Deflationists cannot make sense ofthe notion of referential indeterminacybecause they deny the existence of substantivereference. One way for them to make sense ofthe objective existence of linguisticindeterminacy is by embracing theworldly (or objectual) view ofindeterminacy, the view that indeterminacyexists not in reference relations but in the(non-linguistic) world itself. On this view,the entire world is divided into precisified worlds, just as it is dividedinto temporal slices and (arguably) alethicpossible worlds. Supervaluationism proves tobe neutral with respect to the debate betweenthe worldly view (...) and the referential view ofindeterminacy. (shrink)

On two standard views of vagueness, vagueness as to whether Harry is bald entails that nobody knows whether Harry is bald—either because vagueness is a type of missing truth, and so there is nothing to know, or because vagueness is a type of ignorance, and so even though there is a truth of the matter, nobody can know what that truth is. Vagueness as to whether Harry is bald does entail that nobody clearly knows that Harry is bald and that (...) nobody clearly knows that Harry is not bald. But it does not entail that nobody knows that Harry is bald or that nobody knows that Harry is not bald. Hence, the two standard views of vagueness are mistaken. (shrink)

I argue, pace Timothy Williamson, that one cannot provide an adequate account of what it is for a case to be borderline by appealing to facts about our inability to discriminate our actual situation from nearby counterfactual situations in which our language use differs in subtle ways. I consider the two most natural ways of using such resources to provide an account of what it is for a case to be borderline and argue that both face crippling defects. I argue (...) that the problems faced by these two accounts point to more general reasons to be skeptical of the claim that facts about semantic indiscriminability provide sufficient resources for an analysis of what it is for a case to be borderline. (shrink)

This book is a must-read for researchers interested in taking a high-level, non-mechanistic approach to answering age-old fundamental questions in the brain ...

In this paper I present two arguments against the thesis that we experience qualia. I argue that if we experienced qualia then these qualia would have to be essentially vague entities. And I then offer two arguments, one a reworking of Gareth Evans' argument against the possibility of vague objects, the other a reworking of the Sorites argument, to show that no such essentially vague entities can exist. I consider various objections but argue that ultimately they all fail. In particular (...) I claim that the stock responses to the Sorites argument fail against my reworking of the argument because they require us to make a distinction between a determinate reality and how that reality appears to us, whereas in the case of qualia we can make no such distinction. I conclude that there can be no such things as qualia. (shrink)

"All the limitative Theorems of metamathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally. Godel's Incompleteness Theorem, Church's Undecidability Theorem, Turing's Halting Problem, Turski's Truth Theorem -- all have the flavour of some ancient fairy tale which warns you that `To seek self-knowledge is to embark on a journey which...will always be incomplete, (...) cannot be charted on a map, will never halt, cannot be described. " - Douglas R. Hofstadter.. (shrink)

The problem addressed is that of finding a sound characterization of ambiguity. Two kinds of characterizations are distinguished: tests and definitions. Various definitions of ambiguity are critically examined and contrasted with definitions of generality and indeterminacy, concepts with which ambiguity is sometimes confused. One definition of ambiguity is defended as being more theoretically adequate than others which have been suggested by both philosophers and linguists. It is also shown how this definition of ambiguity obviates a problem thought to be posed (...) by ambiguity for truth theoretical semantics. In addition, the best known test for ambiguity, namely the test by contradiction, is set out, its limitations discussed, and its connection with ambiguity's definition explained. The test is contrasted with a test for vagueness first proposed by Peirce and a test for generality propounded by Margalit. (shrink)

This paper presents an new epistemicist account of vagueness, one that avoids standard arbitrariness worries by exploiting a plenitudinous metaphysic.

The idea that the phenomenon of vagueness might be modelled by a paraconsistent logic has been little discussed in contemporary work on vagueness, just as the idea that paraconsistent logics might be fruitfully applied to the phenomenon of vagueness has been little discussed in contemporary work on paraconsistency. This is prima facie surprising given that the earliest formalisations of paraconsistent logics presented in Ja´skowski (1948) and Halldén (1949) were presented as logics of vagueness. One possible explanation for this is that, (...) despite initial advocacy by pioneers of paraconsistency, the prospects for a paraconsistent account of vagueness are so poor as to warrant little further consideration. In this paper we look at the reasons that might be offered in defence of this negative claim. As we shall show, they are far from compelling. Paraconsistent accounts of vagueness deserve further attention. (shrink)

Classical logic rests on the assumption that there are two mutually exclusive and jointly exhaustive truth values. This assumption has always been surrounded by philosophical controversy. Doubts have been raised about its legitimacy, and hence about the legitimacy of classical logic. Usually, the assumption is stated in the form of a general principle, namely the principle that every proposition is either true or false. Then, the philosophical controversy is often framed in terms of the question whether every proposition is either (...) true or false. The main purpose of the paper is to show that there is something wrong in this way of putting things. The point is that the common way of understanding the controversial assumption is misconceived, as it rests on a wrong picture of propositions. In the first part of the paper I outline this picture and I argue against it. In the second part I sketch a different picture of propositions and I suggest how this leads to conceive the issue of classical logic in different terms. (shrink)

