A Proof-Theoretic Study of Modal Logic
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910231989059584 |
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| author | Kushida, Hirohiko |
| author_facet | Kushida, Hirohiko |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18043 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Proof-Theoretic Study of Modal Logic Kushida, Hirohiko Logic in Computer Science 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. |
| title | A Proof-Theoretic Study of Modal Logic |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2605.18043 |