Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements (
s -maps), are studied. It is shown, that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our investigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using
s -maps and
j -maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.
Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements (s-maps), are studied. It is shown that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our invesigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using s-maps and j-maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.
According to quantum mechanics in its current form, quantum randomness is an intrinsic property of the physical world. The legitimacy of such interpretation is the essence of the Einstein-Bohr debate. Bell’s theorem implies that local realism should be responsible for the experimental invalidation of Bell inequalities. In this paper, it is shown that quantum randomness is due to the unattainability of precise space and time coordinates, and the failure of Bell inequalities is irrelevant to local realism. Furthermore, by reconsidering Bell’s theorem in the context of the mathematical setting for quantum physics, namely, Hilbert space, it is shown that, whether a measurement is performed on a system or not, the logical relation between orthogonal vectors corresponding to mutually exclusive properties of the system must be disjunction (“or”) rather than conjunction (“and”). Moreover, if the unattainability of precise space and time coordinates is taken into account, and if disjunction is used to serve as the logical relation between the orthogonal vectors, then it is possible to render quantum mechanics complete in the sense considered by Einstein while not essentially modifying the mathematical setting for quantum physics.