Upgrading Probability via Fractions of Events

Frič, Roman ; Papčo, Martin

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

Résumé

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