Borderline Cases and Bivalence

Abstract

It is generally agreed that vague predicates like 'red', 'rich', 'tall', and 'bald', have borderline cases of application. For instance, a cloth patch whose color lies midway between a definite red and a definite orange is a borderline case for 'red', and an American man five feet eleven inches in height is (arguably) a borderline case for 'tall'. The proper analysis of borderline cases is a matter of dispute, but most theorists of vagueness agree at least in the thought that borderline cases for vague predicate '4' items whose satisfaction of '1' is in some sense unclear or problematic: it is unclear whether or not the patch is red, unclear whether or not the man is tall.1 For example, Lynda Burns cites a widespread view as holding that borderline cases are not definitely within the positive or negative extension of the predicate. ... Borderline cases seen as falling within a gap between the cases of definite application of the predicate and cases of definite application of its negation (1995, 30). Michael Tye writes that the of a borderline case is the concept of a case that is neither definitely in nor definitely out (1994b, 18). Reflecting this common view, the standard philosophical analysis defines borderline cases for vague predicate '4' as items that neither definitely (clearly, determinately) 0 nor definitely not q. A borderline case for '4' is then also a borderline case for 'not V': being borderline not D consists in being neither definitely not ( nor definitely not not (D, which is equivalent to being borderline (. This standard analysis is usually meant to express a semantic conception of borderline cases-that is, a conception of them as arising from semantic features of a vague predicate. In that case the definiteness operator is interpreted so that the sentence 'x is definitely V' is true if and only if the sentence 'x is V' is true, and false if and only if 'x is V' is not true, where being not true consists in being either false or neither true nor false.2 Where xis a borderline case, then, the sentences 'x is V' and 'x is not V' not true; they also not false, and so neither true nor false. Various nonclassical semantics have been introduced to capture the meanings of sentences that neither true nor false, including such devices as supervaluations, indefinite or indeterminate values, truth-value gaps, and degrees of truth. Some of these systems invalidate Excluded Mid-