Construction of canonical models for K45 and weaker systems

Abstract In this paper we explore the construction of canonical models for K45 and weaker systems by extending Wittgenstein’s picture theory of language to a family of locally tabular modal logics. After discussing the philosophical implications of revisiting Wittgenstein’s concept of logical space, especially for K45, we turn to the mathematical side, establishing soundness and completeness results for K$4^{k}!5^{-}$. These theorems pave the way for the description of canonical models, which we illustrate for the systems K45, K5, and K$4^{2}!5^{-}$ in the case where a single proposition is considered. The purpose of these illustrations is to gain insight into these systems through the combinatorial structure of their canonical models.