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Interval-Valued Algebras and Fuzzy Logics

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35 Years of Fuzzy Set Theory

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 261))

Abstract

In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter.

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References

  1. Alcalde, C., Burusco, A., Fuentes-González, R.: A constructive method for the definition of interval-valued fuzzy implication operators. Fuzzy Sets and Systems 153, 211–227 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bustince, H.: Indicator of inclusion grade for interval-valued fuzzy sets. application to approximate reasoning based on interval-valued fuzzy sets. International Journal of Approximate Reasoning 23, 137–209 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chang, C.: Algebraic analysis of many valued logics. Transactions of the American Mathematical Society 88(2), 467–490 (1958)

    MATH  MathSciNet  Google Scholar 

  4. Chang, C.: A new proof of the completeness of the lukasiewicz axioms. Transactions of the American Mathematical Society 93(1), 74–80 (1959)

    MATH  MathSciNet  Google Scholar 

  5. Cignoli, R., Esteva, F., Godo, L., Torrens, A.: Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Computing 4, 106–112 (2000)

    Article  Google Scholar 

  6. Cintula, P., Esteva, F., Gispert, J., Godo, L., Montagna, F., Noguera, C.: Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Annals of Pure and Applied Logic 160(1), 53–81 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cornelis, C., Deschrijver, G., Kerre, E.: Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application. International Journal of Approximate Reasoning 35, 55–95 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cornelis, C., Deschrijver, G., Kerre, E.: Advances and challenges in interval-valued fuzzy logic. Fuzzy Sets and Systems 157(5), 622–627 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Deschrijver, G.: The Łukasiewicz t-norm in interval-valued fuzzy and intuitionistic fuzzy set theory. In: Atanassov, K.T., Kacprzyk, J., Krawczak, M., Szmidt, E. (eds.) Issues in the Representation and Processing of Uncertain and Imprecise Information. Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets, and Related Topics, Akademicka Oficyna Wydawnicza EXIT, pp. 83–101 (2005)

    Google Scholar 

  10. Deschrijver, G., Cornelis, C., Kerre, E.: On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Transactions on Fuzzy Systems 12, 45–61 (2004)

    Article  Google Scholar 

  11. Deschrijver, G., Kerre, E.: Classes of intuitionistic fuzzy t-norms satisfying the residuation principle. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11, 691–709 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dubois, D.: On ignorance and contradiction considered as truth values. Personal communication (2006)

    Google Scholar 

  13. Dubois, D., Prade, H.: Can we enforce full compositionality in uncertainty calculi? In: Proceedings of the 11th National Conference on Artificial Intelligence (AAAI 1994), Seattle, Washington, pp. 149–154 (1994)

    Google Scholar 

  14. Dummett, M.: A propositional calculus with denumerable matrix. The Journal of Symbolic Logic 24(2), 97–106 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  15. Esteva, F., Garcia-Calvés, P., Godo, L.: Enriched interval bilattices and partial many-valued logics: an approach to deal with graded truth and imprecision. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2(1), 37–54 (1994)

    Article  MathSciNet  Google Scholar 

  16. Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124, 271–288 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Esteva, F., Godo, L., Garcia-Cerdaña, A.: On the hierarchy of t-norm based residuated fuzzy logics. In: Fitting, M., Orlowska, E. (eds.) Beyond Two: Theory and Applications of Multiple Valued Logic, pp. 251–272. Physica-Verlag, Heidelberg (2003)

    Google Scholar 

  18. Font, J.: Beyond rasiowa’s algebraic approach to non-classical logics. Studia Logica. 82(2), 172–209 (2006)

    Article  MathSciNet  Google Scholar 

  19. Font, J., Jansana, R., Pigozzi, D.: A survey of abstract algebraic logic. Studia Logica. 74, 13–79 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Font, J., Rodriguez, A., Torrens, A.: Wajsberg algebras. Stochastica 8, 5–31 (1984)

    MATH  MathSciNet  Google Scholar 

  21. Gehrke, M., Walker, C., Walker, E.: Some comments on interval-valued fuzzy sets. International Journal of Intelligent Systems 11, 751–759 (1996)

    Article  MATH  Google Scholar 

  22. Gödel, K.: Zum intuitionistischen aussagenkalkül. In: Anzeiger der Akademie der Wissenschaften in Wien, pp. 65–66 (1932)

    Google Scholar 

  23. Gottwald, S.: Mathematical fuzzy logic as a tool for the treatment of vague information. Information Sciences 172, 41–71 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hájek, P.: Metamathematics of Fuzzy Logic. In: Trends in Logic—Studia Logica Library. Kluwer Academic Publishers, Dordrecht (1998)

