Vagueness as Arbitrariness: Outline of a Theory of Vagueness (Synthese Library, 436) 3030667804, 9783030667801

Table of contents :
Preface
Contents
Chapter 1: Introduction
References
Chapter 2: Problem of Vagueness
2.1 Sorites
2.1.1 A Little Bit of History
2.1.2 Sorites (Conditional Version)
2.1.3 Sorites (Quantified Version)
2.1.4 Sorites (Line-Drawing Version)
2.1.5 Ways to Solve the Paradox
2.2 Criteria of Adequacy
2.2.1 Problem of Precisification
2.2.2 Three Criteria of Adequacy
2.2.3 The Foundational Problem of Precisification
2.3 Intuitions About Vagueness
2.3.1 How Not to Refute a Theory of Vagueness
2.4 Problem of Vagueness
References
Chapter 3: Imprecise Predicates
3.1 Criterion of Precisification
3.1.1 Naïve Many-Valued Picture of Imprecise Predicates
3.1.2 Hierarchical Higher-Order Imprecision
3.1.3 Imprecise Predicates
3.1.4 Criterion of Precisification
3.2 Epistemic Sense of Imprecision
3.2.1 Linguistic Imprecision Versus Epistemic Imprecision
3.2.2 Columnar Higher-Order Imprecision
3.2.3 Linguistic Imprecision Versus Epistemic Imprecision (II)
References
Chapter 4: Theories of Vagueness
4.1 Three-Valued Theory
4.1.1 Solution to the Sorites
4.1.2 Criteria of Adequacy
4.1.3 Intuitions
4.1.4 Further Objections
4.2 Degrees-of-Truth Theory
4.2.1 Solution to the Sorites
4.2.2 Criteria of Adequacy
4.2.3 Intuitions
4.2.4 Further Objections
4.3 Supervaluationism
4.3.1 Solution to the Sorites
4.3.2 Criteria of Adequacy
4.3.3 Intuitions
4.3.4 Further Objections
4.4 Other Theories
4.4.1 Epistemicism
4.4.2 Incoherentism
4.4.3 Contextualism
References
Chapter 5: Vagueness as Arbitrariness
5.1 Minimal Constraints
5.1.1 Tolerance: Problem of Unrestricted Application
5.1.2 Tolerance: Criterion of Precisification
5.1.3 Tolerance: Positive Case Against Unrestricted Application
5.1.4 Ideal Cases
5.1.5 Ideal Cases Versus Clear Cases
5.1.6 Three Kinds of Vague Predicates
5.1.7 Ideal Cases and Boundaries
5.2 Theory of Vagueness as Arbitrariness
5.2.1 Thesis of Arbitrariness
5.2.2 Sainsbury’s Many-Boundaries Approach
5.2.3 TA and Truth-Conditional Theories of Vagueness
5.2.4 Theory of Vagueness as Arbitrariness
5.2.5 Optimism, Pessimism and Nihilism
5.2.6 Nihilism and Pessimism
5.2.7 Expressivism: Do Vague Sentences Express Plans?
5.2.8 VA and Contextualism
5.2.9 Sorites: Rejecting the Principle of Tolerance
5.2.10 Arbitrariness Versus Tolerance
5.2.11 Intuitions
5.2.12 Some Advantages of VA
5.3 Objections
5.3.1 Do We Need a Clear-Case Constraint?
5.3.2 Counterexample to VA
5.3.3 Vague Sentences and Truth-Conditions
References
Index

Citation preview

Synthese Library 436 Studies in Epistemology, Logic, Methodology, and Philosophy of Science

Sagid Salles

Vagueness as Arbitrariness Outline of a Theory of Vagueness

Synthese Library Studies in Epistemology, Logic, Methodology, and Philosophy of Science

Volume 436 Editor-in-Chief Otávio Bueno, Department of Philosophy, University of Miami, Coral Gables, USA Editorial Board Berit Brogaard, University of Miami, Coral Gables, USA Anjan Chakravartty, University of Notre Dame, Notre Dame, USA Steven French, University of Leeds, Leeds, UK Catarina Dutilh Novaes, VU Amsterdam, Amsterdam, The Netherlands Darrell P. Rowbottom, Department of Philosophy, Lingnan University, Tuen Mun, Hong Kong Emma Ruttkamp, Department of Philosophy, University of South Africa, Pretoria, South Africa Kristie Miller, Department of Philosophy, Centre for Time, University of Sydney, Sydney, Australia

The aim of Synthese Library is to provide a forum for the best current work in the methodology and philosophy of science and in epistemology. A wide variety of different approaches have traditionally been represented in the Library, and every effort is made to maintain this variety, not for its own sake, but because we believe that there are many fruitful and illuminating approaches to the philosophy of science and related disciplines. Special attention is paid to methodological studies which illustrate the interplay of empirical and philosophical viewpoints and to contributions to the formal (logical, set-theoretical, mathematical, information-theoretical, decision-theoretical, etc.) methodology of empirical sciences. Likewise, the applications of logical methods to epistemology as well as philosophically and methodologically relevant studies in logic are strongly encouraged. The emphasis on logic will be tempered by interest in the psychological, historical, and sociological aspects of science. Besides monographs Synthese Library publishes thematically unified anthologies and edited volumes with a well-defined topical focus inside the aim and scope of the book series. The contributions in the volumes are expected to be focused and structurally organized in accordance with the central theme(s), and should be tied together by an extensive editorial introduction or set of introductions if the volume is divided into parts. An extensive bibliography and index are mandatory.

More information about this series at http://www.springer.com/series/6607

Sagid Salles

Vagueness as Arbitrariness Outline of a Theory of Vagueness

Sagid Salles Philosophy and Social Sciences Institute Federal University of Rio de Janeiro Rio de Janeiro, RJ, Brazil

ISSN 0166-6991     ISSN 2542-8292 (electronic) Synthese Library ISBN 978-3-030-66780-1    ISBN 978-3-030-66781-8 (eBook) https://doi.org/10.1007/978-3-030-66781-8 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my mother

Preface

Some years ago I was talking to a friend and said I found all traditional theories of vagueness equally implausible. He replied to the contrary that those theories were in fact equally plausible. These two claims can be viewed as equivalent. After all, two theories are equally implausible if and only if they are equally plausible. But we had different things in mind. I intended to express my pessimism about those theories, while he intended to express his optimism about them. Since the theories we were talking about are not mutually consistent, we could at least agree on one thing: there is a problem in deciding for any particular theory of vagueness. There are many different theories, and we couldn’t tell which was the best alternative. In this sense, the existence of many equally good theories is as problematic as the existence of many equally bad theories. This situation may be due to various different factors. First, there is no consensus about what kind of phenomenon vagueness is. It is not clear whether vagueness is primarily a linguistic, metaphysical or epistemic phenomenon. Second, although we may agree that the most salient feature of vagueness is that it appears to be the source of the sorites paradox, there is no agreement about how vagueness is to be initially defined. There have been many different intuitions about the nature of vagueness. These intuitions, which may not even be consistent with one another, lead to different solutions to the sorites paradox. Third, theories of vagueness usually have drastic consequences. It is as if there were a curse on theories of vagueness, so that sooner or later they lead to counterintuitive consequences. The upshot is that the context favors the emergence of many theories, while it makes it difficult to decide for any one of them. In any case, I came to the conclusion that the main source of the problem lies elsewhere. An ideal theory of vagueness should satisfy three criteria of adequacy, which I call the “criterion of sorites,” “criterion of coherence,” and “criterion of precisification.” These criteria emerge as soon as we begin to think about the sorites paradox, and they can be found either explicitly or implicitly throughout the history of the philosophy of vagueness. It turns out, however, that it is not clear how a theory of vagueness could jointly satisfy these criteria of adequacy. The result is that the main theories of vagueness violate at least one of them, and this was the source vii

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of my initial pessimism about them. These criteria play a central role in this book. First, they led me to a particular interpretation of the problem of vagueness as the problem of explaining vague predicates in a way that satisfies all three criteria of adequacy and systematizes the relevant intuitions about the nature of vagueness. Second, my main objection to Three-Valued Theory, Degrees-of-Truth Theory, Supervaluationism, Epistemicism, Incoherentism, and some versions of Contextualism is that they all fail to satisfy some of the relevant criteria. Finally, my main argument in favor of my positive account, which I call the “Theory of Vagueness as Arbitrariness,” is that it satisfies all three criteria. In sum, these criteria of adequacy are at the heart of the discussion presented in this book. In philosophy, theories are seldom completely rejected. We may reject some aspects of some theories while accepting others. My main objection to the aforementioned theories is that they fail to satisfy at least one criterion of adequacy, but this does not mean that they must be completely rejected. Even if we still do not have a totally satisfactory explanation of the phenomenon of vagueness, we can certainly say that much light has been cast on this phenomenon. Supervaluationism, for example, is based on correct intuitions about the nature of vagueness, especially the idea of semantic indecision. I think my positive account can be considered a suitable interpretation of this idea. Moreover, my interpretation of semantic indecision recognizes that two core theses advanced by contextualists are correct. The theory of vagueness presented in Chap. 5 may or may not be considered a version of Contextualism, depending on whether or not those two theses are sufficient for a theory to be contextualist. Nonetheless, my main influences in this book are certainly Sainsbury, Raffman, Braun, and Sider. The first part of my positive account, which I call the “Thesis of Arbitrariness,” incorporates some theses endorsed by Raffman in her Unruly Words and almost all the theses advanced by Sainsbury in his “Lessons for Vagueness from Scrambled Sorites.” Curiously, Sainsbury thinks his ideas commit him to reject the criterion of precisification. I disagree with him in this respect, but argue that the Thesis of Arbitrariness should be augmented with further assumptions in order to reach a final definition of vague predicates. The result is my Theory of Vagueness as Arbitrariness, which defines a vague predicate as an arbitrary predicate that must be precisified in order to contribute to a sentence that has truth-conditions. This theory implies Semantic Nihilism, which has been made a viable alternative by Braun and Sider’s “Vague, So Untrue.” Thus, my positive account is to a great extent based on the works of these philosophers. I am indebted to many persons. Thanks to Guido Imaguire, who supervised my doctoral thesis on vagueness, and Otávio Bueno, who kindly agreed to advise me during my stay at the University of Miami. In fact, this book is the result of several discussions with Imaguire, Bueno, and also Iago Bozza. I would also like to thank to Alessandro Bandeira Duarte, Célia Teixeira, Roberto Horácio de Sá Pereira, and Marco Ruffino for helpful comments on a first draft of this book. I have also benefited enormously from the remarks of an anonymous referee. Thanks to André Pontes and Juliana Faccio Lima for reading portions of this book and making numerous valuable suggestions. Many colleagues have helped me with insightful

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remarks in private discussions and conferences, especially Luiz Helvécio Marques Segundo, Pedro Merlussi, Rodrigo Reis Lastra Cid, Eduardo Pereira Monteiro, Gean de Melo da Silva, Frank Wyllys Cabral Lira, and Paula Caroline Costa de Oliveira. I am also grateful to James Stuart Brice for proofreading the text. This book would have been impossible without the support of CAPES.  Finally, I am especially indebted to my wife, Elizielly Martins, who has helped me during every stage of preparing this book. Rio de Janeiro, Brazil  Sagid Salles

Contents

1 Introduction����������������������������������������������������������������������������������������������    1 References��������������������������������������������������������������������������������������������������    8 2 Problem of Vagueness������������������������������������������������������������������������������    9 2.1 Sorites ����������������������������������������������������������������������������������������������    9 2.1.1 A Little Bit of History����������������������������������������������������������    9 2.1.2 Sorites (Conditional Version)������������������������������������������������   11 2.1.3 Sorites (Quantified Version)��������������������������������������������������   13 2.1.4 Sorites (Line-Drawing Version)��������������������������������������������   14 2.1.5 Ways to Solve the Paradox����������������������������������������������������   16 2.2 Criteria of Adequacy ������������������������������������������������������������������������   17 2.2.1 Problem of Precisification����������������������������������������������������   17 2.2.2 Three Criteria of Adequacy��������������������������������������������������   20 2.2.3 The Foundational Problem of Precisification ����������������������   22 2.3 Intuitions About Vagueness��������������������������������������������������������������   25 2.3.1 How Not to Refute a Theory of Vagueness��������������������������   29 2.4 Problem of Vagueness ����������������������������������������������������������������������   30 References��������������������������������������������������������������������������������������������������   30 3 Imprecise Predicates��������������������������������������������������������������������������������   33 3.1 Criterion of Precisification����������������������������������������������������������������   34 3.1.1 Naïve Many-Valued Picture of Imprecise Predicates�����������   34 3.1.2 Hierarchical Higher-Order Imprecision��������������������������������   41 3.1.3 Imprecise Predicates ������������������������������������������������������������   47 3.1.4 Criterion of Precisification����������������������������������������������������   52 3.2 Epistemic Sense of Imprecision��������������������������������������������������������   53 3.2.1 Linguistic Imprecision Versus Epistemic Imprecision����������   53 3.2.2 Columnar Higher-Order Imprecision������������������������������������   58 3.2.3 Linguistic Imprecision Versus Epistemic Imprecision (II) ��������������������������������������������������������������������   62 References��������������������������������������������������������������������������������������������������   64

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4 Theories of Vagueness������������������������������������������������������������������������������   65 4.1 Three-Valued Theory������������������������������������������������������������������������   66 4.1.1 Solution to the Sorites����������������������������������������������������������   70 4.1.2 Criteria of Adequacy ������������������������������������������������������������   72 4.1.3 Intuitions ������������������������������������������������������������������������������   77 4.1.4 Further Objections����������������������������������������������������������������   79 4.2 Degrees-of-Truth Theory������������������������������������������������������������������   81 4.2.1 Solution to the Sorites����������������������������������������������������������   83 4.2.2 Criteria of Adequacy ������������������������������������������������������������   85 4.2.3 Intuitions ������������������������������������������������������������������������������   93 4.2.4 Further Objections����������������������������������������������������������������   96 4.3 Supervaluationism����������������������������������������������������������������������������   98 4.3.1 Solution to the Sorites����������������������������������������������������������  101 4.3.2 Criteria of Adequacy ������������������������������������������������������������  102 4.3.3 Intuitions ������������������������������������������������������������������������������  110 4.3.4 Further Objections����������������������������������������������������������������  112 4.4 Other Theories����������������������������������������������������������������������������������  114 4.4.1 Epistemicism������������������������������������������������������������������������  114 4.4.2 Incoherentism������������������������������������������������������������������������  117 4.4.3 Contextualism ����������������������������������������������������������������������  120 References��������������������������������������������������������������������������������������������������  126 5 Vagueness as Arbitrariness ��������������������������������������������������������������������  129 5.1 Minimal Constraints�������������������������������������������������������������������������  130 5.1.1 Tolerance: Problem of Unrestricted Application������������������  132 5.1.2 Tolerance: Criterion of Precisification����������������������������������  135 5.1.3 Tolerance: Positive Case Against Unrestricted Application����������������������������������������������������������������������������  137 5.1.4 Ideal Cases����������������������������������������������������������������������������  140 5.1.5 Ideal Cases Versus Clear Cases��������������������������������������������  144 5.1.6 Three Kinds of Vague Predicates������������������������������������������  145 5.1.7 Ideal Cases and Boundaries��������������������������������������������������  147 5.2 Theory of Vagueness as Arbitrariness ����������������������������������������������  148 5.2.1 Thesis of Arbitrariness����������������������������������������������������������  148 5.2.2 Sainsbury’s Many-Boundaries Approach�����������������������������  153 5.2.3 TA and Truth-Conditional Theories of Vagueness����������������  159 5.2.4 Theory of Vagueness as Arbitrariness����������������������������������  161 5.2.5 Optimism, Pessimism and Nihilism��������������������������������������  167 5.2.6 Nihilism and Pessimism��������������������������������������������������������  172 5.2.7 Expressivism: Do Vague Sentences Express Plans? ������������  178 5.2.8 VA and Contextualism����������������������������������������������������������  180 5.2.9 Sorites: Rejecting the Principle of Tolerance������������������������  182 5.2.10 Arbitrariness Versus Tolerance����������������������������������������������  186 5.2.11 Intuitions ������������������������������������������������������������������������������  187 5.2.12 Some Advantages of VA ������������������������������������������������������  191

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5.3 Objections ����������������������������������������������������������������������������������������  193 5.3.1 Do We Need a Clear-Case Constraint? ��������������������������������  193 5.3.2 Counterexample to VA����������������������������������������������������������  195 5.3.3 Vague Sentences and Truth-Conditions��������������������������������  196 References��������������������������������������������������������������������������������������������������  198 Index������������������������������������������������������������������������������������������������������������������  201

Chapter 1

Introduction

The word “vague” is ordinarily used in various different ways. To begin with, suppose you ask me when Eubulides was born and I reply that he was born between the fifth and thirdth centuries B.C. It might be said that my information is vague, in the sense that it is not specific about the day Eubulides was born. In this case, “vague” means non-specificity (Eklund 2005: 27). Now, consider the sentence “Mary has only a vague memory of her childhood”. One who says this is usually saying that Mary does not have a clear memory of her childhood. In this case, “vague” is used to mean unclear. Notwithstanding the variety of uses of “vague”, here I am interested in one specific phenomenon. The most salient characteristic of this phenomenon is that it allegedly is the source of the sorites paradox. I turn next to an informal exposition of this paradox. Sorites  Suppose that the hair on the head of an arbitrary person, let us call him “John”, grows in a very specific way, one strand at a time. At a first moment, John has 0 hairs on his head, and then 1, 2, 3, 4, and so on. Now, consider the following sequence, each line representing a different moment. John has 0 hairs on his head. John has 1 hair on his head. John has 2 hairs on his head. (…) John has 5000 hairs on his head. John has 5001 hairs on his head. (…) John has 10,000 hairs on his head. Take the first line of the above sequence, which represents the moment John has no hair on his head, and ask yourself: is John bald at that moment? The obvious answer is “yes”. Now, turn to the second line and ask yourself the same question: is John bald at that moment? The answer is “yes”. Suppose you have to answer that question again and again, one line at a time. When will the answer change from © Springer Nature Switzerland AG 2021 S. Salles, Vagueness as Arbitrariness, Synthese Library 436, https://doi.org/10.1007/978-3-030-66781-8_1

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1 Introduction

“yes” to “no”? Intuitively, the addition of a single strand of hair is not sufficient to turn a bald person into a not bald person. If this is true, then you cannot stop answering “yes”, and hence you will have to say that John is bald at the moment when he has 10,000 hairs on his head. This result is absurd. Let us now consider the above sequence from the bottom up. Take first the last line, which represents the moment when John has 10,000 hairs on his head, and ask yourself: is John bald at that moment? The obvious answer is “no”. Turn next to line 9999. Is John bald at that moment? Again, the answer is “no”. If you keep doing this, when will the answer change from “no” to “yes”? Intuitively, the removal of just one strand of hair is not sufficient to turn a not bald person into a bald person. Thus, you cannot change your answer, and hence you have to say that John is not bald even when he has no hair. This result is also absurd. In the first case, we began with the uncontroversial supposition that a person with 0 hairs on his head is bald and ended with the unacceptable conclusion that at every step in the sequence – even when the person has 10,000 hairs – he is bald. In the second case, we began with the uncontroversial supposition that a person with 10,000 hairs on his head is not bald and reached the unacceptable conclusion that at each step in the sequence – even when he has no hair – he is still not bald. The kind of argument we used to go from those intuitive suppositions to the absurd conclusions is known as the sorites paradox. Together, the two arguments above can be used to show that the same person is bald and is not bald at the same time. Needless to say, this is absurd. Why does this matter? It should be noted that this is a paradox in both the weak and the strong senses of the term. In the weak sense, a paradox is an apparently valid argument with apparently true premises and an apparently false conclusion. Each direction of the above argument is a paradox in the weak sense. In the strong sense, a paradox is an apparently valid argument with apparently true premises and a contradictory conclusion. If we take jointly the two directions of the argument, then we have a paradox in the strong sense. Paradoxes are important because they show where something could work out wrong with either certain of our beliefs or certain of our inference rules. Philosophers try to solve the sorites paradox in order to discover what, if anything, can go wrong. Another reason for interest in the sorites paradox is that it is quite ubiquitous. We have seen a version of it for bald, but there are versions of the sorites for thin, fat, tall, short, big, small, mature, child, adult, rich, poor, strong, weak, etc. First, consider a version for tall. Let Mary be an arbitrary adult human being and think of the following sequence. Mary is 5 feet tall. Mary is 5 feet and 0.5 inches tall Mary is 5 feet and 1 inch tall. Mary is 5 feet and 1.5 inches tall (...) Mary is 6 feet and 5 inches tall.

1 Introduction

3

Take the first line of the above sequence and ask yourself: is Mary tall? The answer is “no”. Go to the second line: is Mary tall? Again, the answer is “no”. When will the answer change from “no” to “yes”? Intuitively, the addition of a mere 0.5 inches is not sufficient to turn a not tall person into a tall person. Consequently, you have to keep answering “no” and you will be forced to accept that Mary is not tall even when she is 6 feet and 5 inches tall. In other words, you will have to accept that at each step in the sequence she is not tall. Moreover, if you consider the sequence from the bottom up and apply the converse version of the sorites argument, the result will be that at each step in the sequence she is tall. The upshot is that at each step in the sequence she is both tall and not tall. Turn now to the following sequence. John is 0 years old. John is 0 years +1 second old. John is 0 years +2 seconds old. John is 0 years +3 seconds old. (…) John is 0 years +3,153,600,000 seconds old. (approximately 100 years old). We can easily formulate a version of the sorites to show that at each step of the above sequence John is a child and is not a child at the same time, is mature and is not mature at the same time, etc. This is to say that each of the aforementioned notions is sorites susceptible, because for each of them we can formulate a version of the sorites that takes us from intuitive premises to contradictions. The sorites paradox is a quite ubiquitous one. Things can get even worse. First of all, with a little bit of ingenuity we can formulate versions of the sorites paradox for notions that are not obviously vague. Graham Priest (2003: 9) formulated a version that proves that you are a scrambled egg, and Peter Unger (1979) applied the sorites to notions for ordinary objects, such as stone, table, lake, planet, etc. Moreover, there is a justified suspicion that vagueness and sorites affect some notions of central importance to philosophy, such as truth, identity, existence, justice, etc. Finally, the sorites paradox leads us to some very counterintuitive philosophical theses. It can be used to show that everyone is a child. If we add the plausible premise that children are not morally responsible for their actions, we end up with the very counterintuitive conclusion that no one is ever morally responsible. Vagueness probably affects most of our notions and surely leads us to unacceptable conclusions. Unfortunately, it is not easy to come up with an explanation of the phenomenon, not to say one that solves the sorites. What Is the Problem of Vagueness About?  The problem of vagueness is the problem of explaining the phenomenon of vagueness. It can be said that to explain this phenomenon is to answer questions such as “What is the nature of vagueness?” or “What does vagueness consist in?”. Vague notions are the source of the sorites paradox, and any account of them should be accompanied by a solution to the

4

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sorites.1 Nonetheless, there is more to be said. So far, it is not even clear whether the problem of vagueness concerns our language or the world. I have been using the phrase “vague notions”, instead of the more precise “vague expressions” or “vague objects/properties”, and I spoke about the notion of bald without specifying whether I was talking about the expression “bald” or about the property of being bald. It is not clear whether the problem of vagueness concerns expressions or objects/properties. At any rate, I will take the second path.2 From now on, the problem of vagueness will be understood as a problem concerning vague expressions. For the moment, it can be formulated as the problem of explaining what it means for a vague expression to be vague, in such a way as to solve the sorites paradox. It should be noted that in so understanding the problem I am not ruling out the possibility that vagueness is not (primarily) a property of expressions. I am not ruling out, for instance, the metaphysical conception of vagueness. According to the metaphysical conception, vagueness is (primarily) a feature of objects/properties. One who defends this conception could solve the above problem in two ways. (i) One could say that although there could be vague expressions, their vagueness is explained in terms of the vagueness of the relevant objects and properties; perhaps a vague expression is vague because it stands for a vague object/property. (ii) One could say that there could be nothing vague but objects and properties, and hence there is no problem of explaining what it means for an expression to be vague. One who defends the metaphysical conception of vagueness will either have to explain the vagueness of linguistic expressions in terms of the vagueness of objects/properties or reject the thesis that there could be vague expressions at all. Either way, we will end up with a solution to (or a dissolution of) the above formulation of the problem of vagueness. Similar observations apply to the epistemic conception of vagueness. According to the epistemic conception, vagueness is a matter of ignorance about where the boundaries of vague predicates lie. One who defends such a conception could either explain the vagueness of vague expressions in terms of epistemic vagueness or reject the thesis that there could be any vague expressions. Someone who chooses the first option might say that what makes a vague expression vague is precisely the fact that we are ignorant about where its boundary is. To sum up, neither the metaphysical nor the epistemic conception of vagueness is ruled out by a linguistic formulation of the problem. In any case, I intend to make it clear that the objects of this book are vague expressions, not vague properties or objects (if there are any). Vague Predicates  Even if we focus only on the case of vague expressions, the phenomenon of vagueness is too broad. After all, vagueness is not a feature of one very specific category of expressions, it rather affects different categories, for not only adjectives (“bald”, “tall”), but also nouns (“poverty”, “wealth”), verbs (“run”, “shout”), etc. can be vague. These different grammatical categories are arguably  I am not suggesting that a solution of the sorites must imply the rejection of its conclusion. See Sect. 2.1.5. 2  For criticisms of the metaphysical conception of vagueness, see Eklund (2008). For criticisms of the linguistic conception of vagueness, see Bacon (2018: Chap. 5). 1

1 Introduction

5

subsumed under the logical category of predicates. However, vagueness can also be found in other logical categories, such as singular terms and quantifiers. To begin with, let us consider the case of quantifiers. Although the universal and existential quantifiers do not seem to be vague, they can at least be derivatively vague. Roughly speaking, an expression is derivatively vague when it is vague by virtue of the vagueness of some other expression(s). The universal and the existential quantifiers can be vague by virtue of the vagueness of the predicates that specify their domain. Whether or not “every” is vague, “every bald person” is. Since we can use the sorites argument to show that each person is bald and is not bald at the same time, we can use the sorites to show that each person both is and is not an element of the domain of “every bald person”. It should be noted that this does not happen when we use a precise predicate to specify a domain for the quantifier. The sorites paradox cannot be formulated, for example, to show that each number both is and is not an element of the domain of “every natural number”. The difference between the first and the second case is that “bald”, but not “natural number”, is vague. The vagueness of “bald” results in the vagueness of “every bald person”. Thus, quantifiers can be at least derivatively vague. Moreover, there are also examples of quantifiers that are vague in more than just a derivative way. The adverbs “many” and “few” are possible examples of this. It seems that we can use them as quantifiers, given that we can use them to say that a property is instantiated many/few times by objects in a domain (“few oranges are rotten”, “many philosophers are smart”). However, such expressions are vague by their own merits and not by virtue of any other expressions. Let us now turn to the case of singular terms, that is, expressions that are meant to refer to a particular object. The indexical “here” is at least sometimes used as a vague singular term. Suppose John notices that his dog is sniffing a squirrel but he (John) can’t see where the squirrel is. In this context, John says (1). 1. There is a squirrel here. It seems that the indexical is used in (1) to pick out some specific place. But which specific place is the referent of “here”? One possible answer is that the referent of “here” in this case is the denotation of an allegedly precise definite description which John somehow associates with the indexical. In other words, the referent of “here” in (1) is fixed by a precise definite description that John associates with the indexical. Even if it is plausible to believe that the referent of “here” might be determined in such a descriptive manner, it is fair enough to assume that it does not have to be so determined. In any event, let a be the referent of “here” in (1), and suppose it is not determined in such a descriptive manner. It is intuitive that (1) is true if and only if there is a squirrel in a. Now think of the following sequence: The squirrel is 1 foot away from John. The squirrel is 1 foot and 0.5 inches away from John. The squirrel is 1 foot and 1 inch away from John. (…) The squirrel is 30,000 miles away from John. Etc.