This paper is an attempt to extend the meaning of the concept of indeterminacy for the human sciences. The authors do this by coining the term methodological indeterminacy and arguing that indeterminacy is better understood when linked to specific methodological techniques. Paradoxically, while specific research techniques demonstrate that the issue of indeterminacy is complex, yielding the possibility of types and degrees, it does not eliminate the problem of translation first raised by Quine. However, the authors go on to argue that, (...) from a research perspective, indeterminacy can and must be approached in such a way that it is possible to reduce cases of it, even though never completely eliminating it in the human sciences. (shrink)

This paper explores the relationships between Davidson's indeterminacy of interpretation thesis and two semantic properties of sentences that have come to be recognized recently, namely semantic incompleteness and semantic indecision.1 More specifically, I will examine what the indeterminacy thesis entails for sentences of the form 'By sentence S (or word w), agent A means that m' and 'Agent A believes that p.' My primary goal is to shed light on the indeterminacy thesis and its consequences. I will distinguish two kinds (...) of indeterminacy that have very different sources and very different consequences. But this does not purport to be an exhaustive study: there may well be other forms of indeterminacy that this .. (shrink)

Abstract In Causing Actions, Pietroski defends a distinctive view of the relationship between mind and body which he calls Personal Dualism. Central to his defence is the Argument from Differential Vagueness. It moves from the claim that mental events have different vagueness of spatiotemporal boundaries from neural events to the claim that mental events are not identical to neural events. In response, I argue that this presupposes an ontological account of vagueness that there is no reason to believe in this (...) context. I further argue that Pietroski's reasons for rejecting the possibility that mental events are vaguely constituted from neural events are inadequate. I go on to show how Pietroski's Personal Dualism is ill-equipped to deal with the problem of mental causation because of its apparently necessary appeal to ceteris paribus laws. (shrink)

Ordinary intuitions that vague predicates are tolerant, or cannot have sharp boundaries, can be formalized in first-order logic in at least two non-equivalent ways, a stronger and a weaker. The stronger turns out to be false in domains that have a significant central gap for the predicate in question, i.e. where a sufficiently large middle segment of the ordering relation (such as taller for ‘tall’) is uninstantiated. The weaker principle is true in such domains, but does not in those domains (...) induce the sorites conclusion. (shrink)

The goal of this paper is to develop a theory of content for vague language. My proposal is based on the following three theses: (1) language-mastery is not rulebased— it involves a certain kind of decision-making; (2) a theory of content is to be thought of instrumentally—it is a tool for making sense of our linguistic practice; and (3) linguistic contents are only locally defined—they are only defined relative to suitably constrained sets of possibilities. CiteULike Connotea Del.icio.us What's this?

My goal is to defend the indeterminist approach to vagueness, according to which a borderline vague utterance is neither true nor false. Indeterminism appears to contradict bivalence and the disquotational schema for truth. I agree that indeterminism compels us to modify each of these principles. Kit Fine has defended indeterminism by claiming that ordinary ambiguous sentences are neither true nor false when one disambiguation is true and the other is false. But even if Fine is right about sentences, his point (...) does not seem to generalize to utterances. What the indeterminist needs -- and what ordinary ambiguity does not provide -- is an ambiguous utterance where what is being said is indeterminate between two different propositions. I will show that such cases exist. These cases imply that the modifications that indeterminism makes to bivalence and the disquotational schema are required independently of indeterminism, in fact independently of vagueness. (shrink)

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P , then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of (...) it, which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases. (shrink)

The problem of vagueness and the sorites paradox arise because we try to treat natural language as if it were a unitary formal system. In fact, natural language contains a large variety of representational ontologies that serve different purposes and which cannot be united formally, but which can intuitively be taken as ways of seeing a common basic ontology. Using this framework, we can save classical logic from vagueness and avoid the sorites.

Determining whether the law of excluded middle requires bivalence depends upon whether we are talking about sentences or propositions. If we are talking about sentences, neither side has a decisive case. If we are talking of propositions, there is a strong argument on the side of those who say the excluded middle does require bivalence. I argue that all challenges to this argument can be met.

