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

This article focuses on the minimal structural logic, denoted |$\textbf{M}_{\textbf{S}}$|, a sequent calculus including all structural rules but excluding operational ones, and investigates its semantic properties. It establishes that |$\textbf{M}_{\textbf{S}}$|, despite its minimal expressive power, shares with intuitionistic and certain modal logics the absence of sound and complete finitely-valued deterministic semantics, a phenomenon related to Gödel-Dugundji-style impossibility results. The paper further demonstrates that |$\textbf{M}_{\textbf{S}}$| admits sound and complete finitely-valued non-deterministic semantics based on maximally permissive non-deterministic matrices (Nmatrices). Additionally, the study extends these results to subsystems of |$\textbf{M}_{\textbf{S}}$|, including the structural kernels of the strict-tolerant (|$\textbf{ST}$|) and tolerant-strict (|$\textbf{TS}$|) logics, showing that while some subsystems lack finitely-valued deterministic semantics even under variable designated value frameworks, others admit such semantics, highlighting nuanced differences within substructural logics.