Timothy Williamson traces the history of the problem of vagueness from discussions of the heap-of-sand paradox in ancient Greece to modern formal approaches such as fuzzy logic. His is the first comprehensive treatment of this increasingly important topic in metaphysics and the philosophy of logic and language. Students in these fields and researchers in artificial intelligence will find Williamson's non-technical approach extremely helpful and rewarding. Amazon Reviews
- “Vagueness provides a complete and lucid account of one of the hottest topics in philosophy of language and philosophy of logic ... drives epistemicism to a new level of depth and distinction.”
→ Roy Sorensen, New York University
- “Nothing should henceforth be written on vagueness which fails to learn from this book. It should not be read only for its contribution to vagueness, but also for what it says about knowledge; for the purity of its style. . . and as an example of philosophy at its best.”
→ Mark R. Sainsbury, British Journal of the Philosophy of Science
- The Early History of the Sorite Paradoxes
→ 1.1 The First Sorites
→ 1.2 Chrysippan Silence
→ 1.3 Sorites Arguments and Stoic Logic
→ 1.4 The Sorites in Later Antiquity
→ 1.5 The Sorites after Antiquity
- The Ideal of Precision
→ 2.1 The Emergence of Vagueness
→ 2.2 Frege
→ 2.3 Peirce
→ 2.4 Russell
- The Rehabilitation of Vagueness
→ 3.1 Vagueness and Ordinary Language
→ 3.2 The Black-Hempel Debate
→ 3.3 Family Resemblances
→ 3.4 Open Texture
- Many-Valued Logic and Degrees of Truth
→ 4.1 Overview
→ 4.2 Truth-functionality
→ 4.3 Three-valued Logic: Beginnings
→ 4.4 Three-valued Logic: Hallden
→ 4.5 Three-valued Logic: Korner
→ 4.6 Three-valued Logic: Second-order Vagueness
→ 4.7 Continuum-valued Logic: A Rational
→ 4.8 Continuum-valued Logic: Truth-tables
→ 4.9 Fuzzy Sets and Fuzzy Logic
→ 4.10 Degree-theoretic Treatments of the Sorites Paradoxes
→ 4.11 Comparatives and Modifiers
→ 4.12 Vague Degrees of Truth
→ 4.13 Non-numerical Degrees of Truth
→ 4.14 Degree-functionality
→ 4.15 Appendix: Axiomatizations of Continuum-valued Logic
→ 5.1 Incomplete Meanings
→ 5.2 Origins
→ 5.3 Logic and Semantics
→ 5.4 The Elusiveness of Supertruth
→ 5.5 Supervaluational Degrees of Truth
→ 5.6 Supervaluations and Higher-order Vagueness
→ 5.7 Truth and Supertruth
→ 6.1 Despair
→ 6.2 Global Nihilism
→ 6.3 Local Nihilism: Appearances
→ 6.4 Local Nihilism: Colours
- Vagueness as Ignorance
→ 7.1 Bivalence and Ignorance
→ 7.2 Bivalence and Truth
→ 7.3 Omniscient Speakers
→ 7.4 The Supervenience of Vagueness
→ 7.5 Meaning and Use
→ 7.6 Understanding
→ 7.7 Decidable Cases
- Inexact Knowledge
→ 8.1 The Explanatory Task
→ 8.2 The Crowd
→ 8.3 Margins for Error
→ 8.4 Conceptual Sources of Inexactness
→ 8.5 Recognition of Vague Concepts
→ 8.6 Indescribable Differences
→ 8.7 Inexact Beliefs
- Vagueness in the World
→ 9.1 Supervenience and Vague Facts
→ 9.2 Determinacy in the World
→ 9.3 Unclarity de re
Appendix: The Logic of Clarity
"Williamson (Timothy) - Vagueness: Preface + Introduction"
Source: Williamson (Timothy) - Vagueness
Preface (Truncated full text)
[ … Snip … provenance + acknowledgements … ] Introduction (Full text)
- This book originated in my attempts to refute its main thesis: that vagueness consists in our ignorance of the sharp boundaries of our concepts, and therefore requires no revision of standard logic. For years I took this epistemic view of vagueness to be obviously false, as most philosophers do. In 1988 Simon Blackburn, then editor of the journal Mind, asked me to review Roy Sorensen's intriguing book Blindspots, which includes a defence of the epistemic view. It did not persuade me; I could not see what makes us ignorant, and Sorensen offered no specific explanation. An alternative treatment of vagueness, supervaluationism, looked more or less adequate — unlike other popular alternatives, such as three-valued and fuzzy logic, which on technical grounds have always looked like blind alleys. However, I continued to think about the epistemic view, for the standard objections to it did not seem quite decisive. It was not clear that they did not assume a suspect connection between what is true and what we can verify. It then struck me that the notion of a margin