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Take the first line of the above sequence and ask yourself: is (1) true in that case? The answer is “yes”. This implies that when the squirrel is 1 foot away from John, it is in a. Turn next to the second line: is (1) true in that case? Once again, the answer is “yes”, and hence when the squirrel is 1 foot and 0.5 inches away from John, it is in a. If you keep asking the same question, when will the answer change from “yes” to “no”? Intuitively, you will never be justified in changing your answer, and hence you will have to keep answering “yes”. This entails that independently of the distance between John and the squirrel, you will have to accept that the latter is in a. This ultimately means that a must be a very large place, perhaps the entire Earth or even the Universe. This is an absurdity. There are also proper names that seem to be vague. One example of this is “Mount Everest”. What exactly is the referent of “Mount Everest”? Suppose someone unrealistically tries to answer this question by drawing a line L surrounding what she claims to be the referent of the name. We could then reply that if “Mount Everest” refers to the object surrounded by L, then it also refers to the object surrounded by L’, where L’ is a line that is just slightly larger than L and that surrounds L. After many steps, this will lead to the conclusion that “Mount Everest” refers to the entire Earth or even the Universe. This is absurd. The upshot is that not only predicates or general terms, but also quantifiers and singular terms, can be vague.3 These are very different kinds of expressions, each one posing specific problems for a general theory of vagueness. The situation is similar to that of theories of singular reference, those that explain the reference of singular terms. The existence of different kinds of singular terms makes it difficult to come up with a general theory for them. This happens because the explanation of the reference of one kind of singular term might have to involve elements that the explanation of the others does not have to involve. For instance, the explanation of the reference of indexicals, such as “today” and “I”, has to accommodate the fact that their referents can shift from context to context. According to a standard view, (pure) indexicals are associated with a rule (their linguistic meaning or character) that determines a referent for each context of use; the rule for “I”, for instance, determines that its referent in each context is the speaker in that context. By  In any case, predicates provide us with the most intuitive example of vagueness. In fact, it is easier to formulate the sorites paradox for predicates than for quantifiers or singular terms. A version of the sorites for a name like “Mount Everest” would sound more artificial than a version for a predicate like “bald”. Because of this, there may be some suspicion about whether other logical categories of expressions really can be vague. Besides, even if we accept that vagueness affects different logical categories of expressions, there seem to be conflicting intuitions with respect to them. For example, it might be said that while the vagueness of predicates (and perhaps quantifiers) is intuitively linguistic, the vagueness of singular terms is intuitively metaphysical. The name “Mount Everest” does not seem to involve any problem: it refers to one, and only one, entity. If this name is vague, this is arguably because the entity to which it refers is itself vague. Although I can accept that the intuition with respect to proper names favors a metaphysical conception of vagueness, I do not think the same applies to all singular terms. It does not sound plausible to say that the indexical “here”, in the aforementioned context, is vague only because it refers to a vague entity. Come what may, neither the metaphysical conception of vagueness nor the category of singular terms will be discussed throughout this book. 3

1 Introduction

7

contrast, the explanation of the reference of proper names does not have to involve any such rules, for their referents do not shift from context to context. Just as the existence of different kinds of singular terms is an obstacle for a general theory of singular reference, the existence of different kinds of vague expressions is an obstacle for a general theory of vagueness. To be sure, the case of vagueness is even worse, for vague expression is a broader and more heterogeneous notion than singular term. In light of this, I will restrict my investigation to the case of vague predicates, leaving aside expressions of any other categories. I do hope, however, that my proposal can be further enlarged to account for all other kinds of vague expressions. Problem of Vagueness and the Theory of Vagueness as Arbitrariness  From what I said above, the problem of vagueness can be understood as the problem of defining what a vague predicate is. This is still a provisional formulation, since I will dedicate Chap. 2 to a more precise formulation of this problem. At any rate, one of my main goals in this book is to define vague predicates. The starting point of my positive account is the Thesis of Arbitrariness (TA). This thesis provides an interpretation of the following intuition: all admissible precisifications of a vague predicate are equally arbitrary. Although this intuition is already well known, it is sometimes stated as if it were only an alternative formulation of the principle of tolerance. As far as I know, Supervaluationists were the first to accept it in its own sense, so as to make an important contribution to the philosophy of vagueness. Nonetheless, I believe that even supervaluationists did not take this intuition as seriously as they should have. In order to properly understand it we have to understand what it means for an admissible precisification to be arbitrary. I do this by distinguishing two senses of arbitrariness: semantic arbitrariness and pragmatic arbitrariness. The result is in line with Sainsbury’s Many-Boundaries Approach (Sainsbury 2013) and with some of the main theses advanced by Raffman (2014). I do not think TA tells the whole story about vagueness. On the contrary, we need to augment it with further assumptions in order to reach a final definition of vague predicates. The result is the Thesis of Vagueness as Arbitrariness (VA). VA defines a vague predicate as an arbitrary predicate that must be precisified (and hence be made not vague) in order to contribute to a sentence that has truth-conditions. VA implies that vague sentences do not have truth-conditions, and this commits me to Semantic Nihilism. Braun and Sider’s account of how vagueness can be harmlessly ignored in ordinary contexts has made Semantic Nihilism a viable alternative (Braun and Sider 2007). Yet, Braun and Sider’s proposal implies the existence of a clear-­ case constraint. For reasons that will become clear later, I think this is an undesired consequence. By virtue of this, I propose some modifications in their account in order to avoid this negative consequence. All in all, my positive account of vague predicates will appear only in Chap. 5. Before discussing any positive account, we need a more precise formulation of the problem of vagueness. I propose this formulation in the next chapter. This formulation will be at the heart of my further discussion. On the one hand, my main argument against some alternative theories of vagueness is that they fail to solve this

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problem in an adequate manner (Chap. 4). On the other hand, my main argument in favor of VA is that it adequately solves the problem of vagueness as interpreted here. So, let us begin by examining the problem more carefully.

References Bacon, A. (2018). Vagueness and thought. Oxford: Oxford University Press. Braun, D., & Sider, T. (2007). Vague, so untrue. Noûs, 41(2), 133–156. Eklund, M. (2005). What vagueness consists in. Philosophical Studies, 125(1), 27–60. Eklund, M. (2008). Deconstructing ontological vagueness. Canadian Journal of Philosophy, 38(1), 117–140. Priest, G. (2003). A site for sorites. In J. Beall (Ed.), Liars and heaps (pp. 09–23). Oxford: Oxford University Press. Raffman, D. (2014). Unruly words: A study of vague language. Oxford: Oxford University Press. Sainsbury, M. (2013). Lessons for vagueness from scrambled sorites. Metaphysica, 14(2), 225–237. Unger, P. (1979). There are no ordinary things. Synthese, 41(2), 117–154.

Chapter 2

Problem of Vagueness

2.1  Sorites 2.1.1  A Little Bit of History If we believe the famous biographer of Greek philosophers, Diogenes Laertius, Eubulides of Miletus (fourth cent. B.C) really deserves to be considered the past master of paradox (Burnnyeat 1982: 315). According to the biographer (D.L.  II, 108), Eubulides framed not only the sorites paradox, but also many others, such as the well-known liar paradox. The word “sorites” derives from “soros”, a Greek word that means the same as “heap”; and the name “sorites paradox” has to do with the fact that it was first formulated for the notion of heap. There are doubts as to whether Eubulides really understood how general and serious the consequences of the sorites were. In fact, this paradox may have been initially formulated as an argument concerning the specific notion of heap, and it is not clear when exactly we became conscious of its generality and power. All in all, it is safe to say that the sorites became well known in antiquity. The Stoics, for instance, paid considerable attention to it. Chrysippus wrote Of the Sorites Argument as applied to Uttered Words, in three books, and, in two books, Of the Argument from Small Increments. The argument from small increments is more often called the “little-by-little argument”, which, in turn, is just another name for the sorites argument (D.L.  VII, 192 and 197). Sextus Empiricus (Outlines of Pyrrhonism, II, 253) and Galen (On Medical Experience XVI-XVII) also thought about the sorites paradox; the latter being perhaps the most remarkable discussion that has come down to us. Moreover, it did not take long for it to become known that the sorites can be applied to other notions than heap. Eubulides applied the sorites argument to the notion of bald and, in a handbook of Stoic logic, it was applied to few. It seems clear that Galen was aware of the fact that the sorites affects a wide variety of notions, and he does not treat this as something new.

© Springer Nature Switzerland AG 2021 S. Salles, Vagueness as Arbitrariness, Synthese Library 436, https://doi.org/10.1007/978-3-030-66781-8_2

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2  Problem of Vagueness According to what is demanded by the logos, 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 flock of sheep and herd of cattle, the nation and the crowd. And the doubt and confusion introduced by the logos leads to contradiction of fact in the transition of man from one stage of his life to another, and in the changes of time, and the changes of seasons. For in the case of the boy one is uncertain and doubtful as to when the actual moment arrives for his transition from boyhood to adolescence, and in the case of the youth when he enters the period of manhood, also in the case of the man in his prime when he begins to be an old man. And so it is with the seasons of the year when winter begins to change and merges into spring, and spring into summer, and summer into autumn. (Galen, On Medical Experience XVI, p.114–115)

The reason why the logos entails that there are no such things as heaps of grain, mountains, strong love, cities, etc. is that the sorites argument shows that all these notions are incoherent. In short, there has been a growing recognition of the generality of the sorites paradox. The earliest formulations of sorites, however, were usually different from today’s most common ones. Apparently, it was not originally formulated as a paradox (Hyde 2011: 11), but as a puzzle formed by a set of questions. A beautiful example of such a formulation can be found in Galen: Wherefore I say: tell me, do you think that a single grain of wheat is a heap? Thereupon you say: No. Then I say: What do you say about 2 grains? For it is my purpose to ask you questions in succession, and if you do not admit that 2 grains are a heap then I shall ask you about 3 grains. Then I shall proceed to interrogate you further with respect to 4 grains, then 5 and 6 and 7 and 8, and you will assuredly say that none of these makes a heap. Also 9 and 10 and 11 grains are not a heap. (Galen, On Medical Experience XVII, p.115).

It should be noted that, in this formulation, the sorites is not a paradox, it is not an apparently valid argument with apparently true premises and an apparently false or contradictory conclusion; it is not an argument at all. Nonetheless, this is a very hard puzzle to solve. Intuitively, in Galen’s succession of questions, there is no adjacent pair that is such that one could answer “no” to the first question and “yes” to the second. One who did this would be committed to the thesis that a single grain of wheat is sufficient to turn a not heap into a heap or, in other words, that there is a sharp boundary between things that are heaps of wheat and those that are not. This would be an absurdity. Moreover, one could not answer all questions with “no” (“yes”), because this would result in obvious falsehoods. In short, it is far from clear how to solve this puzzle. Whether or not the sorites first appeared as a puzzle, it appeared as a genuine paradox already in antiquity. In the aforementioned handbook of Stoic logic, it was formulated as a well-structured argument. It cannot be that if two is few, three is not so likewise, nor that if two or three are few, four is not so and so on up to ten. But two is few, therefore so also is ten. (DL, VII, 82).

In fact, it is possible to argue, as Barnes (1982: 28–29) did, that with the above formulation the Stoics had in mind what we will further call the “conditional version of the sorites” (see Sect. 2.1.2). Mignucci (1993) claims that this formulation is also

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11

at least implicit in Galen. At any rate, the aforementioned puzzle clearly lies at the origin of what is currently known as the sorites, and it does not take much to turn it into an argument. In the introduction, for example, I began with a puzzle to formulate informal versions of the paradox. In any case, today we have many different and precise formulations of the sorites, which means that there are different versions of it. These different versions might differ in their premises, conclusion or in the inference rules employed in the derivation. This is important because we want to solve all versions of the sorites, but that probably is not easy to do. Suppose one tries to solve the sorites by rejecting a specific inference rule. If there is a version of the paradox that does not depend on this rule, then the solution will not apply to it. In this case, we would need different solutions to different versions of the sorites. Of course the ideal would be a theory that provides us with a uniform solution, but, in principle, we do not have any guarantee that this is even possible. In the next three sections, I present three versions of the sorites paradox.

2.1.2  Sorites (Conditional Version) A person with 0 hairs on her head is bald. If a person with 0 hairs on her head is bald, then a person with 1 hair on her head is bald too. Therefore, a person with 1 hair on her head is bald. If a person with 1 hair on her head is bald, a person with 2 hairs on her head is bald too. Therefore, a person with 2 hairs on her head is bald… Repeat this process 10,000 times and you will end up with the absurd conclusion that a person with 10,000 hairs on her head is bald. We can easily formulate an argument in the converse direction. A person with 10,000 hairs on her head is not bald. If a person with 10,000 hairs on her head is not bald, then a person with 9999 hairs on her head is not bald too. Therefore, a person with 9999 hairs on her head is not bald. If a person with 9999 hairs on her head is not bald, a person with 9998 hairs on her head is not bald too. Therefore, a person with 9998 hairs on her head is not bald… Repeat this process 10,000 times and you will end up with the absurd conclusion that a person with 0 hairs on her head is not bald. The above arguments are instances of the conditional version of the sorites paradox. Let “B” be the predicate “bald”, “a” an arbitrary name of a person, and let the numbers on the lower right indicate the number of hairs on the head of a. The argument, in both directions above, can be formulated as follows:

12 (SB) Ba0 Ba0 → Ba1 Ba1 Ba1 → Ba2 Ba2 (…) Ba9, 999 → Ba10, 000 Ba10, 000

2  Problem of Vagueness (¬SB) ¬Ba10, 000 ¬Ba10, 000 →  ¬Ba9, 999 ¬Ba9, 999 ¬Ba9, 999 →  ¬Ba9, 998 ¬Ba9, 998 (…) ¬Ba1 →  ¬Ba0 ¬Ba0

Each argument begins with what seems to be an indisputable premise: Ba0 in the first case and ¬Ba10,000 in the second one. This first premise is usually called the “categorical premise”. The arguments also contain a set of conditional premises, which, for obvious reasons, are called “conditional premises”. All other premises are obtained by means of Modus Ponens, and I follow Mignucci (1993: 236) in calling them “intermediate premises”. The conditional version of the sorites is formed by one categorical premise and several conditional ones, the conclusion (and the intermediate premises) being derived by Modus Ponens. In this formulation, the sorites is a genuine paradox and not a mere puzzle. If we consider each column in isolation, then it is an apparently valid argument with apparently true premises and an apparently false conclusion. If we consider jointly the two columns, then we have an apparently valid argument with apparently true premises and a contradictory conclusion. The upshot is that, for any person, we can use an argument of this kind to show that she is bald and also to show that she is not bald. The predicate “bald” is incoherent. The conditional premises seem to be especially problematic, for it is not clear where they come from. The categorical premise is a very plausible assumption, while each intermediate one is derived from the categorical and a conditional premise, or from a conditional and an intermediate premise, by Modus Ponens. But where do the conditional premises come from? More precisely, what justifies the introduction of the conditional premises? To answer this question, we should look at what would happen if we denied one of the conditional premises. Suppose, for example, that the conditional Ba5,000 → Ba5,001 of (SB) is false. This implies that Ba5,000 is true and Ba5,001 is false: Ba5,000 ʌ ¬Ba5,001. In other words, a person with 5000 hairs is bald and a person with 5001 hairs is not bald. This is absurd. It does not seem that the addition of a single strand of hair can make a bald person into a not bald one. The idea that someone could be made not bald by the addition of a single strand of hair is hard to swallow. It should be noted that the above problem has nothing to do with the particular conditional we chose to deny. Had we chosen any other conditional premise of (SB) as an example, the problem would remain the same. This is what justifies the set of conditional premises of (SB): to deny any of them would amount to the absurd conclusion that the addition of a single strand of hair can make a bald person into a not bald one. Similarly, what justifies the conditional premises of (¬SB) is that to deny

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any of them would amount to the absurd conclusion that removing a single strand of hair is sufficient to make a not bald person into a bald one. Crispin Wright (1975: 334) has famously expressed this point by saying that predicates like “bald” are tolerant. A predicate is tolerant when very slight changes respecting its standard(s) of application cannot make a difference in its application. If the predicate applied to an object before the change, then it will still apply to the object after the change; and if it did not apply to the object before the change, it will not apply to the object after the change. At least one standard for the application of “bald” is the number of hairs on the head. As we have seen, very slight differences with respect to the number of hairs on a person’s head do not make a difference in the application of the predicate “bald” to her. If a person is bald when she has n hairs on her head, then she is bald when she has n + 1 hairs on her head; and if she is not bald when she has n hairs on her head, then she is not bald when she has n – 1 hairs on her head. The addition or removal of a single strand of hair does not alter the justice – to use Wright’s expression – with which the predicate is applied. We can represent the tolerance of “bald” with the following principle:

( PTB) : ∀n ( Ban → Ban +1 )



This means that, for every natural number n, if a person with n hairs on her head is bald, then a person with n + 1 hairs on her head is bald too. This principle justifies the conditional premises of (SB). In fact, each conditional premise of (SB) is an instance of this principle, while each conditional of (¬SB) is an instance of the logically equivalent ∀n (¬Ban + 1 → ¬Ban). Finally, recall that the sorites paradox is not restricted to the notion of bald. The conditional version can be easily formulated for a wide range of other notions, such as heap, fat, thin, short, many, few, shout, etc. Each of them can be shown to be incoherent by means of the conditional version of the sorites.

2.1.3  Sorites (Quantified Version) Once we have seen the principle of tolerance, we can formulate a simpler and more direct version of the sorites paradox for the predicate “bald”. Here are two instances of it: (SB) (1) Ba0 (2) ∀n(Ban → Ban + 1) (3) Ba10, 000

(¬SB) (1) ¬Ba10, 000 (2) ∀n (¬Ban + 1 →  ¬Ban) (3) ¬Ba0

In less formal language, here is what the above version of (SB) says: 1. A person with 0 hairs on her head is bald.

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2. For every natural number n, if a person with n hairs on her head is bald, then a person with n + 1 hairs on her head is bald too. 3. A person with 10,000 hairs on her head is bald. (1) is the categorical premise of each case. This time, however, we do not have a set of conditional premises, but just (2), which is the principle of tolerance. The conclusion is derived by successive applications of Universal Instantiation and Modus Ponens. Once again, we have an apparently valid argument with apparently true premises and an apparently false or contradictory conclusion. The quantified version of the sorites, as well the conditional version, affects a wide variety of ordinary predicates. Here is an informal example for “heap of salt”. 1 . 0 grains of salt is not a heap. 2. For every natural number n, if n grains of salt are not a heap, then n + 1 grains of salt are also not a heap. 3. Therefore, 10,000 grains of salt are not a heap. This version of the sorites has some advantages over the previous one. It is not only simpler than the conditional version, but also makes it explicit how the argument depends on the principle of tolerance. While in the conditional version this principle is only an implicit justification for the conditional premises, in the quantified version it appears as a premise of the argument. Because of this, I will take the quantified version of the sorites as the standard one, mentioning the others directly only when this is relevant to the discussion.

2.1.4  Sorites (Line-Drawing Version) I will follow Hyde (2011: 14) in calling the next version of the sorites the “line-­ drawing version”. Without further ado, here is an example for “bald”. (1) Ba0 (2) ¬ Ba10, 000 ____________ (3) ∃n (BanΛ ¬Ban + 1)

In this version, the sorites is a paradox in the weak sense of the term. The idea of the argument is very simple. The first premise says that a person with 0 hairs on her head is bald. The second says that a person with 10,000 hairs on her head is not bald. Intuitively, both (1) and (2) are true, but this seems to imply that there is a number n such that a person with n hairs is bald and a person with n + 1 hairs is not bald. This conclusion is clearly false. The principle of tolerance does not appear in the line-drawing version of the sorites, not even implicitly. We do not have to assume that “bald” is tolerant in order to accept that Ba0 and ¬Ba10,000 imply that there is a sharp boundary between bald

2.1 Sorites

15

and not bald persons. Yet the conclusion of this version is logically equivalent to the negation of the principle, so it is arguable that it is because we intuit that the latter is true that the conclusion sounds absurd to us. Nonetheless, not everyone would agree that the highly counterintuitive aspect of (3) should be explained in terms of the highly intuitive aspect of the principle of tolerance. In fact, it would be plausible to suggest that (3) is by itself counterintuitive. Wright (2009: 530–531) went one step further and claimed that the thesis that vague predicates do not have sharp boundaries is a more fundamental intuition than the thesis that they are tolerant. Taking “bald” as an example, this means that ¬Ǝn (Banʌ ¬Ban + 1) is a more fundamental intuition than ∀n (Ban → Ban + 1). If this is correct, we should avoid appealing to the principle of tolerance to explain why the conclusion of the line-drawing version of the sorites sounds so absurd. I will come back to the relevant intuitions about vagueness at the end of this chapter. In short, we have seen four versions of the sorites, three of them being a genuine paradox. The conditional and the quantified versions can be a paradox in both the strong and weak senses of the term, while the line-drawing version is a paradox only in the weak sense. Although these are the most common versions, they are by no means the only ones.1 Nevertheless, in this book I will focus on the four above mentioned versions and, more specifically, on the quantified one. One reason for this is that to take all versions of the sorites into account would make each step of the discussion too complex. Moreover, there are very specific motivations for

 There is a common version of the sorites that will not be considered here. An instance of this version for the predicate “bald” would be like this: 1

1 . The number 0 is such that every person with that number of hairs on her head is bald. 2. For every natural number n, if n is such that a person with n hairs on her head is bald, then n + 1 is such that a person with n + 1 hairs on her head is bald. 3. Every natural number n is such that a person with n hairs on her head is bald. This is the inductive version of the sorites. It is formed by a base premise, (1), and an inductive step, (2). The conclusion is derived by Mathematical Induction. The base premise states that a certain property holds for the number 0 – the property of being such that any person with that number of hairs on her head is bald. The inductive step states that if that property holds for a natural number n, then it holds for its successor n + 1. From this it follows that the property holds for any natural number, which means that any natural number is such that a person who has that number of hairs on her head is bald. Although this version of the sorites is common, it sounds a bit artificial, since we are talking about numbers with odd properties such as to be such that every person with that number of hairs is bald. It is not clear that numbers can have such properties. Moreover, theories that solve the quantified version of the sorites will usually supply us with a solution to the inductive version too. For example, Epistemicism solves the quantified version by denying the quantified premise, and hence it will solve the inductive version by denying the inductive premise. Three-Valued Theory, in turn, rejects the truth of the quantified version, though it does that without implying that this premise is false. As a consequence, it solves the inductive version by rejecting its inductive premise. Degrees-of-Truth Theory renders the quantified version not completely true, and the argument invalid. In the same way, the inductive premise of the inductive version would not be completely true, and the argument would not be valid. For present purposes, we do not need to go into details here. The solutions of these theories for the three aforementioned versions of the sorites will be discussed in Chap. 4.

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discussing certain versions, which involve problems that are far beyond my goals in this book. Dummett (1975) and Wright (1975, 1976), for example, paid special attention to observational predicates and the notion of indiscriminability, so as to give rise to particular versions of the sorites related to these specific issues. A more detailed presentation of different versions of sorites can be found in Hyde (2011).

2.1.5  Ways to Solve the Paradox A paradox is an apparently valid argument with apparently true premises and an apparently false or contradictory conclusion. There are three ways to solve a paradox: (a) Reject one of its premises. (b) Reject one of the inference rules employed in the derivation of its conclusion. (c) Accept its conclusion. Let us take an instance of the quantified version of the sorites as an example: (SB) (1) Ba0 (2) ∀n (Ban → Ban + 1) _______________ (3) Ba10, 000

If we adopt strategy (a), we have to reject (1) or (2) of the above argument. The rejection of (1) entails the absurdity that a person with no hair is not bald. I will return to premise (2) later on, so let us put it aside for now. If we adopt strategy (b), we have to reject Modus Ponens or Universal Instantiation. Since these seem to be very basic inference rules, this path is not better than the previous one. Furthermore, not all versions of the sorites paradox employ Modus Ponens and Universal Instantiation. The line-drawing version does not employ any of these rules, and the conditional version does not employ Universal Instantiation. Finally, strategy (c) only could be taken as a last resort. The remaining option is to return to strategy (a) and reject premise (2) of the argument. In other words, we could reject the principle of tolerance for “bald”. On the one hand, this principle is not as intuitive as the belief that a person with no hair is bald. On the other hand, rejecting this principle does not seem as drastic as rejecting Modus Ponens, Universal Instantiation or accepting the conclusion of the sorites. Even though the principle of tolerance is intuitive, its rejection seems to be more plausible than any other possible path. The rejection of the principle of tolerance also provides us with a solution to the other versions of the sorites paradox. The solution to the conditional version would be to reject some conditional premise. Given that each conditional premise is an

2.2  Criteria of Adequacy

17

instance of the principle of tolerance, if the principle is false, some conditional premise must be false too. The solution to the line-drawing version would be to accept its conclusion. Given that the conclusion of this version is the negation of the principle of tolerance, and the latter is false, the former must be true. At first sight, the rejection of the principle of tolerance is the most plausible way to solve the sorites paradox. Unfortunately, even this option faces a serious problem. It is with this problem that we begin the next section.

