Studia Logica

, Volume 61, Issue 1, pp 7–33 | Cite as

Labeled Calculi and Finite-Valued Logics

  • Matthias Baaz
  • Christian G. Fermüller
  • Gernot Salzer
  • Richard Zach


A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite-valued logic if the labels are interpreted as sets of truth values (sets-as-signs). Furthermore, it is shown that any finite-valued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic in the number of truth values, and it is shown that this bound is tight.

finite-valued logic labeled calculus signed formula sets-as-signs 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Avron, A., ‘Natural 3-valued logics — characterization and proof theory’, J. Symbolic Logic 56(1):276–294, 1991.Google Scholar
  2. [2]
    Baaz, M., C. G. FermÜller, and R. Zach, ‘Dual systems of sequents and tableaux for many-valued logics’, Bull. EATCS 51:192–197, 1993 (paper read at 2nd Workshop on Tableau-based Deduction, Marseille, April 1993).Google Scholar
  3. [3]
    Baaz, M., C. G. FermÜller, and R. Zach, ‘Elimination of cuts in first-order finite-valued logics’, J. Inform. Process. Cybernet. EIK 29(6):333–355, 1994.Google Scholar
  4. [4]
    Borowik, P., ‘Multl-valued n-sequential propositional logic’ (Abstract), J. Symbolic Logic 52:309–310, 1985.Google Scholar
  5. [5]
    Carnielli, W. A., ‘Systematization of finite many-valued logics through the method of tableaux’, J. Symbolic Logic 52(2):473–493, 1987.Google Scholar
  6. [6]
    Carnielli, W. A., ‘On sequents and tableaux for many-valued logics’, J. Non-Classical Logic 8(1):59–76, 1991.Google Scholar
  7. [7]
    Chang, C.-L. and R. C.-T. Lee, Symbolic Logic and Mechanical Theorem Proving, Academic Press, London, 1973.Google Scholar
  8. [8]
    HÄhnle, R., Automated Deduction in Multiple-Valued Logics, Oxford University Press, Oxford, 1993.Google Scholar
  9. [9]
    HÄhnle, R., ‘Commodious axiomatization of quantifiers in multiple-valued logic’, in 26th Int. Symp. on Multiple-Valued Logics, Santiago de Compostela, Spain, p. 118–123, IEEE Press, Los Alamitos, May 1996.Google Scholar
  10. [10]
    Leitsch, A., The Resolution Calculus, Springer, 1997.Google Scholar
  11. [11]
    Rousseau, G., ‘Sequents in many valued logic I’, Fund. Math. 60:23–33, 1967.Google Scholar
  12. [12]
    Salzer, G., ‘Optimal axiomatizations for multiple-valued operators and quantifiers based on semi-lattices’, in 13th Int. Conf. on Automated Deduction (CADE'96), LNCS (LNAI) 1104, p. 688–702, Springer, 1996.Google Scholar
  13. [13]
    SchrÖter, K., ‘Methoden zur Axiomatisierung beliebiger Aussagen-und Prädikatenkalküle’, Z. Math. Logik Grundlag. Math. 1:241–251, 1955.Google Scholar
  14. [14]
    Smullyan, R., First-order Logic, Springer, New York, 1968.Google Scholar
  15. [15]
    Sperner, E., ‘Ein Satz über Untermengen einer endlichen Menge’, Math. Z. 27:544–548, 1928.Google Scholar
  16. [16]
    Takahashi, M., ‘Many-valued logics of extended Gentzen style I’, Sci. Rep. Tokyo Kyoiku Daigaku Sect A 9(231):95–110, 1967.Google Scholar
  17. [17]
    Takeuti, G., Proof Theory, Studies in Logic 81, North-Holland, Amsterdam, 2nd edition, 1987.Google Scholar
  18. [18]
    Zach, R., Proof Theory of Finite-valued Logics, Diplomarbeit, Technische Universität Wien, Vienna, Austria, 1993.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Matthias Baaz
    • 1
  • Christian G. Fermüller
    • 1
  • Gernot Salzer
    • 1
  • Richard Zach
    • 1
  1. 1.Institut für Algebra und Diskrete MathematikTechnische Universität WienViennaAustria

Personalised recommendations