A Logic for Vagueness

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 non-observational 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 deep-seated 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.