2.2  Criteria of Adequacy 2.2.1  Problem of Precisification Think of predicates like “bald”, “heap”, “tall”, “big”, etc. Intuitively, these predicates do not establish sharp divisions between objects to which they apply and ones to which they don’t. This is what differentiates them from precise predicates like “even” and “odd”. Obviously, a vague predicate could be precisified. For example, we may estipulate that “bald” applies to a person if and only if she has at most n hairs on her head, thereby establishing a sharp division between persons to whom this predicate applies and ones to whom it doesn’t. In principle, nothing prevents us from doing this. Similarly, economists may determine that “poor” applies to all and only all persons whose annual income is at most n dollars. Without such a precisification it would be more difficult to evaluate people’s situation and public policies. Yet we would not classify the results of such stipulations as vague predicates, we would not say that “bald” and “poor” are vague predicates in the aforementioned contexts. Rather, we would classify them as precise predicates. The intuition under consideration here is that a predicate is vague only if it is not precise. The notions of being vague and being precise are inconsistent with each other, so that nothing can be vague and precise at the same time. It may help to think in terms of soritical sequences. According to Barnes (1982: 31), a sequence is soritical, relative to an ordered set of subjects  and a predicate F, if it satisfies the following three conditions: (i) Fa0 seems to be true (false); (ii) Fan seems to be false (true); (iii) given any adjacent pair of subjects in the sequence, ai and ai + 1, it seems that either both Fai and Fai + 1 are true or both are false. Now consider the following sequence:

a0 , a1 , a2 ,⊃ , a5,000 , a5,001⊃ , a10,000

If we let “a” be an arbitrary name of a person, and the numbers on the lower right indicate the number of hairs on the head of a, then the above sequence is a soritical sequence with respect to “bald”. The predicate “bald” applies to a0 and does not apply to a10,000. Furthermore, it seems that for any adjacent pair of subjects in the sequence, “bald” either applies to both or applies to neither of them. In other words,

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there is no sharp division between cases to which “bald” applies and cases to which it doesn’t. If there were such a division, “bald” would be a precise predicate, not a vague one. Let us in turn suppose that “a10,000” names a clear sample of heap formed by 10,000 grains of wheat, “a9,999” names the result of removing a single grain of a10,000, and so on. In this case, the predicate “heap of wheat” applies to a10,000 but not to a1. However, there is no sharp division between cases to which “heap of wheat” applies and cases to which it doesn’t. If there were such a division, “heap of wheat” would be a precise predicate, not a vague one; and the same goes for all other vague predicates. Again, it is possible to precisify “bald”, “heap” or any other vague predicate by establishing a sharp boundary between objects to which they apply and ones to which they don’t. But the resulting precisified predicates would be no longer vague. Nothing can be vague and precise at the same time. The notions of precise and imprecise predicate will be accurately defined only in the next chapter. For the moment, I want to draw attention to the following two intuitions: (i) a predicate is vague only if it is imprecise, and (ii) a predicate is imprecise only if there is no sharp boundary between objects to which it applies and ones to which it does not apply. Intuitions (i) and (ii) point to at least one reason why it is difficult to solve the sorites paradox by denying the principle of tolerance. We have seen that the initially most plausible strategy to solve the paradox is to reject the principle of tolerance; any other strategy being harder to accept. Unfortunately, a closer examination will reveal that the rejection of this principle implies that vague predicates are precise predicates; and this also seems to be absurd. Consider, once again, the principle for “bald”: ∀n (Ban → Ban + 1). If this principle is false, then at least one of its instances is false. Suppose this instance is Ba5,000 → Ba5,001, so that the antecedent of this conditional is true and the consequent is false. This would entail that a person with 5000 hairs on her head is bald and a person with 5001 hairs on her head is not bald; which means that there is a sharp boundary between bald persons and not bald ones. Those persons who have at most 5000 hairs are bald, those who have more than 5000 are not. As a consequence, “bald” is a precise predicate, a predicate that establishes a sharp division between persons to whom it applies and ones to whom it doesn’t. But if “bald” is precise, then it is not vague. Therefore, we cannot deny the principle of tolerance for “bald” without rejecting that “bald” is vague, and the same holds for all other vague predicates. Consequently, neither the quantified version of the sorites can be solved by simply denying the quantified premise, nor the conditional version can be solved by denying some conditional premise.2 A further consequence is that we cannot solve the line-drawing version of the sorites by simply accepting its conclusion (Ǝn (Banʌ  ¬Ban  +  1)), which is logically equivalent to the negation of the principle of  This problem was partially anticipated when we thought about the reason why it would be implausible to deny any conditional premise of the conditional version of the sorites. We have seen that the rejection of any conditional premise would commit us to the counterintuitive thesis that vague predicates have sharp boundaries. We can see now that the problem goes beyond the fact that this thesis is counterintuitive. The main problem is that the rejection of any conditional premise would amount to the conclusion that the relevant predicate is a precise predicate, not a vague one. 2

2.2  Criteria of Adequacy

19

tolerance (¬∀n (Ban → Ban + 1)). The conclusion of the line-drawing version says that there is a number n such that a person with n hairs on her head is bald and a person with n + 1 hairs on her head is not bald; which means that there is a sharp boundary between persons to whom “bald” applies and those to whom it doesn’t.3 Again, the result is that “bald” is a precise predicate, not a vague one. There have been many theories trying to solve the sorites by rejecting the principle of tolerance. When opting for one of those theories, one may just bite the bullet and reject the claim that vague predicates are imprecise. Curiously, however, some of these theories have been proposed as a way to reconcile the rejection of the principle of tolerance with the idea that vague predicates are imprecise. This is certainly the case of Three-Valued Theory, Degrees-of-Truth Theory and Supervaluationism. Despite this, Horgan (1994) has plausibly accused all them of treating vague predicates as if they were precise predicates, and in Chap. 4 I will argue for a similar conclusion. On the one hand, the rejection of the principle of tolerance seems to be the initially most plausible way to solve the paradox. On the other hand, it entails that vague predicates are precise predicates and thus are not really vague. The upshot is that there is no clear way to solve the paradox. The problem of explaining vague predicates without implying that they are precise predicates – that is, without precisifying them – is (at least part of) what Terence Horgan (1994: 162) called the “problem of arbitrary precisification”. For my purposes in this book, the shorter expression “the problem of precisification” will be better.4 As we have seen, the relevant intuition concerning the imprecision of vague predicates is that all vague predicates are imprecise; that is, a predicate is vague only if it is imprecise. Whether or not a vague predicate can be precisified, the intuition is that after a precisification it is no longer vague. Along these lines, the problem of

3  It should be noted that in itself Ǝn (Ban ʌ ¬Ban + 1) does not imply that there is a boundary between persons who are bald and those who are not bald, in the sense that there is a pair of adjacent subjects such that the first is the last bald person and the following is the first not bald person in the relevant sequence. In order to see this, consider the sequence of natural numbers. There plainly is a number n such that n is odd and n + 1 is not odd, but from this it does not follow that there is a pair of adjacent numbers such that the first is the last odd number and the following is the first not odd number in the sequence. When I say that such a consequence follows from Ǝn (Ban ʌ ¬Ban + 1), I am implicitly assuming some obvious constraints on the application of “bald”. About these constraints, see Sect. 5.1. I thank Iago Bozza for this point. 4  I prefer the shorter expression for two reasons. First, I suspect that what Horgan called the “problem of arbitrary precisification” is a mix of two different problems, which I respectively call the “problem of precisification” and the “foundational problem of precisification”. Given that I treat these problems separately, I give them different names. Second, in order to argue for this problem, Horgan appeals to at least three different intuitions about vagueness, which I will respectively call the “lack of a sharp boundary intuition”, “no fact of the matter intuition” and “arbitrariness of the boundary intuition” (see Sect. 2.3). It seems that the third one justifies the inclusion of “arbitrary” in “the problem of arbitrary precisification”. In fact, it is not clear whether Horgan does or does not differentiate these intuitions; for from page 162 to 163 he speaks of all of them as if they were the same. Nonetheless, I do differentiate them, and I believe that the problem of precisification is more intimately related to the lack of a sharp boundary intuition, while the foundational problem of precisification is more intimately related to the arbitrariness of the boundary intuition.

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precisification may be understood as follows: the problem of explaining vague ­predicates without implying that there could be a predicate that is vague and precise at the same time. This problem is more difficult than it may seem at first sight. Again, in Chap. 4 I will argue that some theories that are supposed to solve this problem indeed fail in this task. Contrary to what may appear to be the case, ThreeValued Theory, Degrees-of-Truth Theory and Supervaluationism do not meet this goal. In fact, I think this is one of the most pressing problems concerning vagueness. In any case, the difficulty of reconciling the rejection of the principle of tolerance with the imprecision of vague predicates might suggest that we should look for alternative ways to solve the sorites paradox. One option would be to maintain the principle of tolerance and the conclusion that vague predicates are incoherent. Horgan (1994), Wright (1975, 1976) and Eklund (2005) believed that the principle is at least in some sense correct and that vague predicates are indeed incoherent. Notwithstanding the fact that this kind of solution is initially implausible  – for, despite the sorites, vague predicates do not seem to be incoherent – it might be the only way to avoid the problem of precisification. In Chap. 5, however, I will develop a theory of vagueness that reconciles the rejection of the principle of tolerance with the imprecision of vague predicates.

2.2.2  Three Criteria of Adequacy Theories of vagueness must explain the phenomenon of vagueness. It is desirable that they do this in a way that solves the sorites paradox. After all, this paradox is the most salient characteristic of the phenomenon. Moreover, it is desirable that they not imply the incoherence of vague predicates. The conclusion that “bald”, “heap”, “tall”, “mature”, “big” and so many other predicates are incoherent sounds really counterintuitive, not to say plainly absurd. Finally, it is desirable that they not precisify vague predicates, not imply that vague predicates are precise or that a predicate could be vague and precise at the same time. All this leads us to three criteria of adequacy for an ideal theory of vagueness. (a) Criterion of sorites: explain vague predicates in a way that solves the sorites paradox. (b) Criterion of coherence: explain vague predicates without implying that they are incoherent predicates. (c) Criterion of precisification: explain vague predicates without implying that there could be a predicate that is vague and precise at the same time. It is important to note that these criteria do not arise from sophisticated theorizing about vagueness; they come up as soon as we begin to think about the sorites and the initial ways to solve it. In addition, each criterion stated above has been present in the literature about vagueness ever since ancient Greek and Roman philosophy, either explicitly or implicitly. The two first criteria are more intuitive and, I suppose, no one is likely to doubt that it would be desirable for any theory of

2.2  Criteria of Adequacy

21

vagueness to satisfy (a) and (b). But there could be some suspicion about the criterion of precisification. To be sure, I think no one would deny (c) as it stands above. But I have interpreted this criterion in terms of the intuition that a vague predicate admits of no sharp boundary between objects to which it applies and ones to which it doesn’t. In so doing, I am assuming a linguistic sense of imprecision. (This linguistic sense of imprecision is presupposed also in the next section). Nonetheless, this sense of imprecision gives rise to at least two questions. First, it is not totally clear how exactly we should understand it. As we will see, there are different interpretations on the table, and they are not equally plausible. Second, there are alternative and non-linguistic interpretations of the notion of imprecision, and one might appeal to this fact in order to claim that this notion should not be linguistically interpreted. I will address these problems in the next chapter. For the time being, suffice it to say that vagueness indeed seems to be a phenomenon concerning boundaries and, more precisely, lack of sharp boundaries. As we have seen, this is part of the reason why it is not so easy to solve the paradox by denying the principle of tolerance. Furthermore, the intuition that vague predicates do not establish sharp boundaries has a long history. One example of this is the previously quoted Stoic formulation of the sorites: It cannot be that if two is few, three is not so likewise, nor that if two or three are few, four is not so and so on up to ten. But two is few, therefore so also is ten. (DL, VII, 82).

It is clear that the intuition that there is no sharp boundary between numbers to which “few” applies and those to which it doesn’t is at least implicit in this formulation of the sorites. Galen also accepts this kind of intuition, as we can see in the following passage: If you do not say with respect to any of the numbers, as in the case of the 100 grains of wheat for example, that it now constituted a heap, but afterwards when a grain is added to it, you say that a heap has now been formed, consequently this quantity of corn became a heap by the addition of the single grain of wheat, and if the grain is taken away the heap is eliminated. And I know of nothing worse and more absurd than that the being and not-being of a heap is determined by a grain of corn. (Galen, On Medical Experience XVII, p.116).

Galen speaks in metaphysical (rather than linguistic) terms, and he is probably exaggerating when he says “I know nothing worse and more absurd than that the being and not-being of a heap is determined by a grain of corn”. At any rate, there is no doubt he would consider absurd the idea that vague predicates establish sharp boundaries between things to which they apply and things to which they don’t. Of course he would not deny that one could precisify “heap” or many other vague predicates by stipulating a sharp boundary for them. There would be no absurdity in such stipulations. A plausible suggestion is that the absurdity lies in the idea that any predicate of a certain class – in modern terms, the class of vague predicates – is precise. If this is correct, Galen would agree that no vague predicate is precise, that a predicate is vague only if it is imprecise. It therefore comes as no surprise that a violation of the criterion of precisification is usually believed to be a problem for any theory of vagueness. While some believe that a violation of this criterion is a

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sufficient condition for rejecting a theory of vagueness (Horgan 1994; Wright 1975, 1976; Tye 1994: 193; Sainsbury 1990; Fodor and Lepore 1996: 523; etc.), others believe it can be tolerated. Even among the latter group, however, it is usually believed that the burden of proof rests on whoever violates the criterion; that is to say, any violation of it requires justification (Fara 2000; Priest 2003: 11; etc.).5 I do not believe that a violation of (a)–(c) is unacceptable in the sense that it is a sufficient condition for rejecting a theory of vagueness. We do not even know whether it is possible to jointly satisfy (a)–(c). As we will see in Chap. 4, this is not easy to do, and some of our main theories of vagueness fail in this respect. When I said that these are criteria for an ideal theory of vagueness, I meant that it is desirable that any theory not violate them. The violation of any of these criteria will be seen as a disadvantage to the theory, and the satisfaction of any of them will be seen as an advantage to it. In this context, any theory of vagueness that satisfies (a)-(c) will have an important advantage over those that don’t.

2.2.3  The Foundational Problem of Precisification The criterion of precisification was so far based on the intuition that a predicate is vague only if it is not precise, that nothing can be vague and precise at the same time. It is possible for a theory to violate this criterion even if it does not imply that some existing vague predicate is precise. It is possible to claim that although no actual vague predicate is precise, this is a mere accident. Any theory committed to such a claim would violate the criterion of precisification, thereby rejecting the above intuition. In general, however, theories that violate this criterion not only imply the possibility of some predicate being both vague and precise, they imply that some, many or all existing vague predicates are indeed precise predicates. (I return to this in Chap. 4, where I consider whether or not, and in which sense, different theories of vagueness violate this criterion). Theories that have this latter consequence involve a stronger violation of the criterion of precisification, and in so doing they give rise to what I call the “foundational problem of precisification”: if some, many or all existing vague predicates are precise, them it will be difficult to explain how their boundaries are determined. This problem provides us with a partial argument in favor of the criterion of precisification. I say “partial” because it provides us with a reason to avoid the stronger violation of the criterion, and holds only for those theories that violate, in this stronger sense, the relevant criterion.

 Of course I am not suggesting that those philosophers explicitly say that they accept the criterion of precisification, for they do not use this expression. In any case, those of the first group do think it is not acceptable to treat vague predicates as if they were precise predicates, that vagueness “cannot be reconciled with any precise dividing lines” (Tye 1994: 193). Those of the second group accept the weaker thesis that although it is acceptable to treat vague predicates as precise predicates, it is not desirable. 5

2.2  Criteria of Adequacy

23

Partial or not, however, in Chap. 4 we will see that this argument presents a challenge to many traditional theories of vagueness. Many natural language predicates apply to at least some objects. There are objects to which they apply and objects to which they do not apply. An example of this is the predicate “bald”. This predicate applies to persons who have 0 hairs on their head, but not to persons who have 10,000 hairs on their head. We all can agree that this is not a matter of magic. We do not utter sentences containing these predicates and they magically do or do not come to apply to things. There must be some mechanism determining success (or failure) in the application of “bald” and other vague predicates. This must be explained by a theory of reference or a theory of application for vague predicates. A theory of this kind should explain how the aforementioned mechanism works, how it determines that a predicate applies to whatever it applies to. Suppose that at least some existing vague predicates are precise, that there is a sharp boundary between objects to which they apply and objects to which they don’t. This sharp boundary must be determined by the referential mechanism  – whatever it is – of vague predicates. We can now ask how this mechanism works. Those who believe that some vague predicates are precise predicates have to explain how their sharp boundaries are determined. But it is not clear how one could do that. What makes it especially difficult to explain this is that for any vague predicate, there will be many cut-off points with an equal right to the status of the boundary of that predicate. Suppose one suggests that “bald” applies to a person if and only if she has at most n hairs on her head. We can express this by saying that the boundary of “bald” is in n (n is the last case to which “bald” applies in the kind of soritical sequence we discussed in Sect. 2.2.1). In this case, we could easily ask how the relevant mechanism determined n instead of, for example, n + 1 or n – 1, as the boundary of “bald”. In short, why n instead of n + 1 or n – 1? What is so special about the former? It is difficult to see how one could plausibly answer this question. The same problem arises for any vague predicate. In conclusion, those who violate (in the strong sense of violation) the criterion of precisification have to explain how the sharp boundaries of the relevant vague predicates are determined, and this is no easy task. It is worth illustrating the point in a more detailed manner. It is not difficult to imagine how one could determine a sharp boundary for a vague predicate. This could be done, for example, by means of stipulations. One could simply stipulate that a person is bald if and only if she has at most n hairs on her head; or that a person is bald if and only if she has at most the same number of hairs as John, etc. It is hard to believe, however, that the boundary of “bald” is, in general, determined in this manner. Even though we could determine sharp boundaries for vague predicates by means of stipulations, we usually do not do this. The relevant problem here is how the boundaries of the relevant vague predicates are actually determined, not how they could be determined. A natural response to this problem would be as follows. The predicate “horse” applies to horses, and only to them. In other words, “horse” applies to things that have the property of being a horse, and only to them. Since all horses, and only all

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horses, have the property of being a horse, we can now say that this property determines the extension of “horse”. Similarly, one could suggest, the predicate “bald” applies to all persons who have the property of being bald, and only to them. This means that the boundary of “bald” is determined by the property of being bald. The core idea is that the boundary of a vague predicate F is determined by the property of being F. The main problem with this explanation is that there are many properties with an equal right to be the property of being bald, and any choice of a particular property would be arbitrary. There is the property of having at most 0 hairs on the head, the property of having at most 1 hair on the head, the property of having at most 2 hairs on the head, and so on. Why would one of them be a better candidate for the property of being bald than all others? In order for that view to be plausible, there must be something that makes one of those properties special with respect to all others. Unfortunately, it seems that for any property you choose, there will be some others that seem to have an equal right to be the property of being bald. One could go a step further and complete the above view with a Causal Theory of Reference, such as that propounded by Putnam (1975) and Devitt and Sterelny (1999). Roughly speaking, causal theories say that a (use of a) predicate F applies to x if and only if there is an appropriate causal link between x and (the use of) F. Things seem to work well for words like “water”. At some moment in time, someone pointed to an amount of water and introduced the word “water” to speak about that amount and everything that has the same nature. After that, the word was diffused from person to person, forming an extensive causal chain related to that amount and everything with the same nature. Given that the nature of that first amount was its chemical composition, H2O, the predicate “water” (as used by a member of the relevant causal chain) applies to that amount and everything with the same chemical composition, H2O. According to causal theories of reference, the introduction of a predicate involves two elements: an ostensive and a nature element (Devitt and Sterelny 1999: 88). In order to introduce the predicate “water”, for example, someone first had to be in perceptual contact with an amount of water (this is the ostensive element). Yet this is not enough, for this predicate was not introduced to speak only about that specific amount, it was not a proper name of that amount. It was rather introduced to speak of that amount and everything with the same nature as it (this is the nature element). As a consequence, if this mechanism is supposed to work, then there must be a common nature shared by all things to which the predicate “water” applies. Can we use this theory to explain the sharp boundaries of vague predicates? Let us say that, at some moment in time, someone pointed to a bald person and introduced the predicate “bald” to speak about that person and everyone with the same relevant property. After that, the predicate “bald” was diffused from speaker to speaker, forming an extensive causal chain related to the original person and everyone with the same relevant property. The relevant property, of course, is the property of being bald. Uttered by a member of this chain, “bald” applies to all, and only all, persons who have the property of being bald. Whether or not this sounds plausible

2.3  Intuitions About Vagueness

25

with respect to “water”, I will argue that it does not work for vague predicates. In fact, this view is susceptible to the same objection that I laid down above. If Causal Theory is supposed to work for “bald”, then there must be a property that has a special role in the determination of the boundary of “bald”, the property of being bald. Nonetheless, there are many properties with an equal right to be the property of being bald, and any choice of a particular property would be arbitrary. We can clarify this by thinking of the following sequence. property of having at most n – 1 hairs on the head property of having at most n hairs on the head property of having at most n + 1 hairs on the head

Whatever the value of n is, ask yourself: what would make one of the above properties more relevant than the others to the determination of the boundary of “bald”? In line with Causal Theory of Reference, how could one of them have a special causal role in the chain of users of “bald”? We need some reason to believe that one of them is special with respect to the others. What seems to be easy with respect to “water” is very difficult with respect to “bald” and other vague predicates. To sum up, anyone who violates (in the strong sense) the criterion of precisification has to accept that some existing vague predicates are precise predicates, which means that they have sharp boundaries. Those who accept this have to explain how the alleged sharp boundaries are determined. Therefore, anyone who violates this criterion has to explain how the alleged sharp boundaries of vague predicates are determined. We have seen that this is a difficult problem. My goal here was not to show that it is impossible to solve this problem, but only to highlight the fact that there is a problem and it is a difficult one. Before moving on, it is important to keep in mind the difference between the problem of precisification and the foundational problem of precisification. The former is the problem of explaining vague predicates without implying that a predicate could be both vague and precise, and it rests on the intuition that a predicate is vague only if it is not precise. This is a problem for any theory of vagueness, and it is a reason for the criterion of precisification. Not all theories that violate this criterion raise the foundational problem. This latter problem arises only for those theories that imply that some existing vague predicate is indeed precise, and so it may be considered a partial reason in favor of the criterion of precisification.

2.3  Intuitions About Vagueness Hitherto, the phenomenon of vagueness has been mostly related to the sorites paradox. Each criterion of adequacy for an ideal theory of vagueness emerged in the context of this paradox and its general solutions. We have seen that an ideal theory of vagueness should explain vague predicates in a way that solves the sorites (the criterion of sorites), without implying that vague predicates are incoherent (the criterion of coherence) and, finally, without implying that a predicate could be vague

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and precise at the same time (the criterion of precisification). It seems clear that vagueness is intimately related to the sorites paradox and that (the search for) a solution to the latter is at least a strong motivation to look for a theory of vagueness. It should be noted, however, that the discussion of vagueness goes beyond the issue of the sorites. The discussion of vagueness, like most philosophical discussions, begins with initial characterizations of the phenomenon to be explained. Unfortunately, there have been many different initial characterizations of vagueness, and it is not even clear whether they are mutually consistent. This is a problem because, as far as we know, these different initial characterizations might not capture the same phenomenon. Besides, even if they all capture the same phenomenon, we still do not know which of them are correct. After all, it is possible that some initial characterizations capture only less important aspects of the phenomenon, leaving aside its most important features. These initial characterizations represent the most common intuitions about what vagueness consists in. Let us say that an intuition A is fundamental with respect to the phenomenon of vagueness if any theory of vagueness should accommodate A. Furthermore, A is more fundamental than an (fundamental) intuition B if B is explained in terms of A. Any theory of vagueness has to say something about the relevant intuitions. Should some of them be rejected? If yes, why should they be rejected? What are the (most) fundamental intuitions? In this section, I will present a generic formulation of six common intuitions about the nature of vagueness. The formulation is generic because the relevant intuitions will receive a rigorous formulation only in the context of theories of vagueness, and different theories often involve different formulations of them. Thus, the exposition here must be generic enough to allow the same intuition to be differently interpreted by different theories. Finally, each intuition will be formulated as an intuition about language, that is, about vague predicates; but nothing prevents them from being interpreted as intuitions about the world, that is, about vague properties. (i) Lack of a Sharp Boundary Intuition We have already seen this intuition throughout our discussion. According to it, vague predicates do not have sharp boundaries. More precisely, they do not establish a sharp division between objects to which they apply and ones to which they don’t. In Sect. 2.2.2 I claimed that this intuition has a long history. It is at least implicit in some passages of Galen’s On Medical Experience and also in a handbook of Stoic logic quoted by Diogenes Laertius. More recently, Keefe (2000: 153) and Tye (1994: 193) took it as a fundamental intuition about vagueness, while Horgan (1994) and Wright (2009: 530) took it as the most fundamental one. (ii) No Fact of the Matter Intuition There is no fact of the matter about where the boundary of a vague predicate lies. It is not clear what that means. It might be interpreted as an alternative way to say that vague predicates do not have sharp boundaries. In this case, however, this intuition could not be the most fundamental, because it would be explained in terms of the lack of a sharp boundary intuition. We also might understand the non-existence of a fact of the matter about the location of the boundary in terms of indeterminacy of the boundary.

2.3  Intuitions About Vagueness

27

The proponents of the open future, for example, defend the view that it is now indeterminate whether it will rain tomorrow. Similarly, one could claim, the location of the boundary of any vague predicate is in some sense indeterminate. Yet, the kind of indeterminacy involved in vagueness should be different from that concerning the open future, for the latter is not related to the sorites paradox (Williams 2008: 767). Keefe (2000: 153) and Tye (1994: 193) took this intuition as fundamental, while Field (2003: 457) took it as the most fundamental one. (iii) Tolerance Intuition This is another intuition we have already seen throughout our discussion. According to it, vague predicates are tolerant to very slight changes. In other words, very slight changes do not make a difference in the application of vague predicates. The addition of a single strand of hair does not make a bald person into a not bald person, and the removal of a single strand of hair does not make a not bald person into a bald one; the addition of a single grain of wheat does not make a not heap of wheat into a heap, just as the removal of a single grain does not make a heap of wheat into a not heap. This intuition also has a long history, being explicit or implicit in most discussions of vagueness. It was Wright (1975, 1976), however, who developed the notion of tolerance and, at that moment, took it as the most fundamental intuition about vagueness. (iv) Borderline Cases Intuition This intuition has not yet been considered here, but it probably is one of the most important intuitions about vagueness. The core idea is that vague predicates admit of borderline cases, which means that there are cases to which it is not clear whether the predicate does or does not apply. Think of the predicate “bald”. It clearly applies to a person who has no hair, and it clearly does not apply to a person who has 10,000 hairs on her head. But what should we say about a person who has 80 percent of her scalp covered by hair? Does “bald” apply to this person? Since there is no clear answer to this question, this person is a borderline case of “bald”. Many philosophers believe that the intuition that vague predicates admit of borderline cases is fundamental or even the most fundamental. The classical example is Kit Fine (1975), another example is M. Richard (2009: 465–467). (v) Arbitrariness of the Boundary Intuition For any vague predicate, there are many cut-off points with an equal right to the status of the boundary of that predicate. For any cut-off point one chooses as the boundary of a vague predicate, we can ask: why this cut-off point instead of any other? Given that there seems to be no plausible answer to this question, any choice will seem totally arbitrary. We can sum this up by saying that all precisifications of a vague predicate are equally arbitrary. Supervaluationism is usually based on this intuition, whether or not it is taken as the most fundamental one. Horgan (1994) and Soames (1999: 206) took it as at least a fundamental intuition. In Chap. 5, I will argue that this is the most fundamental intuition about the nature of vagueness. (vi) Unknowability of the Boundary Intuition Whether or not vague predicates have sharp boundaries, it seems clear that we couldn’t know their boundaries.

