Vagueness and Blurry Sets

Abstract

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 S ₁ to the effect that ‘Bob is bald’ is x true, for x in [0,1], there should be a further statement S ₂ which tells us how true S ₁ 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.