Advertisement

Generalized tableau systems for intermediate propositional logics

  • Alessandro Avellone
  • Ugo Moscato
  • Pierangelo Miglioli
  • Mario Ornaghi
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1227)

Abstract

Given an intermediate propositional logic L (obtained by adding to intuitionistic logic INT a single axiom-scheme), a pseudo tableau system for L can be given starting from any intuitionistic tableau system and adding a rule which allows to insert in any line of a proof table suitable T-signed instances of the axiom-scheme. In this paper we study some sufficient conditions from which, given a well formed formula H, the search for these instances can be restricted to a suitable finite set of formulae related to H. We illustrate our techniques by means of some known logics, namely, the logic D of Dummett, the logics PR k (k≥1) of Nagata, the logics FIN m (m≥1), the logics G n (n≥1) of Gabbay and de Jongh, and the logic KP of Kreisel and Putnam

Keywords

intermediate propositional logics tableau systems duplications filtration techniques 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 57(3):795–807, 1992.Google Scholar
  2. 2.
    M. Ferrari and P. Miglioli. Counting the maximal intermediate constructive logics. Journal of Symbolic Logic, 58(4):1365–1401, 1993.Google Scholar
  3. 3.
    M. Ferrari and P. Miglioli. A method to single out maximal intermediate propositional logics with the disjunction property I. Annals of Pure and Applied Logic, 76:1–46, 1995.Google Scholar
  4. 4.
    M. Ferrari and P. Miglioli. A method to single out maximal intermediate propositional logics with the disjunction property II. Annals of Pure and Applied Logic, 76:117–168, 1995.Google Scholar
  5. 5.
    M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969.Google Scholar
  6. 6.
    D.M. Gabbay. The decidability of Kreisel-Putnam system. Journal of Symbolic Logic, 35:431–437, 1970.Google Scholar
  7. 7.
    D.M. Gabbay. Semantical Investigations in Heyting's Intuitionistic Logic. Reidel, Dordrecht, 1981.Google Scholar
  8. 8.
    D.M. Gabbay and D.H.J. de Jongh. A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property. Journal of Symbolic Logic, 39:67–78, 1974.Google Scholar
  9. 9.
    J. Hudelmaier. An o(n log n)-space decision procedure for intuitionistic propositional logic. Journal of Logic and Computation, 3(1):63–75, 1993.Google Scholar
  10. 10.
    G. Kreisel and H. Putnam. Eine Unableitbaxkeitsbeweismethode für den Intuitionistischen Aussagenkalkül. Archiv für Mathematische Logik und Grundlagenforschung, 3:74–78, 1957.Google Scholar
  11. 11.
    J. Lukasiewicz. On the intuitionistic theory of deduction. Indagationes Mathematicae, 14:69–75, 1952.Google Scholar
  12. 12.
    P. Miglioli. An infinite class of maximal intermediate propositional logics with the disjunction property. Archive for Mathematical Logic, 31(6):415–432, 1992.Google Scholar
  13. 13.
    P. Miglioli, U. Moscato, and M. Ornaghi. How to avoid duplications in a refutation system for intuitionistic logic and Kuroda logic. In K. Broda, M. D'Agostino, R. Goré, R. Johnson, and S. Reeves, editors, Proceedings of 3rd Workshop on Theorem Proving with Analytic Tableaux and Related Methods. Abingdon, U.K., May 4–6, 1994. Imperial College of Science, Technology and Medicine TR-94/5, 1994, pp. 169–187.Google Scholar
  14. 14.
    P. Miglioli, U. Moscato, and M. Ornaghi. An improved refutation system for intuitionistic predicate logic. Journal of Automated Reasoning, 12:361–373, 1994.Google Scholar
  15. 15.
    P. Miglioli, U. Moscato, and M. Ornaghi. Refutation systems for propositional modal logics. In P. Baumgartner, R. Hähnle, and J. Posegga, editors, Theorem Proving with Analytic Tableaux and Related Methods: 4th International Workshop, Schloss Rheinfels, St. Goar, Germany, volume 918 of LNAI, pages 95–105. Springer-Verlag, 1995.Google Scholar
  16. 16.
    P. Miglioli, U. Moscato, and M. Ornaghi. Avoiding duplications in tableau systems for intuitionistic and Kuroda logics. L.J. of the IGPL, 5(1):145–167, 1997.Google Scholar
  17. 17.
    P. Minari. Indagini semantiche sulle logiche intermedie proposizionali. Bibliopolis, 1989.Google Scholar
  18. 18.
    S. Nagata. A series of successive modifications of Peirce's rule. Proceedings of the Japan Academy, Mathematical Sciences, 42:859–861, 1966.Google Scholar
  19. 19.
    H. Ono. Kripke models and intermediate logics. Publications of the Research Institute for Mathematical Sciences, Kyoto University, 6:461–476, 1970.Google Scholar
  20. 20.
    K. Sasaki. The simple substitution property of the intermediate propositional logics on finite slices. Studia Logica, 52:41–62, 1993.Google Scholar
  21. 21.
    C.A. Smorynski. Applications of Kripke models. In A.S. Troelstra, editor, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, volume 344 of Lecture Notes in Mathematics. Springer-Verlag, 1973.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Alessandro Avellone
    • 1
  • Ugo Moscato
    • 1
  • Pierangelo Miglioli
    • 1
  • Mario Ornaghi
    • 1
  1. 1.Dipartimento di Scienze dell'InformazioneUniversità degli Studi di MilanoMilanoItaly

Personalised recommendations