Using Functor Substitutability for the basis of creating an intuitionist interpretation of the results of that subscribe. Laying the Grounds for Challenging Goedel's Incompleteness Theorems with the Aspects of the Poincare Group. Proceeding with: Propositional Quantum Logic, Transitioning “Non-Commensurable” Smoothly to Functor-Auxiliary Substitutability and Detachment.
Using Functor Substitutability for the basis of creating an intuitionist interpretation of the results of that subscribe. Laying the Grounds for Challenging Goedel's Incompleteness Theorems with the Aspects of the Poincare Group. Proceeding with: Propositional Quantum Logic, Transitioning “Non-Commensurable” Smoothly to Functor-Auxiliary Substitutability and Detachment.
It has been shown by Birkhoff and v. Neumann (1936) and by Jauch and Piron (1963,1964,1968) that the subspaces of Hilbert space constitute an orthocomplemented quasi-modular lattice Lq, if one considers between two subspaces (elements) a, b the relation a⊆b and the operations a∩b, a∪b, a⊥. Furthermore, since the subspaces can be interpreted as quantum mechanical propositions, and since the operations ∩,∪ ,⊥ have some similarity with the logical operations ⋀ (and), ⋁ (or) and ⌝ (not), the question has been raised already by Birkhoff and v. Neumann, whether the lattice of subspaces of Hilbert space can be interpreted as a propositional calculus, sometimes called quantum logic. There are many kinds of lattices which can be interpreted as a propositional or logical calculus. A Boolean lattice LB of propositions corresponds to the calculus of classical logic and an implicative (Birkhoff, 1961) lattice Li has as a model the calculus of effective (intuitionistic) logic.
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-
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.