Online research in philosophy

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.

It is common among philosophers who take an interest in the phenomenon of vagueness in natural language not merely to acknowledge higher-order vagueness but to take its existence as a basic datum— so that views that lack the resources to account for it, or that put obstacles in the way, are regarded as deficient just on that score. My main purpose in what follows is to loosen the hold of this deeply misconceived idea. Higher-order vagueness is no basic datum but an illusion, fostered by misunderstandings of the nature of (ordinary, if you will ‘first-order’) vagueness itself. To see through the illusion is to take a step that is prerequisite for a correct understanding of vagueness, and for any satisfying dissolution of its attendant paradoxes.

A number of arguments purport to show that vague properties determine sharp boundaries at higher orders. That is, although we may countenance vagueness concerning the location of boundaries for vague predicates, every predicate can instead be associated with precise knowable cut-off points deriving from precision in their higher order boundaries.

I argue that this conclusion is indeed paradoxical, and identify the assumption responsible for the paradox as the Brouwerian principle B for vagueness: that if p then it's completely determinate that either it's vague whether p, or p. Other paradoxes which do not appear to turn on B turn instead on some subtle issues concerning the relation between assertion, belief and higher order vagueness.

In this paper a B-free picture of assertion, knowledge and logic is outlined which is completely free of higher order precision. A class of realistic models containing counterexamples to B and a number of weakenings of B are introduced and its logic is shown to support vagueness at every order. A novel framework for thinking about the semantic apparatus in the presence of metalinguistic vagueness is also developed. In this framework the vague metatheoretic vocabulary is part of the object language and can readily be applied to itself.

Stewart Shapiro’s book develops a contextualist approach to vagueness. It’s chock-full of ideas and arguments, laid out in wonderfully limpid prose. Anyone working on vagueness (or the other topics it touches on—see below) will want to read it. According to Shapiro, vague terms have borderline cases: there are objects to which the term neither determinately applies nor determinately does not apply. A term determinately applies in a context iff the term’s meaning and the non-linguistic facts determine that they do. The non-linguistic facts include the “external” context: “comparison class, paradigm cases, contrasting cases, etc.” (33) But external-contextsensitivity is not what’s central to Shapiro’s contextualism. Even fixing external context, vague terms’ (anti-)extensions exhibit sensitivity to internal context: the decisions of competent speakers. According to Shapiro’s open texture thesis, for each borderline case, there is some circumstance in which a speaker, consistently with the term’s meaning and the non-linguistic facts, can judge it to fall into the term’s extension and some circumstance in which the speaker can judge it to fall into the term’s anti-extension: she can “go either way.” Moreover, borderline sentences are Euthyphronically judgment- dependent: a competent speaker’s judging a borderline to fall into a term’s (anti- )extension makes it so. For Shapiro, then, a sentence can be true but indeterminate: a case left unsettled by meaning and the non-linguistic facts (and thus indeterminate, or borderline) may be made true by a competent speaker’s judgment. Importantly, among the non-linguistic facts that constrain speakers’ judgments (at least in the cases Shapiro cares about) is a principle of tolerance: for all x and y, if x and y differ marginally in the relevant respect (henceforth, Mxy), then if one competently judges Bx, one cannot competently judge y in any other manner in the same (total) context.1 This does not require that one judge By: one might not consider the matter at all..

Stewart Shapiro's ambition in Vagueness in Context is to develop a comprehensive account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary according to their context: a person can be tall with respect to male accountants and not tall (even short) with respect to professional basketball players. The key feature of Shapiro's account is that the extensions of vague terms also vary in the course of conversations and that, in some cases, a competent speaker can go either way without sinning against the meaning of the words or the non-linguistic facts. As Shapiro sees it, vagueness is a linguistic phenomenon, due to the kinds of languages that humans speak; but vagueness is also due to the world we find ourselves in, as we try to communicate features of it to each other.

The paper presents a new theory of higher-order vagueness. This theory is an improvement on current theories of vagueness in that it (i) describes the kind of borderline cases relevant to the Sorites paradox, (ii) retains the ‘robustness’ of vague predicates, (iii) introduces a notion of higher-order vagueness that is compositional, but (iv) avoids the paradoxes of higher-order vagueness. The theory’s central building-blocks: Borderlinehood is defined as radical unclarity. Unclarity is defined by means of competent, rational, informed speakers (‘CRISPs’) whose competence, etc., is indexed to the scope of the unclarity operator. The unclarity is radical since it eliminates clear cases of unclarity and, that is, clear borderline cases. This radical unclarity leads to a (bivalence-compatible, non-intuitionist) absolute agnosticism about the semantic status of all borderline cases. The corresponding modal system would be a non-normal variation on S4M.

It is generally supposed that borderline cases account for the tolerance of vague terms, yet cannot themselves be sharply bounded, leading to infinite levels of higher order vagueness. This higher order vagueness subverts any formal effort to make language precise. However, it is possible to show that tolerance must diminish at higher orders. The attempt to derive it from indiscriminability founders on a simple empirical test, and we learn instead that there is no limit to how small higher order tolerance may become. That means there is no limit to how precisely we may draw the boundaries of borderline cases, thus forestalling any requirement for higher order vagueness.

To get to grips with what Shapiro does and can say about higher-order vagueness, it is first necessary to thoroughly review and evaluate his conception of (first-order) vagueness, a conception which is both rich and suggestive but, as it turns out, not so easy to stabilise. In Sections I–IV, his basic position on vagueness (see Shapiro [2003]) is outlined and assessed. As we go along, I offer some suggestions for improvement. In Sections V–VI, I review two key paradoxes of higher-order vagueness, while in Section VII, I explore a possible line of response to such paradoxes given by Keefe [2000]. In Section VIII, I assess whether which Shapiro might adapt Keefe’s response to combat both paradoxes.

Higher-order vagueness is widely thought to be a feature of vague predicates that any adequate theory of vagueness must accommodate. It takes a variety of forms. Perhaps the most familiar is the supposed existence, or at least possibility, of higher-order borderline cases—borderline borderline cases, borderline borderline borderline cases, and so forth. A second form of higherorder vagueness, what I will call ‘prescriptive’ higher-order vagueness, is thought to characterize complex predicates constructed from vague predicates by attaching operators having to do with the predicates’ proper application. For example, the predicates ‘mandates application of “old”’ and ‘can competently be called “old”’ are prescriptively higher-order vague. Higher-order vagueness appears in other guises as well,1 but these two have been of particular interest to philosophers and will be my target here. I want to expose some misconceptions about them. If I am right, higher-order vagueness is less prevalent, and less important theoretically, than is usually supposed.2 In what follows I am going to assume that vagueness is a semantic feature of natural language. For the most part I won’t discuss epistemic or pragmatic views, and I will say nothing about so-called metaphysical vagueness.

ABSTRACT: Stewart Shapiro recently argued that there is no higher-order vagueness. More specifically, his thesis is: (ST) ‘So-called second-order vagueness in ‘F’ is nothing but first-order vagueness in the phrase ‘competent speaker of English’ or ‘competent user of “F”’. Shapiro bases (ST) on a description of the phenomenon of higher-order vagueness and two accounts of ‘borderline case’ and provides several arguments in its support. We present the phenomenon (as Shapiro describes it) and the accounts; then discuss Shapiro’s arguments, arguing that none is compelling. Lastly, we introduce the account of vagueness Shapiro would have obtained had he retained compositionality and show that it entails true higher-order vagueness.