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Motivation for a new semantics for vagueness
 EWIC (Irish Workshop on Formal Methods
, 1999
"... Vagueness is the phenomenon that natural language predicates have borderline regions of applicability and that the boundaries of the borderline region are not determinable. A theory is presented which argues that vagueness is due to the fact that we are computationally bound by Church’s Thesis. Synt ..."
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Vagueness is the phenomenon that natural language predicates have borderline regions of applicability and that the boundaries of the borderline region are not determinable. A theory is presented which argues that vagueness is due to the fact that we are computationally bound by Church’s Thesis. Syntactic and semantic models motivated by the theory are introduced. Each disallows the use of classical negation, capturing the fact that it is generally only possible to semidecide but not decide our interpretations of natural language predicates. The role of negation is filled, for each predicate R, by the existence of a dual predicate nonR that acts as if it is the negation, although its interpretation is generally, at best, only an approximation to the complement of R. Multiple “levels ” of vagueness are modeled using concepts from recursion theory.
ON EVANS’S VAGUE OBJECT FROM SET THEORETIC VIEWPOINT
"... Abstract. Gareth Evans proved that if two objects are indeterminately equal then they are different in reality. He insisted that this contradicts the assumption that there can be vague objects. However we show the consistency between Evans’s proof and the existence of vague objects within classical ..."
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Abstract. Gareth Evans proved that if two objects are indeterminately equal then they are different in reality. He insisted that this contradicts the assumption that there can be vague objects. However we show the consistency between Evans’s proof and the existence of vague objects within classical logic. We formalize Evans’s proof in a set theory without the axiom of extensionality, and we define a set to be vague if it violates extensionality with respect to some other set. There exist models of set theory where the axiom of extensionality does not hold, so this shows that there can be vague objects. §1. Introduction. In his short paper “Can There Be Vague Objects? ” [5], Gareth Evans questioned a consistency between two assumptions: Vagueness is ‘a necessary feature of any true description ’ of the world, and ‘amongst the statements which may not have a determinate truth value as a result of their vagueness are identity statements’. He defined vague objects as having vague
A Logic for Vagueness
, 2010
"... In Dummett’s important paper [1] on the sorites paradox it is suggested that the vagueness of observational predicates such as ‘...is red ’ or more obviously ‘...looks red ’ generates an apparent incoherence: their use resembles a game governed by inconsistent rules. A similar incoherence is seen by ..."
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In Dummett’s important paper [1] on the sorites paradox it is suggested that the vagueness of observational predicates such as ‘...is red ’ or more obviously ‘...looks red ’ generates an apparent incoherence: their use resembles a game governed by inconsistent rules. A similar incoherence is seen by Wright [18, 17] as a real and serious threat to very ordinary ideas of how language works. Wright argues not that the use of vague predicates is incoherent but that it would be if the use of language were a practice in which the admissibility of moves were determined by rules whose general properties are discoverable by appeal to nonbehavioural notions. But unless we do move from anecdotes about behaviour to just such rules, how are we to reason at all? The incoherence in question is an outcome of the vagueness or tolerance of observational locutions, and would seem if established for them to spread to nonobservational vague expressions like ‘...is water ’ or ‘...is a test tube’, 1 thus vitiating almost all of our attempts to use language consistently, even in science. On the face of it, vagueness is everywhere, whence such deepseated incoherence would upset even such fragile understanding of semantics as we have gleaned from a century’s work. The argument connecting vagueness to incoherence, therefore, strikes at the heart of logic: every philosophical logician is called upon to respond to it. The sorites paradox, the “slippery slope argument ” or the “paradox of the heap”, is old and famous and wears an air of sophistry. One feels that the problem will vanish on exposure of the trivial trick involved. What is surprising is that it is so deep and difficult after all. An example or two will help focus the discussion. 1 How long and thin must a piece of glassware be, or how polluted may a liquid be, before it no longer counts as a test tube or as water? John Slaney, “A Logic for Vagueness”, Australasian Journal of Logic (8) 2010, 100–134http://www.philosophy.unimelb.edu.au/ajl/2010 101 example 1 Imagine a long coloured strip shading gradually from scarlet at one end to lemonyellow at the other, divided into 250 thin segments. 2
American Research Press
"... F1 � ({1}�F2) �F2 � ({1} � F1) �F1 � F2 � ({1}�F1) � ({1}�F2)). NL(A1 � A2) = ( {1}�T1�T1�T2, {1}�I1�I1�I2, {1}�F1�F1�F2). NL(A1 � A2) = ( ({1}�T1�T1�T2) � ({1}�T2�T1�T2), ({1} � I1 � I1 � I2) � ({1}�I2 � I1 � I2), ({1}�F1�F1 � F2) � ({1}�F2�F1 � F2)). ..."
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F1 � ({1}�F2) �F2 � ({1} � F1) �F1 � F2 � ({1}�F1) � ({1}�F2)). NL(A1 � A2) = ( {1}�T1�T1�T2, {1}�I1�I1�I2, {1}�F1�F1�F2). NL(A1 � A2) = ( ({1}�T1�T1�T2) � ({1}�T2�T1�T2), ({1} � I1 � I1 � I2) � ({1}�I2 � I1 � I2), ({1}�F1�F1 � F2) � ({1}�F2�F1 � F2)).