for error could be used to give a specific explanation of ignorance of the sharp boundaries of our concepts, and the epistemic view began to look more plausible. A limited version of it was tentatively proposed in my book Identity and Discrimination (Oxford, Blackwell, 1990). The more closely the objections to it were analysed, the weaker they seemed. The next step was to focus on the fact that the meaning of vague expressions can be stated only in a language into which those expressions can be translated; it is a mistake to treat the language in which one theorizes about vagueness as though it were precise. Mark Sainsbury's inaugural lecture at King's College London, ‘Concepts without Boundaries1', helped to bring the significance of this point home to me, although we used it in quite different ways. It permits the formulation of arguments against a wide range of non-epistemic views, including the supervaluationism that had previously looked adequate (my objection to it, however, is not the one made in Sainsbury's lecture). The balance of arguments seemed to have moved firmly onto the side of the epistemic view. A book-length treatment was clearly needed. This is the result.
- Logicians are often accused of treating language as though it were precise, and ignoring its vagueness. Their standards of valid and invalid reasoning are held to be good enough for artificial precise languages, but to break down when applied to the natural vague languages in which we actually reason about the world that we experience. A perfectly precise language for such reasoning is an idealization never to be realized. Although we can make our language less vague, we cannot make it perfectly precise. If we try to do so by stipulating what our words are to mean, our stipulations will themselves be made in less than perfectly precise terms, and the reformed language will inherit some of that vagueness.
- The problem is not confined to logic. Attempts to describe the semantics of natural languages in formal terms are also frequently supposed to ignore vagueness, and therefore to misdescribe the meanings of ordinary expressions. Of course, a theory might ignore vagueness and remain a useful approximation for some purposes, but it is also legitimate to ask what changes of theory are needed to take vagueness into account.
- At the core of classical (i.e. standard) logic and semantics is the principle of bivalence, according to which every statement is either true or false. This is the principle most obviously threatened by vagueness. When, for example, did Rembrandt become old? For each second of his life, one can consider the statement that he was old then. Some of those statements are false; others are true. If all of them are true or false, then there was a last second at which it was false to say that Rembrandt was old, immediately followed by a first second at which it was true to say that he was old. Which second was that? We have no way of knowing. Indeed, it is widely felt to be just silly to suppose that there was such a second. Our use of the word ‘old' is conceived as too vague to single one out. On such grounds, the principle of bivalence has been rejected for vague languages. To reject bivalence is to reject classical logic or semantics.
- At some times, it was unclear whether Rembrandt was old. He was neither clearly old nor clearly not old. The unclarity resulted from vagueness in the statement that Rembrandt was old. We can even use such examples to define the notion of vagueness. An expression or concept is vague if and only if it can result in unclarity of the kind just exemplified. Such a definition does not pretend to display the underlying nature of the phenomenon. In particular, it does not specify whether the unclarity results from the failure of the statement to be true or false, or simply from our inability to find out which. The definition is neutral on such points of theory. Just as we might agree to define the term ‘light', or ‘poetry', by examples, in order not to talk past each other when disagreeing about the nature of light, or poetry, so we can agree to define the term 'vagueness' by examples, in order not to talk past each other when disagreeing about the nature of vagueness.
