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Linear Kripke frames and Gödel logics

Published online by Cambridge University Press:  12 March 2014

Arnold Beckmann  and
Norbert Preining
Arnold Beckmann
Affiliation:
Department of Computer Science, University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, UK. E-mail: [email protected]
Norbert Preining
Affiliation:
Dipartimento di Scienze Matematiche, Università di Siena, 53100 Siena, Italy. E-mail: [email protected]
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Abstract

We investigate the relation between intermediate predicate logics based on countable linear Kripke frames with constant domains and Gödel logics. We show that for any such Kripke frame there is a Gödel logic which coincides with the logic defined by this Kripke frame on constant domains and vice versa. This allows us to transfer several recent results on Gödel logics to logics based on countable linear Kripke frames with constant domains: We obtain a complete characterisation of axiomatisability of logics based on countable linear Kripke frames with constant domains. Furthermore, we obtain that the total number of logics defined by countable linear Kripke frames on constant domains is countable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 1 , March 2007 , pp. 26 - 44
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Baaz, M., Infinite-valued Gödel logics with 0-1-projections and relativizations, Proceedings Gödel '96, Logic foundations of mathematics, computer science and physics—Kurt Gödel's legacy (Hájek, P., editor), Lecture Notes in Logic, vol. 6, Springer, 1996, pp. 23–33.Google Scholar
[2]Baaz, M., Leitsch, A., and Zach, R., Completeness of a first-order temporal logic with time-gaps, Theoretical Computer Science, vol. 160 (1996), no. 1–2, pp. 241–270.CrossRefGoogle Scholar
[3]Baaz, M., Leitsch, A., and Zach, R., Incompleteness of an infinite-valued first-order Gôdel logic and of some temporal logics of programs, Computer science logic (Börger, E., editor), Springer, Berlin, 1996, Selected papers from CSL '95, pp. 1–15.Google Scholar
[4]Baaz, M., Preining, N., and Zach, R., First-order Gödel logics, Annals of Pure and Applied Logic, (2006), Accepted. Preprint: http://arxiv.org/math.L0/0601147.Google Scholar
[5]Baaz, M. and Zach, R., Compact propositional Gödel logics, 28th International Symposium on Multiple-valued Logic. May 1998, Fukuoka, Japan. Proceedings, IEEE Press, Los Alamitos, 1998, pp. 108–113.Google Scholar
[6]Baaz, M. and Zach, R., Hypersequent and cut-elimination for intuitionistic fuzzy logic, Computer Science Logic, Proceedings of the CSL 2000 (Clote, P. G. and Schwichtenberg, H., editors), Lecture Notes in Computer Science 1862, Springer, 2000, pp. 178–201.Google Scholar
[7]Beckmann, A., Goldstern, M., and Preining, N., Continuous Fraïssé conjecture, Order. Submitted. Preprint: http://arxiv.org/math.L0/0411117.Google Scholar
[8]Birkhoff, G., Lattice theory, third ed., American Mathematical Society, 1967.Google Scholar
[9]Chagrov, A. and Zakharyaschev, M., Modal logic, Oxford Logic Guides, vol. 35, The Clarendon Press Oxford University Press, New York, 1997, Oxford Science Publications.CrossRefGoogle Scholar
[10]Davey, B. A., On the lattice of subvarieties, Houston Journal of Mathematics, vol. 5 (1979), pp. 183–192.Google Scholar
[11]Dummett, M., A propositional logic with denumerable matrix, this Journal, vol. 24 (1959), pp. 96–107.Google Scholar
[12]Esakia, L. L., Heyting algebra, I, Duality theory, Metsniereba, Tbilisi, 1985, in Russian.Google Scholar
[13]Fitting, M. C., Intuitionistic logic, model theory and forcing, Studies in Logic and the Foundation of Mathematics, North-Holland, Amsterdam, 1969.Google Scholar
