To the memory of John D. Trimmer This paper assesses the impact of disjunctive facts on the quantum logic read off procedure. The purpose of the procedure is to transfer a significant quantum structure to a set of propositions; its first step is an attempt to discover that structure. Here I propose that disjunctive facts as traditionally conceived have blocked the procedure at its first step and have therefore subverted the best-known attempts to read off quantum logic. Recently however Allen Stairs has proposed a view of disjunctive facts which re-establishes the possibility of reading off quantum logic. Both the traditional conception and Stairs’ revision of disjunctive facts are interesting in their own right, independent of quantum propositional logic. Too many things are called ‘quantum logic’. The term is disentangled and notation is fixed in this preliminary step which relies on basic lattice theory.
To the memory of John D. Trimmer This paper assesses the impact of disjunctive facts on the quantum logic read off procedure. The purpose of the procedure is to transfer a significant quantum structure to a set of propositions; its first step is an attempt to discover that structure. Here I propose that disjunctive facts as traditionally conceived have blocked the procedure at its first step and have therefore subverted the best-known attempts to read off quantum logic. Recently however Allen Stairs has proposed a view of disjunctive facts which re-establishes the possibility of reading off quantum logic. Both the traditional conception and Stairs’ revision of disjunctive facts are interesting in their own right, independent of quantum propositional logic. Too many things are called ‘quantum logic’. The term is disentangled and notation is fixed in this preliminary step which relies on basic lattice theory.
The purpose of this paper is to defend ‘quantum logic’ against a vigorous attack of J.H. McGrath (1978). The case is serious, as he allegedly discredits quantum propositional logic showing it unmotivated and useless. The object of his criticism is the procedure of “taking a well-known theory and distilling its logic” (Finkelstein 1969), or “reading off quantum logic from quantum mechanics” (McGrath 1978). This procedure is indeed the corner stone of the quantum logic philosophy, so if it were be “discredited in a most general way” and shown to be “fundamentally misconceived” (McGrath 1978) then the idea of quantum logic has broken down. I am going to show, on the contrary, that McGrath's account of the ‘reading off’ procedure is not sufficiently precise resulting in the failure of his ‘acrobatic’ counterexample.
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.
The purpose of this paper is to prove that the construction of a quantum logic is a task of specific interest for non-classical logic, but that it is relevant neither to the methodology of quantum mechanics nor to the formalisation of the alternative yes-no process. The main argument is that quantum structures (for example, orthocomplemented partially ordered sets, orthomodular lattices and partial algebras) are not systems of logic proper, but rather are perfectly formalizable structures in classical logic. In § 1, the orthodox formulation of Jauch and Piron are analyzed. I affirm that the objections made by these authors against “quantum logic” are not eliminated by the introduction of a conditional in the lattice structures, since the true reason why quantum “logic” is not really logic lies in its formulation, which is not that of a propositional system, but rather that of a specific theory whose model is these lattices. It is argued that the collection of the properties satisfied by certain objects of a structure does not justify their presentation as purely logical properties, nor does the presence of certain operations of the least upper bound and greatest lower bound permit us to call symbols introduced by analogy in propositionaI logic “connectives”. In any case, if that formalization is indeed possible, it does not imply the adoption of a new logic as a formal instrument for quantum mechanics. In § 2, aspects of quantum theories which are especially semantic in nature are analyzed. I agree with van Fraassen’s statement in the sense that that semantic formulation makes specific concepts of the theory clearer, but it does not attempt to construct its own logic. I deduce that it is non-classical logic which can actually take advantage of quantum semantics, as a model which, eventually, could throw light on the metalogical properties of certain special propositional systems. Finally, by using Benioff’s profound set theory analysis, I advance the hypothesis that classical logic is possibly the only one that can account for the richness of the meanings that appear in quantum mechanics.[Summary by Carlos Lungarzo]
This talk is about quantum logic, and before we engage in the subject proper let me say how much I appreciate the invitation to deliver it that was extended to me by the Japanese Society for the Philosophy of Science.In particular because I feel that such a meeting is really the place that is most adequate for the discussion of these matters.Let us introduce the subject by mentioning a few general points.First of all, logics deals with propositions.Hence we shall be concerned with propositions.But in fact our interest will be exclusively focused on what is-or what should be the simplest type of propositions: the propositions by means of which a physicist expresses what he knows about a physical system.A priori, our subject should then be very simple.Indeed the elementary books about logic tell us that the rules concerning these propositions can be formalized.That, for example, the propositions can be combined by means of two rules, corresponding to and and to or, and that the resulting structure is just that of a Boolean Algebra.We would expect this to hold for any system.However, as you know, a major difficulty already creeps in at this point when we want to deal with atomic systems.This is because a very simple word such as "and" may well have no obvious meaning in that case.And when we try to force a meaning to it was sometimes encounter quite unpleasant surprises, exemplified by the Heisenberg relations.Among the founders of quantum physics the reactions to such a fact were quite varied as is well known.Some of them sort of hoped that the difficulty just came from our using the wrong concepts.For example, it is quite obvious that if we try to describe as a particle something which is really a wave, we are heading for trouble and we are bound to formulate some meaningless propositions.However, this remark did not lead to a satisfactory solution in the end.Other physicists notably the Copenhagen group took a very different standpoint.Substantially, what they said boiled to this: since we cannot "see" the elementary particles, let us refrain from formulating directly propositions bearing on them.Ultimately, there fore, all their propositions bear on the instruments; or should be interpreted in terms of such propositions.This is -it turned out a very good recipe.Of course, propositions bearing on instruments are propositions of classical, macroscopic physcis.As such, they fall -57-
Summary Foundational studies in quantum theory have consisted in large part of attempts to avoid paradoxical situations in describing properties of a physical system. Quantum logic addresses some of the perplexing problems that plague interpretations of quantum theory by invoking a non-classical propositional calculus for the set of propositions of a quantum system. Historically, quantum logic derives from von Neumann’s observation that the set of projection operators on a Hilbert space constitute a ‘logic’ of experimental propositions. More than two decades passed from the pioneering work of Birkho and von Neumann before interest in quantum logic was re-ignited by Mackey’s probabilistic analysis of quantum theory. Piron’s axiomatization provided a signicant
This paper deals with the foundations of quantum mechanics. We start by outlining the characterisation, due to Birkhoff and Von Neumann, of the logical structures of the theories of classical physics and quantum mechanics, as boolean and modular lattices respectively. We then derive these descriptions from what we claim are basic properties of any physical theory - i.e. the notion that a quantity in such a theory may be analysed into parts and that the results of this analysis may be treated in languages with an underlying boolean structure. We shall see that in the course of constructing a model of a theory with these properties different indistinguishable possibilities will arise for how the elements of the model may be named, that is to say different possibilities arise for how they can be associated with points from Set. Taking a particular collection of possibilities gives the usual boolean lattice of the propositions of classical physics. Taking all possibilities - in a sense, the set of all things that may be described by physical theories - gives the lattice of quantum mechanical propositions. This gives an interpretation of quantum mechanics as the complete set of such possible descriptions, the complete physical description of the world.