Adequacy Results for Some Priorean Modal Propositional Logics

Abstract

Standard possible world semantics for propositional modal languages ignore truth-value gaps. However, simple considerations suggest that it should not be so. In Section 1, I identify what I take to be a correct truth-clause for necessity under the assumption that some possible worlds are incomplete (i.e., "at" which some propositions lack a truth-value). In Section 2, I build a world semantics, the semantics of TV-models, for standard modal propositional languages, which agrees with the truth-clause for necessity previously identified. Sections 3-5 are devoted to systematic concerns. In particular, in Section 4, Prior's system $Q$ (propositional version) is given a TV-models semantics and proved adequate (i.e., sound and complete) with respect to it.