueness of exempli¿cation of the property in question. So too where the sun is a borderline case for being a hot star. It is a resilient borderline case with its borderline status remaining unaffected by precisi¿cation since the heat-difference between the sun and any resulting precisi¿cation is too small to make a difference. Contrast this with Rolf’s case of vagueness surrounding whether or not the sun has a diameter of 1.39 × 109 metres. We are not confronted with a vague property in such a case but, instead, a mereologically vague object that is a borderline case of the precise spatial property referred to. The exempli¿cation of the spatial property is sensitive to variation in the spatial boundaries of the relevant object, with various precisi¿cations resulting in objects that do and objects that do not exemplify the property in question. The vagueness here can be ‘blamed upon’ the object itself, the sun, which, though a borderline case, is not a resilient borderline case.49 Of course, what goes for properties here might as easily apply to relations. A twoplace relation R is vague if and only if there are or could be object-pairs 〈a, b〉 for which it is indeterminate whether they stand in the relation R and there are no two precisi¿cations of the object-pair (i.e. no two ways of precisifying one or both of the 49

See Rolf (1980) and Sainsbury (1989).

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objects) such that one precisi¿cation results in objects that determinately stand in the relation and the other results in objects that determinately do not. And so on for nplace relations generally – their vagueness consists in their having resilient borderline cases. Of particular interest is the identity relation. We have seen that, sometimes at least, vague objects make for de re vague identities. One might then wonder whether the identity relation itself is vague or whether the vagueness is always attributable to the objects involved. In fact, as with spatial properties ascribing precise dimensions to objects, the identity relation is precise. It is easy to see in the foregoing examples of de re vague identity that in each case the borderline object-pair is not a resilient borderline case for identity; while there is vagueness as to whether identity holds, this is because of vagueness in the objects, not the relation. It is vague whether Tibbles is identical to Tibbles′ but there is a precisi¿cation of Tibbles such that the resulting object and Tibbles′ are identical and another precisi¿cation such that they are distinct. The object-pair 〈Tibbles, Tibbles′〉 is a non-resilient borderline case for identity. So too with the other cases considered. Object-pairs that are borderline cases for identity always include at least one vague object and there will always be two distinct precisifications of the borderline object-pair, one instantiating (determinate) identity and the other instantiating (determinate) difference. De re vagueness of identity occurs but the vagueness is always attributable to the vagueness of at least one of the objects involved and is not due to the vagueness of the identity relation itself. So we see that the identity relation is precise, as is the two-place identity predicate; the predicate precisely designates a precise relation. But there can also be cases of imprecisely designated properties or relations. Not all vague predicates that refer refer precisely to vague properties. There can, of course, be de dicto vagueness here, where it is vague which property is picked out. Consider predicates like “is the greatest ruler”, “being the same height as the most intelligent basketball player”, or “having a shape like that of the highest hill”. Each such predicate is vague, resilient borderline cases being possible. However, the vagueness has its source in indeterminacy concerning just what property is denoted thereby. Phrases like “the greatest ruler”, “the most intelligent basketball player” and “the highest hill” are all vague and arguably merely representationally vague. When speaking of ‘the highest hill’ there is simply no fact of the matter which hill from amongst a range of candidate hills we are talking about. Similar indeterminacy in respect of denotation arises when speaking of ‘the greatest ruler’ and ‘the most intelligent player’. The representational vagueness encountered is then the source of the vagueness of the predicates in which they ¿gure. Some predicate-vagueness is representational. With an account of vague objects and vague properties to hand, talk of vague states of affairs is not far away. In fact some, like Parsons (2000), will speak of indeterminacy in the world only in the sense of there being vague states of affairs, eschewing any attempt to theorize about ontological indeterminacy in a more ¿negrained way. While we have theorized about the vagueness or otherwise of objects and properties, it is easy to see that states of affairs will sometimes be vague. For example, suppose it is indeterminate whether the sun is a hot star, the vague object that is the sun neither determinately possesses nor determinately does not possess the property of being a hot star. Then the state of affairs that is the sun instantiating the

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property of being a hot star neither determinately obtains nor determinately does not obtain. Let us say that a state of affairs is vague if and only if it neither determinately holds nor determinately does not hold. Then the state of affairs that is the sun’s being a hot star is vague. The relevant sentence expressing the obtaining of the vague state of affairs, “The sun is a hot star”, will accordingly itself be vague, and thus be neither true nor false. The vagueness of the sentence, then, is accounted for by the vagueness of the state of affairs that is the sentence’s truth-maker. Objects, properties and the states of affairs they may combine to form can all be vague. 5.6

Vagueness ‘in the World’ or ‘of the World’?

When all is said and done, the reader may be left with the lingering feeling that, as described, ‘ontological vagueness’ is indeed a coherent and ef¿cacious theory to adopt in the face of vagueness but only by virtue of our being rather too liberal as to what to count as properly ontological. To be sure, you might respond, clouds, mountains and cats might be vague but the ‘objects’ they represent are vague only in an anodyne sense. The postulated vagueness ‘comes into the world only with the use of representations; it is not the world in itself that is vague’ (Copeland 1995: 83).50 While there is vagueness ‘in the world’ (e.g. cats are vague objects in the world) it is not ‘of the world’. It is not due to the world but rather is the product of the representational scheme, and vague language in particular, that we bring to the world considered in itself. Such a feeling, the sense that ontological vagueness properly considered is somehow deeper and less super¿cial than the kinds of ontological vagueness discussed so far, might then be further elaborated upon in a number of ways. One way is to suggest that a thesis of ontological vagueness would need to establish that the world as scienti¿cally conceived and described is vague. It is only if vagueness arises at this supposedly ontologically fundamental level that we can really say that the world is vague in a suf¿ciently deep sense. Any other vagueness is super¿cial or inessential in the sense that it is eliminable, reducible to or supervenient upon this fundamental level. We have already discussed proposals such as this in Chapter 3 and found them unconvincing. The assumption that all that there is, the nature of the world in itself, can be inferred from commitments at this level is highly contentious, at best. Moreover, the idea that when we come to assess the extent of vagueness at this supposedly fundamental level we ¿nd none, all vagueness is merely super¿cial, was also found wanting. Another sense in which ontological vagueness as described so far might be thought super¿cial is that the postulated vagueness might be thought the product of our representations whereas ontological vagueness properly considered is a matter of ‘the world being a certain way before we ¿nd it. Our job is to fashion concepts to mirror it. Because it contains vague objects we ¿nd vague objects, and fashion vague concepts to match’ (Sainsbury 1995b: 79). But, as Sainsbury goes on to point out, there is no sense to the idea that the preconceptual world contains vague objects 50

Burgess (1990: §7), Sainsbury (1995b: §VII) and Copeland (1995: §1) all raise the issue.

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because there is no intelligible notion of the world independently of our concepts. If this is what one means by the claim that there is ontological vagueness properly considered, then it is little wonder that one is sceptical of the veracity of such a claim and dismissive of ontological theories of vagueness. Ontological vagueness conceived of in this way is, as Copeland points out, ‘a particularly dark piece of metaphysics’ indeed. But the problem does not lie with an ontological theory of vagueness but, instead, with the high rede¿nition of what such a theory amounts to. Any metaphysics committed to discussion of a ‘preconceptualized world’ will be dark. Vagueness in the world understood in the only way that makes good sense – the world of cats, dogs, clouds, mountains and molecules – is neither trivial nor impossible, but to commit to it is to commit to a thesis concerning the nature of objects, properties and states of affairs and the vague language we use to talk about them that has much to recommend it.

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Chapter 6

Vague Individuation and Counting As we saw earlier, there are strong prima facie reasons in favour of the view that identity may be de re vague, and apparent examples are not hard to
nd. Attempts to show the incoherence of such a view have proceeded primarily by way of the Evans Argument and have so far proved unsuccessful. For all that has been said so far, de re vague identity is coherent and, for the reasons outlined in the previous chapter, ought to be endorsed by those advocating an ontological approach to vagueness. But attacks on the coherence of de re vague identity are not con
ned to the Evans Argument. There are other challenges. Ensuing problems concerning individuation and counting of vague objects, in particular, threaten. As we shall see, the common-sense idea promoted in the previous chapter that vague singular terms might precisely designate or individuate a unique vague object is challenged by the commitment to de re vague identity. Terms
guring in such identities cannot, it is argued, possess such straightforward reference, uncomplicated by vagueness, since any supposed vagueness in an object precludes our being able to uniquely individuate that object. The commitment will then threaten to undermine initial promises that an acceptance of ontological vagueness, and vague objects in particular, underwrites a common-sense response to Russell’s Problem of the Multiple Denotation and the Problem of the Many. Avoidance of the inscrutability of reference associated with a representationalist approach and a resulting solution to the Problem of the Many are inconsistent with an acceptance of vague identity, so it is argued, yet a commitment to de re vague identity follows from a metaphysical approach to vagueness. In seeing just where such a challenge goes astray we shall clarify and further defend the semantic and metaphysical picture being proposed. 6.1

Vagueness and Individuation

An initial set of concerns centres on the idea that an analysis of vague identity as de re undermines the very presuppositions required for the claim in question to be counted an identity because the terms in question fail to properly individuate anything at all. Sainsbury (1995b) voices such a concern, offering reasons for thinking apparent cases of de re vague identity are not what they seem, explaining how, despite appearances, there are no such cases nor can there be. The explanation is offered to bolster the Evans position, which Sainsbury (1995b) endorses. We have shown cause to doubt that position but, given its rejection, the new explanation offered may be recast as an independent justi
cation of the impossibility of de re vague identity, consequently bolstering the representationalist position, and thus is deserving of scrutiny.

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Sainsbury invites us to take seriously the idea that identity might be de re vague. Suppose, for example, that there really is no fact of the matter as to whether a pile of rubbish currently in front of me, b, is identical to the one I saw on the road yesterday morning, a. It follows then that there is no fact of the matter whether b endured the hailstorm that passed over yesterday afternoon (say). Similarly, there is no fact of the matter whether b is of recent origin, beginning with today’s clean-up, or has a more distant origin lying in the mists of early yesterday morning along with a. But individuation of an object requires that the act of individuation determine an origin. Since, on the foregoing suppositions, no origin has been determined, no object has been individuated when we speak of ‘b’ despite what might be mistaken beliefs to the contrary. The very conditions required for the coherence of de re vague identity preclude our being able to properly individuate objects, individuation that is necessary for our being able to say that there are objects x and y concerning which it is vague whether they are one and the same object. In other words, the very conditions required undermine the possibility of de re vague identity (Sainsbury1995b: §V). But such reasoning risks equivocation. Certainly no precise origin has been determined, just as when we describe something as red no precise colour has been ascribed. But in the latter case a colour has been ascribed nonetheless and, similarly, we may say in the former case that an origin has been determined (namely, one that lies vaguely in the time interval between yesterday morning and today). To argue then that no object has been individuated since no origin has been identi
ed is to make a false assumption. To argue, on the other hand, that no object has been individuated since no precise origin has been determined simply begs the question, and such an inference will be contested. The implied constraint on objectindividuation is obviously too strict. To be sure, we might be justi
ed in concluding that since no precise origin has been determined no precise object has been individuated, but this will hardly count as news in the circumstances. Recognizing the strictness of the constraint, Sainsbury himself suggests there may be some indeterminacy concerning exactly when an object comes into being. In fact, this will be true of almost any material object. To demand strict determinacy of origin would preclude all such ‘things’, thus eliminating most currently accepted as wellindividuated objects. However, to maintain in the face of this ubiquitous indeterminacy of origins that some limited indeterminacy be accepted while the more extended indeterminacy considered above be counted as grounds for denying that any object has properly been individuated appears arbitrary. What extent of indeterminacy, for example, will count as acceptable and why only that extent? What basis is there for maintaining that some leeway be permitted as to the exact moment when a pile of rubbish came into being, but denying that this leeway be extended to whether it began yesterday or today, or even, under extreme circumstances, whether it began last decade or today? No principled justi
cation is offered of the allowable leeway and it is dif
cult to imagine any that would not threaten to extend beyond the time frame envisaged. Accounts that differentiate between 10 seconds and 10 hours or 10 seconds and 10 years on grounds sensitive to the interests and/or time frame of human lives imbue ontology with unacceptably anthropocentric elements.1 1 One might think that Lewis’s (1993) defence of almost-identity, rather than identity, as a criterion for counting in everyday contexts is of some use here in defending Sainsbury’s

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A more promising variation on Sainsbury’s explanation of the incoherence of de re vague identity might focus less on the extent of the indeterminacy involved and more on the kind of indeterminacy. More speci
cally, it might be supposed that the admissibility or inadmissibility of indeterminacy when individuating objects may not be a matter of the extent of the indeterminacy but, rather, might depend on whether or not the indeterminacy in question precludes proper individuation by giving rise to indeterminacy of count. Proper individuation is, on this view, precluded only where it does indeed give rise to such indeterminacy of count. So, for example, indeterminacy surrounding which exact moment to count as the origin of, say, a single pile of rubbish might be admitted as unproblematic where it gives rise to no indeterminacy of count – it may be vague when the pile came into being yet remain nonetheless true that there is but one pile. On the other hand, the indeterminacy generated by the supposedly de re vague identity between a pile of rubbish currently in front of me, b, and the one I saw on the road yesterday morning, a, might be thought to be just such a case of indeterminacy that gives rise to indeterminacy of count. Intuitively, there is no fact of the matter whether a and b are one or two piles of rubbish.2 To have individuated an object about which there may remain some residual indeterminacy is one thing, but to think that one can do so in such a way as to leave it indeterminate whether one or more objects is thereby individuated is entirely another. In the latter case, it might now be argued, individuation fails and so too the idea that there can be de re vague identities, since it is precisely in such supposed circumstances that indeterminacy of count arises. Burgess advocates just such a view. Focusing less on the Kripkean requirement of determinacy of origin and more on a Fregean requirement of determinateness (or de
niteness) of sense, Burgess explains the incoherence of de re vague identities by directly appealing to the idea that where indeterminacy gives rise to indeterminacy of count ‘there is no (unique) thing’ which is picked out (1990: 271). As in Sainsbury, the purported explanation is intended to account for a prohibition on de re vague identity, but it is also extended to cover de dicto cases as well.3 Burgess concentrates on an example involving spatial indeterminacy – one encountered earlier, at the beginning of Chapter 5. He supposes that a linguistic position. Might we not say that where the indeterminacy surrounding what to count as the origin of a putative object is limited and the alternatives differ only marginally, in everyday contexts we will count the object as having but one unique candidate for its origin whereas where the indeterminacy concerns alternatives that differ markedly we must admit a plurality of alternatives even in the everyday sense? Such a thought, though perhaps initially tempting, will be of little use. At best it would only provide a defence of the claim that a folk-ontology of objects for everyday contexts remains intact. One would still have to concede that in fact such everyday objects do indeed lack a unique precise origin and therefore fail to be properly individuated. 2 A rigorous account of how to count in the presence of ontological vagueness will be pursued in §3. 3 Some remarks in Sainsbury (1995b), especially in §II, suggest that the Evans Proof ought extend to de dicto vague identities as well, in particular the idea that abstraction (or ‘importation’ as he calls it) ought be accepted by representationalists, but he is not explicit about this.

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community under investigation has been found to have named a local mountain “Aphla” and a local mountain “Ateb”. They are unaware of the fact that the visible peak they refer to as “Aphla” and the visible peak they refer to as “Ateb” are separated by a saddle whose dimensions in relation to the peaks is such that there is no fact of the matter whether the saddle separates two mountains or is part of a single, topologically complex mountain including two peaks joined by the saddle. It would seem then that there is no fact of the matter whether Aphla is Ateb. Were one to grant that “Aphla” and “Ateb” are indeed names individuating objects, then, Burgess concedes, the relevant identity claim would be indeterminate. However, appearances are deceiving in such a case and fail to recognize that such names are, as it turns out, vacuous. For, if there is no fact of the matter whether Aphla is Ateb (i.e. if it is vague whether Aphla is Ateb), then, it is claimed, there is no unique thing which counts as Aphla. In such a case, ‘the sense of “Aphla” is not suf
ciently de
nite to determine a unique object for Aphla to be’, as a consequence of which the name is vacuous. According to this objection, then, de re vague identities give rise to cases of indeterminacy of count. For example, in the case where it is supposedly vague whether Aphla is Ateb, are there two mountains or one? There is, presumably, no fact of the matter. It is indeterminate. Similarly, it is being suggested, there is no fact of the matter whether “Aphla” denotes one thing or not. Consequently, on the view under consideration, de re vague identity claims, by their very nature, involve names that fail to uniquely denote anything. This, in turn, is taken as evidence of the fact that the sense of such names is inde
nite or indeterminate, and so defective and vacuous. Not only would the soundness of such an argument undermine the claim that Aphla, say, is vaguely identical to Ateb, but also even claims of self-identity involving “Aphla” and “Ateb”. The argument for vacuity is not convincing, though. Such names are vague and their senses may be admitted as vague or indeterminate to the extent that there is no precise thing that counts as Aphla. But this absence of determinacy ought not to be confused with the absence of uniqueness. The (true) claim that there is no precise thing designated by “Aphla” must be distinguished from the (false) claim that there is no one thing that is precisely designated by “Aphla”. There is no reason for denying that there is a unique thing that “Aphla” precisely designates. Admittedly there is no fact of the matter as to whether or not it includes Ateb, just as there is no fact of the matter as to whether Ateb includes Aphla. But neither absence of fact precludes uniqueness, as we shall see more clearly in the next section. Admittedly, as a consequence of the vagueness of identity, I think we ought to agree that there is no fact of the matter whether keen climbers, having scaled the thing called “Aphla”, have thereby scaled the thing called “Ateb”. Nonetheless there is a clear sense in which they have scaled but one mountain and not two – a sense according to which, when the thing named “Aphla” has been scaled, only one thing has been scaled just in case there is but one thing determinately denoted by “Aphla”. (Of course, it is a fact that in merely scaling one peak the climbers have determinately not scaled the other; the peaks themselves are determinately distinct and climbing both is determinately not achieved by climbing one. What is indeterminate, recall, is whether the two distinct peaks comprise one mountain or two.)

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If there were no unique thing that counted as Aphla, then we might wonder whether any sense attaching to the name “Aphla” was defective and the name vacuous. However, all that one can claim in the case under consideration is that there is no (unique) precise thing which Aphla is – i.e. that we cannot precisely individuate the thing that is Aphla. And this does not imply that the sense of “Aphla” is not suf
ciently de
nite to determine a unique thing unless we suppose that there being no unique precise thing determined by its sense (i.e. its not precisely designating a precise object – not being a precise name, that is) entails that there is no unique thing which it is (i.e. its not being a precise designator). This cannot be supposed without begging the question against believers in de re vague identity according to whom there can be vague names precisely designating vague objects.4 Burgess is right that there are issues concerning non-uniqueness lurking in the background as a consequence of the claimed indeterminacy of count generated by de re vague identity. Such indeterminacy of count will, as we shall see, leave open how many things there are when we speak of Aphla and Ateb, but it remains clear that Aphla is but one thing and Ateb but one thing. Whether they are collectively one and the same thing or two is indeterminate, but each individually is one thing even if what that thing is is also indeterminate. Allied problems may be thought to arise from a commitment to de re vague identity. Van Inwagen (1988) points to an apparent consequence of a commitment to such identities. If there are such identities, then, he maintains, the semantical relation between name and thing named must sometimes also be vague. ‘If, for example “Alpha” de
nitely [i.e. determinately] names x, and it is neither de
nitely true nor de
nitely false that x = y, then it seems inevitable to suppose that it is neither de
nitely true nor de
nitely false that “Alpha” names y’ (1988: 259).5 More generally, the principle being appealed to by van Inwagen, van Inwagen’s principle, is the following: (VIP) If it is de re vague whether x = y and “a” determinately names x, then it is vague whether “a” names y. This principle may be thought to undermine the coherence of such identities and the supposedly precise designators that appear therein. Parsons thinks as much and consequently goes on to reject VIP. Consider a situation where “Samantha’s Pride” determinately names a ship that left port on Tuesday, and “Kim4Ever” determinately names the ship that docks on Wednesday; moreover there has been disruption between departure and arrival (i.e. repair/assembly reminiscent of the processes discussed in relation to Theseus’s Ship) such that it is indeterminate whether Samantha’s Pride is Kim4Ever. Parsons’s concern is that if it is indeed accepted as inevitable that it is therefore indeterminate whether “Kim4Ever” names Samantha’s Pride, then we would be 4

And therewith Cowles’s (1994: 153) criticism of the idea of determinate reference in the absence of precise individuation is also defused. 5 Van Inwagen is considering the case of vague identity between persons, his favourite disrupted-persons example. The case is essentially similar in relevant respects to the Aphla/Ateb example discussed by Burgess, and Parsons’s disrupted-ship example below.