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We couldn’t know the exact cut-off point between “bald” and “not bald”, “heap” and “not heap”, “tall” and “not tall”, etc. It might be that we couldn’t know this because there is indeed no cut-off point to be known. Yet, from the fact that we couldn’t know the boundary of vague predicates it does not follow that they lack a sharp boundary. In short, it is strongly intuitive that we do not know and we could not know the exact location of the relevant cut-off points. Timothy Williamson (1992, 1994, 1997, etc.) took this as the most fundamental intuition about vagueness. Moral of the Story Here is a list of the aforementioned intuitions about the nature of vagueness: (i) Lack of a sharp boundary intuition (ii) No fact of the matter intuition (iii) Tolerance intuition (iv) Borderline cases intuition (v) Arbitrariness of the boundary intuition (vi) Unknowability of the boundary intuition Notwithstanding the fact that there have been other initial characterizations of vagueness, I think this list is fair enough. Although it is not exhaustive, it includes the most commonly accepted intuitions about vagueness.6 (Ronzitti (2011: v) and Smith (2008: 1), for example, include only three intuitions in their lists). I now turn to two important points. First, we have no guarantee that intuitions (i)–(vi) are mutually consistent. For all we know, it might be that they cannot all be true at the same time. Second, we have to face the problem of how to systematize these intuitions, by which I mean that we have to (a) identify which intuitions are accepted and which ones are rejected, (b) explain why the rejected intuitions seem to be true, and (c) determine how the accepted intuitions are related to each other, that is, which are the most fundamental ones and how the others are explained in terms of them. My own solution to this problem will be guided by the three aforementioned criteria of adequacy for an ideal theory of vagueness. I believe (v) is the most fundamental intuition about the phenomenon, mainly because it provides us with a theory that satisfies all three criteria of adequacy. Obviously, this is not to say that I  Wright once said that the phenomenon of permissible disagreement at the margins “is of the very essence of vagueness, and that to leave it out of account is merely to miss the subject matter.” (1994: 138). It seems that he is taking what we can call the “permissible disagreement intuition” as the most fundamental intuition about vagueness. The idea is that if F is a vague predicate, then it admits of borderline cases and at least some of its borderline cases are such that a speaker S1 could say that it is F and a speaker S2 say that it is not F, while both are in some sense correct. “If Jones is on the borderline of baldness, that will, of course, make it allowable if we each judge that he is borderline, but also allowable – at least in very many cases – if you regard him as bald and if I do not.” (Wright 1994: 138). I believe Wright is correct about this point. In Chap. 5, I will explain this intuition in terms of the arbitrariness of the boundary of vague predicates. Anyway, I did not include the permissible disagreement intuition on my list because I am not sure whether this is a common intuition about vagueness. 6

2.3  Intuitions About Vagueness

29

reject all other intuitions. I think intuitions (i), (ii), and (vi) can be explained in terms of (v). Although intuition (iv) is correct, it is especially problematic and requires an independent explanation. Nonetheless, I reject intuition (iii), which says that vague predicates are tolerant to very slight changes. In due time, however, I will explain this rejection.

2.3.1  How Not to Refute a Theory of Vagueness There is an important lesson to be learned from the previous section: we should be cautious with respect to one specific kind of objection to theories of vagueness. Let T be a theory of vagueness, one that explains what it means for a predicate to be vague. It might be argued that T is incorrect either because it is too narrow or because it is too broad. In the first case, it is said that T wrongly implies that a vague predicate is not vague, while in the second it is said that T wrongly implies that a not vague predicate is vague. Although this kind of objection is legitimate, in the case of vagueness it requires special attention. As I said above, the phenomenon of vagueness is related to many different intuitions and it is not even clear whether they can all be true at the same time. At the end of the day, it might be that every theory of vagueness has to reject at least one relevant intuition. If that is so – and I think it is – then no theory of vagueness will be able to accommodate all intuitions about the nature of vagueness. In order to refute a theory of vagueness by means of a counterexample, we must be sure we are not appealing to some intuition that has already been properly rejected. Here is how we could easily refute a theory of vagueness. Suppose T is not consistent with intuition (i) about the nature of vagueness. In this case, we can construct a positive counterexample to T as follows. We first construct an artificial predicate F that satisfies intuition (i), but not the conditions laid down by T. Now we claim that T is too narrow because F is vague – given that it satisfies intuition (i) – but T implies that it is not vague. Similarly, we can construct a negative counterexample as follows. We first construct an artificial predicate F that satisfies the conditions laid down by T, but not intuition (i). Now we claim that T is too broad because F is not vague – given that it does not satisfy (i) – but T implies that it is vague. The problem is that the relevant intuition might be only implicit in the above kind of argument. It might be claimed that a vague predicate is not included by T, or that a not vague predicate is included by T, without there being any explicit mention to which intuition about the nature of vagueness is relevant to the argument. In this case, there is a risk that the relevant intuition is one that has already been properly rejected by the proponents of T. I will get back to this kind of argument later (in Sect. 5.3.2 I will consider an objection of this kind to the Theory of Vagueness as Arbitrariness).

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2.4  Problem of Vagueness I turn now to my final formulation of the problem of vagueness: • Problem of vagueness: the problem of explaining vague predicates (expressions) in a way that systematizes intuitions (i)–(vi) and, if possible, satisfies all three criteria of adequacy for an ideal theory of vagueness (criterion of sorites, criterion of coherence and criterion of precisification). With this in mind, I can clarify what I am looking for in this book. I am looking for a theory of vagueness that solves the above problem or, more precisely, a theory that satisfies all three criteria of adequacy and systematizes all the relevant intuitions. The above formulation of the problem of vagueness also clarifies which kinds of difficulties we can find along the way: there might be problems with respect to the satisfaction of some criteria of adequacy or with respect to some relevant intuition.

References Barnes, J. (1982). Medicine, experience and logic. In J.  Barnes, J.  Brunschwig, M.  Burnyeat, & M.  Schofield (Eds.), Science and speculation: Studies in Hellenistic theory and practice (pp. 24–68). Cambridge: Cambridge University Press. Burnnyeat, M. F. (1982). Gods and heaps. In M. Schofield & M. C. Nussbaum (Eds.), Language and logos (pp. 315–338). Cambridge: Cambridge University Press. Devitt, M., & Sterelny, K. (1999). Language and reality: An introduction to the philosophy of language. Oxford: Blackwell. Dummett, M. (1975). Wang’s paradox. Synthese, 30, 301–324. Eklund, M. (2005). What vagueness consists in. Philosophical Studies, 125(1), 27–60. Empiricus, S. Outlines of pyrrhonism (R.  G. Bury, Trans.). Cambridge: Harvard University Press, 1933. Fara, D. G. (2000). Shifting sands: An interest-relative theory of vagueness. Philosophical Topics, 28(1), 45–81. Field, H. (2003). No fact of the matter. Australasian Journal of Philosophy, 81(4), 457–480. Fine, K. (1975). Vagueness, truth and logic. Synthese, 30, 265–300. Fodor, J. A., & Lepore, E. (1996). What cannot be evaluated cannot be evaluated and it cannot be supervalued either. The Journal of Philosophy, 96(10), 516–535. Galen, C.  On medical experience. In Three treatises on the nature of science (R.  Walzer, & M. Frede, Trans.). Indianapolis: Hackett Publishing Company, 1985. Horgan, T. (1994). Robust vagueness and the forced-march sorites paradox. Philosophical Perspectives, 8, 159–188. Hyde, D. (2011). The sorites paradox. In G.  Ronzitti (Ed.), Vagueness: A guide (pp.  1–17). Dordrecht: Springer. Keefe, R. (2000). Theories of vagueness. Cambridge: Cambridge University Press. Laertius, D. Lives of the eminent philosophers (R. H. Hicks, Trans. Vol. 2). London: The Loeb Classical Library, 1925. Mignucci, M. (1993). The stoic analysis of the sorites. Proceedings of the Aristotelian Society, 93, 231–245.

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Priest, G. (2003). A site for sorites. In J. Beall (Ed.), Liars and heaps (pp. 09–23). Oxford: Oxford University Press. Putnam, H. (1975). The meaning of “meaning”. In Philosophical papers: Volume 2 (pp. 215–271). Cambridge: Cambridge University Press. 1979. Richard, M. (2009). Indeterminacy and truth value gaps. In D. Richard & S. Moruzzi (Eds.), Cuts and clouds: Vagueness, its nature, and its logic (pp. 464–481). Oxford: Oxford University Press. Ronzitti, G. 2011. Vagueness and... In: G. Ronzitti (Ed.), Vagueness: A guide (pp. v–x). Dordrecht: Springer. Sainsbury, M. 1990. Concepts without boundaries. Inaugural lecture delivered at King’s College London, 1990. In: R. Keefe & P. Smith (Eds.), Reprinted in Vagueness: A reader (pp. 294–316). Cambridge: MIT Press. Smith, J. J. N. (2008). Vagueness and degrees of truth. Oxford: Oxford University Press. Soames, S. (1999). Understanding truth. Oxford: Oxford University Press. Tye, M. (1994). Sorites paradoxes and the semantics of vagueness. Philosophical Perspectives, 8, 189–206. Williams, J. R. (2008). Ontic vagueness and metaphysical indeterminacy. Philosophy Compass, 3(4), 763–788. Williamson, T. (1992). Vagueness and ignorance. Proceedings of the Aristotelian Society, 66, 145–177. Williamson, T. (1994). Vagueness. London: Routledge. Williamson, T. (1997). Imagination, stipulation and vagueness. Philosophical Issues, 8, 215–228. Wright, C. (1975). On the coherence of vague predicates. Synthese, 30, 325–365. Wright, C. (1976). Language-mastery and the sorites paradox. In G. Evans & J. McDowell (Eds.), Truth and meaning: Essays in semantics (pp. 223–247). Oxford: Oxford University Press. Wright, C. (1994). The epistemic conception of vagueness. The Southern Journal of Philosophy, XXXIII, 133–159. Wright, C. (2009). The illusion of higher-order vagueness. In D. Richard & S. Moruzzi (Eds.), Cuts and clouds: Vagueness, its nature, and its logic (pp. 523–549). Oxford: Oxford University Press.

Chapter 3

Imprecise Predicates

I have presented three criteria of adequacy for an ideal theory of vagueness: the criterion of sorites, the criterion of coherence and the criterion of precisification. We have seen that all are intuitive, but my interpretation of the latter is controversial. I suppose everyone would agree with this: a predicate is vague only if it is imprecise. Yet, I have assumed a linguistic sense of imprecision, according to which a vague predicate is imprecise only if there is no sharp boundary between objects to which it applies and ones to which it doesn’t. This gives rise to two questions: How exactly should we interpret this linguistic sense of imprecision? Is there any reason to reject this sense of imprecision? In Sect. 3.1, I address the first question. I begin by considering a natural definition of imprecise predicates, according to which a predicate is imprecise if and only if it admits a penumbra between its positive and negative extensions. I call this the “naïve many-valued picture of imprecision”, and I claim that it is unsatisfactory because it does not provide a sense of imprecision that conforms to the phenomenon of vagueness. I then consider a way to enrich this naïve many-valued picture by appealing to hierarchical higher-order imprecision and claiming that vague predicates are radically higher-order imprecise. I present two objections to this proposal. First, radically imprecise higher-order vague predicates are not imprecise in the sense required by vagueness. Second, not all vague predicates are radically higher-­ order imprecise. My diagnosis is that both approaches fail because they imply that there could be a vague predicate with a sharp boundary between objects to which its application yields some particular truth-value and objects to which its application does not yield that value. In this context, I propose that a predicate is imprecise if and only if it admits of no such boundary. I argue that this definition avoids all problems raised by the naïve many-valued picture, whether or not in its enriched version. With the definition of imprecision in hand, I provide a more accurate formulation of the criterion of precisification. This is my final formulation of this criterion, the one I will presuppose in the remainder of this book. There remains the question of whether or not we should interpret imprecision in linguistic terms. This question is addressed in Sect. 3.2. Although the linguistic © Springer Nature Switzerland AG 2021 S. Salles, Vagueness as Arbitrariness, Synthese Library 436, https://doi.org/10.1007/978-3-030-66781-8_3

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conception of imprecision is intuitive, it is not the only one on the table. In particular, imprecision might be interpreted epistemically. The existence of an epistemic sense is important, because it apparently allows one to accept that a predicate is vague only if it is imprecise, while rejecting my final formulation of the criterion of precisification. This, in turn, would undermine some of the main theses and arguments presented in this book. I argue that the two relevant senses of imprecision are mutually consistent, and the mere existence of the epistemic sense does not raise any relevant problem for my views. A problem arises only if one claims in addition that vague predicates are not linguistically imprecise. Since this claim requires justification, the burden of proof is on the side of its proponents. I then consider a way to enrich the epistemic conception by appealing to columnar higher-order imprecision, in addition to the thesis that vague predicates are radically higher-order imprecise in the columnar sense. As before, I object that the only circumstance in which this enriched picture poses a problem for my views is the one that leads its proponents to implausible results that require justification. In conclusion, independently of whether or not it is interpreted according to the columnar model, the existence of an epistemic sense of imprecision poses no problem for my views. I do not have to refute this sense of imprecision in order to advance any of my points.

3.1  Criterion of Precisification 3.1.1  Naïve Many-Valued Picture of Imprecise Predicates It has been said that a predicate is precise if there is a sharp boundary between cases to which it applies and cases to which it doesn’t. Accordingly, a predicate is imprecise only if there is no such boundary. In this section, I consider and reject a possible interpretation of this linguistic sense of (im)precision. I begin by briefly considering what we can call the “classical picture of predicates”. Since predicates that fit into this picture are clearly precise in the above sense, it may help us to find a definition of precise predicates, which in turn may help us to define imprecise predicates. The relevant question is this: what are the features postulated in the classical picture by virtue of which the predicates that fit into this picture are precise? The answer to this will lead us to what I call the “naïve many-valued picture” of imprecision.1 This picture is evaluated according to whether or not it provides us with a satisfactory account of the sense in which vague predicates are imprecise, and my conclusion is that it doesn’t.

 “Naïve” is not intended to have a pejorative sense. On the contrary, it is intended to indicate that the picture under consideration is an intuitive model, many details of which are not developed here. Those details will become important when we evaluate theories of vagueness, how some of them accommodate the naïve picture above and try to avoid the problems raised by it. This is a topic for the next chapter. 1

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Before continuing, two points should be noted. First, the pictures of precision and imprecision under consideration in this and the next two sections are not supposed to accurately represent one or another specific theory of vagueness. They are definitions of imprecision, not theories of vagueness. In fact, the pictures under consideration here are intuitive models that may come to our mind before we begin to theorize about vagueness. To be sure, there might be important relations between theories of vagueness and the relevant definitions of imprecision. For example, some important theories of vagueness are based on the naïve many-valued picture. However, I will postpone to the next chapter the discussion of theories of vagueness. Second, in what follows, unless otherwise stated, I will consider only predicates whose domain can be organized into a soritical sequence. Recall that I have adopted Barnes’s definition of soritical sequence, according to which a sequence is soritical relative to an ordered set of subjects and a predicate F, if it satisfies the following three conditions: (i) Fa0 seems to be true (false); (ii) Fan seems to be false (true); (iii) given any adjacent pair of subjects in the sequence, ai and ai + 1, it seems that either both Fai and Fai + 1 are true or both are false (Barnes 1982: 31). The reason for this is that this is a book on vagueness, and my main interest in this chapter is to understand the claim that a predicate is vague only if it is imprecise. Although there might be doubt about whether or not the domain of application of all vague predicates can be organized into a soritical sequence, the paradigmatic examples of vague predicates do indeed have this property.2 Moreover, the idea of imprecision, and particularly the idea of a predicate having or lacking a sharp boundary, becomes clearer when we frame the discussion in terms of soritical sequences. For example, we have seen that a problem with rejecting the principle of tolerance was that it implies that vague predicates are precise, and this may be easily noted if we organize the domain of application of a vague predicate into a soritical sequence. In so doing, it becomes clear that, if one denies this principle, there will be a last case to which the predicate applies, immediately followed by a first case to which it doesn’t. This means that the predicate is precise, and hence not vague. That said, let us turn to our main topic. According to the classical picture, the domain of application of a predicate is divided into two parts: the positive extension and the negative extension. All members of the domain are in either the negative or positive extension of the predicate. The application of the predicate to objects that are in its positive extension yields sentences that express a true proposition, while its application to objects that are in its negative extension yields sentences that express a falsehood. Natural kind terms, such as “water” and “tiger”, arguably fit into this picture. Now, let us assume that vague predicates also fit into this picture. If this is the case, then a soritical sequence for “bald” is divided into two parts, which can be represented as follows:  Soames (1999: 217), Weatherson (2009: 80) and Shapiro (2006: 4) cast doubt on the claim that every vague predicate is a soritical predicate, and I agree with them in this respect. My own definition of imprecision, presented in Sect. 3.1.3, allows for a non-soritical predicate to be precise/ imprecise. In Sect. 5.2.11 I present Weatherson’s example of an apparently vague predicate that is clearly not soritical, and my diagnosis is that this predicate is really vague, and hence really imprecise. 2

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positive ext. negative ext. |a0, a1, a2, a3 ..., an | an + 1, an + 2..., a10,000| true false This picture obviously entails the existence of a sharp boundary between persons to whom “bald” applies and ones to whom it doesn’t. There will be a last person in the soritical sequence to whom “bald” applies, an, and a first person to whom it doesn’t, an + 1. This is to say that “bald” is a precise predicate. In order to recognize that “bald”, and any other vague predicate, is imprecise in the desired sense, we need to reject this picture. Why does the classical picture precisify vague predicates? What are the properties that make “bald” precise in this picture? A first hypothesis is that “bald” is precise in the classical picture because there is a sharp boundary between its positive and negative extensions. In addition, one might be tempted to claim that the existence of such boundary is not only sufficient, but also necessary, for a predicate to be precise. As a result, a predicate is precise if and only if there is a sharp boundary between its positive and negative extensions. Accordingly, a predicate is imprecise if and only if there is no such sharp boundary. But how exactly should we understand this latter claim? While it is easy to understand the above idea of precision, the idea of imprecision is less clear. It is not clear how we are supposed to understand the claim that there is no sharp boundary between a predicate’s positive and negative extensions. The above idea of imprecision is often understood in terms of the borderline cases intuition (Sect. 2.3), according to which vague predicates admit of borderline cases. Borderline cases are those to which the predicate neither clearly applies nor clearly does not apply. Let us call “penumbra” the part of the domain of application of a vague predicate that is formed by all and only all borderline cases of that predicate. The core idea is that there is no sharp boundary between a predicate’s positive and negative extensions because, if we organize the objects in its domain of application into a soritical sequence, there will be a penumbra between them. This leads us to the following definition of precise and imprecise predicates. • Naïve many-valued picture of imprecision A predicate is precise if and only if there is no penumbra between its positive and negative extensions. Accordingly, a predicate is imprecise if and only if there is a penumbra between its positive and negative extensions. The result is that, instead of being divided into two parts, the domain of application of an imprecise predicate is divided into three and only three parts: the positive extension, the negative extension, and the penumbra. Correspondingly, imprecise predicates admit of three and only three categories: clear positive cases, clear negative cases and borderline cases.3 Taking “bald” as an example, the resulting picture can be represented as follows.  As we will see in this and the next chapter, the idea that imprecise predicates admit of only three categories of cases is problematic, and the naïve picture has been modified in different ways in order to accommodate the rejection (in some sense of “rejection”) of this claim. 3

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positive ext. penumbra negative ext. |a0, a1…, an | an + 1..., an + m | an + m + 1..., a10,000| true ? false

As before, the application of “bald” to persons in its positive extension yields a truth, and the application of it to persons in its negative extension yields a falsehood. How about the objects in the penumbra? There have been many philosophical accounts of the situation of the objects in a predicate’s penumbra. Here are some of them: (a) application of the predicate to objects in its penumbra yields sentences that fail to express a proposition, (b) application of the predicate to objects in its penumbra yields sentences that express a proposition that is neither true nor false, (c) application of the predicate to objects in its penumbra yields sentences that express a proposition that is neither totally true nor totally false (that is, a proposition with some intermediate truth-value). I will remain neutral about these options by just saying that the application of “bald” to borderline cases yields sentences that are neither (totally) true nor (totally) false. Here is the question: does the naïve many-valued picture provide a satisfactory account of precise and imprecise predicates? Unfortunately, the answer is “no”. In fact, the attempt to fit vague predicates into this picture raises problems that are similar to those raised by attempts to fit them into the classical one. Let us consider them one by one. Sharp Boundaries  The naïve many-valued picture does not deny that vague predicates admit of sharp boundaries, it denies that they admit of sharp boundaries between their positive and negative extensions. The main difference with respect to the classical picture is that vague predicates now have at least two sharp boundaries. Note that an is the last person to whom the application of “bald” yields a (totally) true proposition, and an + 1 is the first one to whom its application yields a sentence that is neither (totally) true nor (totally) false. This is to say that there is a last clear positive case immediately followed by a first borderline case of “bald”, which means that there is a sharp boundary between its positive extension and its penumbra. Furthermore, an + m is the last person to whom the application of “bald” yields a sentence that is neither (totally) true nor (totally) false, and an + m + 1 is the first one to whom its application yields a (totally) false proposition. This is to say that there is a last borderline case immediately followed by a first clear negative case of “bald”, which means that there is a sharp boundary between its penumbra and its negative extension. It is indeed intuitive that there is no sharp boundary between the positive and negative extensions of a vague predicate. Nonetheless, just as this is intuitive, it is intuitive that there is no sharp boundary between the positive extension and the penumbra of a vague predicate, and it is intuitive that there is no sharp boundary between the penumbra and the negative extension. As Sainsbury (1991: 168) correctly notes, the non-existence of a sharp boundary between the positive (negative) extension and the penumbra of a vague predicate is as intuitive as the nonexistence of a sharp boundary between its positive and negative extensions. In this

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context, the first problem with the naïve many-valued picture is that it explains the non-­existence of a certain boundary at the cost of postulating at least two new sharp boundaries that are equally implausible. In so doing, it can hardly provide us with a satisfactory account of the imprecision of vague predicates. The idea of an imprecise predicate was introduced in connection with the sense in which vague predicates lack sharp boundaries, and it was supposed to express this sense. Since a vague predicate admits of no sharp boundary between its positive (negative) extension and its penumbra, an appropriate definition of imprecise predicates should not imply such boundary. Principle of Tolerance  In Chap. 2, I said that the initially most plausible strategy to solve the sorites paradox is to reject the principle of tolerance. We have seen, however, that this strategy is not as easy as it may seem at first sight. The problem is that the rejection of this principle implies that vague predicates are precise, in which case they are not really vague. As before, consider the principle of tolerance for “bald”: ∀n (Ban→Ban + 1). The negation of this principle is equivalent to this: Ǝn (Ban  ʌ  ¬Ban  +  1). The latter commits us to the existence of a sharp boundary between the positive and negative extensions of “bald”, which means that this predicate is not imprecise. Now, it is not difficult to see that the new picture will commit us to the rejection – in some sense of “rejection” – of the principle of tolerance. According to this picture, vague predicates admit of clear positive and negative cases, and so we can expect that not all instances of this principle will be (totally) true, and hence the principle itself will not be (totally) true. Exactly how this works depends on details irrelevant for present purposes (but see the next chapter). Nonetheless, the rejection of this principle by the naïve many-valued picture does not imply the existence of a sharp boundary between the positive and negative extensions of vague predicates. After all, there is a penumbra between them. Since vague predicates admit of borderline cases, there will be no sharp boundary between persons to whom the application of “bald” yields a truth and ones to whom its application yields a falsehood. In other words, there will be no sharp boundary between the positive and negative extensions of “bald”, and hence it cannot be (totally) true that Ǝn (Ban  ʌ ¬Ban + 1). Therefore, this picture apparently allows us to reject the principle of tolerance while keeping the thesis that vague predicates are imprecise. Unfortunately, to the extent to which the many-valued picture implies the existence of a sharp boundary between the positive (negative) extension and the penumbra of vague predicates, it raises the same kind of problem raised by attempts to deny the principle of tolerance. In order to see this, note that we can come up with new versions of this principle that (i) are as plausible as the original one, and (ii) are rendered false by this picture. Here is a possible example: for every natural number n, if the application of “bald” to an yields a (totally) true proposition, then the application of “bald” to an + 1 also yields a (totally) true proposition. To put it differently: for every natural number n, if “bald” clearly applies to a person with n hairs on her head, then it clearly applies to a person with n + 1 hairs on her head. Notwithstanding the fact that this version is as intuitive as the original one, it is rendered false by the many-valued picture, given that the latter implies the existence of a last clear case

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immediately followed by a first borderline case of “bald”. As a consequence, the many-valued picture can hardly provide us with a satisfactory account of the imprecision of vague predicates. The idea of imprecision was introduced in connection with the fact that the rejection of the principle of tolerance leads to the above kind of problem. It is because vague predicates are imprecise that we cannot just deny the original version of the principle of tolerance. By parity of reasoning, it is because they are imprecise predicates that we cannot just deny the new version of the principle. Since the many-valued picture is not able to avoid this problem, it is not able to accommodate the sense in which vague predicates are imprecise predicates. Sorites Paradox  A straightforward consequence of the previous point is that the many-valued picture gives rise to new versions of the sorites paradox. Without further ado, here is an instance of the quantified version for “bald”. 1. The application of “bald” to a0 yields a sentence that expresses a (totally) true proposition. 2. For every natural number n, if the application of “bald” to an yields a sentence that expresses a (totally) true proposition, then the application of “bald” to an + 1 yields a sentence that expresses a (totally) true proposition. 3. Therefore, the application of “bald” to a10,000 yields a sentence that expresses a (totally) true proposition. This version of the sorites is as plausible as the original one. The solution implied by the many-valued picture consists in denying (2). As I said above, this solution can hardly be reconciled with the imprecision of vague predicates. Foundational Problem of Precisification  One who claimed that ordinary vague predicates fit into the classical picture would face the foundational problem of precisification. She would have to explain how the sharp boundary between the positive and negative extensions of vague predicates is determined. In Sect. 2.2.3, we saw that this is no easy task. What makes this problem especially difficult is that for any vague predicate, there will be many cut-off points with an equal right to the status of the boundary of that predicate. The many-valued picture not only gives rise to the same problem, it makes it more difficult to solve. According to this picture, vague predicates admit of at least two sharp boundaries. If ordinary vague predicates fit into this picture, we now have to explain how each of the boundaries is determined. How is the boundary between the positive extension and the penumbra determined? How is the boundary between the penumbra and the negative extension determined? This is a difficult problem, because for any vague predicate, there will be many cut-­ off points with an equal right to be the boundary between the positive (negative) extension and the penumbra of that predicate. Suppose one claims that the boundary between the positive extension and the penumbra of “bald” is in n, so that an is the last clear positive case of this predicate, and an + 1 is the first borderline case. What makes n, instead of n – 1 or n + 1, the better candidate for the boundary in this case? There seems to be no reason that justifies one particular choice here. Irrespective of the value of n, it is as arbitrary to claim that the boundary between the positive extension and the penumbra of “bald” is in n as it would be to claim that it is in n – 1

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or n + 1. The same holds for the boundary between the negative extension and the penumbra. The result is that we now have to explain how two, instead of only one, boundaries are determined. The more boundaries a theory postulates for vague predicates, the more difficult it makes the foundational problem of precisification. Just as the foundational problem of precisification provides us with a reason against the claim that an ordinary vague predicate is such that there is a sharp boundary between its positive and negative extensions, it provides us with a reason against the claim that an ordinary vague predicate is such that there is a sharp boundary between its positive (negative) extension and its penumbra. If the existence of this problem is a reason against the claim that ordinary vague predicates are precise, then “precise” here must include both predicates that fit into the classical picture and ones that fit into the many-valued picture. The latter does not provide a satisfactory account of the sense in which vague predicates are imprecise. In sum, to the extent that the notion of imprecise predicate is related to vagueness – and particularly to the sense in which vague predicates do not admit of sharp boundaries, the problem with rejecting the principle of tolerance, the sorites paradox and the foundational problem of precisification – the naïve many-valued picture is not a correct account of what imprecision consists in. Predicates that satisfy this picture are not thereby imprecise. Whether or not the existence of a penumbra is a necessary condition for a predicate to be imprecise, it is not a sufficient condition. Consequently, whether or not the non-existence of a sharp boundary between a predicate’s positive and negative extensions is a necessary condition for it to be imprecise, it is not a sufficient condition. It might be objected that one can accept the naïve many-valued picture by rejecting that a definition of precise and imprecise predicates must be related to the phenomena under consideration here. In other words, one might accept that a predicate is imprecise if and only if there is a penumbra between its positive and negative extensions, but claim that imprecision in this sense is not related to (a) the sense in which vague predicates do not admit of sharp boundaries, (b) the problem with rejecting the principle of tolerance, (c) the consequent difficulty in solving the sorites paradox and (d) the foundational problem of precisification. The core point is that (a)–(d) are phenomena regarding specifically vagueness, while precision and imprecision are broader notions that should not be supposed to account for particularities with respect to vagueness. I concede that one might do this. Yet, in this case one would need a new phrase, perhaps “boundaryless” (to use Sainsbury’s (1991) expression), in order to speak of that property that is related to (a)–(d). Furthermore, it would remain the case that vague predicates seem to be boundaryless, and that the following hold: (a) they are boundaryless in a sense that is not captured by the classical or many-valued picture, (b) this makes it difficult to reject the principle of tolerance (in the original version or not), (c) this, in turn, makes it difficult to solve the sorites paradox, and (d) the claim that some ordinary vague predicates are not boundaryless leads to what I called the “foundational problem of precisification”. Similarly, there would remain a problem of defining the property of being boundaryless, and Def.1 would not provide us with a satisfactory definition of this property. Now, I have chosen to call this property “imprecision”, and this is neither an

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arbitrary nor a mandatory choice. In any case, what matters is not what one calls it, but that vague predicates intuitively have this property, and this underlies a set of philosophical problems. Let us illustrate the point by means of an example. Imagine a possible world in which (i) no one has thought about the sorites paradox, and (ii) it is generally agreed that ordinary vague predicates fit into the naïve many-valued picture. In this world, there is a philosopher of language, Mary, who is worried about vague predicates. She notes that they do not seem to establish the cut-off points posited by the naïve many-valued picture. Imagining (what we would call) a soritical sequence for “bald”, she realizes that there seems to be no last clear positive case of “bald” immediately followed by a first borderline case, and no last borderline case immediately followed by a clear negative case. She may express this by saying that vague predicates are boundaryless in a sense that can hardly be accommodated by the standard picture (which, in this world, is the many-valued picture). Eventually, she comes to the sorites paradox, a version of which is presented above. She then considers the option of simply denying premise (2) (the principle of tolerance for that version), but notes that this would result in implausible sharp boundaries. Such boundaries are implausible not only because vague predicates seem to be boundaryless, but also because it is especially difficult to explain how they are determined. To sum up, Mary found that vague predicates seem to be boundaryless, and that this is related to a set of interesting and difficult philosophical problems. Finally, suppose Mary presents her ideas in a well-written paper and publishes it in a leading journal of philosophy. Mary’s philosophical community would be justified in taking her paper very seriously. The fact that the many-valued picture is generally accepted in Mary’s philosophical community does not make her discoveries less challenging. Now, what Mary would call a “boundaryless predicate” I call an “imprecise predicate”. As before, what matters is not what one prefers to call it, but that vague predicates indeed seem to have this property, and this is related to a set of difficult philosophical problems. In conclusion, the naïve many-valued picture does not provide us with a satisfactory account of imprecision, because it fails to explain the sense in which vague predicates are imprecise.

