University of Texas at Austin dever{at}mail.utexas.edu ' + u + '@' + d + ' '//--> Chris Pappas University of Texas at Austin Abstract Supervaluational accounts of vagueness have come under assault from Timothy Williamson for failing to provide either a sufficiently classical logic or a disquotational notion of truth, and from Crispin Wright and others for incorporating a notion of higher-order vagueness, via the determinacy operator, which leads to contradiction when combined with intuitively appealing ‘gap principles’. We argue that (...) these criticisms of supervaluation theory depend on giving supertruth an unnecessarily central role in that theory as the sole notion of truth, rather than as one mode of truth. Allowing for the co-existence of supertruth and local truth, we define a notion of local entailment in supervaluation theory, and show that the resulting logic is fully classical and allows for the truth of the gap principles. Finally, we argue that both supertruth and local truth are disquotational, when disquotational principles are properly understood. CiteULike Connotea Del.icio.us What's this? (shrink)

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. (shrink)

Recently a bold and admirable interpretation of Chrysippus’ position on the Sorites has been presented, suggesting that Chrysippus offered a solution to the Sorites by (i) taking an epistemicist position1 which (ii) made allowances for higher-order vagueness.2 In this paper I argue (i) that Chrysippus did not take an epistemicist position, but − if any − a non-epistemic one which denies truth-values to some cases in a Sorites-series, and (ii) that it is uncertain whether and how he made allowances for (...) higher-order vagueness, but if he did, this was not grounded on an epistemicist position. (shrink)

A short follow-up to Anna Mahtani’s paper ‘Can Vagueness Cut Out at Any Order?’. I describe a model implementing Mahtani's idea of “variable accessibility ranges” in which, for every n, there is a sentence that is nth-order vague but n+1th-order precise, in the sense of Williamson’s paper ‘On the Structure of Higher-Order Vagueness’.

The conclusion of this chapter is that higher-order vagueness is universal: no sentence whatsoever is definitely true, definitely definitely true, definitely definitely definitely true, and so on ad infinitum. The argument, of which there are several versions, turns on the existence of Sorites sequences of possible worlds connecting the actual world to possible worlds where a given sentence is used in such a way that its meaning is very different. The chapter attempts to be neutral between competing accounts of the (...) nature of vagueness and definiteness. (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)

Philosophers disagree about whether vagueness requires us to admit truth-value gaps, about whether there is a gap between the objects of which a given vague predicate is true and those of which it is false on an appropriately constructed sorites series for the predicate—a series involving small increments of change in a relevant respect between adjacent elements, but a large increment of change in that respect between the endpoints. There appears, however, to be widespread agreement that there is some sense (...) in which vague predicates are gappy which may be expressed neutrally by saying that on any appropriately constructed sorites series for a given vague predicate there will be a gap between the objects of which the predicate is definitely true and those of which it is definitely false. Taking as primitive the operator ‘it is definitely the case that’, abbreviated as ‘D’, we may stipulate that a predicate F is definitely true (or definitely false) of an object just in case ‘DF (a)’, where a is a name for the object, is true (or false) simpliciter.1 This yields the following conditional formulation of a ‘gap principle’: (DΦ(x) ∧ D¬Φ(y)) → ¬R(x, y). Here ‘Φ’ is to be replaced with a vague predicate, while ‘R’ is to stand for a sorites relation for that predicate: a relation that can be used to construct a sorites series for the predicate—such as the relation of being just one millimetre shorter than for the predicate ‘is tall’. Disagreements about the sense in which it is correct to say that vague predicates are gappy can then be recast as disagreements about how to understand the definitely operator. One might give it, for example, a pragmatic construal such as ‘it would not be misleading to assert that’; or an epistemic construal such as ‘it is known that’ or ‘it is knowable that’; or a semantic construal such as ‘it is true that’. (shrink)