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required to admit that when we speak using “Kim4Ever” to say that it is indeterminate whether Kim4Ever left port on Tuesday, we are also ‘speaking indeterminately’ of Samantha’s Pride. But ‘are we then speaking indeterminately about Samantha’s Pride? ... Are we saying indeterminately that it is indeterminate whether Samantha’s Pride left port on Tuesday [as van Inwagen requires]?’ Surely not, urges Parsons. Since it is determinate that Samantha’s Pride left on Tuesday, it is determinately not indeterminate. Though it is true to say that it is indeterminate whether Kim4Ever left port on Tuesday, it is false to say that it is indeterminate whether Samantha’s Pride left port on Tuesday, yet van Inwagen’s principle would require that it not be false and so must be rejected. In using the name “Kim4Ever” ‘I am not speaking of Samantha’s Pride at all, even indeterminately’ (2000: 151). More exactly, Parsons’s concern can be brought out as follows. Suppose we accept the following principle – what we might call the naive principle of indeterminate reference: (NPIR) If it is vague whether “a” denotes y, then necessarily if “Fa” is true then it is not false that y is F, for any name “a”, object y and predicate “F”.6 Then the existence of cases where we are prepared to accept “Fa” as true yet accept it as false that b is F (e.g. a case like that just described above concerning Kim4Ever and Samantha’s Pride where the predicate “F” involves talk of indeterminacy) shows that, contra van Inwagen’s VIP, it is not vague whether “a” denotes b despite the fact that in such a case it is vague whether a is b and “a” determinately denotes a. The application of NPIR is taken to establish the falsity of van Inwagen’s principle. By way of further example, suppose it to be vague whether “a” names b. Then, by NPIR, in accepting the truth of “It is vague whether a is b” we would be committed to its being not false that it is indeterminate whether b is b, and yet it is determinate that b is b and so false to suggest that it is indeterminate. It is accordingly not vague whether “a” names b, and again van Inwagen’s principle appears to be false. Parsons briey considers a Leibnizian defence of VIP. If, contrary to the principle, we deny that it is vague whether “a” denotes b, insisting instead that “a” does not denote b despite “a” denoting a, then, by DD, a and b must be said to differ. And this contradicts its being vague whether a is b. The defence is declared fallacious though because, it is suggested, it supposes wrongly that “N refers to [or denotes] x” is a legitimate context for Leibnizian substitution, wrongly being counted a predicate that describes a property of x. “‘N refers to x” is a paradigm case of a predicate that does not stand for a property of x’ (2000: 152). However, property or not, it seems perfectly plausible to suppose that if N refers to x and x is one and the same thing as y, then N refers to y, and so where N refers to one but not the other 6

One might be tempted to attribute to Parsons a stronger principle: if it is vague whether “a” denotes y, then necessarily if “Fa” is true then it is vague whether y is F, for any y and F. But to use “a” to speak vaguely of b (say) does not preclude b from being (determinately) F where “Fa” is true, since both a and b may be F irrespective of the relation between “a” and b.

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they are distinct. Reference is surely transparent. Parsons’s rejection of VIP looks untenable. A much more plausible resolution of the problem now before us – the problem of endorsing both VIP and NPIR – is readily to hand. We might reasonably restrict NPIR and commit to a less naive principle of indeterminate reference: (PIR) If it is vague whether “a” denotes y, then necessarily if “Fa” is true then it is not false that y is F, for any name “a”, object y and property F. Such a principle is compelling in its own right and constrains what can legitimately be inferred from the fact that some name vaguely denotes an object. Where “Fa” is true and “F ” denotes a property F, the denotation of “a” is F. Were it false that b (say) is F, it would follow that b is determinately not the denotation of “a”. So where it is vague whether “a” denotes b, it must not be false that b is F. Predicates involving indeterminacy or determinacy, such as those considered by Parsons in arguing against van Inwagen’s principle, are admittedly problematic contexts but serve only to remind us, once again, that such predicates do not denote properties. Where it is vague whether “Kim4Ever” denotes Samantha’s Pride (e.g. where it is vague whether Kim4Ever, the thing that “Kim4Ever” determinately denotes, is Samantha’s Pride), if it is true that Kim4Ever docked in port on Wednesday, then it is indeed not false that Samantha’s Pride docked in port on that day; in fact, it is vague whether she did. Similarly, if Kim4Ever is up for sale, then it is not false but vague whether Samantha’s Pride is up for sale. But it does not follow from these limited claims concerning the attribution of a property to the thing referred to by “Kim4Ever” that, when we say that it is indeterminate whether Kim4Ever left port on Tuesday, we are saying indeterminately that it is indeterminate whether Samantha’s Pride left port on Tuesday. As with considerations of the Evans Proof, when we speak of its being indeterminate whether Kim4Ever left port on Tuesday, we are not speaking of some property of Kim4Ever which is vaguely attributed to Samantha’s Pride. Rather, we are denying that there is a fact of the matter in relation to Kim4Ever’s having left port on that day. For all that has been said so far, then, de re vague identity is not challenged by VIP. But the principle does, once again, throw Burgess’s concerns into sharp relief. In fact, it can be seen as an articulation of exactly that supposed consequence of a commitment to de re vague identity that leads Burgess to claim that unique reference to objects
guring in such an identity relation is impossible. If, for example, we suppose that Kim4Ever is determinately denoted by any name at all, “Kim4Ever”, say, then by VIP that name will also vaguely refer to Samantha’s Pride (under the conditions described in the foregoing example). How then can we suppose that the name refers to one thing and one thing only, as required for it to count as a singular term? To provide a de
nitive answer to such a question we must turn to a detailed consideration of how to count in the presence of vague objects and vague identity.7

7 It might also seem that a commitment to VIP requires that where there is de re indeterminacy of identity there is also de dicto indeterminacy, i.e. imprecision in designation. But it does not. Imprecision in designation is a matter of there being no object from amongst

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6.2

Vagueness of Count

When discussing vague denotation above, it was suggested that de re vague identities give rise to indeterminacy of count. In the example of Aphla and Ateb it was said that there was no fact of the matter as to whether there are two mountains or one. This and pressing concerns centring on uniqueness of denotation raise the general question of how to count items in the context of ontological vagueness. Taking our cue from Parsons (2000), a satisfactory extension of the classical theory of counting can be developed, one capable of underwriting the foregoing response to Burgess’s criticism. As we shall see later on, Parsons’s own formal theory is problematic as it stands; nonetheless an improved variation is available. The theory to be proposed will, like Parsons’s theory, be truth-functional, i.e. a strongly paracomplete theory. To be sure, Keefe’s (1995) diagnosis of a aw in the Evans Proof (Chapter 5, §2.1) is available from within the non-truth-functional approach encapsulated by supervaluationism. Advocates of such a weakly paracomplete approach are thus in a position to accept the coherence of de re vague identity and can embrace the proposed ontological approach to vagueness.8 But, for the reasons laid out in Chapter 4, we would do well to try to maintain a truthfunctional semantics, and so we shall. Of course, if the anomalies faced by such a semantics turn out to exceed those faced by supervaluationism, then, despite the representationalist defence of the theory being unavailable, we might follow Keefe (2000) and advocate a pragmatic defence. However, not only have we already witnessed the failure of attempts to defend against counterintuitive aspects of nontruth-functionality incumbent upon supervaluationism (Chapter 4), but alleged anomalies that are said to beset a truth-functional approach are exaggerated, as we shall see. A range of stock examples will be useful in developing and assessing our theory of counting in the presence of vagueness, so let us turn,
rstly, to a catalogue of the variety of non-classical situations to be accounted for. 6.2.1

Some examples

As we have already seen, there are what we shall call type-1 cases where disruption to an object, e.g. a ship or person, may be such that it is vague whether the disrupted thing is one and the same as the original. Each thing individually may be counted a person, say, yet there may be no fact of the matter whether the original person is one the candidate referents that is determinately picked out; denotation is one–many. Given VIP, on the other hand, de re indeterminacy of identity requires that we admit cases where, though something is determinately individuated from the range of candidates, there is an object concerning which it is vague whether it is individuated because it is vague whether that object is identical to the thing that is individuated. 8 Keefe (2000: 160) acknowledges this possibility of advocating a supervaluationist semantics without the inscrutablity of reference that follows from a representational approach. Such a possibility is declared to be not ‘within the spirit of supervaluationism’ and I think she is quite right about this. Nonetheless it remains an option.

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and the same as the disrupted person. We see the original person and are reassured that there is at least one person. Then we later encounter the disrupted person and, though we are con
dent that they too are a person, we simply cannot say whether they are a second person or not. There simply is no fact of the matter. There is at least one person and no more than two (it is determinately not the case that there are at least three people in the situation described, for example), yet there is no fact of the matter whether there is at most one person or not. Such cases of de re vague identity present situations in which it seems there is no fact of the matter as to exactly how many persons there are, one or two. Similarly, in the case outlined above concerning Samantha’s Pride and Kim4Ever, though each is undoubtedly a ship there is no fact of the matter as to exactly how many ships there are, one or two. In general, then, there may be cases where objects x and y are such that they are both F, yet it is vague whether the one F is identical to the other and thus there is no fact of the matter exactly how many Fs there are. Thus vague identity can give rise to vagueness of count, and for that reason there may be no determinate answer to the question “How many Fs are there?” Such vagueness of count can also arise, however, in what we shall call type-2 cases where there may be no vagueness in individuating objects but it is nonetheless vague whether some object is to count as F at all. Thus we may be confronted by an object which is not vague itself and enters into no vague identity relations, yet it is indeterminate whether it is red, say. In a situation that includes a clear example of a red car, another determinately distinct car that is just such a borderline case of redness, and nothing else even remotely red, there is no answer to the question “How many red cars are there?” As before, there is at least one and no more than two, but it is simply indeterminate whether there is at most one (and therefore exactly one) or not (and therefore exactly two). There is no fact of the matter whether there are exactly one or two red cars. In general, then, there may be cases where objects x and y are such that they are determinately distinct and thus exactly two objects, yet x is determinately F and y is vaguely F and thus there is no fact of the matter exactly how many Fs there are. Thus vague property exempli
cation can give rise to vagueness of count, and for that reason there may be no determinate answer to the question “How many Fs are there?” There are also type-3 cases, hybrid cases, where indeterminacy of identity generates borderline cases which, in turn, give rise to vagueness of count. Consider, for example, the situation described above involving Samantha’s Pride and Kim4Ever. How many ships should we say left port? Samantha’s Pride determinately did. Yet, since it is indeterminate whether Kim4Ever is Samantha’s Pride, it is indeterminate whether Kim4Ever left port. Thus there is no fact of the matter as to how many ships left port. This may seem a little surprising, so let us spell it out in more detail. Notice,
rstly, that since there is no fact of the matter as to whether Samantha’s Pride is Kim4Ever and since Samantha’s Pride truly left port on Tuesday, it is not false that Kim4Ever left port on Tuesday. This follows from the fact that, since Samantha’s Pride left port on Tuesday, if it were the case that Kim4Ever determinately did not, then Samantha’s Pride and Kim4Ever would, by DD, be determinately distinct. Therefore, since they are not determinately distinct, it cannot be false that Kim4Ever left port.

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More generally, the following principle holds – the principle of indeterminate identity:9 (PII) If x has property F and it is vague whether x = y, then y cannot lack property F. Its plausibility follows from the combined force of van Inwagen’s principle and PIR discussed above, and also follows directly from DD. If x has property F yet y determinately does not, then x and y must, by DD, be determinately distinct. By assumption, however, they are not determinately distinct; it is vague whether they are identical and thus there is no fact of the matter as to whether they are identical or not. Thus, assuming x has property F and it is vague whether x = y, it cannot be that y lacks F.10 Returning to our example, then, notice, secondly, that it is not true that Kim4Ever left port on Tuesday, since in the scenario described if that were the case then Kim4Ever would be identical to Samantha’s Pride which, ex hypothesi, they cannot be said to be. (Of course, if one took the view that Kim4Ever did leave port on Tuesday, then the situation would simply revert to that of the previously described case of indeterminacy of count, where Samantha’s Pride and Kim4Ever determinately left port on Tuesday but it is vague whether they are one or two objects.) This, combined with the
rst point made above, then, points to its being vague whether Kim4Ever left port on Tuesday. In the example under consideration, then, it is indeterminate whether Samantha’s Pride is Kim4Ever; Samantha’s Pride exempli
es the property of having left port and Kim4Ever is a borderline case for such a property. In such circumstances where indeterminacy of identity generates borderline cases, how many ships should we suppose left port? In fact, there can be no determinate answer to this question. To see this, consider the following truism, one which Parsons (2000) implicitly supposes and Pinillos (2003) dubs the principle for counting with indeterminacy (PCI): (PCI) If, for some x, it is indeterminate whether x has property F, then there is no determinate answer as to exactly how many things have property F. Pinillos cites this as a principle of ‘great plausibility’ and goes on to show how the required commitment to such a principle undermines the coherence of vague identity. We shall return to this argument later; for now it is merely the principle itself that is of interest. To use an example from Pinillos (2003: 39), if it is indeterminate whether Harry is bald, then there is no determinate answer as to exactly how many bald things there are: ‘It will be indeterminate whether or not to count Harry as someone who is bald.’ Similarly, if it is indeterminate whether 9

Pinillos (2003: 43). As Pinillos points out, sometimes it will be indeterminate whether y is F (e.g. in the example currently being considered where Samantha’s Pride left port and it is vague whether it is Kim4Ever, then, as we shall see, it is vague whether Kim4Ever left port) but not always. Sometimes y will be F in its own right, as it were (e.g. where a is a person and through a case of disruption it is vague whether a = b, then it remains true that b is a person – it is merely vague whether it is the same person). 10

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something counts as red, or as a person, then there is no determinate answer as to exactly how many red things there are, nor how many persons there are. Consequently, returning to the question of Samantha’s Pride and Kim4Ever, since it is indeterminate whether Kim4Ever left port, there is no determinate answer as to exactly how many ships left port. At least one did, namely Samantha’s Pride, and at most two did, namely Samantha’s Pride and Kim4Ever. But there is no precise fact of the matter whether one or two did. We are faced once again with indeterminacy of count. 6.2.2

The ambiguity of count

At this point many will feel that the theory being advanced here is reduced to absurdity. Surely, it will be protested, nothing could be clearer in the example described than that exactly one ship left port and any theory not delivering such an answer is thereby refuted. But the objection fails to take into account complexities surrounding the counting of objects that arise here and elsewhere, placing too heavy a weight on our intuitions which are, in the situation at hand, quite malleable indeed.11 While there is, in the sense just outlined, no determinate answer to the question as to how many ships left port, in another sense the answer ‘Determinately one’ is delivered by the foregoing theory. As with counting in other situations, there are a number of distinct ways of counting, some of which accord with our pretheoretical intuitions and others which do not. Consider ways in which we count roads, for example. In following a path across the countryside I
nd myself at one point standing on a piece of asphalt that extends away from me in either direction, vehicles passing by. I am asked how many roads I am standing on and I reply with the obvious response, “One”. Surely nothing could be clearer than that I am currently standing on but one road. Yet, as Lewis (1976: 27) points out, the asphalt might be part of two roads, the M1 and M3, that overlap at that point, merging into a single stretch of asphalt. I am, in a sense, standing on two roads – after all, I am standing on the M1 and I am standing on the M3, and each of these is a distinct road with one leading north to Cairns and the other leading west to Alice Springs. Of course, there is also a sense in which I am standing on one road. I am indeed standing on the M1 and standing on the M3, but they are not locally distinct, or, as Lewis puts it, they are not distinct along my path. Thus if we count using identity, given that the M1 and M3 are not identical, I am standing on each of two roads. Yet, if we count using identity-along-my-path, then, given that the M1 and M3 are identical-along-my-path by virtue of their overlapping where I cross them, I am standing on but one road. There are two ways of counting roads, each delivering a different answer to count-questions and only one of which accords with common intuition in the situation described. So too with counting in the presence of indeterminacy. In such circumstances there will always be two distinct ways of counting which may result in distinct answers to count-questions, only one of which will accord with our intuitions in the relevant circumstances. In the example of a type-3 case above, the equivocation 11 What follows owes much to Parsons’s response to the count problem under discussion; see Parsons (2000: 136f).

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arises by virtue of the confusion between the asking how many ships left port and how many ships determinately left port. When, such questions are explicitly differentiated in this way, we can see that though the former is not properly answered by “Exactly one”, the latter does indeed have that answer. Samantha’s Pride determinately left on Tuesday so at least one ship did, and Kim4Ever did not determinately leave so at most one did.12 The admittedly strong intuition that the simple question “How many ships left port?” has the answer “One” derives from our conating the question with what Parsons describes as its ‘adorned’ counterpart involving determinacy. In many contexts there is no distinction to be drawn between being F and determinately being F, but sometimes there is and this is just such a situation. In normal circumstances we take determinacy for granted, and there is no relevant difference in our assertions between “F ” and “determinately F ”. Also, since we aim to assert what is (determinately) true, there is no difference between asserting “a” and asserting “determinately a”. But when a clause is embedded inside an assertion, and when there is a possibility of indeterminacy, then we need to distinguish whether we mean “determinately F” or just “F”. This is the case when we assert “there are n Fs” in a situation of indeterminacy. In the case at hand, there is a difference between counting the ships that left port, or the ships that determinately left port. (2000: 136–7)

The intuition that, contrary to the answer to the count-question outlined above, exactly one ship left port ignores the difference. It is indeed initially surprising that the number of ships that left port is not precisely one, just as it is initially surprising in the example mentioned above that the number of roads I am standing on is not precisely one. But just as we can come to see how it might be that I am standing on two roads and recognize that my intuitions to the contrary are explained by counting in an alternative sense, so too here we can come to see how it might be that there is no determinate answer to how many ships left port and recognize that our intuitions to the contrary are explained by counting in an alternative sense, the adorned sense. The same can be said of other type-3 cases where indeterminacy of identity generates borderline cases which, in turn, give rise to vagueness of count. In light of a problem to be discussed below, the Pinillos Argument, cases of so-called ‘split indeterminate identity’ are of particular interest. Such cases, remember (see Chapter 5, §1), involve objects x, y and z where there is no fact of the matter as to whether x = y, nor whether y = z, yet it is determinately the case that x ≠ z. For example, consider a case where Samantha’s Pride, the ship that left port on Tuesday, is itself the product of a disruptive process applied to a ship dubbed “One4All”, which docked in port on Monday. Again, as with the disruptive process applied to Samantha’s Pride (which resulted in there being no fact of the matter whether it was the one and the same thing as Kim4Ever – the ship that resulted from the process), suppose there to be no fact of the matter whether One4All is Samantha’s Pride. Nonetheless, suppose the disruptive processes applied to One4All and Samantha’s 12 Of course, the fact that Samantha’s Pride and Kim4Ever (determinately) differ as regards determinately leaving port does not establish them as determinately distinct since “determinately leaving port” is not a property, as we saw in Chapter 5, §2.

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Pride to be such that the ship that docked on Monday, One4All, is determinately not the ship that docked on Wednesday. (We are simply faced with a case where changes to a thing, One4All, result in a determinately different thing, Kim4Ever, but at some point in the process we are confronted by a thing, Samantha’s Pride, which is a borderline case for being either of them.) In the example under consideration, then, despite One4All being determinately distinct from Kim4Ever, it is indeterminate whether One4All is Samantha’s Pride and indeterminate whether Samantha’s Pride is Kim4Ever. Moreover, it is clearly the case that One4All and Kim4Ever determinately docked in port (though on different days), and clear that Samantha’s Pride determinately left port. How many ships should we then suppose left port and how many docked? The answer in each case is that there is no fact of the matter, though, again, it is clear that one ship determinately left port and two determinately docked. By PCI (as we saw above) it is not determinately not the case that Kim4Ever left port, and for the same reason it is not determinately not the case that One4All left port. Nor, in the given situation, is it determinately the case for either ship that it left port on the Tuesday in question. It is indeterminate for each of One4All and Kim4Ever whether they left port, since for each of them it is indeterminate whether they are Samantha’s Pride. Consequently, by PII, there is no fact of the matter as to exactly how many ships left port. At least one left port, namely Samantha’s Pride, and at most three did, but there is no fact of the matter exactly how many ships left. Similarly, it is indeterminate whether Samantha’s Pride docked and thus, by PII, there is no fact of the matter exactly how many ships docked. At least two did, One4All on Monday and Kim4Ever on Wednesday, and at most three did, but there is no fact of the matter exactly how many ships docked, all told. In such circumstances where indeterminacy of identity generates borderline cases, there is simply no fact of the matter how many ships left port and how many docked. In spite of this, however, we can recognize that exactly one ship determinately left and two determinately docked. So much for counting in type-3 cases. What of other situations? In some cases the distinction between ways of counting will not result in any difference. Type-1 cases are such that the distinction has no bearing. Whether we count things that are F or those that are determinately F will make no difference since the things being counted, the Fs, are, given the nature of such cases, determinately F anyway. Thus either way of counting will yield the kinds of answers outlined earlier for such cases. Both the adorned and unadorned analyses yield the intuitively correct result. In some other cases, e.g. type-2 cases, the distinct ways of counting will admit of distinct answers and our intuitions seem to be best explained by the unadorned reading. As we saw above, in such cases there may be no vagueness in individuating objects but it is nonetheless vague whether some object is to count as F at all. For example, we may be confronted by an object which is not itself vague yet it is indeterminate whether it is red. In a situation that includes a clear example of a red car, another determinately distinct car that is just such a borderline case, and nothing else even remotely red, there is no answer to the question “How many red cars are there?” As before, there is at least one and no more than two, but it is simply indeterminate whether there is at most one (and therefore exactly one) or not (and therefore exactly two). There is no fact of the matter whether there are exactly one

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or two red cars. This seems the right answer, yet the adorned reading yields the answer “One”. Counting in the presence of vagueness is simply ambiguous and, while the ambiguity makes no difference in some situations (e.g. type-1 cases), in other situations it does and our intuitions seem sometimes to accord with one disambiguation and sometimes another. It may be objected here that the theory proposed is simply ad hoc. In different types of cases our intuitions are consistent with different analyses of how to count, and we can only claim the theory to be consistent with our intuitions by choosing, in an unprincipled way, distinct analyses in different types of cases. But the charge misunderstands the position. What is not being proposed is that there is, in any given type of case, only one unambiguously correct analysis of counting (and, consequently, that a distinct analysis must be proposed for distinct kinds of cases). Rather, there are a number of distinct ways of analysing count-claims in any given type of case and only some of them accord with our intuitions. There is one special situation, however, where the count-ambiguity cannot be tolerated. Surely when a vague singular term “a” is employed it must be supposed that it uniquely refers, i.e. it must be supposed that the answer to the question “How many things does ‘a’ denote?” is “One”, precisely one. It was exactly Burgess’s concern that terms
guring in de re vague identities could not uniquely refer that led him to argue that such identities are incoherent. So can the overall theory being advanced – that there are de re vague identities involving precise designators that uniquely denote vague objects – be underwritten by the theory of counting as described and constrained by VIP, PIR, PII and PCI? This question is of particular interest given the revealed ambiguity of count just discussed. 6.2.3

Uniqueness of reference

The problem is this. If “a” is a precise designator (determinately) denoting an object x and it is (de re) vague whether x = y then, by VIP, it is vague whether “a” denotes y. There will be no fact of the matter whether “a” denotes y because, though it (determinately) denotes x, there is no fact of the matter whether x itself is y. Given PCI, then, considering the property of being denoted by “a”, there can be no determinate answer as to how many objects are denoted by “a”. Thus it is not true that exactly one object is denoted by “a”. If the term precisely designates an object standing in a vague identity relation, then it fails to be properly singular in reference. We thus seem precluded from naming vague objects that
gure in vague identities. Worse still, you might think, given the Vague Identity Thesis, any vague object
gures in vague identities. So we seem precluded from naming vague objects tout court. (Obviously, those more sparing in attributions of de re vague identity will be proportionately spared this preclusion, and limited possibilities for naming vague objects will present themselves.) “Tibbles”, for example, does not uniquely refer to a vague object. Reference would appear to be inscrutable after all, and Burgess’s criticism with which we began this chapter might seem valid. But as we have already seen in type-3 cases, there is ambiguity when it comes to counting in such circumstances. Indeterminacy of count arises when counting using the unadorned reading, while on the adorned reading no indeterminacy arises.

