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PSA Proceedings of the Biennial Meeting of the Philosophy of Science Association

The Modal Interpretation of Quantum Mechanics

Gary M. Hardegree

Published Jan 1, 1976• 10 citations
Publisher

Abstract

In the present paper I describe a general formal semantic scheme for the interpretation of quantum mechanics (QM) , and on the basis of this scheme I examine the modal interpretation of QM — both the Copenhagen and the anti-Copenhagen variants — proposed by van Fraassen [19, 20, 21], This is intended to be a fragment of a larger work [12] which additionally investigates a number of closely related interpretations, including ones proposed by Bub and Demopoulos [1, 2, 3, 4], Fine [5, 6], and Krips [16, 17]. Formal semantically speaking (see, e.g., Thomason [18]), a logic L may be characterized as an ordered pair <SYN,SEM>, where SYN is the underlying syntax (language) of L, and SEM is the semantics of L, which consists of a class of semantic assignments on SYN.

DOI
10.1086/psaprocbienmeetp.1976.1.192353

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The modal interpretation explained here construes the mathematical formalism of quantum mechanics in terms of physical properties (``be-ables,'' ``existents'') in such a way that the property attribution reflects the mathematical structure as much as possible---no additional structure is superimposed on the formalism. We generalize the interpretation by specifying a dynamics of physical properties, and also extend it while replying to recent objections by Albert and Loewer 2 [Proceedings of the 1990 Biennial Meeting of the Philosophy of Science Association, edited by A. Fine, M. Forbes, and L. Wessels (Philosophy of Science Association, East Lansing, MI, 1990), Vol. 1, p. 277], and by Elby [Found. Phys. Lett. 6, 5 (1993)]. I also discuss questions relating to relativistic aspects and casuality.

