Compositions Consistent with the Modus Ponens Property Used in Approximate Reasoning

  • Barbara PȩkalaEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 643)


In this paper it is investigated when some kinds of aggregation functions satisfy the Modus Ponens with respect to other aggregation function, or equivalently, when they are \(\mathcal {A}\)-conditionals. Moreover, some operation connected with \(\mathcal {A}\)-conditionals is examined and used to algorithm of approximate reasoning.


Interval-valued fuzzy relation Modus Ponens property Approximate reasoning 


  1. 1.
    Aczél, J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966)zbMATHGoogle Scholar
  2. 2.
    Asiain, M.J., Bustince, H., Bedregal, B., Takáć, Z., Baczyński, M., Paternain, D., Dimuro, G.: About the Use of Admissible Order for Defining Implication Operators, pp. 353–362. Springer International Publishing, Cham (2016)Google Scholar
  3. 3.
    Asiain, M.J., Bustince, H., Mesiar, R., Kolesarova, A., Takac, Z.: Negations with respect to admissible orders in the interval-valued fuzzy set theory. IEEE Trans. Fuzzy Syst. doi: 10.1109/TFUZZ.2017.2686372
  4. 4.
    Beliakov, G., Bustince, H., Calvo, T.: A practical Guide to Averaging Functions. In: Studies in Fuzziness and Soft Computing, vol. 329. Springer (2016)Google Scholar
  5. 5.
    Bentkowska, U., Bustince, H., Jurio, A., Pagola, M., Pȩkala, B.: Decision making with an interval-valued fuzzy preference relation and admissible orders. Appl. Soft Comput. 35, 792–801 (2015)CrossRefGoogle Scholar
  6. 6.
    Bustince, H., Burillo, P.: Interval-valued fuzzy relations in a set structures. J. Fuzzy Math. 4(4), 765–785 (1996)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bustince, H., Burillo, P.: Mathematical analysis of interval-valued fuzzy relations: application to approximate reasoning. Fuzzy Sets Syst. 113, 205–219 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bustince, H., Fernandez, J., Kolesárová, A., Mesiar, R.: Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets Syst. 220, 69–77 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bustince, H., Barrenechea, E., Pagola, M., Fernandez, J., Xu, Z., Bedregal, B., Montero, J., Hagras, H., Herrera, F., De Baets, B.: A historical account of types of fuzzy sets and their relationships. IEEE Trans. Fuzzy Syst. (2015). doi: 10.1109/TFUZZ.2015.2451692
  10. 10.
    Deschrijver, G.: Quasi-arithmetic means and OWA functions in interval-valued and Atanassov’s intuitionistic fuzzy set theory. In: EUSFLAT-LFA 2011, Aix-les-Bains, France, pp. 506–513 (2011)Google Scholar
  11. 11.
    Elkano, M., Sanz, J.A., Galar, M., Pȩkala, B., Bentkowska, U., Bustince, H.: Composition of interval-valued fuzzy relations using aggregation functions. Inform. Sci. doi: 10.1016/j.ins.2016.07.048
  12. 12.
    Komorníková, M., Mesiar, R.: Aggregation functions on bounded partially ordered sets and their classification. Fuzzy Sets Syst. 175, 48–56 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mas, M., Mayor, G., Torrens, J.: The modularity condition for uninorms and t-operators. Fuzzy Sets Syst. 126, 207–218 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mas, M., Monserrat, M., Ruiz-Aguilera, D., Torrens, J.: On generalization of the Modus Ponens: U-conditionality. In: Carvalho, J.P., et al. (eds.) IPMU 2016, Part I. CCIS, vol. 610, pp. 387–398 (2016)Google Scholar
  15. 15.
    Mas, M., Monserrat, M., Ruiz-Aguilera, D., Torrens, J.: RU and (U, N)-implications satisfying Modus Ponens. Int. J. Approximate Reasoning 73, 123–137 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    De Miguel, L., Bustince, H., Fernandez, J., Induráin, E., Kolesárová, A., Mesiar, R.: Construction of admissible linear orders for intervalvalued Atanassov intuitionistic fuzzy sets with an application to decision making. Inform. Fus. 27, 189–197 (2016)CrossRefGoogle Scholar
  17. 17.
    Pradera, A., Beliakov, G., Bustince, H., De Baets, B.: A review of the relationships between implication, negation and aggregation functions from the point of view of material implication. Inf. Sci. 329, 357–380 (2016)CrossRefGoogle Scholar
  18. 18.
    Sambuc, R.: Fonctions \(\phi \)-floues: Application á l’aide au diagnostic en pathologie thyroidienne. Ph.D. Thesis, Universit\(\acute{e}\) de Marseille, France (1975)Google Scholar
  19. 19.
    Saminger, S., Mesiar, R., Bodenhoffer, U.: Domination of aggregation operators and preservation of transitivity. Int. J. Unc. Fuzz. Knowl. Based Syst. 10, 11 (2002).
  20. 20.
    Zadeh, L.A.: A theory of approximate reasoning. In: Hayes, J.E., Michie, D., Mikulich, L.I. (eds.) Machine Intelligence 9, pp. 149–194 (1979)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Interdisciplinary Centre for Computational Modelling, Faculty of Mathematics and Natural SciencesUniversity of RzeszówRzeszówPoland

Personalised recommendations