Vagueness provides the first comprehensive examination of a topic of increasing importance in metaphysics and the philosophy of logic and language. Timothy Williamson traces the history of this philosophical problem from discussions of the heap paradox in classical Greece to modern formal approaches such as fuzzy logic. He illustrates the problems with views which have taken the position that standard logic and formal semantics do not apply to vague language, and defends the controversial realistic view that vagueness is a kind (...) of ignorance--that there really is a grain of sand whose removal turns a heap into a non-heap, but we cannot know which one it is. (shrink)

The core of Tony Brueckner’s critique in ‘Knowledge, Evidence, and Skepticism according to Williamson’ is his claim in section 5 that my account of perceptual knowledge has an unacceptable consequence. My reply will concentrate on that claim and largely ignore the rest of Brueckner’s interesting discussion, for it is easy to check that the claim is essential to Brueckner’s argument against my analysis of skepticism and evidence.

The core of Tony Brueckner’s critique in ‘Knowledge, Evidence, and Skepticism according to Williamson’ is his claim in section 5 that my account of perceptual knowledge has an unacceptable consequence. My reply will concentrate on that claim and largely ignore the rest of Brueckner’s interesting discussion, for it is easy to check that the claim is essential to Brueckner’s argument against my analysis of skepticism and evidence. The alleged consequence at issue concerns a case in which Brueckner knows by seeing (...) that his cup is red. According to Brueckner, I am committed to the implausible view that ‘[his] belief of the proposition that [his] cup is red is justified in virtue of [his] belief of the proposition that [his] cup is red!’ Why does he take my account of.. (shrink)

Discussions of higher-order vagueness rarely define what it is for a term to have nth-order vagueness for n>2. This paper provides a rigorous definition in a framework analogous to possible worlds semantics; it is neutral between epistemic and supervaluationist accounts of vagueness. The definition is shown to have various desirable properties. But under natural assumptions it is also shown that 2nd-order vagueness implies vagueness of all orders, and that a conjunction can have 2nd-order vagueness even if its conjuncts do not. (...) Relations between the definition and other proposals are explored; reasons are given for preferring the present proposal. (shrink)

Every philosopher should wish for commentators as thoughtful and rigorous as Mario Gómez-Torrente and Delia Graff. Their papers contribute significantly to our understanding of the fine structure of vagueness, but not by undermining its epistemicist interpretation.

Every philosopher should wish for commentators as thoughtful and rigorous as Mario Gómez-Torrente and Delia Graff. Their papers contribute significantly to our understanding of the fine structure of vagueness, but not by undermining its epistemicist interpretation.

Consider an object or property a and the predicate F. Then a is vague if there are questions of the form: Is a F? that have no yes-or-no answers. In brief, vague properties and kinds have borderline instances and composite objects have borderline constituents. I'll use the expression "borderline cases" as a covering term for both. ;Having borderline cases is compatible with precision so long as every case is either borderline F, determinately F or determinately not F. Thus, in addition (...) to borderline cases, vagueness requires "fuzzy borders". A has fuzzy borders if there is no sharp line between a's borderline cases, a's determinate cases and a's determinate non-cases. To illustrate, loudness is vague because in a continuum of gradually increasing volume, there is no fact about where the last definite loud noise is and where the first borderline noise is. ;Actual borderline cases are not required for vagueness. Suppose actual noises are either ear-shatteringly loud or silent. Even so, loudness is vague since it could have borderline instances. Just so for objects. Suppose a table is cleanly bounded such that there is a sharp break between the table and its surroundings. Still, that table is vague since there are worlds at which it is indeterminate whether some wood molecule a is, or is not, part of that table. ;There are three views about which things can be vague. On the linguistic view, vagueness is solely a feature of expressions. On the nihilistic view, nothing is vague because vagueness is incoherent. Pitted against these views is the ontological conception of vagueness according to which borderline instances and constituents arise due to genuine indeterminacy in nature. ;My dissertation defends the ontological conception. I argue against the adequacy of the theory's rivals, disarm a damaging objection and demonstrate the causal superiority of vague events. Anyone familiar with the literature on vagueness will notice that only a subset of issues are discussed. What I offer is preliminary--I hope only to make the theory of ontological vagueness attractive enough to merit future work. (shrink)

" In 'Vagueness and Alternative Logic' (Realism and Reason, Cambridge 1983, pp. 271-86, especially 285-6), Hilary Putnam puts forward a suggestion for a formal treatment of the logic of vagueness. … Putnam admits that, at the time of writing, he had not thought this idea through. What will already be apparent to the alert reader is that, in order to disclose serious difficulties for the proposal, Putnam would not have had to think far.".

