A Proof-Theoretic Study of Modal Logic
For proof theorists and modal logicians, this provides a unified syntactic framework for cut-elimination in major modal logics, though the results are incremental as hypersequent calculi for these logics already exist.
This paper develops a hypersequent calculus framework for modal logics including S5 and its subsystems, proving cut-elimination for most systems except KB, KDB, and KTB, and extending the framework to quantified versions.
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.