The naive theory of vagueness holds that the vagueness of an expression consists in its failure to draw a sharp boundary between positive and negative cases. The naive theory is contrasted with the nowadays dominant approach to vagueness, holding that the vagueness of an expression consists in its presenting borderline cases of application. The two approaches are briefly compared in their respective explanations of a paramount phenomenon of vagueness: our ignorance of any sharp boundary between positive and negative cases. These (...) explanations clearly do not provide any ground for choosing the dominant approach against the naive theory. The decisive advantage of the former over the latter is rather supposed to consist in its immunity to any form of sorites paradox. But another paramount phenomenon of vagueness is higher-order vagueness: the expressions introduced in order to express in the object language the vagueness of the object language are themselves vague. Two highly plausible claims about higher-order vagueness are articulated and defended: the existence of “definitely ω ” positive and negative cases and the “radical” character of higher-order vagueness itself. Using very weak logical principles concerning vague expressions and the ‘definitely’-operator, it is then shown that, in the presence of higher-order vagueness as just described, the dominant approach is subject to higher-order sorites paradoxes analogous to the original ones besetting the naive theory, and therefore that, against the communis opinio, it does not fare substantially better with respect to immunity to any form of sorites paradox. (shrink)

This paper is a tribute to Delia Graff Fara. It extends her work on failures of meta-rules for validity as truth-preservation under a supervaluationist identification of truth with supertruth. She showed that such failures occur even in languages without special vagueness-related operators, for standards of deductive reasoning as materially rather than purely logically good, depending on a context-dependent background. This paper extends her argument to: quantifier meta-rules like existential elimination; ambiguity; deliberately vague standard mathematical notation. Supervaluationist attempts to qualify the (...) meta-rules impose unreasonable cognitive demands on reasoning and underestimate her challenge. (shrink)

In their paper “Vagueness, Ignorance, and Margins for Error” Kenton Machina and Harry Deutsch criticize the epistemic theory of vagueness. This paper answers their objections. The main issues discussed are: the relation between meaning and use; the principle of bivalence; the ontology of vaguely specified classes; the proper form of margin for error principles; iterations of epistemic operators and semantic compositionality; the relation or lack of it between quantum mechanics and theories of vagueness.

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.

The idea of higher-order vagueness is usually associated with conceptions of vagueness that focus on the existence of borderline cases. What sense can be made of it within a conception of vagueness that focuses on tolerance instead? A proposal is offered here. It involves understanding ‘definitely’ not as a sentence operator but as a predicate modifier, and more precisely as an intensifier, that is, an operator that shifts the predicate extension along a scale. This idea is combined with the author’s (...) earlier approach to the semantics of vague expressions, which builds on the idea of a central gap associated with a predicate. The central gap approach is generalized to handle arbitrarily many iterations of ‘definitely’. (shrink)

In this paper we compare different models of vagueness viewed as a specific form of subjective uncertainty in situations of imperfect discrimination. Our focus is on the logic of the operator “clearly” and on the problem of higher-order vagueness. We first examine the consequences of the notion of intransitivity of indiscriminability for higher-order vagueness, and compare several accounts of vagueness as inexact or imprecise knowledge, namely Williamson’s margin for error semantics, Halpern’s two-dimensional semantics, and the system we call Centered semantics. (...) We then propose a semantics of degrees of clarity, inspired from the signal detection theory model, and outline a view of higher-order vagueness in which the notions of subjective clarity and unclarity are handled asymmetrically at higher orders, namely such that the clarity of clarity is compatible with the unclarity of unclarity. (shrink)

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.

Timothy Williamson has shown that the B axiom for 'definitely' (α → Δ¬Δ¬α) guarantees that if a sentence is second-order vague in a Kripke model, it is nth order vague for every n. More recently, Anna Mahtani has argued that Williamson's epistemicist theory of vagueness does not support the B axiom, and conjectured that if we consider models in which the “radius of accessibility” varies between different points, we will be able to find sentences that are nth-order vague but (n+1)th-order (...) precise, for any n. This paper bolsters Mahtani's argument, shows her conjecture to be true, and shows that imposing certain further natural constraints on "variable radius" models does not change the situation. (shrink)

In chapter 5 of Knowledge and its Limits, T. Williamson formulates an argument against the principle (KK) of epistemic transparency, or luminosity of knowledge, namely “that if one knows something, then one knows that one knows it”. Williamson’s argument proceeds by reductio: from the description of a situation of approximate knowledge, he shows that a contradiction can be derived on the basis of principle (KK) and additional epistemic principles that he claims are better grounded. One of them is a reflective (...) form of the margin for error principle defended by Williamson in his account of knowledge. We argue that Williamson’s reductio rests on the inappropriate identification of distinct forms of knowledge. More specifically, an important distinction between perceptual knowledge and non-perceptual knowledge is wanting in his statement and analysis of the puzzle. We present an alternative account of this puzzle, based on a modular conception of knowledge: the (KK) principle and the margin for error principle can coexist, provided their domain of application is referred to the right sort of knowledge. (shrink)

This paper is about three of the most prominent debates in modern epistemology. The conclusion is that three prima facie appealing positions in these debates cannot be held simultaneously. The first debate is scepticism vs anti-scepticism. My conclusions apply to most kinds of debates between sceptics and their opponents, but I will focus on the inductive sceptic, who claims we cannot come to know what will happen in the future by induction. This is a fairly weak kind of scepticism, and (...) I suspect many philosophers who are generally anti-sceptical are attracted by this kind of scepticism. Still, even this kind of scepticism is quite unintuitive. I’m pretty sure I know (1) on the basis of induction. (1) It will snow in Ithaca next winter. Although I am taking a very strong version of anti-scepticism to be intuitively true here, the points I make will generalise to most other versions of scepticism. (Focussing on the inductive sceptic avoids some potential complications that I will note as they arise.) The second debate is a version of rationalism vs empiricism. The kind of rationalist I have in mind accepts that some deeply contingent propositions can be known a priori, and the empiricist I have in mind denies this. Kripke showed that there are contingent propositions that can be known a priori. One example is Water is the watery stuff of our acquaintance. (‘Watery’ is David Chalmers’s nice term for the properties of water by which folk identify it.) All the examples Kripke gave are of propositions that are, to use Gareth Evans’s term, deeply necessary (Evans, 1979). It is a matter of controversy presently just how to analyse Evans’s concepts of deep necessity and contingency, but most of the controversies are over details that are not important right here. I’ll simply adopt Stephen Yablo’s recent suggestion: a proposition is deeply contingent if it could have turned out to be true, and could have turned out to be false (Yablo, 2002)1. Kripke did not provide examples of any deeply contingent propositions knowable a priori, though nothing he showed rules out their existence.. (shrink)