Intransitivity and Vagueness

There are many examples in the literature that suggest that indistinguishability is intransitive, despite the fact that the indistinguishability relation is typically taken to be an equivalence relation (and thus transitive). It is shown that if the uncertainty perception and the question of when an agent reports that two things are indistinguishable are both carefully modeled, the problems disappear, and indistinguishability can indeed be taken to be an equivalence relation. Moreover, this model also suggests a logic of vagueness that seems to solve many of the problems related to vagueness discussed in the philosophical literature. In particular, it is shown here how the logic can handle the sorites paradox.Comment: A preliminary version of this paper appears in Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference (KR 2004

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. Using a single example where there is a conflict of interest between speaker and listener, we start by treating game-theoretic rationales for vagueness, and for the related concepts of generality and ambiguity. We argue that the most plausible application of these models to vagueness in natural language is one where the listener only imperfectly observes the context in which the speaker makes her utterances. We next look at a rationale for vagueness when there is no conflict between speaker and listener, and which is an application of Horn's rule. Further, we tackle the Sorites paradox. This paradox apparently violates standard axioms of rational behaviour. Yet, once it is taken into account that vague language is used in an appropriate context, these axioms are no longer violated. We end with a behavioural approach to vagueness, where context directly enters agents. preferences. In an application of prospect theory, agents think in terms of gains and losses with respect to a reference point. Vague predicates now allow agents to express their subjective valuations, without necessarily specifying the context.Vagueness, signalling games, decision theory, prospect theory

Arguments Whose Strength Depends on Continuous Variation

Both the traditional Aristotelian and modern symbolic approaches to logic have seen logic in terms of discrete symbol processing. Yet there are several kinds of argument whose validity depends on some topological notion of continuous variation, which is not well captured by discrete symbols. Examples include extrapolation and slippery slope arguments, sorites, fuzzy logic, and those involving closeness of possible worlds. It is argued that the natural first attempts to analyze these notions and explain their relation to reasoning fail, so that ignorance of their nature is profound

The Sorites Paradox in Practical Philosophy

The first part of the chapter surveys some of the main ways in which the Sorites Paradox has figured in arguments in practical philosophy in recent decades, with special attention to arguments where the paradox is used as a basis for criticism. Not coincidentally, the relevant arguments all involve the transitivity of value in some way. The second part of the chapter is more probative, focusing on two main themes. First, I further address the relationship between the Sorites Paradox and the main arguments discussed in the first part, by elucidating in what sense they rely on (something like) tolerance principles. Second, I briefly discuss the prospect of rejecting the respective principles, aiming to show that we can do so for some of the arguments but not for others. The reason is that in the latter cases the principles do not function as independent premises in the reasoning but, rather, follow from certain fundamental features of the relevant scenarios. I also argue that not even adopting what is arguably the most radical way to block the Sorites Paradox – that of weakening the consequence relation – suffices to invalidate these arguments

Insignificant differences : the paradox of the heap

This study investigates six theoretical approaches offered as solutions to the paradox of the heap (sorites paradox), a logic puzzle dating back to the ancient Greek philosopher Eubulides. Those considered are: Incoherence Theory, Epistemic Theory, Supervaluation Theory, Many-Valued Logic, Fuzzy Logic, and Non-Classical Semantics. After critically examining all of these, it is concluded that none of the attempts to explain the sorites are fully adequate, and the paradox remains unresolved.Philosophy, Practical and Systematic TheologyM.A. (Philosophy

“Almost Identical with Itself” : A Search for a Logic of Fuzzy Identity

This thesis grows out of a fascination with the vagueness of natural language, its manifestation in the ancient Sorites paradox, and the way in which the paradox is dealt with in fuzzy logic. It is an attempt to resolve the tension between two versions of the paradox, and the related problem of whether identity can be fuzzy. If it can be fuzzy, then the most popular argument against vague objects is mistaken, which would be great news for those who hold that there can be vagueness in the world independently of our representation or knowledge of it. The standard Sorites is made up of conditionals about an ordinary predicate (e.g. “heap”) by the rule of modus ponens. It is typically solved in fuzzy logic by interpreting the predicate as a fuzzy relation and showing that the argument fails as a result. There is another, less known version of the paradox, based on the identity predicate and the rule of substitutivity of identicals. The strong analogy between the two versions suggests that their solutions might be analogical as well, which would make identity just as vague as any relation. Yet the idea of vague identity has traditionally been rejected on both formal and philosophical grounds. Even Nicholas J. J. Smith, who is known for his positive attitude toward fuzzy relations in general, denies that identity could be fuzzy. The opposite position is taken by Graham Priest, who argues for a fuzzy interpretation of identity as a similarity relation. Following Priest, I aim to show that there is a perfectly sensible logic of fuzzy identity and that a fuzzy theoretician of vagueness therefore cannot rule out fuzzy identity on logical grounds alone. I compare two fuzzy solutions to the identity Sorites: Priest’s solution, based on the notion of local validity, and B. Jack Copeland’s solution, based on the failure of contraction in sequent calculus. I provide a synthesis of the two solutions, suggesting that Priest’s local validity counts as a genuine kind of validity even if he might not think so himself. The substitutivity of identicals is not locally valid in Priest’s logic, however; his solution only applies to a special case with the rule of transitivity. Applying L. Valverde’s representation theorem and other mathematical results, I lay the foundation for a stronger logic where the substitutivity rule is locally valid and the two Sorites merge into one paradox with one solution. Finally, I defend fuzzy identity against Gareth Evans’ argument that vague identity leads to contradiction, and Smith’s argument that vague identity is not really identity. The former relies on a fallacious application of the substitutivity rule; to the latter, my principal response is to question Smith’s understanding of identity and argue for a broader one. I conclude that not only is fuzzy identity logically possible, but it also has potential applicability in metaphysics and elsewhere

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

Being Metaphysically Unsettled: Barnes and Williams on Metaphysical Indeterminacy and Vagueness

This chapter discusses the defence of metaphysical indeterminacy by Elizabeth Barnes and Robert Williams and discusses a classical and bivalent theory of such indeterminacy. Even if metaphysical indeterminacy arguably is intelligible, Barnes and Williams argue in favour of it being so and this faces important problems. As for classical logic and bivalence, the chapter problematizes what exactly is at issue in this debate. Can reality not be adequately described using different languages, some classical and some not? Moreover, it is argued that the classical and bivalent theory of Barnes and Williams does not avoid the problems that arise for rival theories

Neutralism and the Observational Sorites Paradox

Neutralism is the broad view that philosophical progress can take place when (and sometimes only when) a thoroughly neutral, non-specific theory, treatment, or methodology is adopted. The broad goal here is to articulate a distinct, specific kind of sorites paradox (The Observational Sorites Paradox) and show that it can be effectively treated via Neutralism