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Systematization of finite many-valued logics through the method of tableaux

  • Walter A. Carnielli (a1)

Abstract

This paper presents a unified treatment of the propositional and first-order many-valued logics through the method of tableaux. It is shown that several important results on the proof theory and model theory of those logics can be obtained in a general way.

We obtain, in this direction, abstract versions of the completeness theorem, model existence theorem (using a generalization of the classical analytic consistency properties), compactness theorem and Löwenheim-Skolem theorem.

The paper is completely self-contained and includes examples of application to particular many-valued formal systems.

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References

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[0]Borowik, P., On Gentzen's axiomatization of the reducts of many-valued logics, this Journal, vol. 48 (1983), pp. 12241225 (abstract).
[1]Carnielli, W. A., Sobre o método dos tableaux em lógicas polivalentes finitáias, Ph.D. thesis, University of Campinas, Brazil, 1982.
[2]Carnielli, W. A., The problem of quantificational completeness and the characterization of all perfect quantifiers in 3-valued logics, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik (to appear).
[3]D'Ottaviano, I. M. L., Sobre uma teoria de modelos trivalente, Ph.D. thesis, University of Campinas, Brazil, 1982.
[4]Fitting, M., Model-existence theorems for modal and intuitionistic logics, this Journal, vol. 38 (1973), pp. 613627.
[5]Rasiowa, H. and Sikorski, R., The mathematics of metamathematics, PWN, Warsaw, 1970.
[6]Rosenberg, I., The number of maximal closed classes in the set of functions over a finite domain, Journal of Combinatorial Theory Series A, vol. 14 (1973), pp. 17.
[7]Sette, A. M., On the propositional calculus p1, Mathematica Japonicae, vol. 16 (1973), pp. 173180.
[8]Smullyan, R. M., A unifying principle in quantification theory, Proceedings of the National Academy of Sciences of the United States of America, vol. 49 (1963), pp. 828832.
[9]Smullyan, R. M., First order logic. Springer-Verlag, Berlin, 1968.
[10]Surma, S. J., An algorithm for axiomatizing very finite logic, Computer science and multiple-valued logic: theory and applications (Rine, D. C., editor), North-Holland, Amsterdam, 1977, pp. 137143.

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Systematization of finite many-valued logics through the method of tableaux

  • Walter A. Carnielli (a1)

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