Vagueness is currently the subject of vigorous debate in the philosophy of logic and language. Vague terms -- such as 'tall', 'red', 'bald', and 'tadpole' -- have borderline cases ; and they lack well-defined extensions. The phenomenon of vagueness poses a fundamental challenge to classical logic and semantics, which assumes that propositions are either true or false and that extensions are determinate.This anthology collects for the first time the most important papers in the field. After a substantial introduction that surveys (...) the field, the essays form four groups, starting with some historically notable pieces. The 1970s saw an explosion of interest in vagueness, and the second group of essays reprints classic papers from this period. The following group of papers represent the best recent work on the logic and semantics of vagueness. The essays in the final group are contributions to the continuing debate about vague objects and vague identity. (shrink)

Difficulties in arriving at an adequate conception of vagueness have led many writers to describe a phenomenon that has come to be known as "higher-order vagueness". Almost as many have found it to be a problem that needs to be addressed. In what follows I shall argue that, whilst we must acknowledge its presence, it is a pseudo-problem. The crucial point is the vagueness of "vague", which shows the phenomenon to be unproblematic though real enough.

A discussion of Crispin Wright's 'paradox of higher-order vagueness', I suggest that the paradox may be resolved by careful attention to the logical principles used in its formulation. In particular, I focus attention on the rule of inference that allows for the inference from A to 'Definitely A', and argue that this rule, though valid, may not be used in subordinate deductions, e.g., in the course of a conditional proof. Wright's paradox uses the rule (or its equivalent) in this way.

Vagueness is given a philosophically neutral definition in terms of an epistemic notion of tolerance. Such a notion is intended to capture the thesis that vague terms draw no known boundary across their range of signification and contrasts sharply with the semantic notion of tolerance given by Wright (1975, 1976). This allows us to distinguish vagueness from superficially similar but distinct phenomena such as semantic incompleteness. Two proofs are given which show that vagueness qua epistemic tolerance and vagueness qua borderline (...) cases (when properly construed to exclude terms which are stipulated to give rise to borderline cases) are in fact conceptually equivalent dimensions of vagueness, contrary to what might initially be expected. It is also argued that the common confusion of tolerance and epistemic tolerance has skewed the vagueness debate in favour of indeterminist over epistemic conceptions of vagueness. Clearing up that confusion provides an indirect argument in favour of epistemicism. Finally, given the equation of vagueness with epistemic tolerance, it is shown that there must be radical higher-order vagueness, contrary to what many authors have argued. (shrink)

I argue that, contrary to widespread philosophical opinion, phenomenal indiscriminability is transitive. For if it were not transitive, we would be precluded from accepting the truisms that if two things look the same then the way they look is the same and that if two things look the same then if one looks red, so does the other. Nevertheless, it has seemed obvious to many philosophers (e.g. Goodman, Armstrong and Dummett) that phenomenal indiscriminability is not transitive; and, moreover, that this (...) non-transitivity is straightforwardly revealed to us in experience. I show this thought to be wrong. All inferences from the character of our experience to the non-transitivity of indiscriminability involve either a misunderstanding of continuity, a mistaken interpretation of the idea that we have limited powers of discrimination, or tendentious claims about what our experience is really like; or such inferences are based on inadequately supported premisses, which though individually plausible are jointly implausible. (shrink)

