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.
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.
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.
The non-classical nature of quantum theory has led to a number of efforts to formalize the logic of statements whose universe of discourse is the subatomic (see, for example, [3], [6], [8], and [10]).It might be hoped that ideally the fruit of such efforts would be a formalism not totally divorced from the classical logic, but incorporating or incorporated in it in such a manner as to unify the logic of the micro-and the macro-worlds.Indeed, this has been the general direction taken by these efforts, including that which we principally examine and expand upon here.*Josef Jauch, [6], pp.67-89, developed a "propositional calculus of quantum mechanics".He objected to calling a system like his a "logic of quantum mechanics"; however, our use of 'logic' here is not, I think, the use to which Jauch objected.A logic, for our present purposes, is a mathematical formalism having a reasonable interpretation as a theory of deduction; its actual application as normative of argumentation in reality is a matter of empirical judgment.This seems close to the way Jauch wished to use "propositional calculus" but more general, allowing for extension to a quantification theory of quantum mechanics, for example.Although Jauch's presentation of a propositional calculus of quantum mechanics is mathematically interesting and suggestive, there are a number of problems connected with it.It is our purpose to engage some of these problems and to present a rigorously defined formalism intended to come close to doing the job intended by Jauch for his logic.The success or lack of success of this effort as related to actual application of this formalism in the foundations of quantum theory is, I think, a matter for the physicists to decide; the empirical judgment involved in this application is not in the scope of this paper.If a formalism is to have a reasonable interpretation as a theory of deduction, it seems clear that it must include a relation which, when it holds between two elements of the formalism, may be reasonably interpreted as expressing that one of the elements 'follows from' the other, in •The author wishes to thank Robert Piziak for advice and comments in the course of research for this paper.
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.
In the present work, I discuss the interpretation of quantum mechanics (QM) from the viewpoint of quantum logic (QL). I regard the objects of QL to be the possible (accidental) properties that can be ascribed to a quantum system SYS. The basic idea is that, at any given moment t, SYS actualizes some properties, but not others. The micro-state of SYS at time t is identified with the set of all properties that SYS actualizes at time t. One thing an interpretation of QM is supposed to do, I believe, is delineate the admissible quantum micro-states. Since the characterization of quantum micro-states is intimately related to the characterization of quantum logical consistency, the interpretation of QM is intimately tied to the interpretation of QL. Two kinds of interpretations are discussed. Strict interpretations are based on the assumption that the properties of a system are individuated by the projection operators on the associated Hilbert space.
The book investigates the ontology and logic of quantum physics. The first part discusses the relationship of theory and observation and different views on the ontological status of scientific theories. It introduces the fundamentals of quantum mechanics and some of its interpretations and their compatibility with various ontological positions. In the second part, implications of quantum mechanics on classical logic, especially on the distributive law and bivalence, as discussed by Garrett Birkhoff & John von Neumann (1936) and Hilary Putnam (1968), and their counterarguments are reconstructed and discussed. It is concluded that classical logic is sufficient for dealing with quantum mechanical propositions.