The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, , of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in as a (...) fuzzy subset of [0, 1].Since is not closed under the operations of negation, conjunction, disjunction and implication, the result of an operation on truth-values in requires, in general, a linguistic approximation by a truth-value in . As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very, quite, more or less, etc. (shrink)

Vagueness provides the first comprehensive examination of a topic of increasing importance in metaphysics and the philosophy of logic and language. Timothy Williamson traces the history of this philosophical problem from discussions of the heap paradox in classical Greece to modern formal approaches such as fuzzy logic. He illustrates the problems with views which have taken the position that standard logic and formal semantics do not apply to vague language, and defends the controversial realistic view that vagueness is a kind (...) of ignorance--that there really is a grain of sand whose removal turns a heap into a non-heap, but we cannot know which one it is. (shrink)

This paper defends the idea that there might be vagueness or indeterminacy in the world itself---as opposed to merely in our representations of the world---against the charges of incoherence and unintelligibility. First we consider the idea that the world might contain vague *properties and relations*; we show that this idea is already implied by certain well-understood views concerning the semantics of vague predicates (most notably the fuzzy view). Next we consider the idea that the world might contain vague *objects*; we (...) argue that an object is indeterminate in a certain respect (colour, size, etc.) just in case it is a borderline case of a maximally specific colour (size, etc.) property. Finally we consider the idea that the *world as a whole* might be indeterminate; we argue that the world is indeterminate just in case it lacks a determinate division into determinate objects. (shrink)

In VAGUENESS AND DEGREES OF TRUTH, Nicholas Smith develops a new theory of vagueness: fuzzy plurivaluationism. -/- A predicate is said to be VAGUE if there is no sharply defined boundary between the things to which it applies and the things to which it does not apply. For example, 'heavy' is vague in a way that 'weighs over 20 kilograms' is not. A great many predicates -- both in everyday talk, and in a wide array of theoretical vocabularies, from law (...) to psychology to engineering -- are vague. -/- Smith argues, based on a detailed account of the defining features of vagueness, that an accurate theory of vagueness must involve the idea that truth comes in degrees. The core idea of degrees of truth is that while some sentences are true and some are false, others possess intermediate truth values: they are truer than the false sentences, but not as true as the true ones. Degree-theoretic treatments of vagueness have been proposed in the past, but all have encountered significant objections. In light of these, Smith develops a new type of degree theory. Its innovations include a definition of logical consequence that allows the derivation of a classical consequence relation from the degree-theoretic semantics, a unified account of degrees of belief and their relationships with degrees of truth and subjective probabilities, and the incorporation of semantic indeterminacy -- the view that vague statements need not have unique meanings -- into the degree-theoretic framework. -/- As well as being essential reading for those working on vagueness, Smith's book provides an excellent entry-point for newcomers to the area -- both from elsewhere in philosophy, and from computer science, logic and engineering. It contains a thorough introduction to existing theories of vagueness and to the requisite logical background. (shrink)

A formal theory of truth, alternative to tarski's 'orthodox' theory, based on truth-value gaps, is presented. the theory is proposed as a fairly plausible model for natural language and as one which allows rigorous definitions to be given for various intuitive concepts, such as those of 'grounded' and 'paradoxical' sentences.

Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Godel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from the realm of (...) philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gode1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of computable. When they all turned out to be equivalent, there was a collective realization that this was indeed the right notion. Kleene played a key role in this process. One could say that he was there at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Godel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education.". (shrink)

This paper defends the idea that there might be vagueness or indeterminacy in the world itself--as opposed to merely in our representations of the world--against the charges of incoherence and unintelligibility. First we consider the idea that the world might contain vague properties and relations ; we show that this idea is already implied by certain well-understood views concerning the semantics of vague predicates (most notably the fuzzy view). Next we consider the idea that the world might contain vague objects (...) ; we argue that an object is indeterminate in a certain respect (colour, size, etc.) just in case it is a borderline case of a maximally specific colour (size, etc.) property. Finally we consider the idea that the world as a whole might be indeterminate; we argue that the world is indeterminate just in case it lacks a determinate division into determinate objects. (shrink)

This paper deals with the truth-Conditions and the logic for vague languages. The use of supervaluations and of classical logic is defended; and other approaches are criticized. The truth-Conditions are extended to a language that contains a definitely-Operator and that is subject to higher order vagueness.