- 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.
- The phenomenon of vagueness is deep as well as broad. It would be shallow if it could be adequately described in precise terms. That is not generally possible. The difficulties presented by the question ‘When did Rembrandt become old?' are also presented by the question ‘When did Rembrandt become clearly old?'. At some times, it was unclear whether it was unclear whether Rembrandt was old. The limits of vagueness are themselves vague. The same difficulties are presented by the question ‘When did Rembrandt become clearly clearly old?'; the point reiterates ad infinitum. This is the phenomenon of higher-order vagueness. It means that the meta-language in which we describe the vagueness of a vague language will itself be vague.
- The use of non-classical systems of logic or semantics has been advocated for vague languages. New and increasingly complex systems continue to be invented. What none has so far given is a satisfying account of higher-order vagueness. In more or less subtle ways, the meta-language is treated as though it were precise. For example, classical logic is said to be invalid for vague languages, and is then used in the meta-language. Such proposals underestimate the depth of the problem.
- The problem is not solved by the pessimistic idea that no system of logic or semantics, classical or non-classical, is adequate for a vague language. That idea still permits one to ask for perspicuous descriptions of vagueness in particular cases. No one has given a satisfying and perspicuous description of higher-order vagueness without use of classical logic. Of course, the nature of vagueness might be to defy perspicuous description, but that counsel of despair should prevail only if there is good evidence that it does not overestimate the depth of the problem.
- The thesis of this book is that vagueness is an epistemic phenomenon. As such, it constitutes no objection to classical logic or semantics. In cases of unclarity, statements remain true or false, but speakers of the language have no way of knowing which. Higher-order vagueness consists in ignorance about ignorance.
- At first sight, the epistemic view of vagueness is incredible. We may think that we cannot conceive how a vague statement could be true or false in an unclear case. For when we conceive that something is so, we tend to imagine finding out that it is so. We are uneasy with a fact on which we cannot attain such a first-personal perspective. We have no idea how we ever could have found out that the vague statement is true, or that it is false, in an unclear case; we are consequently unable to imagine finding out that it is true, or that it is false; we fallaciously conclude that it is inconceivable that it is true, and inconceivable that it is false. Such fallacies of the imagination must be laid aside before the epistemic view can be adequately assessed.
- Most work on the problem of vagueness assumes that the epistemic view is false, without seriously arguing the point. If the epistemic view is true, that work is fundamentally mistaken. Even if the epistemic view is false, that work is ungrounded until cogent arguments against the view have been found. The assessment of the epistemic view is therefore one of the main tasks facing the study of vagueness. This book contributes to that task.
- The assessment of the epistemic view has ramifications far beyond the study of vagueness. As already noted, classical logic and semantics are at stake. More generally, the epistemic view implies a form of realism, that even the truth about the boundaries of our concepts can outrun our capacity to know it. To deny the epistemic view of vagueness is therefore to impose limits on realism; to assert it is to endorse realism in a thoroughgoing form.
- The first part of the book is historical. It traces the slow and intermittent recognition of vagueness as a distinct and problematic phenomenon, up to the origins of the theories of vagueness that have been popular over the last two decades. This part is also critical. It argues that none of the extant non-epistemic theories of vagueness is adequate. Not only do they abandon classical logic or semantics for alternatives of doubtful coherence; those sacrifices are not rewarded by adequate insight into the nature of vagueness. The second part of the book is constructive. It develops and applies an epistemic view of vagueness, finds the standard objections to it fallacious, and concludes that the epistemic view provides the best explanation of the phenomenon of vagueness.
- The Greeks introduced the problem of vagueness into philosophy, in the guise of the original sorites paradox: if the removal of one grain from a heap always leaves a heap, then the successive removal of every grain still leaves a heap. Chapter 1 sketches the history of this paradox and its variants from their invention to the nineteenth century. Stoic logicians are interpreted as taking an epistemic view of sorites paradoxes.
