Truth tables for modal logics T and S4, by using three-valued non-deterministic level semantics

Abstract

Abstract Novel three-valued non-deterministic level semantics for modal logics $\textbf {T}$ and $\textbf {S4}$ are presented. A criterion for partial level valuations is given, making it possible to create truth tables. Additionally, semantics and truth tables for $\textbf {0}$ (defined as $\textbf {PC}$ plus rule of necessitation) and $\textbf {0T}$ with only two values are based on Ivlev’s work. We need Kearns’ notion of level valuations: a generalization of Dugundji’s theorem shows that there is no non-deterministic semantics for modal logics up to $\textbf {S5}$, containing the rule of necessitation.