Studia Logica 101 (5):901-913 (2013)

Abstract
In this article von Neumann’s proposal that in quantum mechanics projections can be seen as propositions is followed. However, the quantum logic derived by Birkhoff and von Neumann is rejected due to the failure of the law of distributivity. The options for constructing a distributive logic while adhering to von Neumann’s proposal are investigated. This is done by rejecting the converse of the proposal, namely, that propositions can always be seen as projections. The result is a weakly Heyting algebra for describing the language of quantum mechanics
Keywords Quantum logic  Intuitionistic logic  Weakly Heyting algebras
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DOI 10.1007/s11225-012-9401-3
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Quantum Logic, Realism, and Value Definiteness.Allen Stairs - 1983 - Philosophy of Science 50 (4):578-602.
Bounded Distributive Lattices with Strict Implication.Sergio Celani & Ramon Jansana - 2005 - Mathematical Logic Quarterly 51 (3):219-246.
Quantum Logic in Intuitionistic Perspective.Bob Coecke - 2002 - Studia Logica 70 (3):411-440.

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