  We address the basic meaning of apparent contradictions of quantum theory and
probability frameworks as expressed by Bell's inequalities. We show that these
contradictions have their origin in the incomplete considerations of the
premisses of the derivation of the inequalities. A careful consideration of
past work, including that of Boole and Vorob'ev, has lead us to the formulation
of extended Boole-Bell inequalities that are binding for both classical and
quantum models. The Einstein-Podolsky-Rosen-Bohm gedanken experiment and a
macroscopic quantum coherence experiment proposed by Leggett and Garg are both
shown to obey the extended Boole-Bell inequalities. These examples as well as
additional discussions also provide reasons for apparent violations of these
inequalities.
