Classical logic without bivalence
Tor Sandqvist · Analysis · 2009
Semantic justifications of the classical rules of logical inference typically make use of a notion of bivalent truth, understood as a property guaranteed to attach to a sentence or its negation regardless of the prospects for speakers to determine it as so doing. For want of a convincing alternative account of classical logic, some philosophers suspicious of such recognition-transcending bivalence have seen no choice but to declare classical deduction unwarranted and settle for a weaker system; intuitionistic logic in particular, buttressed by assertion-conditional semantics, is often considered to enjoy a degree of meaning-theoretical respectability unattainable by classical logic. The decision to forgo the classical inference rules is not always made lightly. Thus, Dummett: In the resolution of the conflict between [the view that generally accepted classical modes of inference ought to be theoretically accommodated, and the demand that any such accommodation be achieved without recourse to bivalence] lies, as I see it, one of the most fundamental and intractable problems in the theory of meaning; indeed, in all philosophy. (1978: 318)