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This paper considers the connections between semantic shiftiness (plasticity), epistemic safety and an epistemic theory of vagueness as presented and defended by Williamson (1996a, b, 1997a, b). Williamson explains ignorance of the precise intension of vague words as rooted in insensitivity to semantic shifts: one’s inability to detect small shifts in intension for a vague word results in a lack of knowledge of the word’s intension. Williamson’s explanation, however, falls short of accounting for ignorance of intension.

No one can know the threshold of a vague predicate. For instance, given that ten is the last small number, no one can know that ten fits the description `the last small number’.1 Knowledge that ten is the last small number requires knowledge of the weaker proposition that ten is a small number. Knowledge of thresholds entails knowledge at thresholds. Timothy Williamson ingeniously argues that this weaker knowledge is always impossible. On that basis he claims to have explained our ignorance (...) of the thresholds of vague predicates. (shrink)

The first part of this paper presents asympathetic and critical examination of the approachof Shahid Rahman and Walter Carnielli, as presented intheir paper The Dialogical Approach toParaconsistency. In the second part, possibleextensions are presented and evaluated: (a) top-downanalysis of a dialogue situation versus bottom-up, (b)the specific role of ambiguities and how to deal withthem, and (c) the problem of common knowledge andbackground knowledge in dialogues. In the third part,I claim that dialogue logic is the best-suitedinstrument to analyse paradoxes of the (...) Sorites type.All these considerations lead to philosophicallyrelevant observations concerning principles of charityon the one hand, and compactness on the other. (shrink)

This paper defends a pragmatical approach to vagueness. The vagueness-adaptive logic VAL is a good reconstruction of and an excellent, instrument for human reasoning processes in which vague predicates are involved. Apart from its proof-theory and semantics, a Sorites-treating model based on it is presented, disarming the paradox. The paper opens perspectives with respect to the construction of theories by means of vague predicates.

There is now a widespread accord among philosophers that the vagueness of natural language gives rise to some particularly deep and perplexing problems and paradoxes. It was not always so. For most of the first century of analytical philosophy, vagueness was generally regarded as a marginal, slightly irritating phenomenon, —receiving some attention, to be sure, in parts of the Philosophical Investigations and in the amateur linguistics enjoyed by philosophers in Oxford in the 1950s, but best idealised away in any serious (...) theoretical treatment of meaning, understanding and valid inference. Frege, as is well known, had come to be thoroughly mistrustful of vagueness, supposing that a language fit for the purpose of articulating scientific and mathematical knowledge would have to be purified of it. Later trends in philosophical logic and semantics followed his lead, not indeed in setting about the (futile) task of expurgating vagueness from natural language but by largely restricting theoretical attention to artificial languages in whose workings vagueness was assigned no role. (shrink)

According to extension-shifting theories of vagueness, the extensions of vague predicates have sharp boundaries, which shift as a function of certain psychological factors. Such theories have been claimed to provide an attractive explanation of the appeal of soritical reasoning. I challenge this claim: the demand for such an explanation need not constrain the semantics of vague predicates at all.

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)

The reason, according to the contextualist, that precise boundaries for expressions like ‘heap’ or ‘tall for a basketball player’ are so difficult to detect is that when two entities are sufficiently similar (or saliently similar), we tend to shift the interpretation of the vague expression so that if one counts as falling in the extension of the property expressed by that expression, so does the other. As a conse- quence, when we look for the boundary of the extension of a (...) vague expression in its penumbra, our very looking has the effect of chang- ing the interpretation of the vague expression so that the boundary is not where we are looking. This accounts for the persuasive force of sorites arguments. (Stanley 2003: 269). (shrink)

