Unification of Two Approaches to Quantum Logic: Every Birkhoff - von Neumann Quantum Logic is a Partial Infinite-Valued Lukasiewicz Logic

In the paper it is shown that every physically sound Birkhoff - von Neumann quantum logic, i.e., an orthomodular partially ordered set with an ordering set of probability measures can be treated as partial infinite-valued Lukasiewicz logic, which unifies two competing approaches: the many-valued, and the two-valued but non-distributive, which have co-existed in the quantum logic theory since its very beginning.