Abstract
We present a general framework that allows to construct systematically analytic calculi for a large family of (propositional) many-valued logics — called projective logics — characterized by a special format of their semantics. All finite-valued logics as well as infinite-valued Gödel logic are projective. As a case-study, sequent of relations calculi for Gödel logics are derived. A comparison with some other analytic calculi is provided.
Research supported by EC Marie Curie fellowship HPMF-CT-1999-00301.
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References
Avellone, A., Ferrari, M. and Miglioli, P.: Duplication-free tableau calculi together with cut-free and contraction free sequent calculi for the interpolable propositional intermediate logics. Logic J. of the IGPL, 7, (4) (1999) 447–480
Avron, A.: Hypersequents, logical consequence and intermediate logics for con-currency. Annals of Mathematics and Artificial Intelligence, 4 (1991) 225–248
Avron, A.: The method of hypersequents in the proof theory of propositional nonclassical logics. In Logic: from Foundations to Applications, European Logic Colloquium. Oxford Science Publications. Clarendon Press. Oxford (1996) 1–32
Avron, A.: A tableau system for Gödel-Dummett logic based on a hypersequential calculus. In Automated Reasoning with Tableaux and Related Methods (Tableaux’2000), volume 1847 of Lectures Notes in Artificial Intelligence (2000) 98–112
Avron, A. and Konikowska, B.: Decomposition proof systems for Gödel logics. Studia Logica, 69 (2001) 197–219
Baaz, M.: Infinite-valued Gödel logics with 0–1-projections and relativizations. In Gödel 96. Kurt Gödel’s Legacy, volume 6 of LNL (1996) 23–33
Baaz, M., Ciabattoni, A. and Fermüller, C.: Cut-elimination in a sequentsof-relations calculus for Gödel logic. In International Symposium on Multiple Valued Logic (ISMVL’2001). IEEE (2001) 181–186
Baaz, M. and Fermliller, C.: Analytic calculi for projective logics. In Automated Reasoning with Tableaux and Related Methods (Tableaux’99), volume 1617 of Lectures Notes in Artificial Intelligence (1999) 36–51
Baaz, M., Fermüller, C. and Salzer, G.: Automated deduction for many-valued logic. In Handbook of Automated Reasoning. Elsevier (2001)
Baaz, M., Fermüller, C. and Zach, R • Elimination of cuts in first-order finite-valued logics. J. Inform. Process. Cybernet. EIK, 29, (6) (1994) 333–355
Bachmair, L. and Ganzinger, H.: Ordered chaining for total orderings. In CADE`94, volume 814 of Lecture Notes in Computer Science (1994) 435–450
Bachmair, L. and Ganzinger, H.: Ordered chaining calculi for first-order theories of transitive relations. J. of the ACM, 45, (6) (1998) 1007–1049
Ciabattoni, A. and Ferrari, M.: Hypersequent calculi for some intermediate logics with bounded Kripke models. J. of Logic and Computation, 2, (11) (2001) 283–294
Dummett, M.: A propositional logic with denumerable matrix. J. of Symbolic Logic, 24 (1959) 96–107
Dunn, J.M. and Meyer, R.K.: Algebraic completeness results for Dummett’s LC and its extensions. Z. Math. Logik Grundlagen Math, 17 (1971) 225–230
Dyckhoff, R.: A deterministic terminating sequent calculus for Gödel Dummett logic. Logic Journal of the IGPL, 7 (1999) 319–326
Gentzen, G.: Untersuchungen über das logische Schliessen I, II. Mathematische Zeitschrift, 39 (1934) and (1935) 176–210 and 405–431
Gödel, K.: Zum intuitionistischen Aussagenkalkül. Anz. Akad. Wiss. Wien, 69 (1932) 65–66
Hâjek, P.: Metamathematics of Fuzzy Logic. Kluwer (1998)
Hâjek, P., Godo, L. and Esteva, F.: A complete many-valued logic with product-conjunction. Archive for Mathematical Logic, 35 (1996) 191–208
Lukasiewicz, J.: Philosophische Bemerkungen zu mehrwertigen Systemen der Aussagenlogik. Comptes Rendus de la Societé des Science et de Lettres de Varsovie (1930) 51–77
Moisil, G.: Essais sur les logiques non chrysipiennes. Editions de l’Academie de la Republique Socialiste de Roumanie, Bucarest (1972)
Rousseau, G.: Sequents in many valued logic I. Fund. Math., 60 (1967) 23–33
Schütte, K.: Proof Theory. Springer, Berlin and New York (1977)
Sonobe, O.: A Gentzen-type formulation of some intermediate propositional logics. J. of Tsuda College, 7 (1975) 7–14
Takahashi, M.: Many-valued logics of extended Gentzen style I. Science Reports of the Tokyo Kyoiku Daigaku, 9 (1967) 95–116
Takeuti, G. and Titani, T.: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. J. of Symbolic Logic, 49 (1984) 851–866
Troelstra, A.S. and Schwichtenberg, H.: Basic Proof Theory. Cambridge University Press (1996)
Visser, A.: On the completeness principle: a study of provability in Heyting’s Arithmetic. Annals of Math. Logic, 22 (1982) 263–295
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Baaz, M., Ciabattoni, A., Fermüller, C.G. (2003). Sequent of Relations Calculi: A Framework for Analytic Deduction in Many-valued Logics. In: Fitting, M., Orłowska, E. (eds) Beyond Two: Theory and Applications of Multiple-Valued Logic. Studies in Fuzziness and Soft Computing, vol 114. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1769-0_6
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