3.1.2  Hierarchical Higher-Order Imprecision Arguably, the problems for the naïve many-valued picture arose because it implies a sharp boundary between the positive (negative) extension and the penumbra of vague predicates. It might be argued that this consequence can be avoided by appealing to the notion of higher-order imprecision. The core idea is that a certain conception of higher-order imprecision, in addition to the claim that vague predicates are radically imprecise, allows us to enrich the naïve many-valued picture in a way that precludes any undesired sharp boundaries.

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In order to see how the idea of higher-order imprecision might help, consider the following definitions.4 • Imprecisen predicate An imprecisen predicate is a predicate that is imprecise in an order n. A predicate is imprecise in an order n if it divides its domain of application into 2n + 1 parts, being compatible with the existence of 2n sharp boundaries between those parts. • Higher-order imprecise predicate A higher-order imprecise predicate is one that is imprecisen for some n > 1. • Radically higher-order imprecise predicate A predicate that is imprecisen for any n. According to these definitions, a first-order imprecise predicate, an imprecise1 predicate, must divide its domain of application into at least three parts, which is in line with the naïve many-valued picture previously discussed. Nonetheless, since we are distinguishing among different orders of imprecision, we should now be careful not to conclude that an imprecise1 predicate divides its domain into three and only three parts, so as to establish three and only three categories of cases, with two sharp boundaries between them. This would happen only if the predicate were merely imprecise1. If a predicate is merely imprecise1, then it is not imprecise2 (and hence not higher-order imprecise). If a predicate is merely imprecise2, then it is (higher-order imprecise but) not imprecise3. Generalizing, if a predicate is merely imprecisen, then it is not imprecisen + 1, and it has 2n sharp boundaries. This suggests that, in order to avoid the postulation of undesired boundaries, we must accept that for any n, vague predicates are not merely imprecise n. This is to say that vague predicates are radically higher-order imprecise. Since radically imprecise predicates do not admit of any problematic kind of sharp boundaries, and the naïve many-­ valued picture might easily be modified in order to recognize that vague predicates are radically imprecise, the latter still holds. In order to conform the naïve many-valued picture to the above definitions, we first need to give up the idea that an imprecise predicate must divide its domain into only three parts. An imprecise1 predicate F divides its domain of application into three categories: one including objects that are clearly F (CF), one including objects that are clearly not F (C¬F), and the last one including borderline cases (BF). These three categories establish a hierarchy of clarity: the objects in BF are not as clearly F as those in CF, but are more clearly F than those in C¬F. If the predicate is merely imprecise1, then these are the only relevant categories. As a consequence, there will be two sharp boundaries between them. Three categories, two boundaries. But an imprecise1 predicate does not need to be merely imprecise1. It might also be  The below definitions of higher-order imprecise and radically higher-order imprecise predicates correspond to Sainsbury’s definitions of higher-order vague and radically higher-order vague predicates. There is a minor difference concerning the definition of an imprecisen predicate. I distinguish between a predicate that is imprecisen and one that is merely imprecisen, and Sainsbury’s definition of the former, at least as it appear in Sainsbury (1991: 169), corresponds more exactly to my definition of the latter. 4

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imprecise2. An imprecise2 predicate divides its domain into at least five categories, two of them being new in relation to those found in the first order. The first new category includes objects that are between CF and BF, but that are neither in CF nor in BF. We now have to distinguish between objects that are clearly clear F (C(CF)) and ones that are clearly borderline F (C(BF)), in order to recognize the existence of objects that are borderline clear F (B(CF)). The second includes objects that are neither in BF nor in C¬F, but are between them. We now have to distinguish between objects that are clearly borderline F (C(BF)) and ones that are clearly clear not F (C(C¬F)), in order to recognize the existence of objects that are borderline clear not F (B(C¬F)). Note that B(CF) and B(C¬F) are new in the sense that they classify objects not previously classified, and each introduces a new element into the previously established hierarchy of clarity. If F is merely imprecise2, then there are five and only five relevant categories, and four sharp boundaries between them. Five categories, four boundaries. As before, however, an imprecise2 predicate does not need to be merely imprecise2. It might also be imprecise3, imprecise4, and so on. Those predicates that are imprecisen for any n, thereby establishing an infinite hierarchy of clarity in their domains, are radically imprecise. If (a) radically imprecise predicates do not admit of any problematic kind of sharp boundaries and (b) vague predicates are radically imprecise, then the enriched naïve many-valued picture does not raise the problems of the previous section. A predicate that is merely imprecise3 may be represented as follows. C(F) B(F) C(¬F) C(CF) B(CF) C(BF) B(C¬F) C(C¬F) C(CCF) B(CCF) C(BCF) B(BCF) C(CBF) B(CB¬F) C(BC¬F) B(BC¬F) C(CC¬F)

Each line represents an order of imprecision. Note that categories occupying different lines in the same column are co-extensional. For example, an object is in BF if and only if it is in C(BF), and it is in C(BF) if and only if it is in C(CBF). By virtue of this, only the bold formulas represent really new categories of borderlineness. Each new category introduced in an order n classifies only objects that have not been classified in order n – 1, and each introduces a new element into the hierarchy of clarity. New categories of borderlineness are always introduced between two preexisting categories, and they include only objects that occupy an intermediate position in the hierarchy established by the latter ones. In fact, an order n will require a new category between each pair of neighboring categories in n – 1. The existence of empty spaces between two categories Ф and Ψ indicates the existence of a gap between them, which means that some objects occupy an intermediate position in the hierarchy established by Ф and Ψ, in which case they are neither in Ф nor in Ψ. The existence of empty spaces in a line indicates that the classification made by the categories in that line is not exhaustive, so that new categories are required for a more complete classification. The fact that F is merely imprecise3 is indicated by the absence of empty spaces in the third line. A radically higher-order predicate is such that, for any line n, there are empty spaces in the kind of table above. In fact,

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given the hierarchical ordering of the categories, a radically imprecise predicate is such that, for every line n, and for every category pair Ф and Ψ in n, there is an empty space between Ф and Ψ. This highlights the sense in which radically higher-order predicates have no sharp boundaries. For every order n, and every category pair Φ and Ψ, there is a gap between Φ and Ψ in n. In soritical terms, there will be no pair of adjacent objects such that the first is the last object in Φ and the second is the first in Ψ. It seems that this proposal avoids not only the postulation of sharp boundaries between the positive and negative extensions, but also the postulation of any other problematic boundaries. If we add the thesis that vague predicates are radically imprecise, then we have a picture of imprecision that conforms perfectly to vagueness. Note that the above strategy assumes a hierarchical model of higher-order imprecision. It is called “hierarchical” because, according to it, imprecise predicates admit of a set of mutually exclusive categories of clarity, and each new order of imprecision introduces new categories into the previously established hierarchy of clarity. If a predicate is radically higher-order imprecise, then it admits of an infinite hierarchy of categories of clarity in its domain. Since vague predicates are radically imprecise, they permit such an infinite hierarchy of clarity. There are two possible objections to this proposal. First, one might cast doubt on the claim that radically imprecise predicates are imprecise in the sense required by vagueness. Second, one might deny that vague predicates are radically imprecise.5 Let us assume that “bald” is radically imprecise, so that there is an infinite hierarchy between a person with no hair (a clear positive case) and a person with 10,000 hairs on her head (a clear negative case). It is still true that there is no infinite hierarchy of clarity between the clear positive and clear negative cases in the following soritical sequence for “bald”: . I suppose a proponent of the enriched approach would accept that a0 is a clear positive case of “bald”, and a10,000 a clear negative case. Since a10,000 is a clear negative case, any person with more than 10,000 hairs on her head will be a negative case too.6 If we remain neutral about the status of the other persons in the sequence, we are left with  As we will see below, the present discussion of higher-order imprecision is often framed as a discussion of higher-order vagueness. What I call the “hierarchical model of higher-order imprecision” corresponds to the model of higher-order vagueness criticized by Sainsbury (1991) and Bobzien (2013, 2015). It should be noted, however, that Sainsbury and Bobzien do not have exactly the same model in mind. The latter, but not the former, describes this model in such a way as to exclude the borderline clear cases and collapse the distinction between borderline clear positive and borderline clear negative cases (Nader 2017: 11–12). In this respect, my exposition is closer to Sainsbury’s model. Moreover, my graphical representation of the hierarchical model is inspired by (but not identical to) Bobzien’s representation (2013: 3). The main difference is that my representation is intended to represent, by means of empty spaces, the gap between different categories of clarity. In order to maintain the pyramidal form of the set of categories that Bobzien’s representation is intended to represent, I have represented the new categories of borderlineness in each order with bold formulas. Note that the bold formulas form a pyramid, in contrast to what happens in the columnar model of higher-order vagueness (Sect. 3.2.2). 6  Obviously, I am making an idealization here, for I am ignoring the multi-dimensionality of “bald”. See  Chap. 5, n. 2 for an explanation of why this idealization is harmless for present purposes. 5

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9999 possibilities with respect to borderline cases of “bald”. Suppose that at least one of them is a borderline case. We are left with a maximum of 9998 unclassified possibilities. Suppose at least one of them is a borderline clear positive case, and at least one is a borderline clear negative case. Now, the maximum number of unclassified possibilities is 9996. As the argument goes, the number of possibilities decreases, and given that this number is finite, sooner or later we will be left with no unclassified possibilities anymore. This is to say that “bald” divides the above soritical sequence into n parts (for some n) and establishes n  – 1 sharp boundaries between them. Note that the crucial point here is not that “bald” in itself does not admit of an infinite hierarchy of clarity between its positive and negative cases, but that, whether or not it does, there is no infinite hierarchy of clarity between the positive and negative cases in the above soritical sequence. In a similar way, the point is not that “bald” in itself establishes unacceptable sharp boundaries in its domain of application, but that it establishes unacceptable sharp boundaries when its domain is restricted to that of the soritical sequence above. The same holds for virtually every ordinary vague predicate. It is not difficult to see why such boundaries are unacceptable. In the final analysis, they amount to the conclusion that there is a last person in the sequence to whom the application of “bald” yields a (totally) true proposition, immediately followed by a first to whom its application does not yield a totally true proposition.7 If we imagine the soritical sequence represented in the kind of table above, we will note that the categories of clarity in the first column to the left of each line represent the clearest cases of “bald”, those to which the application of “bald” yields a (total) truth. Since in the last line there will be a sharp boundary between the clearest cases and the first category of borderlineness, there will be a sharp boundary between persons to whom the application of “bald” yields a (total) truth and ones to whom it doesn’t. As far as I can see, there is no more reason to accept this latter claim than there is to accept the sharp boundaries implied by the classical picture. In line with this, the enriched many-valued picture will give rise to versions of the principle of tolerance that (a) are as plausible as the original one, and (b) are rendered false by it. A possible example is this: for every natural number n, if the application of “bald” to an yields a (totally) true proposition, then the application of “bald” to an + 1 also yields a (totally) true proposition. Recall that the idea of imprecision was introduced in connection with the principle of tolerance. It is because vague predicates are imprecise that we cannot just deny the original version of the principle of tolerance. By parity of reasoning, it is because vague predicates are imprecise that we cannot simply deny the corresponding alternative versions of the principle of tolerance.

 Arguably, one could avoid this consequence by interpreting the hierarchy of clarity in terms of a hierarchy of metalanguages, and rejecting the view that each new category of borderlineness corresponds to a new truth-value. I will analyze this kind of strategy regarding vagueness in Sect. 4.3.2, in which I discuss Rosanna Keefe’s (2000) response to the objection that Supervaluationism violates the criterion of precisification. I will argue that, on the one hand, Keefe’s proposal does not satisfy her own criterion of adequacy for a semantic theory, and on the other hand, it does not really avoid the aforementioned problems. 7

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Consequently, the enriched picture leads us to alternative versions of the sorites paradox that are as problematic for it as the original version is for the classical picture. Finally, we have to explain how the referential mechanism of “bald” determines a certain number n, instead of n – 1 or n + 1, as the boundary between the clearest and borderline cases of that sequence; and here the foundational problem of precisification seems as striking as before. In sum, we are back to the four problems discussed in the previous section.8 Things can get even worse for the enriched naïve many-valued picture. After all, even if we concede that radically higher-order imprecise predicates are imprecise in the sense required by vagueness, it is just not the case that vague predicates are radically imprecise. On the contrary, it seems clear that some vague predicates are not radically imprecise. We have seen that when we restrict the domain of application of “bald” to the above soritical sequence it does not admit of an infinite hierarchy of clarity. Yet, it seems that no domain of application of “bald” would admit of an infinite hierarchy of clarity. If we leave aside any remaining vagueness in what counts as a person, a strand of hair, etc., it seems that no domain of application of “bald” would admit of an infinite hierarchy of clarity.9 Another example is “heap”. As far as I can see, there is not an infinity hierarchy of categories of clarity between the alleged clear positive and clear negative cases of “heap”. Therefore, neither “bald” nor “heap” is a radically higher-order imprecise predicate. On the one hand, the notion of radical higher-order imprecision does not provide us with a satisfactory account of the sense in which vague predicates are imprecise. On the other hand, not all vague predicates are radically higher-order imprecise. The enriched naïve many-valued picture is not correct. Before moving on, note that the discussion of higher-order imprecision is often framed as a discussion of higher-order vagueness. Here is a rough description of how this can be done. First, one might take the borderline cases intuition as the most fundamental intuition concerning the nature of vagueness, and define vagueness in  Sainsbury observes that the enriched picture does not avoid the three-fold classification implied by the naïve many-valued picture. This is because we still can distinguish among three relevant sets: the set of objects to which the application of the predicate yields a truth, the set to which it yields a falsehood and the set formed by “the union of the remaining sets associated with the predicate” (Sainsbury 1991: 169). These sets establish an exhaustive distinction in the predicate’s domain of application. They establish an exhaustive distinction between objects to which the application of the predicate yields a (total) truth and ones to which it does not yield a (total) truth, and an exhaustive distinction between objects to which the application of the predicate yields a (total) falsehood and ones to which it does not yield a (total) falsehood. Sainsbury thinks that such exhaustive distinctions cannot be reconciled with vagueness, and I agree in that I think they cannot be reconciled with imprecision. See Sect. 3.1.3. 9  The imprecision of “person”, “strand of hair”, etc. can at most make “bald” derivatively imprecise. Nevertheless, “bald” is imprecise in more than just a derivative way, it is imprecise by its own merits. This predicate remains imprecise even if “person”, “strand of hair”, etc. are precisified. If the enriched many-valued picture holds for “bald”, it must recognize that this predicate is radically imprecise even under this condition, and the fact that no domain of application of “bald” would admit of an infinite hierarchy of clarity under this condition is thus relevant. This is why I can leave aside any remaining imprecision in “person”, “strand of hair”, etc. 8

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terms of borderlineness. This kind of strategy leads us to the postulation of undesired sharp boundaries and all the consequences that result from it. In this context, one might be tempted to distinguish between predicates that are merely vaguen and predicates that are radically higher-order vague. The final step is to claim that vagueness, at least as we know it from ordinary language, is indeed radical higher-­ order vagueness, thereby hoping to avoid the postulation of undesired sharp boundaries and related problems. This rough description might vary in many aspects depending on the details of the theory of vagueness under consideration. I will investigate some examples of this strategy in the next chapter, where I address some theories of vagueness. For now, it suffices to say that, to the extent that this proposal mirrors the enriched naïve many-valued theory, it raises the same problems and must have the same fate.

3.1.3  Imprecise Predicates As we have seen in Sect. 3.1.1, notwithstanding the fact that the naïve many-valued picture recognizes that imprecise predicates do not admit of a sharp boundary between their positive and negative extensions, it implies the existence of a sharp boundary between objects to which the application of the predicate yields some particular truth-value and ones to which it does not yield that truth-value. For example, it implies the existence of a sharp boundary between objects to which the application of “bald” yields a (total) truth and ones to which it does not yield a (total) truth, just as it implies the existence of a sharp boundary between objects to which the application of “bald” yields a (total) falsehood and ones to which it doesn’t. In Sect. 3.1.2, I argued that this problem cannot be solved by appealing to higher-order imprecision and claiming that vague predicates are radically higher-order imprecise. First, the thesis that a predicate F is radically imprecise does not avoid undesired sharp boundaries, because it is compatible with the existence of a last case in a soritical sequence for F to which the application of F yields a (total) truth, immediately followed by a first to which its application does not yield a (total) truth. This is as unacceptable as the kind of boundaries postulated in the classical picture. Second, vague predicates such as “bald” and “heap” are not radically imprecise, and hence do not admit of an infinite hierarchy of clarity between their clear positive and clear negative cases. This amounts to the implausible conclusion that they divide their domains so to establish a cut-off point between objects to which their application yields a (total) truth and ones to which it doesn’t. In this case, however, the diagnosis of the enriched naïve many-valued picture would be that these predicates are not really vague. This diagnosis is initially implausible, given that “bald” and “heap” are usually taken as paradigmatic examples of vagueness. We would do better with a definition of imprecision that includes “bald” and “heap”, thereby being compatible with the intuition that they are vague. Whether or not it is enriched by the notion of higher-order imprecision, the naïve many-valued picture shares the following property with the classical picture: it

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implies the existence of a sharp boundary between cases to which the application of a vague predicate yields some particular truth-value and ones to which it doesn’t. My suggestion is that this property is the source of all four previously discussed problems. This is why none of these pictures can account for the sense in which vague predicates are imprecise. In this context, a definition of imprecise predicates must forbid not only a sharp boundary between the positive and negative extensions, but also any sharp boundaries between objects to which the application of the predicate yields a certain truth-value and objects to which it doesn’t. This leads us to the following alternative definition. • Def.1 A predicate is precise if and only if there is a sharp boundary between cases to which the application of the predicate yields some particular truth-value and ones to which it does not yield that truth-value. Accordingly, a predicate is imprecise if and only if there is no sharp boundary between cases to which its application yields some particular truth-value and ones to which it doesn’t. Def.1 implies that imprecise predicates do not admit of a sharp boundary between a positive and a negative extension. But it also implies that they do not admit of a sharp boundary between a positive extension and a penumbra, between a negative extension and a penumbra, and so on. For short, we may express this by just saying that an imprecise predicate does not admit of any sharp boundaries between any kinds of extensions. This is in line with Sainsbury’s proposal that vague predicates are boundaryless, and that a boundaryless predicate “draws no boundary between its positive and its negative cases, between its positive cases and its borderline cases, between its positive cases and those which are borderline cases of borderline cases” (Sainsbury 1991: 179). In fact, if we understand the notions of clear and borderline cases as we have been doing, then Sainsbury’s claim is a consequence of Def.1. Just below the above passage Sainsbury outlines a description of what a predicate should be like in order to be boundaryless in his sense. I will not discuss his description here, but I am not convinced that it is the correct description of a predicate that is imprecise in the sense of Def.1. In fact, I think the correct description is to be found in Sainsbury (2013) rather than Sainsbury (1991) (see Sect. 5.2.2 of this book). Let  be a soritical sequence for a vague and imprecise predicate F, and consider a corresponding sequence of questions such as “Is a0 F?”, “Is a1 F?”, “Is a3 F?”, etc. Def.1 implies that there is no pair of objects that have the following properties: (a) they are adjacent in a soritical sequence for F, and (b) the application of F to one yields some particular truth-value, and the application of F to the other does not yield that value. Thus, there is a sense in which objects that are adjacent in the above sequence cannot be contrasted. This is in line with Kit Fine’s (2008: 115, 2020: 19) incompatibility requirement, which states that the (global) indeterminacy claim  – according to which F is indeterminate in its application to a range of objects  – is incompatible with the existence of a sharp response to a series of questions like the one above. A series of responses is sharp “if it draws a contrast between at last two neighboring cases” (Fine 2020: 20). Following Sainsbury, he assumes that the indeterminacy claim is incompatible not only with any response that draws a boundary between positive and negative cases,

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but also with responses that draw a boundary between positive and borderline cases, between borderline and negative cases, etc. In sum, just as Def.1 implies that F’s imprecision is incompatible with there being a pair with properties (a) and (b), Fine’s incompatibility requirement implies that F’s indeterminacy is incompatible with there being any such pair. In fact, it seems that, in Fine’s sense of indeterminacy, the indeterminacy of a predicate implies that it is imprecise in my sense. Despite this, I am not sure whether Fine’s concept of indeterminacy corresponds exactly to my concept of imprecision. Although F’s (global) indeterminacy implies that F satisfies Def.1, it is not clear whether or not the converse hold. Does the fact that F satisfies Def.1 implies that F is indeterminate in its application to a range of objects? The answer depends on what more Fine thinks that is involved in the notion of indeterminacy, and we do not need to worry about this here. Def.1 avoids all four previously mentioned problems. First, if F is imprecise, then there will be no sharp division – whether in its domain of application or in a soritical sequence for it – between objects to which its application yields some particular truth-value and ones to which it doesn’t. By virtue of this, no predicate will be such that (i) it is imprecise, but (ii) it establishes intuitively unacceptable sharp boundaries between kinds of extensions. Def.1 thus recognizes the connection between the notion of imprecision and the sense in which vague predicates lack sharp boundaries. Second, Def.1 is consistent with the claim that a core problem of simply denying the principle of tolerance is that this would amount to the further claim that they are precise. What is more, this holds not only for the original version of this principle, but also for alternative ones. Consider the aforementioned alternative version of this principle: for every natural number n, if the application of “bald” to an yields a sentence that expresses a (totally) true proposition, then the application of “bald” to an + 1 yields a sentence that expresses a (totally) true proposition. The negation of this principle would be this: there is a number n such that the application of “bald” to an yields a sentence that expresses a (totally) true proposition and the application of “bald” to an + 1 yields a sentence that does not express a (totally) true proposition (it does not express a proposition, or it expresses a proposition that is neither true nor false, or it expresses a proposition with some intermediate truth-­ value). According to Def.1, the latter implies that “bald” is a precise predicate. Along these lines, there need not be a difference in the reason one cannot simply deny the original version and the reason one cannot simply deny the present version of the relevant principle. In both cases, the problem – or at least a central problem – is that this would entail that vague predicates are precise. Correspondingly, one cannot solve the original version of the paradox by simply denying the principle of tolerance for the same reason one cannot solve alternative versions of the former by simply denying alternative versions of the latter: this would entail that vague predicates are precise. Def.1 keeps the connection between the notion of imprecision and the difficulty in rejecting the principle of tolerance and the sorites paradox. Finally, since Def.1 does not commit us to any boundaries between any kinds of extensions, it does not give rise to the foundational problem of precisification. In conclusion, Def.1 avoids all problems raised by the previously considered pictures.

