Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avron, A., ‘Natural 3-valued logics — characterization and proof theory’, J. Symbolic Logic 56(1):276–294, 1991.
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).
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.
Borowik, P., ‘Multl-valued n-sequential propositional logic’ (Abstract), J. Symbolic Logic 52:309–310, 1985.
Carnielli, W. A., ‘Systematization of finite many-valued logics through the method of tableaux’, J. Symbolic Logic 52(2):473–493, 1987.
Carnielli, W. A., ‘On sequents and tableaux for many-valued logics’, J. Non-Classical Logic 8(1):59–76, 1991.
Chang, C.-L. and R. C.-T. Lee, Symbolic Logic and Mechanical Theorem Proving, Academic Press, London, 1973.
HÄhnle, R., Automated Deduction in Multiple-Valued Logics, Oxford University Press, Oxford, 1993.
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.
Leitsch, A., The Resolution Calculus, Springer, 1997.
Rousseau, G., ‘Sequents in many valued logic I’, Fund. Math. 60:23–33, 1967.
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.
SchrÖter, K., ‘Methoden zur Axiomatisierung beliebiger Aussagen-und Prädikatenkalküle’, Z. Math. Logik Grundlag. Math. 1:241–251, 1955.
Smullyan, R., First-order Logic, Springer, New York, 1968.
Sperner, E., ‘Ein Satz über Untermengen einer endlichen Menge’, Math. Z. 27:544–548, 1928.
Takahashi, M., ‘Many-valued logics of extended Gentzen style I’, Sci. Rep. Tokyo Kyoiku Daigaku Sect A 9(231):95–110, 1967.
Takeuti, G., Proof Theory, Studies in Logic 81, North-Holland, Amsterdam, 2nd edition, 1987.
Zach, R., Proof Theory of Finite-valued Logics, Diplomarbeit, Technische Universität Wien, Vienna, Austria, 1993.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Baaz, M., Fermüller, C.G., Salzer, G. et al. Labeled Calculi and Finite-Valued Logics. Studia Logica 61, 7–33 (1998). https://doi.org/10.1023/A:1005022012721
Issue Date:
DOI: https://doi.org/10.1023/A:1005022012721