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Set Theory and Arithmetic in Fuzzy Logic

  • Libor BěhounekEmail author
  • Zuzana Haniková
Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 6)

Abstract

This chapter offers a review of Petr Hájek’s contributions to first-order axiomatic theories in fuzzy logic (in particular, ZF-style fuzzy set theories, arithmetic with a fuzzy truth predicate, and fuzzy set theory with unrestricted comprehension schema). Generalizations of Hájek’s results in these areas to MTL as the background logic are presented and discussed.

Keywords

Fuzzy set theory Fuzzy logic Naive comprehension Non-classical arithmetic 

Notes

Acknowledgments

The work was supported by grants No. P103/10/P234 (L. Běhounek) and P202/10/1826 (Z. Haniková) of the Czech Science Foundation.

References

  1. Běhounek, L. (2010). Extending Cantor-Łukasiewicz set theory with classes. In P. Cintula, E. P. Klement & L. N. Stout (Eds.), Lattice-valued logic and its applications: Abstracts of the 31st Linz seminar on fuzzy set theory (pp. 14–19). Linz.Google Scholar
  2. Běhounek, L., & Cintula, P. (2005). Fuzzy class theory. Fuzzy Sets and Systems 154,(1):34–55Google Scholar
  3. Běhounek, L., Cintula, P., & Hájek, P. (2011). Introduction to mathematical fuzzy logic. In P. Cintula, P. Hájek & C. Noguera (Eds.), Handbook of mathematical fuzzy logic (pp. 1–101). London: College Publications.Google Scholar
  4. Cantini, A. (2003). The undecidability of Gris̆in’s set theory. Studia Logica, 74, 345–368.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Chang, C. C. (1963). The axiom of comprehension in infinite valued logic. Mathematica Scandinavica, 13, 9–30.MathSciNetGoogle Scholar
  6. Fenstad, J. E. (1964). On the consistency of the axiom of comprehension in the Łukasiewicz infinite valued logic. Mathematica Scandinavica, 14, 65–74.MathSciNetGoogle Scholar
  7. Gottwald, S. (1976a). Untersuchungen zur mehrwertigen Mengenlehre I. Mathematica Nachrichten, 72, 297–303.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Gottwald, S. (1976b). Untersuchungen zur mehrwertigen Mengenlehre II. Mathematica Nachrichten, 74, 329–336.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gottwald, S. (1977). Untersuchungen zur mehrwertigen Mengenlehre III. Mathematica Nachrichten, 79, 207–217.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gottwald, S. (2006). Universes of fuzzy sets and axiomatizations of fuzzy set theory. Part I: Model-based and axiomatic approaches. Studia Logica, 82(2), 211–244.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Grayson, R. J. (1979). Heyting-valued models for intuitionistic set theory. In M. Fourman, C. Mulvey & D. S. Scott (Eds.), Application of sheaves. Lecture notes in computer science (Vol. 743, pp. 402–414). Berlin: Springer.Google Scholar
  12. Grishin, V. N. (1982). Predicate and set-theoretic calculi based on logic without contractions. Mathematics of the USSR-Izvestiya, 18, 41–59.Google Scholar
  13. Hájek, P. (1998). Metamathematics of fuzzy logic. Dordercht: Kluwer.Google Scholar
  14. Hájek, P. (2000). Function symbols in fuzzy logic. In Proceedings of the East-West Fuzzy Colloquium, pp. 2–8, Zittau/Görlitz, IPM.Google Scholar
  15. Hájek, P., Paris, J., & Shepherdson, J. C. (2000). The liar paradox and fuzzy logic. Journal of Symbolic Logic, 65(1), 339–346.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hájek, P. (2001). Fuzzy logic and arithmetical hierarchy III. Studia Logica, 68(1), 129–142.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Hájek, P., & Haniková, Z. (2003). A development of set theory in fuzzy logic. In M. Fitting & E. Orlowska (Eds.), Beyond two: Theory and applications of multiple-valued logic (pp. 273–285). Heidelberg: Physica-Verlag.Google Scholar
  18. Hájek, P. (2005). On arithmetic in Cantor-Łukasiewicz fuzzy set theory. Archive for Mathematical Logic, 44(6), 763–782.MathSciNetCrossRefzbMATHGoogle Scholar