Here’s a false generalization with manifestly false consequences: all people who differ by one millimeter in height from a short person are themselves short. Why are we inclined to believe it? Boundary-shifters, usually lumped together under the heading “contextualists”, say that we believe the false generalization because when we consider any instance, that instance is true at the time of our consideration. Critics complain that the explanation is no good, for (i) if it were, then fallacious inferences would be rampant; (...) (ii) if it were, then we would always generalize from instances that we knew to be true at the time we considered them. But, they say, fallacious inferences are not rampant and we do not always generalize from the instances. Responses: (i) boundary-shifters require only that weakly fallacious inferences are rampant, and indeed they are; (ii) generalizing from the instances is both natural and explanatory, albeit defeasible. (shrink)

Draft (Version 1.1, October 2007): (PDF file) A reply to Jason Stanley’s Analysis criticism of my interest-relative view on vagueness.

I propose that the meanings of vague expressions render the truth conditions of utterances of sentences containing them sensitive to our interests. For example, 'expensive' is analyzed as meaning 'costs a lot', which in turn is analyzed as meaning 'costs significantly greater than the norm'. Whether a difference is a significant difference depends on what our interests are. Appeal to the proposal is shown to provide an attractive resolution of the sorites paradox that is compatible with classical logic and semantics.

According to the view I christen sharpism, when Joe says to his daughter in a perfectly ordinary context ‘The Earth is super-duper old’, his claim has an incredibly discriminating truth condition: although it’s true if the Earth is over 347,342,343 years, 2 days, and 17 nanoseconds old, if the Earth is even a nanosecond younger then his claim has some status other than “just plain true”—but we leave open what that new status might be: false, indeterminate, indeterminately indeterminate, meaningless, just (...) under 100% true, or whatnot. The material point is that the claim changes in truth status (“alethic status”) with a nanosecond change in the Earth’s age. The sharp cutoff might not be a cutoff separating the true from the false, but it is a sharp alethic cutoff nonetheless. It has this sensitive truth condition even though Joe has never made any relevant linguistic stipulations and doesn’t even know what a nanosecond is. Another example: when I say to a visiting speaker, ‘The auditorium where you’ll give your lecture is a short walk from here’, my claim is true if the auditorium is no more than 123 meters, 6 centimeters, and 16 nanometers away. Hence, if it turns out that we were 123 meters, 6 centimeters, and 17 nanometers away from the auditorium, my claim had some status other than true. (shrink)

Jason Stanley has criticized a contextualist solution to the sorites paradox that treats vagueness as a kind of indexicality. His objection rests on a feature of indexicals that seems plausible: that their reference remains fixed in verb phrase ellipsis. But the force of Stanley’s criticism depends on the undefended assumption that vague terms, if they are a special sort of indexical, must function in the same way that more paradigmatic indexicals do. This paper argues that there can be more than (...) one sort of indexicality, that one term might easily have both sorts, and that therefore, and despite Stanley’s worries, vagueness might easily be assimilated to one form. (shrink)

Saul Kripke pointed out that whether or not an utterance gives rise to a liar-like paradox cannot always be determined by checking just its form or content.1 Whether or not Jones’s utterance of ‘Everything Nixon said is true’ is paradoxical depends in part on what Nixon said. Something similar may be said about the sorites paradox. For example, whether or not the predicate ‘are enough grains of coffee for Smith’s purposes’ gives rise to a sorites paradox depends at least in (...) part on what Smith’s purposes are. If Smith’s purpose is to make some coffee to drink, so that he can wake up and start his day, then we would be inclined to accept, and would find it strange to deny the following sorites sentence. (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)

Contextualism about vagueness (hereafter ‘Contextualism’) is the view that vagueness consists in a particular species of context-sensitivity and that properly accommodating this fact into our semantic theory will yield a plausible solution to the sorites paradox.[1],[2] But Contextualism, as many commentators have noted, faces the following immediate objection: if we hold the context fixed, vagueness still remains, therefore vagueness is not a species of context-sensitivity. Call this ‘the simple objection’.[3] Absent a convincing reply to the simple objection, Contextualism is in (...) very bad shape. Oddly enough, defenders of Contextualism have said very little in reply. Proponents of the objection have tended to assume that this is because no reply is in the offing—the simple objection is taken to be unassailable. In this short paper, we sketch two replies to the simple objection which result in two very different kinds of Contextualism: Epistemicist Contextualism and Radical Contextualism. With these two theories in hand, the simple objection loses its force. (shrink)