Let us say that the proposition that p is transparent just in case it is known that p, and it is known that it is known that p, and it is known that it is known that it is known that p, and so on, for any number of iterations of the knowledge operator ‘it is known that’. If there are transparent propositions at all, then the claim that any man with zero hairs is bald seems like a good candidate. (...) We know that any man with zero hairs is bald. And it also does not seem completely implausible that we know that we know it, and that we know that we know that we know it, and so on. (shrink)

This paper presents some difficulties for Timothy Williamson's epistemicist view of vagueness and for an argument he gives in its defense. First, I claim that the argument, which uses the notion of an "omniscient speaker", is question-begging. Next, I argue that some presumably true scientific hypotheses, which postulate certain relations between everyday vague predicates and scientific predicates, make the central theses of epistemicism highly implausible. Finally, I show that the "margin for error principles" used by Williamson to explain away the (...) kind of ignorance conjectured by epistemicism lead to new sorites-like arguments with unacceptable conclusions. (shrink)

Timothy Williamson’s potentially most important contribution to epistemicism about vagueness lies in his arguments for the basic epistemicist claim that the alleged cut-off points of vague predicates are not knowable. His arguments for this are based on so-called ‘margin for error principles’. This paper argues that these principles fail to provide a good argument for the basic claim. Williamson has offered at least two kinds of margin for error principles applicable to vague predicates. A certain fallacy of equivocation seems to (...) underlie his justification for both kinds of principles. Besides, the margin for error principles of the first kind can be used in the derivation of unacceptable consequences, while the margin for error principles of the second kind can be shown to be compatible with the falsity of epistemicism, under a number of assumptions acceptable to the epistemicist. (shrink)

The goal of this paper is a comprehensive analysis of basic reasoning patterns that are characteristic of vague predicates. The analysis leads to rigorous reconstructions of the phenomena within formal systems. Two basic features are dealt with. One is tolerance: the insensitivity of predicates to small changes in the objects of predication (a one-increment of a walking distance is a walking distance). The other is the existence of borderline cases. The paper shows why these should be treated as different, though (...) related phenomena. Tolerance is formally reconstructed within a proposed framework of contextual logic, leading to a solution of the Sorites paradox. Borderline-vagueness is reconstructed using certain modality operators; the set-up provides an analysis of higher order vagueness and a derivation of scales of degrees for the property in question. (shrink)

This paper deals with the truth-Conditions and the logic for vague languages. The use of supervaluations and of classical logic is defended; and other approaches are criticized. The truth-Conditions are extended to a language that contains a definitely-Operator and that is subject to higher order vagueness.

I wish to present a proof that vagueness is impossible. Of course, vagueness is possible; and so there must be something wrong with the proof. But it is far from clear where the error lies and, indeed, all of the assumptions upon which the proof depends are ones that have commonly been accepted. This suggests that we may have to radically alter our current conception of vagueness if we are to make proper sense of what it is.

It is a paradox, whose importance familiarity fails to diminish, that the most highly developed and useful scientific theories are ostensibly expressed in terms of objects never encountered in experience. The line traced by a draughtsman, no matter how accurate, is seen beneath the microscope as a kind of corrugated trench, far removed from the ideal line of pure geometry. And the “point-planet” of astronomy, the “perfect gas” of thermodynamics, or the “pure species” of genetics are equally remote from exact (...) realization. Indeed the unintelligibility at the atomic or sub-atomic level of the notion of a rigidly demarcated boundary shows that such objects not merely are not but could not be encountered. While the mathematician constructs a theory in terms of “perfect” objects, the experimental scientist observes objects of which the properties demanded by theory are and can, in the very nature of measurement, be only approximately true. As Duhem remarks, mathematical deduction is not useful to the physicist if interpreted rigorously. It is necessary to know that its validity is unaltered when the premise and conclusion are only “approximately true.” But the indeterminacy thus introduced, it is necessary to add in criticism, will invalidate the deduction unless the permissible limits of variation are specified. To do so, however, replaces the original mathematical deduction by a more complicated mathematical theory in respect of whose interpretation the same problem arises, and whose exact nature is in any case unknown. (shrink)

A selective background -- Broadly classical approaches -- Paracompleteness -- More on paracomplete solutions -- Paraconsistent dialetheism.