Fuzzy quantum logics as a basis for quantum probability theory

Representation of an abstract quantum logic with an ordering set of states S in the form of a family L (S) of fuzzy subsets of S which fulfils conditions analogous to Kolmogorovian conditions imposed on a sigma-algebra of random events allows us to construct quantum probability calculus in a way completely parallel to the classical Kolmogorovian probability calculus. It is shown that the quantum probability calculus so constructed is a proper generalization of the classical Kolmogorovian one. Some indications for building a phase-space representation of quantum mechanics free of the problem of negative probabilities are given.