Contextualism concerning vagueness (hereafter ‘CV’) is a popular response to the puzzle of vagueness.[1] The goal in this paper is to uncover in what ways vagueness may be a particular species of context-sensitivity. The most promising form of CV turns out to be a version of socalled ‘Non-Indexical Contextualism’.[2] In §2, we sketch a generic form of CV (hereafter ‘GCV’). In §3, we distinguish between Truth CV and Content CV. A non-indexical form of CV is a form of Truth CV, (...) while an indexical form of CV is a form of Content CV. In §4, we argue that the theory of vagueness given in Fara (2000) is crucially incomplete but is best seen as a non-indexical form of CV. In §5, we set forth four kinds of error-theory to which CV might be committed. It turns out that Non-Indexical CV is committed to a weaker, and more plausible, error-theory than Indexical CV. In §6, we address a challenge posed by Keefe (2007) to the effect that CV entails that any speech report of what has been said by a particular vague utterance, where the context of utterance and the reporting context are relevantly different, will almost always be inaccurate. While this challenge is prima facie effective against Indexical CV it proves to be less effective against Non-Indexical CV. In §7, we look at two tests for context-sensitivity and assess whether they can be employed against CV. These tests, if cogent, reveal that the only workable form of CV is Non-Indexical CV. (shrink)

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.. (shrink)

Drawing upon research in philosophical logic, linguistics and cognitive science, this study explores how our ability to use and understand language depends upon our capacity to keep track of complex features of the contexts in which we converse.

Read as a comment on Crispin Wright's \"Vagueness: A Fifth Column Approach\", this paper defends a form of supervaluationism against Wright's criticisms. Along the way, however, it takes up the question what is really wrong with Epistemicism, how the appeal of the Sorities ought properly to be understood, and why Contextualist accounts of vagueness won't do.

In this paper I offer a critique of the recent popular strategy of giving a contextualist account of vagueness. Such accounts maintain that truth-values of vague sentences can change with changes of context induced by confronting different entities (e.g. different pairs through a sorites series). I claim that appealing to context does not help in solving the sorites paradox, nor does it give us new insights into vagueness per se. Furthermore, the contextual variation to which the contextualist is committed is (...) problematic in various ways. For example, it yields the consequence that much of our everyday (non-soritical) reasoning is fallacious, and it renders us ignorant of what we and others have said. (shrink)

The present offers a pragmatic solution of the Heap Paradox, based on the idea that vague predicates are “indexical” in the sense that their denotation does not only depend on the context of their use, but it is a function of the context. The analysis is based on the following three claims. The borderlines of vague terms are undetermined in the sense that though they may be determined in some contexts, they may differ from one context to the next. Vagueness (...) serves the important communicative function, enabling speakers to identify entities as objects (as things we can talk about) in terms of some quantitative differences between the “object” and its background in the context. Thus, in some contexts we can naturally partition the group of men uniquely so as to distinguish the bald from the not-bald. Whether a man with a given hair number is among the bald in a given context depends not only on his own hair number but also on the hair number of others in that context. This provides the background for the claim that when we assert that John is bald, we presuppose that there is a unique demarcation to the bald in that context. I consider the truth of the Paradox’s statements in contexts where the presupposition is true and in contexts where it is false. The analysis yields that the contradiction is avoided because though each of the statements is often true, never are all the sentences in the Paradox true together. (shrink)

