Kolmogorovian versus Non-Kolmogorovian Probabilities in Contextual Theories

Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow for an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical probabilities we show that a class [Formula: see text] of theories can be selected in which probabilities are introduced as classical averages of Kolmogorovian probabilities over sets of (microscopic) contexts, which endows them with an epistemic interpretation. The conditions characterizing [Formula: see text] are compatible with classical mechanics (CM), statistical mechanics (SM), and QM, hence we assume that these theories belong to [Formula: see text]. In the case of CM and SM, this assumption is irrelevant, as all of the notions introduced in them as members of [Formula: see text] reduce to standard notions. In the case of QM, it leads to interpret quantum probability as a derived notion in a Kolmogorovian framework, explains why it is non-Kolmogorovian, and provides it with an epistemic interpretation. These results were anticipated in a previous paper, but they are obtained here in a general framework without referring to individual objects, which shows that they hold, even if only a minimal (statistical) interpretation of QM is adopted in order to avoid the problems following from the standard quantum theory of measurement.