Characterizations of classical and quantum logics

In classical logic (Boolean algebras) probability systems involving correlations are fully characterized by the system of generalized Bell inequalities. On the other hand, probability systems with pairwise correlations on orthomodular lattices (OML) representing quantum logics are so general that the only inequalities that hold universally are the trivial inequalities 0p_i1, 0p_ijmin {p_i,p_j}. In this paper it is shown that every correlation sequence p=(p₁,...,p_n,...,P_ij,...) satisfying the above inequalities can be represented by a probability measure on an orthomodular latticeL admitting a full set of {0,1}-valued probability measures with the additional property that isL ortho-Arguesian.