On an Intuitionistic Fuzzy Probability Theory

  • Čunderlíková Katarína
  • Riečan BeloslavEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 401)


We present some basic facts about a probability theory on IF-events. It is based on the Lukasiewicz operations and on the corresponding probability theory. We present a representation theorem originally published, as reported by Riečan (Soft Methodology and Random Information Systems, pp 243–248, [21]). We also show that the probability IF algebra can be embedded to a probability MV-algebra.


Lukasiewicz Operations Random Information Systems Soft Methodology Lukasiewicz Conjunction Kolmogorov Sense 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The support of the grant VEGA 1/0621/1 is kindly announced.


  1. 1.
    Atanassov, K.T.: Intuitionistic Fuzzy Sets: Theory and Applications. Studies in Fuzziness and Soft Computing. Physica-Verlag, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  2. 2.
    Atanassov, K.T., Riečan, B.: On two new types of probaility on IF-events. Notes IFS (2007)Google Scholar
  3. 3.
    Cignoli, L., D’Ottaviano, M., Mundici, D.: Algebraic Foundations of Many-valed Reasoning. Kluwer, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  4. 4.
    Foulis, D., Bennet, M.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1325–1346 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Georgescu, G.: Bosbach states on fuzzy structures. Soft Comput. 8, 217–230 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gerstenkorn, T., Manko, J.: Probabilities of intuitionistic fuzzy events. In: Hryniewicz, O. et al. (ed.) Issues on Intelligent Systems Paradigms, pp. 63–68 (2005)Google Scholar
  7. 7.
    Grzegorzewski, P., Mrówka, E.: Probabilitty on intuitionistic fuzzy events. In: Grzegoruewsko, P. et al. (ed.) Soft Methods in Probability, Statistics and Data Analysis, pp. 105–115 (2002)Google Scholar
  8. 8.
    Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca 44, 21–34 (1994)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Krachounov M.: Intuitionistic probability and intuitionistic fuzzy sets. In: El-Darzi et al. (eds.) First International Workshop on IFS, pp. 714–717 (2006)Google Scholar
  10. 10.
    Lendelová, K.: Convergence of IF-observables. In: Issues in the Representation and Processing of Uncertain and Imprecise Information—Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalzed Nets, and related TopicsGoogle Scholar
  11. 11.
    Lendelová, K.: IF-probability on MV-algebras. Notes IFS 11, 66–72 (2005)Google Scholar
  12. 12.
    Lendelová, K.: A note on invariant observables. Int. J. Theor. Physics 45, 915–923 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lendelová K.: Conditional IF-probability. In: Advances in Soft Computing Methods for Integrated Uncertainty Modeling, pp. 275–283 (2006)Google Scholar
  14. 14.
    Lendelová, K., Petrovičovā, J.: Representation of IF-probability for MV-algebras. Soft Comput. A 10, 564–566 (2006)CrossRefzbMATHGoogle Scholar
  15. 15.
    Lendelová, K., Riečan, B.: Weak law of large numbers for IF-events. In: de Baets, B. et al. (ed.) Current Issues in Data and Knowledge Ebgineering, pp. 309–314 (2004)Google Scholar
  16. 16.
    Lendelová, K., Riečan, B.: Probability in triangle and square. In: Proceedings of the Eleventh International Conference IPMU, pp. 977–982. Paris (2006)Google Scholar
  17. 17.
    Lendelová, K., Riečan, B.: Strong law of large numbers for IF-events. In: Proceedings of the Eleventh International Conference IPMU, pp. 2363–2366. Paris (2006)Google Scholar
  18. 18.
    Mazúreková, Riečan B.: A measure ectension theorem. Notes on IFS 12, 3–8 (2006)Google Scholar
  19. 19.
    Renčová, M., Riečan, B.: Probability on If sets: an elmentary approach. In: First International Workshop on IFS, Genralized Nets and Knowledge Systems, pp. 8–17 (2006)Google Scholar
  20. 20.
    Riečan, B.: A descriptive definition of probability on intutionistic fuzzy sets. In: Wagenecht, M., Hampet, R. (eds.) EUSFLAT’2003, pp. 263–266 (2003)Google Scholar
  21. 21.
    Riečan, B.: Representation of probabilities on IFS events. In: Dieaz, L. et al. (ed.) Soft Methodology and Random Information Systems, pp. 243–248 (2004)Google Scholar
  22. 22.
    Riečan, B.: On a problem of Radko Mesiar: general form of IF-probabilities. Fuzzy Sets Syst. 152, 1485–1490 (2006)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Riečan, B., Mundici, D.: In: Pap, E. (ed.) Probability in MV-algebras. Handbook of Measure Thoery. Elsevier, Heidelberg (2002)Google Scholar
  24. 24.
    Riečan, B., Neubrunn, T.: Integral, Measure, and Ordering. Kluwer, Dordrecht (1997)zbMATHGoogle Scholar
  25. 25.
    Riečan, B.: Probability theory on intuitionistic fuzzy events. A volume in Honour of Daniele Mundici‘s 60th Birthday Lecture Notes in Computer Science (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Faculty of Natural SciencesMatej Bel UniversityBanská BystricaSlovakia
  2. 2.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia

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