- What makes a sorites paradox paradoxical is the vagueness of its central term, e.g. 'heap'. Historically, however, such paradoxes were identified by their form. Vagueness as such became a topic of philosophical discussion only at the start of the twentieth century, when it presented an obstacle to the ideal of a logically perfect language associated with the development of modern logic. Only with difficulty was the phenomenon of unclear boundaries separated from other phenomena, such as lack of specificity, to which the term ‘vagueness' is also applied in everyday usage. Chapter 2 discusses three stages in the emergence of the philosophical concept of vagueness, in the work of Frege, Peirce and Russell.
- As philosophical attention turned to ordinary language, vagueness acquired a more positive image. It was seen no longer as a deviation from an ideal norm, but as the real norm itself. As such, it was described by Black, Wittgenstein and Waismann. Their work is discussed in Chapter 3.
- Formal treatments of vagueness have become common only in the last few decades. One main approach relies on many-valued logic, which replaces the dichotomy of truth and falsity by a manyfold classification. Chapter 4 follows the development of its application to the problem of vagueness, from the use of three-valued logic to the growth of ‘fuzzy logic' and other logics based on infinitely many values, and then of more sophisticated accounts appealing to a qualitative conception of degrees of truth. These views are criticized on several grounds. None has adequately treated higher-order vagueness; degrees of truth are not connected with vagueness in the requisite way; the generalization from two-valued logic to many-valued logic has highly counter-intuitive consequences when applied to natural languages.
- A technically subtler approach to vagueness is supervaluationism, with which Chapter 5 is concerned. It preserves almost all of classical logic, at the expense of classical semantics, by giving a non-standard account of truth. It also treats higher-order vagueness in a promising way. However, it is argued that the treatment of higher-order vagueness undermines the non-standard account of truth, making supervaluationism as a whole unmotivated.
- On a more pessimistic view, vagueness is a form of incoherence. If this view is taken globally, Chapter 6 suggests, all rational discourse is subverted, for vagueness is ubiquitous. However, local forms of nihilism might be coherent. They have been defended in the special case of concepts used to describe perceptual appearances, on the grounds that such concepts cannot differentiate between perceptually indiscriminable items, yet perceptually discriminable items can be linked by a sorites series of which each member is perceptually indiscriminable from its neighbours. However, careful attention to the structure of the relevant concepts shows that the paradoxical arguments are unsound. In particular, they falsely assume that appearances are just what they appear to be.
- Chapter 7 defends the epistemic view of vagueness. First, it argues that it is incoherent to deny the principle of bivalence for vague statements in unclear cases. It then questions our ability to think through coherently the consequences of a non-epistemic view of vagueness. Obvious objections to the epistemic view are analysed and shown to be fallacious. A picture of linguistic understanding is sketched, on which we can know that a word has a given meaning without knowing what the boundaries of that meaning are in conceptual space.
- Chapter 8 develops the epistemological background to the epistemic view. It gives independent justification for principles about knowledge on which the ignorance postulated by the view was only to be expected as a special case of a much wider phenomenon, inexact knowledge. Nevertheless, the case is special, for the source of the inexactness is distinctive in being conceptual. Higher-order vagueness has a central place in this account, for a central feature of inexact knowledge is that one can know something without being in a position to know that one knows it; when the inexactness takes the form of vagueness, this becomes unclarity about unclarity. The epistemology of inexact knowledge is then used to analyse in greater depth the phenomena of indiscriminability to which the nihilist appeals.
- It is controversial whether in any sense the world itself, as opposed to our representations of it, can be vague. Chapter 9 examines the issue. It argues that the epistemic view permits objects to be vague in a modest sense, for the impossibility of knowing their boundaries may be independent of the way in which the objects are represented.
- The Appendix identifies the formal system uniquely appropriate for the logic of clarity and unclarity on the epistemic view of vagueness.