Several philosophers offer explanations of linguistic vagueness by appealing to the referential context-dependence of vague terms. Timothy Williamson argues pre-emptively that any such approach must fail, on the grounds that context-dependence is neither necessary nor sufficient for vagueness. He supports this claim, in turn, by example. This paper argues that his examples fail to show that context-dependence is either unnecessary or insufficient for vagueness, and hence that he has failed by his own lights to show that it cannot explain vagueness.

accounts in general, contrary to what he seems to think. Stanley’s discussion concerns the dynamic or ‘forced march’ version of the sorites, viz. the version framed in terms of the judgments that would be made by a competent speaker who proceeds step by step along a sorites series for a vague predicate ‘F’. According to Stanley, the contextualist treatment of the paradox is based on the idea that the speaker shifts the content of the predicate whenever necessary to make it (...) the case that each successive pair of adjacent items are category-identical – in other words, either both items satisfy ‘F’ or neither does. These adjustments allow the speaker to progress from a clear case for ‘F’ to a clear case for ‘not-F’ without breaking the seeming continuity of the series. (shrink)

It is a commonplace that the extensions of most, perhaps all, vague predicates vary with such features as comparison class and paradigm and contrasting cases. My view proposes another, more pervasive contextual parameter. Vague predicates exhibit what I call open texture: in some circumstances, competent speakers can go either way in the borderline region. The shifting extension and anti-extensions of vague predicates are tracked by what David Lewis calls the “conversational score”, and are regulated by what Kit Fine calls penumbral (...) connections, including a principle of tolerance. As I see it, vague predicates are response-dependent, or, better, judgement-dependent, at least in their borderline regions. This raises questions concerning how one reasons with such predicates. In this paper, I present a model theory for vague predicates, so construed. It is based on an overall supervaluationist-style framework, and it invokes analogues of Kripke structures for intuitionistic logic. I argue that the system captures, or at least nicely models, how one ought to reason with the shifting extensions (and anti-extensions) of vague predicates, as borderline cases are called and retracted in the course of a conversation. The model theory is illustrated with a forced march sorites series, and also with a thought experiment in which vague predicates interact with so-called future contingents. I show how to define various connectives and quantifiers in the language of the system, and how to express various penumbral connections and the principle of tolerance. The project fits into one of the topics of this special issue. In the course of reasoning, even with the external context held fixed, it is uncertain what the future extension of the vague predicates will be. Yet we still manage to reason with them. The system is based on that developed, more fully, in my Vagueness in Context , Oxford, Oxford University Press, 2006, but some criticisms and replies to critics are incorporated. (shrink)

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. (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)

In this book, Scott Soames illuminates the notion of truth and the role it plays in our ordinary thought, as well as in our logical, philosophical, and scientific theories. Part I addresses crucial background issues, including the identification of the bearers of truth, the basis for distinguishing truth from other notions (like certainty, with which it is often confused), and the formulation of positive responses to well-known forms of philosophical skepticism about truth. Part II explicates the formal theories of Alfred (...) Tarski and Saul Kripke, including their treatments of the Liar paradox, and evaluates the philosophical significance of their work. Part III extends important lessons drawn from Tarski and Kripke into new domains: vague predicates, the Sorites paradox, and the development of a larger, deflationary perspective on truth. Soames aims to integrate and deepen the most significant insights on truth from a variety of sources. He powerfully brings together the best technical work an the most important philosophical reflection on truth and shows how each can illuminate the other. (shrink)

According to what I will call a contextualist solution to the sorites paradox, vague terms are context-sensitive, and one can give a convincing dissolution of the sorites paradox in terms of this context-dependency. The reason, according to the contextualist, that precise boundaries for expressions like “heap” or “tall for a basketball player” are so difficult to detect is that when two entities are sufficiently similar (or saliently similar), we tend to shift the interpretation of the vague expression so that if (...) one counts as falling in the extension of the property expressed by that expression, so does the other. As a consequence, when we look for the boundary of the extension of a vague expression in its penumbra, our very looking has the effect of changing the interpretation of the vague expression so that the boundary is not where we are looking. This accounts for the persuasive force of sorites arguments. (shrink)

