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On theories and models in fuzzy predicate logics

Published online by Cambridge University Press:  12 March 2014

Petr Hájek
Affiliation:
Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod VodÁrenskou Věží 2, Prague, 182 07, Czech Republic.E-mail:[email protected]:http://www.cs.cas.cz/~hajek/
Petr Cintula
Affiliation:
Institute of Computer Science, Academy of Sciences of The Czech Republic, Pod VodÁrenskou Věží 2, Prague, 182 07, Czech Republic.E-mail:[email protected]:http://www.cs.cas.cz/~cintula/

Abstract

In the last few decades many formal systems of fuzzy logics have been developed. Since the main differences between fuzzy and classical logics lie at the propositional level, the fuzzy predicate logics have developed more slowly (compared to the propositional ones). In this text we aim to promote interest in fuzzy predicate logics by contributing to the model theory of fuzzy predicate logics. First, we generalize the completeness theorem, then we use it to get results on conservative extensions of theories and on witnessed models.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Baaz, Matthias, Infinite-valued Gödel logic with 0-1-projections and relativisations, Gödel'96: Logical foundations of mathematics, computer science, and physics (Hájek, Petr, editor), Lecture Notes in Logic, vol. 6, Springer-Verlag, 1996, pp. 2333.Google Scholar
[2]Cintula, Petr, Weakly implicative (fuzzy) logics I: Basic properties, Archive for Mathematical Logic, 2006, to appear.Google Scholar
[3]Cintula, Petr, Advances in the ŁΠ and ŁΠ½ logics, Archive for Mathematical Logic, vol. 42 (2003), no. 5, pp. 449468.CrossRefGoogle Scholar
[4]Dummett, Michael, A propositional calculus with denumerable matrix, this Journal, vol. 27 (1959), pp. 97106.Google Scholar
[5]Esteva, Francesc, Gispert, Joan, Godo, Lluís, and Noguera, Carles, Adding truth-constants to continuous t-norm based logics: Axiomatization and completeness results, submitted, 2005.Google Scholar
[6]Esteva, Francesc and Godo, Lluís, Monoidal t-norm based logic: Towards a logic for leftcontinuous t-norms, Fuzzy Sets and Systems, vol. 123 (2001), no. 3, pp. 271288.CrossRefGoogle Scholar
[7]Esteva, Francesc, Godo, Lluís, Hájek, Petr, and Navara, Mirko, Residuated fuzzy logics with an involutive negation, Archive for Mathematical Logic, vol. 39 (2000), no. 2, pp. 103124.CrossRefGoogle Scholar
[8]Esteva, Francesc, Godo, Lluís, and Montagna, Franco, The ŁΠ and ŁII½ logics: Two complete fuzzy systems joining Łukasiewicz and product logics, Archive for Mathematical Logic, vol. 40 (2001), no. 1, pp. 3967.CrossRefGoogle Scholar
[9]Esteva, Francesc, Godo, Lluís, and Noguera, Carles, On rational Weak Nilpotent Minimum logics, Journal of Multiple-Valued Logic and Soft Computing, vol. 12 (2006), pp. 932.Google Scholar
[10]Font, Josep Maria, Jansana, Ramon, and Pigozzi, Don, A survey of Abstract Algebraic Logic, Studia Logica, vol. 74 (2003), no. 1–2, pp. 1397.CrossRefGoogle Scholar
[11]Gottwald, Siegfried and Hájek, Petr, Triangular norm based mathematical fuzzy logic, Logical, algebraic, analytic and probabilistic aspects of triangular norms (Klement, Erich Petr and Mesiar, Radko, editors), Elsevier, Amsterdam, 2005, pp. 275300.CrossRefGoogle Scholar
[12]Hájek, Petr, Metamathematics of fuzzy logic, Trends in Logic, vol. 4, Kluwer, Dordercht, 1998.CrossRefGoogle Scholar
[13]Hájek, Petr, Fuzzy logic and arithmetical hierarchy III, Studia Logica, vol. 68 (2001), no. 1, pp. 129142.CrossRefGoogle Scholar
[14]Hájek, Petr, Fuzzy logic and arithmetical hierarchy IV, First-order logic revised (Hendricks, V., Neuhaus, F., Pedersen, S. A., Schffler, U., and Wansing, H., editors), Logos Verlag, Berlin, 2004, pp. 107115.Google Scholar
[15]Hájek, Petr, Arithmetical complexity of fuzzy predicate logics—a survey, Soft Computing, vol. 9 (2005), no. 12, pp. 935941.CrossRefGoogle Scholar
[16]Hájek, Petr, Making fuzzy description logic more general, Fuzzy Sets and Systems, vol. 154 (2005), no. 1, pp. 115.CrossRefGoogle Scholar
[17]Hájek, Petr and Cintula, Petr, Triangular norm based predicate fuzzy logics, Proceedings of Linz Seminar 2005, to appear.Google Scholar
[18]Hájek, Petr, Paris, Jeff, and Shepherdson, John C., The liar paradox and fuzzy logic, this Journal, vol. 65 (2000), no. 1, pp. 339346.Google Scholar
[19]Hájek, Petr, Paris, Jeff, and Shepherdson, John C., Rational Pavelka logic is a conservative extension of Łukasiewicz logic, this Journal, vol. 65 (2000), no. 2, pp. 669682.Google Scholar
[20]Hájek, Petr and Shepherdson, John C., A note on the notion of truth in fuzzy logic, Annals of Pure and Applied Logic, vol. 109 (2001), pp. 6569.CrossRefGoogle Scholar
[21]Horčík, Rostislav and Cintula, Petr, Product Łukasiewicz logic, Archive for Mathematical Logic, vol. 43 (2004), no. 4, pp. 477503.CrossRefGoogle Scholar
[22]Montagna, Franco, Three complexity problems in quantified fuzzy logic, Studia Logica, vol. 68 (2001), no. l,pp. 143152.CrossRefGoogle Scholar
[23]Rasiowa, Helena, An algebraic approach to non-classical logics, North-Holland, Amsterdam, 1974.Google Scholar
[24]Takeuti, Gaisi and Titani, Satoko, Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, this Journal, vol. 49 (1984), no. 3, pp. 851866.Google Scholar