A certain conception of Hell is inconsistent with God’s traditional attributes, or so I will argue. My argument is novel in focusing on considerations involving vagueness. The target doctrine of Hell is part of a “binary” conception of the afterlife, by which I mean one with the properties of dichotomy, badness, non-universality, and divine control. Dichotomy: there are exactly two states in the afterlife, Heaven and Hell. After death each person will come to be, determinately, in exactly one of these (...) states. (The doctrine of Purgatory does not violate dichotomy provided everyone in Purgatory eventually ends up in Heaven.) My argument does not apply to a continuous conception of the afterlife, which to my mind is more defensible than the usual binary doctrine. Badness: Hell is very, very bad. Or at least, Hell is much worse than Heaven; for most of the argument this weaker premise will suf ce. More carefully, the premise is that everyone in Heaven is much, much better off than everyone in Hell. Non-universality: some people go to Heaven, and some people go to Hell. I have no objection to Universalists, according to whom everyone goes to Heaven. Nor does my argument apply to those who uphold universal damnation. Divine control: God is in control of the institution of divine judgment, in control of the mechanism or criterion that determines who goes to Heaven and who goes to Hell. This is not to say that God is solely responsible for the fate of created beings, for the divinely mandated criterion might contain a role for free choices. Nor is it to say that God is vindictive. The requirement makes no assumptions about the nature of the criterion, beyond that it is in God’s control. The argument proceeds as follows. Given dichotomy, the only possibilities in the afterlife are determinate membership in either Heaven or Hell; given badness, the second is far worse the rst; and given non-universality, each is populated. Divine control requires that God be in control of the criterion determining these populations, and thus that God’s choice of a criterion be consistent with his attributes. (shrink)

We use the interpretation of vague scalar predicates like small as an illustration of how systematic semantic models of dialogue context enable the derivation of useful, fine-grained utterance interpretations from radically underspeci- fied semantic forms. Because dialogue context suffices to determine salient alternative scales and relevant distinctions along these scales, we can infer implicit standards of comparison for vague scalar predicates through completely general pragmatics, yet closely constrain the intended meaning to within a natural range.

THERE IS OFTEN UNCERTAINTY ABOUT WHETHER SOME PREDICATE APPLIES TO SOME PHYSICAL OBJECT OR STATE. THIS UNCERTAINTY MAY HAVE ANY OF THREE SOURCES - VAGUENESS OF A TERM, INEXACTNESS OF A CONCEPT, OR PRACTICAL DIFFICULTY IN DETERMINING ITS APPLICABILITY. VARIOUS WAYS IN WHICH CONCEPTUAL INEXACTNESS OR PRACTICAL DIFFICULTY MAY PRODUCE UNCERTAINTY ARE DISTINGUISHED. NEITHER TERMINOLOGICAL VAGUENESS, NOR PRACTICAL DIFFICULTY IN DETERMINING THE APPLICABILITY OF A CONCEPT ARE NECESSARY FEATURES OF EVERY LANGUAGE IN EVERY PHYSICAL WORLD, BUT CONCEPTUAL INEXACTNESS IS A (...) NECESSARY FEATURE OF EMPIRICAL CONCEPTS. THESE RESULTS ARE UTILIZED TO DETERMINE THE SOURCES AND NECESSITY OF IMPRECISION IN MEASUREMENT. (shrink)

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some extra (...) set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true , etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided. (shrink)

If one believes that vagueness is an exclusively representational phenomenon, one faces the problem of the many. In the vicinity of Kilimanjaro, there are many many ‘mountain candidates’ all, apparently, with more-or-less equal claim to be mountains. David Lewis has defended a radical claim: that all the billions of mountain candidates are mountains. This paper argues that the supervaluationist about vagueness should adopt Lewis’ proposal, on pain of losing their best explanation of the seductiveness of the sorites.

This paper addresses three problems: the problem of formulating a coherent relativism, the Sorites paradox and a seldom noticed difficulty in the best intuitionistic case for the revision of classical logic. A response to the latter is proposed which, generalised, contributes towards the solution of the other two. The key to this response is a generalised conception of indeterminacy as a specific kind of intellectual bafflement - Quandary. Intuitionistic revisions of classical logic are merited wherever a subject matter is conceived (...) both as liable to generate Quandary and as subject to a broad form of evidential constraint. So motivated, the distinctions enshrined in intuitionistic logic provide both for a satisfying resolution of the Sorites paradox and a coherent outlet for relativistic views about, e.g., matters of taste and morals. An important corollary of the discussion is that an epistemic conception of vagueness can be prised apart from the strong metaphysical realism with which its principal supporters have associated it, and acknowledged to harbour an independent insight. (shrink)

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. (shrink)

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. (shrink)

A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of this paper is (...) to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

ABSTRACT: 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)

This paper discusses Fara's so-called 'Paradox of Higher-Order Vagueness' concerning supervaluationism. In the paper I argue that supervaluationism is not committed to global validity, as it is largely assumed in the literature, but to a weaker notion of logical consequence I call 'regional validity'. Then I show that the supervaluationist might solve Fara's paradox making use of this weaker notion of logical consequence. The paper is discussed by Delia Fara in the same volume.

