Abstract
It has been shown by Birkhoff and v. Neumann (1936) and by Jauch and Piron (1963, 1964, 1968) that the subspaces of Hilbert space constitute an orthocomplemented quasi-modular lattice L q, if one considers between two subspaces (elements) a, b the relation a⊆b and the operations a∩b, a∪b, a ⟂. Furthermore, since the subspaces can be interpreted as quantum mechanical propositions, and since the operations ∩, ∪, ⟂ have some similarity with the logical operations ⋀ (and), ⋁ (or) and ┐ (not), the question has been raised already by Birkhoff and v. Neumann, whether the lattice of subspaces of Hilbert space can be interpreted as a prop-ositional calculus, sometimes called quantum logic.
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References
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Mittelstaedt, P. (1978). Quantum Logic. In: Hooker, C.A. (eds) Physical Theory as Logico-Operational Structure. The University of Western Ontario Series in Philosophy of Science, vol 7. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9769-1_5
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