Vague Theory Research Articles

Higher Order Vagueness in Geographical Information: Empirical Geographical Population of Type n Fuzzy Sets

Fuzzy set theory has been suggested as a means for representing vague spatial phenomena, and is widely known for directly addressing some of the issues of vagueness such as the sorites paradox. Higher order vagueness is widely considered a necessary component of any theory of vagueness, but it is not so well known that it too is competently modelled by Type n Fuzzy sets. In this paper we explore the fuzzy representation of higher order vagueness with respect to spatial phenomena. Initially we relate the arguments on philosophical vagueness to Type n Fuzzy sets. As an example, we move on to an empirical generation of spatial Type 2 Fuzzy sets examining the spatial extent of mountain peaks in Scotland. We show that the Type 2 Fuzzy sets can be populated by using alternative parameterisations of a peak detection algorithm. Further ambiguities could also be explored using other parameters of this and other algorithms. We show some novel answers to interrogations of the mountain peaks of Scotland. The conclusion of this work is that higher order vagueness can be populated for Type 2 and higher fuzzy sets. It does not follow that it is always necessary to examine these higher order uncertainties, but a possible advantage in terms of the results of spatial inquiry is demonstrated.

Intransitivity and Vague Preferences

This paper is concerned with intransitivity in normative rational choice. It focuses on a class of intransitivities which have received little attention, those involving vague preferences. “Vague preferences” are defined in terms of vague predicates such as “red” or “bald.” Such preferences appear common, and intransitive indifference is argued to be an unavoidable feature of them. Such preferences are argued to undermine intransitive strict preference also. Various formal theories of vagueness are applied to an example of vague preferences, but none of them provide a justification for the transitivity axiom.

Fuzzy Grammar: A Reader (review)

Reviewed by: Fuzzy grammar: A reader ed. by Bas Aarts, David Denison, Evelien Keizer, and Gergana Popova Kleanthes K. Grohmann Fuzzy grammar: A reader. Ed. by Bas Aarts, David Denison, Evelien Keizer, and Gergana Popova. (Oxford linguistics.) Oxford: Oxford University Press, 2004. Pp. ix, 526. ISBN 0199262578. $50. The stated aim of the editors of this ‘carefully selected [collection of] key texts’ is ‘to bring together a number of classical and recent writings from philosophy and linguistics in the areas of vagueness and fuzziness’ (1). In my opinion, this aim has been achieved wonderfully. The volume contains five parts, presented below in the order of their arrangement with a brief indication of the source for each contribution. The editors’ ‘Introduction: The nature of grammatical categories and their representation’ (1–28) lays out the motivation behind the collection and the different parts, and they summarize the main points of interest and/or contention and briefly sketch more recent approaches, offering their thoughts on future prospects as well. Part 1 offers some classical writings forming the ‘philosophical background’ of this collection. It contains excerpts from Aristotle’s ‘Categories’ (31–32; the original paragraphs 4b20, 4b22, 1006b28, 1008b31 of Metaphysics, 350 BC/1963 and 1971), Gottlob Frege’s ‘Concepts’ (33; an English translation of §56 from Grundgesetze der Arithmetik, vol. 2, 1903/1970), and Ludwig Wittgenstein’s ‘Family resemblances’ (41–44; remarks 66–78 from Philosophical investigations, 1953/1968), as well as Bertrand Russell’s article ‘Vagueness’ (35–40; Australasian Journal of Philosophy and Psychology 1, 1923) and Rosanna Keefe’s chapter ‘The phenomena of vagueness’ (45–64; from her Theory of vagueness, 2000). Part 2, ‘Categories in cognition’, begins with William Labov’s article ‘The boundaries of words and their meanings’ (67–89; from the volume New ways of analyzing variation in English, 1973), followed by Eleanor Rosch’s ‘Principles of categorization’ (91–108; from the volume Cognition and categorization, 1978) and Ray Jackendoff’s ‘Categorization, fuzziness, and family resemblances’ (109–29; ch. 5 of his Semantics and cognition, 1983). The section ‘Discreteness’ (131–37; from his Foundations of cognitive grammar, vol. 1, 1987) by Ronald W. Langacker and George Lakoff’s ‘The importance of categorization’ (139–77; ch. 1 of his Women, fire and dangerous things: What categories reveal about the mind, 1987) round off this part. The broad topic categories in grammar makes up Part 3. Here we find Otto Jespersen’s ‘Parts of speech’ (181–90; ch. 4 of The philosophy of grammar, 1924) and David Crystal’s ‘English word classes’ (191–211; Lingua 17, 1967). John Lyons’s chapter offers ‘A notional approach to the parts of speech’ (213–23; §7.6 from his Introduction to theoretical linguistics, 1968), while John M. Anderson addresses ‘Syntactic categories and notional features’ (225–37; §2.1 from A notional theory of syntactic categories, 1997). The second half of this part contains Langacker’s ‘Bounded regions’ (239–45; §5.2 of the above-mentioned book), Paul J. Hopper and Sandra A. Thompson’s ‘The discourse basis for lexical categories in universal grammar’ (247–91; Language 60, 1984), and John Taylor’s ‘Grammatical categories’ (292–308; ch. 10 of his Linguistic categorization: Prototypes in linguistic theory, 1995). Part 4, ‘Gradience in grammar’, starts off with Dwight Bolinger’s chapter ‘Gradience’ (311–19; from his Generality, gradience, and the all-or-none, 1961) and Noam Chomsky’s ‘Degrees of grammaticalness’ (321–25; a section from his article in Word 17, 1961). These are followed by ‘Descriptive statement and serial relationship’ (327–39; Language 41, 1965) by Randolph Quirk, ‘On the analysis of linguistic vagueness’ (341–49; Travaux linguistique de Prague 2, 1966) by Jiří V. Neustupný, and ‘Nouniness’ (351–422; from the volume Three dimensions of linguistic research, 1973) by John Robert Ross. This part is complemented by ‘The coordination-subordination gradient’ (423–30; §§13.6–13.19 of Randolph Q...

