Supervaluationism and Its Logics

If we adopt a supervaluational semantics for vagueness, what sort of logic results? As it turns out, the answer depends crucially on how the standard notion of validity as truth preservation is recasted. There are several ways of doing that within a supervaluational framework, the main alternative being between ‘global’ construals (e.g., an argument is valid if and only if it preserves truth-under-all-precisifications) and ‘local’ construals (an argument is valid if and only if, under all precisifications, it preserves truth). The former alternative is by far more popular, but I argue in favour of the latter, for (i) it does not suffer from a number of serious objections, and (ii) it makes it possible to restore global validity as a defined notion. Supervaluationism is a mixed bag. It is sometimes described as the ‘standard’ theory of vagueness, at least insofar as vagueness is construed as a semantic phenomenon, but exactly what that standard theory amounts to is far from clear. In fact, it’s pretty clear that there isn’t just one supervaluational semantics out there—there are lots of such semantics; and although it is true that they all exploit the same insight, their relative differences are by no means immaterial. For one thing, a lot depends on how exactly supervaluations are constructed, that is, on how exactly we come to establish the truth-value of a given statement. (And when I say that a lot depends on this I mean to say that different explanations may give rise to different philosophical worries, or justify different reactions.) Secondly, and equally importantly, a lot depends on how a given supervaluationary machinery is brought into play when it comes to explaining the logic of the language, that is, not the notion of truth, or ‘super-truth’, as it applies to individual statements, but the notion of validity, or ‘super-validity’, as it applies to whole arguments. (I am thinking for instance of how different explanations may bear on the question of whether, or to what extent, vagueness involves a departure from classical logic.) Here I want to focus on this second part of the story. However, since the notion of validity depends on the notion of truth—or so one may argue—I also want to comment briefly on the first. 1. Precisifications and Supervaluations I take it that the basic insight of any supervaluationary semantics boils down to the following two thoughts: first, a vague language is one that admits of several