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Labeled sequent calculi for modal logics and implicit contractions

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.

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References

  1. Brünnler K.: Deep sequent systems for modal logic. Arch. Math. Log. 48, 551–577 (2009)

    Article  MATH  Google Scholar 

  2. Brünnler, K.: Nested Sequents. Habilitation Thesis, Bern (2010)

  3. Fitting M.C.: Tableau methods of proof for modal logics. Notre Dame J. Formal Log. 13, 237–247 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  4. Fitting M.C.: Proof Methods for Modal and Intuitionistic Logics. Reidel, Dordrecht (1983)

    Book  MATH  Google Scholar 

  5. 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)

  6. Fitting M.C.: Prefixed tableaus and nested sequents. Ann. Pure Appl. Log. 163, 291–313 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  7. Gabbay D.: Labelled Deductive Systems: Volume 1, Foundations. Oxford University Press, Oxford (1996)

    Google Scholar 

  8. 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)

    Chapter  Google Scholar 

  9. Hill B., Poggiolesi F.: A contraction-free and cut-free sequent calculus for propositional dynamic logic. Stud. Log. 94, 47–72 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  10. 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

  11. 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)

    Google Scholar 

  12. Negri S.: Contraction-free sequent calculi for geometric theories with an application to Barr’s theorem. Arch. Math. Log. 42, 389–401 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  13. Negri S.: Proof analysis in modal logic. J. Philos. Log. 34, 507–544 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  14. 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)

    Google Scholar 

  15. Negri S., Dyckhoff R.: Proof analysis in intermediate logics. Arch. Math. Log. 51, 71–92 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  16. Poggiolesi, F.: Sequent Calculi for Modal Logic. PhD Thesis, Firenze/Paris (2008)

  17. Poggiolesi F.: Gentzen calculi for modal propositional logics. Springer, Berlin (2011)

    Book  Google Scholar 

  18. Viganò L.: Labelled Non-classical Logics. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  19. Wansing H.: Displaying Modal Logic. Kluwer, Dordrecht (1998)

    Book  MATH  Google Scholar 

  20. 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)

    Chapter  Google Scholar 

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Correspondence to Pierluigi Minari.

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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

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  • DOI: https://doi.org/10.1007/s00153-013-0350-y

Keywords

  • Labeled sequent calculi
  • Modal logics
  • Kripke semantics
  • Contraction rule

Mathematics Subject Classification (2000)

  • 03F03
  • 03B45