Is the quantum-logic interpretation dead? Its near total absence from current discussions about the interpretation of quantum theory suggests so. While mathematical work on quantum logic continues largely unabated, interest in the quantum-logic interpretation seems to be almost nil, at least in Anglo-American philosophy of physics. This paper has the immodest purpose of changing that fact. I shall argue that while the quantum-logic interpretation faces challenges, it remains a live option. The usual objections either miss the mark, or admit a reasonable answer, or fail to decide the issue conclusively.
Is the quantum-logic interpretation dead? Its near total absence from current discussions about the interpretation of quantum theory suggests so. While mathematical work on quantum logic continues largely unabated, interest in the quantum-logic interpretation seems to be almost nil, at least in Anglo-American philosophy of physics. This paper has the immodest purpose of changing that fact. I shall argue that while the quantum-logic interpretation faces challenges, it remains a live option. The usual objections either miss the mark, or admit a reasonable answer, or fail to decide the issue conclusively.
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.
Why should anyone think that what is so deeply puzzling about the quantum world is really a matter of logic? Why should anyone think that studying the logic of quantum mechanics should be a route to a proper understanding of quantum mechanics? There is a tradition in the philosophy of quantum mechanics, a tradition stronger among philosophical commentators than among physicists, which sees the subject as the business of understanding quantum mechanical language and the way that language relates to the world. It is an idea more strongly held by philosophers – it comes much more naturally to them – because the analytic tradition, in which most Anglo-American philosophers of science are raised, is generally much exercised about words and how they hook on to things. The analytic tradition in the philosophy of quantum mechanics is not unreasonable. Everyone admits that we cannot picture the microphysical world, that a graphic or iconic representation of quantum systems is impossible. Therefore understanding quantum mechanics must be a matter of understanding the logic of the words and the mathematics of quantum mechanics. If this seems implausible, it is because of an ambiguity in our use of the verb ‘to understand’. We understand the physical world, and we understand physics. But physics is not the physical world. It is something of an entirely different sort, a human product, a way of representing the world.
The epistemological interpretation of quantum mechanics is still in an unacceptable status. This becomes obvious if looking on the variety of interpretations currently under discussion. However, the physical community together with philosophers seem not to be willing to really face the problem and to find an explanation for the oddness of the foundation of nature, as the mathematical treatment of quantum physics can be judged as finished and perfect for the use. The epistemology of quantum physics is often treated as a negligible side effect, not worth to put effort in, as any outcome will not affect the calculation results at all. For the sake of human curiosity and to get a better understanding of the universe we are living in, this attitude should be changed. A possibly interesting new approach is outlined in this article, which may help to adjust the focus.
Quantum logic is the logic of the high-level language in which we describe what is true and false of quantum systems. I use the term ‘highlevel’ in its computational sense. The language whose logic quantum logic is, is inexpressive. It has no machinery for describing probabilities, only truths and falsehoods and perhaps whatever is in between. Quantum logic conceals from its user the underlying details of the formalism of quantum mechanics, and the details (whatever they may be) of the underlying ‘machine’, the physical world. Quantum logic encourages a top-down view of the quantum world. Quantum mechanics, as done by the practising physicist, is bottom-up. The program of feeding back quantum logic into our metalevel discussion of realism fails. Quantum logic does not licence quantummechanical realism. It cannot override or rewrite the bottom-up view. The program itself is odd in that quantum logic is most naturally thought of as expressing quantum-mechanical antirealism, just as quantum mechanics itself is most naturally interpreted antirealistically. Thus quantum logic is consonant with nonlocality, in the sense that it does not allow one to derive the Bell inequalities. The conditional in quantum logic nicely expresses the effect of ideal measurements on quantum systems.
In concluding their major survey, “Quantum Logics,” Greechie and Gudder (1975) listed four outstanding directions for future research, the last of which was, “Explain the meaning of the word ‘logic’ in the title of this paper.” Broadly speaking, there are two radically different points of view on this. There is the view that, taken strictly, ‘logic’ is a misnomer: quantum logic studies certain algebraic structures naturally arising from the formal, apparatus of quantum theory, structures which may afford insights into quantum physics, on a number of levels, but which do not in any way involve giving up or replacing classical logical laws. (Cf., e.g., Jauch (1968, p. 77).) In contrast, there is the view that the non-Boolean structures arising from the Hilbert space formalism should be taken as giving rise to a genuine non-classical logic, in conflict with classical logic, but appropriate to the description of quantum mechanical reality.
What do we expect of a quantum logical interpretation of quantum mechanics worthy of the name? In what sense should it resolve the paradoxes? What scope should it have? Should quantum logic, if it is successful in the micro world, replace classical logic in the macro world and in mathematics? We distinguish between two kinds of quantum logical interpretations of quantum mechanics. An example of the first kind, the activist kind, will do much more than simply rewrite the assertions that quantum mechanics makes about the quantum world in an unfamiliar notation - in the elementary language of quantum mechanics. It will offer a solution to the major puzzles of quantum mechanics. It will show that the paradoxes of quantum mechanics can be resolved if we do away with classical logic and replace it with quantum logic, at least in our descriptions of quantum phenomena. It may defend a form of quantum-mechanical realism, a sort of quantum logical particle view of quantum systems, protecting it from the paradoxes that classical logic would impose on it. The second kind of quantum logical interpretation, the passive or quietist kind, is less ambitious. A quietist interpretation offers a solution to the paradoxes only in the following sense. It will assert that quantum mechanics, when it describes quantum phenomena in the elementary language, need not generate paradox, that the so-called paradoxes are not even formulable in quantum logic.