In ‘Something to Do with Vagueness ...’, Linda Burns defends an analogy between the informational and the borderline-case variety of vagueness. She argues that the latter is in fact less extraordinary and less disastrous than people in the tradition of Michael Dummett and Crispin Wright have told us. However, her account involves presuppositions that cannot be taken for granted. Here is to take a closer look at some of these presuppositions and argue hat they may--when left unguarded--undermine much of Burns’ (...) general account. (shrink)

One of the main reasons for providing formal semantics for languages is that the mathematical precision afforded by such semantics allows us to study and manipulate the formalization much more easily than if we were to study the relevant natural languages directly. Michael Tye and R. M. Sainsbury have argued that traditional set-theoretic semantics for vague languages are all but useless, however, since this mathematical precision eliminates the very phenomenon (vagueness) that we are trying to capture. Here we meet this (...) objection by viewing formalization as a process of building models, not providing descriptions. When we are constructing models, as opposed to accurate descriptions, we often include in the model extra ‘machinery’ of some sort in order to facilitate our manipulation of the model. In other words, while some parts of a model accurately represent actual aspects of the phenomenon being modelled, other parts might be merely artefacts of the particular model. With this distinction in place, the criticisms of Sainsbury and Tye are easily dealt with—the precision of the semantics is artefactual and does not represent any real precision in vague discourse. Although this solution to this problem is independent of any particular semantics a detailed account of how we would distinguish between representor and artefact within Dorothy Edgington's degree-theoretic semantics is presented. (shrink)

Predicates are supposed to slice reality neatly in two halves, one for which the predicate holds, the other for which it fails. Yet far from being razors, predicates tend to be dull knives that mangle reality. If reality is a tomato and predicates are knives, then when these knives divide the tomato, plenty of mush remains unaccounted for. Of course some knives are sharper than others, just as some predicates are less vague than others. “x is water” is certainly sharper (...) than “x is beautiful.” But perfect sharpness, perfect boundaries, and perfect separation seem only to obtain in mathematics. The vagueness inherent in many predicates became particularly evident in the twentieth century. Quantum mechanics, the revival of certain ancient paradoxes, and the philosophy of science all contributed to a growing awareness that vagueness was ineliminable from many predicates. Quantum mechanical superposition seems to allow mutually exclusive simultaneous states. In searching for plausible interpretations of quantum mechanics, some researchers attempted to do away with classical bivalent logic. In its place they substituted multivalent quantum logics. Any logic with more than two values forces its predicates to slice reality into more than two parts. Ancient paradoxes involving heaps and baldness also pointed up the 1 Random Predicate Logic 2.. (shrink)

It might be thought that vagueness precludes the possibility of classical conceptual analysis and, thus, that the classical or definitional view of the nature of complex concepts is incorrect. The present paper argues that classical analysis can be had for concepts expressed by vague language since (1) all of the general theories of vagueness are compatible with the thesis that all complex concepts have classical analyses and also that (2) the meaning of vague expressions can be analyzed by having the (...) degree of vagueness of a given analysandum be mapped onto the vagueness of an analysans. (shrink)

Vagueness is one of the most persistent and challenging topics in the intersection of philosophy and logic. At least five other noteworthy books on vagueness have been written by philosophers since 1991 [2, 6, 11, 12, 15]. A (necessarily incomplete) bibliography that has been compiled for the Arché project Vagueness: its Nature and Logic (2004-2006) of the University of St Andrews lists more than 350 articles and books on vagueness until 2005.1 Many new and interesting contributions have appeared since. The (...) book under review is much more than yet another addition to this prolific discourse. Nicholas Smith manages to tackle two different tasks that are potentially in tension. On the one hand, he provides a comprehensive, systematic and well written account of various approaches to vagueness that have been debated so far. On the other hand, Smith carefully explains and defends his own theory of vagueness, called fuzzy plurivaluationism. Given the complex and almost unsurmountably large amount of relevant literature and the fact that theories of vagueness based on fuzzy logic have almost universally been rejected by philosophers so far this is no simple feat. In the comments below, I will largely follow the structure of book. If along the way I cannot resist to make side remarks or even take issue with some of.. (shrink)

