3 June 2008, Truth values, neither-true-nor-false, and supervaluations, Nuel Belnap

The first part of this essay defends reliance on truth values against those who, on nominalistic grounds, would uniformly substitute a truth predicate. I rehearse some practical advantages of working with truth values in logic. In the second part I look at several cases in which logics involve, as part of their semantics, quantification over a silent parameter, such as modal logic's quantification over worlds. In many cases, this facility produces truth values for sentences which seem neither true nor false by ``supervaluation,'' that is, by ``quantifying out'' the extra, silent argument. Logics that generate truth values for the neither-true-nor-false in this way exhibit striking differences. I consider the following: open sentences in first order logic, vague sentences, ambiguous sentences, paradoxical sentences, and future-tensed sentences in indeterministic tense logic.