Physical propositions and quantum languages

Abstract

The word \textit{proposition} is used in physics with different meanings, which must be distinguished to avoid interpretational problems. We construct two languages ${\mathcal{L}}^{\ast }(x)$ and $\mathcal{L}(x)$ with classical set-theoretical semantics which allow us to illustrate those meanings and to show that the non-Boolean lattice of propositions of quantum logic (QL) can be obtained by selecting a subset of \textit{p-testable} propositions within the Boolean lattice of all propositions associated with sentences of $\mathcal{L}(x)$. Yet, the aforesaid semantics is incompatible with the standard interpretation of quantum mechanics (QM) because of known no-go theorems. But if one accepts our criticism of these theorems and the ensuing SR (semantic realism) interpretation of QM, the incompatibility disappears, and the classical and quantum notions of truth can coexist, since they refer to different metalinguistic concepts (\textit{truth} and \textit{verifiability according to QM}, respectively). Moreover one can construct a quantum language ${\mathcal{L}}_{TQ}(x)$ whose Lindenbaum-Tarski algebra is isomorphic to QL, the sentences of which state (testable) properties of individual samples of physical systems, while standard QL does not bear this interpretation.

15 pages, no figure, standard LateX

References in corpus (5)

• Hidden Variables and the Two Theorems of John Bell
• Locality and measurements within the SR model for an objective interpretation of quantum mechanics
• MGP versus Kochen-Specker condition in hidden variables theories
• A Semantic Approach to the Completeness Problem in Quantum Mechanics
• On the Notion of Proposition in Classical and Quantum Mechanics

Cited by in corpus (5)

• Recovering Quantum Logic within an Extended Classical Framework
• A Pragmatic Interpretation of Quantum Logic
• A Pragmatic Interpretation of Quantum Logic
• Throwing bridges: where and how can classical and quantum view be connected?
• Kolmogorovian versus non-Kolmogorovian probabilities in contextual theories