Upgrading Probability via Fractions of Events

Frič, Roman   ; Papčo, Martin

Communications in Mathematics, Tome 24 (2016) no. 1, pp. 29-41

Abstract (VO)

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 $\{0,1\}$-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.

Détail

Classification : 60A05, 60A86
Keywords: Classical probability theory; upgrading; quantum phenomenon; category theory; D-poset of fuzzy sets; Łukasiewicz tribe; observable; statistical map; duality

Bibliographie

[1] Adámek, J.: Theory of Mathematical Structures. 1983, Reidel, Dordrecht, | MR

[2] Bugajski, S.: Statistical maps I. Basic properties. Math. Slovaca, 51, 3, 2001, 321-342, | MR | Zbl

[3] Bugajski, S.: Statistical maps II. Operational random variables. Math. Slovaca, 51, 3, 2001, 343-361, | MR | Zbl

[4] Chovanec, F., Frič, R.: States as morphisms. Internat. J. Theoret. Phys., 49, 12, 2010, 3050-3060, | DOI | MR | Zbl

[5] Chovanec, F., Kôpka, F.: D-posets. Handbook of Quantum Logic and Quantum Structures: Quantum Structures, 2007, 367-428, Elsevier, Amsterdam, Edited by K. Engesser, D. M. Gabbay and D. Lehmann. | MR | Zbl

[6] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. 2000, Kluwer Academic Publ. and Ister Science, Dordrecht and Bratislava, | MR

[7] Frič, R.: Łukasiewicz tribes are absolutely sequentially closed bold algebras. Czechoslovak Math. J., 52, 2002, 861-874, | DOI | MR | Zbl

[8] Frič, R.: Remarks on statistical maps and fuzzy (operational) random variables. Tatra Mt. Math. Publ, 30, 2005, 21-34, | MR | Zbl

[9] Frič, R.: Extension of domains of states. Soft Comput., 13, 2009, 63-70, | DOI | Zbl

[10] Frič, R.: On D-posets of fuzzy sets. Math. Slovaca, 64, 2014, 545-554, | DOI | MR | Zbl

[11] Frič, R-, Papčo, M.: A categorical approach to probability. Studia Logica, 94, 2010, 215-230, | DOI | MR | Zbl

[12] Frič, R., Papčo, M.: Fuzzification of crisp domains. Kybernetika, 46, 2010, 1009-1024, | MR | Zbl

[13] Frič, R., Papčo, M.: On probability domains. Internat. J. Theoret. Phys., 49, 2010, 3092-3100, | DOI | MR | Zbl

[14] Frič, R., Papčo, M.: On probability domains II. Internat. J. Theoret. Phys., 50, 2011, 3778-3786, | DOI | MR | Zbl

[15] Frič, R., Papčo, M.: On probability domains III. Internat. J. Theoret. Phys., 54, 2015, 4237-4246, | DOI | MR | Zbl

[16] Goguen, J. A.: A categorical manifesto. Math. Struct. Comp. Sci., 1, 1991, 49-67, | DOI | MR | Zbl

[17] Gudder, S.: Fuzzy probability theory. Demonstratio Math., 31, 1998, 235-254, | MR | Zbl

[18] Kolmogorov, A. N.: Grundbegriffe der wahrscheinlichkeitsrechnung. 1933, Springer, Berlin, | MR | Zbl

[19] Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca, 44, 1994, 21-34, | MR

[20] Kuková, M., Navara, M.: What observables can be. Theory of Functions, Its Applications, and Related Questions, Transactions of the Mathematical Institute of N.I. Lobachevsky 46, 2013, 62-70, Kazan Federal University,

[21] Loève, M.: Probability theory. 1963, D. Van Nostrand, Inc., Princeton, New Jersey, | MR | Zbl

[22] Mesiar, R.: Fuzzy sets and probability theory. Tatra Mt. Math. Publ., 1, 1992, 105-123, | MR | Zbl

[23] Navara, M.: Triangular norms and measures of fuzzy sets. Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, 2005, 345-390, Elsevier, | MR | Zbl

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

[25] Navara, M.: Tribes revisited. 30th Linz Seminar on Fuzzy Set Theory: The Legacy of 30 Seminars, Where Do We Stand and Where Do We Go?, 2009, 81-84, Johannes Kepler University, Linz, Austria,

[26] Papčo, M.: On measurable spaces and measurable maps. Tatra Mt. Math. Publ., 28, 2004, 125-140, | MR | Zbl

[27] Papčo, M.: On fuzzy random variables: examples and generalizations. Tatra Mt. Math. Publ., 30, 2005, 175-185, | MR | Zbl

[28] Papčo, M.: On effect algebras. Soft Comput., 12, 2008, 373-379, | DOI | Zbl

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

[30] Riečan, B., Mundici, D.: Probability on MV-algebras. Handbook of Measure Theory, Vol. II, 2002, 869-910, North-Holland, Amsterdam, | MR | Zbl

[31] Riečan, B., Neubrunn, T.: Integral, Measure, and Ordering. 1997, Kluwer Acad. Publ., Dordrecht-Boston-London, | MR

[32] Zadeh, L. A.: Probability measures of fuzzy events. J. Math. Anal. Appl., 23, 1968, 421-427, | DOI | MR | Zbl

[33] Zadeh, L. A.: Fuzzy probabilities. Inform. Process. Manag., 19, 1984, 148-153, | Zbl