Though it is standardly assumed that supervaluationism applied to vagueness is committed to global validity, Achille Varzi (2007) argues that the supervaluationist should take seriously the idea of adopting local validity instead. Varzi’s motivation for the adoption of local validity is largely based on two objections against the global notion: that it brings some counterexamples to classically valid rules of inference and that it is inconsistent with unrestricted higher-order vagueness. In this discussion I review these objections and point out ways (...) to address them not considered in Varzi’s paper. (shrink)

Paraconsistent approaches have received little attention in the literature on vagueness (at least compared to other proposals). The reason seems to be that many philosophers have found the idea that a contradiction might be true (or that a sentence and its negation might both be true) hard to swallow. Even advocates of paraconsistency on vagueness do not look very convinced when they consider this fact; since they seem to have spent more time arguing that paraconsistent theories are at least as (...) good as their paracomplete counterparts, than giving positive reasons to believe on a particular paraconsistent proposal. But it sometimes happens that the weakness of a theory turns out to be its mayor ally, and this is what (I claim) happens in a particular paraconsistent proposal known as subvaluationism. In order to make room for truth-value gluts subvaluationism needs to endorse a notion of logical consequence that is, in some sense, weaker than standard notions of consequence. But this weakness allows the subvaluationist theory to accommodate higher-order vagueness in a way that it is not available to other theories of vagueness (such as, for example, its paracomplete counterpart, supervaluationism). (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)

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)

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)

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)

Supervaluational theories of vagueness have achieved considerable popularity in the past decades, as seen in eg [5], [12]. This popularity is only natural; supervaluations let us retain much of the power and simplicity of classical logic, while avoiding the commitment to strict bivalence that strikes many as implausible. Like many nonclassical logics, the supervaluationist system SP has a natural dual, the subvaluationist system SB, explored in eg [6], [28].1 As is usual for such dual systems, the classical features of SP (...) (typically viewed as benefits) appear in SB in ‘mirror-image’ form, and the nonclassical features of SP (typically viewed as costs) also appear in SB in ‘mirror-image’ form. Given this circumstance, it can be difficult to decide which of two dual systems is better suited for an approach to vagueness.2 The present paper starts from a consideration of these two approaches— the supervaluational and the subvaluational—and argues that neither of them is well-positioned to give a sensible logic for vague language. §2 presents the systems SP and SB and argues against their usefulness. Even if we suppose that the general picture of vague language they are often taken to embody is accurate, we ought not arrive at systems like SP and SB. Instead, such a picture should lead us to truth-functional systems like strong Kleene logic (K3) or its dual LP. §3 presents these systems, and argues that supervaluationist and subvaluationist understandings of language are better captured there; in particular, that a dialetheic approach to vagueness based on the logic LP is a more sensible approach. §4 goes on to consider the phenomenon of higher-order vagueness within an LP-based approach, and §5 closes with a consideration of the sorites argument itself. (shrink)

Los dos grandes problemas del enfoque supervaluacionista para la vaguedad son determinar cuáles son las precisificaciones admisibles y la vaguedad de orden superior ilimitado. Apelando al uso de los términos vagos por la comunidad lingüística competente puede dividirse de forma tajante la extension de un término precisando en qué casos se aplica definidamente, en cuáles se aplica indefinidamente y en cuales es indeterminado si se aplica. Esto produce dos órdenes de vaguedad, con lo que se bloquean los argumentos sorites. Finalmente (...) se indaga qué semántica resulta más adecuada para una vaguedad de segundo orden. La vaguedad se basa en una indeterminación determinada en la aplicación de ciertos términos por parte de los hablantes nativos. The two main problems for the supevaluationationist approach to vagueness are deciding which precifications are admissible and unlimited higher-order vagueness. Resorting to the use of vague terms by the competent linguistic community the extension of a term can be sharply divided, making precise in which cases it applies definitely, when it applies indefinitely and when it is undetermined whether it applies. This produces two orders of vagueness, and as a result sorties arguments are blocked. Finally, it is enquired what semantic fits better to second order vagueness. Vagueness is based on a determined indeteminacy concerning the application of certain terms by native speakers. (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 accommodate vague statements and predicates, I propose an infinite-valued, non-truth-functional interpretation of logic on which the tautologies are exactly the tautologies of classical two-valued logic. iI introduce a determinacy operator, analogous to the necessity operator in alethic modal logic, to allow the definition of first-order and higher-order borderline cases. On the interpretation proposed for determinacy, every statement corresponding to a theorem of modal system T is a logical truth, and I conjecture that every logical truth on the interpretation corresponds (...) to a theorem of T. the interpretation is extended to predicate logic. A borderline case of a predicate 'F’ is neither determinately F nor determinately not-F. Traditional sorites arguments are seen to fall apart early in their gradual stepwise passage from truth to falsity. (shrink)