How effective indeed is present-day mathematics?

We argue that E. Wigner’s well-known claim that mathematics is unreasonably effective in physics (and not in the natural sciences in general, as the title of his article suggests) is only one side of the hill. The other side is the surprising insufficiency of present-day mathematics to capture the uniformities that arise in science outside physics. We describe roughly what the situation is in the areas of (a) everyday reasoning, (b) theory of meaning and (c) vagueness. We make also the point that mathematics, as we know it today, founded on the concept of set, need not be a conceptually final and closed system, but only a stage in a developing subject.

SQUEEZING AND STRETCHING: HOW VAGUENESS CAN OUTRUN BORDERLINENESS

The paper develops a critical dialectic with respect to the nowadays dominant approach in the theory of vagueness, an approach whose main tenet is that it is in the nature of the vagueness of an expression to present borderline cases of application, conceived of as enjoying some kind of distinctive normative status. Borderlineness is used to explain the basic phenomena of vagueness, such as, for example, our ignorance of the location of cut-offs in a soritical series. Every particular theory of vagueness exemplifying the approach makes use, in the vague object language, of a definiteness operator which, however substantially interpreted, unavoidably inherits the vagueness of the expressions on which it operates (‘higher-order vagueness’). It is first argued that finite soritical series force a surprising collapse result concerning a particular set of expressions involving the definiteness operator. It is then shown that, under two highly plausible assumptions about higher-order vagueness (the existence of ‘absolutely definitely’ positive and negative cases and the ‘radical’ character of higher-order vagueness itself), the collapse result implies the inadequacy of the dominant approach as a theory of vagueness, as its main tenet can be, at best, not absolutely definitely true.

The Fool and ‘Stranger Anxiety’: Creative and Destructive Possibilities

“I have a vague theory that one learns most…. from having someone suddenly make fun of something one has taken seriously up to then.

Precise entities but irredeemably vague concepts?

Various arguments have recently been put forward to support the existence of vague or fuzzy objects. Nevertheless, the only possibly compelling argument would support, not the existence of vague objects, but indeterminately existing objects. I argue for the non-existence of any vague entities - either particulars or properties - in the mind-independent world. Even so, many philosophers have claimed that to reduce vagueness to semantics is of no avail, since linguistic vagueness betrays semantic incoherence and this is no less a problem than is ontological incoherence. After spelling out why there are fewer essentially vague concepts than usually thought, I claim that only the linguistic competence of the whole speaking community for each word can draw the sharp boundaries for its concept, even if these are unknowable in practice and still leave a precise range of indetermination. This could explain both the existence of boundaries and our non-removable ignorance of them, fulfilling the intuitions of the epistemic theory of vagueness with the supervaluationist's indeterminacy.

