Partial classical logic, the Lindenbaum property and the hidden variable problem
Orthologic and orthomodular quantum logic are examples of total logics: their language is syntactically closed under the logical connectives and any molecular sentence always has a meaning both in the algebraic and in the Kripkean semantics. We will now investigate a form of quantum logic (first proposed by Kochen and Specker (1965b)) based on a different idea. From the semantic point of view, the crucial relation will be represented by a compatibility relation, that may hold between the meanings of two sentences. As expected, the intended physical interpretation of the compatibility relation will be the following: two sentences α and β will have compatible meanings iff α and β can be simultaneously tested.
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