A New Axiomatization of Unified Quantum Logic

The question as to whether quantum logic can be considered a theory of deduction underlying quantum mechanics has been given many apparently contradictory answers to date. It has been argued that quantum logic is necessarily an empirical logic and that, therefore, it cannot be a theory of a priori valid inferences.[18,Ch.8.6] At the same time, many axiomatic deductive calculuses, all of which have an equational class of orthomodular lattices as their model, have actually been formulated. [Forareviewsee29] The two approaches are only apparently contradictory since they are actually yet another expression of the individual versus the statistical interpretation of quantum mechanics.[18] A difference between the two approaches formulated within the Hilbert space frame of quantum mechanics is given in Refs. [30,32] where we investigate quantum probability equal to unity which characterizes statistically repeatable and predictable measurements of the first kind. In this case probability equal to unity ascribes a unique value of a measured observable to the ensemble of individual systems measured. It does not necessarily ascribe a unique experimental value of the measured observable to all individual systems as it does to an ensemble of them. Whether it does or not, depends on a particular function. The function would exhibit a jump at one end point of the closed probability interval [0,1] if the probability equal to one ascribed the value to all the individual systems. If not, it would stay continuous. We review the result in Sec. 2.