Admissibility of cut in congruent modal logics

Andrzej Indrzejczak



We present a detailed proof of the admissibility of cut in sequent calculus for some congruent modal logics. The result was announced much earlier during the Trends in Logic Conference, Toruń 2006 and the proof for monotonic modal logics was provided already in Indrzejczak [5]. Also some tableau and natural deduction formalizations presented in Indrzejczak [6] and Indrzejczak [7] were based on this result but the proof itself was not published so far. In this paper we are going to fill this gap. The delay was partly due to the fact that the author from time to time was trying to improve the result and extend it to some additional logics by testing other methods of proving cut elimination. Unfortunately all these attempts failed and cut elimination holds only for these logics which were proved to satisfy this property already in 2005.


modal logics; proof methods; sequent calculi; cut elimination

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Bull, R., and K. Segerberg, “Basic Modal Logic”, pages 1–88 in: D. Gabbay, F. Guenthner (eds.), Handbook of Philosophical Logic, vol II, Reidel Publishing Company, Dordrecht 1984.

Chellas, B., Modal Logic, Cambridge University Press, Cambridge 1980.

Dragalin, A., Mathematical Intuitionism: Introduction to Proof Theory, American Mathematical Society, Providence, Rhode Island 1988.

Hansen, H.H., Monotonic Modal Logics, MA thesis, University of Amsterdamm 2003.

Indrzejczak, A., “Sequent calculi for monotonic modal logics”, Bulletin of the Section of logic 34, 3 (2005): 151–164.

Indrzejczak, A., “Labelled tableau calculi for weak modal logics”, Bulletin of the Section of logic 36, 3–4 (2007): 159–173.

Indrzejczak, A., Natural Deduction, Hybrid Systems and Modal Logics, Trends in Logic series, vol 30, Springer Verlag 2010.

Lavendhomme, R., and T. Lucas, “Sequent calculi and decision procedures for weak modal systems”, Studia Logica 65 (2000): 121–145.

Negri, S., and J. von Plato, Structural Proof Theory, Cambridge University Press, Cambridge 2001.

Segerberg, K., An Essay in Classical Modal Logic I-III, Filosofiska Studier no 13, Uppsala Universitet, Uppsala 1971.

Troelstra, A.S., and H. Schwichtenberg, Basic Proof Theory, Oxford University Press, Oxford 1996.

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

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