This is a contribution to the discussion on the role of truth degrees in manyvalued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko’s Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying these in the (...) particular cases of Łukasiewicz’s many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties. (shrink)

Many criticisms of prototype theory and/or fuzzy-set theory are based on the assumption that category representativeness (or typicality) is identical with fuzzy membership. These criticisms also assume that conceptual combination and logical rules (all in the Aristotelian sense) are the appropriate criteria for the adequacy of the above “fuzzy typicality”. The present paper discusses these assumptions following the line of their most explicit and most influential expression by Osheron and Smith (1981). Several arguments are made against the above identification, the (...) most important being that representativeness in prototype theory is exclusively based on element-to-element similarity while fuzzy membership is inherently an element-to-category relationship. Also the above criteria for adequacy are criticized from the viewpoint of both prototype theory and fuzzy-set theory as well as from that of both conceptual and logical combination, and also from that of integration. (shrink)

It has been a vexing question in recent years whether concepts are fuzzy. In this paper several views on the fuzziness of concepts are pointed out to have stemmed from dubious concepts of fuzziness. The underlying notions of the roles feasibly played byprototype, set, andprobability in modeling concepts strongly suggest that the controversy originates from a vague relation between intuitive and mathematical ideas in the cognitive sciences. It is argued that the application of fuzzy sets cannot resolve this vagueness since (...) they are one sided,viz., defined on sets. An alternative definition based on classes (in the sense of axiomatic set theory) is proposed. (shrink)

Prototypes and fuzziness are regarded in this paper as fundamental phenomena in the inherent logic of concepts whose relationship, however, has not been sufficiently clarified. Therefore, modifications are proposed in the definition of both. Prototypes are defined as the elements possessing maximal degree of membership in the given category such thatthis membership has maximal cognitive efficiency in representing theelement. A modified fuzzy set (m-fuzzy set) is defined on aclass (possibly self-contradictory collection) such that its core (the collection of elements with (...) full membership) is aset (self-consistent collection) comprising the finite set of the prototypes. In this scheme of recursive representation the possibility of contradictions, which has been proven to be inevitable in the logic of concepts, islocalized. Finally a related model of the brain''s pattern recognition mechanism is briefly summarized. (shrink)

The goal of this paper is a comprehensive analysis of basic reasoning patterns that are characteristic of vague predicates. The analysis leads to rigorous reconstructions of the phenomena within formal systems. Two basic features are dealt with. One is tolerance: the insensitivity of predicates to small changes in the objects of predication (a one-increment of a walking distance is a walking distance). The other is the existence of borderline cases. The paper shows why these should be treated as different, though (...) related phenomena. Tolerance is formally reconstructed within a proposed framework of contextual logic, leading to a solution of the Sorites paradox. Borderline-vagueness is reconstructed using certain modality operators; the set-up provides an analysis of higher order vagueness and a derivation of scales of degrees for the property in question. (shrink)

Initially proposed as rivals of classical logic, alternative logics have become increasingly important in areas such as computer science and artificial intelligence. Fuzzy logic, in particular, has motivated major technological developments in recent years. Susan Haack's Deviant Logic provided the first extended examination of the philosophical consequences of alternative logics. In this new volume, Haack includes the complete text of Deviant Logic , as well as five additional papers that expand and update it. Two of these essays critique fuzzy logic, (...) while three augment Deviant Logic 's treatment of deduction and logical truth. Haack also provides an extensive new foreword, brief introductions to the new essays, and an updated bibliography of recent work in these areas. Deviant Logic, Fuzzy Logic will be indispensable to students of philosophy, philosophy of science, linguistics, mathematics, and computer science, and will also prove invaluable to experienced scholars working in these fields. (shrink)

PART ONE I 'Alternative' in 'Alternative logic There are many systems of logic — many-valued systems and modal systems for instance - which are non-standard ...

