The Kochen-Specker theorem without a Hilbert lattice (original) (raw)

2019, arXiv: Quantum Physics

The failure of distributivity in quantum logic is motivated by the principle of quantum superposition. However, this principle can be encoded differently, i.e., in different logical-algebraic objects. As a result, the logic of experimental quantum propositions might have various semantics: e.g., a semantics, in which the distributive law of propositional logic fails (quantum logic), or a semantics, in which this law holds but the valuation relation -- i.e., the function from atomic propositions into the set of two objects, true and false -- is not total (called supervaluationism). Consequently, closed linear subspaces of a Hilbert space (representing experimental quantum propositions) could be organized in different structures -- i.e., a Hilbert lattice (or its generalizations) identified with quantum logic, or a collection of invariant-subspace lattices (Boolean blocks of contexts) identified with the supervaluational semantics. Then again, because one can verify simultaneously onl...