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Answers to the question “How many Fs are there?” – e.g. “How many objects are denoted by ‘a’?” – will depend on which analysis one adopts. On the adorned reading we must count the objects determinately denoted by “a” and that is x and x alone, not y since y is not determinately denoted by the term. The adorned reading accords with the requirement for uniqueness of reference. What we can say, then, is that a vague name “a” can truly be said to precisely designate a unique vague object in the sense that it determinately denotes that one thing. Of course, there is also a sense (expressed by the unadorned reading) in which this is not true. So it seems fair to ask why one should consider it a victory that the theory can
nd an analysis (adorned) that not only accords with general intuitions about counting but also, in the case of names, with the requirement of uniqueness of reference when the same theory also makes available another analysis at odds with these intuitions and the uniqueness requirement in the case of names. Why consider the ambiguous count-analysis a satisfactory dissolution of the problems faced rather than simply providing the means for an ad hoc response which promotes one analysis and ignores the other? The answer is simply that where there is ambiguity in a context and only one disambiguation makes sense of the context, then that disambiguation is forced by the context. Examples abound. If I start talking of ‘the man over there’ when in fact there are two, one short and the other tall, the phrase is, as it stands, ambiguous. But if I go on to say that he is tall, then the audience will naturally interpret the ambiguous phrase so as to refer to the tall man. Similarly, when only one disambiguation of the ambiguous count-analysis is able to underwrite my use of a singular term like “Kim4Ever”, then the context supports that disambiguation, namely the adorned reading, according to which the term is indeed a singular term that precisely designates a unique object. There is nothing ad hoc about such a proposal since context provides the principled basis for choosing the adorned reading over the unadorned reading, just as in the case of talk about ‘the tall man’ the context provides a principled basis for choosing one interpretation over another. When using singular terms to refer to objects, the things referred to are those determinately denoted thereby, and the vagueness of the terms and things referred to in no way undermines the terms’ function as precise designators of unique objects. What is precisely designated is that unique thing determinately denoted. The use of language supposing uniqueness is thus charitably accommodated by the use of that analysis of count which accords with the presupposed uniqueness. Where the designator is used as if it is a precisely designating singular term yet it is vague qua name or description, then it must be presumed that ‘the referent’ be the thing determinately denoted. Any other analysis makes a nonsense of the presuppositions underlying the use of such language. “Aphla” precisely designates exactly one vague object Aphla (i.e. the thing determinately denoted by the term) and “Ateb” designates exactly one vague object Ateb (i.e. the thing determinately denoted by that term), but there is no fact of the matter whether Aphla is one and the same thing as Ateb. Vague objects are uniquely referred to by precise designators, designators that are vague singular terms capable of
guring in de re vague identities. When all is said and done, the counting of objects is contentious, as is well known from other examples, and the presence of vague objects and de re indeterminacy

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generates its own peculiarities. But this ought not to come as a surprise given problems elsewhere. 6.3

Formalizing Count-phrases

To make rigorous an account of the foregoing analysis of counting in the context of ontological vagueness, some formal commitments will have to be made. As declared at the close of Chapter 4, we shall pursue a paracomplete approach. With the previous chapter’s analysis of the Evans Argument as fallacious and a coherent account of ontological vagueness made available, the route to a defence of a weakly paracomplete, supervaluationist approach by way of metaphysical precision is undermined. The alternative, pragmatic defence detailed by Keefe (2000) is, as we shall see in due course, also unsuccessful. This will only emerge, however, once a strongly paracomplete rival has been described and assessed (a task begun here but only completed in the next chapter). With this in mind, let us begin with that fragment of the logic required to make rigorous the foregoing view of how to count with vague objects. (The phenomenon of higher-order vagueness will induce some added logical complexity, but we can leave this until the next chapter.) The commonly proposed conditional-free fragment of the logic as modelled by both Kleene’s strong logic K3 and the three-valued Łukasiewicz logic L3 (both fragments of the logic of First Degree Entailment FDE) is accepted as a suitable approximation to natural-language negation, conjunction and disjunction (respectively).13 Thus: “~A” is true if “A” is false, false if “A” is true, and indeterminate if “A” is indeterminate. “A & B” is true if both “A” and “B” are true, false if either “A” or “B” is false, and indeterminate if neither is false but not both are true. “A V B” is true if either “A” or “B” is true, false if both “A” and “B” are false, and indeterminate if neither is true but not both are false.

If we supplement the propositional fragment with quanti
ers ∀ and ∃, understood respectively as generalizations of disjunction and conjunction, then we can say that: “∀xFx” is true if all instances are true, false if some instance is false, and indeterminate if no instance is false but not all are true. “∃xFx” is true if some instance is true, false if every instance is false, and indeterminate if no instance is true but not all are false.

The semantics outlined so far admits of two interpretations. On the
rst there are only the two classical truth-values, “true” and “false”, but sentences that are indeterminate in truth-value simply lack a value – a strictly ‘truth-value-gap’ approach, favoured by supervaluationists and endorsed here. On the second approach, sentences that are indeterminate are on a par with sentences that are true 13

See Tye (1990, 1994), Parsons (2000) and Field (2003).

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(or false) in the sense that they take a truth-value, but one other than “true” or “false”. Thus on this view there are three values, “true”, “false” and “neither true nor false” or “indeterminate”. It is frequently said that there is no real difference between the two approaches (e.g. Keefe 2000: 90) and both supervaluationism and truth-functional approaches are often presented as being three-valued. For formal purposes I too shall present the logic as being three-valued but do not advocate a three-valued logic (or many-valued logic) in the strict sense. There are only two truth-values but there are sentences taking neither value and they can conveniently be represented as taking a third value. (As we shall see when we come to defend the truth-functional approach against criticisms arising from higher-order vagueness in the next chapter, the difference between the two interpretations may be far more signi
cant for the truth-functionalist than is typically supposed.) Representing the semantic values then as 1 (true), @ (indeterminate), and 0 (false) and taking them to be ordered in the usual way with top value 1, negation corresponds to an order-inversion operation with
xed point @, the value of a conjunction is simply the minimum value of the conjuncts, and the value of a disjunction is the maximum value of the disjuncts: A

~A

A

B

A&B

AVB

1 @ 0

0 @ 1

1 @ 0 1 @ 0 1 @ 0

1 1 1 @ @ @ 0 0 0

1 @ 0 @ @ 0 0 0 0

1 1 1 1 @ @ 1 @ 0

A semantics for the determinacy operator “determinately”, “D”, is also naturally forthcoming: “DA” is true if “A” is true, and false if “A” is not true. A

DA

1 @ 0

1 0 0

The matter of an adequate account of the conditional “if … then” is more contentious. Tye (1990) proposes retaining the classical connective, “⊃”, extended in the obvious way to preserve the classical equivalence between “A ⊃ B” and “~A V B” (as modelled in K3, as we shall see in the next chapter). Parsons (2000) argues for an analysis in terms of the Łukasiewicz conditional, “→”, characterized below (as modelled in L3, as we shall see in the next chapter):

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A

B

A ⊃ C

1 @ 0 1 @ 0 1 @ 0

1 1 1 @ @

1 1 1 @ @ 1 0 @ 1

0 0 0

A→C 1 1 1 @ 1 1 0 @ 1

Whether either of these is in fact an adequate analysis of conditionality need not concern us at present. As we shall see, we can provide an analysis of count-phrases without invoking a conditional. An analysis of conditionality can be postponed until we come to give an account of the sorites paradox in the next chapter. With the foregoing semantics in hand we can now make good our earlier aim of formalizing counting in the presence of metaphysical vagueness. The following is a standard way of analysing claims of the form “there are at least n Fs”: There is at least one F: ∃xFx There are at least two Fs: ∃x∃y(x ≠ y & Fx & Fy) There are at least three Fs: ∃x∃y∃z(x ≠ y & x ≠ z & y ≠ z & Fx & Fy & Fz) Parsons (2000) offers the following analysis of “there are at most n Fs”: There is at most one F: There are at most two Fs:

∀x∀y((Fx & Fy) → x = y) ∀x∀y∀z((Fx & Fy & Fz) → (x = y V y = z V x = z))

etc. For example, the claim that there is at most one F is analysed by the locution “Any x and y, if both F, are one and the same thing” where “if ... then” is modelled by “→”. But this cannot be right. It seems undeniable that the claims “there are at most n Fs” and “there are at least n + 1 Fs” are contradictories. That is to say, one is true if and only if the other is false, and indeterminate otherwise (at least in a non-empty universe). Yet, on Parsons’s analysis, given the proposed semantics, this turns out not to be the case.14 For example, it may be true that there is at most one F, yet not false that there are at least two Fs. (Simply let “Fa” be true, and each of “Fb” and “a = b” be indeterminate, taking the value @.14 For example, let “a” be Samantha’s Pride, “b” be Kim4Ever, where “F ” is “is a ship that left port on Tuesday”. It is then true that there is at most one ship that left port on Tuesday, yet indeterminate whether at least two did.) Accepting the analysis of “there are at least n Fs”, we should reject the analysis offered for “at most”. 14 Though not recognized as such, it is this anomaly that lies behind the criticism of Parsons’ analysis of count-phrases in Pinillos (2003: 41ff).

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An obvious alternative which avoids the need to settle disputes surrounding analyses of conditionality, suggested by the foregoing criticism and equivalent to the foregoing analysis in situations where indeterminacy does not arise, is the following: There is at most one F: There are at most two Fs:

~∃x∃y(x ≠ y & Fx & Fy) ~∃x∃y∃z(x ≠ y & x ≠ z & y ≠ z & Fx & Fy & Fz)

etc. Or equivalently: There is at most one F: There are at most two Fs:

∀x∀y((Fx & Fy) ⊃ x = y) ∀x∀y∀z((Fx & Fy & Fx) ⊃ (x = y V y = z V x = z))

etc. We need not dwell on whether this adds weight to an analysis of conditionality in terms of the generalized material conditional, “⊃”.15 The foregoing analysis can be adopted irrespective of one’s views concerning conditionality since it is formalizable in the conditional-free fragment of the formal language being proposed. We shall, of course, return to considerations of the conditional when we discuss the sorites paradox in the next chapter, and conditional forms of the paradox in particular but, for now, the matter is best put aside. Combining the at-least analysis with the foregoing at-most analysis, an obvious analysis is forthcoming for claims of the form “There are exactly n Fs”.16 There is exactly one F: There are exactly two Fs:

∃xFx & ~∃x∃y(x ≠ y & Fx & Fy) ∃x∃y(x ≠ y & Fx & Fy) & ~∃x∃y∃z(x ≠ y & x ≠ z & y ≠ z & Fx & Fy & Fz)

etc. Given this account, then, what of the range of stock examples described earlier? 15 To show that anything relevant to conditionality follows from this analysis using the material conditional, it would need to be shown that “at most” phrases involve conditionality in some way. 16 The proposed analysis conicts with an analysis of “exactly one” proposed by Pinillos (2003: 44) according to which there is exactly one F just in case there is an F and nothing other than it is F. For example, it is said to be a ‘truism’ that there is exactly one ship that docked just in case there is a ship that docked and nothing other than it is a ship that docked. But given that this ‘truism’ conicts with the equally obvious (and classically equivalent) analysis proposed below (which entails the falsity of “there is exactly one ship” in the One4All, Samantha’s Pride and Kim4Ever scenario described earlier, as opposed to the indeterminacy of the claim as entailed by the ‘truism’), this response appears to beg the question.

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Note,
rstly, that where no actual indeterminacy arises, since the logic described coincides with classical logic on classical values, contingent count-truths will remain unaffected. Thus, where we have objects determinately individuated from one another, which either determinately are or determinately are not F, questions of the form “Exactly how many Fs are there?” have determinate answers: either none, or one, or two, etc. With this in mind, we need only check those more controversial cases. In the type1 example concerning Samantha’s Pride and Kim4Ever, though each is undoubtedly a ship, it was said that there was no fact of the matter as to exactly how many ships there are, one or two. And indeed this is the answer delivered by the formal analysis, as is easily veri
ed. Similarly in the case that includes a clear example of a red car and another (determinately distinct) car that is a borderline case of redness, and nothing else even remotely red. There is no fact of the matter as to exactly how many red cars there are, one or two, and this, again, is the answer delivered by the analysis offered. Similarly, considering One4All, Samantha’s Pride and Kim4Ever, the formal analysis offered delivers the expected answer that there is no fact of the matter as to exactly how many ships left port. At least one left port (Samantha’s Pride) and at most three did, but there is no fact of the matter exactly how many ships left. Similarly, there is no fact of the matter exactly how many ships docked. At least two did (One4All on Monday and Kim4Ever on Wednesday) and at most three did, but there is no fact of the matter exactly how many ships docked. And how many ships can we say determinately left port and how many ships determinately docked? Exactly one determinately left (Samantha’s Pride), and exactly two determinately docked (One4All and Kim4Ever). These are the answers determined by the foregoing analysis. 6.4

De¿nite Descriptions

Whatever one’s preferred way of handling definite descriptions prior to considerations arising from vagueness, with the foregoing analysis in place it is easy to see how de
nite descriptions are to be handled when vagueness presents itself. To be sure, vagueness in identity will introduce an ambiguity in respect of countquestions here too, just as arose with vague names. However, just as uniqueness of reference for precisely designating vague names was secured by appeal to an obvious disambiguation, so too with de
nite descriptions and the requirement for uniqueness. No special dif
culties present themselves, and de
nite descriptions are easily accommodated. Consider an example, “the ship that left port”. As we have seen, there is an ambiguity in respect of count. In one sense (the adorned sense) there is but one ship that left, and in another sense (the unadorned sense) there is no fact of the matter whether there is one or two. In answer to the question “How many ships left port?”, there are two ways of counting, each delivering a different answer. But this is not really surprising. Consider the counting of roads, again. When standing on a stretch of asphalt which is part of the M1 and also part of the overlapping M3, there is a similar ambiguity as to how many roads I am standing on. As is well known, it

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depends on how we count roads. But the ambiguity here does not threaten talk of ‘the road’ and its bumpiness (say). We easily disambiguate in the context so as to invoke that way of counting which makes sense of the talk, i.e. that sense which delivers uniqueness in respect of count so as to underwrite the presuppositions required for proper use of a de
nite description. So too when talking (ambiguously) of ‘the ship that left port’. Where uniqueness is supposed, as it obviously is when invoking de
nite descriptions, we disambiguate in favour of talk of ‘the ship that determinately left port’. There are two broadly distinct approaches to such descriptions, a Russellian approach and an approach that treats such descriptions as denoting phrases. On the Russellian approach such descriptions are incomplete symbols whose analysis is given only in the context of a complete sentence. So when we say, for example, that the ship that left port was in need of repair, this is to be analysed as the claim that there is exactly one ship that left port and that (unique) thing was in need of repair. With ambiguity of count there is an interpretation of the foregoing claim (the unadorned reading) according to which the claim is not true since there is no fact of the matter exactly how many ships left port. However, there is also an adorned reading according to which exactly one ship did indeed leave port, Samantha’s Pride, and the claim is therefore true just in case Samantha’s Pride was indeed in need of repair. On this reading the claim is analysed as: ∃x[x is a ship that determinately left port &~∃x∃y(x ≠ y & x is a ship that determinately left port & y is a ship that determinately left port) & x was in need of repair]. Given that the ship that left port was Samantha’s Pride, it is therefore true that the ship that (determinately) left port was in need of repair if and only if Samantha’s Pride was in need of repair and false otherwise. There is no mystery here. What of theories that treat such de
nite descriptions as denoting phrases? We should simply admit ambiguity again, with an obvious disambiguation that interprets such phrases as uniquely denoting. On this approach we can say, in general, that the phrase “ιxFx” will be taken to denote a if and only if a is F and at most one thing is F, i.e. a is that unique thing that is F. In the context of vagueness there will sometimes be ambiguity between talk of the Fs and the determinate Fs, and when speaking of the ship that left port (for example), under the circumstances envisaged, “the ship that left port” will denote Samantha’s Pride and only Samantha’s Pride given the adorned reading. Samantha’s Pride, is a ship that determinately left port and there is no other thing that is a ship that determinately left port. 6.5

The Pinillos Argument

So far, so good. Vague predicates might precisely designate (i.e. directly refer to) vague properties, and vague singular terms and de
nite descriptions might precisely designate vague objects whose identity conditions might sometimes be indeterminate. There can, accordingly, sometimes also be cases of indeterminacy of count. There is, however, one
nal challenge to the account of ontological vagueness focusing again on the admission of de re vague identity. Concentrating on claims concerning counting and sets, Pinillos (2003) presents a new argument against vague identity, distinct from that offered a quarter of a century

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ago by Evans (1978), which is taken to tell against those who would interpret such identity as de re. The Pinillos argument focuses on the inconsistency between supposed facts concerning counting in the presence of de re indeterminacy and cases of split indeterminate identity (SII) to which de re vague identity theorists are supposedly committed. Pinillos invites us to consider the puzzle of Theseus’s Ship. ‘[S]uppose that we repair a wooden ship by replacing its planks one by one with new ones while at the same time reconstructing it using the discarded planks. Some defenders of vague or indeterminate identity claim that: (1) although the reconstructed ship is distinct from the repaired ship, it is indeterminate whether the original ship is the reconstructed ship and indeterminate whether it is the repaired ship’; thus we have a case of SII. With the additional claim that: (2) the indeterminacy is de re, being ‘due to the world and not just an imprecision in the language used to describe the situation’, Pinillos argues that such a description is incoherent. For reasons outlined earlier (Chapter 5, §1), one might think that the particular case of split indeterminate identity described, a case of synchronous split indeterminate identity involving Theseus’s Ship, is a controversial example of SII. Nonetheless, other examples of SII are available by means of which Pinillos’s general argument will have equal force. Suppose that instead of a
ssion-like case centring on Theseus’s Ship, we consider the case described in the previous section involving One4All, Samantha’s Pride and Kim4Ever. Although One4All is distinct from Kim4Ever, it is indeterminate whether Samantha’s Pride is One4All and indeterminate whether it is Kim4Ever. Thus we have a case of SII. Moreover, it is one that I and others like Parsons take to involve de re indeterminacy. Why think such a case to be ‘incoherent’, then, as Pinillos suggests? Pinillos argues that: (i) if exactly one ship left port and exactly two ships docked, then the described case of (de re) split indeterminate identity does not obtain; moreover, (ii) exactly one ship did leave port and exactly two ships did dock. Hence the described case of (de re) split indeterminate identity does not obtain. Assuming: (iii) if SII is not true for the ship case, then it is not true in the general case (remembering that for those who might take issue with the original case proposed by Parsons and the target of Pinillos’s argument we have an alternative example which is more clearly a case of SII if anything is), we can infer that no case of (de re) split indeterminate identity can obtain (2003: 36). Furthermore: (iv) a position that endorses cases of de re indeterminate identity but not split indeterminate identity ‘is unprincipled since puzzles that motivate [split indeterminate identity] are the same kinds of puzzles that motivate all indeterminate identity claims’ (2003: 50). Consequently, then, there can be no case of (de re) indeterminate identity. The argument is, I think, ultimately unsuccessful. Nonetheless, it articulates a subtle and testing worry about the coherence of de re indeterminate identity and there is, accordingly, much to be learned from its analysis. Much of its force derives from set-theoretic considerations which are taken to preclude the possibility of SII when such considerations are evaluated in the context of a strongly paracomplete logic. Pinillos takes the (not uncommon) view that, where the indeterminacy in question is due to the world rather than mere semantic (i.e. de dicto or representational) indeterminacy, weakly paracomplete supervaluational responses are inappropriate. Consequently, the argument that is aimed, like that of Evans, at

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establishing the incoherence of de re vague identity attributes to the de re theorist a formal semantics that is strongly paracomplete – consistent with the foregoing conditional-free semantics and, more particularly from Pinillos’s point of view, equivalent to the semantics proposed by Parsons (2000). The explicit representative target for the view being criticized is Parsons (2000), the currently best-articulated defence of the position of the de re theorist. This much having been said, a defence of (i) then proceeds initially by way of Parsons’s explicit commitment to the set-theoretic Axiom of Extensionality: Extensionality ∀z(z ∈ X ↔ z ∈ Y) ↔ X = Y, for any sets X, Y and object z.17 A commitment to this principle, dubbed ‘Set Essence’ by Parsons,18 is said, in part, to establish the fact that if exactly one ship left port (e.g. Samantha’s Pride) and exactly two docked (e.g. One4All and Kim4Ever), then that which left is distinct from one or the other of those that docked. As a consequence, then, one cannot describe such situations as cases of split indeterminate identity (SII). There must indeed be a fact of the matter as to whether that which left is identical to either or both of those that docked – it is (determinately) distinct from at least one of them. The more general point being argued for here is that not only must we agree that where some objects b and c are distinct (e.g. two ships which docked in port), then no object a (e.g. the ship which left port) can be identical to both – a commitment to Leibniz’s Law contra Prior (1968) – but we must also agree to the stronger claim that where b and c are distinct, then any object a must be distinct from (at least) one of them. To postulate cases of split indeterminate identity is to deny this latter claim, agreeing to Leibniz’s Law but drawing on the distinction between not being identical and being distinct, yet Pinillos argues that this denial is incoherent (eventually going on to claim, more strongly, that the underlying distinction cannot be maintained to admit of cases of indeterminate identity at all). So how exactly does the argument for (i) proceed? Well, suppose that exactly one ship, a, say, did indeed leave port and exactly two ships docked, say b and c. Then, by the Axiom of Comprehension: Comprehension ∃X∀z(z ∈ X ↔ Fz), for any predicate F,19 17 Note that as yet we have not provided any formal semantics for the conditional “↔”, and nor does Pinillos prior to detailing the argument that invokes it in the formal expression of Set Essence. As it turns out, little hangs on this. We need only suppose the conditional to be such that it satis
es modus tollens, and the derivative biconditional “↔” to be such that ~(A ↔ B) entails (A ↔ ∼B). Whether these are indeed acceptable principles remains to be seen. For the moment let us simply accept such principles. (The version of Set Essence endorsed by Parsons invokes the L3 conditional which does satisfy these principles.) 18 Parsons (1987), Woodruff and Parsons (1999 and 2000). 19 Again, as with Extensionality, minimal assumptions are made regarding the semantics of “↔”. Parsons assumes an analysis in terms of the L3 biconditional (true if and only each both sides have equal value) and also restricts F to predicates that denote properties. Nothing here of note hangs on my more liberal statement of the Axiom except that given this more liberal principle of set existence we cannot assume set membership to be a property relevant to the individuation of objects via Leibniz’s Law. This will re-emerge below and will be discussed when it arises.