- Most of the book keeps technicalities to a minimum. The gain in intelligibility will, it is hoped, outweigh the loss in rigour. There is also a philosophical reason for minimizing technicality. 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. Since vague terms are meaningful, this is an expressive limitation on the meta-language. It is not an innocent one. The argument in Chapter 7 for the incoherence of denials of bivalence in unclear cases can be stated only in a language into which the relevant vague terms can be translated. To deny bivalence is in the end to treat vague utterances as though they said nothing. Vagueness can be understood only from within.
COMMENT: Annotated printout filed in "Various - Papers on Identity Boxes: Vol 19 (W)".
In-Page Footnotes ("Williamson (Timothy) - Vagueness: Preface + Introduction")
Footnote 1: See "Sainsbury (Mark) - Concepts Without Boundaries".
"Williamson (Timothy) - The Early History of the Sorite Paradoxes"
Source: Williamson (Timothy) - Vagueness, Chapter 1
1.1 The First Sorites
1.2 Chrysippan Silence
1.3 Sorites Arguments and Stoic Logic
1.4 The Sorites in Later Antiquity
1.5 The Sorites after Antiquity
"Williamson (Timothy) - The Ideal of Precision"
Source: Williamson (Timothy) - Vagueness, Chapter 2
2.1 The Emergence of Vagueness
"Williamson (Timothy) - The Rehabilitation of Vagueness"
Source: Williamson (Timothy) - Vagueness, Chapter 3
3.1 Vagueness and Ordinary Language
3.2 The Black-Hempel Debate
3.3 Family Resemblances
3.4 Open Texture
"Williamson (Timothy) - Many-Valued Logic and Degrees of Truth"
Source: Williamson (Timothy) - Vagueness, Chapter 4
4.3 Three-valued Logic: Beginnings
4.4 Three-valued Logic: Hallden
4.5 Three-valued Logic: Korner
4.6 Three-valued Logic: Second-order Vagueness
4.7 Continuum-valued Logic: A Rational
4.8 Continuum-valued Logic: Truth-tables
4.9 Fuzzy Sets and Fuzzy Logic
4.10 Degree-theoretic Treatments of the Sorites Paradoxes
4.11 Comparatives and Modifiers
4.12 Vague Degrees of Truth
4.13 Non-numerical Degrees of Truth
4.15 Appendix: Axiomatizations of Continuum-valued Logic
"Williamson (Timothy) - Supervaluations"
Source: Williamson (Timothy) - Vagueness, Chapter 5
5.1 Incomplete Meanings
5.3 Logic and Semantics
5.4 The Elusiveness of Supertruth
5.5 Supervaluational Degrees of Truth
5.6 Supervaluations and Higher-order Vagueness
5.7 Truth and Supertruth
"Williamson (Timothy) - Nihilism"
Source: Williamson (Timothy) - Vagueness, Chapter 6
6.2 Global Nihilism
6.3 Local Nihilism: Appearances
6.4 Local Nihilism: Colours
"Williamson (Timothy) - Vagueness as Ignorance"
Source: Williamson (Timothy) - Vagueness, Chapter 7
7.1 Bivalence and Ignorance
7.2 Bivalence and Truth
7.3 Omniscient Speakers
7.4 The Supervenience of Vagueness
7.5 Meaning and Use
7.7 Decidable Cases
"Williamson (Timothy) - Inexact Knowledge"
Source: Williamson (Timothy) - Vagueness, Chapter 8
8.1 The Explanatory Task
8.2 The Crowd
8.3 Margins for Error
8.4 Conceptual Sources of Inexactness
8.5 Recognition of Vague Concepts
8.6 Indescribable Differences
8.7 Inexact Beliefs
"Williamson (Timothy) - Vagueness in the World"
Source: Williamson (Timothy) - Vagueness, Chapter 9
9.1 Supervenience and Vague Facts
9.2 Determinacy in the World
9.3 Unclarity de re
"Williamson (Timothy) - The Logic of Clarity"
Source: Williamson (Timothy) - Vagueness, Appendix
Text Colour Conventions (see disclaimer)
- Blue: Text by me; © Theo Todman, 2018
- Mauve: Text by correspondent(s) or other author(s); © the author(s)