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Although Horgan (1994) did not present an explicit definition of precise and imprecise predicates, I think Def.1 is in line with his claim that vague predicates are imprecise. Horgan’s argument to the effect that some theories of vagueness treat vague predicates as if they were precise rests on the claim that they postulate the existence of “a fact of the matter about the truth-value transitions in sorites sequences” (Horgan 1994: 163). By this he means that these theories imply the existence of a sharp boundary between cases to which the application of a vague predicate yields a certain truth-value and ones to which it doesn’t. Three-Valued Theory, for example, implies the existence of a number n such that Ban is true and Ban + 1 is not (it is neither true nor false). Degrees-of-Truth Theory implies the existence of a number n such that Ban is true to degree 1 and Ban + 1 is not true to degree 1 (it is true to a degree less than 1). Supervaluationists will accept that there is a number n such that Ban is supertrue (true in all precisifications) and Ban + 1 is not supertrue (it is true in at least one precisification and false in at least one precisification). In this context, Horgan clearly accepts that a predicate is imprecise only if there is no sharp boundary between objects to which its application yields some particular truth-value and objects to which it doesn’t. What he takes as a necessary condition for a predicate to be imprecise, Def.1 takes as a necessary and sufficient condition. Before moving on, it is important to clarify the notion of sharp boundary. With regard to vagueness, it is natural to understand the claim that a predicate F admits of a sharp boundary in terms of there being a pair of objects that have the following properties: (a) they are adjacent in a soritical sequence for F, and (b) the application of F to one yields some particular truth-value, and the application of F to the other does not yield that value. (As far as I can see, it is in this sense that Horgan understands this notion). Although this manner of speaking is in general useful and harmless, it is not totally correct. This is because a predicate might be precise – in which case it admits of a sharp boundary in the relevant sense – without there being any pair of objects with the relevant properties. This should come as no surprise, since a predicate does not need to be soritical in order to be precise. “Natural number” is precise, but it is not a soritical predicate. If this is correct, then it would be incorrect to speak of a pair of objects in a soritical sequence for “natural number”, and so there is no pair of objects in this predicate’s domain of application that satisfies (a). Furthermore, consider predicates that apply to nothing (empty) or everything (universal) in the relevant domain. The application of an empty or a universal predicate to any object in its domain yields the same truth-value. By virtue of this, even if it makes sense to speak of a soritical sequence for an empty or universal predicate, there will be no pair of objects in this sequence that satisfies (b). The upshot is that, although the existence of a pair of objects with the relevant properties is sufficient for a predicate to be precise, it is not necessary. In order to properly understand the idea of a sharp boundary, we must clarify in what sense “natural number”, or empty and universal predicates, admit of a sharp boundary. I think they admit of a sharp boundary in the sense that they establish an exhaustive distinction between objects to which their application yields some particular truth-value and ones to which their application does not yield that truth-­ value. In this context, the notion of a sharp boundary must include not only soritical predicates that admit of a pair of adjacent objects with the above properties, but also

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any predicate that establishes an exhaustive distinction between objects to which its application yields a certain truth-value and ones to which it doesn’t.10 Imagine that Φ is a box containing the objects to which the application of a predicate F yields a certain truth-value v, and that non-Φ is a box containing all other objects. Now, there are two ways in which F may come to have a sharp boundary. First, this happens when there is a pair of adjacent objects in a soritical sequence for F such that one is in Φ and the other is in non-Φ. Second, this happens because there is an exhaustive distinction between objects that are in Φ and ones that are in non-Φ. The fact that the relevant division cannot be represented by a pair of objects with properties (a)–(b) should not prevent us from saying that there is indeed a sharp division between objects that are in Φ and ones that aren’t. It is in this sense that “natural number”, empty and universal predicates have sharp boundaries.11 It is suspicious to speak of empty or universal predicates as establishing some sorts of exhaustive distinctions. It might be objected that empty and universal predicates do not establish exhaustive distinctions, on the grounds that they fail to establish distinctions of any kind. A predicate that applies to everything/nothing does not draw a distinction between objects to which it applies and ones to which it doesn’t. In the same way, if the application of a predicate to every object in its domain yields the same truth-value, then there is no distinction, much less an exhaustive distinction, between objects to which the application of the predicate yields that truth-­ value and objects to which it doesn’t. Instead of saying that empty and universal predicates establish distinctions of this or that kind, we should say that they fail to establish distinctions of any kinds. It seems to me this objection rests on a metaphysical understanding of distinction. The core point is that we should restrict the idea of distinction to predicates that express properties that ground objective similarities, predicates that carve out the joints of reality (possibly predicates that express what David Lewis (1986: 59–60) called “sparse properties”). Now, the argument goes, since an empty/universal predicate applies to everything/nothing, it cannot express a property that grounds genuine similarities between objects in the world, and hence they do not draw distinctions of any kinds.

 This is why I said that I agree with Sainsbury’s point in footnote 8 above.  The relation between these different senses of sharp boundary is not totally clear. To begin with, a predicate might admit of a sharp boundary in the second sense without thereby admitting a sharp boundary in the first (and soritical) sense. This is the case of empty and universal predicates, or predicates such as “natural number” and “real number”. It is not clear, however, whether or not a predicate could have a boundary in the first sense without thereby having a boundary in the second one. Suppose that there is a number n such that the application of “bald” to an yields a value v1 and the application of it to an + 1 does not yield v1. Does it follow from this that there is an exhaustive distinction between objects to which the application of “bald” yields v1 and ones to which it doesn’t? The answer depends on how we interpret the situation of persons who have less than n hairs on their head. This, in turn, depends on how many truth-values we accept and how they are structured. Suppose, for example, that there are only two truth-values, v1 and v2, and that they are mutually exclusive. In addition, suppose that if the application of “bald” to an yields v1, then the application of “bald” to an - 1 yields v1 too, and if the application of “bald” to an yields v2, then the application of it to an  +  1 also yields v2. Under these suppositions, “bald” will admit of a sharp boundary in the second sense. 10 11

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I concede that one could understand the notion of distinction in this sense, and I would not object to one who wants to understand it only in this sense. It should be noted, however, that this would not affect my point. The distinction between precise and imprecise predicates should not be restricted to predicates that express properties grounding objective similarities. On the contrary, predicates with totally miscellaneous extensions might be clearly precise. Let us define F in such a way that its positive extension includes the Eiffel Tower, Socrates and the number 3, while the negative extension includes everything else. F is clearly precise, but it does not express a property that carves out the joints of the world. In fact, there are indefinitely many precise predicates with miscellaneous extensions. Now, if one wants to restrict the notion of distinction to predicates that correspond to objective similarities, then one might say that F draws an exhaustive division between objects to which its application yields a certain truth-value and ones to which it does not yield that value; where the notion of division is clarified by the metaphor of the box, which also applies to empty and universal predicates. Finally, Def.1 is not totally clear about what an imprecise predicate consists in. It is not totally clear how we are supposed to understand the claim that there are no sharp boundaries between kinds of extensions of a certain predicate. What such a predicate should be like? The answer to this question will appear only in Chap. 5, where I propose a picture of vague predicates that conforms to Def.1. For the moment, it suffices to say that any picture of vague predicates that conforms to the intuition that they are imprecise predicates must be in accord with Def.1.

3.1.4  Criterion of Precisification Recall that I have presented three criteria of adequacy for an ideal theory of vagueness. (a) Criterion of sorites: explain vague predicates in a way that solves the sorites paradox. (b) Criterion of coherence: explain vague predicates without implying that they are incoherent predicates. (c) Criterion of precisification (CPI): explain vague predicates without implying that there could be a predicate that is vague and precise at the same time. CPI is the initial formulation of the criterion of precisification. Now that we have a definition of precise and imprecise predicates, I am in the position to offer a more accurate formulation of it. Here it is. (d) Criterion of precisification (CPF): explain vague predicates without implying that there could be a predicate that is vague and precise at the same time; that is, without implying that a predicate could be vague at the same time that there is a sharp boundary between objects to which its application yields some particular truth-value and ones to which it doesn’t. CPF is my final formulation of the criterion of precisification.

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3.2  Epistemic Sense of Imprecision 3.2.1  Linguistic Imprecision Versus Epistemic Imprecision Let us summarize some important points advanced so far. I proposed that the following is a criterion of adequacy for an ideal theory of vagueness. • CPI Explain vague predicates without implying that there could be a predicate that is both vague and precise. CPI was based on the intuition that a predicate is vague only if it is not precise. I have explained this intuition by assuming a linguistic conception of imprecision, according to which a predicate is imprecise only if it admits of no sharp boundary between cases to which it applies and ones to which it doesn’t. In other words, I have explained I1 in terms of I2. • I1 A predicate is vague only if it is imprecise. • I2 A predicate is vague only if it admits of no sharp boundary between objects to which it applies and ones to which it doesn’t. I then argued that the linguistic conception of imprecision should be interpreted as the claim that a predicate is imprecise iff there is no sharp boundary between any kinds of extensions of the predicate. The result is that we should interpret I2 as I3. • I3 A predicate is vague only if there is no sharp boundary between objects to which the application of the predicate yields some particular truth-value and ones to which it does not yield that value. All this led us to the following final formulation of the criterion of precisification. • CPF Explain vague predicates without implying that a predicate could be vague and precise at the same time; that is, without implying that a predicate could be vague at the same time that there is a sharp boundary between objects to which its application yields some particular truth-value and ones to which it doesn’t. I suppose CPI is not controversial, and both I1 and I2 are intuitively true. Besides, I have already argued that I2 should be interpreted in terms of I3. But I have provided no argument to the effect that the idea of imprecision should be interpreted in linguistic terms, and thus no argument in favor of interpreting I1 in terms of I2. This is important, because if one rejects this step of the reasoning, one might reject (I2, I3 and) CPF. What is more, one might reject it while accepting both CPI and I1. To put this in intuitive terms, one might accept that a predicate is vague only if it is imprecise, and that all theories should recognize this, while rejecting the claim that a predicate is vague only if it admits of no sharp boundaries between kinds of extensions. Is there any reason to reject the interpretation of I1 in terms of I2? Obviously, this interpretation depends on the linguistic conception of imprecision. Namely, it depends on the supposition that imprecision has to do with the non-existence of sharp boundaries between objects to which a predicate applies and ones to which it

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does not apply. But one might reject this supposition by appealing to a non-­linguistic sense of imprecision. As far as I can see, the most plausible candidate here would be the epistemic sense of imprecision, to which I turn next.12 There is no doubt that Epistemicism violates CPF. According to this view, vagueness is a kind of ignorance, it is an epistemic phenomenon. The vagueness of vague predicates is not explained in terms of the non-existence of sharp boundaries, but in terms of the fact that it is not possible for us to know where those boundaries lie. There is a sharp boundary between a positive and a non-positive extension of “bald”, but this boundary is unknowable. (For a more detailed description of Epistemicism, see Sect. 4.4.1). This theory clearly implies that vague predicates are precise in my sense of the term, and hence it violates CPF. Despite this, Williamson, one of the main proponents of Epistemicism, accepts I1 and CPI. The phenomenon of vagueness is broad. Most challenges to classical logic or semantics depend on special features of a subject matter: the future, the infinite, the quantum mechanical. For all such a challenge implies, classical logic and semantics apply to statements about other subject matters. Vagueness, in contrast, presents a ubiquitous challenge. It is hard to make a perfectly precise statement about anything. If classical logic and semantics apply only to perfectly precise languages, then they apply to no language that we can speak. (Williamson 1994: 2). The emphasis on formal systems has encouraged the illusion that vagueness can be studied in a precise meta-language. It has therefore caused the significance of higher-order vagueness to be underestimated. Indeed, to use a supposedly precise meta-language in studying vague terms is to use a language into which, by hypothesis, they cannot be translated (Williamson 1994: 6–7).

In both passages Williamson is assuming that vague languages are imprecise. The point is not that our vague languages just happen to be imprecise, although there could be a language that is vague but not imprecise. If there could be such a language, then there would be no room for the claim that vague languages cannot be translated into precise ones. The point is thus that to be imprecise is at least a necessary condition for a language to be vague; that is, a language is vague only if it is imprecise. In the same way, if there could be a term that is vague but not imprecise, then there would be no room for the claim that vague terms cannot be translated into  There are other candidates, such as the metaphysical conception of imprecision. As far as I can see, one who accepts a metaphysical conception of imprecision will also accept the linguistic conception, and in particular I2. For example, suppose one claims that imprecision is primarily a property of the semantic values of predicates, for instance the sets or properties that they pick out. Let a set be imprecise if it does not draw a sharp boundary between objects that are in it and ones that are outside it, and a property be imprecise if there is no sharp boundary between objects that have it and ones that do not have it. Independently of whether “bald” picks out the set of bald persons or the property of being bald, there will be no sharp boundary between objects to which “bald” applies and ones to which it doesn’t. The above metaphysical conception implies a corresponding linguistic conception of imprecision, which will lead us to I2 and, in the final analysis, also to I3 and CPF. It is by no means clear how a metaphysical conception of imprecision would pose a problem for my views. In fact, many who endorse a metaphysical conception of vagueness indeed intend to accommodate the fact that vague predicates are linguistically imprecise. (See, for example, the discussion of Tye’s theory of vagueness in the next chapter, Sect. 4.1). 12

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a precise language. A term – whether a name, a predicate or the like – is vague only if it is imprecise. Therefore, Williamson accepts I1. Since he accepts that a predicate is vague only if it is imprecise, he naturally would agree that any theory of vagueness should recognize this fact, in which case he should accept CPI too. (In fact, part of his arguments against alternative theories of vagueness appeal to the claim that they fail to accommodate the imprecision of vague predicates). The upshot is that Williamson accepts I1 and CPI at the same time as he accepts that a vague predicate might admit of a sharp boundary between a positive and a non-positive extension. It is thus evident that he cannot understand imprecision in the same way I do. Of course, he may appeal to a definition of imprecision which I have provided no reason to question. Namely, a predicate is imprecise if and only if it is impossible for us to know its boundary. This definition might be enriched by an explanation of why it is impossible for us to know the boundary of a vague predicate, or an account of the sense in which we are ignorant about the relevant boundary. For the moment, however, it suffices to say that it provides us with an epistemic sense of imprecision, according to which the latter has to do with a certain kind of unknowability of sharp boundaries. According to this epistemic conception, a predicate might be imprecise even if it admits of a sharp boundary between a positive and a non-positive extension. It is this epistemic notion of imprecision that allows Williamson to accept I1 and CPI, while rejecting I2 and CPF. With the epistemic sense of imprecision in hand, he might claim that a predicate is vague only if it is imprecise in this sense. Epistemicism does not violate this intuition. According to this theory, a predicate is vague only if its boundary is unknowable, and it is clear that nothing can be vague in this sense without being epistemically imprecise. As a result, Williamson’s view accepts I1 and satisfies CPI. But since a predicate might be epistemically imprecise without being linguistically so, nothing prevents Williamson from rejecting I2 and CPF. My understanding of precise and imprecise predicates and the corresponding formulation of the criterion of precisification (CPF) are at the heart of many of the arguments and theses advanced in this book. My main objection to some theories of vagueness is that they entail rejection of the intuition that a predicate is vague only if it is linguistically imprecise (I2 interpreted as I3), and hence they violate the criterion of precisification as I understand it (CPF). My main argument in favor of the positive theory outlined in Chap. 5 is that it satisfies all three criteria of adequacy for an ideal theory of vagueness, including CPF. If my sense of imprecision and the corresponding formulation of the criterion of precisification can be rejected, them some of the core theses and arguments in this book might be rejected, and my whole project here is undermined. A not totally satisfactory way to avoid the problem is to turn the main negative and positive theses of this book into conditionals. Instead of saying that a certain theory is implausible because it does not satisfy CPF, I could say something like what follows: if my sense of imprecision and the corresponding formulation of the criterion of precisification, CPF, are to be accepted, them this theory is implausible to the extent that it violates CPF. Instead of saying that an advantage of my theory is that it satisfies CPF, I could say the following: if my sense of imprecision and CPF

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are to be accepted, them an advantage of my theory is that it satisfies CPF. This strategy would not make the theses and arguments in this book arbitrary or trivial. Whether or not there are non-linguistic understandings of imprecision, neither I2 nor CPF is initially implausible. Without independent arguments against I2 and CPF, one cannot accuse a theory of being arbitrary merely because it assumes them. Furthermore, since some traditional theories of vagueness are often supposed to satisfy CPF, both the negative thesis that they fail to do this and the positive thesis that some theory satisfies it are far from trivial. In any case, this strategy is limited in the sense that it makes many of the theses and arguments of this book plausible only to those who endorse I2 and CPF. An alternative strategy is to present positive arguments against the epistemic sense of imprecision. In philosophy, a common way of disputing a definition is by means of counterexamples. Weatherson (2009) presents an interesting example in order to show that to be epistemically imprecise is not a sufficient condition for being imprecise. Here it is. It is possible that a kind of mysterianism about ethics is true, and we cannot know whether good is vague or precise. For a concrete example, let’s assume it is knowable that some kind of divine command theory is true, but it is unknowable whether to be good one must obey all of God’s commands or merely enough of them, where it is vague what counts as enough of them. In fact morality requires obeying all God’s commands, but this is not knowable— for all we know the satisficing version is the true moral theory. If this is the case then good will be epistemically tolerant, for we cannot know that a small difference in how many of God’s commands you obey makes a difference to whether you are good, or determinately good etc. But in fact good is precise, for it precisely means obeying all of God’s commands. Earlier I objected to Eklund’s theory because semantic competence does not require knowing parameters of application, especially as such. This is the converse objection—I claim that a term’s being precise does not imply that we know, or even could know, that it applies in virtue of a precise condition. All that matters is that it does apply in virtue of a precise condition. (Weatherson 2009: 83)

In Weatherson’s imagined example, the predicate “good” is epistemically imprecise, since it is not possible for us to know its boundary. Nonetheless, the argument goes, we would not call this predicate “imprecise”, we would call it “precise”. The upshot is that to be epistemically imprecise is not a sufficient condition for being imprecise. A proponent of the epistemic conception might reply to this in different ways. For example, one might insist that “good”, in the above example, is indeed imprecise, and that there is no legitimate sense in which it is precise. In this case, the proponent of the epistemic sense of imprecision would have to explain away the intuition that “good” is precise. As far as I can see, however, the most plausible strategy would be to reject the claim that “good”, in the above example, is epistemically imprecise. Recall that the definition of epistemic imprecision might be enriched by an explanation of why it is impossible for us to know the boundary of a vague predicate, or an account of the sense in which we are ignorant about the relevant boundary. According to Williamson, for example, our ignorance concerning the boundaries of vague predicates is explained by the idea of inexact knowledge with a conceptual source (1994, Sect. 8.4). Along these lines, it might be claimed

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that the unknowability of a predicate’s boundary is not sufficient for it to be epistemically imprecise; it is also necessary that the unknowability rests on inexact knowledge with a conceptual source. Given that nothing in Weatherson’s example suggests that this is the case for “good”, it cannot be regarded as a counterexample to the epistemic conception of imprecision. Whether or not Weatherson’s example is plausible, I do not think I have to refute the epistemic sense of imprecision. This is because the only situation in which the existence of an epistemic sense would pose a problem for my views leads its proponents to implausible results that require justification, so that the burden of proof is on their side. In order to see this, some points must be clarified. For a start, consider (I2) and (I4). • I2 A predicate is vague only if it does not admit of a sharp boundary between objects to which it applies and ones to which it doesn’t. • I4 A predicate is vague only if it is not possible for us to know any alleged sharp boundaries between objects to which it applies and ones to which it doesn’t. I think no one will disagree that both I2 and I4 are intuitively true. Note in addition that I2 implies that I4 is in some sense correct. If there is no sharp boundary between objects to which a vague predicate applies and ones to which it doesn’t, then it is not possible for us to know where any such boundary lies. In fact, I2 explains I4, so that the reason why it is not possible for us to know the boundary of a vague predicate is that there is no boundary to be known. Since the linguistic conception of imprecision and the corresponding formulation of the criterion of precisification, CPF, conform to I2, and the latter implies that I4 is correct, they also conform to I4. This is exactly what we want, given that both I2 and I4 are intuitively true.13 The moral is that the two senses of imprecision are not mutually inconsistent, and we may accept that vague predicates are both epistemically and linguistically imprecise, that both I2 and I4 are true. I indeed accept this, and so I do not deny that vague predicates are epistemically imprecise. The existence of an epistemic sense of imprecision poses no problem for my views. A problem arises only if one insists, along with Williamson, that there could be a vague predicate that draws a sharp boundary between objects to which it applies and ones to which it doesn’t, in which case I2 would be false. This is because only in this case the epistemic sense of imprecision provides a reason to reject CPF.  Yet, since I2 is intuitively true, its rejection requires justification. In conclusion, the burden of proof is on those who endorse an epistemic sense of imprecision in order to reject the view that vague predicates must be linguistically imprecise.  One could object that I4 cannot be interpreted in terms of I2 on the grounds that the former should be interpreted in terms of the following conjunction: vague predicates have sharp boundaries and it is not possible for us to know where they lie. If I4 is interpreted in this way, then its truth implies the falsehood of I2, and hence no one could accept both of them. I concede that the above interpretation is available, but that we should opt for it is exactly what the proponents of the epistemic conception must show. 13

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3.2.2  Columnar Higher-Order Imprecision The epistemic conception of imprecision might be enriched by a columnar model of higher-order imprecision. This kind of model has been developed by Susanne Bobzien (2013, 2015, 2019), who plausibly argues that it has some important advantages over the hierarchical model. In this section, I briefly present the columnar model and argue that, whether or not it is the correct model of epistemic imprecision, it poses a problem for my views only if we reject I2. As before, this means that the burden of proof is on the side of the proponents of the columnar picture. According to the hierarchical model of imprecision, imprecise predicates admit of a hierarchy of mutually exclusive categories of clarity; that an object is in a certain category implies that it is in none of the others. In this model, something is borderline F only if it can be identified as neither F nor ¬F, so that B(F) is a category including only objects that are neither in F nor in ¬F. This is the distinguishing feature of what Bobzien (2013: 5) calls the “classificatory sense of borderlineness”. The hierarchical model presupposes this classificatory sense. However, borderlineness also admits of an epistemic sense, according to which “borderlineness is the result of an epistemic inaccessibility” (Nader 2017: 9). This sense is better understood if we take borderlineness as contrastive. Instead of saying that a is borderline F, we should say that it is borderline F/¬F, thereby meaning that it is borderline whether Fa or whether ¬Fa. Epistemically interpreted, this means that a can be identified neither as F nor as ¬F; which in turn might be interpreted as meaning that it is not possible for us to know whether a is F, and it is not possible for us to know whether a is ¬F.14 As a result: B(F/¬F)a→[(¬◊KFa) ˄ (¬◊K¬Fa)]. An object a fails to be a borderline F/¬F if it is possible to know that it is F, or if it is possible to know that it is ¬F. In the first case, a is clearly F (CFa), and in the second, it is clearly ¬F (C¬Fa). This is to say that a is borderline F/¬F only if it is neither clearly F nor clearly ¬F: B(F/¬F)a→[(¬CFa) ˄ (¬C¬Fa)]. In this respect, the epistemic and classificatory senses are similar to each other, for something is a borderline F in the latter sense only if it is neither clearly F nor clearly ¬F. Nevertheless, there are some important differences between those two senses of borderlineness. First, in the epistemic (but not in the classificatory) sense, the fact that a is borderline F/¬F is compatible with both a being F and a being ¬F. Second, the fact that a predicate admits of borderline cases in the epistemic sense is compatible with the existence of a sharp boundary between objects to which F applies and objects to which ¬F applies, and in particular with a boundary between objects to which the application of F yields a truth and ones to which it yields a falsehood (Bobzien 2013: 9, see also n. 20). Furthermore, epistemic borderlineness is compatible with there being no such things as borderline clear cases and clear borderline cases. Nothing prevents us from saying that if a is a clear case of F, then it is clearly a clear case of F. This  For simplicity, I am assuming a particular interpretation of “identify” here, while Bobzien remains neutral about the different epistemic interpretations (2013: 8–9). 14

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would only amount to say that if it is possible for us to know that a is F, then it is possible for us to know that it is possible for us to know that a is F. In fact, nothing prevents us from saying that if a is a clear case of F, then it is clear that it is clear that it is clear that… a is F. If this is the case, then there are no borderline clear cases of F. Correspondingly, nothing prevents us from saying that if a is borderline F/¬F, then a is a borderline borderline F/¬F. This would only amount to say that if it is not possible for us to know whether Fa or whether ¬Fa, then it is not possible for us to know whether or not it is possible for us to know whether Fa or whether ¬Fa. In fact, nothing prevents us from saying that if a is borderline F/¬F, then a is borderline borderline borderline… F/¬F. If this is the case, then there are no clear borderline cases. To sum up, the epistemic sense of borderlineness is consistent with the following two statements. • CF→CnF for any n • B(F/¬F)→Bn(F/¬F) for any n Where the upper number indicates the number of occurrences of the relevant operator (B or C). In line with this, let us say that F is imprecisen if there are borderlinen cases of F/¬F. As before, F is higher-order imprecise if n > 1, and it is radically higher-order imprecise if it is imprecisen for any n. With these definitions in mind, the above formulas imply that if F is imprecisen, then F is radically higher-order imprecise. Consequently, there are no merely imprecisen predicates. A further consequence is that there are only three possibilities: clear cases of F, clear cases of ¬F and borderline F/¬F. In ascending to a new order we will not create new categories that include objects not previously included by the old ones. The objects that are clear in the first order correspond exactly to those that are clear in the second, which in turn correspond exactly to those that are clear in the third, and so on indefinitely. Similarly, the objects that are borderline in the first order correspond to those that are borderline in the second, third, and so on indefinitely. No matter the order under consideration, there will be only the three aforementioned possibilities, formed by two classes of clear cases and one of borderline cases. Importantly, there is a single class of borderlineness, B(F/¬F), and hence no distinction between objects that are borderline F and ones that are borderline ¬F. All this summarizes the distinguishing features of the columnar picture of higher-order imprecision. Although Bobzien speaks in terms of vagueness rather than imprecision, the picture is clear enough in the following passage. Columnar higher-order vagueness differs from hierarchical higher-order vagueness in that, extensionally, it contains just one kind of borderline cases, and that each borderline case is radically higher-order, or radically borderline, i.e. borderline borderline …, ad infinitum. Columnar higher-order vagueness also maintains that if something is a clear case, it is radically clear, i.e. clearly clearly … clearly clear, and that if there is something that is borderline, it is borderline that this is so. As a result, there are no clear borderline cases and no borderline clear cases, and it is not clear whether there are any borderline cases. (Bobzien and Keefe 2015: 63)

Here is an example of a graphical representation of three orders of imprecision by the columnar model.