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the set of ships that left, L, has exactly one member, a, i.e. L = {a}. The set of ships that docked, D, has exactly two members, b and c, i.e. D = {b, c}. Since L has exactly one member and D has exactly two they are (determinately) distinct sets, i.e. L ≠ D. Thus, by Extensionality, they must (determinately) differ on some member, i.e. there must be some member z such that (determinately) z ∈ L ↔ z ∉ D.20 So either a is (determinately) not in D or one of b and c is (determinately) not in L. Furthermore, since the predicates used to characterize each of the sets in question denote properties, (determinate) difference in respect of membership of either D or L amounts to (determinate) difference in respect of a property. Hence, by the law of de
nite difference (DD), either (determinately) a ≠ b or (determinately) a ≠ c (or both). Since this contradicts there being no fact of the matter as to whether a is identical to either of b or c, it follows that (i) – if exactly one ship left port and exactly two ships docked, then a case of (de re) split indeterminate identity does not obtain. As it stands, of course, this does not rule out the existence of cases of split indeterminate identity. That stronger claim would require af
rmation of (ii) – the antecedent of the now accepted, foregoing conditional, (i). Pinillos does af
rm this antecedent, but we have seen that it would be wrong to do so – it is not true that exactly one ship left port and exactly two docked. Consequently, despite the truth of (i), One4All and Kim4Ever, though distinct ships, are neither of them (determinately) distinct from Samantha’s Pride nor (determinately) identical to it. There is simply no fact of the matter in either case, and we are indeed confronted with a case of split indeterminate identity. Of course, we can af
rm (ii′): exactly one ship determinately left port and exactly two determinately docked. Now, however, we are unable to modify the foregoing proof of (i) to establish (i′): if exactly one ship determinately left port and exactly two ships determinately docked, then a case of (de re) split indeterminate identity does not obtain. So cases of split indeterminate identity can be shown not to obtain by means of the foregoing argument only by equivocating. The failure of the modi
ed set-theoretic proof of (i′) follows from the fact that membership of the modi
ed sets formed using Comprehension does not amount to (determinate) difference in respect of some property. Thus the application of DD invoked in the foregoing proof of (i) breaks down in the context of the modi
ed proof of (i′). While the predicates used to characterize set membership in the proof of (i) (“is a ship that left port” and “is a ship that docked”) denote properties, the modi
ed predicates used to characterize set membership in the proof of (i′) (“is a ship that determinately left port” and “is a ship that determinately docked”) do not denote properties for reasons already canvassed in Chapter 5, §2. Thus, as with the Evans Proof, DD is inapplicable. To be sure, by Comprehension there is a set L′ with exactly one member (Samantha’s Pride) and a set D′ with exactly two members (One4All and Kim4Ever) so that L′ ≠ D′. Hence, by Extensionality, these sets (determinately) differ on some member. For example, Samantha’s Pride is 20 Hawley (2002: 133) rejects Extensionality. The Pinillos Argument will then be unpersuasive, but there is no relief from the overall conclusion that de re vague identity is incoherent since her rejection depends upon an antecedent acceptance of the soundness of the Evans Proof. Out of the frying pan into the
re.

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(determinately) not in D′ since it is false to say that it determinately docked (there is simply no fact of the matter as to whether it docked or not). Similarly, neither One4All nor Kim4Ever are in L′ since it is false to say of either that they determinately left port. But unlike the argument for (i), (determinate) difference in respect of membership of either D′ or L′ does not amount to (determinate) difference in respect of a property. Hence the law of de
nite difference does not apply, and we cannot infer that either Samantha’s Pride (determinately) differs from One4All or that Samantha’s Pride (determinately) differs from Kim4Ever.21 Pinillos offers a further defence of the view that cases of SII are incoherent, a ‘philosophical argument’ in defence of (i) which when coupled with the supposed truth of its antecedent is taken to establish the incoherence of SII. As with the settheoretic argument just considered, this philosophical argument can succeed only by equivocation. Claim (i) can again be defended but its antecedent, again, cannot be af
rmed. In defence of (i), assume that exactly one ship, a, say, did indeed leave port and exactly two ships, b and c, say, docked. Suppose further that a case of SII obtains in the manner already canvassed earlier, namely that b and c, are such that there is no fact of the matter as to whether a = b, nor as to whether a = c. Since a left port and it is vague whether a = b, it would then follow by PII that it is not false that b left port. Similarly for c. Nor could it be true that b left port since were it true, then, given symmetry considerations arising from the described situation, c would also have to be counted a ship that left port. But now, since b and c are distinct, we would be forced to admit that more than one ship left port, contrary to our original assumption. Consequently, we would be forced to admit that there is no fact of the matter as to whether or not b left port (and similarly for c). By PCI, then, there would be no determinate answer as to exactly how many things left port, contrary to our assumption that exactly one ship left port. Hence if exactly one ship left and exactly two docked, then a case of SII does not obtain. Accepting this proof of (i), however, is consistent with the case in question being a case of SII since, as we have seen, it is not true that exactly one ship left and exactly two docked; i.e. (ii) is not true. Again, as with the set-theoretic argument already considered, it is true that (ii′): exactly one ship determinately left and exactly two determinately docked, but this is not suf
cient to rule out the obtaining of a case of SII since it is simply not true that exactly one ship determinately left and exactly two determinately docked only if a case of SII does not obtain, i.e. (i′) is not true. The foregoing ‘philosophical’ defence of (i) cannot be modi
ed in defence of (i′) since PII and PCI are no longer applicable. Their applicability would require treating “is a ship that determinately left port” and “is a ship that determinately docked” as predicates that denote properties and, again, they do not for reasons canvassed in Chapter 5, §2 when considering the Evans Proof. 21 We could, of course, restrict the Axiom of Comprehension so that the characterizing feature F must be a property. Then objects (determinately) differing with regard to setmembership would (determinately) differ on some property and thus, by DD, be (determinately) different. The attempt to undermine cases of split indeterminate identity would now fail since there would be no set of things that determinately left port nor a set of things that determinately docked.

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Thus, to return the argument from (i), (ii), (iii) and (iv) to the incoherence of de re vague identity, we see that only by equivocation can the argument of Pinillos (2003) succeed in establishing its conclusion. De re vague identity is, for all the ink that has been spilt on attacking it, coherent. 6.6

Summary

We have before us now an account of vagueness according to which vague predicates might precisely designate (i.e. directly refer to) vague properties, and vague singular terms and de
nite descriptions might precisely designate vague objects whose identity conditions might sometimes be indeterminate. Thus vagueness is not purely representational, and any lingering thought that anomalies that beset the weakly paracomplete supervaluationist approach to vagueness might be offset by appeal to the purely representational nature of vagueness is undermined. For all that has been said so far, the truth-functional logic coupled with this ontological approach holds its own against supervaluationism in respect of a pragmatic choice of logical theory. But to remain the preferred theory, the strongly paracomplete logic needs much closer scrutiny and must be defended against some serious charges of inadequacy. It is to this we now turn.

Chapter 7

The Logic of Vagueness As suggested in the last chapter, the paracomplete approach to be pursued in modelling vagueness will, initially, be that strongly paracomplete, truth-functional logic L3. Vagueness generates truth-value gaps and it is sometimes indeterminate whether a sentence is true or false. Classical logic is extended to cover such cases and a logic for vagueness results. With a number of well-established objections facing the proposed truth-functional approach, we will then turn to consider the primary case against truth-functionality – namely, that it results in sentences receiving truth-values (as a function of the values of their parts) that are counterintuitve. When the commonly reported intuitions that underlie this objection to truth-functionality are subjected to scrutiny, they are far from compelling in my view. A truth-functional-gap approach does not face the high costs often claimed for it. Higher-order vagueness will add some additional complexity and will be considered in §4. With a subsequent account of higher-order vagueness we are ¿nally able to see our way out of the puzzlement generated by vagueness. For now, let us turn to the basic system of ¿rst-order vagueness. 7.1

First-order Vagueness

In what immediately follows the basic logic is described and the sorites paradox disarmed. We shall assume that the language to be modelled is free of semantic anomalies like ambiguity and reference failure. We shall suppose that the language contains a full complement of referring names, predicates and variables, to which are added the logical connectives, quanti¿ers, a determinacy operator, and identity. 7.1.1

The basic system

As is frequently acknowledged with three-valued systems, we can formally represent the logic being advocated as ‘three-valued’ either in the sense of claiming that sentences that are neither true nor false take some new third value or in the sense that, as with the approach advocated here and typical in presentations of supervaluationism (e.g. in Keefe 2000), such sentences are said merely to exhibit a truth-value gap. On this latter view there is, strictly speaking, no third value and the logic is not a ‘three-valued’ logic at all but one with only two truth-values – the classical values “true” and “false” – in which vague sentences are characterized by their taking no value at all.1 We shall suppose just such a truth-value gap approach. 1

An analogous point arises in the case of paraconsistent logics. We can either view sentences that are evaluated as true and false as taking a non-classical value (e.g. “both”),

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Despite this rejection of a third value, properly speaking, the logic of such a truthvalue-gap approach is conveniently modelled by invoking three values, now to be understood as ‘truth-values’ only in an instrumental sense, useful for the calculation of logical consequence relations, etc. Atomic sentences will be true if the object referred to (determinately) satis¿es the predicate, false if it (determinately) does not, and indeterminate if it is a borderline case. Where ν is a valuation function mapping sentences onto the truth-set {1, @, 0}, the values of negations, conjunctions and disjunctions are then determined as follows:

ν (~A) = 1 – ν (A). ν (A & B) = min{ν (A),ν (B)}. ν (A V B) = max{ν (A),ν (B)}. Quanti¿ed sentences are evaluated in the obvious way. Where D is the domain of quanti¿cation:

ν (∀xFx) = min{ν (Fx): x ∈ D}. ν (∃xFx) = max {ν (Fx): x ∈ D}. To allow for the expression of indeterminacy in the object-language, the language is extended to include a determinacy operator, “determinately”, a one-place sentence operator represented as “D”. Its semantics is also naturally forthcoming:

ν (DA) = 1 if ν (A) = 1; 0 otherwise. Thus, to express the fact that it is indeterminate whether A, we shall say that it is neither determinately the case that A nor determinately not the case, i.e. IA = ~DA & ~D~A. Hence:

ν (IA) = 1 if and only if ν (A) = @. Validity is de¿ned as truth-preservation, i.e.: Σ = Γ =df for all ν, if ν (B) = 1 for all B ∈ Σ then ν (A) = 1 for all A ∈ Γ. More familiar is the special case where the consequence-relation is de¿ned for single conclusions, i.e.: Σ = A if and only if for all ν, if ν (B) = 1 for all B ∈ Σ then ν (A) = 1. or as taking both of the two classical values “true” and “false”, thus resulting in truth-value gluts. On the glut approach truth is now a relation which a sentence may bear to more than one truth-value, as opposed to a function with sentences taking unique values.

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It is easy to show then that: (Det) If = A then = DA. The matter of an adequate account of the conditional “if … then” is more contentious. Tye (1990) proposes retaining the classical connective “⊃”, extended in the obvious way to preserve the classical equivalence between “A ⊃ B” and “~A V B”, as is the case for the strong Kleene system K3. A problem frequently cited as a reason for doubting the adequacy of such a logic, however, centres on the ensuing inability to validate the identity thesis: ≠

A ⊃ A.

To avoid this untoward feature the Łukasiewicz conditional, “→”, is to be preferred. De¿ned:

ν (A → B) = 1 if ν (A) ” ν (B), and = 1– [ν (A) – ν (B)] otherwise it follows that: (Id)

=

A → A.

The two ‘conditionals’ are contrasted below: A

B

A⊃ B

A→B

1 @ 0 1 @ 0 1 @ 0

1 1 1 @ @ @ 0 0 0

1 1 1 @ @ 1 0 @ 1

1 1 1 @ 1 1 0 @ 1

(Notice that they differ only on one case, that which produces the counterexample to the identity thesis (Id) for “⊃ ”, where antecedent and consequent are both indeterminate.) With a conditional in place we can de¿ne a biconditional in the usual way, i.e. A ↔ B =df (A → B) & (B → A), and subsequently de¿ne logical equivalence in the obvious way: A is logically equivalent to B if and only if = A ↔ B. Theoremhood in the logic of vagueness differs markedly from classical theoremhood. Note ¿rstly that, like supervaluationism:

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182 ≠

A, ~A.

Yet, unlike supervaluationism, given subjunction we are committed to: A V ~A= A, ~A. Thus it follows that the law of excluded middle fails: ≠

A V ~A.

Moreover, the law of non-contradiction also fails: ≠

~(A & ~A).

It should come as no surprise that these laws fail. The case against LEM has been considered in earlier chapters and, with the logic validating the De Morgan principles, LNC will similarly fail. In fact, as is well known from K3, there are no theorems in the conditional-free fragment of the language. With the addition of the conditional, however, we get: =

A→A

=

~ ~A ↔ A

=

(A & B) → A

=

A → (A V B)

=

A → (B → A)

=

~A → (A → B)

=

(A → B) → [(B → C) → (A → C)]

=

(~A → ~B) → (B → A)

and so too: =

(A ↔ B) V (A ↔ C) V (A ↔ D) V (B ↔ C) V (B ↔ D) V (C ↔ D).2

The inferences validated by the logic substantially overlap the class of classically valid inferences. For example, modus ponens is valid: A, A → B= B. And so, given the foregoing theorems, the rules for double negation (from ~ ~A infer A and vice versa), simpli¿cation (from A & B infer either or both of A or B), and addition (from A infer A V B) are valid. 2 This last theorem will subsequently be falsi¿ed when we extend to account for higherorder vagueness.

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Adjunction, disjunctive syllogism and ex falso quodlibet are also valid, i.e.: A, B= A & B A, ~A V B= B A & ~A= B However, the failure of LEM does undermine some classically valid inferences. For example: B≠ A V ~A. Nonetheless, in the broadly classical setting adopted, just as an arbitrary sentence B follows from what is impossible (i.e. ex falso quodlibet), it remains the case that what is necessary validly follows from any arbitrary sentence. So, for example: B= A→ A.3 Unsurprisingly, there are other signi¿cant departures from classical inference. In particular, reminiscent of supervaluationism, a number of rules that permit the transformation of one valid argument into another – sometimes referred to as ‘structural rules’ – fail in the extended language including “D”. Consider conditional proof. According to the general statement of this rule, if B follows from a set of assumptions Γ along with A, then “if A then B” follows from assumptions Γ. Taking the conditional to be “→” and letting Γ = ∅, we get the particular instance: If A= B then = A → B. This, in conjunction with modus ponens, would yield both halves of the deduction theorem: A = B if and only if = A → B. However, as with supervaluationism, given that: (*)

A = DA yet clearly: ≠ A → DA,

a familiar counterexample to conditional proof and the deduction theorem presents itself. As in supervaluationism we can offer a modi¿ed version of conditional proof: if B follows from a set of assumptions Γ along with A, then “if DA then B” follows from assumptions Γ, i.e.: (CP′) if Γ, A = B then Γ = DA → B. Notice that despite A → DA not being a theorem, its converse, (T), of course is: (T) = DA → A. 3

This paradoxical result has been frequently remarked upon in the long history of logic and is on a par with other problematic irrelevant inferences like the positive and negative paradoxes of material implication, also valid in the system being proposed. While I think that they present a serious challenge to the adequacy of the system proposed, it would greatly complicate matters to deal with them here, and things are already complicated enough.

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The foregoing counterexample to conditional proof, (*), also points to a counterexample to contraposition: if B follows from a set of assumptions Γ along with A then A follows from Γ along with ~B. For, notice that: (**) A= DA yet:

~DA ≠ ∼A.

So contraposition fails. Again, as in supervaluationism, here too there is a closely analogous principle: if B follows from a set of assumptions Γ along with A, then ~DA follows from Γ along with ~B, i.e.: (Contra′) if Γ, A = B then Γ, ~B = ~DA. Finally, reductio ad absurdum also fails. By the structural rule of reductio, if B & ~B follows from a set of assumptions Γ along with A, then ∼A follows from assumptions Γ. However: (***) B & ~B = B & ~B yet: ≠ ~(B & ~B).4 What does hold, of course, is: (Reductio′) if Γ, A = B & ~B then Γ = ~DA. What remains to be given is an account of identity, “=”. We have seen in Chapters 5 and 6 that identity sentences are taken to admit of truth-value gaps. With this in mind, we shall take identity to be subject to the usual Leibnizian constraint: (LL) = x = y ↔ (Fx ↔ Fy), for objects x and y, and property F. Given the logic described, the law of de¿nite difference then follows: (DD) = ∃F(Fx & ~Fy) → ~(x = y), for objects x and y, and property F. It was this principle that ¿gured in the Evans Proof. We accept that objects are (determinately) identical just in case they (determinately) agree on all properties and (determinately) differ where they (determinately) differ on some property, and admit cases of indeterminate identity otherwise, i.e. where there is no fact of the matter whether there is agreement on all properties. Moreover, the law of self-identity is also valid: (SId) = ∀x(x = x). It is now easy to see that, as claimed in Chapter 1, §4, precision is inherited. Moreover, we can also see that vagueness is not inherited; regardless of the vagueness or otherwise of A, A → A (for example) is always true. 4 This nice counterexample is from Parsons (2000: 25) and differs from that given for the failure of reductio in supervaluationist logic. See Williamson (1994: 152).

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Some of the foregoing is, as already noted, familiar from the rival semantics for a paracomplete approach to vagueness – supervaluationism. Supervaluationists encounter similar non-classical features in their logical response to vagueness, ‘anomalies’ from the perspective of classical logic but necessary weakenings of a reasonable account of inference able to accommodate the truth-value gaps generated by vagueness. However, a commitment to the supervaluationist’s weakly paracomplete approach also brings in its train non-truth-functionality with associated counterintuitive results frequently remarked upon, particularly with regard to the treatment of disjunction and existential quanti¿cation. The usual rules for proof by cases and existential instantiation then also fail, in addition to those already discussed above in relation to the proposed truth-functional logic. As we have already seen in Chapter 4, arguments were put forward to justify their non-truth-functional approach, arguments from ‘penumbral connections’. But these positive arguments were found wanting. The case for non-truth-functionality, though, is also frequently bolstered by a range of negative arguments that seek to show that truth-functionality is objectionable in its own right. These arguments are surprisingly weak, but before turning to consider them let us conclude this section by seeing how the proposed logic deals with the sorites paradox. 7.1.2

The sorites paradox

The strongly paracomplete logic L3 evaluates the standard sorites paradox as unsound due to the non-truth of one of its premises. Modus ponens is valid in L3 yet the paradox has a false conclusion; thus not all its premises can be true. Obviously no premise is false; the categorical premise “Fa1” is accepted as true and no conditional premise “Fan → Fan+1” can be false since that would require a transition from being determinately F to determinately not-F between adjacent members an and an+1 of the series 〈a1, …, ak 〉 with respect to which “F ” is soritical – i.e. it would require a sharp boundary for the vague predicate “F ”. As with the supervaluationist response SpV, the absurdities that follow from either accepting all premises as true or rejecting some premise as false are avoided by the recognition of a third possibility, the possibility of indeterminate truth-value. As we progress along the chain of sorites reasoning, we initially encounter a true categorical premise and true conditional premises, the truth of the latter being guaranteed by the truth of their consequents. Eventually borderline Fs are encountered and, as a first-order approximation, we are confronted with a conditional whose antecedent is true and whose consequent is indeterminate, i.e. an indeterminate conditional. Progressing into the borderline Fs proper, we encounter conditional premises whose antecedent and consequent are indeterminate and which are themselves thus true. In the transition, again, from borderline Fs to determinate non-Fs we are faced with an indeterminate conditional before moving once more to conditionals that are true by virtue of the falsity of their antecedent. First-order boundaries to the region of borderline cases generate non-true conditionals. Of course, no such boundaries are determinately identi¿able since their existence is ruled out by higher-order vagueness; in the march down a sorites sequence there is no sharp point at which we move from determinate cases to borderline cases. The transition is vague and there will be cases for which there is simply no fact of the

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matter whether it is a determinate case or a borderline case, and so recognition of second-order vagueness requires instead that there be conditionals for which there is no fact of the matter whether they are true or indeterminate. As with the diagnosis afforded by the ¿rst-order approximation, such conditionals will not be true. As we shall see when we discuss higher-order vagueness in more detail, second-order vagueness will give way to third-order, and so on. What will remain under the increasing complexity generated by ever higher orders of vagueness, however, is the non-truth of some conditional premise and therewith a solution to the conditional sorites. The mathematical induction sorites is dealt with similarly. Recall that the supervaluationist analysis SpV treats the standard sorites and the closely related form where all the conditional premises are conjoined in distinct ways. As we have seen, the former is said to be unsound by virtue of there being some non-true (conditional) premise. Conjoining the conditional premises, however, results in a paradox whose unsoundness derives from the falsity of this conjunctive premise. Analogously, the mathematical induction sorites is similarly said to be unsound due to the falsity of some premise, the universally quanti¿ed premise. (It is this particular response that has generated considerable doubts as to the adequacy of SpV as a logic of vagueness.) L3, on the other hand, requires that the universally quanti¿ed premise take a value only as low as its lowest instance, and since this is less than true so too will be the quanti¿ed premise, though it will not be false. The line-drawing sorites will, of course, have true premises but a non-true conclusion. No adjacent members an and an+1 of the series 〈a1, …, ak〉 will ever be such that Fan and not-Fan+1, and so the conclusion asserting just such a claim will not be true. Nor will it, of course, be false since its negation – the universally quanti¿ed premise of the mathematical induction form – is not true. Truth-functionality enables us to treat the conditional and mathematical induction forms of the sorites analogously, as one might expect, both being unsound due to the non-truth of some premise. The line-drawing form, though having true premises and being classically valid, is in fact invalid and issues in a non-true conclusion; its implicit reliance on the law of excluded middle or least number principle proving its downfall. 7.2

Objections to Truth-functionality

A range of objections has been levelled at the account of vagueness proposed. In particular, Fine (1975: 269f), Williamson (1994: 135f), Edgington (1997: 304f) and Keefe (2000: 96f) present arguments for the claim that truth-functionality presents insuperable problems for a paracomplete, truth-value-gap approach to vagueness. Problems are said to attend a truth-functional conjunction, disjunction and conditional, and this charge is levelled from a position according to which, in the absence of vagueness, classical logic with its attendant truth-functionality ought to be accepted as theoretically adequate. The charge is therefore that vagueness, in particular, necessitates a departure from an otherwise acceptably truth-functional logic, classical logic, a departure that must abandon truth-functionality.