[14]Fraïssé, R., Sur la comparaison des types d'ordres, Comptes Rendus de l’Académie des Sciences Paris, vol. 226 (1948), pp. 1330–1331.Google Scholar
[15]Gabbay, D. M., Semantical investigations in Heyting's intuitionistic logic, Synthese Library, vol. 148, D. Reidel Publishing Company, 1981.CrossRefGoogle Scholar
[16]Gödel, G., Zum Intuitionistischen Aussagenkalkül, Ergebnisse eines mathematischen Kolloquiums, vol. 4 (1933), pp. 34–38.Google Scholar
[17]Goldblatt, R., Mathematical modal logic: A view of its evolution, Journal of Applied Logic, vol. 1 (2003), no. 5–6, pp. 309–392.CrossRefGoogle Scholar
[18]Hájek, P., Metamathematics of fuzzy logic, Kluwer, 1998.CrossRefGoogle Scholar
[19]Horn, A., Logic with truth values in a linearly ordered Heyting algebra, this Journal, vol. 34 (1969), no. 3, pp. 395–409.Google Scholar
[20]Kechris, A. S., Classical descriptive set theory, Springer, 1995.CrossRefGoogle Scholar
[21]Kremer, P., On the complexity of propositional quantification in intuitionistic logic, this Journal, vol. 62 (1997), no. 2, pp. 529–544.Google Scholar
[22]Kripke, S. A., Semantical analysis of intuitionistic logic. I, Formal systems and recursive functions (Proceedings of the Eighth Logic Colloquium, Oxford, 1963), North-Holland, Amsterdam, 1965, pp. 92–130.Google Scholar
[23]Minari, P., Takano, M., and Ono, H., Intermediate predicate logics determined by ordinals, this Journal, vol. 55 (1990), no. 3, pp. 1099–1124.Google Scholar
[24]Moschovakis, Y. N., Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland, Amsterdam, 1980.Google Scholar
[25]Ono, H., Kripke models and intermediate logics, Publications. Research Institute of Mathematical Sciences, Kyoto University, vol. 6 (1971), pp. 461–476.Google Scholar
[26]Ono, H., A study of intermediate predicate logics, Publications. Research Institute of Mathematical Sciences, Kyoto University, vol. 8 (1972/1973), pp. 619–649.CrossRefGoogle Scholar
[27]Ono, H., On finite linear intermediate predicate logics, Studia Logica, vol. 47 (1988), no. 4, pp. 391–399.CrossRefGoogle Scholar
[28]Preining, N., Gödel logics and Cantor-Bendixon analysis, Proceedings of LPAR'2002 (Baaz, M. and Voronkov, A., editors), Lecture Notes in Artificial Intelligence 2514, Springer, 10 2002, pp. 327–336.Google Scholar
[29]Preining, N., Complete recursive axiomatizability of Gödel logics, Ph. D. thesis, Vienna University of Technology, Austria, 2003.Google Scholar
[30]Raney, G. N., Completely distributive complete lattices. Proceedings of the American Mathematical Society, vol. 3 (1952), pp. 677–680.CrossRefGoogle Scholar
[31]Scarpellini, B., Die Nichtaxiomatisierbarkeit des unendlichwertigen Prädikatenkalkülus von Łukasiewicz, this Journal, vol. 27 (1962), pp. 159–170.Google Scholar
[32]Skvortsov, D., On the superintuitionistic predicate logics of Kripke frames based on denumerable chains, Algebraic and Topological Methods in Non-classical Logics II, Barcelona, 15–18 June, 2005, 2005, pp. 73–74.Google Scholar
[33]Takano, M., Another proof of the strong completeness of the intuitionistic fuzzy logic, Tsukuba Journal of Mathematics, vol. 11 (1987), no. 1, pp. 101–105.CrossRefGoogle Scholar
[34]Takano, M., Ordered sets R and Q as bases of Kripke models. Studia Logica, vol. 46 (1987), pp. 137–148.CrossRefGoogle Scholar
[35]Takeuti, G. and Titani, T., Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, this Journal, vol. 49 (1984), pp. 851–866.Google Scholar
[36]Troelstra, A. S., Aspects of constructive mathematics, Handbook of mathematical logic (Barwise, J., editor), North Holland, 1977, pp. 973–1052.Google Scholar