Some aspects of vagueness as presented in Shapiro’s book Vagueness in Context [23] are analyzed from the point of fuzzy logic. Presented are some generalizations of Shapiro’s formal apparatus.

A theory of vagueness gives a model of vague language and of reasoning within the language. Among the models that have been offered are Degree Theorists’ numerical models that assign values between 0 and 1 to sentences, rather than simply modelling sentences as true or false. In this paper, I ask whether we can benefit from employing a rich, well-understood numerical framework, while ignoring those aspects of it that impute a level of mathematical precision that is not present in the (...) modelled phenomenon of vagueness. Can we ignore apparent implications for the phenomena by pointing out that it is just a model and that the unwanted features are mere artefacts? I explore the distinction between representors and artefacts and criticise the strategy of appealing to features as mere artefacts in defence of a theory. I focus largely on theories using numerical resources, but also consider other, related theories and strategies, including theories appealing to non-linear structures. (shrink)

Degree theories of vagueness build on the observation that vague predicates such as 'tall' and 'red' come in degrees. They employ an infinite-valued logic, where the truth values correspond to degrees of truth and are typically represented by the real numbers in the interval [0,1]. In this paper, the success with which the numerical assignments of such theories can capture the phenomenon of vagueness is assessed by drawing an analogy with the measurement of various physical quantities using real numbers. I (...) argue that degree theories of vagueness are undermined by the failure of the necessary connectedness principle. Moreover, the semantics for the connectives entail that there must be a uniquely correct numerical assignment for the sentences, and this is implausible. Different senses of 'coming in degrees' are then distinguished; I argue that a confusion between them could be the source of the degree theorist's error, and the distinction illuminates the problem cases described earlier in the paper. (shrink)

It is taken for granted in much of the literature on vagueness that semantic and epistemic approaches to vagueness are fundamentally at odds. If we can analyze borderline cases and the sorites paradox in terms of degrees of truth, then we don’t need an epistemic explanation. Conversely, if an epistemic explanation suffices, then there is no reason to depart from the familiar simplicity of classical bivalent semantics. I question this assumption, showing that there is an intelligible motivation for adopting a (...) many-valued semantics even if one accepts a form of epistemicism. The resulting hybrid view has advantages over both classical epistemicism and traditional many-valued approaches. (shrink)

Fuzzy logics are systems of logic with infinitely many truth values. Such logics have been claimed to have an extremely wide range of applications in linguistics, computer technology, psychology, etc. In this note, we canvass the known results concerning infinitely many valued logics; make some suggestions for alterations of the known systems in order to accommodate what modern devotees of fuzzy logic claim to desire; and we prove some theorems to the effect that there can be no fuzzy logic which (...) will do what its advocates want. Finally, we suggest ways to accommodate these desires in finitely many valued logics. (shrink)

This paper began life as a short section of a more general paper about non-classical mereologies. In that paper I had a mereological theory that I wanted to show could be applied to all sorts of different metaphysical positions — notably, to those positions that believe in mereological vagueness in re — in “vague individuals”. To do that I felt I first had to dispatch the leading rival theory of vague individuals, which is due to Peter van Inwa-gen, and holds (...) that the part-whole relation admits of degrees. It seemed to me that this theory had a serious technical problem, or at best a serious gap. I sat down to write a paragraph or two highlighting the gap, preferably showing that it couldn’t be filled. This paper is the result. (shrink)

(No abstract is available for this citation).