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60 CF B(F/¬F) C¬F CCF BB(F/¬F) CC¬F CCCF BBB(F/¬F) CCC¬F

The fact that the table continues indefinitely can be represented as follows. CF CnF for any n

B(F/¬F) Bn(F/¬F) for any n

C¬F Cn¬F for any n

This model is called “columnar” because the borderline cases form a single column between two columns of clear cases. (Compare it with the hierarchical model, in which the new categories of borderlineness form a pyramid).15 As we have seen, a difficult problem for the enriched many-valued picture – that appeals to the hierarchical model of higher-order imprecision and the claim that vague predicates are radically imprecise – is that vague predicates do not seem to be radially imprecise in the predicted sense. Rather, some vague predicates do not admit of an infinite hierarchy of clarity between the categories of clear cases. One important advantage of the columnar model is that there seems to be no problem in claiming that vague predicates are radically higher-order in the columnar sense. What we need to know is whether or not the claim that they are radically imprecise in this latter sense poses any problems for my views. Let us suppose that the columnar model provides us with a correct explanation of the epistemic sense of imprecision, and that vague predicates are radically imprecise in the columnar sense. These two suppositions form what I call the “enriched epistemic picture of imprecision”. In this case, a soritical sequence for “bald”, for example, is such that there is a number n such that it is not possible for us to know whether an is bald or whether an is not bald. By virtue of this, it is not possible for us to know the boundary between bald and not bald persons. This explains I4. Note, however, that the enriched epistemic picture is not required by I4. It explains I4 by interpreting the unknowability of the boundary of a vague predicate in terms of the epistemic sense of borderlineness. Yet, one might accept the unknowability of the boundary without accepting the proposed interpretation, and hence one might accept I4 without accepting the enriched epistemic picture. In fact, the rejection of the latter does not commit us to the rejection of any clearly intuitive thesis (CPI, I1, I2, I4, etc.). Granted that it is possible to interpret the unknowability of the boundary and I4 in terms of epistemic borderlineness, it is not granted that we should do this. We need positive arguments to the effect that these interpretations should be adopted. The enriched picture is clearly consistent with CPI and I1. As before, the main problem concerns I2. This picture does not imply the truth of I2. The claim that vague predicates are imprecise in the columnar sense does not imply that a predicate 15

 See n. 5

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is vague only if there is no sharp boundary between objects to which it applies and objects to which it doesn’t. But let us assume, for the sake of argument, that the enriched epistemic theory is at least consistent with I2. We are left with two options: either accept or reject I2. Suppose we accept I2 in addition to the enriched epistemic picture. In this case, it is not clear how a columnar sense of epistemic imprecision would affect my points. My arguments to the effect that we should interpret I2 in terms of I3, and the consequent formulation of the criterion of precisification, CPF, would still hold. On the other hand, if one tries to accept I2 but not I3 and CPF, then one has to provide an explanation of I2 that is better than that provided by I3. This is no easy task. As we have seen, neither the classical nor the many-valued picture adequately explains the linguistic sense of imprecision, and hence neither adequately explains I2. As before, it seems that a problem for my view would arise only if one rejects I2. If a proponent of the enriched epistemic picture rejects I2, she has reason to reject I3 and CPF as well. But I2 is intuitively true, and its rejection requires justification. Thus, the only circumstance in which the enriched epistemic picture poses a problem for my view is the one that leads its proponents to implausible conclusions that require justification. By contrast, the linguistic sense of imprecision and CPF conform to both I2 and I4 above. This is an advantage of my proposal, and makes it initially superior to its competitor. Again, the burden of proof is on those who endorse the columnar sense of imprecision in order to reject the view that vague predicates must be linguistically imprecise. It has been said that the discussion of higher-order imprecision is often framed in terms of a discussion of higher-order vagueness. As the above passage from Bobzien shows, this also holds for the columnar model.16 As before, one might take the borderline cases intuition as the most fundamental intuition concerning the nature of vagueness, and define vagueness in terms of borderlineness. We then interpret borderlineness in epistemic terms, and explain it by means of the columnar model. The result is that to be vague is to be radically higher-order vague in the columnar sense. This strategy raises the same problem as that considered above. Since we are defining vagueness in this way, we must assume that linguistic imprecision is not a necessary condition for vagueness, that a vague predicate might be linguistically precise. This is to say that I2 is false. Since I2 is intuitively true, its rejection requires justification. To be sure, this is no decisive argument against the enriched epistemic theory or the columnar model. But it highlights that the burden of proof is not on my side, and my adoption of a linguistic interpretation of imprecision is by no means arbitrary or unjustified. Finally, it should be noted that the enriched epistemic picture shares some questionable presuppositions with the enriched many-valued picture. Both take the borderline cases intuition as the fundamental intuition about the nature of imprecision,  See Bobzien (2019) for a solution of the sorites paradox in terms of the columnar model of higher-order vagueness. I will not evaluate Bobzien’s theory of vagueness in the next chapter. But my objection to it, and in fact my objection to any variety of Epistemicism, is that it fails to satisfy CPF. In the next chapter, I briefly discuss Williamson’s version of Epistemicism. 16

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and both assume that some core problems raised by this idea must be solved by distinguishing between predicates that are and those that are not higher-order imprecise, and claiming that vague predicates are radically higher-order imprecise. I reject both presuppositions. First, the definition of imprecision presented in Sect. 3.1.3 does not appeal to the phenomenon of borderlineness. Of course it might be objected that, in the final analysis, we will need to appeal to borderlineness in order to explain what a predicate that satisfies that definition should be like. In due time, we will see that this is not the case. It is the phenomenon of the arbitrariness of the boundary (of vague predicates) that helps us to describe what a predicate should be like in order to be imprecise in the desired sense. Second, my definition of imprecision and the corresponding formulation of the criterion of precisification, CPF, do not require that vague predicates be higher-order imprecise, much less radically higher-order imprecise. Higher-order imprecision plays no special role in the theses I am presenting in this book.17

3.2.3  Linguistic Imprecision Versus Epistemic Imprecision (II) We have seen that the existence of an epistemic sense of imprecision is a problem for my view only if one accepts in addition that there could be a vague predicate with a sharp boundary between objects to which it applies and ones to which it doesn’t. In other words, a problem arises only if one rejects I2. Since I2 is intuitively true, its rejection must be justified. I think the most plausible way to justify this would be by appealing to the obscurity of the idea of linguistic imprecision. Recall that the naïve many-valued picture did not provide us with a satisfactory account of the linguistic sense of imprecision. Although Def.1 is a better alternative, it is not clear about what an imprecise predicate should be like. After all, it is not clear what a predicate that fails to admit of a sharp boundary between any kinds of extensions should be like. The result is that we have no clear picture of linguistically imprecise predicates thus far. In this context, we should be suspicious of the very idea of linguistic imprecision. Since I2 depends on this conception of imprecision, we should be suspicious of it too.

 This is not to say that there is no room for higher-order imprecision, or that higher-order imprecision is impossible. I might say, for example, that higher-order imprecision is nothing but imprecision in a metalanguage. A metalanguage is a language used to speak about another language. Roughly, languages can be ordered on different levels. The language we use to speak of objects is a first-order language; the one we use to speak of the first-order language is a second-order language; the one we use to speak of the second-order language is a third-order language; and so on for a potentially infinite hierarchy of languages. With the exception of the first-order language, all others are metalanguages. Correspondingly, there may be imprecision on each level: first-order imprecision, second-order imprecision, third-order imprecision, etc. With the exception of firstorder imprecision, all others are instances of higher-order imprecision. In this context, to say that a predicate is n-order imprecise would be to say that it is a n-order predicate and it is imprecise. 17

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A possible reply on behalf of linguistic imprecision is that it might be understood in terms of the idea of tolerance. A predicate that satisfies Def.1 is simply one that is tolerant to very slight changes. This would provide a clear definition of imprecise predicates, Def.1, and a clear picture of what they should be like. Unfortunately, it seems that this leads us to undesired consequences. The initially most plausible way to solve the sorites paradox, to reject the principle of tolerance, would not be available anymore. As a result, both the sorites paradox and the consequent incoherence of vague predicates would blow up once again. It is far from clear whether the price of such a picture would be worth paying. Furthermore, in Sect. 5.1.2 I will argue that, contrary to what may appear to be the case, the imprecision of vague predicates cannot be easily accommodated by the thesis that they are tolerant. On the one hand, we have no initially plausible picture of what an imprecise predicate should be like. On the other hand, the idea of unknowability of the boundary does not seem to be obscure, and the epistemic sense of imprecision provides a sufficiently clear picture of what an imprecise predicate should be like. In this context, it would not be implausible to accept I4 and reject I2. There is no better way to reply to this objection than by framing a satisfactory picture of what linguistically imprecise predicates should be like. In Chap. 5, I present a picture of vague predicates according to which they are imprecise in the sense of Def.1, and so I clarify what a predicate that satisfies this definition should be like. Final Remarks I have defined an imprecise predicate as one that admits of no sharp boundary between objects to which its application yields some particular truth-value and ones to which it doesn’t. Accordingly, I proposed that CPI should be formulated as CPF. My main goal in this chapter was to provide this definition and the corresponding final formulation of the criterion of precisification. CPF will play a prominent role in this book. This is not because I think that the criterion of precisification is more important than the other two criteria, but because it is a major problem for most theories of vagueness. As we will see in the next chapter, it is not totally clear how a theory could satisfy this criterion, and this is part of the reason that makes Epistemicism an especially problematic opponent for many theories of vagueness. Although Epistemicism rejects CPF, it has the benefit of theoretical simplicity; and this is an important advantage over theories that not only fail to satisfy CPF, but also complicate our logic and semantics in many ways. Recall that I do not think that a theory that fails to satisfy some criterion of adequacy is unacceptable, and hence the mere fact that Epistemicism rejects CPF does not mean that it is off the table. But I do think that a theory that satisfies all three criteria of adequacy has an important advantage over it. In fact, I suspect a theory of vagueness can hardly offset the theoretical simplicity of Epistemicism, and the plausibility of (at least many) alternative theories will depend on whether or not they have this advantage. My main goal in this book is to outline a theory that indeed has this property, that satisfies all three criteria of adequacy for an ideal theory of vagueness. Finally, hereafter I will often speak of CPI and CPF indiscriminately as the criterion of precisification. If my goal is to explain a certain theory and its apparent

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advantages, it is harmless to speak indiscriminately of them. But if my goal is to evaluate whether or not a certain theory satisfies the criterion of precisification, then I must (and I will) be explicit about how I understand this criterion.

References Bobzien, S. (2013). Higher-order vagueness and borderline nestings: A persistent confusion. Analytic Philosophy, 54(1), 1–43. Bobzien, S. (2019). A generic solution to the Sorites paradox based on the normal modal logic QS4M+BF+FIN.  In A.  Abasnezhad & O.  Bueno (Eds.), The sorites paradox: New essays. Springer. Forthcoming. Bobzien, S., & Keefe, R. (2015). Columnar higher-order vagueness, or vagueness is higher-order vagueness. Proceedings of the Aristotelian Society Supplementary, 89(1), 61–87. Barnes, J. (1982). Medicine, experience and logic. In J.  Barnes, J.  Brunschwig, M.  Burnyeat, & M.  Schofield (Eds.), Science and speculation: Studies in Hellenistic theory and practice (pp. 24–68). Cambridge: Cambridge University Press. Fine, K. (2008). The impossibility of vagueness. Philosophical Perspectives, 8, 111–136. Fine, K. (2020). Vagueness: A global approach. Oxford: Oxford University Press. Horgan, T. (1994). Robust vagueness and the forced-march sorites paradox. Philosophical Perspectives, 8, 159–188. Keefe, R. (2000). Theories of vagueness. Cambridge: Cambridge University Press. Lewis, D. (1986). On the plurality of worlds. Oxford: Blackwell. Nader, K. 2017. On the columnar model for higher-order vagueness. PhD Thesis. Columbia University. Sainsbury, M. (1991). Is there higher-order vagueness? The Philosophical Quarterly, 41(163), 167–182. Sainsbury, M. (2013). Lessons for vagueness from scrambled sorites. Metaphysi-ca, 14(2), 225–237. Shapiro, S. (2006). Vagueness in context. Oxford: Oxford University Press. Soames, S. (1999). Understanding truth. Oxford: Oxford University Press. Weatherson, B. (2009). Vagueness as indetermination. In D. Richard & S. Moruzzi (Eds.), Cuts and clouds: Vagueness, its nature, and its logic (pp. 77–90). Oxford: Oxford University Press. Williamson, T. (1994). Vagueness. London: Routledge.

Chapter 4

Theories of Vagueness

I have formulated the problem of vagueness as follows (Sect. 2.4). • Problem of vagueness: the problem of explaining vague predicates (expressions) in a way that systematizes the six relevant intuitions concerning the nature of vagueness and, if possible, satisfies all three criteria of adequacy for an ideal theory of vagueness (criterion of sorites, criterion of coherence and criterion of precisification). The relevant intuitions were presented in Sect. 2.3, and the three criteria of adequacy in Sect. 2.2.2. In the previous chapter. I argued that CPI, my initial formulation of the criterion of precisification, should be interpreted in terms of CPF, my final formulation of it. • CPI Explain vague predicates without implying that there could be a predicate that is both vague and precise. • CPF Explain vague predicates without implying that a predicate could be vague and precise at the same time; that is, without implying that a predicate could be vague at the same time that there is a sharp boundary between objects to which its application yields some particular truth-value and ones to which it doesn’t. With all this in hand, we can finally begin to evaluate some theories of vagueness. In this chapter, I take into account six theories: Three-Valued Theory, Degrees-­ of-­ Truth Theory, Supervaluationism, Epistemicism, Incoherentism and Contextualism. Each of them will be evaluated with respect to whether it does or does not solve the problem as formulated above. Hereafter, the expression “the problem of vagueness” will refer to that formulation of the problem. Each theory will be supposed to be a solution to this problem. For reasons that will become clear later, I will discuss Epistemicism, Incoherentism and Contextualism in a single section. It is important to note that the details of each theory will be considered only to the extent that they are relevant to my purposes. Because of this, I will ignore some

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usually important issues on these theories.1 For example, both Three-Valued Theory and Degrees-of-Truth Theory are based on many-valued logics, and hence they give rise to many technical problems, which include problems concerning the correct interpretation of connectives, the correct definition of validity, etc. These problems will largely be ignored here. Moreover, the aforementioned theories would deserve a much longer discussion than the following. This becomes all the more evident if we draw attention to the fact that there are different versions of each. To be sure, “Three-Valued Theory”, “Degrees-of-Truth Theory”, “Supervaluationism”, etc. are indeed names of families of theories. Of course family members are connected by a series of similarities, but there may also be important differences among them. A detailed evaluation of these theories should take these differences into account, but I will only occasionally do this. All in all, I do not intend to provide knockdown arguments against these theories of vagueness. However, I will argue to the effect that none of them satisfies all three criteria of adequacy, while two of them (Three-Valued Theory and Supervaluationism) fail to adequately systematize the relevant intuitions.2 This ultimately means that none of them has a totally satisfactory solution to my formulation of the problem of vagueness. In this context, we may naturally raise the following question: is there any theory of vagueness that satisfies all three criteria of adequacy and systematizes the six relevant intuitions? In Chap. 5, I will develop such a theory.

4.1  Three-Valued Theory Three-Valued Theory explains vague predicates in terms of a third truth-value or a gap between the two classical values: true and false. Although they require a three-­ valued logic, not all theories that require this kind of logic are instances of Three-­ Valued Theory of Vagueness. The distinctive feature of these theories is that the third truth-value, or the gap between the two classical values, is placed at the center of the explanation of vague predicates. Both Supervaluationism and some versions of Contextualism (Soames 1999) require a three-valued logic, but they are not instances of Three-Valued Theory; for Supervaluationism places the notions of precisification and supervaluation at the heart of the explanation and, according to Contextualism, what is central to vagueness is some kind of context-sensitivity.  Williamson (1994) and Keefe (2000) dedicate a whole chapter to each different theory, and much more detailed discussions can be found there. 2  The case of Contextualism is indeed problematic. I identify three core theses of Contextualism, and it seems that by themselves they imply neither a violation nor a satisfaction of the three criteria of adequacy. But if they are interpreted in terms of a classical or a many-valued semantics, then Contextualism amounts to a violation of the criterion of precisification. In the next chapter, I argue that at least two of the three core theses of Contextualism are correct and they can be accommodated by a theory that satisfies all three criteria of adequacy for an ideal theory of vagueness. If those two theses are sufficient for a theory to be contextualist, then my proposal is a version of Contextualism, and the latter can satisfy the relevant criteria. 1

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The best way to begin to think about Three-Valued Theory of Vagueness is to appeal to the borderline cases intuition. As we have seen, it seems that vague predicates admit of borderline cases, that is, cases to which it is not clear whether or not the predicate applies. This intuition is an initial support for Three-Valued Theory. In order to see this, consider the following sentence: 1. John is bald. If John has 0 hairs on his head, then the proposition expressed by (1) is true. If John has 100 percent of his scalp covered by hair, then the proposition expressed by (1) is false. But what if he has 80 percent of his scalp covered by hair (if you like, think of a more favorable percentage)? Intuitively, it would be incorrect to say that the proposition expressed by (1) is true in this case, just as it would be incorrect to say that it is false. The simplest explanation for this is that the proposition is neither true nor false or, as Tye (1994) prefers to say, indefinite.3 There are at least two different interpretations of indefinite. It may be interpreted as either a third truth-value or a gap between true and false. One reason for the first interpretation is that it can be easily accommodated to the intuition that (1) expresses a proposition even when John is a borderline case of “bald”. After all, propositions are supposed to be the bearers of truth-values, independently of there being two, three or infinite values. On the other hand, this interpretation gives rise to the metaphysical problem of explaining what this third value, the indefinite, consists in. One reason for the second interpretation is that, since it does not postulate a third truth-­ value, it does not raise the metaphysical question of what it is. On the other hand, it is not clear how this interpretation could accommodate the aforementioned intuition, and it would not be plausible to deny it. As far as possible, I remain neutral as to which interpretation is best. For the sake of simplicity, I will use the expression “indefinite sentence”, and you may read it as either “a sentence that expresses a proposition that has the value indefinite” or “a sentence that fails to express a proposition that has a truth-value” (either because it expresses a proposition that has no truth-value or because it does not express a proposition at all). The question I am concerned with is this: how can a theory of vague predicates be based on the idea of an indefinite sentence? We can find an answer to this question if we look at the constituents of sentences. According to a classical picture, a sentence such as (1) is true if the person referred to by the name “John” is an element of the set picked out by “bald”; otherwise it is false. If this is correct, the domain of application of “bald” is divided into two parts, one formed by all persons to whom the predicate applies, its positive extension, and the other formed by all persons to whom it does not apply, its negative extension. All members of the  Sentences like (1) are also said to be indeterminate, but I prefer to say indefinite rather than indeterminate. The word “indeterminate” is already broadly used to speak of metaphysical indeterminacy, which includes kinds of indeterminacy that are not directly related to vagueness. One paradigmatic example of metaphysical indeterminacy is the case of the open future – it is now indeterminate whether it will rain tomorrow – but this kind of indeterminacy has nothing to do with vagueness. Arguments against the use of “indeterminate” to speak about vagueness can be found in Williams (2008: 767) and Eklund (2005: 29–30). 3

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domain are in either the negative or positive extension of the predicate. According to Three-Valued Theory, vague predicates do not fit into this picture. What is wrong with the above picture is that vague predicates do not divide their domains of application into two parts. If they did this, sentences such as (1) would be either true or false, and there would be no room left for a third truth-value or gap between the two classical ones. This would violate the intuition that (1) is neither true nor false if John is a borderline case of “bald”. In order to accommodate this intuition we must also accept the existence of a penumbra. The penumbra of a vague predicate is formed by all borderline cases of that predicate.4 If John is a member of the penumbra of “bald”, then (1) is neither true nor false. In light of Three-Valued Theory of Vagueness, it is a distinctive feature of vague predicates that, in addition to the positive and negative extensions, they also have a penumbra. There may be disagreement as to whether the explanation for the existence of a penumbra is linguistic or metaphysical. Tye adopts a metaphysical conception. For a start, he assumes that there are vague sets. One important feature of these sets is that, for some objects, “there is no determinate, objective fact of the matter about whether they are in the set or outside it” (1994: 195). If John is a borderline case of the set of bald persons, then there is no fact of the matter about whether or not he is an element of this set. Since the predicate “bald” picks out the set of bald persons, there will be no fact of the matter about whether or not “bald” applies to John. As a consequence, John is in the penumbra of “bald”, in which case (1) would be indefinite. The result is that the predicate “bald” has a penumbra only because the set of bald persons has a penumbra. More generally, vague predicates have a penumbra only because the sets they pick out do.5 By contrast, it is also possible to adopt a linguistic explanation for the existence of a penumbra. The usual strategy is to explain the existence of a penumbra in terms of the conditions for the c­ orrect/incorrect application of vague predicates. Although Scott Soames is not a proponent of  It might not be correct to say that all borderline cases are in the penumbra, for this suggests that the domain of application of a vague predicate is divided into only three parts: a positive extension, a negative extension and a penumbra. Yet this assumption might be rejected. Tye (1994: 195), for example, claims that there is no fact of the matter about whether or not the domain of a vague predicate is divided solely into three parts. I will return to this point in Sect. 4.1.2, where I discuss how Three-Valued Theory deals with the problem of precisification. 5  In addition to vague sets, Tye also accepts vague properties. The definition of vague sets is different from the definition of vague properties. A set is vague if and only if (i) it has members, not members and borderline members, and (ii) there is no fact of the matter about whether these are the only relevant categories (Tye 1994: 195). A property is vague if and only if (i) it could have instances, not instances and borderline instances, and (ii) there is no fact of the matter about whether there could be other categories (Tye 1994: 195–196). If there were only persons who have 0 hairs on their head, the set of bald persons would not be vague anymore, but the property of being bald would remain vague. One could argue that, since the predicate “bald” would remain vague in this case, it would be better to associate vague predicates with vague properties rather than vague sets. So, we should explain the penumbra of vague predicates in terms of the penumbra of vague properties: vague predicates have a penumbra only because the properties they pick out have a penumbra. I have no objection to this. But since this distinction does not affect any of my further points, we can leave it aside. 4

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Three-Valued Theory, his notion of partially defined predicates provides us with a good example of a linguistic conception of the penumbra. Here it is: Our task is to construct a model of partially defined predicates according to which, for certain things, both the claim that these predicates apply to those things and the claim that they do not apply to them must be rejected. The key to this model is the contention that a predicate may be introduced into a language by rules that provide sufficient conditions for it to apply to an object and sufficient conditions for it to fail to apply but no conditions that are both individually sufficient and jointly necessary for it to apply or for it to fail to apply to an object. Since the conditions are mutually exclusive but not jointly exhaustive, there will be objects for which there are no possible grounds for accepting either the claim that the predicate applies to them or the claim that it does not. (Soames 1999: 163).

Roughly, a vague predicate has a penumbra because the conditions for the correct/incorrect application of the predicate are not fully determinate. John is in the penumbra of “bald” if he does not meet the sufficient conditions for the correct application of “bald” and does not meet the sufficient conditions for the incorrect application either. In this case, (1) would be indefinite. Against Soames, one could insist that if an object does not satisfy the sufficient conditions for the correct application of a vague predicate, then nothing prevents it from being in the negative extension of that predicate. If John does not meet the sufficient conditions for the correct application of “bald”, why shouldn’t we just say that he is in the negative extension of “bald”? In order to answer this question, Mark Richard (2009: 466) reformulates Soames’s proposal as follows: a predicate F has a penumbra if (i) there is a set of conditions, C1, that are individually necessary and jointly sufficient for the correct application of F, (ii) there is a set of conditions, C2, that are individually necessary and jointly sufficient for the incorrect application of F, (iii) C1 and C2 are mutually exclusive, and (vi) C1 and C2 are not jointly exhaustive. In this view, a vague predicate has fully determinate conditions for its correct application and fully determinate conditions for its incorrect application, though they are not jointly exhaustive. Now, it might be said that the set of conditions C1 for the correct application of “bald” determines that it is not correct to apply “bald” to John, while the set C2 for the incorrect application of “bald” determines that it is not incorrect to apply “bald” to John. The latter fact is what prevents John from being in the negative extension of “bald”. John would be a borderline case of “bald”, and hence (1) would be indefinite. Criticisms of the very idea of a third truth-value, or a gap between the two classical ones, can be found in Glanzberg (2003). For present purposes, this is enough. In short, vague predicates do not divide their domains of application into only two parts. In addition to the negative and positive extensions, there is also a penumbra formed by all borderline cases. This is what explains the fact that sentences containing vague predicates can be neither true nor false. When a is a borderline F, Fa is indefinite. According to Three-Valued Theory of Vagueness, the distinctive feature of vague predicates is that they fit into the above kind of picture.

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4.1.1  Solution to the Sorites The solution to the sorites paradox will depend on which version of Three-Valued Theory of Vagueness is considered. More precisely, it depends on how we interpret connectives, quantifiers and the notion of validity. The postulation of a third value, or a gap between the two classical values, makes it necessary to reformulate some classical definitions. It is possible to adopt a definition of validity that renders the sorites argument invalid, but the simplest solution is to maintain validity as preservation of truth and reject some of its premises. In the latter case, the only remaining question would concern the interpretation of connectives and quantifiers. The main disagreement with respect to connectives is about whether they should be truth-­ functionally (Tye 1994) or not truth-functionally interpreted (Richard 2009). Furthermore, there are different (not) truth-functional interpretations of connectives (Keefe 2000: 100–104; Williamson 1994: Chap. 4). I will not go into details about these issues here. In what follows, I assume Tye’s interpretation and present his solution to the paradox. Tye adopts one of Kleene’s truth-functional interpretations of connectives (Kleene 1952: 332–340). A T T T I I I F F F

B T I F T I F T I F

¬A F F F I I I T T T

A v B T T T T I I T I F

A ʌ B T I F I I F F F F

A → B T I F T I I T T T

There is a sense in which the above truth-tables are in accordance with classical truth-tables; namely, they satisfy the normality constraint/requirement (Keefe 2000: 86; Machina 1976: 55). This is to say that when all sentences in a compound sentence express propositions with classical truth-values, the classical reading of the latter is correct. For example, if A is false and B true, then ¬A will be true, A v B true, A ʌ B false, A → B true, and so on. Differences with respect to classical truth-­ tables come out only when some sentence (expresses a proposition that) is indefinite. The negation of an indefinite sentence A is itself indefinite. On the one hand, if ¬A were true, A would be false, and hence it would not be indefinite. On the other hand, if ¬A were false, A would be true, and hence it would not be indefinite. If at least one disjunct is true, then the whole disjunction is true; if both disjuncts are false, the disjunction is itself false; otherwise, it is indefinite. If both conjuncts are true, the whole conjunction is true; if some conjunct is false, the conjunction is itself

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false; otherwise, it is indefinite. In regard to the conditional, Tye only observes that A →  B should be equivalent to ¬A  v  B; while A  ↔  B should be equivalent to (A → B) ʌ (B → A) (Tye 1994: 194). The existential quantifier is defined as follows. ƎxFx is true if at least one of its instances is true and false if all instances are false; otherwise, it is indefinite. The intuition behind the definition is that an existential sentence is indefinite when none of its instances is true, and at least one is indefinite. The same holds for the disjunction of all instances: Fa1 v Fa2 v Fa3... v Fan. The universal quantifier is defined as follows. ∀xFx is true if all its instances are true and false if at least one instance is false; otherwise, it is indefinite. The idea is that a universal sentence is indefinite when none of its instances is false, and at least one is indefinite. The same holds for the conjunction of all instances: Fa1 ˄ Fa2 ˄ Fa3... ˄ Fan (Tye 1994: 196). I turn next to Tye’s solution to the sorites paradox. Consider first the quantified version. Basically, the strategy is to reject the principle of tolerance without accepting its negation. Recall the principle for the predicate “bald”: ∀n (Ban → Ban + 1). Given that no instance of the principle is false, the principle itself is not false. Nonetheless, some of its instances are indefinite, for they have an indefinite antecedent and an indefinite consequent. According to Three-Valued Theory, vague predicates have a penumbra, formed by all cases to which it is not clear whether or not they apply. In light of this, it is plausible to suppose that, for some n, both an and an  +  1 are borderline cases of “bald”. As a consequence, both Ban and Ban  +  1 are indefinites. Since a conditional of which both the antecedent and consequent are indefinites is itself indefinite, Ban → Ban + 1 is indefinite. So, there is at least one instance of ∀n (Ban → Ban + 1) that is indefinite. When no instance of a universal sentence is false and some instance is indefinite, the universal sentence is indefinite. Therefore, ∀n (Ban → Ban + 1) is indefinite. Generalizing: the principle of tolerance is indefinite, or, if you like, neither true nor false. This proposal makes room for the solution to the conditional version: at least one of its conditional premises is indefinite. The solution to the line-drawing version of the sorites is especially interesting. This version of the paradox goes from Ba0 and ¬Ba10,000 to the conclusion that Ǝn (Ban ˄ ¬Ban + 1). Tye’s Three-Valued Theory renders this argument invalid. Given that any person with no hair is in the positive extension of “bald”, Ba0 is true. Given that any person who has 10,000 hairs on her head is in the negative extension of “bald”, ¬Ba10,000 is true too. Yet, from this it does not follow that there is a number n between 0 and 10,000 such that Ban is true and Ban + 1 is false. This conclusion would be true only if the predicate “bald” divided its domain of application into only a positive and a negative extension. As we have seen, however, that is not the case; for there is also a penumbra. In fact, for some number n between 0 and 10,000, Ban is indefinite. Therefore, the line-drawing version is invalid.

