The aim of this paper is to present and discuss a probabilistic framework that is adequate for the formulation of quantum theory and faithful to its applications. Contrary to claims, which are examined and rebutted, that quantum theory employs a nonclassical probability theory based on a nonclassical "logic," the probabilistic framework set out here is entirely classical and the "logic" used is Boolean. The framework consists of a set of states and a set of quantities that are interrelated in a (...) specified manner. Each state induces a classical probability space on the values of each quantity. The quantities, so considered, become statistical variables (not random variables). Such variables need not have a "joint distribution." For the quantum theoretic application, there is a uniform procedure that defines and determines the existence of such "joint distributions" for statistical variables. A general rule is provided and it is shown to lead to the usual compatibility-commutivity requirements of quantum theory. The paper concludes with a brief discussion of interference and the misunderstandings that are involved in the false move from interference to nonclassical probability. (shrink)

The two concepts of probability used in physics are analyzed from the formal and the material points of view. The standard theory corresponds toprob 1 (probability of the coexistence of two properties). A general logicomathematical theory ofprob 2 (probability of transition between states) is presented in axiomatic form. The underlying state algebra is neither Boolean nor Birkhoff-von Neumann but partial Boolean. In the Boolean subalgebras,prob 1 theory holds. The theory presented contains the logicomathematical foundations of quantum mechanics and, as degenerate (...) cases, the theories of stochastic games and of Markov chains. (shrink)

Fine has recently proved the surprising result that satisfaction of the Bell inequality in a Clauser-Horne experiment implies the existence of joint probabilities for pairs of noncommuting observables in the experiment. In this paper we show that if probabilities are interpreted in the von Mises-Church sense of relative frequencies on random sequences, a proof of the Bell inequality is nonetheless possible in which such joint probabilities are assumed not to exist. We also argue that Fine's theorem and related results do (...) not impugn the common view that local realists are committed to the Bell inequality. (shrink)

While the Phase Space formulation of quantum mechanics has received considerable attention it has seldom been defended as a viable interpretation. In this paper I expound the Phase Space Picture, use it to provide a quasi-classical 'hidden variables' interpretation of quantum mechanics and offer a defence of it against various objections.

Both physicists and philosophers claim that quantum mechanics reduces to classical mechanics as 0, that classical mechanics is a limiting case of quantum mechanics. If so, several formal and non-formal conditions must be satisfied. These conditions are satisfied in a reduction using the Wigner transformation to map quantum mechanics onto the classical phase plane. This reduction does not, however, assist in providing an adequate metaphysical interpretation of quantum theory.