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In the present paper I describe a general formal semantic scheme for the interpretation of quantum mechanics (QM) , and on the basis of this scheme I examine the modal interpretation of QM — both the Copenhagen and the anti-Copenhagen variants — proposed by van Fraassen [19, 20, 21], This is intended to be a fragment of a larger work [12] which additionally investigates a number of closely related interpretations, including ones proposed by Bub and Demopoulos [1, 2, 3, 4], Fine [5, 6], and Krips [16, 17]. Formal semantically speaking (see, e.g., Thomason [18]), a logic L may be characterized as an ordered pair <SYN,SEM>, where SYN is the underlying syntax (language) of L, and SEM is the semantics of L, which consists of a class of semantic assignments on SYN.
Before considering the particulars of the different versions of the modal interpretation in the next chapter, I first present their common characteristics. Then I list the starting points from which I develop these modal interpretations to fully-fledged descriptions of reality. Finally, I give the criteria I think interpretations should meet and present a number of desiderata I hope they meet. General characteristics The name ‘modal interpretation’ originates with Van Fraassen (1972) who, in order to interpret quantum mechanics, transposed the semantic analysis of modal logics to the analysis of quantum logic. The resulting interpretation was for obvious reasons called the modal interpretation of quantum logic. Since then, the term modal interpretation has acquired a much more general meaning and lost its direct kinship with modal logics. In particular new interpretations of quantum mechanics developed in the 1980s by Kochen (1985), Krips (1987), Dieks (1988), Healey (1989) and Bub (1992) became known as modal interpretations and also older traditions like Bohmian mechanics (Bohm 1952; Bohm and Hiley 1993) were identified as modal ones. But why are all these interpretations still called modal? And what is the present-day meaning of this term? I think part of the answer to the first question has to do with public relations. The name ‘modal’ is short, sounds nice and is rather intriguing. Furthermore, I guess that also Van Fraassen's prestige as a philosopher of science adds a special gloss to the term. But, apart from all this, I believe the name ‘modal interpretation’ is quite suited.
There is no such thing as “The Quantum Logical Interpretation of Quantum Mechanics”. Rather, there is a cluster of interpretations, all of which can be described as “quantum logical”. Here I provide a general framework for discussing interpretations of this kind, and then locate various suggestions within it. The presentation owes much to van Fraassen (see, in particular, van Fraassen 1974 ); his “modal interpretation” is one of those I discuss, along with those of Jauch, Putnam and Kochen. I begin by rehearsing some orthodox quantum theory. Within quantum mechanics we deal with a set 0 of measurable quantities, orobservables(position, momentum, components of spin and so on). Experiment can determine the value of an observable for a given system: the values so determined will be real numbers. A maximal amount of information about what the result would be for a given system, whatever experiment we chose to perform on it, is available once we know thestateof the system.
Bas van Fraassen in [4] has recently tried to use modal logic to solve the measurement problem of quantum mechanics. His model is based on a method of expressing quantum states developed by Hugh Everett [1] called the “relative state formulation.” Unfortunately, Everett's mathematics cannot be generalized as van Fraassen requires. The difficulty itself is elementary enough. But a revision of van Fraassen's postulates (such as the one he adopted on p. 341 after he had been told of the problem) can save the mathematics only on pain of making the whole study irrelevant to the physics. I shall explain the dilemma in this note. In a later paper [5], van Fraassen avoids this dilemma by developing an abstract modal interpretation which does not rely on any expression for quantum states.
We lean upon the usual variant of the logic of quantum mechanics [1]. Here the propositions correspond to the results of the quantum experiments. A beautiful essay [2] may be connected with a conservation of semantics. But we try to broaden the semantics, admitting propositions about the possibility of results of experiments. Doing so, we fulfil the old wish of W. A. Fock, who attracts our attention to the importance of the modal categories for the interpretation of the quantum theory [3]. We begin with the formal description of the modal system Br+. The alphabet contains the signs ~, ∨ and Ↄ (negation, alternative and the sign of necessity), the set V of the prepositional variables and parentheses. The rules of formation are: if A ∈ V, then A is a proposition; if X and Y are propositions, then ~ X, X ∨ Y and Ↄ X are propositions also; there are no other propositions. ℬ will design the set of all propositions. The rules of transformation are the following: . Here X, Y ∈ ℬ. The sign ⊨ denotes the derivability in Br+ (in one step here).
Recognizing that syntactic and semantic structures of classical logic are not sufficient to understand the meaning of quantum phenomena, we propose in this paper a new interpretation of quantum mechanics based on evidence theory. The connection between these two theories is obtained through a new language, quantum set theory, built on a suggestion by J. Bell. Further, we give a modal logic interpretation of quantum mechanics and quantum set theory by using Kripke's semantics of modal logic based on the concept of possible worlds. This is grounded on previous work of a number of researchers (Resconi, Klir, Harmanec) who showed how to represent evidence theory and other uncertainty theories in terms of modal logic. Moreover, we also propose a reformulation of the many-worlds interpretation of quantum mechanics in terms of Kripke's semantics. We thus show how three different theories — quantum mechanics, evidence theory, and modal logic — are interrelated. This opens, on one hand, the way to new applications of quantum mechanics within domains different from the traditional ones, and, on the other hand, the possibility of building new generalizations of quantum mechanics itself.
The modal interpretation explained here construes the mathematical formalism of quantum mechanics in terms of physical properties (``be-ables,'' ``existents'') in such a way that the property attribution reflects the mathematical structure as much as possible---no additional structure is superimposed on the formalism. We generalize the interpretation by specifying a dynamics of physical properties, and also extend it while replying to recent objections by Albert and Loewer 2 [Proceedings of the 1990 Biennial Meeting of the Philosophy of Science Association, edited by A. Fine, M. Forbes, and L. Wessels (Philosophy of Science Association, East Lansing, MI, 1990), Vol. 1, p. 277], and by Elby [Found. Phys. Lett. 6, 5 (1993)]. I also discuss questions relating to relativistic aspects and casuality.