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)

Kamp and Fine presented an influential argument against the use of fuzzy logic for linguistic semantics in 1975. However, the argument assumes that contradictions of the form "A and not A" have semantic value zero. The argument has been recently criticized because sentences of this form are actually not perceived as contradictory by naive speakers. I present new experimental evidence arguing that fuzzy logic still isn't useful for linguistic semantics even if we take such naive speaker judgements at face value. (...) Specifically I show that naive speakers judge "A and not A" in the relevant cases as more true than "B and not A" even when A and B are judged to be equally true. A truth functional semantics such as fuzzy logic cannot account for these intuitions directly. (shrink)

I argue that though a satisfactory semantics for the logic of inexact reference may assign no truth value to some statements, it should not assign truth (or falsity) of various degrees. Well-formed assertions are simply true or not. Inexactness does not “ramify.” I distinguish inexactness from other sorts of vagueness, including nonspecificity. I show that arguments from (i) use of quantifiers, (ii) the existence of properties which can be construed as a series of properties (as, e. g., red can be (...) construed as a set of shades of red), (iii) the constructability of apparently paradoxical sorites arguments, and (iv) the presence of prototypes in the extension of a predicate do not show that there are degrees of truth.Much of the alleged evidence that inexactness ramifies is, in fact, a misreading of the undeniable evidence that there may be uncertainty about the truth value of a claim. In support of my claims, I discuss how cases of deeming that a predicate applies relate to its actually applying. A distinction between predicates of “pure” and “impure” function is essential to this. (shrink)

Philosophers, linguists and others interested in problems concerning natural language frequently employ tools from logic and model theory. The question arises as to the proper interpretation of the formal methods employed—of the relationship between, on the one hand, the formal languages and their set-theoretic models and, on the other hand, the objects of ultimate interest: natural language and the meanings and truth conditions of its constituent words, phrases and sentences. Two familiar answers to this question are descriptivism and instrumentalism. More (...) recently, a third answer has been proposed: the logic as modelling view. This paper seeks to clarify and assess this view of logic. The conclusion is that we can successfully adopt the modelling perspective on a given piece of logical machinery only if we have to hand some other machinery to which we take the descriptive attitude. Thus, logic as modelling is not a full-fledged alternative to the descriptive view— for it cannot stand alone: it can at best be an addition to the descriptive perspective. The paper first presents the argument in a general, abstract form, before working through a detailed case study. The case examined is the one with respect to which the logic as modelling view has been developed in the greatest detail in the literature: the case of fuzzy model theory as an account of vagueness in natural language. (shrink)

The major reason given in the philosophical literature for dissatisfaction with theories of vagueness based on fuzzy logic is that such theories give rise to a problem of higherorder vagueness or artificial precision. In this paper I first outline the problem and survey suggested solutions: fuzzy epistemicism; measuring truth on an ordinal scale; logic as modelling; fuzzy metalanguages; blurry sets; and fuzzy plurivaluationism. I then argue that in order to decide upon a solution, we need to understand the true nature (...) and source of the problem. Two possible sources are discussed: the problem stems from the very nature of vagueness—from the defining features of vague predicates; or the problem stems from the way in which the meanings of predicates are determined—by the usage of speakers together with facts about their environment and so on. I argue that the latter is the true source of the problem, and on this basis that fuzzy plurivaluationism is the correct solution. (shrink)

A number of authors have noted that vagueness engenders degrees of belief, but that these degrees of belief do not behave like subjective probabilities. So should we countenance two different kinds of degree of belief: the kind arising from vagueness, and the familiar kind arising from uncertainty, which obey the laws of probability? I argue that we cannot coherently countenance two different kinds of degree of belief. Instead, I present a framework in which there is a single notion of degree (...) of belief, which in certain circumstances behaves like a subjective probability assignment and in other circumstances does not. The core idea is that one’s degree of belief in a proposition P is one’s expectation of P’s degree of truth. (shrink)

To make the book accessible to non-specialists, Nicholas Smith includes both an introduction to the relevant philosophical literature, and a gentle but thorough ...