The Logic of Vagueness

7.2.1

187

Conjunction and disjunction

Turning ¿rstly to conjunction, why think that truth-functionality is a problem? Taking our cue from Keefe, suppose Tim and Tek are both borderline cases of “is tall”, with Tek a little taller than Tim. Assuming negation to be de¿ned as described earlier, “Tek is tall” and “Tek is not tall” are both indeterminate, taking value @. Thus consider (a) “Tim is tall and Tek is not” and (b) “Tim and Tek are tall”. By truthfunctionality, (a) must be equivalent in truth-value to (b). But, the complaint goes, (a) is surely false; given that Tek is taller than Tim, Tim cannot be tall and Tek not tall. Yet given their borderline status, neither Tim nor Tek is (determinately) not tall so (b) must be non-false; in fact, it should count as indeterminate, taking value @, as indeed it will on the semantics under consideration. The need to distinguish the truth-values of (a) and (b) is clear, yet their component conjuncts are equivalent in truth-value, therefore, it is argued, conjunction cannot be truth-functional. On closer inspection, however, the need to distinguish between the value of (a) and (b) is far from clear. What is clear is that (a) cannot be true; as Keefe says, ‘if Tim is shorter than Tek, then it cannot be that Tim is tall and Tek is not’ (2000: 96). To assert that it can be that Tim is tall and Tek is not is to present as true a claim which is not and, under the circumstances described, cannot be. But this is entirely consistent with the view under consideration, namely that it is not false but indeterminate. Fine too presents examples to drive home the need for a non-truth-functional conjunction to handle vague expressions. Suppose we are confronted with a colourpatch on the border of pink and red, its colour being a borderline case of both “is pink” and “is red”. Consider (a′) “The patch is pink and the patch is red” and (b′) “The patch is pink and the patch is pink”. The conjunction (b′), since equivalent to the one conjunct alone, ought count as indeterminate yet, Fine claims, the conjunction (a′) ‘is false since the predicates “is pink” and “is red” are contraries’ (1975: 269). But what, we might ask, is meant by the notion of contrariness if not simply that predicates are contraries if and only if nothing can satisfy both? On such a view, claims A and B are contraries if and only if they cannot both be true together; thus, given truth-functionality, their conjunction cannot be true. If this is all one means by calling the predicates ‘contraries’, then the theory under scrutiny is not challenged for it can well agree that the conjuncts of (a′) cannot be true together and that their conjunction, (a′) itself, can never be true. The theory can admit that if the one conjunct is true, then the other conjunct is not. More strongly, the theory under consideration can admit that if one is true then the other is not simply not true but, more particularly, is false. For the contrariness of the predicates to entail the falsity of the conjunction (a′), the relation of contrariness must be de¿ned in such a way that the conjunction of contraries is always false. Of course, given bivalence (e.g. in classical logic), the stronger, latter de¿nition follows from the former. But where bivalence fails, as it does here, to assert that the predicates in question are ‘contraries’ in this stronger sense simply begs the question against the truthfunctional account under attack. In a similar vein Williamson imagines someone drifting off to sleep. At a certain point it would seem that the theory under scrutiny will admit to “He is awake” and

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“He is asleep” both being indeterminate in value. By truth-functionality, their conjunction will be similarly indeterminate. But how can that be, Williamson asks? ‘Waking and sleeping by de¿nition exclude each other. “He is awake and he is asleep” has no chance at all of being true’ (1994: 136). And he is right to point out that the conjunction cannot be true. Nor is it. It is indeterminate just as is the conjunction of either simple claim with itself. Truth-functionality is not threatened unless the falsehood of indeterminate conjuncts can sometimes be established. It has not been established. Williamson considers an example where intuitions might be thought to speak most strongly against a truth-functional approach, an example involving contradictories. Consider “He is awake” and its negation. When conjoined with itself it is indeterminate for the commonly cited reason that such a conjunction is equivalent in content to the single conjunct itself, which is indeterminate. Yet when conjoined with its negation it is surely false since it is a contradiction. ‘How can an explicit contradiction be true to any degree other than 0 [i.e. be anything other than false]?’ (ibid.). Answer: By being a conjunction of contradictories neither of which is false. Of course, such a possibility is classically ruled out since, given bivalence, it would amount to each conjunct being true, thus contravening the constraint on negation that it yield a falsehood when appended to a truth. But in the current, nonbivalent context, neither conjunct need be false by virtue of each counting as indeterminate. Admittedly, there appears to be some force behind the insistence that a contradiction must be false (and supervaluationists acquiesce), but what underlies this? To be sure, it would be wrong in the circumstances envisaged to assert the contradiction. We can agree that the situation does not present us with a true contradiction, a dialethia.5 But this is not enough to establish the falsehood of the contradiction in question. Can its falsehood be taken as obvious? Surely not. In the ¿rst instance, we should be careful about reliance on any strong intuition we may have concerning the ‘obvious’ falsehood of contradictions in non-vague circumstances (an intuition not necessarily shared by all; witness dialethic approaches to the Liar, for example). Even granting the accuracy of such an intuition where the conjuncts are either true or false, it remains to be shown that the intuition extends to circumstances involving the conjunction of contradictory claims that are each indeterminate. It is not enough, remember, to point to the intuition that we arguably do have when confronted with such a conjunction, namely that such a claim simply cannot be true. As noted, this fails to establish their falsehood. Secondly, appeal to a direct intuition of falsehood, even if identi¿able, is undermined by the existence of apparently conÀicting intuitions. Given the accepted De Morgan equivalence between claims of the form ~(A & ~A) and A V ∼A, our intuition of the falsehood of the contradiction A & ∼A – i.e. the truth of its negation, ~(A & ~A) – ought to be only as strong as our intuition of the truth of the corresponding excluded, middle-claim A V ~A, yet the truth of excluded-middle claims are frequently contested in the context of vagueness speci¿cally because 5

Recall at the end of Chapter 4 we put aside paraconsistent responses to vagueness, as a dialethic response would be.

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many have a strong intuition that such claims should not be true for vague A.6 Unless one is prepared to reject the standard De Morgan equivalence, any intuitions in favour of the falsehood of contradictions are offset by contrary intuitions of the nontruth of excluded middle claims. So what of such intuitions in respect of the non-truth of instances of the law of excluded middle? They are certainly present, according to many. But despite them, many (supervaluationists among them) will nonetheless reject them on the basis of arguments to the contrary. Such arguments, in seeking to undermine an approach to excluded middle as sometimes non-true, might also then serve to undermine the corresponding view of contradictions as sometimes non-false, thus reinforcing through reason the view of contradictions as always false in the absence of reliable intuitions to this effect. This reasoning, itself intending to support a non-truthfunctional account of disjunction, would thus reinforce Williamson’s argument for the non-truth-functionality of the dual notion of conjunction even given intuitions that push in the opposite direction. However, we have already looked at such arguments in Chapter 4 and found them wanting. No adequate reason has been given to undermine the thought that excluded middle-claims are sometimes non-true. Therewith, by the De Morgan equivalence, we are presented with an intuition that speaks against the necessary falsehood of contradictions in the context of vagueness. When all is said and done, then, to the extent that we do have a strong intuition that contradictory indeterminate conjuncts nonetheless conjoin to produce a falsehood, we face a stand-off. But even this supposedly strong intuition is, I suspect, one illicitly derived from what we are inclined to think in cases where vagueness does not make itself felt. From what has already been said, we can see that arguments for the non-truthfunctionality of disjunction are similarly unconvincing. Analogous to arguments concerning conjunction, cases are presented of indeterminate disjuncts (e.g. “He is awake” and “He is asleep” considered at that point between waking and sleeping when he is a borderline case of each) whose disjunction is nonetheless true, despite examples of disjunctions with indeterminate disjuncts that are themselves indeterminate (e.g. “He is awake” disjoined with itself). As already noted, the appeal to intuition in favour of the truth of disjunctions with indeterminate disjuncts is weak, and the arguments offered in Chapter 4 in defence of the supervaluationist’s appeal to penumbral connections and non-truth-functionality more generally were found wanting. 7.2.2

The conditional

So the case against a truth-functional account of conjunction and disjunction comes down to questionable intuitions and arguments based on a set of central examples that remain unconvincing. Perhaps the toughest case to defend against, however, is a truth-functional account of the conditional in the presence of vagueness. Even Field, going so far as to accept the truth-functional account of negation, conjunction 6 Even supervaluationists, committed to the truth of such claims, often feel uncomfortable with Burgess and Humberstone (1987) going so far as to modify supervaluationism so as to abandon the law of excluded middle.

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and disjunction proposed at the beginning of the chapter, abandons truthfunctionality in respect of the conditional when considering vagueness. ‘[N]o 3valued truth function is remotely adequate’ (2003: 473). The argument for this is articulated by Williamson and Keefe. But before considering this particular aspect of conditionality, let us ¿rstly gather together those features we might consider to be generally acceptable. We immediately run into a morass, given the notoriously contested nature of the conditional. Unlike negation, conjunction and disjunction, there is very little agreement about what to count as an adequate account of conditionality even before the phenomenon of vagueness is brought into focus.7 There will be misgivings from some quarter or other concerning any account of the conditional taken as a startingpoint for further theorizing about complicating aspects (if any) arising from vagueness, and the sorites paradox in particular. Without wishing to generally endorse a classical account, i.e. an account appealing to the classical material conditional, I shall take just such an account as a general starting-point. That is, I shall suppose that the account of conditionality sought should agree with the material conditional where vagueness does not arise. We shall suppose therefore that the connective is functional on the two classical values “true” and “false” in the manner of material implication. Of the four possible combinations of truth and falsity of antecedent and consequent, the conditional will count as false if and only if the antecedent is true and the consequent false. Thus we shall proceed on terms acceptable to the classical logician. I do this for two reasons. Firstly, those arguing that vagueness, in particular, necessitates a nontruth-functional conditional generally do so from a classical perspective. Epistemicists like Williamson and supervaluationists like Keefe, in reproducing arguments for the necessary non-truth-functionality of a three-valued conditional, grant the adequacy of truth-functional, material implication when we restrict ourselves to the two values “true” and “false”. (Williamson ultimately argues that no extension to three values and away from truth-functionality is required; Keefe argues that extension to three values is required and therefore that it will have to abandon truth-functionality.) Thus, in an attempt to engage them on their own terms, the assumption of the adequacy of two-valued classical logic to precise language is granted for the sake of argument. Secondly, granting this classical assumption will, given most readers’ familiarity with the classical account, allow us to focus more clearly on the points at issue arising from the phenomenon-vagueness. The simplicity of formal aspects of the account (e.g. the simple truth-tabular account of material implication) will also keep complications at this level to a minimum. Of course, complications arising from the application of this account (e.g. the paradoxes of material implication) will have to be carefully heeded lest the non-classical logic being developed here be objected to for reasons having their basis in the classical starting-point. So, we shall begin by supposing the conditional to be true if either the antecedent is false or the consequent true, and that it is false if the antecedent is true and the consequent false. What should we say in the remaining ¿ve possible combinations 7 Of course, there may be disagreements centring on aspects of negation, conjunction, and disjunction as well. Nonetheless, there is considerable agreement.

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of the three values, 1, 0 and @? The classicist can hardly object to our claiming that a false antecedent truly implies anything; this is just the classically acceptable negative paradox of material implication. Thus the non-classical possibility of a conditional with false antecedent and indeterminate consequent should result in a true conditional. Similarly, given the positive paradox of material implication, a three-valued extension of classical logic should count a conditional with true consequent and indeterminate antecedent as true. So far, so good. The following, partial, table displays the results so far: A

B

If A then B

1 @ 0 1 @ 0 1 @ 0

1 1 1 @ @ @ 0 0 0

1 1 1 1 0 1

Two of the three remaining cases are typically settled without dispute – lines four and eight of the table above. Conditionals with true antecedents and indeterminate consequents, and those with indeterminate antecedents and false consequents, are taken to be indeterminate in truth-value. Supervaluationists, theorists like Tye (1990) advocating Kleene’s strong K3, and those like Parsons (2000) advocating Łukasiewicz’s L3, all agree here. The remaining case is the one focused on as grounds for non-truth-functionality. What are we to say in cases where conditionals have indeterminate antecedents and indeterminate consequents? Consider, again, Tim and Tek, both borderline cases of “is tall”, with Tek slightly taller than Tim. Given truth-functionality, the conditionals (c) “if Tim is tall then Tek is tall” and (d) “if Tim is tall then Tek is not tall” must take the same value. In the circumstances described, it seems that we should accept (c) as true. Yet ‘(d) is surely false’ (Keefe 2000: 97). The conditionals (e) “if Tim is tall then Tim is tall” and (f) “if Tim is tall then Tim is not tall” must also take the same value given truthfunctionality, namely the value taken by (c) and (d). Given that (e) is an instance of the identity principle “if A then A”, it ought surely to count as true. But then so too should (f), yet intuitively, it is argued, this cannot be. (f), like (d), must be false. The conditional, it is claimed, simply cannot be truth-functional. However, as with the debate concerning conjunction and disjunction, the argument here is too quick and ultimately unsuccessful. We should indeed accept a truth-functional analysis of the conditional, that given by L3, and accept that conditionals with both indeterminate antecedents and consequents are true. Consider, ¿rstly, the status of (c) and (e). Claims for their truth are well founded. Contra Tye and the K3 analysis of the conditional, we should abandon the analysis of the conditional as “⊃” and the associated general equivalence between “if A then

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B” and “~A V B” and accept, in particular, that despite the failure of the law of excluded middle the classical and K3 equivalent “if A then A” is true. And it seems uncontroversial that anyone taller than a tall person is tall, and thus that, like (e), (c) is true. In a similar vein we should agree with Field’s observation that an object’s being blue is inconsistent with its being green and accept that ‘as a matter of conceptual necessity anything that is blue is not green: ■∀x (if Blue(x) then ~Green(x))’ (2003: 473). An analysis of the conditional as “⊃” will not support such a view. The predicate “Blue(x) ⊃ ~Green(x)” applied to a blue/green borderline case will yield a non-truth (in fact, the claim will take the value @). Accepting the L3 analysis using “→” recaptures this truth. So too with the claim that anything in the blue–green region is such that if it is not blue then it is green. Again, “⊃” fails as an adequate analysis where “→” succeeds. Conditionals (c) and (e) are true and the truth-functional analysis offered by L3 accounts for this. Given the truth of (c) and (e), then, truth-functionality is threatened to the extent that conditionals like (d) and (f) are false, but is their falsity as obvious as claimed? We can admit that in circumstances where Tim is (determinately) tall he cannot be not tall, and nor could anyone taller (Tek, say) since they too must be tall. In these circumstances (f) and (d) respectively are indeed false since being tall is inconsistent with being not tall. So too, given the inconsistency between being blue and being green, blue objects cannot be green and so it is false to say of them that if they are blue then they are green; they are not green. But we should not confuse any inconsistency between being F and G (e.g. tall and not tall, or blue and green) with the falsity of “if Fa then Ga” per se. In particular, we should not confuse the inconsistency between being tall and being not tall with the falsity of (f) and (d) per se. Given our assumption that the theory being sought should reduce to the classical theory on classical values, all parties can agree that in circumstances where Tim is (determinately) not tall the conditionals (f) and (d) are in fact true. So their supposed falsity in the circumstances considered above where Tim and Tek count as a borderline case of tallness with Tek taller than Tim – i.e. the supposedly problematic data for advocates of a truth-functional approach to conditionality – cannot follow from their being false in general. The supposed falsity of (f) and (d), then, must derive from the speci¿c circumstances described; that is, from facts surrounding cases where antecedent and consequent are indeterminate. Moreover, the mere intuition or appearance of falsity will be insuf¿cient since the Gricean story that is invoked to explain away the intuition in circumstances where the antecedent is false can be marshalled here.8 In the latter circumstances it is argued that since the conditional’s truth derives from the more informative fact of the falsity of its antecedent, the assertibility of the less informative conditional is undermined by the requirement that one be as informative as possible in cooperative conversational situations. Similarly, we can argue that the intuition of the falsity of 8 That the Gricean defence of the material conditional might fail is beside the point. We are here trying to establish the relative merits of a three-valued truth-functional approach to vagueness as compared to its more classical counterparts. We are trying to avoid the complications incurred by engaging with the more general debate on conditionals. The claim at issue then is whether, given the supposition that the truth-functional material conditional is adequate, vagueness gives reason for abandoning truth-functionality. The view being defended here is that it does not.

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(f) and (d) in circumstances where the antecedent and consequent are indeterminate can be accounted for by pointing to the fact that, even supposing the conditionals true by virtue of the indeterminacy of their parts, they are nonetheless not assertible since their truth then derives from the more informative fact of the indeterminacy of their parts. Thus the intuition of the non-truth of the conditionals in circumstances where their parts are indeterminate is no more a reliable guide to their truth-value than the intuition of their non-truth in circumstances where their antecedents are false. Where a Gricean defence is available for the latter classical case it is also available for the former non-classical cases. In short, were a conditional true by virtue of the indeterminacy of its parts, we could as easily explain the intuition of its falsity as could the classicist in the case where a conditional is true by virtue of the falsity of its antecedent. Thus the values of (f) and (d) remains an open issue for all that has been said so far, despite the inconsistency between antecedent and consequent, and strong intuitions concerning each conditional’s falsity. And there is good reason to claim each conditional as true (despite its non-assertibility). Consider (f). In circumstances where Tim is borderline tall, then, if he counts as tall under such circumstances, by parity of reasoning he should also count as not tall. Equally poised between each by virtue of being equally neither, if he is to count as one he counts as the other. More generally, if borderline Fs are to count as F after all, then they ought also count as not-F. There can be no principled reason for counting a borderline F as one but not the other, thus ‘one in, both in’. Similarly for (d). Were we to count borderline case Tim as tall after all, then, given that Tek is taller, Tek must also count as tall after all. But, by the former reasoning, if borderline case Tek is to count as tall after all, then by parity of reasoning he ought also count as not tall, and this is just what (d) expresses – if (borderline case) Tim is tall (then the taller Tek must also be tall and so), then Tek is not tall. Not only is the supposed falsity of (d) and (f) not obvious; it is not even true. Truth-functionality is not threatened by the few cases offered against it. The minimal modi¿cation of classical logic necessitated by the phenomenon of vagueness results in the truth-functional logic outlined in the previous section. The arguments for non-truth-functionality depend on questionable defences of LEM and LNC, failed appeals to ‘penumbral connections’, and questionable intuitions concerning the conditional. When the widely reported ‘problems’ faced by a truth-functional approach are lined up and squarely faced, they appear to lack the requisite strength to bear the burden asked of them. Truth-functionality cannot be defeated so easily. 7.3

Truth and the T-schema

Problems are also said to attend the abandonment of bivalence. As such, paracomplete responses to vagueness – both weakly paracomplete (e.g. supervaluationism) and strongly paracomplete (e.g. K3 and L3) – face a common objection. Acceptance of Tarski’s T-schema, along with minimal assumptions regarding falsehood and principles governing negation and disjunction, seem to provide suf¿cient grounds for supposing any rejection of bivalence to be absurd. The

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Williamson Argument (1994: ch. 7, §2), as it has become known, clearly presents the challenge. According to the Principle of Bivalence: (PB) For any sentence A, “A” is true or “A” is false.9 To suppose this false by virtue of vagueness is to claim the existence of a vague sentence that is not either true or false (where the negation in question here corresponds to “~”). By De Morgan’s Laws, then, it entails the existence of a vague sentence that is both not true and not false. To then further assume the validity of the T-schema: (T) “A” is true if and only if A, given that the falsity of “A” is equivalent to the truth of “∼A”, it would then follow that ∼A (since “A” is not true) and ~ ~A (since “A” is not false, hence “∼A” is not true). Contradiction ensues. To assume the falsity of bivalence in the face of vagueness would then require that we abandon the T-schema and with it a disquotationalist account of truth. But this is too high a price to pay, or so it is claimed. Contrary to those advocating a view of vagueness as a paracomplete phenomenon, we should infer the truth of bivalence from the foregoing contradiction. In short, it is argued that given the T-schema, any falsi¿cation of bivalence entails a falsi¿cation of the excluded middle which, in turn, leads to contradiction. As a consequence, it is claimed, we should refrain from denials of bivalence and accept the principle. How might the advocate of a strongly paracomplete account of vagueness respond to such an argument? An initial response might accept the absurdity of the joint supposition of the Tschema and the falsity of the principle of bivalence, and go on to deny the falsity of bivalence without accepting its truth. In a three-valued context with a weakened rule of reductio, Reductio′, one cannot infer from the T-schema that the negation of bivalence is itself (determinately) not the case, i.e. that the Principle of Bivalence is (determinately) true, only that its negation is not determinately the case, i.e. that the Principle is not false. As a consequence, then, we should indeed refrain from asserting the negation of bivalence, but it is wrong to suggest that we should refrain from denials of bivalence. We can deny the Principle without asserting its negation.10 The same point is elsewhere put in terms of weak negation (see 9 Williamson expresses the principle as one asserting truth or falsity of any utterance that expresses a unique proposition, as opposed to the truth or falsity of any proposition. He does so to close off a possible line of retreat by a semantic theorist who might accept a true/false classi¿cation of propositions but describe vagueness as an inability to so classify utterances by virtue of vague utterances not expressing a unique proposition. Since this line is not being pursued here, we need only consider the more usual account of bivalence concerning propositions. It is this which is contested and which Williamson will object to in the circumstances. 10 Recall that in paracomplete contexts we must separate denials from the assertion of negations, see Chapter 4, §1.4.

