Studia Logica

, Volume 85, Issue 2, pp 199–214 | Cite as

A Deep Inference System for the Modal Logic S5

  • Phiniki StouppaEmail author


We present a cut-admissible system for the modal logic S5 in a formalism that makes explicit and intensive use of deep inference. Deep inference is induced by the methods applied so far in conceptually pure systems for this logic. The system enjoys systematicity and modularity, two important properties that should be satisfied by modal systems. Furthermore, it enjoys a simple and direct design: the rules are few and the modal rules are in exact correspondence to the modal axioms.


modal logic S5 proof theory deep inference calculus of structures cutadmissibility 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Avron A. (1996). ‘The Method of Hypersequents in the Proof Theory of Propositional Non-classical Logics’. In: Hodges W., Hyland M., Steinhorn C., Truss J. (eds) Logic: From Foundations to Applications. Oxford University Press, Oxford, pp. 1–32Google Scholar
  2. 2.
    Braüner T. ‘A cut-free Gentzen formulation of the modal logic S5’. in the Logic Journal of the Interest Group in Pure and Applied Logics 8(5):629–643, 2000.Google Scholar
  3. 3.
    Braüner, T., ‘Functional completeness for a natural deduction formulation of hybridized S5’, in P. Balbiani, N.-Y. Suzuki, F. Wolter and M. Zakharyaschev (eds.), Advances in Modal Logic, vol. 4, King’s College Publications, 2003, pp. 31–49.Google Scholar
  4. 4.
    Brünnler, K., Deep Inference and Symmetry in Classical Proofs, PhD thesis, Technische Universität Dresden, 2003.Google Scholar
  5. 5.
    Fitting M. (1999). ‘A simple propositional S5 tableau system’. Annals of Pure and Applied Logic 96:107–115CrossRefGoogle Scholar
  6. 6.
    Gentzen G., ‘Investigations into logical deduction’, in M. E. Szabo (ed.), The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam, 1969, pp. 68–131.Google Scholar
  7. 7.
    Guglielmi, A., ‘A System of Interaction and Structure’, ACM Transactions on Computational Logic 8 (1):1–64, 2007. Scholar
  8. 8.
    Guglielmi, A., and L. Strassburger, ‘Non-commutativity and MELL in the Calculus of Structures’, in L. Fribourg (ed.), CSL 2001, LNCS 2142, Springer-Verlag, 2001, pp. 54–68.Google Scholar
  9. 9.
    Hein, R., Geometric Theories and Modal Logic in the Calculus of Structures, Master Thesis, Technische Universität Dresden, 2005.Google Scholar
  10. 10.
    Hughes, G., and M. Cresswell, A New Introduction to Modal Logic, Routledge, 1996.Google Scholar
  11. 11.
    Indrzejczak A. (1998). ‘Cut-free Double Sequent Calculus for S5’. Logic Journal of the Interest Group in Pure and Applied Logics 6(3):505–516Google Scholar
  12. 12.
    Indrzejczak A. (1997). ‘Generalised Sequent Calculus for Propositional Modal Logics’. Logica Trianguli 1:15–31Google Scholar
  13. 13.
    Kahramanoğullari O. ‘Implementing System BV of the Calculus of Structures in Maude’, in L. Alonso i Alemany and P. Égré (eds.), Proceedings of the ESSLLI-2004 Student Session, 2004, pp. 117–127.Google Scholar
  14. 14.
    Kahramanoğullari, O., ‘Reducing Nondeterminism in the Calculus of Structures’, Technical Report WV-06-01, Technische Universität Dresden, 2006.Google Scholar
  15. 15.
    Kahramanoğullari, O., ‘System BV without the Equalities for Unit’, in C. Aykanat, T. Dayar and I. Korpeoglu (eds.), ISCIS’04, LNCS 3280, Springer-Verlag, 2004, pp. 986–995.Google Scholar
  16. 16.
    Kanger S. (1957). Provability in Logic. Almqvist & Wiksell, StockholmGoogle Scholar
  17. 17.
    Lemmon, E., and D. Scott An Introduction to Modal Logic. Oxford:Blackwell, 1977.Google Scholar
  18. 18.
    Mints, G., A Short Introduction to Modal Logic, CSLI Lecture Notes 30, CSLI Publications, Stanford, 1992.Google Scholar
  19. 19.
    Mints, G., ‘Lewis’ systems and system T’, Selected Papers in Proof Theory, Bibliopolis, North-Holland, 1992, pp. 221–294.Google Scholar
  20. 20.
    Negri S. (2005). ‘Proof Analysis in Modal Logic’. Journal of Philosophical Logic 34:507– 544CrossRefGoogle Scholar
  21. 21.
    Ohnishi, M., and K. Matsumoto, ‘Gentzen method in modal calculi’, parts I and II, Osaka Mathematical Journal9:113–130, 1957, and 11:115–120, 1959.Google Scholar
  22. 22.
    Orłowska E. (1996). ‘Relational Proof Systems for Modal Logics’. In: Wansing H. (eds.) Proof Theory of Modal Logic. Kluwer Academic Publishers, Dordrecht, pp. 55–77Google Scholar
  23. 23.
    Pottinger G. (1983). ‘Uniform, Cut-free formulations of T, S4 and S5’, abstract. Journal of Symbolic Logic 48:900Google Scholar
  24. 24.
    Sato, M., ‘A Study of Kripke-type Models for Some Modal Logics by Gentzen’s Sequential Method’, Publications of the Research Institute for Mathematical Sciences, Kyoto University, vol. 13, 1977, pp. 381–468.Google Scholar
  25. 25.
    Shvarts, G. F., ‘Gentzen style systems for K45 and K45D’, in A. R. Meyer and M. A. Taitslin (eds.), Logic at Botik’89, LNCS 363, Springer, Berlin, 1989, pp. 245– 256.Google Scholar
  26. 26.
    Simpson, A., The Proof Theory and Semantics of Intuitionistic Modal Logic, PhD thesis, University of Edinburgh, 1994.Google Scholar
  27. 27.
    Stewart, C., and P. Stouppa ‘A systematic proof theory for several modal logics’ in R. Schmidt, I. Pratt-Hartmann, M. Reynolds and H. Wansing (eds.), Advances in Modal Logic, vol. 5, King’s College Publications, 2005, pp. 309–333.Google Scholar
  28. 28.
    Stouppa, P., The Design of Modal Proof Theories: the case of S5, Master Thesis, Technische Universität Dresden, 2004.Google Scholar
  29. 29.
    Strassburger, L., ‘A Local System for Linear Logic’, in M. Baaz and A. Voronkov (eds.), LPAR 2002, LNAI 2514, Springer-Verlag, 2002, pp. 388–402.Google Scholar
  30. 30.
    Wansing, H., Displaying Modal Logic, Kluwer Academic Publishers, 1998.Google Scholar
  31. 31.
    Wansing H. (1998). ‘Translation of Hypersequents into Display Sequents’. Logic Journal of the Interest Group in Pure and Applied Logics 6 (5):719–733Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Institut für Informatik und Angewandte MathematikUniversity of BernBernSwitzerland

Personalised recommendations