Vagueness, tolerance and nontransitive entailment
"... 1 Tolerance and vagueness Vagueness is standardly opposed to precision. Just as gradable adjectives like ‘tall ’ and a quantity modifier like ‘a lot ’ are prototypical vague expressions, mathematical adjectives like ‘rectangular’, and measure phrases like ‘1.80 meter ’ are prototypically precise. Bu ..."
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1 Tolerance and vagueness Vagueness is standardly opposed to precision. Just as gradable adjectives like ‘tall ’ and a quantity modifier like ‘a lot ’ are prototypical vague expressions, mathematical adjectives like ‘rectangular’, and measure phrases like ‘1.80 meter ’ are prototypically precise. But what does it mean for these latter expressions to be precise? On first thought it just means that they are precise, because they have an exact mathematical definition. However, if we want to use these terms to talk about observable objects, it is clear that these mathematical definitions would be useless: if they exist at all, we cannot possibly determine what are the rectangular objects in the precise geometrical sense, or objects that are exactly 1.80 meters long. For this reason, one allows for a margin of measurement error, or a threshold, in physics, psychophysics and other sciences. Assuming that the predicates we use are observational predicates gives rise to another consequence as well. If statements like ‘the length of stick S is 1.45 meters ’ come with a large enough margin of error, the circumstances in which this statement can be made appropriately (or truly, if you don’t want the notion of truth to be empty) might overlap with the circumstances in which the statement ‘the The main ideas of this paper were first presented in a workshop on vagueness at
Strategic Vagueness, and appropriate contexts
"... This paper brings together several approaches to vagueness, and ends by suggesting a new approach. The common thread in these approaches is the crucial role played by context. In Section 2, we treat gametheoretic rationales for vagueness, and for the related concepts of ambiguity and generality. Co ..."
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This paper brings together several approaches to vagueness, and ends by suggesting a new approach. The common thread in these approaches is the crucial role played by context. In Section 2, we treat gametheoretic rationales for vagueness, and for the related concepts of ambiguity and generality. Common
J Philos Logic DOI 10.1007/s109920109165z Tolerant, Classical, Strict
, 2010
"... Abstract In this paper we investigate a semantics for firstorder 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, theny should be P whenever y is similar enough to x. The semantics, which makes use of in ..."
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Abstract In this paper we investigate a semantics for firstorder 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, theny 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.
Vagueness, Tolerance and NonTransitive
"... Vagueness is standardly opposed to precision. Just as gradable adjectives like ‘tall’ and a quantity modifier like ‘a lot ’ are prototypically vague expressions, mathematical adjectives like ‘rectangular’, and measure phrases like ‘1.80 meters ’ are prototypically precise. But what does it mean for ..."
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Vagueness is standardly opposed to precision. Just as gradable adjectives like ‘tall’ and a quantity modifier like ‘a lot ’ are prototypically vague expressions, mathematical adjectives like ‘rectangular’, and measure phrases like ‘1.80 meters ’ are prototypically precise. But what does it mean for these latter expressions to be precise? On first thought it just means that they have an exact mathematical definition. However, if we want to use these terms to talk about observable objects, it is clear that these mathematical definitions would be useless: if they exist at all, we cannot possibly determine what are the existing (nonmathematical) rectangular objects in the precise geometrical sense, or objects that are exactly 1.80 meters long. For this reason, one allows for a margin of measurement error, or a threshold, in physics, psychophysics and other sciences. The assumption that the predicates we use are observational predicates gives rise to another consequence as well. If statements like ‘the length of stick S is 1.45 meters ’ come with a large enough margin of error, the circumstances in which this statement is appropriate (or true, if you don’t want the notion of truth to be empty) might overlap with the appropriate circumstances for uttering statements like ‘the length of stick S is 1.50 meters’. Thus, although
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"... Tolerant, classical, strict In this paper we investigate a semantics for firstorder 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, w ..."
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Tolerant, classical, strict In this paper we investigate a semantics for firstorder 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. Our aim in this paper is to explore a semantic framework originally proposed by [van Rooij, 2010b] in order to deal with the sorites paradox, and intended to formalize the idea that vague predicates are tolerant. Standardly, the idea of tolerance is expressed
Contextual Effects on Vagueness and the Sorites Paradox: A Preliminary Study
"... Abstract. Context is a prominent theme in accounts of the semantics of vagueness. Contextual theories of vagueness have been motivated in part by a desire to disarm skeptical arguments that make use of vagueness; another motive is the hope that they might help to resolve the Sorites Paradox. This pa ..."
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Abstract. Context is a prominent theme in accounts of the semantics of vagueness. Contextual theories of vagueness have been motivated in part by a desire to disarm skeptical arguments that make use of vagueness; another motive is the hope that they might help to resolve the Sorites Paradox. This paper is intended as part of a larger project, which aims at describing and comparing the leading theories that have been presented, to bring out the ways in which contextual theories have informed the semantic phenomenon of vagueness, and to make strategic assesments and suggestions concerning the crucial case of adjectives. It concentrates on Hans Kamp’s 1975 theory, and two recent contributions by Haim Gaifman and Joseph Halpern. The specific goal of this paper is to examine the role of context in the semantics of certain adjectives, and assess to what extent context can help to solve what I take to be the most challenging version of the Sorites Paradox. 1