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Williamson 1994: 193, for example) or exclusion negation (see Beall 2002). If we de¿ne a weak negation “¬”, corresponding to “~D”, so that “¬ A” is true if and only if “A” is false or indeterminate (i.e. if and only if “A” is not true), then the response being considered distinguishes between those rejections of bivalence that amount to the assertion of its negation and those that amount to the assertion of its weak negation. When bivalence is ‘rejected’ to accommodate vagueness, its weak negation is accepted, i.e. ¬(every sentence is true or false). By the foregoing proof one is only able to conclude that ¬A and ¬∼A; no contradiction follows. This response, articulated most clearly by Beall (2002), is ultimately unsuccessful, however. It shows that, where truth is taken to satisfy the T-schema, a vague sentence A is such that it is also vague whether “A” is true and thus retreats from the originally proposed non-bivalent approach to vagueness according to which a vague sentence is (determinately) not true. Such a retreat in the face of the Williamson Argument might appear to have the virtue of retaining the T-schema and those theories of truth that require it (e.g. redundancy or deÀationist theories), but, as Williamson shows, higher-order vagueness quickly renders this weakened nonbivalent approach untenable. For, just as ¿rst-order vagueness in A now leads us to assert the weak negation of the claim that “A” is true or “∼A” is true (i.e. “A” is false), higher-order vagueness in A will now similarly lead us to assert the weak negation of the claim that “A” is true or “¬A” is true (i.e. “A” is not true) – ¬(“A” is true or “¬A” is true). By reasoning exactly analogous to the Williamson Argument we can show that, given the T-schema, this entails ¬(A or ¬A), so ¬A and ¬¬A. A quick inspection of the de¿nition of “¬” con¿rms the inconsistency of such a claim and so our commitment to even the weak negation of bivalence in the face of vagueness when conjoined with the T-schema results in absurdity. Appeal to a weak negation has only shifted the problem. The problem now encountered can be similarly dealt with by appeal to some weakly weak negation but this will, again, only shift the problem, not solve it.11 Rather than retain (T) and abandon the robustly non-bivalent approach originally proposed according to which the principle of bivalence is false, like the weakly paracomplete supervaluationist response we should reject (T).12 In so doing note, ¿rstly, that we might admit that the T-schema provides a minimum constraint on truth in bivalent circumstances but deny that this covers all circumstances. Thus we might concur with Tarski when he approvingly reports Aristotle as claiming that ‘to say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, or of what is not that it is not, is true’ (1943: 342–3). Keefe (2000: 215) reminds us that such a claim is uncontroversial in current circumstances. Aristotle and Tarski are undeniably correct, but this is not yet suf¿cient to justify (T). It remains to be considered what to say in circumstances where it is simply indeterminate whether something is or is not, i.e. indeterminate whether A. According to Williamson, in such circumstances it remains the case that either A obtains or ∼A obtains (the indeterminacy is a matter of ignorance), and we speak 11 Pelletier and Stainton (2003), while acknowledging Williamson’s point here, seem to underestimate its force. It is not simpy that higher orders of vagueness complicate things, as they claim, but, more strongly, that they lead to a vicious regress. 12 See Keefe (2000: ch. 8, §3) for discussion from a supervaluationist perspective.

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truly if and only if we describe things as they are, in accordance with (T). According to paracomplete theorists, however, in such circumstances “A” is indeterminate in truth-value; its truth-value status is indeterminate and therefore not true. Given the foregoing account of conditionality, then, such an instance of (T) will not be true but will itself be indeterminate. All can agree with Tarski and Aristotle on the evidence they proffer, but it remains equivocal between the truth or otherwise of (T). As has long been recognized, all parties can agree to the weaker principle: (T*) “A” is true entails and is entailed by A.13 This captures those aspects of truth about which both bivalentists and nonbivalentists agree. But what of the idea promoted by Tarski, and widely accepted since, that (T) presents a minimal constraint on truth per se? Surely to reject a minimal constraint on truth, i.e. (T), in favour of a weaker principle, i.e. (T*), is to abandon the ordinary conception of truth. So argues Williamson. But the evidence offered so far in support of (T) as a minimal constraint has already been found to be less than compelling. Further argument fails to ¿ll the gap. Williamson (1994: 190) offers a ‘rationale’ for (T) claiming that what it takes for “A” to be true is just for it to be the case that A and what it takes for “∼A” to be true (i.e. for “A” to be false) is just for it to be the case that ∼A. For example, what it takes for it to be true that TW is thin is just that TW be thin and what it takes for it to be false is just that TW not be thin. But, again, all parties can agree to this claim yet contest (T). The rationale fails to address cases where there simply is no fact of the matter whether A. It does establish, as Keefe points out, the inconsistency associated with, for example, TW being thin and its not being true that TW is thin or TW not being thin and its not being false that TW is thin, but this content is fully captured by (T*). (T*) constrains truth in such a way that “A” is true whenever it is the case that A and “∼A” is true (i.e. “A” is false) whenever it is the case that ~A. Whenever TW is thin it is true that TW is thin and whenever TW is not thin it is false that TW is thin since it is true that TW is not thin. However, the rationale does not speak to the case where there is no fact of the matter whether A. No reason has yet been given for thinking that in such a case there is no fact of the matter whether A is true. There is as yet no objection to the paracomplete approach to vague sentences according to which, in such circumstances, A is not true. Of course, if the only way that it could fail to be true that A is that ∼A, then, given that this is all it takes for “A” to be false, “A” failing to be true would entail its falsity. If, for example, the only way it could fail to be true that TW is thin is that TW is not thin, then in such circumstances we ought to agree that it is false that TW is thin. But there is another way in which it could fail to be true that TW is thin, namely if there is simply no fact of the matter whether TW is thin or not. In such circumstances no objection presents itself to postulating a truth-value gap. As in supervaluationism, we can accept failures of bivalence even for vague sentences. The requirement that we endorse (T) as opposed to (T*) is unjusti¿ed. 13 The point is familiar from van Fraassen’s early work on paracomplete theories to deal with reference failure, e.g. van Fraassen (1966).

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197

Higher-order Vagueness

As we saw in Chapter 1, it is uncontestable that there are higher orders of vagueness at least somewhere. With the common understanding of vagueness suggesting that the lack-of-sharp-boundaries phenomenon must be acknowledged through the existence of border cases, border border cases, and so on, it seems then that the admission of some higher orders of vagueness opens the Àoodgates – with the admission of some higher orders it seems we should go on and accept vagueness at all higher orders. Not only is the extension of “is red” (for example) vague, but so too “is borderline red”, “is borderline borderline red”, and so on. There are borderline cases for all such predicates, the class of red things not being sharply bounded, nor the borderline cases of red things, nor their borderline cases, etc. Given this recognition of higher orders of vagueness, two issues then immediately present themselves. Firstly, can the three-valued logic described above accommodate this phenomenon, and, if so, secondly, what structure is the phenomenon described as having? 7.4.1

How many truth-values?

So, turning to the ¿rst issue, can the three-valued approach to vagueness succeed in the face of higher-order vagueness, in particular in light of the fact that the class of borderline cases for a vague predicate is itself vague and exhibits borderline cases? It is sometimes suggested that it cannot, and many-valued approaches to vagueness are forced to acknowledge in¿nitely many values to account for higher orders of vagueness. The reasoning for this claim is quite straightforward. If, as three-valued approaches suggest, a sentence such as “TW is thin” is to be evaluated as either true, false or neither, then TW’s being a borderline case of thinness is accounted for by evaluating the sentence as neither true nor false but indeterminate. Since there is no fact of the matter as to TW’s thinness, the corresponding sentence claiming he is thin is neither true nor false. So far, so good. But what if TW were a little bit thinner than he actually is, so that he was not yet unquestionably thin but was thin enough to call into doubt his status as a borderline case of thinness? He would then count as a borderline case of a borderline case of thin. Now, just as borderline cases of thinness can be accounted for by saying that there is no fact of the matter as to whether they are thin or not thin, so too the counterfactual situation just described involving higher-order vagueness. It can be accounted for by saying that there is no fact of the matter as to whether TW is determinately thin or not determinately thin. But, it is further argued, this goes only part of the way to describing the situation, for what are we to say in respect of predications of thinness per se, as opposed to predications of determinate thinness? If a third value, “neither (true nor false)”, is required between truth and falsity to accommodate borderline cases, then surely additional values are required to accommodate borderline borderline values – namely values between “true” and “neither”, and similarly between “neither” and “false”. If a new truth-value is required to accommodate the lack of a sharp boundary between the thin and not thin, then further values are required to accommodate the lack of sharp boundaries between the determinately thin and borderline thin, and

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between the borderline thin and determinately not thin. But we cannot rest content with a ¿ve-valued logic, for the absence of sharp boundaries anywhere generates borderline borderline borderline cases, and so on up the hierarchy. To account for the existence of the in¿nitely many kinds of cases, in¿nitely many truth-values are required. Such reasoning to an in¿nite-valued logic is Àawed, however. If TW were a borderline borderline case of thinness, then there would be no fact of the matter whether TW is a borderline case of thinness or not, and a claim to the effect that he was a borderline case would count as neither true nor false. It would then be neither true nor false to suggest that the sentence predicating thinness of him was indeterminate in truth-value. It is not true to say that the sentence is indeterminate since that would wrongly assimilate TW to the borderline cases of thinness, nor can it be false to say that the sentence is indeterminate since that would wrongly assimilate TW to the non-borderline cases (i.e. to the determinate cases or determinate non-cases). The ascription of “neither” to the sentence “TW is thin” is neither true nor false; it is indeterminate whether the sentence takes that value and thus we see how ascriptions of truth-value can themselves be indeterminate. Just as the object-language is vague, so too the metalanguage. Higher-order vagueness does not require the postulation of in¿nitely many truth-values but instead can be accommodated by recognizing that vagueness is a feature not only of the object-language, but of the in¿nite hierarchy of metalanguages as well. In this sense, higher-order vagueness of a language can be accommodated by vagueness in higher-order languages. Such vagueness in the metalanguage should not come as any surprise. We have already seen that some metalinguistic notions are vague; in particular the notion of vagueness itself and the associated notion of a borderline case are vague, as we saw in Chapter 1, §5. There may be no fact of the matter whether some particular predicate P is vague and, as a borderline case of vagueness, it is therefore neither true nor false that P is vague. Similarly, for some particular sentence A it may be neither true nor false that A is true (say), or neither true nor false that A is neither true nor false. But if higher-order vagueness in the object-language can be modelled using a vague higher-order (i.e. meta-) language in a way that does not require additional values to those already invoked to account for ¿rst-order vagueness, it seems reasonable to ask whether we need additional non-classical truth-values to account for even ¿rst-order indeterminacy in the object-language. To put the point another way, if vagueness in the object-language requires the extension of the classical, twovalued truth-set to three values, then it might seem that, iterating up through the hierarchy of vague metalanguages, we must similarly extend the three values to include ever-increasing new values. In this vein, Keefe (2000: 121) suggests that a many-valued theory ‘cannot consistently admit that a metalinguistic sentence assigning some given intermediate value to A itself receives an intermediate value’. The inconsistency arises because in the three-valued case under scrutiny, having invoked a vague metalanguage above to retain a three-valued approach in the face of higher-order vagueness, the resulting possibility of indeterminacy surrounding the ascription of a non-classical truth-value to a sentence is taken to require the introduction of a new truth-value.

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The case has been made repeatedly by Tye. If it might be indeterminate whether a sentence A were indeterminate, then A would have to be said to be ‘indeterminately indeterminate’. Since this value is distinct from any other (it is claimed), such vagueness in the metalanguage can be seen to necessitate the introduction of a new truth-value (1990: 554–5). Similarly, it is claimed that if there are sentences for which it is neither true nor false that they are true, then there must be sentences ‘that are neither true nor false nor indeterminate’ (1994: 203). Thus the initial move described above of avoiding the slide to in¿nitely many values from considerations to do with higher-order vagueness by employing a vague metalanguage is said to be little improvement since the existence of sentences ascribing ‘intermediate values’ themselves taking an ‘intermediate value’ requires the introduction of new values, thus undermining the valency of the original theory proposed. The apparently trivalent theory proves not to be trivalent after all. As noted above, Keefe takes the (Àawed) point Tye makes to be a quite general one, applicable to many-valued theories in general, from three- to in¿nite-valued. But why should one think the point to have force at all? Keefe offers an argument in the special case of degree theories (i.e. in¿nite-valued logics), an argument which is presumed to generalize. The argument begins from the observation that degree theorists postulate degrees of truth to reÀect the fact that “x is F” and “‘x is F’ is true” can be a matter of degree. There can be degrees of redness and degrees of truth, so the two classical truth-values are inadequate and new values must be introduced. Classical logic must be rejected. But accepting that truth-value ascriptions can themselves hold to intermediate degrees ‘would, in the same way, demonstrate the inadequacy of that chosen set of values and hence undermine the logic built on it’ (2000: 118). The argument seems to be then that, where there can be degrees of truth, the set of truth-values deemed adequate prior to the recognition of degrees is rendered inadequate and new values are therefore required. But the reasoning is Àawed. The acceptance of degrees of truth demonstrates the inadequacy of the classical set of truth-values. In accepting vagueness of the objectlanguage as requiring degrees of truth, the classical set of values is abandoned and its logic is said to be captured by a degree theory, but the iteration of the idea of degrees of truth up through the hierarchy of languages requires only the abandonment of classical values up through the hierarchy of metalanguages, i.e. requires only the iterated use of the already-proposed degree theory up through the hierarchy. Keefe’s argument confuses the correct idea that iterated recognition of vagueness in metalanguages requires iterated use of a non-classical truth-set in metametalanguages with the mistaken idea that iterated recognition of vagueness in metalanguages requires iterated extension of the non-classical truth-set in the metametalanguages. The initial recognition of object-language vagueness brings in its wake the acceptance of degrees of truth, but once accepted they are then available for redeployment without revision in modelling the vagueness that similarly appears in the metalanguage and the metametalanguage, and so on. In the case of interest, where the degrees are three in number as opposed to in¿nite, the recognition that “x is F ” and “‘x is F’ is true” can be a matter of degree due to vagueness demonstrates the inadequacy of the simple, classical red/not-red, true/false distinctions and leads to acceptance of the three values true, false and neither. But accepting that truth-ascriptions can themselves hold to intermediate

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degrees only serves once again to highlight the inadequacy of the classical, twovalued, true/false distinction (now in its application to classi¿cation of sentences of the metalanguage) and does not undermine the proposed non-classical extended set of values. The situation exactly replicates that which obtains in respect of the supervaluationist’s own response to higher-order vagueness. The two approaches, weakly and strongly paracomplete, are alike in this respect. As with the view just defended, supervaluationists maintain that a sentence A exhibiting higher-order vagueness may be such that ‘we should not conclude that [it] is true, but nor should we call [it] neither true nor false … The truth-value status of A (whether it is true, false or lacks a value) remains unsettled’ (Keefe 2000: 203). And, just as this indeterminacy surrounding truth-value status gives no cause for thinking that supervaluationists ought to embrace additional truth-values, so too with the truthfunctional, strongly paracomplete approach being advocated. Higher-order vagueness can be accounted for in a ‘three-valued’ framework. Of course, talk of three values here is potentially misleading. As noted earlier, what is proposed is actually a truth-value gap theory. The third ‘value’, the gap, reÀects a distinct semantic category into which sentences may fall. It captures the distinct truth-value status of a vague sentence but is not itself a truth-value. Its use in formal semantics as if it were such a value (e.g. in truth-tables) is a technical convenience for modelling the logical behaviour of complex sentences whose truthvalue status is a function of the truth-value status of its parts. When A is indeterminate in truth-value, so too, for example, is its negation ~A. It is convenient to represent this by way of a truth table for “~” with ‘truth-values’ 1, 0 and @. And so too with other truth-functional relations. The representation of “neither” as a third value is a convenient ¿ction, nothing more. Bearing this in mind, we can see our way clear of another objection Keefe raises to reinforce her argument against manyvalued approaches to vagueness, including the truth-value gap approach being advocated – an objection which echoes concerns expressed earlier by Williamson (1994: 112). The many-valued theorist objects to some classical laws (e.g. LEM) on the grounds that, while they are true in all classical valuations, they are not true in all valuations. From a many-valued perspective classical valuations do not exhaust the possibilities. From a three-valued perspective, for example, two-valued classical valuations, in assigning only 1 and 0, ignore the possible valuation assigning @. Since logical truth is a matter of truth on all valuations, principles that are classically valid may nonetheless fail. Now, when the three-valued theorist (for example) de¿nes logical truth as truth on all three-valued valuations, it is assumed that the truth-value status of sentences is exhausted by considering all and only 1, 0 and @. But, of course, on the approach to higher-order vagueness just considered it might sometimes be the case that no such value is assigned to a sentence and such three-valued valuations can no longer be supposed to exhaust the possibilities. Higher-order gaps present cases where it is indeterminate whether any such value can be assigned to the sentence in question. Thus the three-valued truth-tables can now no longer be said to provide a complete account of the behaviour of the logical connectives, and are no longer suf¿cient for de¿ning logical truth and validity. To the extent that these truth-tables

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are taken to generate a logic, the three-valued theorists’ account of logical truths and of logic in general ‘is incompatible with the claim that the metalanguage is vague. And they cannot modify that account by also requiring that for A to be a logical truth, A must also be true if its [parts] cannot be truly assigned a value from the range. For the system can give no method of calculating the value of A in such a situation’ (Keefe 2000: 119). The way out of this problem is suggested by the foregoing gap theorists’ initial treatment of truth-value gaps. There, in order to secure a complete account of the logic, the absence of a truth-value was modelled by way of a ‘third value’, thus making possible the use of truth-tables to describe the behaviour of the propositional logical constants. So too now, we should model higher-order gaps by way of additional ‘truth-values’. In this way, the truth-value status of any sentence (precise, vague, vaguely vague, etc.) can be treated as if it were a truth-value and the logic’s functionality with respect to truth-value status – i.e. the fact that the truth-value status of a sentence is taken to be a function of the truth-value status of its parts – can be represented as functionality with respect to truth-value. The logic is then truth-functional, with propositional constants (negation, conjunction, etc.) characterized in the standard way by means of a many-valued truth-table. To count A as a logical truth is now a matter of its being true irrespective of the truth-value status of its parts, where we now admit of more than three possibilities. The newly extended truth-set will be in¿nite-valued for reasons analogous to those given earlier which were claimed to push three-valued theorists towards in¿nitely many values. To account for the in¿nite variety in truth-value status that follows on from the in¿nite hierarchy of borderline cases, in¿nitely many distinctions are required, and so in¿nitely many ‘truth-values’ are required. Thus, though a description of higher-order vagueness itself requires only truthvalue gaps and their iteration, functionality with respect to the in¿nite variety in truth-value status does ultimately require that we extend the L3 semantics proposed earlier. What structure should the extended truth-set have? We should look to the structure of higher-order vagueness. Before turning our attention to a detailed analysis of higher-order vagueness, though, let us be clear about what has been argued. Higher-order vagueness gives no reason for abandoning a truth-value-gap approach to vagueness. Like Keefe’s supervaluationist approach, vagueness gives rise to truth-value gaps and higherorder vagueness gives rise to higher-order gaps. In this sense a ‘three-valued’ (i.e. gap) approach is not undermined. That a sentence may be indeterminately indeterminate does not give any more reason for thinking that it has some truth-value distinct from “true”, “false” and “neither” than the fact that a sentence may be indeterminate gives adequate reason for thinking that it has some truth-value distinct from “true” or “false”. What such higher-order vagueness does show is that in addition to sentences that are neither true nor false – sentences exhibiting a truthvalue gap – there are sentences for which it is neither true nor false that they are true, false or neither – sentences exhibiting a gap between being truth-valued and lacking a truth-value. Unlike supervaluationism, however, the approach advocated takes the truth-value status of a complex sentence to be a function of the truth-value status of its parts and so will appeal to an in¿nite set of values to represent this functional dependence.

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7.4.2

The structure of higher-order vagueness

We have been concentrating on higher-order vagueness understood as that aspect of vagueness according to which terms in a language might possess borderline cases and borderline cases of borderline cases, and so on. This is typical of discussions of higher-order vagueness. As Williamson (1999) makes clear, however, the phenomenon as typically discussed is too narrowly construed and underdescribed. Let us turn our attention then to higher-order vagueness and describe its structure from the point of view of the truth-functional approach being advanced here. In so doing we will discharge the requirement to ¿ll out our theory to accommodate the phenomenon, and also see what constraints the phenomenon places on the formal semantics whose extension the phenomenon mandates in the manner just described in the previous section. So far, when we have spoken of higher-order vagueness we have pointed to the possibility of there being, in addition to borderline cases for a vague term (e.g. “thin”), borderline cases of borderline cases, and borderline cases of borderline cases of borderline cases, and so on. Turning our attention to sentences and invoking object-language expression of determinacy and indeterminacy by way of “D” and “I” instead of metalinguistic ascent, we can say that a sentence A has borderline cases just in case there are situations where it is neither determinately the case that A nor determinately not the case that A – i.e. there are situations where ~DA & ~D∼A, i.e. where IA. And A has borderline cases of borderline cases just in case there are situations where it is neither determinately the case that IA nor determinately not the case that IA. More generally, if we de¿ne I 0A = A and I n+1A = ~DI nA & ~D~I nA (n ≥ 0), we can say that A has (n + 1)th-order borderline cases if and only if InA has borderline cases. Now, certainly, the absence of sharp boundaries characteristic of vagueness requires that a vague sentence A have borderline cases of all orders, yet higher-order borderline cases as de¿ned do not exhaust the range of indeterminacies that might manifest themselves at higher orders. Consider, for example, the boundary between situations where it is the case that DDA and those where it is the case that ~DDA. Just as there is no sharp boundary between being determinately determinately thin and not determinately determinately thin (i.e. “determinately determinately thin” is vague), so too should there be no sharp boundary between situations where it is the case that DDA and those where it is the case that ~DDA. And so there will be situations where it is the case that ~DDDA & ~D~DDA, i.e. where IDDA. But such higher-order indeterminacy is not captured within the hierarchy of borderline cases just described. Such a hierarchy then clearly captures only part of the phenomenon of higher-order vagueness. Rather than focusing simply on borderline cases and their borderline cases, etc., we should also consider borderline cases of determinate determinate cases, and more.14 We can initially classify situations or states of affairs in terms of whether it is the case that A or ∼A – a ¿rst-order classi¿cation – and there may sometimes, of course, be no fact of the matter and this ¿rst-order classi¿cation is vague. Thus we encounter ¿rst-order vagueness. A second-order classi¿cation can further classify 14

What follows owes much to Williamson (1999).

