Cut-free Completeness for Modular Hypersequent Calculi for Modal Logics K, T, and D

TL;DR

This paper presents a novel, direct semantical proof of cut-free completeness for modular hypersequent calculi in modal logics K, T, and D, highlighting their structural rule-based modularity and limitations.

Contribution

It provides the first direct semantical cut-free completeness proofs for K, T, and D modal logics using hypersequent calculi, and discusses their modularity and limitations.

Findings

First direct semantical cut-free completeness proofs for K, T, and D.

Modular hypersequent systems share modal rules across different logics.

The method does not extend to B and S4 logics.

Abstract

We investigate a recent proposal for modal hypersequent calculi. The interpretation of relational hypersequents incorporates an accessibility relation along the hypersequent. These systems give the same interpretation of hypersequents as Lellman's linear nested sequents, but were developed independently by Restall for S5 and extended to other normal modal logics by Parisi. The resulting systems obey Dosen's principle: the modal rules are the same across different modal logics. Different modal systems only differ in the presence or absence of external structural rules. With the exception of S5, the systems are modular in the sense that different structural rules capture different properties of the accessibility relation. We provide the first direct semantical cut-free completeness proofs for K, T, and D, and show how this method fails in the case of B and S4.