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.
Citation
Fabrice Correia. "Adequacy Results for Some Priorean Modal Propositional Logics." Notre Dame J. Formal Logic 40 (2) 236 - 249, Spring 1999. https://doi.org/10.1305/ndjfl/1038949539
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