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.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.

arXiv:quant-ph/0603172Conference on the Foundations of Quantum Mechanics: Historical Analysis and Open QuestionsJun 1, 20061 citation

Properties of Quantum Languages

Stanley Gudder, Richard N. Ball

International Journal of Theoretical PhysicsApr 1, 20029 citations

The Metalogic of Quantum Logic

Peter Mittelstaedt

In order to establish the object language of quantum physics, we begin with a quantum mechanical system S (atom, nucleus, elementary particle) and consider propositions A,B,… about this system S which can be proved or disproved by measuring processes. These propositions will be called elementary propositions. We will assume here that elementary propositions are value definite, i.e.,there always exists an experimental testing procedure which decides between truth and falsity of the respective proposition. The logical connectives A ∧ B (A and B), A ∨ B (A or B), A → B (A then B) and ┐ A (not A) will be defined by means of dialogs. A dialog is a formalized kind of discussion between two participants, the proponent P who asserts a certain proposition and the opponent 0 who attempts to refute it.In order to establish the object language of quantum physics, we begin with a quantum mechanical system S (atom, nucleus, elementary particle) and consider propositions A,B,… about this system S which can be proved or disproved by measuring processes. These propositions will be called elementary propositions. We will assume here that elementary propositions are value definite, i.e.,there always exists an experimental testing procedure which decides between truth and falsity of the respective proposition. The logical connectives A ∧ B (A and B), A ∨ B (A or B), A → B (A then B) and ┐ A (not A) will be defined by means of dialogs. A dialog is a formalized kind of discussion between two participants, the proponent P who asserts a certain proposition and the opponent 0 who attempts to refute it.

PSA Proceedings of the Biennial Meeting of the Philosophy of Science AssociationJan 1, 19781 citation

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.