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situations or states of affairs as determinately falling under one of the two ¿rst-order categories, A, ~A, or being a borderline case; thus situations or states of affairs can be classi¿ed according to whether it is the case that DA, D∼A , or IA. And here too there may sometimes be no fact of the matter whether a situation can determinately be classi¿ed under one of these headings and borderline cases may arise. Thus we encounter second-order vagueness. Further classi¿cation of situations or states of affairs according to whether they determinately do or do not fall under one of the three second-order categories might similarly produce borderline cases, thus generating cases of third-order vagueness. In general, we can describe nth-order vagueness of A as vagueness in the nth-order classi¿cation. More exactly, let a classi¿cation be a set of sentences X. An nth-order classi¿cation for X, Cn X, is now de¿ned inductively: C1X = X; Cn+1X = {DA, IA, D~A: A ∈ Cn X}. Note now that where A ∈ Cn X, then DA ∈ Cn+1X, D~A ∈ Cn+1X and IA ∈ Cn+1X. So by means of Cn+1X any state of affairs can be classi¿ed according to whether A determinately holds, determinately does not, or is borderline (i.e. indeterminate), for any A ∈ Cn X. That is to say, Cn+1X is the (smallest) classi¿cation that allows for the expression of any indeterminacy in respect of the classi¿cation Cn X. For example, the second-order classi¿cation C2{A} allows for the expression of indeterminacy that might arise in the ¿rst-order classi¿cation {A}. We shall say that a classi¿cation is precise if and only if all its members are determinate, where any sentence A is determinate if and only if it will not admit of borderline cases: = DA V D~A.15 A classi¿cation then is vague if and only if some member is indeterminate, i.e. admits of borderline cases, so that in some model M, M = IA. Now we can de¿ne what it is for a sentence to be higher-order vague: A is nth-order vague if and only if Cn{A} is vague and A is nth-order precise if and only if Cn{A} is precise.16 With this de¿nition in place, we can now establish a number of results concerning higher-order vagueness. Note ¿rstly that membership of Cn{A} is not closed under logical equivalence. For example, C1{~ ~p} contains ~ ~p but not the logically equivalent p. However, nth-order vagueness is closed under logical equivalence. It is easy to show that if = A↔ B, then every member of Cn{A} is logically equivalent to some member of Cn{B}, and vice versa. Now, since logically equivalent sentences are such that one is indeterminate if and only if the other is (i.e. indeterminacy is closed under logical equivalence), Cn{A} is vague if and only if Cn{B} is and so A is nth-order vague if and only if B is nth-order vague. 15 Williamson (1999) de¿nes a sentence A to be precise if and only if it does not admit of borderline cases, i.e. = DA V D∼A. This de¿nition of vagueness has the result that A is vague (i.e. not precise) if and only if A is ¿rst-order vague (see next paragraph). Thus it commits to the possibility of vagueness without higher-order vagueness, a possibility at odds with the ‘no sharp boundaries’ conception of vagueness discussed earlier in the book. Considered, alternatively, as a de¿nition of ‘determinacy’ rather than ‘precision’, the relation between vagueness and higher-order vagueness is thus left open. 16 The foregoing de¿nition of higher-order vagueness differs from that offered in Williamson (1999). There a classi¿cation is de¿ned as a set of sentences closed under negation, conjunction, disjunction and implication, with an nth-order classi¿cation for X, CnX, now de¿ned inductively as: C1X = CX; Cn+1X = C{DA: A ∈ Cn X}.

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Moreover, A is nth-order vague if and only if Cn{A} is vague, by de¿nition, so A is ¿rst-order vague if and only if C1{A} is vague, i.e. if and only if A is indeterminate. And so we can establish, as expected, that A is ¿rst-order vague if and only if A is indeterminate. Obviously too, despite C1{A} ≠ C1{~A}, since A is indeterminate if and only if ∼A is indeterminate, C1{A} is vague if and only if C1{~A} is vague, and so A is ¿rstorder vague if and only if ∼A is ¿rst-order vague. Additionally, C2{A} = {DA, IA, D~A} and C2{~A} = {D~A, I~A, D~ ~A}, so C2{A} = C2{~A}. Given that Cn{A} = Cn{B} for all n > 1 if C2{A} = C2{B}, it follows then that Cn{A} = Cn{~A} for all n > 1. So, for all n > 1, Cn{A} is vague if and only if Cn{~A} is vague; and so, for all n > 1, A is nth-order vague if and only if ∼A is nth-order vague. Summing up, then, A is nth-order vague if and only if ~A is nth-order vague, for all n. As we saw above, A is ¿rst-order vague if and only if A is indeterminate. To rule out ¿rst-order vagueness in A, then, is to claim that A is determinate – which is to say: = DA V D∼A. As it happens, A is second-order vague if and only if either DA or D∼A is indeterminate, one or the other admitting of borderline cases; so to rule out second-order vagueness in A is to claim that both DA and D∼A are determinate – which is to say: = DDA V D~DA and = DD~A V D~D∼A. To see this, note ¿rstly that A is second-order vague, by de¿nition, if and only if C2{A} is vague, i.e. if and only if some member of {DA, IA, D~A} is indeterminate. So, if either DA or D∼A is indeterminate, then C2{A} is vague. Conversely, if C2{A} is vague, then one of DA, IA, or D∼A must be indeterminate. But if IA is indeterminate, then ~IA is indeterminate. Since ~IA is logically equivalent to (DA V D~A) and indeterminacy is closed under logical equivalence, (DA V D~A) is indeterminate. But a necessary condition for the indeterminacy of the disjunction is the indeterminacy of one or the other disjunct (determinacy is inherited), so it is either indeterminate that DA or indeterminate that D∼A. So, if C2{A} is vague, then DA or D∼A is indeterminate. Summing up, then, C2{A} is vague if and only if either DA or D∼A is indeterminate, and consequently A is second-order vague if and only if either DA or D∼A is indeterminate. To rule out second-order vagueness in A, then, is to claim: = DDA V D~DA and = DD~A V D~D∼A. On the truth-functional approach being proposed, given ¿rst-order precision of A, it is easily shown that: = A → DA. Where A exhibits ¿rst-order vagueness and A is therefore itself indeterminate, then: = A → DA.17 Given the fact that: = DA → A, it follows then that: A is ¿rst-order precise if and only if = A ↔ DA. As we saw in respect of the T-schema, where A is neither true nor false but indeterminate, “D” is non-redundant and A ↔ DA may be non-true. This is unsurprising given that the operator effectively is a truth-predicate. As on the supervaluationist approach, second-order precision of A warrants: = DA → DDA Where A is ¿rst-order precise, necessarily DA or D~A, i.e. A is either true or false. If A is false then A → DA, and if A is true then DA is true and so A → DA. So A → DA is always true. Conversely, A is ¿rst-order vague if and only if A may be neither true nor false, and so A → DA can have indeterminate antecedent and false consequent and so is not always true. 17

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and = ~DA → D~DA, with second-order vagueness of A inconsistent with them.18 Thus: A is second-order precise if and only if = DA ↔ DDA and = ~DA ↔ D~DA. Unlike supervaluationism, we can also show that: A is ¿rst-order precise if and only if = A V ~A, and A is second-order precise if and only if = DA V ~DA and = D~A V ~D∼A. Notice now that if A is ¿rst-order precise, then = A V ∼A and so, by (Det), = D(A V ~A). Since “D” distributes across disjunction, it follows that = DA V D∼A, and since D~A entails ~DA, we can prove by dilemma that = DA V ~DA. Moreover, if A is ¿rst-order precise, then ~A is, so = ~A V ~~A and so, by (Det), = D(~A V ~ ~A). Since “D” distributes across disjunction, it follows that = D~A V D~ ~A, and since D~ ~A entails ~D~A, we can prove by dilemma that = D~A V ~D∼A. So if A is ¿rst-order precise, then = DA V ~DA and = D~A V ~D∼A. And so A is second-order precise. In fact, we can prove more generally that: if A is mth-order precise then A is nth-order precise, for any m ” n. The hierarchy of precision is cumulative. (Williamson 1999 establishes the result for modal approaches.) The proof uses the general principle of the inheritance of determinacy: if A is determinate, then any truth-function, including DA, is determinate. (This principle’s validity follows from the simple fact that, as de¿ned, truth-functions for classical values always yield classical values.) Suppose that A is mth-order precise. Thus, by de¿nition, Cm{A} is precise, i.e. every member of Cm{A} is determinate. Consider an arbitrary member B. Since it is determinate, so too is DB. All members of Cm+1{A} are therefore orthodox truth-functions of determinate sentences and are therefore themselves determinate. Hence Cm+1{A} is precise. So for any n ≥ m, Cn{A} is precise and A is nth-order precise. Obviously a similar result does not hold in respect of the hierarchy of vagueness. It is easy to see that we can have ¿rst-order vagueness without second-order vagueness, i.e. A may be indeterminate though both DA and D∼A are determinate. Can we have second-order vagueness without third-order vagueness? Indeed, can vagueness terminate at any level, or might it collapse in such a way that if there is vagueness at some level m, then there is vagueness at all higher levels? 18 For supervaluationists, the absence of second-order vagueness in A is captured by the following two theses of SpV +: =SpV+ DA → DDA and =SpV+ ~DA → D~DA, the ¿rst thesis being the analogue of the characteristic thesis of the modal logic S4, and the second being the analogue of the characteristic thesis of the modal logic S5. For the supervaluationist, then, the latter thesis is characteristic of determinacy at the second order, with the former thesis a derivable consequence. (See Williamson 1999.) First-order precision of A is captured in SpV+ by the thesis: =SpV+ A → DA.

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Williamson (1999: §3) argues that it remains ‘an open problem’ on a supervaluationist approach, depending crucially on the acceptance of the following as a theorem: DA ↔ D~D~DA. If accepted, then, given second-order vagueness, it follows that there is vagueness at all orders. Such a theorem is provable in SpV + if the B-schema, A → D~D∼A, is accepted. Under these conditions, orders of vagueness higher than the ¿rst collapse. The theorem fails in the logic of vagueness proposed. (When ν (DA) = @, ν (D~D~DA) = 1 and the biconditional fails to be true.) Higher orders of vagueness do not collapse. 7.4.3

Extending to L∞

We can now return to the issue of determining the structure of the extended truthset, capable of underwriting a complete account of the connectives and logical consequence in the face of higher-order vagueness. As we saw earlier, in¿nitely many distinct ‘truth-values’ are required to characterize the in¿nite variety in truthvalue status dictated by higher orders of vagueness. In fact, a dense, totally ordered set of semantic values is required. To see this, note initially that ¿rst-order vagueness, though generating only truthvalue gaps, is formally modelled by extending the two-value truth-set {1, 0} to a three-valued set {1, @, 0} where the natural ordering on the semantic values is presumed, so that 0 < @ < 1; i.e. an indeterminate sentence takes a value less than a true sentence and greater than a false one. This ordering of semantic values is presupposed by the formal semantics, L3. Consider a sorites series 〈a1, … ak〉 with respect to which F is soritical. Given the conditions for soriticality (Chapter 1), the series is ordered with respect to F so that a1 is (determinately) F, ak is (determinately) not-F, and, for any m,n such that a1 ” am < an ” ak, if Fan then Fam. On the proposed semantics for the conditional, ν (Fan → Fam) = 1 if and only if ν (Fan ) ” ν (Fam ). Hence, for any m,n such that a1 ” am < an ” ak, ν (Fan) ” ν (Fam). For some borderline case ah, a1 < ah < ak, so ν (Fah) ” ν (Fa1); i.e. ν (Fah) ” 1. Since it is indeterminate whether Fah, ν (Fah ) ≠ 1 but takes some other value, i, say. Thus i < 1. So too, ν (Fak) ” ν (Fah ), i.e. 0 ” i, yet ν (Fah) ≠ 0, so i ≠ 0 and 0 < i. In summary, 0 < i < 1. A totally ordered three-valued set is required. Similarly, adoption of such a set is presupposed by the semantic clauses for negation, conjunction and disjunction. First-order vagueness thus extends the set of semantic values to a set structurally isomorphic to {1, @, 0}. For simplicity, then, we can just adopt the set {1, @, 0} itself. The challenge, recall, is how to extend this semantics to accommodate higher orders of vagueness. Consider second-order vagueness. Such a case is generated by pointing to the possibility of there being an object that is a borderline case for DF or an object that is a borderline case for D~F. Consider then some borderline DF object, ae say, where af is some borderline case for F (i.e. af satis¿es IF) and ad is some object that is determinately F (i.e. ad satis¿es DF). We know that ν (Faf ) = @ and ν (Fad ) = 1. What truth-value status should we ascribe to ν (Fae )? Again, employing the ordering relation used to order the F-soritical series 〈a1, …, ak〉, “<”, af < ae < ad. So, if Faf , then Fae and if Fae, then Fad. Thus, ν (Faf ) ” ν (Fae) ” ν (Fad ). Since the truth-value status of Fae must be distinct from that of Fad and Faf , ν (Faf ) < ν (Fae) < ν (Fad ); i.e. @ < ν (Fae ) < 1. By similar reasoning, where ag is some object borderline between

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IF and D~F, 0 < ν (Fag) < @. By simply letting ν (Fae) = W and ν (Fag) = $, the required ordering of semantic values is achieved. Second-order vagueness requires a totally ordered truth-set isomorphic to {1, W, @, $, 0}; DDFan iff ν (Fan ) = 1, IDFan iff ν (Fan ) = W, DIFan iff ν (Fan) = @, ID~Fan iff ν (Fan) = $, and DD~Fan iff ν (Fan ) = 0. More generally, as we ascend the hierarchy, increasing orders of vagueness require the successive interposition of further intermediate semantic values between each pair of values in the totally ordered truth-set already required to model vagueness of the previously highest order. Higher orders of vagueness require the recognition of additional categories in the increasingly complex higher-order classi¿cations, the additional categories expressing the possibility of vagueness in respect of the categories that constitute the previously highest-order classi¿cation. In this way, higher orders of vagueness demand the interposition of new semantic values within a totally ordered set. In the absence of any ¿nite limit to higher-order vagueness, then, the truth-set is a totally ordered, dense set. The simplest model is provided by the rational numbers between and including 0 and 1, the rational interval [0, 1]. Where the logic of ¿rst-order vagueness was given by L3, the logic of vagueness (including higher orders of vagueness) is given by L∞. The recourse to an in¿nite-valued semantics in the manner outlined should not be confused with any appeal to degrees of truth. The choice of logic L∞ does not carry with it the problems commonly associated with such in¿nite-valued logics. In particular, such semantics are frequently charged with requiring excessively precise 999993 truth-values. How, for example, can we justify ascribing degree of truth 1000000 to a 999992 sentence as opposed to, say, 1000000 The foregoing account, however, requires only a totally-ordered dense set of values. The choice of a speci¿c value from among the in¿nitely many possible is not a measure of the degree of truth and is arbitrary except in so far as it preserves ordering requirements imposed by the structure of higher-order vagueness. No signi¿cance attaches to the choice of value apart from these ordering requirements. In¿nite-valued semantics are also commonly charged with falling foul of higherorder vagueness. Obviously, in so far as the semantics is a direct consequence of the recognition of the phenomenon, it does not succumb to it. More exactly, the charge is that even supposing ¿rst-order vagueness could be modelled by degrees of truth, there remains the problem that higher-order vagueness renders the truth-set incomplete for just the reasons outlined earlier in respect of a three-valued approach. Just as ¿rst-order vagueness presents cases where a sentence takes neither of the two values appropriate for precise sentences, so too higher-order vagueness presents cases where a sentence takes none of the values appropriate for ¿rst-order vague sentences, and so the in¿nite-valued set of degrees of truth does not exhaust the range of possible semantic values. Since the approach we have taken does not offer the in¿nite-valued set as a model of mere ¿rst-order vagueness, but, rather, vagueness at all orders, the problem is avoided. Another way in which this criticism from higher-order vagueness can be expressed is as follows. If any sentence A takes a value from the rational interval [0, 1], then either ν (A) = 1 or ν (A) < 1. But surely, it is said, the boundary between sentences taking the value 1 and those not taking the value is not sharp; higher-order vagueness points to the existence of sentences for which there is no fact of the matter whether it takes the value 1 or not. This claim is false and the implied criticism

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unwarranted. Higher-order vagueness points to the fact that between any two relevant semantic categories (e.g. the determinate truths and borderline cases, the determinate borderline cases and borderline borderline cases, etc.) there can be no sharp boundary and another intermediate value is required to mark the possibility of borderline cases in respect of these categories. But this, of course, is just to require that the truth-set be dense. Higher-order vagueness points to the existence of sentences that take a value between 1 and any other value in the rational interval [0, 1]. In fact, there is a sharp boundary between sentences taking the value 1 and those taking some other value in the interval [0, 1]. And, surprising though it might seem, we should expect just such a boundary – the boundary between those sentences to which no unclarity or indeterminacy due to vagueness attaches at all and those to which some does attach. Any degree of unclarity, indeterminacy or warranted hesitation arising from vagueness that attaches to a sentence will entail it take some value other than 1. And we cannot suppose the boundary to be vague since, given the way the boundary is de¿ned, it follows analytically that there can be no borderline case for the category; any such borderline case must be such that it take a value less than 1. Vague terms and vague objects are indeed boundaryless, up to a point, any ¿nite point. As we climb the hierarchy of ascending orders of vagueness, all sharp boundaries between categories in any classi¿cation are rejected and the recognition of borderline cases for any such category is manifested by the existence of an appropriate category in the classi¿cation of the next highest order. However, with ever-increasing complexity, in the limit as we ascend the orders approaching in¿nity, we reach an idealized point at which no further complexity is possible. At this idealized point vagueness terminates. There can be no borderline cases for sentences that are determinately, determinately, … (infinitely iterated) … determinately the case and those that are not, but since such an idealized boundary is not ¿nitely expressible, its existence can give rise to no paradox. What then, ¿nally, of the resolution of the sorites paradox in light of higher-order vagueness? Consider the standard (many-conditionals) sorites again. As a ¿rst-order approximation, it was suggested above that the paradox is defused by noticing that some conditionals will be non-true – there will come a point where we move from determinate cases of F to borderline cases of F, and the conditional whose antecedent predicates F of a determinate case while its consequent predicates F of a borderline case will itself be indeterminate. The objection from higher-order vagueness will, at this point, highlight the sharp boundary now drawn between the determinate Fs and the borderline Fs, deeming this as problematic as the shunned boundary between the Fs and not-Fs. The sorites series, so the objection goes, exhibits no such sharp boundary of any kind. While there are determinate Fs and borderline Fs, no two adjacent items in the series with respect to which F is soritical, an and an+1, are such that DFan yet IFan+1 – i.e. no pair of adjacent items is such that ν (Fam ) = 1 and ν (Fam+1) = @. Between any determinate F and borderline F there must be some item in the series, ae , say, that is a borderline case of DF and so neither determinately DF nor determinately not-DF. Consequently ν (Fae ) = W And similar reasoning objecting to the existence of a sharp boundary between borderline Fs and

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determinate not-Fs points to an item in the series, ag , say, such that ν (Fag) = $. No pair of adjacent items can differ in their truth-value status by @ and so no conditional involving adjacent members of the series is indeterminate. The objection is well founded but already contains within it the ingredients for a more adequate diagnosis of the sorites fallacy. Given any ¿nite sorites series, no pair of adjacent items will differ in their truth-value status by @, to be sure, but they will differ. If the series admits of borderline determinate Fs and borderline determinate not-Fs, as just postulated, then no conditional of the standard sorites will take value @ but some will take value W and thus not be true. If this assignment is again objected to by way of higher-order vagueness, undermining any supposed sharp boundary between the determinate determinate Fs and borderline determinate Fs by pointing to the existence in the series of an item that is a borderline case of DDF then this will simply generate an item ac such that ν (Fac ) = F, yielding a conditional which, to be sure, does not take value W but now can be seen to take value F. Each new objection from higher-order vagueness ipso facto points to an increasingly ¿ne-grained difference in the semantic status of adjacent items being considered and consequently points to the fact that the relevant conditional takes some value between 1 and that value postulated prior to the tabling of the new objection. In short, the more ¿ne-grained the variation is between adjacent items in the series, the closer the truth-value status of the relevant conditionals will be to 1 while remaining non-true (i.e. not equal to 1). A very coarse sorites series may move from determinate Fs to borderline Fs and then to determinate not-Fs with conditionals taking value @. A less coarse series will admit of ¿ner variation in truthvalue status of F-predications. Thus the more compelling we ¿nd the sorites, due to the more ¿ne-grained differences between adjacent items in the series with respect to F, the closer (at least some of) the standard sorites conditionals will be to taking value 1 without actually being true. Thus we are able to defuse the standard sorites in a manner that readily explains its capacity to appear paradoxical. The paradox is more and more compelling as the difference between adjacent items in the series with respect to F becomes less and less, and this is simply because the relevant conditionals more and more closely approximate true conditionals. Little wonder then that they are liable to deceive. We confuse conditionals whose truth-value status is very close to 1 with true conditionals. The solution to the mathematical induction form of the paradox, by way of the embedded conditional, is similarly solved – the universally quanti¿ed conditional is non-true, though it approximates truth ever more closely as the relevant difference between adjacent items in the sorites series diminishes. The linedrawing form is, as before invalid, for the reasons already stated in §1.2. The truth-value-gap approach to vagueness, a semantic phenomenon of ontological import, best explains the phenomenon. Moreover, despite persistent claims and arguments to the contrary, such an approach can be developed within a truth-functional framework provided by L∞.

