Skip to main content
Log in

Modular Sequent Calculi for Classical Modal Logics

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

This paper develops sequent calculi for several classical modal logics. Utilizing a polymodal translation of the standard modal language, we are able to establish a base system for the minimal classical modal logic E from which we generate extensions (to include M, C, and N) in a modular manner. Our systems admit contraction and cut admissibility, and allow a systematic proof-search procedure of formal derivations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chellas B., Modal Logic: An Introduction. Cambridge University Press. Cambridge 1980.

  2. Fitting, M., Proof theory for modal logic, in P. Blackburn, J. Van Benthem, and F. Wolter, (eds), Handbook of Modal Logic. Elsevier, Amsterdam, 2007.

  3. Gabbay, D., Labelled Deductive Systems: Volume 1. Clarendon Press, Oxford, 1996.

  4. Gasquet, O., and A. Herzig, From classical to normal modal logics, in H. Wansing, (ed.), Proof Theory of Modal Logic. Kluwer, 1996, pp. 293–311.

  5. Gentzen, G., Investigations into logical deduction, in M. Szabo, (ed.), The Collected Papers of Gerhard Gentzen. North-Holland, 1969, pp. 68–131.

  6. Governatori, L., and A. Luppi, Labelled tableaux for non-normal modal logics, in Sixth Conference of the Italian Association for Artificial Intelligence, AI*IA ’99, 2000, pp. 119–130.

  7. Hansen, H. H., Monotonic modal logics. Master’s thesis, University of Amsterdam, 2003.

  8. Hansen, H. H., Tableau games for coalition logic and alternating-time temporal logic—theory and implementation. Master’s thesis, University of Amsterdam, 2004.

  9. Hansen, H. H., C. Kupke, and E. Pacuit, Neighborhood structures: Bisimilarity and basic model theory. Logical Methods in Computer Science 5:1–38, 2009.

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

  11. Indrzejczak A., Admissibility of cut in congruent modal logics. Logic and Logical Philosophy 21:189–203, 2011.

  12. Kracht, M., and F. Wolter, Normal monomodal logics can simulate all others. The Journal of Symbolic Logic 64:99–138, 1999.

  13. Lavendhomme R., T. Lucas.: Sequent calculi and decision procedures for weak modal systems.. Studia Logica 66, 121–145 (2000)

    Article  Google Scholar 

  14. Montague, R., Universal grammar. Theoria 36:373–398, 1970.

  15. Negri S.: Proof analysis in modal logic. Journal of Philosophical Logic 34, 507–544 (2005)

    Article  Google Scholar 

  16. Negri, S., Kripke completeness revisited, in G. Primiero and S.Rahman, (eds.), Acts of Knowledge—History, Philosophy and Logic. College Publications, 2009, pp. 242–282.

  17. Negri S.: Proofs and countermodels in non-classical logics. Logica Universalis 8, 25–60 (2014)

    Article  Google Scholar 

  18. Negri, S., Proof analysis beyond geometric theories: from rule systems to systems of rules. Journal of Logic and Computation. (in press).

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

  20. Negri, S., and J. von Plato, Proof Analysis: A Contribution to Hilbert’s Last Problem. Cambridge University Press, Cambridge, 2011.

  21. Scott, D., Advice on modal logic, in K. Lambert, (ed.), Philosophical Problems in Logic: Some Recent Developments. Reidel, Dordrecht, 1970.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to David R. Gilbert.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gilbert, D.R., Maffezioli, P. Modular Sequent Calculi for Classical Modal Logics. Stud Logica 103, 175–217 (2015). https://doi.org/10.1007/s11225-014-9556-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-014-9556-1

Keywords

Navigation