Von Neumann's concept of quantum logic and quantum probability

The idea of quantum logic first appears explicitly in the short Section 5 of Chapter III. in von Neumann’s 1932 book on the mathematical foundations of quantum mechanics [31]; however, the real birthplace of quantum logic is commonly identified with the 1936 seminal paper co-authored by G. Birkhoff and J. von Neumann [5]. The aim of this review is to recall the main idea of the Birkhoff-von Neumann concept1 of quantum logic as this was put forward in the 1936 paper. The review is motivated partly by two facts related to quantum logic: one, peculiar, is that the 1936 von Neumann concept is an almost totally neglected2 topic in the enormous quantum logic literature [17]; the other, not very well-known, is that von Neumann was never completely satisfied with how he had worked out quantum logic.