/ Part INTRODUCTION Fuzziness is not a priori an obvious concept and demands some explanation. "Fuzziness" is what Black (NF) calls "vagueness" when ...

The primary purpose of this book is to provide the reader with a comprehensive coverage of theoretical foundations of fuzzy set theory and fuzzy logic, as well as a broad overview of the increasingly important applications of these novel areas of mathematics. Although it is written as a text for a course at the graduate or upper division undergraduate level, the book is also suitable for self-study and for industry-oriented courses of continuing education. No previous knowledge of fuzzy set theory (...) and fuzzy logic is required for understanding the material covered in the book. Although knowledge of basic ideas of classical set theory and classical logic is useful, fundamentals of these subject areas are briefly overviewed in the book. In addition, basic ideas of neural networks, genetic algorithms, and rough sets are also explained. This makes the book virtually self-contained. Throughout the book, many examples are used to illustrate concepts, methods, and generic applications as they are introduced. Each chapter is followed by a set of exercises, which are intended to enhance readers' understanding of the material presented in the chapter. Extensive and carefully selected bibliography, together with bibliographical notes at the end of each chapter and a bibliographical subject index, is an invaluable resource for further study of fuzzy theory and applications. (shrink)

Most expressions in natural language are vague. But what is the best semantic treatment of terms like 'heap', 'red' and 'child'? And what is the logic of arguments involving this kind of vague expression? These questions are receiving increasing philosophical attention, and in this book, first published in 2000, Rosanna Keefe explores the questions of what we should want from an account of vagueness and how we should assess rival theories. Her discussion ranges widely and comprehensively over the main theories (...) of vagueness and their supporting arguments, and she offers a powerful and original defence of a form of supervaluationism, a theory that requires almost no deviation from standard logic yet can accommodate the lack of sharp boundaries to vague predicates and deal with the paradoxes of vagueness in a methodologically satisfying way. Her study will be of particular interest to readers in philosophy of language and of mind, philosophical logic, epistemology and metaphysics. (shrink)

This paper defends the idea that there might be vagueness or indeterminacy in the world itself---as opposed to merely in our representations of the world---against the charges of incoherence and unintelligibility. First we consider the idea that the world might contain vague *properties and relations*; we show that this idea is already implied by certain well-understood views concerning the semantics of vague predicates (most notably the fuzzy view). Next we consider the idea that the world might contain vague *objects*; we (...) argue that an object is indeterminate in a certain respect (colour, size, etc.) just in case it is a borderline case of a maximally specific colour (size, etc.) property. Finally we consider the idea that the *world as a whole* might be indeterminate; we argue that the world is indeterminate just in case it lacks a determinate division into determinate objects. (shrink)

This dissertation uses the Liar paradox to motivate an account of the concept of truth that I call the "inconsistency theory of truth." The Liar paradox is the puzzle that arises when we consider such sentences, known as Liar sentences, that say of themselves that they are not true: whatever truth value we attribute to such a sentence, we seem to be immediately driven to the conclusion that it has the opposite truth value. Examining this puzzle reveals that its source (...) is the following principle, which we may call the disquotational principle: for any sentence 'A', 'A' is true if and only if A. While this principle is very natural, it is also inconsistent, as is shown when the sentence 'A' is a Liar sentence. ;After the introductory Chapter 1, Chapter 2 presents some of the technical work on the Liar. Chapter 3 discusses several accounts based on the idea that Liar sentences lack truth values; there I argue that those accounts fail to adequately address a problem known as the strengthened Liar problem. Next, Chapter 4 examines a family of views based on the idea that the formal contradictions derivable by means of the disquotational principle are not really genuine contradictions: according to these views, the appearance of contradiction results from ignoring a contextually determined hidden parameter. I argue that they, too, do not successfully handle the strengthened Liar problem. ;In Chapter 5 I put forth a view based on the idea that the concept of truth itself is inconsistent. Namely, the view I defend is that the linguistic conventions governing the use of the word 'true' commit us to the disquotational schema which, as I said, is inconsistent. I conclude in Chapter 6 by considering some of the implications of believing the inconsistency theory. (shrink)