Toward a More Natural Expression of Quantum Logic with Boolean Fractions

Abstract

The fundamental algebraic concepts of quantum mechanics, as expressed by many authors, are reviewed and translated into the framework of the relatively new non-distributive system of Boolean fractions (also called conditional events or conditional propositions). This system of ordered pairs (A|B) of events A, B, can express all of the non-Boolean aspects of quantum logic without having to resort to a more abstract formulation like Hilbert space. Such notions as orthogonality, superposition, simultaneous verifiability, compatibility, orthoalgebras, orthocomplementation, modularity, and the Sasaki projection mapping are translated into this conditional event framework and their forms exhibited. These concepts turn out to be quite adequately expressed in this near-Boolean framework thereby allowing more natural, intuitive interpretations of quantum phenomena. Results include showing that two conditional propositions are simultaneously verifiable just in case the truth of one implies the applicability of the other. Another theorem shows that two conditional propositions (a|b) and (c|d) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b=d, that their conditions are equivalent. Some concepts equivalent in standard formulations of quantum logic are distinguishable in the conditional event algebra, indicating the greater richness of expression possible with Boolean fractions. Logical operations and deductions in the linear subspace logic of quantum mechanics are compared with their counterparts in the conditional event realm. Disjunctions and implications in the quantum realm seem to correspond in the domain of Boolean fractions to previously identified implications with respect to various naturally arising deductive relations.