What does Quantum Logic Explain?
Quantum logic originated with the Birkhoff-von Neumann paper of 1936.2 The authors argued that the transition from classical to quantum mechanics involves the transition from a propositional calculus forming a Boolean algebra to a propositional calculus which constitutes “a sort of projective geometry”.* For von Neumann, the significance of this thesis for the foundational problems of the theory lay in the possibility of regarding the projective structure as the logic of a quantum set theory -- a non-Boolean theory of sets for properties of quantum mechanical systems in which dimensionality plays the role of cardinality and transition probabilities between events are induced by automorphisms of the logic.** Thus, probabilities are logical: they are not defined, as in the classical case, by measures over events represented as sets of maximal truth value assignments, parametrized by the points of a phase space. Indeed, the quantum sets have no “points” in the general case; there are no “hidden variables” underlying the statistics associated with a quantum logic or quantum set theory. This is a radically different account of the “irreducibility” of the quantum statistics from that provided by the Copenhagen interpretation.
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