Vagueness as closeness

This paper presents and defends a definition of vagueness, compares it favourably with alternative definitions, and draws out some consequences of accepting this definition for the project of offering a substantive theory of vagueness. The definition is roughly this: a predicate ‘F’ is vague just in case for any objects a and b, if a and b are very close in respects relevant to the possession of F, then ‘Fa’ and ‘Fb’ are very close in respect of truth. The definition is extended to cover vagueness of many-place predicates, of properties and relations, and of objects. Some of the most important advantages of the definition are that it captures the intuitions which motivate the thought that vague predicates are tolerant, without leading to contradiction, and that it yields a clear understanding of the relationships between higher-order vagueness, sorites susceptibility, blurred boundaries, and borderline cases. The most notable consequence of the definition is that the correct theory of vagueness must countenance degrees of truth.

Epistemicism and the Combined Spectrum

AbstractDerek Parfit's combined‐spectrum argument seems to conflict with epistemicism, a viable theory of vagueness. While Parfit argues for the indeterminacy of personhood, epistemicism denies indeterminacy. But, we argue, the linguistically based determinacy that epistemicism supports lacks the sort of normative or ontological significance that concerns Parfit. Thus, we reformulate his argument to make it consistent with epistemicism. We also dispute Roy Sorensen's suggestion that Parfit's argument relies on an assumption that fuels resistance to epistemicism, namely, that ‘the magnitude of a modification must be proportional to its effect.’

Theories of Vagueness (review)

Reviewed by: Theories of vagueness by Rosanna Keefe Laura Daniliuc and Radu Daniliuc Theories of vagueness. By Rosanna Keefe. Cambridge: Cambridge University Press. 2000. Pp. 233. ISBN 0521650674. $60 (Hb). A concise survey of theories of vagueness and a defense of the supervaluationism theory, this book is carefully organized into eight chapters that can be grouped into three distinct parts. The first part, consisting of Chs. 1 and 2, covers the fundamentals of vagueness and the central features of [End Page 349] vague expressions. It offers a concise description of the different types of vague expressions, a classification of vagueness theories according to their logic of vagueness and semantics of vagueness, and according to their position with respect to the sorites paradox. In addition, this part also offers a discussion of various methodological principles and perspectives used in the analyses of the vagueness theories. The second part, made up of the next four chapters, is dedicated to the nonsuperevaluationist theories. Ch. 3 is intended as an attack against the epistemic theory, as represented by Timothy Williamson’s version. Chs. 4 and 5 question the area of many-valued logics. Keefe uses Michael Tye’s three-valued logic theory and Kenton Machina’s degree theory as models of degree of truth theories for an analysis of the semantics of connectives and of validity in many valued logics. Ch.5 examines the measurement theory and the representation and uniqueness theorems, with the declared aim to ‘undermine the prospects for degree theories of vagueness’ (125). K concludes that degree theories work only if they deny important features of different classes of vague expressions. Ch. 6 examines the pragmatic approach that defines vagueness as a feature of the relation between users and language. K investigates David Lewis’s philosophy of language and considers his two alternative approaches of vagueness within the cluster of precise languages. She argues that either the pragmatic view is wrong or it becomes identical with superevaluationism, depending upon how ‘the proposal is understood’. The last part, the remaining two chapters, is a thorough analysis of superevaluationism and a defense of K’s own version of it. Ch. 7 provides a historical perspective of superevaluationism and answers to a series of objections against this theory. It also includes a discussion of Kit Fine’s theory and of Lewis’s semantic approach and an analysis of the role of classical logic within superevaluationism. The book concludes with Ch. 8 offering K’s own version of superevaluationism. Overall, the book is concise and well presented. K offers strong and persuasive arguments for her own theory within the broad range of vagueness theories analyzed in the book. The philosophical contribution of the book is a valuable addition to the study of the linguistic implications of vagueness. Laura Daniliuc and Radu Daniliuc Australian National University Copyright © 2004 Linguistic Society of America

Vagueness and Blurry Sets

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

Heaps and gluts: Paraconsistent logic applied to vagueness

This paper is an attempt to show that the subvaluation theory is not a good theory of vagueness. It begins with a short review of supervaluation and subvaluation theories and procedes to evaluate the subvaluation theory. Subvaluationism shares all the main short-comings of supervaluationism. Moreover, the solution to the sorites paradox proposed by subvaluationists is not satisfactory. There is another solution which subvaluationists could avail themselves of, but it destroys the whole motivation for using a paraconsistent logic and is not different from the one offered by supervaluationism.