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Index admissible delimitation 21 Akiba, K. 64n9, 89n14, 105, 120n25, 142 almost-identity 154n1 Alston, W. 6, 16, 18–19, 36 ambiguity, of count 163–66 see also count, vagueness of Analysis of Mind, The (Russell) 35 Aristotle 195 Arruda, A. 76, 94, 95, 96 Austin, J.L. 36 Axiom of Comprehension 175–76, 177 Axiom of Extensionality 175, 177–78 Barber paradox 13n14 Barnes, J. 9, 9n7, 11n9, 12, 57n4 Beall, J.C. 98, 100n27, 195 Benjamin, A.C. 66, 68 bivalence 13, 14, 76, 78, 81, 91 Black, M. 5, 40 borderline cases accuracy 44 ambiguity 40, 44, 45n8 borderline borderline cases 28, 29–30, 41n5, 197–98, 202 generality, and vagueness 40–41 high-order vagueness 27–28 predicate vagueness 43–45, 95 representation, one-manyness of 41–42, 43–44, 45 resilient 22–23, 131 speci¿city, lack of 40 structure-preserving correlation 42–43 what they are 3–6 Broome, J. 109n4 Burgess, J.A. 64n9, 105, 110n6, 113n15, 124n31, 127, 142, 143n45, 150, 150n50, 155–57, 157n5, 166, 189n6 Burks, A.W. 17, 66, 105 Burns, L. 30 Burnyeat, M.F. 9 Cargile, J. 14n15 Carnap, R. 52, 53, 54, 147

Church, A. 5 Colyvan, M. 98 composition material versus meriological 127–29 unrestricted mereological 143–44 vague 136–37 Cook, M. 110n6, 145n47 Copeland, J. 54n1, 109, 150, 150n50, 151 Copi, I. 11n10 count, vagueness of conditionality 169–71 count-ambiguity, non-toleration of 166 counting, ways of 163–66 examples 160–63 First Degree Entailment (FDE) 168 indeterminacy 163–64 Kleene’s strong logic K3 168 law of de¿nite difference (DD) 160–63 metaphysical vagueness 170 non-truth-functional approach 160 paracomplete approach 168 Pinillos Argument 164 principle for counting with indeterminacy (PCI) 162–63, 165, 177 principle of indeterminate identity (PII) 162, 177 reference, uniqueness of 166–68 semantics 168–70 supervaluationism 168–69 three-valued Lukasiewicz logic L3 168 truth-functional approaches 168–69 type-1 cases 160–61, 165 type-2 cases 161, 165 type-3 cases 160–63, 164–65 counterfactual sentences 25 Cowles, D.W. 157n4 ‘crispness postulate’ 54n1 Da Costa, N.C.A. 94 DD see law of de¿nite difference (DD) Deas, R. 29n24 de¿nitional reduction 59–60

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degree versus combinatory vagueness 16–17, 18 denoting phrases de¿nite descriptions 173 non-denoting phrases 23–24 predicate vagueness 46–47, 48 derivational reduction 60 descriptive completeness elimination, of vague terms 51–52, 54 scienti¿c language 54, 55, 58, 59, 68–70 Diogenes Laërtius 7 discussive implication 98n24 discussive logic 94–95, 98–99 Doria, F.A. 94 Duhem, P. 36 Dummett, M. 13, 25n20, 49, 64n9, 74, 76n5, 81, 105 Edgington, D. 86–87, 186 elimination, of vague terms comparative descriptions 53–54 descriptive completeness 51–52, 54 ideal language approach 51, 52–53, 54–59 reduction, against 59–60 supervenience, against 60–66 ‘what me worry’ response 52 Endicott, T. 8n6 Engel, R.E. 54 Eubulides of Miletus 8, 9, 56 Euclidian geometry 58, 141 Evans Argument 83, 109, 129, 168, 174 abstraction inference from (1) to (2) 115–16, 119, 126–27 designators, precise/imprecise 116–17 exposition of 114–15 reduction ad absurdum proof 114, 115, 122–23 Vague Identity Thesis 141–42 vagueness, de re and de dicto 117–20 Evans Proof 113, 144n46, 176, 177 abstraction 124–25 de dicto vague identities 155 de re vagueness 124 Grelling’s paradox 124 impredicativity, and paradox 124–25 law of de¿nite difference (DD) 120–21, 123 ▼ as dual of ▲ 121–22 Russell’s paradox 124

SpV, and weak paracomplete logic 123–24 existence, vague 137–39 Field, H. 168n13, 192 Fine, K. 3n2, 7, 74, 80, 82, 100, 101, 186, 187 First Degree Entailment (FDE) 168 ¿rst-order vagueness adjunction, disjunctive syllogism, ex falso quodlibet, validity of 183 classical inference, departures from 183 conditional proof 183–84 conditionals, de¿ning 181–82 consequence-relation, single conclusion 180 contraposition, failure of 184 determinacy operator, D 180 formal modelling of 206 identity, account of 184 law of de¿nite difference (DD) 184 law of non-contradiction (LNC) 182 law of self-identity (SId) 184 law of the excluded middle (LEM) 182 logical equivalence 181 modus ponens, validity of 183 negations, conjunctions, disjunctions, determination of 180 penumbral connections 185 quanti¿ed sentences, evaluation of 180 reductio ad absurdum, failure of 184 sorites paradox 185–86 supervaluationism (SpV) 181–82, 185 three-valued logic 179 truth-value-gap approach 179–80 validity, de¿ned as truth-preservation 180 four-dimensionalism 132, 133, 134–35 Frege, G. 12, 15, 155 Galen 57 Gallois, A. 113n13 Garrett, B.J. 119n21, 141 Geach, P.T. 74n1, 107 geometries Euclidian geometry 58, 141 non-Euclidian 141 Glymour, C. 69n16 Greenough, P. 15n16 Grelling’s paradox 124 Grim, P, 55n2

Index

221

de¿nite descriptions 172–73 Evans position 153 indeterminacy of count 155 naive principle of indeterminate reference (NPIR) 158–59 origins, indeterminacy of 154 Pinillos Argument 173–78 principle of indeterminate reference (PIR) 158–59, 162 Problem of the Many 153 van Inwagen’s principle (VIP) 155–56, 158–59, 159, 162, 166 inductive inference 57 infection theory, of vagueness 25–26 inheritance of precision 23–25 Inquiry into Meaning and Truth, An (Russell) 35

Haack, S. 12, 52, 54, 67 Hart, H.L.A. 8n6 Hawley, K. 137–39, 144n46, 176n20 Heck, R.G. 119n22, 122, 123n28 high-order vagueness 179, 186 accounts, mixed, of vagueness 32 borderline borderline cases 28, 29–30, 197–98, 202 borderline cases 27–28 classi¿cation, and precision 202–3 extending to L¥ ¿rst-order vagueness, formal modelling of 206 in¿nite-valued semantics 207–8 second-order vagueness 206–7 sorites paradox 208 ¿rst- and second-order vagueness 203–4 ‘gratuitous complications’ 29, 30 inheritance of determinacy principle 205 logical equivalence 203 metalanguage, vagueness in 198–99 natural language, cause/source of vagueness in 30–31 structure of 202–6 supervaluationism (SpV) 200–201 truth-value gap theory 200–201 truth-values, numbers of 197–201 vague predicates 30 vaguely vague predicates 29 vagueness, homological nature of 27–28 Hudson, H. 107 Humbersone, I.L. 189n6 Hyde, D. 28–29, 29n24, 95n18, 97n20, 97n22, 99n26

Kamp, J.A.W. 74 Keefe, R. 15n17, 30, 74, 74n1, 76, 80, 82, 83–84, 86, 87, 88, 90n15, 90n16, 91n17, 92, 99, 101–2, 108, 108n2, 126, 160, 168, 169, 179, 187, 190, 195, 198, 199, 200, 201 Kim, J. 63n8, 65, 65n13 Kleene’s strong K3 181, 182, 191 Kohl, M. 40 Körner, S. 67 Koslicki, K. 133n38 Kripke, S. 111n8, 155 Krips, H. 69n16

ideal language ‘deep structure’ 34–35 elimination, of vague terms 51, 52–53, 54–59 logical analysis of vagueness 49 science 54–59 identity, vague split indeterminate identity (SII) 111–14 vague diachronic identity 109–10 vague synchronic identity 110 vague transworld identities 110–11 see also Vague Identity Thesis individuation, and vagueness application vagueness of 18–19 de dicto vague identities 155 de re vague identity 153–54, 155–56, 160

L¥, and high-order vagueness ¿rst-order vagueness 206 in¿nite-valued semantics 207–8 second-order vagueness 206–7 sorites paradox 208 law of de¿nite difference (DD) 176, 184 count, vagueness of 160–63 Evans Proof 120–21, 123 ¿rst-order vagueness 184 law of non-contradiction (LNC) ¿rst-order vagueness 182 subvaluationism (SbV) 96, 103 supervaluationism (SpV) 86 law of self-identity (SId) ¿rst-order vagueness 184

Jackson, F. 64n11 Jaskowski, S. 73, 93, 94–95, 98–99

222

Vagueness, Logic and Ontology

law of the excluded middle (LEM) falsehood, intuition of 189 ¿rst-order vagueness 182 subvaluationism (SbV) 103 supervaluationism (SpV) 85–86, 87–88, 92–93 Leibniz’s Law 112, 118n19, 175 LEM see law of the excluded middle (LEM) Lewis, D. 25, 30, 64n10, 65, 74, 89n14, 106, 107, 108, 108n2, 110n7, 114, 117, 119n22, 119n23, 122, 133n38, 154n1, 163 linear versus multi-dimensional vagueness 17–18 LNC see law of non-contradiction (LNC) Lockwood, M. 69 logic, of vagueness ¿rst-order vagueness 179–86 high-order vagueness 179, 186, 197–209 paracomplete truth-functional approach, L3 179, 186 truth, and the T-schema 193–96 truth-functionality, objections to 186–93 Long, A.A. 9, 57n4 Lov-Warsaw School of Philosophy 73, 93 Lowe, E.J. 70, 108 Lukasiewicz’s L3 76, 94, 168, 181, 191–92 Margalit, A. 34n2, 49, 58n6 Marxist philosophy 94 McGee, V. 106, 110n7, 147 McGill, V.J. 94 McLaughlin, B. 106, 110n7, 147 meaninglessness 4 Mehlberg, H. 73 Menges, G. 7 mereological vagueness 127–33, 135–36, 142–44, 146 metalanguage, vagueness in 198–99 Mignucci, M. 9, 31n26 Miller, K. 136n40 Milosz, C. 94 modal vagueness (ModV) 135–36, 141 modi¿ers, vagueness of 24–25 modus ponens 12, 13, 75, 79, 96, 98, 100, 103, 182, 183 Mooreian gambit 57–58 Morreau, M. 64n9, 105, 108n2, 127, 134, 136, 142, 143, 143n45

Nagel, E. 60 naive principle of indeterminate reference (NPIR) 158–59 name-vagueness 21–23 natural language cause/source of vagueness in 30–31 classical logical theory 35 truth and falsity 74 vagueness 1 non-Euclidian geometries 141 Noonan, H. 119n23 objects, vague ‘immodest vague existence’ 138 material versus mereological composition 127–29 modal vagueness (ModV) 135–36 MV, and mereological vagueness 129–33 P, and mereological vagueness 128–29, 131 P′, and temporal vagueness 133 simplicity 140–41 spatial indeterminacy 127–28, 134–35 Special Composition Question 136, 138 vague composition 136–37 vague existence 137–39 van Inwagen’s ‘virus’ 138–39 On Medical Experience (Galen) 57 one-manyness designators, precise/imprecise 116–17 representation 41–42, 43–44, 45 resilient difference 46 supervaluationism (SpV) 76–77 ontological vagueness complete scienti¿c description 108–9 early proposals 105 Problem of the Many 107–8 reasons in favour of 106 representationalist approach 106–7 resistance to 106 vague identity 109–14 Vague Identity Thesis 141–47 vague objects 127–41 vague properties 147–50 vague states of affairs 149–50 vagueness ‘in the World’ or ‘of the World’ 150–51 verbalism, Russell’s fallacy of 105 see also Evans Argument

Index paracomplete truth-functional approach, L3 179, 186 Par¿t, D. 110n5 Parry, W.T. 94 Parsons, T. 64n9, 91n17, 105, 109n4, 110n5, 111–13, 124, 134, 142, 145n47, 149, 157n5, 160, 162, 168n13, 169, 170, 175, 175n18 PB see Principle of Bivalence (PB) PCI see principle for counting with indeterminacy (PCI) Peacocke, C. 53, 54, 61–63, 64n9 Peirce, C.S. 5 Pelletier, J. 195n11 Peña, L. 95 penumbral connections 76, 86, 102–3, 185 pervasiveness, of vagueness 7–8 Philosophical Foundations of Physics (Carnap) 54 physicalism 61 PII see principle of indeterminate identity (PII) Pinillos Argument 164 Axiom of Comprehension 175–76, 177 Axiom of Extensionality 175, 177–78 de re vague identity 173–74 set theory, and indeterminacy 174–75 split indeterminate identity (SII) 174, 176–78 Pinillos, N. 111n8, 162, 170n14, 171n16 PIR see principle of indeterminate reference (PIR) Plekhanov, G. 34n2, 94, 105 predicate incompleteness 5–6 predicate vagueness borderline cases 43–45 comparative descriptions 53–54 denoting phrases 46–47, 48 high-order vagueness 30 sentence vagueness 20–21, 22–23 simpliciter versus traditional 45–46 what it is 2, 4–5 Priest, G. 96, 98n24, 100n27 Principia Mathematica (Russell) 35 principle for counting with indeterminacy (PCI) 162–63, 165, 177 Principle of Bivalence (PB) 85, 91, 194–95 principle of indeterminate identity (PII) 162, 177 principle of indeterminate reference (PIR) 158–59, 162

223

Prior, A.N. 112, 175 Przelecki, M. 74 Putman, H. 59 quantum mechanics (QM), and descriptive completeness 68–70 Quine, W.V.O. x, 5, 13, 18, 35, 52, 54, 55, 57, 58, 59, 60–61, 67, 68, 69, 110 Redhead, M. 69 reduction, against 59–60 Relativity Theory 58 representational theory (Russell) ideal language, and ‘deep structure’ 34–35 logic 48–50 Marxist theory 34n2 natural language, and classical logical theory 35 ontological indeterminacy 33–34 representations, and vagueness 34 semantic phenomenon, vagueness as 33, 48–49 speci¿city, lack of 36–37 vagueness, Russell’s de¿nition of borderline-case vagueness 40–45 extension of 45–48 non-denoting phrases 46–47 one-manyness, and resilient difference 46 predicate vagueness 45–47, 48 representation, essential properties of 38–39 Rolf’s, interpretations 38–40 vagueness, source of 35–36 representationalism, descriptive appearance/reality distinction 51 de¿nition of 51 elimination, of vague terms 51, 52–59 representation/represented distinction 51 scienti¿c language, precision of 66–70 resilient borderline cases 22–23, 131 Restall, G. 91 Riley, T.J. 75 Rolf, B. 13, 14n15, 15, 17, 20–21, 23n19, 25, 33, 38–40, 40, 41, 59, 60, 64n9, 67, 68, 82n12, 88, 89, 105, 147n48, 148n49 Rosen, G. 105, 127, 147n48 Rosenberg, A. 25, 36 Routley, R. 98n24

224

Vagueness, Logic and Ontology

Russell, B. x, 5, 6, 12, 19, 25–26, 30, 31, 35, 67, 71, 73, 76–77, 78, 79n4, 83, 89, 106, 116, 117, 125 Sainsbury, R.M. 5, 82n12, 100n27, 116, 118n19, 120, 120n25, 127–28, 128n32, 129n34, 130, 131, 133, 141, 144n46, 147n48, 148n49, 150, 150n50, 153–54 Salmon, N,. 113n14 Sceptics 9, 18 scienti¿c language descriptive completeness 54, 55, 58, 59, 68–70 language regimentation programme 54, 55–57 ontological vagueness 108–9 precision conceptual innovation 67–68 empirical language, vagueness of 66–67 ostensively de¿ned terms 67 tolerance 67 Quinean defence, and cost-bene¿t equation 58, 59 Unger and Wheeler, sceptical arguments of 55–58 second-order vagueness 203–4, 206–7 Sedley, D.N. 9, 57n4 semantic phenomenon, vagueness as 3–4, 31–32, 33, 48–49 sentence vagueness 20–21, 22–23 set theory, and indeterminacy 174–75 Shapiro, S. 15 Shoesmith, D.J. 75 SID see law of self-identity (SId) Sider, T. 132n36, 133n38, 136n40 SII see split indeterminate identity (SII) Skala, H.J. 7 Smith, N.J.J. 105, 108, 127, 147n48 Sorensen, R. 2n1, 13, 27, 28, 29, 31, 37, 44n7, 59 sorites paradox classical logic 12–14 conditional premises 185–86 epistemic view, of vagueness 13 ¿rst-order vagueness 185–86 high-order vagueness 208–9 line-drawing sorites 14–15 logical puzzles 8–9 mathematical induction sorites 14, 186 modus ponens, validity of 185

multi-premise syllogisms 11n10 ‘no-item’ strategy 12 paracomplete logic L3 185 paradoxes 9–10 semantic view, of vagueness 31–32 sorites-of-decomposition example 55–57 subvaluationism (SbV) 97–98 supervaluationism (SpV) 80–85 tolerance 11 split indeterminate identity (SII) 174, 176–78 count, vagueness of 164–65 identity, vague 111–14 Leibniz’s Law 112 synchronous split indeterminate identity 111 Stability Condition 100 Stainton, R. 195n11 Stoics 9, 9n8, 18, 31 subvaluationism (SbV) adjunction, failure of 101–2 admissible precisi¿cation 95 classical logic 93, 96–97 consistency/inconsistency 93–94 determinacy/indeterminacy 100 discussive logic 94–95, 98–99 inconsistency, non-triviality of 94 indeterminacy, as overdetermination 95–96 law of non-contradiction (LNC) 96, 103 law of the excluded middle (LEM) 103 meaning, under-determination of 101 modus ponens, validity of 100–101, 103 multiple-conclusion consequence relation 97 paraconsistency 94–95, 96, 99–101 penumbral connections 102–3 precisi¿cations 100 sorites paradox 97–98, 102 SpV SbV duality 95, 99 strongly paracomplete approach 102–3 strongly paraconsistent approach 102–3 strongly paraconsistent logic 96 truth, gap-inducing notion of 101 truth-value gluts 93, 101 vague predicates, borderline cases 95 weakly paracomplete approach 102 weakly paraconsistent logic 96, 101–2 supervaluationism (SpV) acceptance/rejection patterns 91–92 admissible precisi¿cations 77–78

Index classical supervaluationism 74 denial/assertion distinction 91–92 extending (SpV+) determinacy/indeterminacy 88–89 imprecise designators 89–90 modal theory 89 sorites paradox 88 (super)truth, admission of into objectlanguage 88, 90 ¿rst-order vagueness 181–82, 185 high-order vagueness 200–201 incomplete logic 75 inference 92 law of non-contradictory (LNC) 86 law of the excluded middle (LEM) 76–80, 85–86, 87–88, 92–93 multiple-conclusion consequence relation 75n2 natural language, and truth and falsity 74 non-truth-functional disjunction 86–87 paracomplete logic 73, 75, 76 penumbral connections 76, 86 presuppositional languages 73 representationalism 80, 87–88 Russell’s ‘one-manyness’ of denotion 76–77 sorites validity, in defence of 80–85 T-schema (T), rejection of 195–96 tension, sources of 80, 85–86 three-valued logic 76 truth-value gaps 75, 91 weak paracomplete logic 76, 123–24 supervenience mental-physical relation 60–61 ontological precision 64 reducibility 63 see also elimination, of vague terms Tarski, A. 88, 193, 195, 196 Teller, P. 69 temporal vagueness 133 Thomason, R. 89n14, 116n17 three-dimensionalism 133, 134 three-valued Lukasiewicz logic L3 168 truth, and the T-schema De Morgan’s Laws 194 Principle of Bivalence (PB) 194–95 T-schema (T) 194, 195–96 T-schema (T*), weaker principle 196 TW, thinness of 196–98 Williamson Argument 194, 195

225

truth-functionality, objections to conditionals antecedents and consequents, indeterminate 191–92 classical logic, three-valued extension of 190–91 contested nature of 190 falsity, intuition of 192–93 material conditional, Gricean defence of 192 conjunction/disjunction conjunction 187–89 contrariness 187 De Morgan equivalence 189 disjunction 189 falsity, intuition of 188–89 Tye, M. 22, 105, 106, 127–28, 128n32, 129n33, 142, 143n45, 147n48, 169, 199 Unger, P. 13, 55–58, 107 Vague Identity Thesis de re vague identity 141, 146 Evans Argument 141–42 mereological identity 143 mereological precisi¿cation 143 mereological vague objects 142–43, 146 modally vague objects 141 naming, and precision 145 spatial vagueness 144–45 temporally vague persisting objects 146 transitivity 143 uniformity of indeterminate parts 143 unrestricted mereological composition 143–44 vagueness, what it is ambiguity 4, 5 borderline/penumbral cases 3–6 context-sensitivity 3, 4 extensional precision 3 extensional vagueness 3 higher-order vagueness 6–7 incomplete predicates 5–6 inexactness 2 intensional vagueness 3 meaninglessness 4 natural language, and vagueness 2 necessary precision 3 predicate vagueness 2, 4–5 semantic indeterminacy 3–4 sorites puzzles 5