In order to establish the object language of quantum physics, we begin with a quantum mechanical system S (atom, nucleus, elementary particle) and consider propositions A,B,… about this system S which can be proved or disproved by measuring processes. These propositions will be called elementary propositions. We will assume here that elementary propositions are value definite, i.e.,there always exists an experimental testing procedure which decides between truth and falsity of the respective proposition. The logical connectives A ∧ B (A and B), A ∨ B (A or B), A → B (A then B) and ┐ A (not A) will be defined by means of dialogs. A dialog is a formalized kind of discussion between two participants, the proponent P who asserts a certain proposition and the opponent 0 who attempts to refute it.
In order to establish the object language of quantum physics, we begin with a quantum mechanical system S (atom, nucleus, elementary particle) and consider propositions A,B,… about this system S which can be proved or disproved by measuring processes. These propositions will be called elementary propositions. We will assume here that elementary propositions are value definite, i.e.,there always exists an experimental testing procedure which decides between truth and falsity of the respective proposition. The logical connectives A ∧ B (A and B), A ∨ B (A or B), A → B (A then B) and ┐ A (not A) will be defined by means of dialogs. A dialog is a formalized kind of discussion between two participants, the proponent P who asserts a certain proposition and the opponent 0 who attempts to refute it.
The term proposition usually denotes in quantum mechanics (QM) an element of (standard) quantum logic (QL). Within the orthodox interpretation of QM the propositions of QL cannot be associated with sentences of a language stating properties of individual samples of a physical system, since properties are nonobjective in QM. This makes the interpretation of propositions problematical. The difficulty can be removed by adopting the objective interpretation of QM proposed by one of the authors (semantic realism, or SR, interpretation). In this case, a unified perspective can be adopted for QM and classical mechanics (CM), and a simple first order predicate calculus L(x) with Tarskian semantics can be constructed such that one can associate a physical proposition (i.e., a set of physical states) with every sentence of L(x). The set Pf of all physical propositions is partially ordered and contains a subset PTf of testable physical propositions whose order structure depends on the criteria of testability established by the physical theory. In particular, PTf turns out to be a Boolean lattice in CM, while it can be identified with QL in QM. Hence the propositions of QL can be associated with sentences of L(x), or also with the sentences of a suitable quantum language LTQ(x), and the structure of QL characterizes the notion of testability in QM. One can then show that the notion of quantum truth does not conflict with the classical notion of truth within this perspective. Furthermore, the interpretation of QL propounded here proves to be equivalent to a previous pragmatic interpretation worked out by one of the authors, and can be embodied within a more general perspective which considers states as first order predicates of a broader language with a Kripkean semantics.
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.
Scholars have wondered for a long time whether quantum mechanics (QM) subtends a quantum concept of truth which originates quantum logic (QL) and is radically different from the classical (Tarskian) concept of truth. We show in this paper that QL can be interpreted as a pragmatic language of pragmatically decidable assertive formulas, which formalize statements about physical systems that are empirically justified or unjustified in the framework of QM. According to this interpretation, QL formalizes properties of the metalinguistic concept of empirical justification within QM rather than properties of a quantum concept of truth. This conclusion agrees with a general integrationist perspective, according to which nonstandard logics can be interpreted as theories of metalinguistic concepts different from truth, avoiding competition with classical notions and preserving the globality of logic. By the way, some elucidations of the standard concept of quantum truth are also obtained.
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]
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.
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-