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Sufficient triangular norms in many-valued logics with standard negation

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Abstract

In many-valued logics with the unit interval as the set of truth values, from the standard negation and the product (or, more generally, from any strict Frank t-norm) all measurable logical functions can be derived, provided that also operations with countable arity are allowed. The question remained open whether there are other t-norms with this property or whether all strict t-norms possess this property. We give a full solution to this problem (in the case of strict t-norms), together with convenient sufficient conditions. We list several families of strict t-norms having this property and provide also counterexamples (the Hamacher product is one of them). Finally, we discuss the consequences of these results for the characterization of tribes based on strict t-norms.

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References

  1. Barbieri, G., Navara, M., Weber, H.: Characterization of T-measures. Soft Computing 8, 44–50 (2003)

    Google Scholar 

  2. Barbieri, G., Weber, H.: A representation theorem and a Lyapunov theorem for T s -measures: The solution of two problems of Butnariu and Klement. J. Math. Anal. Appl. 244, 408–424 (2000)

    Article  Google Scholar 

  3. Butnariu, D., Klement, E.P.: Triangular Norm-Based Measures and Games with Fuzzy Coalitions. (Kluwer Academic Publishers, Dordrecht 1993)

  4. Butnariu, D., Klement, E.P., Zafrany, S.: On triangular norm-based propositional fuzzy logics. Fuzzy Sets and Systems 69, 241–255 (1995)

    Article  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. Cignoli, R., D'Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Publishers, Dordrecht, 2000

  7. Cignoli, R., Torrens, A.: An algebraic analysis of product logic. Mult.-Valued Log. 5, 45–65 (2000)

    Google Scholar 

  8. Cintula, P.: About axiomatic systems of product fuzzy logics. Soft Computing 5, 243–244 (2001)

    Article  Google Scholar 

  9. Di Nola, A., Navara, M.: The σ-complete MV-algebras which have enough states. Submitted for publication

  10. Esteva, F., Godo, L., Hájek, P., Navara, M.: Residuated fuzzy logics with an involutive negation. Arch. Math. Logic 39, 103–124 (2000)

    Article  Google Scholar 

  11. Frank, M.J.: On the simultaneous associativity of F(x,y) and x+y-F(x,y). Aequationes Math. 19, 194–226 (1979)

    Article  Google Scholar 

  12. Gottwald, S.: A Treatise on Many-Valued Logic. (Research Studies Press, Baldock 2001)

  13. Hájek, P.: Metamathematics of Fuzzy Logic. (Kluwer Academic Publishers, Dordrecht 1998)

  14. Hájek, P., Godo, L., Esteva, F.: A complete many-valued logic with product-conjunction. Arch. Math. Logic 35, 191–208 (1996)

    Google Scholar 

  15. Halmos, P.R.: Measure Theory. (Springer, Berlin 1974)

  16. Klement, E.P.: Construction of fuzzy σ-algebras using triangular norms. J. Math. Anal. Appl. 85, 543–565 (1982)

    Article  Google Scholar 

  17. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. (Kluwer Academic Publishers, Dordrecht 2000)

  18. Klement, E.P., Navara, M.: A characterization of tribes with respect to the łukasiewicz t-norm. Czechoslovak Math. J. 47 (122), 689–700 (1997)

    Article  Google Scholar 

  19. Klement, E.P., Navara, M.: A survey on different triangular norm-based fuzzy logics. Fuzzy Sets and Systems 101, 241–251 (1999)

    Article  Google Scholar 

  20. Lowen, R.: Fuzzy Set Theory. Basic Concepts, Techniques and Bibliography. (Kluwer Academic Publishers, Dordrecht 1996)

  21. McNaughton, R.: A theorem about infinite-valued sentential logic. J. Symb. Logic 16, 1–13 (1951)

    Google Scholar 

  22. Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. U.S.A. 8, 535–537 (1942)

    Google Scholar 

  23. Mesiar, R., Navara, M.: T s -tribes and T s -measures. J. Math. Anal. Appl. 201, 91–102 (1996)

    Article  Google Scholar 

  24. Mesiar, R., Navara, M.: Diagonals of continuous triangular norms. Fuzzy Sets and Systems 104, 35–41 (1999)

    Article  Google Scholar 

  25. Mesiar, R., Novák, V.: Open problems from the 2nd International Conference on Fuzzy Sets Theory and Its Applications. Fuzzy Sets and Systems 81, 185–190 (1996)

    Article  MathSciNet  Google Scholar 

  26. Montagna, F.: An algebraic approach to propositional fuzzy logic. J. Logic Lang. Inf. 9, 91–124 (2000)

    Article  Google Scholar 

  27. Navara, M.: A characterization of triangular norm based tribes. Tatra Mt. Math. Publ. 3, 161–166 (1993)

    Google Scholar 

  28. Navara, M.: Nearly Frank t-norms and the characterization of T-measures. In: D. Butnariu, E.P. Klement, (eds.), Proceeedings of the 19th Linz Seminar on Fuzzy Set Theory (Linz, 1998) pp. 9–16

  29. Navara, M.: Characterization of measures based on strict triangular norms. J. Math. Anal. Appl. 236, 370–383 (1999)

    Article  Google Scholar 

  30. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. (North-Holland, New York 1983)

  31. Trillas, E.: Sobre funciones de negación en la teoría de conjuntas difusos. Stochastica 3, 47–60 (1979)

    Google Scholar 

  32. Zadeh, L.A.: Fuzzy sets. Inform. and Control 8, 338–353 (1965)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Haifa, 31 905, Haifa, Israel

    Dan Butnariu

  2. Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, 4040, Linz, Austria

    Erich Peter Klement

  3. Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, 81 368, Bratislava, Slovakia

    Radko Mesiar

  4. Center for Machine Perception, Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University, 166 27, Prague 6, Czech Republic

    Mirko Navara

  5. Institute of Information Theory and Automation, Czech Academy of Sciences, 182 08, Prague 8, Czech Republic

    Radko Mesiar

Authors
  1. Dan Butnariu
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  2. Erich Peter Klement
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  3. Radko Mesiar
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  4. Mirko Navara
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Correspondence to Dan Butnariu.

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Butnariu, D., Klement, E., Mesiar, R. et al. Sufficient triangular norms in many-valued logics with standard negation. Arch. Math. Logic 44, 829–849 (2005). https://doi.org/10.1007/s00153-004-0267-6

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  • DOI: https://doi.org/10.1007/s00153-004-0267-6

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