Combining many-valued and intuitionistic tableaux

  • Matthias Baaz
  • Christian G. Fermüller
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1071)

Abstract

We combine a tableau based calculus for intuitionistic logic with the system of tableau calculi for all finite-valued propositional logics. It is shown that these new calculi correspond to a family of logics that arise if we generalize Kripke models for intuitionistic logic to many-valued evaluations. We thus obtain a unique “intuitionistic counterpart” for each finite-valued logic, if one truth value is distinguished in a certain way. We also show that these new logics themselves are not finite-valued, except for trivial cases.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baaz, M., C. G. Fermüller, and R. Zach. [1993] Dual systems of sequents and tableaux for many-valued logics. Bull. EATCS, 51, 192–197.Google Scholar
  2. Beth, E. [1959] The Foundations of Mathematics. North-Holland Publishing Co.Google Scholar
  3. Carnielli, W. A. [1987] Systematization of finite many-valued logics through the method of tableaux. J. Symbolic Logic, 52(2), 473–493.Google Scholar
  4. Carnielli, W. A. [1991] On sequents and tableaux for many-valued logics. J. Non-Classical Logic, 8(1), 59–76.Google Scholar
  5. Fitting, M. [1983] Proof Methods Modal and Intuitionistic Logics, Synthese Library 169. Synthese Library. D. Reidel Publishing Company, Dordrecht, Boston, Lancaster.Google Scholar
  6. Gentzen, G. [1934] Untersuchungen über das logische Schließen I–II. Math. Z., 39, 176–210, 405–431.Google Scholar
  7. Gödel, K. [1932] Zum intuitionistischen Aussagenkalkül. Anzeiger der Akademie der Wissenschaften in Wien, 69, 65–66.Google Scholar
  8. Hähnle, R. [1991] Uniform notation of tableaux rules for multiple-valued logics. In Proc. IEEE International Symposium on Multiple-valued Logic, pp. 238–245.Google Scholar
  9. Hähnle, R. [1993] Automated Proof Search in Multiple-Valued Logic. Oxford University Press, Oxford.Google Scholar
  10. Kripke, S. [1963] Semantical analysis of intuitionistic logics. In Formal Systems and Recursive Functions, J. Crossley and M. Dummet, editors, pp. 92–129, North-Holland, Amsterdam.Google Scholar
  11. Miglioli, P., U. Moscato, and M. Ornaghi. [1994a] How to avoid duplications in refutation systems for intuitionistic logic and Kuroda logic. In Theorem Proving with Analytic Tableaux and Related Methods, 3rd intl. Workshop, TABLEAUX 94, Abington, UK, K. Broda, M. D'Agostino, R. Gore, R. Johnson, and S. Reeves, editors, Imperial College of Science, Technology and Medicine TR-94/5, Imperial College of Science, Technology and Medicine, pp. 169–187.Google Scholar
  12. Miglioli, P., U. Moscato, and M. Ornaghi. [1994b] An improved refutation system for intuitionistic predicate logic. Journal of Automated Reasoning, 13, 361–373.Google Scholar
  13. Miglioli, P., U. Moscato, and M. Ornaghi. [1995] Refutation systems for propositional modal logics. In Theorem Proving with Analytic Tableaux and Related Methods, 4th intl. Workshop, TABLEAUX 95, Schloss Rheinfels, P. Baumgartner, R. Hähnle, and J. Posegga, editors, LNAI 918, pp. 95–105, Springer, Berlin.Google Scholar
  14. Takeuti, G. [1987] Proof Theory. Studies in Logic 81. North-Holland, Amsterdam, 2nd edition.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Matthias Baaz
    • 1
  • Christian G. Fermüller
    • 2
  1. 1.Inst. f. Algebra u. Diskrete MathematikTechnische Universität WienVienna
  2. 2.C.S.L.I.Stanford UniversityStanford

Personalised recommendations