Birkoff-von Neumann's contention that observational propositions in Quantum Mechanics do not obey the Boolean logic, but the modular logic was suggested by the structure of the Hilbert space, and only indirectly related to the nature of observation in physics. Husimi was the first to undertake the task of establishing directly experimental foundation of quantum logic. The present paper has the same aim as Husimi's early paper, but adopts a different approach. A features of the present paper is its methodology of extracting an algebraic structure from a probabilistic structure. A concrete observational interpratation is given for the first time to the conjunction of two incompatible propositions. A side benefit of this work is clarification of the origin of the Booleanity of the ordinary logic, which has to be sought outside the implicational lattice.
Birkoff-von Neumann's contention that observational propositions in Quantum Mechanics do not obey the Boolean logic, but the modular logic was suggested by the structure of the Hilbert space, and only indirectly related to the nature of observation in physics. Husimi was the first to undertake the task of establishing directly experimental foundation of quantum logic. The present paper has the same aim as Husimi's early paper, but adopts a different approach. A features of the present paper is its methodology of extracting an algebraic structure from a probabilistic structure. A concrete observational interpratation is given for the first time to the conjunction of two incompatible propositions. A side benefit of this work is clarification of the origin of the Booleanity of the ordinary logic, which has to be sought outside the implicational lattice.
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.
Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.
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.
In this paper we provide a structural derivation of Birkhoff–von Neumann quantum logic starting from an extended statistical interpretation of quantum mechanics. Quantum logic is not introduced as an independent logical hypothesis, but is shown to arise naturally from the non-classical structure of quantum probability, in particular from interference, non-additivity, and the priority of the whole over its alternatives. We demonstrate that the failure of Boolean logic in quantum mechanics is not a semantic or philosophical anomaly, but a direct consequence of the probabilistic structure encoded in Hilbert space. Within this framework, quantum logic appears as the logical form of quantum probability itself. This interpretation clarifies the physical meaning of quantum logic and highlights its contemporary relevance in quantum information and quantum computation.
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.
Abstract Birkhoff and von Neumann make it clear what the goal of their October, 1936 paper, “The logic of quantum mechanics”, is: “The object of the present paper is to discover what logical structure one may hope to find in physical theories which, like quantum mechanics, do not conform to classical logic. Our main conclusion, based on admittedly heuristic arguments, is that one can reasonably expect to find a calculus of propositions which is formally indistinguishable from the calculus of linear subspaces with respect to set products, linear sums, and orthogonal complements-and resembles the usual calculus of propositions with respect to and, or, and not.” (Birkhoff and von Neumann, 1936, p.823; underline added). Specifically, “Up to now, we have only discussed formal features of logical structure which seem to be common to classical dynamics and the quantum theory. We now turn to the central difference between them-the distributive identity of the propositional calculus: L6: a U (b ꓵ c) = (a U b) ꓵ (a U c) and a ꓵ (b U c) = (a ꓵ b) U (a ꓵ c) which is a law in classical, but not in quantum mechanics.” (Birkhoff and von Neumann, 1936, p.830; underline added) Continuing, Birkhoff and von Neumann comment: “From a deeper mathematical viewpoint, L6 is the characteristic property of set-combination. More precisely, every "field" of sets is isomorphic with a Boolean algebra, and conversely.21 This throws new light on the well-known fact that the propositional calculi of classical mechanics are Boolean algebras.” (Birkhoff and von Neumann, 1936, p.831). Birkhoff and von Neumann then show on p.831 that the distributive law breaks down in quantum mechanics and that the generalized distributive law of logic breaks down in the related field “in the quotient algebra of the field of Lebesgue measurable sets by the ideal of sets of Lebesgue measure 0, which is so fundamental in statistics and the formulation of the ergodic principle.” (ibid., p.831). However, it must be noted that Birkhoff and von Neumann are talking about the Jevons-Pierce-Schroder interpretation (two -valued) of Boole’s original approach (four -valued), which Boole completely rejected. (See Hailperin, 1986, 1996), The reason for this rejection is that the Jevons-Peircean approach required strict complementation, which translated into additivity and linearity in probability assessments, as opposed to Boole’s realization that real world decision making required sub additivity and non-linearity, Thereafter, we will refer to the Jevons-Pierce-Schroder approach as Jevons’s approach to the interpretation of Boolean logic, which uses standard classical logic to underpin discussions of standard probability logic. Jevons’s approach removed the basic algebra which Boole combined with his new relational, propositional logic, so as to deal with indeterminate, imprecise probabilities, which are nonstandard. Jevons’s approach can’t deal with the non-standard probability logics used by Boole in chapters XVIII-XX of The Laws of Thought (1854) and by Keynes in chapters XII, XV-XVII of the A Treatise on Probability (1921) in order to deal with indeterminate, interval valued probabilities.