Propositional logic and modal logic—A connection via relational semantics

Abstract In this paper, by slightly generalizing an observation of Dalla Chiara and Giuntini in their chapter on quantum logic in Handbook of Philosophical Logic, we propose a relational semantics for propositional language with negation and conjunction, which unifies the relational semantics of intuitionistic logic and that of ortho-logic. We study the semantic and syntactic consequence relations and prove the soundness and completeness theorems for five propositional logics: $\textbf{BL}$, $\textbf{PL}$, $\textbf{IL}$, $\textbf{OL}$ and $\textbf{CL}$. Moreover, we prove that they can be translated into the modal logics $\textbf{K}$, $\textbf{T}$, $\textbf{S4}$, $\textbf{KTB}$ and $\textbf{S5}$, respectively, and thus establish a systematic connection between propositional logics and modal logics. The paper ends with a discussion about the possibility and difficulty of incorporating disjunction into our framework.