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4.1.2  Criteria of Adequacy Three-Valued Theory of Vagueness apparently provides us with a solution to the sorites, and it seems clear that it does not imply the incoherence of vague predicates. So, we are left with the question of whether or not it satisfies the criterion of precisification. In the aforementioned classical view, a vague predicate divides its domain of application into a positive and a negative extension, so that every object is in either one or the other. According to this picture, each person is in either the positive or the negative extension of “bald”. A soritical sequence for “bald” would thus be divided in such a way that there would be a number n such that a person with n hairs on her head is bald and a person with n + 1 is not bald. The result can be represented as follows: positive ext.     negative ext. |a0, a1, a2, a3…, an | an + 1, an + 2…, a10, 000| true            false

That view manifestly implies that “bald” is a precise predicate. There is a last person in the sequence to whom “bald” applies immediately followed by a first person to whom it does not apply. In other words, there is a sharp boundary between bald and not bald persons. Three-Valued Theory avoids this consequence by positing a penumbra between the positive and negative extensions of “bald”. The result can be represented as follows: positive ext.   penumbra   negative ext. |a0, a1…, an | an + 1…, an + m | an + m + 1…, a10, 000| true   indefinite   false The application of “bald” to a0, a1…, an yields sentences that express a true proposition, while the application of “bald” to an  +  m  +  1..., a10,000 yields sentences that express a false proposition. Let us call the former group “positive cases of ‘bald’” and the latter “negative cases of ‘bald’”. In the above picture there is indeed a last positive case, an, and a first negative case, an + m + 1, of “bald”. Between them, however, there is a set of persons, an + 1..., an + m, to whom the application of “bald” yields sentences that express an indefinite proposition or fail to express a proposition with a truth-value. This third group is formed by all borderline cases of “bald”. The existence of borderline cases entails that there is no sharp boundary between positive and negative cases of “bald”. As a consequence, “bald” is not a precise predicate.

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Since all vague predicates fit into this picture, Three-Valued Theory of Vagueness satisfies the criterion of precisification. This proposal gives rise to the same kind of problems raised by the naïve many-­ valued picture of imprecision (Sect. 3.1.1). Note that an is the last positive case and an + 1 is the first borderline case of “bald”. Thus, there is a sharp boundary between positive and borderline cases. This is to say that there is a sharp boundary between persons to whom the application of “bald” yields a truth and ones to whom it does not yield a truth. On the other hand, an + m is the last borderline case and an + m + 1 is the first negative case of “bald”. Thus, there is also a sharp boundary between borderline and negative cases. This is to say that there is a sharp boundary between persons to whom the application of “bald” yields a falsehood and ones to whom it does not yield a falsehood. Consequently, there is a sharp boundary between persons to whom the application of “bald” yields some particular truth-value and ones to whom its application does not yield that truth-value. Similar results hold for any other vague predicate. CPF is violated. Vague predicates are not imprecise. As it stands above, Three-Valued Theory raises the problem of precisification. If we assume that the notions of being vague and being precise are inconsistent with each other, Three-Valued Theory implies that the alleged vague predicates – “bald”, “tall”, “short”, etc. – are not really vague. Even if we deny this intuition, however, the foundational problem of precisification will remain unsolved. This is the problem of explaining how the alleged precise boundaries of vague predicates are determined. Given that Three-Valued Theory implies that ordinary vague predicates admit of two sharp boundaries, its proponents must explain how each of them is determined. How is the boundary between positive and borderline cases determined? How is the boundary between borderline and negative cases determined? In Sect. 2.2.3 I said that what makes it especially difficult to answer questions like these is that for any vague predicate, there will be many cut-off points with an equal right to be the boundary of that predicate. No matter what the value of n is, it is hard to see why an would be a better candidate for the last positive case of “bald” than an - 1 or an + 1. In the same way, it is hard to see why an + m would be a better candidate for the last borderline case of “bald” than an + m - 1 or an + m + 1. As we have seen, this is a difficult problem. Let us turn next to two possible replies on behalf of Three-Valued Theory. In my argument I am assuming that the domain of application of a vague predicate is divided into only three parts, so that each object is in the positive extension, the penumbra or the negative extension of the predicate. The first strategy is to reject this supposition. The idea is not to accept that the domain of application of a vague predicate is divided into more than three parts. On the contrary, the idea is to reject the thesis that there are only three parts without accepting that there are more than three parts. According to Tye (1994: 195), for instance, we should accept the following: • A vague predicate admits of positive cases, borderline cases and negative cases; and there is no fact of the matter about whether these are the only cases.

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The above thesis does not imply that the domain of a vague predicate is divided into more than three parts. But it prevents us from supposing that the domain is divided into only three parts. Since the argument against Three-Valued Theory of Vagueness rests on that supposition, it is not sound. Tye goes on to argue in favor of the above thesis in the following manner. We have seen that vague predicates admit of at least three categories of cases: positive, borderline and negative cases. Now ask yourself: are there only three categories? Suppose the answer is “yes”. In this case, vague predicates are precise predicates. Suppose the answer is “no”. In this case, there must be a fourth category of cases, and the relevant question would be this: are there only four categories? If yes, then vague predicates are precise. If not, then there must be a fifth category. Are there only five categories? If yes, vague predicates are precise. If not, then there must be a sixth category. Are there only six categories? And so on indefinitely. On the one hand, it is not true that there are only three categories, because this entails that vague predicates are precise predicates. On the other hand, it is not false that there are only three categories, because this would commit us to “gratuitous metaphysical complications” (Tye 1994: 195) such as, for example, the postulation of many further truth-­ values. Consequently, there is no fact of the matter about whether there are only three categories of cases. We can concisely express this by saying that it is indefinite that there are only three categories of cases. Keefe (2000: 121–122) has formulated a compelling objection to the above argument. Tye’s argument can be summed up like this: 2 . It is not true that there are only three categories. 3. It is not false that there are only three categories. 4. It is indefinite that there are only three categories. How does (4) follow from (2) and (3)? From the fact that a sentence is not true or false it follows that it is indefinite only if indefinite is the only remaining option. If there were a fourth value, or a second gap, indefinite*, the most we could conclude is that the relevant sentence is indefinite or indefinite*. This is to say that (4) follows from (2) and (3) only if there are only three values: true, false and indefinite. But if there are only three truth-values – or two values and a gap – then there are only three categories of cases, each category corresponding to a particular value or gap. The upshot is that in order to show that it is indefinite that there are only three categories, Tye assumes that there are only three categories. In the final analysis, he is relying on the very supposition he intends to reject. Mark Richard (2009: 476) adopts a different strategy to solve the problem. He thinks that although vague predicates indeed admit of three categories of cases, it would be wrong to say that there is a last positive case, a first/last borderline case and a first negative case. Richard’s rebuttal depends on how he understands the notions of penumbra and indefinite.6 As we saw in Sect. 4.1, he adopts a linguistic

 Richard prefers “indeterminate” rather than “indefinite”. For the reason I mentioned earlier (n.3), I prefer “indefinite”. 6

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explanation for the existence of a penumbra. Roughly speaking, a vague predicate has a set of fully determinate conditions, C1, for its correct application and a set of fully determinate conditions, C2, for its incorrect application, although C1 and C2 are not jointly exhaustive. In order for John to be bald, he must satisfy C1; in order for him to be not bald, he must satisfy C2. Yet, given that C1 and C2 are not exhaustive, it might be that they say nothing about John’s situation, which would mean that it is not settled whether or not John is bald. When this happens, John is a borderline case of “bald”. Finally, Richard does not understand indefinite as a third truth-value, but as a kind of gap between true and false. If it is not settled whether or not John is bald, then it is not settled whether (1) is true or whether (1) false. We can express this by saying that it is indefinite whether (1) is true or false. There are only two truth-values, true and false, but sometimes it is indefinite whether a sentence expresses a true proposition or a false one. Once this is in place, we can formulate Richard’s argument in the following way. By definition, an + 1 is a borderline case of “bald”, which here means that it is not settled whether the application of “bald” to her yields a true proposition or whether it yields a false proposition. As a consequence, (a) it is not settled whether or not the application of “bald” to an  +  1 yields a true proposition, and (b) it is not settled whether or not the application of “bald” to an  +  1 yields a false proposition. Nonetheless, let us suppose that an is the last person to whom the application of “bald” yields a truth. In this case, it is settled that the application of “bald” to an + 1 does not yield a true proposition. Once this result contradicts (a), the above supposition is not true. This is not to say that the supposition is false. It is not! Suppose that an is not the last person to whom the application of “bald” yields a truth. In this case, the application of “bald” to an + 1 also yields a truth, and hence it is settled that it does not yield a falsehood. Once this result contradicts (b), it must be rejected too. Therefore, it is indefinite whether or not an is the last person to whom the application of “bald” yields a truth. A similar argument could be used to show that it is indefinite whether or not an + m + 1 is the first person to whom the application of “bald” yields a falsehood. In conclusion, Richard’s proposal indeed satisfies CPF. There are two truth-values, but no sharp boundary between objects to which the application of the predicate yields a certain value and ones to which it does not yield that value. Unfortunately, things are not as clear as they seem to be at first glance. Note that Richard’s proposal does not avoid the existence of sharp boundaries. On the contrary, it implies the existence of a boundary between objects to which the application of the predicate is settled and ones to which it is not. There is a last person such that it is settled whether or not the application of “bald” to her yields a true proposition (an), and a first person such that it is not settled whether or not the application of “bald” to her yields a true proposition (an + 1). Similarly, there will be a last person such that it is not settled whether or not the application of “bald” to her yields a false proposition (an + m), and a first person such that it is settled that the application of “bald” to her yields a false proposition (an + m + 1). Again, we have two sharp boundaries. This is problematic in different ways. It has been said that the notion of sharp boundary might be understood in two different senses (Sect. 3.1.3). First, the claim

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that a predicate F admits of a sharp boundary might be understood in terms of there being a pair of objects that have the following properties: (a) they are adjacent in a soritical sequence for F, and (b) the application of F to one yields some particular truth-value and the application of F to the other does not yield that value. Second, it might be understood in terms of there being an exhaustive distinction between objects to which the application of F yields some particular truth-value and ones to which it does not yield that value. As far as I can see, the above postulation implies that vague predicates have sharp boundaries in both senses. Richard distinguishes between two uses of the expression “not” (2009: 467–468). First, “not” is used as a truth-functional operator that might be understood according to Kleene’s table above. Second, “not” is used as a speech act sui generis – the denial – which is not to be defined in terms of assertion. Let us indicate the second use by “not” (in italics). The denial of p, not p, is appropriate in two and only two cases: when p is false and when p is indefinite. With this in mind, consider the following. By supposition, the application of “bald” to an yields a truth. Yet, it is either false or indefinite that the application of “bald” to an + 1 yields a truth; because if this were true, then an + 1 would not be a borderline case of “bald”. Consequently, it is appropriate to deny that the application of “bald” to an + 1 yields a truth. Therefore: the application of “bald” to an yields a truth and the application of “bald” to an + 1 does not yield a truth. Therefore, there is a pair of adjacent persons in a soritical sequence for “bald” such that the application of this predicate to one yields a particular truth-value and the application of it to the other does not yield that truth-­ value. “Bald” has a sharp boundary in the first sense above, and the same holds for any other vague predicate. The criterion of precisification is violated. Now, note that Richard accepts that there is an exclusive and exhaustive distinction between three categories of objects in the domain of application of vague predicates. Those who think that if Jo is borderline bald, then the claim that he is bald is without truth value often hold that vague predicates – indeed, all predicates – trisect their domains, in the sense that the sets – those of which the predicate is true; those of which it is false; the rest — are exclusive and exhaustive. I myself think this (Richard 2009: 464).

As a result, there is an exhaustive distinction between objects to which the application of the predicate yields a truth and ones to which it does not yield a truth. This is to say that vague predicates admit of a sharp boundary in the second sense above. The criterion of precisification is violated. It might be objected that the existence of sharp boundaries is not a problem if we use “not” in the aforementioned sense. There would be no problem with the claim that there is a sharp boundary between persons to whom the application of “bald” yields a truth and ones to whom it does not yield a truth. A problem arises only if we use “not” in its usual sense (whatever it is). In response, I note that the existence of a boundary between persons to whom “bald” applies and ones to whom it does not apply is as counterintuitive as the existence of a boundary between persons to whom “bald” applies and ones to whom it does not apply. If the notion of imprecision is supposed to express the sense in which vague predicates are boundaryless, it must

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forbid boundaries in the above two senses. Moreover, the foundational problem of precisification seems as striking as before. Why n, instead of n – 1 or n + 1, is the boundary of “bald”, so that the application of this predicate to an yields a truth but the application of it to an + 1 does not yield a truth? In response to this kind of objection, Richard (2009: 477) turns to an epistemic strategy. Although there are indeed (two) cut-off points in the domain of application of a vague predicate, we often do not know where they lie. As I said in Sect. 2.3, even if vague predicates have sharp boundaries, it is intuitive that we can hardly know their exact cut-off points. Come what may, this latter strategy does not preclude the violation of the criterion of precisification; it rather assumes that vague predicates are precise. In conclusion, it is not clear how Three-Valued Theory of Vagueness could satisfy the criterion of precisification.

4.1.3  Intuitions It is time to evaluate how Three-Valued Theory deals with the six relevant intuitions about vagueness (see Sect. 2.3). It has been said that any theory of vagueness should systematize those intuitions, in the sense that they should (a) identify which intuitions are accepted and which ones are rejected, (b) explain why the rejected intuitions seem to be true, and (c) determine how the accepted intuitions are related to each other, that is, which are the most fundamental ones and how the others are explained in terms of them. It is not clear to me how Three-Valued Theory could do this in a plausible way. The best strategy is to take the borderline cases intuition as the most fundamental one and explain the others in terms of it. I will argue that if we take this approach, we are left with no plausible way of systematizing the relevant intuitions. The borderline cases intuition can be easily explained by Three-Valued Theory of Vagueness: borderline cases of a vague predicate are those that are in the penumbra of the predicate. With this in mind, we can explain three other relevant intuitions. First, the lack of a sharp boundary intuition – (i) vague predicates do not have sharp boundaries – might be interpreted in terms of the non-existence of a sharp boundary between the positive and negative extensions of vague predicates. This, in turn, might be used to explain the unknowability of the boundary intuition – (vi) we couldn’t know the exact cut-off point of the boundaries of vague predicates. Given that there is no sharp boundary between positive and negative cases of vague predicates, we obviously couldn’t know the cut-off point between them (there is nothing to be known). Finally, we can explain the arbitrariness of the boundary intuition – (v) all precisifications of a vague predicate are equally arbitrary – by saying that any admissible way to draw a sharp boundary between the positive and negative extensions of the predicate would be equally arbitrary. A precisification would thus be understood as an admissible distribution of all borderline cases between the positive and negative extensions of a predicate, so to determine which side each borderline case is on.

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The above proposal interprets our intuitions concerning the non-existence, unknowability and arbitrariness of the boundary of a vague predicate in terms of the non-existence, unknowability and arbitrariness of the boundary between its positive and negative extensions. Nothing has been said about the boundary between the positive extension and the penumbra, or the boundary between the penumbra and the negative extension. Yet, if the boundary between the positive and negative extensions of a vague predicate is intuitively non-existent/unknowable/arbitrary, so is the boundary between the positive extension and the penumbra, as well as the boundary between the penumbra and the negative extension. Just as it is intuitive that there is no sharp boundary between the positive and negative extensions of “bald”, it is intuitive that there is no sharp boundary between the positive (negative) extension and the penumbra of this predicate. The idea that we can know the boundary between the positive (negative) extension and the penumbra also does not sound better than the idea that we can know the boundary between the positive and negative extensions. Finally, as I said in the previous section, it is intuitive that there are many different and equally arbitrary ways to draw a boundary between the positive (negative) extension and the penumbra. All these intuitions must be explained, but it is not clear how one could do this. A tentative solution to this problem would consist in claiming that those intuitions about the boundary between the positive (negative) extension and the penumbra are to be explained in epistemic terms. The proposal would consist, first, in taking the unknowability of the boundary intuition at face value. The reason why it is intuitive that we cannot know the two relevant boundaries of vague predicates is that they are indeed unknowable. Accordingly, the intuitions about the non-­existence and arbitrariness of the boundary might be explained away in terms of the unknowability intuition. There are different ways of explaining the relevant unknowability and interpret the other two intuitions in terms of it. In Sect. 4.4.1, where I briefly discuss Epistemic Theory of Vagueness, I consider at least one way to do that. In any case, I think this proposal is not plausible. First, the intuitions about the non-­ existence/unknowability/arbitrariness of the boundary between the positive and negative extensions of a vague predicate seem to be of the same kind as the intuitions about the non-existence/unknowability/arbitrariness of the boundary between the positive (negative) extension and the penumbra of a vague predicate. The above strategy, however, would require a different kind of explanation for each group of intuitions. While the intuitions of the first group would be explained in terms of the borderline cases intuition, those of the second would be explained in epistemic terms. This sounds simply wrong. Second, it would not be clear why, in this case, we shouldn’t prefer Epistemicism to Three-Valued Theory of Vagueness. Epistemicism is probably the simplest theory of vagueness; it provides us with a simple solution to the sorites paradox and does not commit us to any revision of classical logic or semantic theory. Its main sin, however, is that it bites the bullet and assumes that vague predicates are precise predicates. Three-Valued Theory, in turn, not only violates the criterion of precisification, but also (a) requires important revisions of classical logic and semantic theory, and (b) explains at least three important intuitions about the nature of vagueness in a way that is typical of Epistemicism.

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There remain two intuitions to be considered. The tolerance intuition  – (iii) vague predicates are tolerant to very slight changes – must be rejected. Vague predicates are not really tolerant. This does not mean that the principle of tolerance is false. Since none of its instances is false, the principle itself is not false. The fact that it is not false allegedly explains why it is intuitive. Besides, the fact that vague predicates have a penumbra explains at least partially why very slight changes cannot make a difference in the application of a vague predicate. A positive case of “bald”, for instance, will not become a negative case by the addition of a single strand of hair. Unfortunately, this cannot be the whole story. After all, the idea that very slight changes cannot make a difference in the application of a vague predicate seems to imply that very slight changes cannot turn a positive case of the predicate into a case of any other kind. The addition of a single strand of hair, for instance, cannot make a positive case of “bald” into a borderline case of “bald”. This latter intuition also requires an explanation. Once again, one could appeal to an epistemic strategy, but this would lead us to the aforementioned problems. Finally, it is not clear to me exactly how the no fact of the matter intuition – (ii) there is no fact of the matter about where the boundary of a vague predicate lies – would be explained. Perhaps we could interpret it in terms of the notion of indefinite: when it is indefinite whether or not a person is bald, then there is no fact of the matter about whether or not she is bald. In this interpretation, the no fact of the matter intuition would be an intuition about whether or not particular individuals are bald, not one about the location of the boundary of “bald”. Although I am not sure about the plausibility of this interpretation, I must concede that this intuition is in itself unclear. In conclusion, Three-Valued Theory of Vagueness can barely systematize intuitions (i)-(vi) in an adequate manner. The intuitions concerning the non-existence, unknowability and arbitrariness of the boundaries of vague predicates, as well the tolerance intuition, remain unexplained.

4.1.4  Further Objections My goal in this chapter is to evaluate some theories of vagueness with respect to whether or not they satisfy all three criteria of adequacy and systematize intuitions (i)–(vi) in an adequate manner. Because of this, I have ignored some important objections to Three-Valued Theory. It is worth mentioning two of them here. First, Three-Valued Theory leads to counterintuitive results respecting the truth-­ values of some sentences. In what follows, I will focus on Tye’s truth-functional interpretation of connectives, but the problem also arises for other truth-functional interpretations (Williamson 1994: Chap. 4). For a start, it is intuitive that every sentence of the form A ʌ ¬A expresses a false proposition, and every sentence of the form A v ¬A expresses a true proposition. According to Tye’s interpretation of connectives, however, both intuitions are wrong. Since Tye’s interpretation satisfies the normality constraint, everything goes well when A expresses a proposition with one

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of the classical values. But if A (expresses a proposition that) is indefinite, then neither will A ʌ ¬A express a false proposition nor will A v ¬A express a true proposition. The most Tye can retain of our intuitions is that sentences of the first form cannot express true propositions, and sentences of the second form cannot express false propositions. This is to say that A ʌ ¬A is a quasi-contradiction and A v ¬A a quasi-tautology (Tye 1994: 194). Now, suppose that (a) John and Mark are borderline cases of “tall” and (b) John is taller than Mark. Intuitively, the conditional “if Mark is tall, then John is tall” expresses a true proposition. In Tye’s interpretation, by contrast, this sentence (expresses a proposition that) is indefinite. Examples of this kind can easily be multiplied (Keefe 2000: 96–97; Santos 2015: 5–6). The second objection concerns the notion of higher-order vagueness (see Sects. 3.1.2 and 3.2.2). Three-Valued theory explains the first-order vagueness of “bald” in terms of the existence of borderline cases. Because this theory implies that vague predicates trisect their domain, it would not be able to accommodate the phenomenon of higher-order vagueness. In the present context, this phenomenon might be understood in terms of higher-order borderlineness. Just as there seem to be borderline cases between positive and negative cases of “bald”, there seems to be a second-­ order category of borderlineness between the positive and borderline cases of this predicate. This is to say that we should distinguish between persons who are clearly clear bald and ones who are clearly borderline bald, in order to accommodate those who are borderline clear bald. Similarly, there seems to be a third-order category of borderlineness between persons who are clearly clear bald and ones who are borderline clear bald. Now we will have to distinguish between persons who are clearly clear clear bald and ones who are clearly borderline clear bald, in order to accommodate those who are borderline clear clear bald. Arguably, the process continues indefinitely. Note, in addition, that we can easily formulate versions of the sorites for metalinguistic expressions such as “positive case of…”, “negative case of…” and “borderline case of…”. Here is an example. 5 . A person who has 0 hairs on her head is a positive case of “bald”. 6. For every natural number n, if a person who has n hairs on her head is a positive case of “bald”, then a person who has n + 1 hairs on her head is a positive case of “bald”. 7. A person who has 10,000 hairs on her head is a positive case of “bald”. This suggests that those metalinguistic expressions are vague, which would mean that they admit of borderline cases, which in turn leads us to higher-order borderlineness. It is not clear how Three-Valued Theory can account for this fact. If, on the one hand, the fact that Three-Valued Theory implies a trisection casts doubt on its ability to accommodate higher-order vagueness, on the other hand, it is possible to appeal to the very phenomenon of higher-order vagueness in order to avoid the trisection. A common way to avoid the violation of the criterion of precisification is by appealing to the thesis that vague predicates are indeed radically higher-order vague. This proposal is available to many-valued theories in general, and I have already indicated why I think it is not plausible (Sect. 3.1.2). In the next

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two sections, I will consider attempts to save both Degrees-of-Truth Theory and Supervaluationism by appealing to higher-order vagueness, and argue in detail that neither is plausible.7

4.2  Degrees-of-Truth Theory Like Three-Valued Theory, Degrees-of-Truth Theory of Vagueness appeals to more than two truth-values in order to explain vagueness. Because of this, it also requires a many-valued logic. In fact, these two theories are the main many-valued accounts of vagueness. One fundamental difference between them is that the latter goes far beyond the third truth-value or gap between the two classical ones, usually accepting infinite truth-values. The motivation for postulating infinite truth-values is that allegedly there are infinite degrees of truth. Both the idea of degrees of truth and infinite-valued logic might be used to explain other phenomena in addition to vagueness. The distinctive feature of Degrees-of-Truth Theory of Vagueness is that it places the notion of degrees of truth at the heart of the explanation of vague predicates. There might be different reasons for postulating degrees of truth. First, it seems that ordinary assignments of truth and falsehood sometimes commit us to the existence of degrees of truth. This is the case when we say, for instance, that a proposition p is to some extent true, that there is a certain amount of truth in p, that p is truer than q, etc. (Sainsbury 2009: 57). Second, our pre-theoretical understanding of truth as correspondence to reality is at first sight well suited to the idea of degrees of truth. What we say presumably might correspond to a higher or lower degree to reality, which means that there are different degrees of correspondence between propositions and reality. In line with this, and assuming a correspondence theory of truth, there should also be different degrees of truth. A proposition is true to the extent that it corresponds to reality and false to the extent that it fails to correspond to reality (Machina 1976: 54). I have said that the best way to begin to think about Three-Valued Theory is by appealing to the borderline cases intuition. The same holds for Degrees-of-Truth Theory. In fact, the latter provides us with a more plausible account of borderline cases than the former. Three-Valued Theory is able to accommodate the intuition that borderline cases of vague predicates have a special status, but it fails to accommodate the intuition that not all borderline cases have the same status. Smith and Mark might be borderline cases of “tall”, for instance, without being equally tall. Instead, let us suppose that Smith is much taller than Mark. While Three-Valued Theory would be indifferent to this supposition, Degrees-of-Truth Theory can accommodate it by saying that Smith is tall to a greater degree than Mark, and hence  In Sect. 4.3.2, I evaluate Keefe’s proposal to use higher-order vagueness in order to conform Supervaluationism to the criterion of precisification. Her strategy could be used by a proponent of Three-Valued Theory, and my arguments against it would hold equally well in this case. 7

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the proposition expressed by “Smith is tall” is truer than the proposition expressed by “Mark is tall”. In sum, two persons might be borderline cases of “tall” without being tall to the same degree.8 Degrees-of-truth theorists must say something about how degrees of truth are to be understood. Let us leave aside metaphysical problems concerning the nature of degrees of truth – What kind of entities are they? Is it really possible to fit them into a correspondence theory of truth? Etc.  – and focus on two questions: how many degrees of truth are there? How are they structured? Intuitively, if an object a is F to degree n, then the sentence “a is F” expresses a proposition that is true to degree n (for short, n true). Also intuitively, some properties admit of infinite degrees. Possible examples are the properties of being tall, being big, being strong, etc. These two facts, together, suggest that we should accept infinite degrees of truth. Furthermore, sometimes “a number of borderline cases arrange themselves in a natural ordering with respect to the degree to which they are F’s” (Machina 1976: 60), which means that the set of degrees of truth should be ordered. Finally, it is arguable that some properties admit of a continuum of borderline cases. A property F admits of a continuum of borderline cases when for every x and y, if x is F to a degree n and y is F to a degree m: m