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On Some Subclasses of the Fodor-Roubens Fuzzy Bi-implication

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7456)

Abstract

The paper deals with fuzzy versions of the classical bi-implication, that is, extensions of classical bi-implication to the canonical domain of mathematical fuzzy logics, the real-valued unit interval [0,1]. Our approach to fuzzy bi-implication may be summarized as follows: first, we recall a well-known approach to bi-implications, by Fodor and Roubens, via the direct axiomatization of the properties of the corresponding class of operators; next, we investigate a particular defining standard of bi-implication in terms of t-norms and r-implications. We study four prospective classes of bi-implications based on such defining standard, by varying the properties of its composing operators, and show that these classes collapse into precisely two increasingly weaker subclasses of the Fodor-Roubens bi-implication.

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References

  1. Baczyński, M., Jayaram, B.: Fuzzy Implications. STUDFUZZ. Springer (2008)

    Google Scholar 

  2. Bedregal, B.R.C., Cruz, A.P.: A characterization of classic-like fuzzy semantics. Logic Journal of the IGPL 16(4), 357–370 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Bodenhofer, U., De Baets, B., Fodor, J.: General Representation Theorems for Fuzzy Weak Orders. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds.) TARSKI II. LNCS (LNAI), vol. 4342, pp. 229–244. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  4. Bodenhofer, U., De Baets, B., Fodor, J.: A compendium of fuzzy weak orders: Representations and constructions. Fuzzy Sets and Systems 158(8), 811–829 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Bustince, H., Barrenechea, E., Pagola, M.: Restricted equivalence functions. Fuzzy Sets and Systems 157(17), 2333–2346 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Bustince, H., Burillo, P., Soria, F.: Automorphisms, negations and implication operators. Fuzzy Sets and Systems 134, 209–229 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Ćirić, M., Ignjatović, J., Bogdanović, S.: Fuzzy equivalence relations and their equivalence classes. Fuzzy Sets and Systems 158, 1295–1313 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Theory and Decision Library. Kluwer (1994)

    Google Scholar 

  9. Hájek, P.: Metamathematics of fuzzy logic. Trends in Logic. Kluwer (1998)

    Google Scholar 

  10. Kitainik, L.: Fuzzy decision procedures with binary relations: towards a unified theory. Theory and Decision Library: System Theory, Knowledge Engineering, and Problem Solving. Kluwer (1993)

    Google Scholar 

  11. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Trends in Logic: Studia Logica Library. Kluwer (2000)

    Google Scholar 

  12. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper I: basic analytical and algebraic properties. Fuzzy Sets and Systems 143(1), 5–26 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper II: general constructions and parameterized families. Fuzzy Sets and Systems 145(3), 411–438 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper III: continuous t-norms. Fuzzy Sets and Systems 145(3), 439–454 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Mesiar, R., Novák, V.: Operations fitting triangular-norm-based biresiduation. Fuzzy Sets and Systems 104, 77–84 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Moser, B.: On the T-transitivity of kernels. Fuzzy Sets and Systems 157(13), 1787–1796 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Novák, V., De Baets, B.: EQ-algebras. Fuzzy Sets and Systems 160(20), 2956–2978 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Recasens, J.: Indistinguishability Operators: Modelling fuzzy equalities and fuzzy equivalence relations. STUDFUZZ. Springer (2010)

    Google Scholar 

  19. Yager, R.: On the implication operator in fuzzy logic. Information Sciences 31(2), 141–164 (1983)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Zadeh, L.A.: Fuzzy sets. Information and Control 8(3), 338–353 (1965)

    CrossRef  MathSciNet  MATH  Google Scholar 

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Callejas, C., Marcos, J., Callejas Bedregal, B.R. (2012). On Some Subclasses of the Fodor-Roubens Fuzzy Bi-implication. In: Ong, L., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2012. Lecture Notes in Computer Science, vol 7456. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32621-9_15

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  • DOI: https://doi.org/10.1007/978-3-642-32621-9_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32620-2

  • Online ISBN: 978-3-642-32621-9

  • eBook Packages: Computer ScienceComputer Science (R0)