Summary
We present an axiomatic approach to the probability theory on IF-sets = intuitionistic fuzzy sets = Atanassov sets (Intuitionistic Fuzzy Sets: Theory and Applications. Physica, New York, 1999). Starting with two constructive definitions (Issues in Intelligent Systems: Paradigms, EXIT, Warszawa 2005, pp. 63–58; Soft Methods in Probability, Statistics and Data Analysis, Physica, New York 2002, pp. 105–115) we consider a theory including not only the special cases but also the general form of the probabilities on IF-sets. Moreover an embedding of the theory to the probability theory on MV-algebras is given. This fact enables to use the well developed probability theory on MV-algebras for constructing probability theory on IF-events.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Atanassov, K.: Intuitionistic Fuzzy Sets: Theory and Applications. Physica, New York 1999.
Cignoli, R. L. O., D’Ottaviano I. M. L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht 2000.
Gerstenkorn, T., Manko, J.: Probabilities of intuitionistic fuzzy events. In: Issues in Intelligent Systems: Paradigms (O. Hryniewicz et al. eds.). EXIT, Warszawa 2005, pp. 63–58.
Grzegorzewski, P., Mrowka, E.: Probability of intuitionistic fuzzy events. In: Soft Methods in Probability, Statistics and Data Analysis (P. Grzegorzewski et al. eds.). Physica, New York 2002, pp. 105–115.
Krachounov, M.: Intuitionostic probability and intuitionistic fuzzy sets. In: First Intern. Workshop on IFS, Generalized Nets and Knowledge Engineering (E. El-Darzi, K. Atanassov, P. Chountas eds.). Univ. of Westminster, London 2006, pp. 18–24, 714–717.
Lendelová, K.: Measure theory on multivalued logics and its applications, Ph.D. thesis. M.Bel University Banská Bystrica 2005.
Lendelová, K.: Conditional probability on L-poset with product. Proc. Eleventh Int. Conf. IPMU, Paris 2006, pp. 946–951.
Lendelová, K., Michalíková, A., Riečan, B.: Representation of probability on triangle. Issues in Soft Computing – Decision and Operation research (O. Hryniewicz, J. Kaczprzyk, D. Kuchta eds.) Warszawa, EXIT 2005, pp. 235–242.
Montagna, F.: An algebraic approach to propositional fuzzy logic. J. Logic, Lang. Inf. 9, 2000, 91–124.
Renčová, M., Riečan, B.: Probability on IF-sets: an elementary approach. In: First Intern. Workshop on IFS, Generalized Nets and Knowledge Engineering (E. El-Darzi, K. Atanassov, P. Chountas eds.). Univ. of Westminster, London 2006, pp. 8–17.
Riečan, B.: On the product MV-algebras. Tatra Mt. Math. Publ. 16, 1999, 143–149.
Riečan, B.: A descriptive definition of the probability on intuitionistic fuzzy sets. In: Proc. EUSFLAT’2003 (Wagenecht, M. and Hampet, R. eds.), Zittau - Goerlitz Univ. Appl. Sci, Dordrecht 2003, pp. 263–266.
Riečan, B.: Representation of probabilities on IFS events. Advances in Soft Computing, Soft Methodology and Random Information Systems (M. Lopez - Diaz et al. eds.) Springer, Berlin Heidelberg New York 2004, pp. 243–246.
Riečan, B.: On the entropy on the Lukasiewicz square. Proc. Joint EUSFLAT - LFA 2005, Barcelona, pp. 330–333.
Riečan, B.: On the entropy of IF dynamical systems. In: Proceedings of the Fifth International workshop on IFS and Generalized Nets, Warsaw, Poland 2005, pp. 328–336.
Riečan, B.: On a problem of Radko Mesiar: general from of IF probabilities. Fuzzy Sets and Systems 152, 2006, 1485–1490.
Riečan, B.: Probability theory on IF-events. In: Algebraic and Proof-theoretic spects of Non-classical Logics. Papers in honour of daniele Mundici on the occasion of his 60th birthday. Lecture Notes in Computer Science, Vol. 4460, Springer, Berlin Heidelberg New York 2007.
Riečan, B.: On some contributions to quantum structures inspired by fuzzy sets. Kybernetika 2007, 43(4), pp. 481–490.
Riečan, B., Mundici, D.: Probability on MV-algebras. In: Handbook of measure Theory (E. Pap ed.) Elsevier, Amsterdam 2002, pp. 869–909.
Riečan, B., Neubrunn, T.: Integral, Measure, and Ordering. Kluwer, Dordrecht 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Riečan, B. (2008). On the Probability Theory on the Atanassov Sets. In: Chountas, P., Petrounias, I., Kacprzyk, J. (eds) Intelligent Techniques and Tools for Novel System Architectures. Studies in Computational Intelligence, vol 109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77623-9_23
Download citation
DOI: https://doi.org/10.1007/978-3-540-77623-9_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77621-5
Online ISBN: 978-3-540-77623-9
eBook Packages: EngineeringEngineering (R0)