    Google Scholar 

  25. Heyting, A.: Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der preuszischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 42–56, 57–71, 158–169 (1930)

    Google Scholar 

  26. Höhle, U.: Commutative, residuated l-monoids. In: Höhle, U., Klement, E. (eds.) Non-classical Logics and their Applications to Fuzzy Subsets: a Handbook of the Mathematical Foundations of Fuzzy Set Theory, pp. 53–106. Kluwer Academic Publishers, Dordrecht (1995)

    Google Scholar 

  27. Huntington, E.: Sets of independent postulates for the algebra of logic. Transactions of the American Mathematical Society 5, 288–309 (1904)

    MATH  MathSciNet  Google Scholar 

  28. Jenei, S.: A more efficient method for defining fuzzy connectives. Fuzzy Sets and Systems 90, 25–35 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  29. Jenei, S., Montagna, F.: A proof of standard completeness for Esteva and Godo’s logic MTL. Studia Logica. 70, 1–10 (2002)

    Article  MathSciNet  Google Scholar 

  30. Łukasiewicz, J., Tarski, A.: Untersuchungen über den Aussagenkalkül. Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, 1–21 (1930)

    Google Scholar 

  31. Ohnishi, M., Matsumoto, K.: Gentzen method in modal calculi, part I. Osaka Mathematical Journal 9, 113–130 (1957)

    MATH  MathSciNet  Google Scholar 

  32. Ohnishi, M., Matsumoto, K.: Gentzen method in modal calculi, part II. Osaka Mathematical Journal 11, 115–120 (1959)

    MATH  MathSciNet  Google Scholar 

  33. Rasiowa, H.: An algebraic approach to non-classical logics. Studies in Logic and the Foundations of Mathematics 78 (1974)

    Google Scholar 

  34. Shoenfield, J.: Mathematical Logic. Addison-Wesley, Reading (1967)

    MATH  Google Scholar 

  35. Turunen, E.: Mathematics behind Fuzzy Logic. Physica-Verlag, Heidelberg (1999)

    MATH  Google Scholar 

  36. Van Gasse, B.: Interval-valued algebras and fuzzy logics. Ph. D. Thesis, Ghent University, Belgium (2010)

    Google Scholar 

  37. Van Gasse, B., Cornelis, C., Deschrijver, G., Kerre, E.: Filters of residuated lattices and triangle algebras. Information Sciences 180(16), 3006–3020 (2010)

    Article  MATH  Google Scholar 

  38. Van Gasse, B., Cornelis, C., Deschrijver, G., Kerre, E.: The standard completeness of interval-valued monoidal t-norm based logic (submitted)

    Google Scholar 

  39. Van Gasse, B., Cornelis, C., Deschrijver, G., Kerre, E.: Triangle algebras: towards an axiomatization of interval-valued residuated lattices. In: Greco, S., Hata, Y., Hirano, S., Inuiguchi, M., Miyamoto, S., Nguyen, H.S., Słowiński, R. (eds.) RSCTC 2006. LNCS (LNAI), vol. 4259, pp. 117–126. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  40. Van Gasse, B., Cornelis, C., Deschrijver, G., Kerre, E.: A characterization of interval-valued residuated lattices. International Journal of Approximate Reasoning 49, 478–487 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  41. Van Gasse, B., Cornelis, C., Deschrijver, G., Kerre, E.: Triangle algebras: A formal logic approach to interval-valued residuated lattices. Fuzzy Sets and Systems 159, 1042–1060 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  42. Van Gasse, B., Cornelis, C., Deschrijver, G., Kerre, E.: The pseudo-linear semantics of interval-valued fuzzy logics. Information Sciences 179, 717–728 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  43. Zadeh, L.: The concept of a linguistic variable and its application to approximate reasoning - i. Information Sciences 8, 199–249 (1975)

    Article  MathSciNet  Google Scholar 

  44. Zalta, E.: Basic concepts in modal logic (1995), http://mally.stanford.edu/notes.pdf

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Van Gasse, B., Cornelis, C., Deschrijver, G. (2010). Interval-Valued Algebras and Fuzzy Logics. In: Cornelis, C., Deschrijver, G., Nachtegael, M., Schockaert, S., Shi, Y. (eds) 35 Years of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 261. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16629-7_4

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  • DOI: https://doi.org/10.1007/978-3-642-16629-7_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-16628-0

  • Online ISBN: 978-3-642-16629-7

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