Base-extension Semantics for Modal Logic
Timo Eckhardt, David J. Pym
TL;DR
This work develops a base-extension semantics (B-eS) for classical propositional modal logics $K$, $KT$, $K4$, and $S4$ with $\square$ as the primary operator, adapting the classical B-eS to include a modal relation on bases and proving soundness and completeness with respect to Kripke semantics and Hilbert systems. It introduces a formal modal relation $\mathfrak{R}$ on bases to define the truth conditions of $\square$ within the base-extension framework, and establishes the duality between $\square$ and $\lozenge$ via standard modal translations. The paper also proves that the B-eS as developed is not complete for Euclidean modal logics, highlighting limits of the current approach and suggesting directions for extending proof-theoretic semantics to broader frame conditions. Overall, the work provides a rigorous, relational, proof-theoretic account of modal validity that mirrors classical Kripke semantics while preserving a base-centric, inference-based interpretation of meaning. $K$, $KT$, $K4$, and $S4$ modal logics are thus given a unified B-eS treatment with duality and a clear boundary regarding Euclidean completeness, paving the way for further integration with reasoning frameworks and applied modal systems.
Abstract
In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a `base' of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems K, KT , K4, and S4, with $\square$ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between $\square$ and a natural presentation of $\lozenge$. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases.