In this chapter, by ‘logic’ we mean Logica Magna (Great Logic), involving higher-order systems and set theories. We revise some of the main reasons provided by quantum theory to question the standard connectives, quantifiers, identity, and the notion of set ascribed by classical logic. In the end, we discuss the interesting fact that even if these argumentations are feasible, quantum theories (as done by the physicists) continue to be developed within standard frameworks, and we ask for the reasons. We presuppose that the reader is more acquainted with logic than with quantum mechanics, so we deserve more space for explanations in this field.
In this chapter, by ‘logic’ we mean Logica Magna (Great Logic), involving higher-order systems and set theories. We revise some of the main reasons provided by quantum theory to question the standard connectives, quantifiers, identity, and the notion of set ascribed by classical logic. In the end, we discuss the interesting fact that even if these argumentations are feasible, quantum theories (as done by the physicists) continue to be developed within standard frameworks, and we ask for the reasons. We presuppose that the reader is more acquainted with logic than with quantum mechanics, so we deserve more space for explanations in this field.
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 purpose of this dissertation is to construct a radically new type of mathematics whose underlying logic differs from the ordinary classical logic used in standard mathematics, and which we feel may be more natural for applications in quantum mechanics. Specifically, we begin by constructing a first order quantum logic, the development of which closely parallels that of ordinary (classical) first order logic --- the essential differences are in the nature of the logical axioms, which, in our construction, are motivated by quantum theory. After showing that the axiomatic first order logic we develop is sound and complete (with respect to a particular class of models), this logic is then used as a foundation on which to build (axiomatic) mathematical systems --- and we refer to the resulting new mathematics as "quantum mathematics." As noted above, the hope is that this form of mathematics is more natural than classical mathematics for the description of quantum systems, and will enable us to address some foundational aspects of quantum theory which are still troublesome --- e.g. the measurement problem --- as well as possibly even inform our thinking about quantum gravity.\n\nAfter constructing the underlying logic, we investigate properties of several mathematical systems --- e.g. axiom systems for abstract algebras, group theory, linear algebra, etc. --- in the presence of this quantum logic. In the process, we demonstrate that the resulting quantum mathematical systems have some strange, but very interesting features, which indicates a richness in the structure of mathematics that is classically inaccessible. Moreover, some of these features do indeed suggest possible applications to foundational questions in quantum theory. \n\nWe continue our investigation of quantum mathematics by constructing an axiomatic quantum set theory, which we show satisfies certain desirable criteria. Ultimately, we hope that such a set theory will lead to a foundation for quantum mathematics in a sense which parallels the foundational role of classical set theory in classical mathematics. One immediate application of the quantum set theory we develop is to provide a foundation on which to construct quantum natural numbers, which are the quantum analog of the classical counting numbers. It turns out that in a special class of models, there exists a 1-1 correspondence between the quantum natural numbers and bounded observables in quantum theory whose eigenvalues are (ordinary) natural numbers. This 1-1 correspondence is remarkably satisfying, and not only gives us great confidence in our quantum set theory, but indicates the naturalness of such models for quantum theory itself. We go on to develop a Peano-like arithmetic for these new "numbers," as well as consider some of its consequences. Finally, we conclude by summarizing our results, and discussing directions for future work.
In this article we try to justify a way of looking for an alternative approach to quantum mechanics based on a non-classical logic. The main motivation is both to accommodate a metaphysical view according to which quantum objects are non-individuals and to introduce, so to say, these objects within the formalism, for as we know, the usual formalism treats of states of quantum systems and not (directly) of quantum systems properly speaking. We consider two specific questions related to quantum theory, namely, entanglement and the indiscernibility of quanta. Our way of looking to these concepts motivate us to construct a metaphysical view according to which quantum objects do not present identity criterion (they are ‘non-individuals’), and then we try to construct a quantum theory based on such a view, which is grounded on a non-classical logic, termed non-reflexive. From the formal point of view, the main motivation comes from the fact that in using classical logic and classical set theory (such as ZFC), we are necessarily committed to individuals, and non-individuals can be considered only by assuming ‘strong’ symmetry principles, which mask the basic idea involving non-individuality. In the core of the article, due to questions of space, only the main philosophical arguments are presented, but in two appendixes we outline, firstly, quasi-set theory, the non-reflexive mathematical framework that sustain our view, and a bottom-top construction of a quantum mechanics based on quasi-set theory.
