On a characterization of classical and nonclassical probabilities

Abstract

In this paper a comparison of various approaches to the problem of characterization of probabilities is presented. The notion of multidimensional probability or S-probability is introduced and it is shown that this notion can be profitably used for characterization of probability systems. The main results are given in Theorems 3 and 4. In Theorem 3 the conditions are considered under which a system of S-probabilities admits a model on a generalized event space, while in Theorem 4 the conditions under which the system admits a classical model are considered. An interpretation for S-probabilities is discussed and it is shown that Theorems 3 and 4 allow one to derive the logical structure of events from the observed probabilities.

On Bounded Posets Arising from Quantum Mechanical Measurements

“…An algebra of S-probabilities (cf. [1] and [10]) is a subset P of [0, 1] S satisfying (P1), (P2) and (P4):…”

“…These structures have been intensely studied, in particular from the point of quantum structures (cf. [1][2][3][4][5][6][7][8][9][10]). …”

“…Helmut Länger helmut.laenger@tuwien.ac.at 1 Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8/104, 1040 Vienna, Austria Let S be set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state s ∈ S. Taking into account p(s) for all s ∈ S we obtain a function from S to [0, 1], which is called a multidimensional probability or, more precisely, an S-probability, or sometimes, more generally, a numerical event (cf. [1,2] and [10]).…”

“…[1][2][3][4] and [6][7][8][9][10]) and there are also some results about a generalization of these structures (cf. [5]), namely so-called generalized fields of events.…”

“…A classical physical system can be identified by the fact that the poset of S-probabilities is a Boolean algebra (cf. [1]). …”