A discussion on the origin of quantum probabilities

•Several recent works use a derivation similar to that of R.T. Cox to obtain quantum probabilities.•We apply Cox’s method to the lattice of subspaces of the Hilbert space.•We obtain a derivation of quantum probabilities which includes mixed states.•The method presented in this work is susceptible to generalization.•It includes quantum mechanics and classical mechanics as particular cases.

We study the origin of quantum probabilities as arising from non-Boolean propositional-operational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorovian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).