On the Notion of Proposition in Classical and Quantum Mechanics

Abstract

The term proposition usually denotes in quantum mechanics (QM) an element of (standard) quantum logic (QL). Within the orthodox interpretation of QM the propositions of QL cannot be associated with sentences of a language stating properties of individual samples of a physical system, since properties are nonobjective in QM. This makes the interpretation of propositions problematical. The difficulty can be removed by adopting the objective interpretation of QM proposed by one of the authors (semantic realism, or SR, interpretation). In this case, a unified perspective can be adopted for QM and classical mechanics (CM), and a simple first order predicate calculus L(x) with Tarskian semantics can be constructed such that one can associate a physical proposition (i.e., a set of physical states) with every sentence of L(x). The set $P^{f}$ of all physical propositions is partially ordered and contains a subset $P^{f}_{T}$ of testable physical propositions whose order structure depends on the criteria of testability established by the physical theory. In particular, $P^{f}_{T}$ turns out to be a Boolean lattice in CM, while it can be identified with QL in QM. Hence the propositions of QL can be associated with sentences of L(x), or also with the sentences of a suitable quantum language $L_{TQ}(x)$, and the structure of QL characterizes the notion of testability in QM. One can then show that the notion of quantum truth does not conflict with the classical notion of truth within this perspective. Furthermore, the interpretation of QL propounded here proves to be equivalent to a previous pragmatic interpretation worked out by one of the authors, and can be embodied within a more general perspective which considers states as first order predicates of a broader language with a Kripkean semantics.