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Communications in Mathematics

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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
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  • 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
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Published Online: 2016-08-20 | DOI: https://doi.org/10.1515/cm-2016-0004


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


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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, Volume 24, Issue 1, Pages 29–41, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2016-0004.

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

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