Michael Tye responds to the problem of higher-order vagueness for his trivalent semantics by maintaining that truth-value predicates are “vaguely vague”: it’s indeterminate, on his view, whether they have borderline cases and therefore indeterminate whether every sentence is true, false, or indefinite. Rosanna Keefe objects (1) that Tye’s argument for this claim tacitly assumes that every sentence is true, false, or indefinite, and (2) that the conclusion is any case not viable. I argue – contra (1) – that Tye’s argument (...) needn’t make that assumption. A version of her objection is in fact better directed against other arguments Tye advances, though Tye can absorb this criticism without abandoning his position’s core. On the other hand, Keefe’s second objection does hit the mark: embracing ‘vaguely vague’ truth-value predicates undermines Tye’s ability to support validity claims needed to defend his position. To see this, however, we must develop Keefe’s remarks further than she does. (shrink)

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. (shrink)

Is 'vague' vague? Is the meaning of 'true' vague? Is higher-order vagueness unavoidable? Is it possible to say precisely what it is to say something precisely? These questions, deeply interrelated and of fundamental importance to logic and semantics, have been addressed recently by Achille Varzi in articles focused on an ingenius attempt by Roy Sorensen ("An Argument for the Vagueness of 'Vague'") to demonstrate that 'vague' is vague.

Varzi has recently joined a thread of arguments originating in an attempt by Sorensen (1985) to demonstrate that the predicate ‘vague’ is itself vague. Sorensen's conclusion is significant in that it has provided the basis for a subsequent effort by Hyde (1994) to defend the legitimacy of supposing higher-order vagueness. Varzi's contribution to this debate is twofold. First, contra earlier criticism by Deas (1989), he claims that Sorensen's result is sound so far as it goes. Second, he argues that despite (...) this it cannot be used as Hyde wishes on pain of circularity. I am not interested in the latter argument—it is examined in Hyde (2003)—but rather wish to defend and elaborate Deas's criticism of Sorensen against Varzi's repudiation. (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.

Could a sentence be, say, 3rd order vague, but 4th order precise? In Williamson 1999 we find an argument that seems to show that this is impossible: every sentence is either 1st order precise, 2nd order precise, or infinitely vague. The argument for this claim is unpersuasive, however, and this paper explains why.

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. (shrink)

It is generally agreed that vague predicates like ‘red’, ‘rich’, ‘tall’, and ‘bald’, have borderline cases of application. For instance, a cloth patch whose color lies midway between a definite red and a definite orange is a borderline case for ‘red’, and an American man five feet eleven inches in height is (arguably) a borderline case for ‘tall’. The proper analysis of borderline cases is a matter of dispute, but most theorists of vagueness agree at least in the thought that (...) borderline cases for vague predicate ‘ ’ are items whose satisfaction of ‘ ’ is in some sense unclear or problematic: it is unclear whether or not the patch is red, unclear whether or not the man is tall.1 For example, Lynda Burns cites a widespread view as holding that borderline cases “are not definitely within the positive or negative extension of the predicate. … Border- line cases are seen as falling within a gap between the cases of definite application of the predicate and cases of definite application of its negation” (1995, 30). Michael Tye writes that the “concept of a border- line case is the concept of a case that is neither definitely in nor defi- nitely out” (1994b, 18). (shrink)

If there are vague numbers, it would be easier to use numbers as semantic values in a treatment of vagueness while avoiding precise cut-off points. When we assign a particular statement a range of values (less than 1 and greater than 0) there is no precise sharp cut-off point that locates the greatest lower bound or the least upper bound of the interval, I should like to say. Is this possible? “Vague Numbers” stands for awareness of the problem. I do (...) not present a serious theory of vague numbers. I sketch some reasons for using a many-value semantics. These reasons refer to my earlier treatments of determinacy and definitions of higher-order borderline cases. I also sketch how definitions of independence use the determinacy operator. The distinction between actually assigned values and values whose assignments are acceptable helps avoid unwanted precise cut-off points. (shrink)

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. (shrink)

