  The orthocomplemented modular lattice of subspaces L[H(d)], of a quantum
system with d- dimensional Hilbert space H(d), is considered. A generalized
additivity relation which holds for Kolmogorov probabilities, is violated by
quantum probabilities in the full lattice L[H(d)] (it is only valid within the
Boolean subalgebras of L[H(d)]). This suggests the use of more general (than
Kolmogorov) probability theories, and here the Dempster-Shafer probability
theory is adopted. An operator D(H1,H2), which quantifies deviations from
Kolmogorov probability theory is introduced, and it is shown to be intimately
related to the commutator of the projectors P(H1),P(H2), to the subspaces
H1,H2. As an application, it is shown that the proof of CHSH inequalities for a
system of two spin 1/2 particles, is valid for Kolmogorov probabilities, but it
is not valid for Dempster- Shafer probabilities. The violation of these
inequalities in experiments, supports the interpretation of quantum
probabilities as Dempster-Shafer probabilities.
