  Various inequalities (Boole inequality, Chung-Erd\"os inequality, Frechet
inequality) for Kolmogorov (classical) probabilities are considered. Quantum
counterparts of these inequalities are introduced, which have an extra `quantum
correction' term, and which hold for all quantum states. When certain
sufficient conditions are satisfied, the quantum correction term is zero, and
the classical version of these inequalities holds for all states. But in
general, the classical version of these inequalities is violated by some of the
quantum states. For example in bipartite systems, classical Boole inequalities
hold for all rank one (factorizable) states, and are violated by some rank two
(entangled) states. A logical approach to CHSH inequalities (which are related
to the Frechet inequalities), is studied in this context.It is shown that CHSH
inequalities hold for all rank one (factorizable) states, and are violated by
some rank two (entangled) states. The reduction of the rank of a pure state by
a quantum measurement with both orthogonal and coherent projectors, is studied.
Bounds for the average rank reduction are given.