After a brief account of the problem of higher-order vagueness, and its seeming intractability, I explore what comes of the issue on a linguistic, contextualist account of vagueness. On the view in question, predicates like ‘borderline red’ and ‘determinately red’ are, or at least can be, vague, but they are different in kind from ‘red’. In particular, ‘borderline red’ and ‘determinately red’ are not colours. These predicates have linguistic components, and invoke notions like ‘competent user of the language’. On my (...) view, so-called ‘higher-order vagueness’ is actually ordinary, first-order vagueness in different predicates. I explore the possibility that, nevertheless, a pernicious regress ensues. (shrink)

This paper presents a new theory of vagueness, which is designed to retain the virtues of the fuzzy theory, while avoiding the problem of higher-order vagueness. The theory presented here accommodates the idea that for any statement S1 to the effect that 'Bob is bald' is x true, for x in [0,1], there should be a further statement S2 which tells us how true S1 is, and so on---that is, it accommodates higher-order vagueness---without resorting to the claim that the metalanguage (...) in which the semantics of vagueness is presented is itself vague, and without requiring us to abandon the idea that the logic---as opposed to the semantics---of vague discourse is classical. I model the extension of a vague predicate P as a BLURRY SET, this being a function which assigns a degree of membership or DEGREE FUNCTION to each object o, where a degree function in turn assigns an element of [0,1] to each finite sequence of elements of [0,1]. The idea is that the assignment to the sequence (0.3,0.2), for example, represents the degree to which it is true to say that it is 0.2 true that o is P to degree 0.3. The philosophical merits of my theory are discussed in detail, and the theory is compared with other extensions and generalisations of fuzzy logic in the literature. (shrink)

A theory of higher-order vagueness for partially-defined, context-sensitive predicates like is blue is offered. According to the theory, the predicate is determinately blue means roughly is an object o such that the claim that o is blue is a necessary consequence of the rules of the language plus the underlying non-linguistic facts in the world. Because the question of which rules count as rules of the language is itself vague, the predicate is determinately blue is both vague and partial in (...) a sense analogous (though not completely identical) to the sense in which is blue is vague and partial. However, higher-order vagueness stops there. Although the anti-extension of is determinately blue differs from that of is blue, further iterations of determinately have no semantic effect. The paper closes with a discussion of the need to posit sharp and precise lines dividing the range of potential application of a vague predicate into semantically distinct categories. It is argued that this is less of a problem for the present theory than for many others, because the categories into which the range is divided by the present theory -- like the category of counting as blue because the rules of the language offer speakers no discretion, as opposed to the category of counting as blue because speakers have invoked an absolute minimum of discretion in adjusting the contextually sensitive boundaries of the predicate -- are not those that the linguistic competence of ordinary speakers equips them to make highly accurate and reliable judgments about. (shrink)

Is higher-order vagueness a real phenomenon? Dominic Hyde (1994) claims that it is, and that it is part and parcel of vagueness itself. According to Hyde, any genuinely vague predicate must also be higher-order vague. His argument for this view is unsound, however. The purpose of this note is to expose the fallacy, and to make some related observations on the vague, the higher-order vague, and the vaguely vague.

A rejoinder to G. Hull’s reply to my Mind 2003. Hull argues that Sorensen’s purported proof that ‘vague’ is vague--which I defended against certain familiar objections--fails. He offers three reasons: (i) the vagueness exhibited by Sorensen’s sorites is just the vagueness of ‘small’; (ii) the general assumption underlying the proof, to the effect that predicates which possess borderline cases are vague, is mistaken; (iii) the conclusion of the proof is unacceptable, for it is possible to create Sorensen-type sorites even for (...) predicates that are paradigmatically precise. I argue that each of these points involves fallacious reasoning. (shrink)

R. Sorensen’s argument to the effect that ’vague’ is a vague predicate has been used by D. Hyde to infer that vague predicates suffer from higher-order vagueness. M. Tye has objected (convincingly) that this is too strong: all that follows from Sorensen’s result is that there are some border border cases, but not necessarily border border cases of every vague predicate. I argue that this is still too strong: Sorensen’s proof presupposes the existence of border border cases, hence cannot be (...) used to establish that fact on pain of circularity. (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)

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. (shrink)