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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 (such as ‘borderline’ and ‘definitely’) 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.

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.

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.

The purpose of this paper is to challenge some widespread assumptions about the role of the modal axiom S4 in a theory of vagueness. In the context of vagueness, S4 usually appears as the principle ‘If it is clear (determinate, definite) that A, then it is clear (determinate, definite) that it is clear (determinate, definite) that A’, or, more formally, CA → CCA. In this paper, we argue first that, contrary to common opinion, higher-order vagueness and S4 are perfectly compatible. This is in response to claims like that by Williamson that, if vagueness is defined with the help of a clarity operator that obeys S4, higher-order vagueness disappears. Second, we argue that, contrary to common opinion, (i) bivalence-preservers (e.g. epistemicists) can without contradiction condone S4, and (ii) bivalence-discarders (e.g. open-texture theorists, supervaluationists) can without contradiction reject S4. To this end, we show how in the debate over S4 two different notions of clarity are in play and what their respective functions are in accounts of higher-order vagueness. Third, we rebut a number of arguments that have been produced by opponents of S4, in particular those by Williamson.

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.

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.

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.

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.

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.