Abstract
The paper settles an open question concerning Negri-style labeled sequent calculi for modal logics and also, indirectly, other proof systems which make (more or less) explicit use of semantic parameters in the syntax and are thus subsumed by labeled calculi, like Brünnler’s deep sequent calculi, Poggiolesi’s tree-hypersequent calculi and Fitting’s prefixed tableau systems. Specifically, the main result we prove (through a semantic argument) is that labeled calculi for the modal logics K and D remain complete w.r.t. valid sequents whose relational part encodes a tree-like structure, when the unique rule which contains an harmful implicit contraction—by which the condition that the premises be less complex than the conclusion is violated—is modified into a contraction-free one respecting the latter condition, thus making the proof-search space finite.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brünnler K.: Deep sequent systems for modal logic. Arch. Math. Log. 48, 551–577 (2009)
Brünnler, K.: Nested Sequents. Habilitation Thesis, Bern (2010)
Fitting M.C.: Tableau methods of proof for modal logics. Notre Dame J. Formal Log. 13, 237–247 (1972)
Fitting M.C.: Proof Methods for Modal and Intuitionistic Logics. Reidel, Dordrecht (1983)
Fitting, M.C.: Modal proof theory. In: Blackburn, P., van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic, pp. 85–138. Elsevier, Amsterdam (2007)
Fitting M.C.: Prefixed tableaus and nested sequents. Ann. Pure Appl. Log. 163, 291–313 (2012)
Gabbay D.: Labelled Deductive Systems: Volume 1, Foundations. Oxford University Press, Oxford (1996)
Goré R.: Tableau methods for modal and temporal logics. In: D’Agostino, M., Gabbay, D., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods, pp. 297–396. Kluwer, Dordrecht (1999)
Hill B., Poggiolesi F.: A contraction-free and cut-free sequent calculus for propositional dynamic logic. Stud. Log. 94, 47–72 (2010)
Maffezioli, P., Naibo, A., Negri, S.: The Church–Fitch knowability paradox in the light of structural proof theory. Synthese (2012). doi:10.1007/s11229-012-0061-7
Massacci F.: Strongly analytic tableaux for normal modal logics. In: Bundy, A. (ed.) Proceedings of CADE 12, Lecture Notes in Artificial Intelligence, vol. 814, pp. 723–737. Springer, Berlin (1994)
Negri S.: Contraction-free sequent calculi for geometric theories with an application to Barr’s theorem. Arch. Math. Log. 42, 389–401 (2003)
Negri S.: Proof analysis in modal logic. J. Philos. Log. 34, 507–544 (2005)
Negri S.: Kripke completeness revisited. In: Primiero, G., Rahman, S. (eds.) Acts of Knowledge—History, Philosophy and Logic, pp. 247–282. College Publicatons, London (2009)
Negri S., Dyckhoff R.: Proof analysis in intermediate logics. Arch. Math. Log. 51, 71–92 (2012)
Poggiolesi, F.: Sequent Calculi for Modal Logic. PhD Thesis, Firenze/Paris (2008)
Poggiolesi F.: Gentzen calculi for modal propositional logics. Springer, Berlin (2011)
Viganò L.: Labelled Non-classical Logics. Kluwer, Dordrecht (2000)
Wansing H.: Displaying Modal Logic. Kluwer, Dordrecht (1998)
Wansing H.: Sequent systems for modal logics. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 8, 2nd edn, pp. 61–145. Kluwer, Dordrecht (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Minari, P. Labeled sequent calculi for modal logics and implicit contractions. Arch. Math. Logic 52, 881–907 (2013). https://doi.org/10.1007/s00153-013-0350-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-013-0350-y