This paper treats some of the issues raised by Putnam's discussion of, and claims for, quantum logic, specifically: that its proposal is a response to experimental difficulties; that it is a reasonable replacement for classical logic because its connectives retain their classical meanings, and because it can be derived as a logic of tests. We argue that the first claim is wrong(1), and that while conjunction and disjunction can be considered to retain their classical meanings, negation cruciallydoes not.The argument is conductedviaa thorough analysis of how the meet, join and complementation operations are defined in the relevant logical structures, respectively Boolean- and ortholattices(3). Since Putnam wishes to reinstate a realist interpretation of quantum mechanics, we ask how quantum logic can be a logic of realism. We show that it certainly cannot be a logic ofbivalencerealism (i.e., of truth and falsity), although itisconsistent with some form ofontologicalrealism(4). Finally, we show that while a reasonable explication of the idealized notion of test yields interesting mathematical structure, it by no means yields the rich ortholattice structure which Putnam (following Finkelstein) seeks.
Standard Boolean logic is based on ideas taken from classical, first-order set theory. The first-order logic of quantum mechanics introduced by von Neumann is not a classical one. It is therefore natural to ask whether it is possible to formulate the corresponding theory of quantum sets, their (quantum) topology and higher-order structures. Once we have quantum sets and quantum topologies we can try to quantize essentially anything: phase space, groups, spacetime, etc. In his book Finkelstein reports results of some 40 years of studies in this direction. The book is not addressed to specialists, but to `thoughtful students of physics, mathematics and philosophy', which may justify the fact that this review is written by a person who is not actively working in the field of quantum logic. From this layman's position I must say that the book does not read very well. First of all, the standard language of physics is not naturally suited for speaking about such problems, so Finkelstein invents a new one. Obviously, it is not easy to read a difficult text which, in addition, is written in a language we do not know. It seems that the author should have tried to depart as little as possible from the standard terminology, but he could not resist coining new terms even for things we already know under different names. When I first looked into the table of contents I felt both curiosity and some intimidation since I did not understand the words. To give an example, we find there spinors, cospinors, anti-spinors, anti-cospinors, ambi-spinors, ambi-cospinors, etc. It turned out that they are simply upper- and lower-index, primed or unprimed ordinary spinors, or their second-rank tensor products. Fortunately higher-rank spinors were not needed. The notation also does not help to understand the text. For example, on p 148 the symbol denotes a spacetime index, but three lines later, in equation (11), it is a gauge group parameter, and no comment is made on this. On the other hand, the same object is often denoted in different ways (see, for instance, equation (1) on p 44 and (7) on p 47) and if we add a number of typographical errors the first reading of the book is really tiring. A general philosophy advocated in the book is to use a kind of two-time description which replaces continuous dynamics with discrete algebraic `activity'. The construction begins with a bi-modal logic of initial and final actions and the preclusion (or `occlusion' ) relation ` ' (essentially iff a transition between and is forbidden). A Hilbert space and a field of coefficients are derived from . Discretization of time goes hand-in-hand with the idea that spacetime should be quantized and, finally, an outline of its construction is given. An intriguing consequence of the Finkelstein formalism is the appearance of internal local gauge groups of the standard model just from the spacetime structure. It seems that some results presented in the book are subject to limitations which are not mentioned by the author. For example, the construction based on will not work correctly if the dynamics between the initial and final acts is nonlinear. This is so because mutually precluding acts do not have to correspond to orthogonal vectors if we use nonlinear eigenvalue problems. So details of evolution between initial and final acts may be more important than one would expect naively. The same concerns the proof that transition probabilities must be of the form . Such subtleties can be ignored in standard quantum mechanics but here we aim at generalization. I find some parts of the text misleading. A canonical example illustrating mathematical and physical meanings of the Finkelstein formalism is a Malusian experiment (i.e. the one involving photons and polarizers), but the Bell theorem given on p 251 cannot distinguish between its single-photon and two-photon versions. Physically the two systems are essentially different and a purely classical Aerts model that gives the Malus law for a single particle is known, even though the theorems and discussion on pp 251 - 3 suggest that this is impossible. Perhaps the original work the theorems were taken from clarifies this point but a reader will not find it for two reasons. First, the name of its author is misprinted. Second, there is no reference to the paper. This is quite typical of the book; the results which are regarded as inessential from the viewpoint of the main argument are oversimplified and quoted without references to original works. The clarity of the presentation would be improved if the author did not try to briefly summarize what he knows about physics (and certainly he knows a lot) but, instead, were more detailed about the material that is really new. In spite of all these minor critical remarks I think the book may be inspiring and opens an entirely new perspective on physics. Many of the problems discussed by Finkelstein remain unsolved but the strategy he proposes looks at least promising and, hopefully, some readers will join the author in his brave attack on the most difficult questions of modern science.
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.