Fundamentals of fuzzy probability theory

Stanley Gudder

U. Mohrhoff

Murat Ozcek

“Fuzzy probability theory” appeared as a smooth extension of classical probability theory in 1995. It was expected that it will be of great importance in quantum mechanics, but the theory doesn’t keep its development as it was expected. This necessitates revising some of its fundamental basic concepts. We argue that if quantum probability theory should have less constrained than classical probability theory as can be seen in the case of joint random variables, we surely need to weaken the definition of the intersection operation. In this paper, discuss the definition validity in quantum probability theory and to discuss the consistency of the given definitions with the whole theory and the possibility to have a more suitable definition.

Jarosław Pykacz

Bell-type inequalities, used in mathematical physics as a criterion to check whether a physical situation allows description in terms of classical (Kolmogorovian) or quantum probability calculus are applied to various fuzzy probability models. It occurs that the standard set of Bell-type inequalities does not allow to distinguish Kolmogorovian probabilities from fuzzy probabilities based on the most frequently used Zadeh intersection or probabilistic intersection, but it allows to distinguish all these models from fuzzy probability models based on Giles (Łukasiewicz) intersection. It is proved that if we use fuzzy set intersections pointwisely generated by Frank's fundamental triangular norms Ts(x,y), then the borderline between fuzzy probability models that can be distinguished from Kolmogorovian ones and these fuzzy probability models that cannot be distinguished is for .

Martin Plávala

Andrei Yu. Khrennikov