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Journal of Symbolic Logic

On the structure of quantum logic

Peter D. Finch

Published Jul 25, 1969• 49 citations
Publisher

Abstract

In the axiomatic development of the logic of nonrelativistic quantum mechanics it is not difficult to set down certain plausible axioms which ensure that the quantum logic of propositions has the structure of an orthomodular poset. This can be done in a number of ways, for example, as in Gunson [2], Mackey [4], Piron [5], Varadarajan [7] and Zierler [8], and we summarise one of these ways in §2 below.

DOI
10.2307/2271104

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In the axiomatic development of the logic of nonrelativistic quantum mechanics it is not difficult to set down certain plausible axioms which ensure that the quantum logic of propositions has the structure of an orthomodular poset. This can be done in a number of ways, for example, as in Gunson [2], Mackey [4], Piron [5], Varadarajan [7] and Zierler [8], and we summarise one of these ways in §2 below.
Since Birkhoff and von Neumann [2] a new area of logical investigation has grown up under the name of quantum logic. However it seems to me that many authors have been inclined to discuss algebraic semantics as such (mainly lattices of a certain kind) almost directly without presenting any axiomatic system, far from developing any proof theory of quantum logic. See, e.g., Gunson [9], Jauch [10], Varadarajan [15], Zeirler [16], etc. In this sense many works presented under the name of quantum logic are algebraic in essence rather than genuinely logical, though it is absurd to doubt the close relationship between algebraic and logical study on quantum mechanics. The main purpose of this paper is to alter this situation by presenting an axiomatization of quantum logic as natural and as elegant as possible, which further proof-theoretical study is to be based on. It is true that several axiomatizations of quantum logic are present now. Several authors have investigated the so-called material implication α → β ( = ¬α∨(α ∧ β)) very closely with due regard to its importance. See, e.g., Finch [5], Piziak [11], etc. Indeed material implication plays a predominant role in any axiomatization of a logic in Hilbert-style. Clark [4] has presented an axiomatization of quantum logic with negation ¬ and material implication → as primitive connectives. In this paper we do not follow this approach. First of all, this approach is greatly complicated because orthomodular lattices are only locally distributive.
Sentences Associated with a Quantum System and Their Truth Values In a sense this chapter is marginal to the content of this volume; the reader can skip it without prejudice to the comprehension of chapters to follow. Quantum logic is a discipline that branched off from the 1936 paper of Birkhoff and von Neumann and has the orthomodular-poset structures encountered in previous chapters as basic mathematical carriers: that is why we think it worthwhile to devote some pages to the subject. But the main interests quantum logic calls into play are in the territory of logicians and philosophers: that is why we (who are neither logicians nor philosophers) shall confine ourselves to a very simplified introduction to the subject. Roughly, the starting question is whether the propositions of a quantum system can be associated with, or can be interpreted as, sentences of a language (or propositional calculus) and which rules this language inherits from the ordered structure of propositions. In raising this question one has in mind the fact that when the physical system is classical its propositions form a Boolean algebra, and Boolean algebras are the algebraic models of the calculus of classical logic. Thus the question above can also be phrased as follows: when a Boolean algebra is relaxed into an orthomodular nondistributive lattice, which logic is it the model of? “Quantum logic” is the name that designates the answer, but there are several views about the content of this name. Here we sketch one approach to a quantum logic.
Since 1933, when Kolmogorov laid the foundations for probability and statistics as we know them today [1], it has been recognized that propositions asserting that such and such an event occurred as a consequence of the execution of a particular random experiment tend to band together and form a Boolean algebra. In 1936, Birkhoff and von Neumann [2] suggested that the so-called logic of quantum mechanics should not be a Boolean algebra, but rather should form what is now called a modular ortholattice [3]. Presumably, the departure from Boolean algebras encountered in quantum mechanics can be attributed to the fact that in quantum mechanics, one must consider more than one physical experiment, e.g., an experiment measuring position, an experiment measuring charge, an experiment measuring momentum, etc., and, because of the uncertainty principle, these experiments need not admit a common refinement in terms of which the Kolmogorov theory is directly applicable. Mackey's Axioms I–VI for quantum mechanics [4] imply that the logic of quantum mechanics should be a σ-orthocomplete orthomodular poset [5]. Most contemporary practitioners of quantum logic seem to agree that a quantum logic is (at least) an orthomodular poset [6], [7], [8], [9], [10] or some variation thereof [11]. P. D. Finch [12] has shown that every completely orthomodular poset is the logic arising from sets of Boolean logics, where these sets have a structure similar to the structures generally given to quantum logic. In all of these versions of quantum logic, a fundamental relation, the relation of compatibility or commutativity, plays a decisive role.
The concept of an operational logic has been developed by Randall and Foulis ([l]-[4], [10], [11]) as a part of a larger effort to obtain a formalism suitable for expressing, comparing, and evaluating various approaches to empirical science, statistics, and in particular, quantum mechanics. The structure of these logics is similar to that of an orthomodular partially ordered set which is often used as a model for quantum logics. However, the operational logic is a more general structure which, among other features, allows for the creation of a tensor product of logics to represent the coupling of physical systems. Randall and Foulis have shown that, given certain reasonable physical constraints, such a product is not possible within the category of orthomodular posets [12].