IntroductionThe well-known spectral theorem for self-adjoint operators on a Hilbert space can be formulated as follows:Let H be a complex separable Hilbert space with dim H > 2 and let L(H) denote the orthomodular lattice (shortly, OML) of all orthogonal projections from H onto closed linear subspaces of H. Let O denote the set of all self-adjoint linear operators on H and {m Q \ a (E S} the set of all pure probability measures on L(H). Then for every A 6 O there exists a unique Z(//)-valued measure (spectral measure) HA on B(R) such that for every a E S the composed mapping m a o is a probability measure on B(R). (Here and in the following B(R) denotes the Boolean a-algebra of all Borel sets of the real line.) Hence, the spectral theorem determines a doubly indexed family (PA,a)Aeo,aes of probability measures on B(R) such that each Pa,o can be decomposed in the form p A ,a = f^a 0 Pa where /i^ is an £(#)-valued measure on B(R) and m a is a pure probability measure on L(H). The family (PA,a)Aeo,aes can be interpreted as the spectral family of probability measures on B(R) corresponding to O and S. Now, by the inverse spectral theorem we may understand the following problem:Given a doubly indexed family (PA,a)Aeo,aes of probability measures on B(R), what conditions are to be put on O and S in order that every This paper is a result of the collaboration of the three authors within the framework of the Partnership Agreement between the University of Technology Vienna and the Warsaw University of Technology. The authors are grateful to both universities for providing financial support which has made this collaboration possible.

In this paper a comparison of various approaches to the problem of characterization of probabilities is presented. The notion of multidimensional probability or S-probability is introduced and it is shown that this notion can be profitably used for characterization of probability systems. The main results are given in Theorems 3 and 4. In Theorem 3 the conditions are considered under which a system of S-probabilities admits a model on a generalized event space, while in Theorem 4 the conditions under which the system admits a classical model are considered. An interpretation for S-probabilities is discussed and it is shown that Theorems 3 and 4 allow one to derive the logical structure of events from the observed probabilities.

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A partial solution to the problem of generalizing Bell’s inequalities to arbitrary numbers of physical properties is proposed. It is first assumed that the considered sets of probabilities correspond to events which satisfy a postulate ensuring that they form an orthomodular partially ordered set admitting a full set of states. In this framework a theorem generalizing Bell’s inequalities to an arbitrary finite number of events is proven. An interpretation of these results in Hilbert space is indicated. Conditions characterizing the classicality of so-called correlation probabilities are then found, and a method for verifying inequalities involving measurable correlations is discussed.