A Gödel-Dugundji-style theorem for the minimal structural logic

Abstract

This paper introduces a sequent calculus, $\textbf{M}_{\textbf{S}}$, the minimal structural logic, which includes all structural rules while excluding operational ones. Despite its limited calculus, $\textbf{M}_{\textbf{S}}$ unexpectedly shares a property with intuitionistic logic and modal logics between $\textsf{S1}$ and $\textsf{S5}$: it lacks sound and complete finitely-valued (deterministic) semantics. Mirroring Gödel’s and Dugundji’s findings, we demonstrate that $\textbf{M}_{\textbf{S}}$ does possess a natural finitely-valued non-deterministic semantics. In fact, we show that $\textbf{M}_{\textbf{S}}$ is sound and complete with respect to any semantics belonging to a natural class of maximally permissive non-deterministic matrices. We close by examining the case of subsystems of $\textbf{M}_{\textbf{S}}$, including the “structural kernels” of the strict-tolerant and tolerant-strict logics $\textbf{ST}$ and $\textbf{TS}$, and strengthen this result to also preclude finitely-valued deterministic semantics with respect to variable designated value frameworks.