# Article

MSC: 60A05, 60A86
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Keywords:
Classical probability theory; upgrading; quantum phenomenon; category theory; D-poset of fuzzy sets; Łukasiewicz tribe; observable; statistical map; duality
Summary:
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.
References:
[1] Adámek, J.: Theory of Mathematical Structures. 1983, Reidel, Dordrecht, MR 0735079
[2] Bugajski, S.: Statistical maps I. Basic properties. Math. Slovaca, 51, 3, 2001, 321-342, MR 1842320 | Zbl 1088.81021
[3] Bugajski, S.: Statistical maps II. Operational random variables. Math. Slovaca, 51, 3, 2001, 343-361, MR 1842321 | Zbl 1088.81022
[4] Chovanec, F., Frič, R.: States as morphisms. Internat. J. Theoret. Phys., 49, 12, 2010, 3050-3060, DOI 10.1007/s10773-009-0234-4 | MR 2738063 | Zbl 1204.81011
[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 2408886 | Zbl 1139.81005
[6] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. 2000, Kluwer Academic Publ. and Ister Science, Dordrecht and Bratislava, MR 1861369
[7] Frič, R.: Łukasiewicz tribes are absolutely sequentially closed bold algebras. Czechoslovak Math. J., 52, 2002, 861-874, DOI 10.1023/B:CMAJ.0000027239.28381.31 | MR 1940065 | Zbl 1016.28013
[8] Frič, R.: Remarks on statistical maps and fuzzy (operational) random variables. Tatra Mt. Math. Publ, 30, 2005, 21-34, MR 2190245 | Zbl 1150.60304
[9] Frič, R.: Extension of domains of states. Soft Comput., 13, 2009, 63-70, DOI 10.1007/s00500-008-0293-0 | Zbl 1166.28006
[10] Frič, R.: On D-posets of fuzzy sets. Math. Slovaca, 64, 2014, 545-554, DOI 10.2478/s12175-014-0224-8 | MR 3227755 | Zbl 1332.06005
[11] Frič, R-, Papčo, M.: A categorical approach to probability. Studia Logica, 94, 2010, 215-230, DOI 10.1007/s11225-010-9232-z | MR 2602573 | Zbl 1213.60021
[12] Frič, R., Papčo, M.: Fuzzification of crisp domains. Kybernetika, 46, 2010, 1009-1024, MR 2797424 | Zbl 1219.60006
[13] Frič, R., Papčo, M.: On probability domains. Internat. J. Theoret. Phys., 49, 2010, 3092-3100, DOI 10.1007/s10773-009-0162-3 | MR 2738067 | Zbl 1204.81012
[14] Frič, R., Papčo, M.: On probability domains II. Internat. J. Theoret. Phys., 50, 2011, 3778-3786, DOI 10.1007/s10773-011-0855-2 | MR 2860035 | Zbl 1254.60009
[15] Frič, R., Papčo, M.: On probability domains III. Internat. J. Theoret. Phys., 54, 2015, 4237-4246, DOI 10.1007/s10773-014-2471-4 | MR 3418298 | Zbl 1329.81095
[16] Goguen, J. A.: A categorical manifesto. Math. Struct. Comp. Sci., 1, 1991, 49-67, DOI 10.1017/S0960129500000050 | MR 1108804 | Zbl 0747.18001
[17] Gudder, S.: Fuzzy probability theory. Demonstratio Math., 31, 1998, 235-254, MR 1623780 | Zbl 0984.60001
[18] Kolmogorov, A. N.: Grundbegriffe der wahrscheinlichkeitsrechnung. 1933, Springer, Berlin, MR 0494348 | Zbl 0007.21601
[19] Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca, 44, 1994, 21-34, MR 1290269
[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 0203748 | Zbl 0108.14202
[22] Mesiar, R.: Fuzzy sets and probability theory. Tatra Mt. Math. Publ., 1, 1992, 105-123, MR 1230469 | Zbl 0790.60005
[23] Navara, M.: Triangular norms and measures of fuzzy sets. Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, 2005, 345-390, Elsevier, MR 2165242 | Zbl 1073.28015
[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 2086282 | Zbl 1112.06005
[27] Papčo, M.: On fuzzy random variables: examples and generalizations. Tatra Mt. Math. Publ., 30, 2005, 175-185, MR 2190258 | Zbl 1152.60302
[28] Papčo, M.: On effect algebras. Soft Comput., 12, 2008, 373-379, DOI 10.1007/s00500-007-0171-1 | Zbl 1127.06003
[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 10.2991/eusat.2013.10
[30] Riečan, B., Mundici, D.: Probability on MV-algebras. Handbook of Measure Theory, Vol. II, 2002, 869-910, North-Holland, Amsterdam, MR 1954631 | Zbl 1017.28002
[31] Riečan, B., Neubrunn, T.: Integral, Measure, and Ordering. 1997, Kluwer Acad. Publ., Dordrecht-Boston-London, MR 1489521
[32] Zadeh, L. A.: Probability measures of fuzzy events. J. Math. Anal. Appl., 23, 1968, 421-427, DOI 10.1016/0022-247X(68)90078-4 | MR 0230569 | Zbl 0174.49002
[33] Zadeh, L. A.: Fuzzy probabilities. Inform. Process. Manag., 19, 1984, 148-153, Zbl 0543.60007

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