Temporal Externalism and Epistemic Theories of Vagueness

Temporal Externalism and Epistemic Theories of Vagueness

Vagueness unlimited: In defence of a pragmatical approach to sorites paradoxes

As far as ‘modern’ logical theories of vagueness are concerned, a main distinction can be drawn between ‘semantical’ ones and ‘pragmatical’ ones. The latter are defended here, because they tend to retake into account important contextual dimensions of the problem abandoned by the former. Their inchoate condition seems not alarming, since they are of surprisingly recent date. This, however, could very well be an accidental explanation. That is, the true reason for it might sooner or later turn out to be bearing exactly on the fundamental human limitations, when it comes to theorizing, that these approaches are urging us to appreciate.

Epistemicism, Parasites, and Vague Names

John Burgess has recently argued that Timothy Williamson's attempts to avoid the objection that his theory of vagueness is based on an untenable metaphysics of content are unsuccessful. Burgess's arguments are important, and largely correct, but there is a mistake in the discussion of one of the key examples. In this note I provide some alternative examples and use them to repair the mistaken section of the argument.

Theories of Vagueness

Theories of Vagueness

Unsolved Problems with Numbers: Reply to Smith

Can we have a successful theory of vagueness that employs degrees of truth, so that ascriptions of tallness to borderline tall men are true to different degrees, measured by the real numbers from o to 1? Are such assignments of numbers possible and can they be true to the phenomena of vagueness? In Keefe 1998 (and Keefe 2000, Ch. 5), I argued that no such assignment of numbers is compatible with the phenomena of vagueness. I raised several problems for degree theorists and their assignments of numbers, for example, problems arising from the multidimensionality typical of many vague predicates. I finished with a suggested diagnosis of the appeal of degree theories. It is tempting to try to build a theory of vagueness on the idea of degrees of because the application of a vague predicate is usually closely related to some attribute (for example, height or temperature) that comes in degrees in the sense that it is a measurable quantity. Smith (2003) tackles this last part of my paper, apparently taking this to constitute my entire argument against degree theories, rather than as a possible diagnosis of the attraction of a theory facing the problems I uncovered in the bulk of the paper. He claims that degree theorists need not always fall into the suggested confusion between different senses of coming in degrees. He argues that they can assign numbers to borderline case predications of tallness, without confusing their assignments with measures of height. His defence, showing how numbers can be assigned, merely illustrates the standard degree-theoretic approach. It ignores the main arguments in Keefe 1998, which show that degree theorists cannot assign numbers compatibly with the nature of vagueness that they seek to capture. Now, I can agree that not all degree theorists are guilty of the confusion Smith concentrates on: whether or not they are guilty of it, they face the problems I raised. They have no way of capturing vagueness by assigning numbers.

Supervaluations and the Problem of the Many

Supervaluational treatments of vagueness are currently quite popular among those who regard vagueness as a thoroughly semantic phenomenon. Peter Unger’s ‘problem of the many’ may be regarded as arising from the vagueness of our ordinary physical-object terms, so it is not surprising that supervaluational solutions to Unger’s problem have been offered. I argue that supervaluations do not afford an adequate solution to the problem of the many. Moreover, the considerations I raise against the supervaluational solution tell also against the solution to the problem of the many which is suggested by adherents of the epistemic theory of vagueness.

Reply to Machina and Deutsch on vagueness, ignorance, and margins for error

In their paper “Vagueness, Ignorance, and Margins for Error” Kenton Machina and Harry Deutsch criticize the epistemic theory of vagueness. This paper answers their objections. The main issues discussed are: the relation between meaning and use; the principle of bivalence; the ontology of vaguely specified classes; the proper form of margin for error principles; iterations of epistemic operators and semantic compositionality; the relation or lack of it between quantum mechanics and theories of vagueness.

Vagueness, ignorance, and margins for error

We argue that the epistemic theory of vagueness cannot adequately justify its key tenet-that vague predicates have precisely bounded extensions, of which we are necessarily ignorant. Nor can the theory adequately account for our ignorance of the truth values of borderline cases. Furthermore, we argue that Williamson’s promising attempt to explicate our understanding of vague language on the model of a certain sort of “inexact knowledge” is at best incomplete, since certain forms of vagueness do not fit Williamson’s model, and in fact fit an alternative model. Finally, we point out that a certain kind of irremediable inexactitude postulated by physics need not be-and is not commonly-interpreted as epistemic. Thus, there are aspects of contemporary science that do not accord well with the epistemicist outlook.