  19. Hájek, P., & Cintula, P. (2006). On theories and models in fuzzy predicate logics. Journal of Symbolic Logic, 71(3), 863–880.MathSciNetCrossRefzbMATHGoogle Scholar
  20. Hájek, P. (2013a). On equality and natural numbers in Cantor-Łukasiewicz set theory. Logic Journal of the Interest Group of Pure and Applied Logic, 21(1), 91–100.Google Scholar
  21. Hájek, P. (2013b). Some remarks on Cantor-Łukasiewicz fuzzy set theory. Logic Journal of the Interest Group of Pure and Applied Logic, 21(2), 183–186.Google Scholar
  22. Hájek, P., & Haniková, Z. (2013). Interpreting lattice-valued set theory in fuzzy set theory. Logic Journal of the Interest Group of Pure and Applied Logic, 2(1), 77–90.Google Scholar
  23. Hájek, P., & Pudlák, P. (1993). Metamathematics of first-order arithmetic. Berlin: Springer.Google Scholar
  24. Haniková, Z. (2004). Mathematical and metamathematical properties of fuzzy logic. PhD thesis, Charles University in Prague, Faculty of Mathematics and Physics.Google Scholar
  25. Kaye, R. (1991). Models of Peano arithmetic. Oxford: Oxford University Press.Google Scholar
  26. Klaua, D. (1965). Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsb Deutsch Akad Wiss Berlin, 7, 859–867.MathSciNetzbMATHGoogle Scholar
  27. Klaua, D. (1966). Über einen zweiten Ansatz zur mehrwertigen Mengenlehre. Monatsb Deutsch Akad Wiss Berlin, 8, 782–802.MathSciNetzbMATHGoogle Scholar
  28. Klaua, D. (1967). Ein Ansatz zur mehrwertigen Mengenlehre. Mathematische Nachr, 33, 273–296.MathSciNetCrossRefzbMATHGoogle Scholar
  29. Montagna, F. (2001). Three complexity problems in quantified fuzzy logic. Studia Logica, 68(1), 143–152.MathSciNetCrossRefzbMATHGoogle Scholar
  30. Petersen, U. (2000). Logic without contraction as based on inclusion and unrestricted abstraction. Studia Logica, 64, 365–403.MathSciNetCrossRefzbMATHGoogle Scholar
  31. Powell, W. C. (1975). Extending Gödel’s negative interpretation to ZF. Journal of Symbolic Logic, 40(2), 221–229.MathSciNetCrossRefzbMATHGoogle Scholar
  32. Restall, G. (1995). Arithmetic and truth in Łukasiewicz’s infinitely valued logic. Logique et Analyse, 36, 25–38.MathSciNetGoogle Scholar
  33. Skolem, T. (1957). Bemerkungen zum Komprehensionsaxiom. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 3, 1–17.MathSciNetCrossRefzbMATHGoogle Scholar
  34. Skolem, T. (1960). Investigations on comprehension axiom without negation in the defining propositional functions. Notre Dame Journal of Formal Logic, 1, 13–22.MathSciNetCrossRefzbMATHGoogle Scholar
  35. Takeuti, G., & Titani, S. (1984). Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. Journal of Symbolic Logic, 49(3), 851–866.MathSciNetCrossRefzbMATHGoogle Scholar
  36. Takeuti, G., & Titani, S. (1992). Fuzzy logic and fuzzy set theory. Archive for Mathematical Logic, 32, 1–32.MathSciNetCrossRefzbMATHGoogle Scholar
  37. Terui, K. (2004). Light affine set theory: A naive set theory of polynomial time. Studia Logica, 77(1), 9–40.MathSciNetCrossRefzbMATHGoogle Scholar
  38. Titani, S. (1999). A lattice-valued set theory. Archive for Mathematical Logic, 38, 395–421.MathSciNetCrossRefzbMATHGoogle Scholar
  39. White, R. B. (1979). The consistency of the axiom of comprehension in the infinite-valued predicate logic of Łukasiewicz. Journal of Philosophical Logic, 8, 509–534.MathSciNetCrossRefzbMATHGoogle Scholar
  40. Yatabe, S. (2007). Distinguishing non-standard natural numbers in a set theory within Łukasiewicz logic. Archive for Mathematical Logic, 46, 281–287.MathSciNetCrossRefzbMATHGoogle Scholar
  41. Yatabe, S. (2009). Comprehension contradicts to the induction within Łukasiewicz predicate logic. Archive for Mathematical Logic, 48, 265–268.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Computer ScienceAcademy of Sciences of the Czech RepublicPragueCzech Republic
  2. 2.Institute for Research and Applications of Fuzzy ModelingNSC IT4Innovations, Division University of OstravaOstravaCzech Republic

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