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S12c. What is the role of quantum logic?
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Quantum logic is a variant of logic often thought to be
appropriate for the foundations of quantum mechanics.
A good exposition is given in
K. Svozil,
Quantum Logic,
Springer, Singapore 1998.
The book is nice and useful for its material on hidden-variable
related arguments.
However, all that is commonly argued in textbooks about QM is argued
in terms of classical logic. An even cursory look at the large
quantum mechanical literature reveals that quantum logic only has
a marginal spectator role in QM, while all proofs of all properties
of quantum systems have always been discussed using the familiar
classical logic. Even in Svozil's book, one can see that quantum
logic is argued in terms of classical logic, and that it has
essentially no role in the analysis of actual physical situations
(apart from those used for testing the foundations).
Beyond a certain point, quantum logic is sterile, which is the reason
it never figures in textbooks (except perhaps in passing).
All one ever needs to know about quantum logic (unless one wants to
specialize in it) is summarized in Sections 6 and 7 of my paper
Int. J. Mod. Phys. B 17 (2003), 2937-2980 = quant-ph/0303047.