Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Communications in Mathematics

Editor-in-Chief: Rossi, Olga

2 Issues per year


Mathematical Citation Quotient (MCQ) 2016: 0.28

Open Access
Online
ISSN
2336-1298
See all formats and pricing
More options …

Upgrading Probability via Fractions of Events

Roman Frič
  • Corresponding author
  • Roman Friè, Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01 Košice, Slovak Republic
  • Catholic University in Ružomberok, Hrabovská cesta 1, 034 01 Ružomberok, Slovak Republic
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Martin Papčo
  • Catholic University in Ružomberok, Hrabovská cesta 1, 034 01 Ružomberok, Slovak Republic
  • Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-08-20 | DOI: https://doi.org/10.1515/cm-2016-0004

Abstract

The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.

Keywords: Classical probability theory; upgrading; quantum phenomenon; category theory; D-poset of fuzzy sets; Łukasiewicz tribe; observable; statistical map; duality

References

  • [1] J. Adámek: Theory of Mathematical Structures. Reidel, Dordrecht (1983).Google Scholar

  • [2] S. Bugajski: Statistical maps I. Basic properties. Math. Slovaca 51 (3) (2001) 321-342.Google Scholar

  • [3] S. Bugajski: Statistical maps II. Operational random variables. Math. Slovaca 51 (3) (2001) 343-361.Google Scholar

  • [4] F. Chovanec, R. Friè: States as morphisms. Internat. J. Theoret. Phys. 49 (12) (2010) 3050-3060.Google Scholar

  • [5] F. Chovanec, F. Kôpka: D-posets. Handbook of Quantum Logic and Quantum Structures: Quantum Structures(2007) 367{428. Edited by K. Engesser, D. M. Gabbay and D. LehmannGoogle Scholar

  • [6] A. Dvurečenskij, S. Pulmannová: New Trends in Quantum Structures. Kluwer Academic Publ. and Ister Science, Dordrecht and Bratislava (2000).Google Scholar

  • [7] R. Friè: Lukasiewicz tribes are absolutely sequentially closed bold algebras. Czechoslovak Math. J. 52 (2002) 861-874.Google Scholar

  • [8] R. Frič: Remarks on statistical maps and fuzzy (operational) random variables. Tatra Mt. Math. Publ 30 (2005) 21-34.Google Scholar

  • [9] R. Frič: Extension of domains of states. Soft Comput. 13 (2009) 63{70.CrossrefWeb of ScienceGoogle Scholar

  • [10] R. Frič: On D-posets of fuzzy sets. Math. Slovaca 64 (2014) 545{554.CrossrefWeb of ScienceGoogle Scholar

  • [11] R- Frič, M. Papèo: A categorical approach to probability. Studia Logica 94 (2010) 215-230.Google Scholar

  • [12] R. Frič, M. Papèo: Fuzzi_cation of crisp domains. Kybernetika 46 (2010) 1009-1024.Google Scholar

  • [13] R. Frič, M. Papèo: On probability domains. Internat. J. Theoret. Phys. 49 (2010) 3092-3100.Google Scholar

  • [14] R. Frič, M. Papèo: On probability domains II. Internat. J. Theoret. Phys. 50 (2011) 3778-3786.Google Scholar

  • [15] R. Frič, M. Papèo: On probability domains III. Internat. J. Theoret. Phys. 54 (2015) 4237-4246.Google Scholar

  • [16] J. A. Goguen: A categorical manifesto. Math. Struct. Comp. Sci. 1 (1991) 49-67.CrossrefGoogle Scholar

  • [17] S. Gudder: Fuzzy probability theory. Demonstratio Math. 31 (1998) 235-254.Google Scholar

  • [18] A. N. Kolmogorov: Grundbegriffe der wahrscheinlichkeitsrechnung. Springer, Berlin (1933).Google Scholar

  • [19] F. Kôpka, F. Chovanec: D-posets. Math. Slovaca 44 (1994) 21-34.Google Scholar

  • [20] M. Kuková, M. Navara: What observables can be. In: R.K. Gubaidullina (ed.): Theory of Functions, Its Applications, and Related Questions, Transactions of the Mathematical Institute of N.I. Lobachevsky 46. Kazan Federal University (2013) 62-70.Google Scholar

  • [21] M. Loève: Probability theory. D. Van Nostrand, Inc., Princeton, New Jersey (1963).Google Scholar

  • [22] R. Mesiar: Fuzzy sets and probability theory. Tatra Mt. Math. Publ. 1 (1992) 105-123.Google Scholar

  • [23] M. Navara: Triangular norms and measures of fuzzy sets. In: E.P. Klement, R. Mesiar (eds.): Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms. Elsevier (2005) 345-390.Google Scholar

  • [24] M. Navara: Probability theory of fuzzy events. In: E. Montseny, P. Sobrevilla (eds.): Fourth Conference of the European Society for Fuzzy Logic and Technology and 11 Rencontres Francophones sur la Logique Floue et ses Applications. Universitat Polit ecnica de Catalunya, Barcelona, Spain (2005) 325-329.Google Scholar

  • [25] M. Navara: Tribes revisited. In: U. Bodenhofer, B. De Baets, E.P. Klement, S. Saminger-Platz (eds.): 30th Linz Seminar on Fuzzy Set Theory: The Legacy of 30 Seminars, Where Do We Stand and Where Do We Go?. Johannes Kepler University, Linz, Austria (2009) 81-84.Google Scholar

  • [26] M. Papčo: On measurable spaces and measurable maps. Tatra Mt. Math. Publ. 28 (2004) 125-140.Google Scholar

  • [27] M. Papčo: On fuzzy random variables: examples and generalizations. Tatra Mt. Math. Publ. 30 (2005) 175-185.Google Scholar

  • [28] M. Papčo: On effect algebras. Soft Comput. 12 (2008) 373{379.CrossrefGoogle Scholar

  • [29] M. Papčo: Fuzzification of probabilistic objects. In: 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013), doi:10.2991/eusat.2013.10. (2013) 67-71.Google Scholar

  • [30] B. Riečan, D. Mundici: Probability on MV-algebras. In: E.Pap (ed.): Handbook of Measure Theory, Vol. II. North-Holland, Amsterdam (2002) 869-910.Google Scholar

  • [31] B. Riečan, T. Neubrunn: Integral, Measure, and Ordering. Kluwer Acad. Publ., Dordrecht-Boston-London (1997).Google Scholar

  • [32] L. A. Zadeh: Probability measures of fuzzy events. J. Math. Anal. Appl. 23 (1968) 421-427.CrossrefGoogle Scholar

  • [33] L. A. Zadeh: Fuzzy probabilities. Inform. Process. Manag. 19 (1984) 148-153.Google Scholar

About the article

Received: 2016-06-29

Accepted: 2016-07-17

Published Online: 2016-08-20

Published in Print: 2016-08-01


Citation Information: Communications in Mathematics, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2016-0004.

Export Citation

© 2016. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Roman Frič and Martin Papčo
Fuzzy Sets and Systems, 2017
[2]
Roman Frič and Martin Papčo
International Journal of Theoretical Physics, 2017
[3]
Peter Eliaš and Roman Frič
International Journal of Theoretical Physics, 2017

Comments (0)

Please log in or register to comment.
Log in