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Purely Syntactic Methods

  • Francesca PoggiolesiEmail author
Chapter
Part of the Trends in Logic book series (TREN, volume 32)

Abstract

In the early 1980s, the failures of the search for a sequent calculus for modal logic gave rise to the idea that the standard Gentzen calculus could only account for classical and intuitionistic logics and should therefore be enriched. Logicians thus started creating methods capable of generating extensions of the sequent calculus and hence suitable for providing modal logic, as well as other logics, with computational tools. These methods can be divided into two groups: the first group consists of methods that generate purely syntactic sequent calculi, while the second group include methods that extend the standard sequent calculus by adding explicit semantic elements. In this chapter we present and discuss the main calculi belonging to the first group.

Keywords

Modal Logic Structural Connective Logical Rule Sequent Calculus Frame Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 6.
    A. Avron. The method of hypersequents in the proof theory of propositional non-classical logic. In W. Hodges, M. Hyland, C. Steinhorn, and J. Strauss, editors, Logic: from Foundations to Applications, pp. 1–32. Oxford University Press, Oxford, 1996.Google Scholar
  2. 8.
    N. D. Belnap. Display logic. Journal of Philosophical Logic, 11:375–417, 1982.Google Scholar
  3. 9.
    N. D. Belnap. Linear logic displayed. Notre Dame Journal of Formal Logic, 31:14–25, 1990.CrossRefGoogle Scholar
  4. 10.
    N. D. Belnap. The display problem. In H. Wansing, editor, Proof Theory of Modal Logic, pp. 79–92. Kluwer Academic Publisher, Dordrecht, 1996.Google Scholar
  5. 12.
    S. Blamey and L. Humberstone. A perspective on modal sequent logic. Publications of the Research Institute for Mathematical Sciences, Kyoto University, 27:763–782, 1991.CrossRefGoogle Scholar
  6. 21.
    C. Cerrato. Modal sequents for normal modal logics. Mathematical Logic Quarterly, 39:231–240, 1993.CrossRefGoogle Scholar
  7. 22.
    C. Cerrato. Modal sequents. In H. Wansing, editor, Proof Theory of Modal Logic, pp. 141–166. Kluwer Academic Publisher, Dordrecht, 1996.Google Scholar
  8. 24.
    B. F. Chellas. Modal Logic. Cambridge University Press, Cambridge, 1980.Google Scholar
  9. 27.
    H. B. Curry. The elimination theorem when modality is present. Journal of Symbolic Logic, 17:249–265, 1952.CrossRefGoogle Scholar
  10. 28.
    H. B. Curry. Foundations of Mathematical Logic. Dover, New York, 1977.Google Scholar
  11. 31.
    K. Došen. Sequent-systems for modal logic. Journal of Symbolic Logic, 50:149–159, 1985.CrossRefGoogle Scholar
  12. 38.
    M. Fitting. Proof Methods for Modal and Intuitionistic Logics. Reidel, Dordrecht, 1983.Google Scholar
  13. 48.
    R. Goré. Substructural logics on display. Logic Journal of the IGPL, 6:669–694, 1998.CrossRefGoogle Scholar
  14. 52.
    S. Gottwald. Mehrwertige Logik. Akademie, Berlin, 1989.Google Scholar
  15. 53.
    A. Guglielmi. A system of interaction and structure. ACM Transactions on computational Logic, 8:1–64, 2007.CrossRefGoogle Scholar
  16. 63.
    A. Indrezejczak. Generalised sequent calculus for propositional modal logics. Logica Trianguli, 1:15–31, 1997.Google Scholar
  17. 64.
    A. Indrezejczak. Cut-free double sequent calculus for S5. Logic Journal of the IGPL, 6:505–516, 1998.CrossRefGoogle Scholar
