Quantum logic as it is presented in Birkhoff-von Neumann’s 1936 paper arose within the framework of the mathematical elaboration of the foundations of quantum mechanics. Birkhoff-von Neumann’s axiomatic was an alternative of the traditional formulation of foundations of quantum mechanics proceeding from the technique of the Hilbert space. It provided more economic formulation of the foundations of the theory. In Mackay’s writings it has been transformed into the axiomatic of quantum theory of probability. However, Mackay came back to the concept of Hilbert space as the fundamental for quantum mechanics. He also adopted the rules of super-selections in his presentation of the foundations of quantum mechanics. The contemporary approach to Birkhoff-von Neumann’s axiomatic is twofold: the criticism of their too rigid construction from the point of view of physics and the development in connection with mathematical intuitionism.
In Mathematical Foundations of Quantum Mechanics (1932) von Neumann made separability of Hilbert space an axiom. Subsequent work in mathematics (some of it by von Neumann himself) investigated non-separable Hilbert spaces, and mathematical physicists have sometimes made use of them. This note discusses some of the problems that arise in trying to treat quantum systems with non-separable spaces. Some of the problems are "merely technical" but others point to interesting foundations issues for quantum theory, both in its abstract mathematical form and its applications to physical systems. Nothing new or original is attempted here. Rather this note aims to bring into focus some issues that have for too long remained on the edge of consciousness for philosophers of physics.