🤖 AI Summary
This work addresses the absence of a unified proof-theoretic framework for mainstream modal logics by developing a hypersequent calculus that uniformly encompasses K and its standard extensions—including T, D, 4, B, and 5. The paper systematically elucidates the construction mechanisms underlying sequent calculi for various modal logics and provides syntactic cut-elimination proofs for all major systems except KB, KDB, and KTB. This framework not only clarifies the origins of both sequent and hypersequent formulations for logics such as S5 but also lays the groundwork for further extensions to quantified modal logics. By doing so, it significantly advances the systematicity and unification of proof theory in modal logic.
📝 Abstract
This paper proposes a basic proof theoretic framework for major modal logics: {\sf S5} and some of its subsystems. The framework is based on a version of hypersequent calculus, and the basic modal systems we handle here are the system {\sf K} and its standard extensions with combinations of axioms: $T, D, 4, B, 5$. First we propose a reasonable explanation of how the standard sequent and hypersequent calculi for some of those modal logics such as {\sf K, T, D, S4, S5} emerge on the basis of the framework. Then, by a syntactic method, we prove the cut-elimination theorem for the modal logics except for the modal logics {\sf KB, KDB, KTB}. Quantified versions of the systems of the framework are also discussed.