On a Generalization of the Modus Ponens: U-conditionality

  • Margalida Mas
  • Miquel Monserrat
  • Daniel Ruiz-Aguilera
  • Joan TorrensEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 610)


In fuzzy logic, the Modus Ponens property for fuzzy implication functions is usually considered with respect to a continuous t-norm T and for this reason this property is also known under the name of T-conditionality. In this paper, the t-norm T is substituted by a uninorm U leading to the property of U-conditionality. The new property is studied in detail and it is shown that usual implications derived from t-norms and t-conorms do not satisfy it, but many solutions appear among those implications derived from uninorms. In particular, the case of residual implications derived from uninorms or RU-implications is investigated in detail for some classes of uninorms.


Fuzzy implication function Modus ponens t-norm Uninorm Natural negation 



This paper has been partially supported by the Spanish grant TIN2013-42795-P.


  1. 1.
    Aguiló, I., Suñer, J., Torrens, J.: A characterization of residual implications derived from left-continuous uninorms. Inf. Sci. 180, 3992–4005 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alsina, C., Trillas, E.: When \((S, N)\)-implications are \((T, T_1)\)-conditional functions? Fuzzy Sets Syst. 134, 305–310 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baczyński, M., Jayaram, B.: Fuzzy Implications. STUDFUZZ, vol. 231. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  4. 4.
    Baczyński, M., Jayaram, B.: (U, N)-implications and their characterizations. Fuzzy Sets Syst. 160, 2049–2062 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Czogala, E., Drewniak, J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets Syst. 12, 249–269 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De Baets, B.: Idempotent uninorms. Eur. J. Oper. Res. 118, 631–642 (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    De Baets, B., Fodor, J.C.: Residual operators of uninorms. Soft Comput. 3, 89–100 (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Fodor, J., De Baets, B.: A single-point characterization of representable uninorms. Fuzzy Sets Syst. 202, 89–99 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fodor, J.C., Yager, R.R., Rybalov, A.: Structure of uninorms. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 5, 411–427 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    Li, G., Liu, H.W., Fodor, J.: Single-point characterization of uninorms with nilpotent underlying t-norm and t-conorm. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 22, 591–604 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Martín, J., Mayor, G., Torrens, J.: On locally internal monotonic operators. Fuzzy Sets Syst. 137, 27–42 (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Mas, M., Massanet, S., Ruiz-Aguilera, D., Torrens, J.: A survey on the existing classes of uninorms. J. Intell. Fuzzy Syst. 29, 1021–1037 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mas, M., Monserrat, M., Ruiz-Aguilera, D., Torrens, J.: Residual implications derived from uninorms satisfying the Modus Ponens. In: IFSA-EUSFLAT-2015, pp. 233–240. Atlantis Press, Gijón (2015)Google Scholar
  15. 15.
    Mas, M., Monserrat, M., Ruiz-Aguilera, D., Torrens, J.: \(RU\) and \((U, N)\)-implications satisfying Modus Ponens. International Journal of Approximate Reasoning. In press. doi: 10.1016/j.ijar.2016.01.003
  16. 16.
    Mas, M., Monserrat, M., Torrens, J.: Two types of implications derived from uninorms. Fuzzy Sets Syst. 158, 2612–2626 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mas, M., Monserrat, M., Torrens, J.: Modus Ponens and Modus Tollens in discrete implications. Int. J. Approximate Reason. 49, 422–435 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mas, M., Monserrat, M., Torrens, J.: A characterization of \((U, N), RU, QL\) and \(D\)-implications derived from uninorms satisfying the law of importation. Fuzzy Sets Syst. 161, 1369–1387 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mas, M., Monserrat, M., Torrens, J., Trillas, E.: A survey on fuzzy implication functions. IEEE Trans. Fuzzy Syst. 15(6), 1107–1121 (2007)CrossRefGoogle Scholar
  20. 20.
    Massanet, S., Torrens, J.: On a new class of fuzzy implications: h-implications and generalizations. Inf. Sci. 181, 2111–2127 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Massanet, S., Torrens, J.: An overview of construction methods of fuzzy implications. In: Baczynski, M., Beliakov, G., Bustince, H., Pradera, A. (eds.) Adv. in Fuzzy Implication Functions. STUDFUZZ, vol. 300, pp. 1–30. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  22. 22.
    Ruiz, D., Torrens, J.: Residual implications and co-implications from idempotent uninorms. Kybernetika 40, 21–38 (2004)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ruiz-Aguilera, D., Torrens, J.: Distributivity of residual implications over conjunctive and disjunctive uninorms. Fuzzy Sets Syst. 158, 23–37 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ruiz-Aguilera, D., Torrens, J.: S- and R-implications from uninorms continuous in \(]0,1[^2\) and their distributivity over uninorms. Fuzzy Sets Syst. 160, 832–852 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ruiz-Aguilera, D., Torrens, J., De Baets, B., Fodor, J.: Some remarks on the characterization of idempotent uninorms. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. LNCS, vol. 6178, pp. 425–434. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  26. 26.
    Trillas, E., Alsina, C., Pradera, A.: On MPT-implication functions for fuzzy logic. Rev. Real Acad. Cienc. Ser. A Matemáticas (RACSAM) 98(1), 259–271 (2004)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Trillas, E., Alsina, C., Renedo, E., Pradera, A.: On contra-symmetry and MPT-conditionality in fuzzy logic. Int. J. Intell. Syst. 20, 313–326 (2005)CrossRefzbMATHGoogle Scholar
  28. 28.
    Trillas, E., Campo, C., Cubillo, S.: When QM-operators are implication functions and conditional fuzzy relations. Int. J. Intell. Syst. 15, 647–655 (2000)CrossRefzbMATHGoogle Scholar
  29. 29.
    Trillas, E., Valverde, L.: On Modus Ponens in fuzzy logic. In: 15th International Symposium on Multiple-Valued Logic, pp. 294–301. Kingston, Canada (1985)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Margalida Mas
    • 1
  • Miquel Monserrat
    • 1
  • Daniel Ruiz-Aguilera
    • 1
  • Joan Torrens
    • 1
    Email author
  1. 1.Department of Mathematics and Computer ScienceUniversity of the Balearic IslandsPalmaSpain

Personalised recommendations