Bell-type inequalities in fuzzy probability calculus. (English) Zbl 1113.03344

Summary: 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 Kolmogorovian probabilities to be distinguished from fuzzy probabilities based on the most frequently used Zadeh intersection or probabilistic intersection, but it allows all these models to be distinguished 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 \(T_{s}(x,y)\), then the borderline between fuzzy probability models that can be distinguished from Kolmogorovian ones and those fuzzy probability models that cannot be distinguished is for \( s=9 +4\sqrt 5\).


03E72 Theory of fuzzy sets, etc.
81P15 Quantum measurement theory, state operations, state preparations
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