  18. 65.
    A. Indrezejczak. Modal hybrid logic. Logic and Logical Philosophy, 16:147–257, 2007.Google Scholar
  19. 69.
    M. Kracht. Highway to the danger zone. Journal of Logic and Computation, 5:93–109, 1995.CrossRefGoogle Scholar
  20. 70.
    M. Kracht. Power and weakness of the modal display calculus. In H. Wansing, editor, Proof Theory of Modal Logic, pp. 93–121. Kluwer Academic Publisher, Dordrecht, 1996.Google Scholar
  21. 77.
    A. Masini. 2-sequents calculus: A proof theory of modalities. Annals of Pure and Applied Logic, 58:229–246, 1992.CrossRefGoogle Scholar
  22. 94.
    F. Paoli. Substructural Logics: a Primer. Kluwer Academic Publisher, Dordrecht-Boston-London, 2002.Google Scholar
  23. 115.
    G. Restall. Displaying and deciding substructural logics. I: Logics with contraposition. Journal of Philosophical Logic, 27:179–216, 1998.CrossRefGoogle Scholar
  24. 117.
    G. Restall. Proofnets for S5: sequents and circuits for modal logic. In C. Dimitracopoulos, L. Newelski, and D. Normann, editors, Logic Colloquium 2005, pp. 151–172. Cambridge University Press, Cambridge, 2007.CrossRefGoogle Scholar
  25. 118.
    G. Rousseau. Sequents in many-valued logic I. Fundamenta Mathematicae, 60:23–131, 1967.Google Scholar
  26. 120.
    G. Sambin, G. Battilotti, and C. Faggian. Basic logic: Reflection, symmetry, visibility. Journal of Symbolic Logic, 65:979–1013, 2000.CrossRefGoogle Scholar
  27. 123.
    M. Sato. 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, 13:381–468, 1977.CrossRefGoogle Scholar
  28. 127.
    M. Schroeter. Methoden zur axiomatisierung beliebiger aussagen und praedikatenkalkuele. Zeitschrift fuer mathematische Logik und Grundlagen der Mathematik, 1:214–251, 1955.Google Scholar
  29. 128.
    K. Sergerberg. An Essay in Classical Modal Logic. Filosofiska Studier (13), Uppsala, 1971.Google Scholar
  30. 131.
    R. M. Smullyan. First-Order Logic. Springer, Berlin, 1968.Google Scholar
  31. 132.
    C. Stewart and P. Stouppa. A systematic proof theory for several modal logics. In R. Schmidt, I. Pratt-Hartmann, M. Reynolds, and H. Wansing, editors, Advances in Modal Logic, Vol 5, pp. 309–333. College Publications, London, 2004.Google Scholar
  32. 133.
    P. Stouppa. The Design of Modal Proof Theories: the Case of S5. Master thesis, Technische Universitat Dresden, 2004.Google Scholar
  33. 145.
    H. Wansing. Sequent systems for normal modal propositional logics. Journal of Logic and Computation, 4:125–142, 1994.CrossRefGoogle Scholar
  34. 146.
    H. Wansing. Displaying as temporalizing. Sequent systems for subintuitionistic logics. In S. Hakama, editor, Logic, Language and Computation, pp. 159–178. Kluwer Academic Publisher, Dordrecht, 1997.Google Scholar
  35. 147.
    H. Wansing. Displaying Modal Logic. Kluwer Academic Publisher, Dordrecht, 1998.Google Scholar
  36. 149.
    H. Wansing. Sequent systems for modal logics. In Handbook of Philosophical Logic, Vol 8, pp. 61–145. Kluwer, Dordrecht, 2002.Google Scholar
  37. 150.
    J.J. Zeman. Modal logic. The Lewis-Modal Systems. Oxford University Press, Oxford, 1973.Google Scholar
  38. 34.
    A. G. Dragalin. Mathematical Intuitionism. Introduction to Proof Theory. American Mathematical Society, Providence, 1988.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.VRIJE UNIVERSITEIT BRUSSEL CLWF/LWBrusselsBelgium

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