As the logic of propositions of quantum measurements,G.Birkhoff and J.von Neumann's quantum logic has been regarded as including Orthogonal complement that is weaker than classic negation,quantum disjunction that is neither classical logic nor distributive,and poset that is a non-Boolean algebra.Based on the unitary operator functional analyses,this paper points out that those apparent nonclassic features of quantum logic are only classic functional nature,and the non-distribution ∩(x ∪ x_) = y ∩1 = y,but(y ∩ x) ∪(y ∩ x_) = ∪ 0 = x at the benzene ring can not be correct.
As the logic of propositions of quantum measurements,G.Birkhoff and J.von Neumann's quantum logic has been regarded as including Orthogonal complement that is weaker than classic negation,quantum disjunction that is neither classical logic nor distributive,and poset that is a non-Boolean algebra.Based on the unitary operator functional analyses,this paper points out that those apparent nonclassic features of quantum logic are only classic functional nature,and the non-distribution ∩(x ∪ x_) = y ∩1 = y,but(y ∩ x) ∪(y ∩ x_) = ∪ 0 = x at the benzene ring can not be correct.
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.
We propose a semantic representation of the standard quantum logic QL within a classical, normal modal logic, and this via a lattice-embedding of orthomodular lattices into Boolean algebras with one modal operator. Thus our classical logic is a completion of the quantum logic QL. In other words, we refute Birkhoff and von Neumann's classic thesis that the logic (the formal character) of Quantum Mechanics would be non-classical as well as Putnam's thesis that quantum logic (of his kind) would be the correct logic for propositional inference in general. The propositional logic of Quantum Mechanics is modal but classical, and the correct logic for propositional inference need not have an extroverted quantum character. One normal necessity modality suffices to capture the subjectivity of observation in quantum experiments, and this thanks to its failure to distribute over classical disjunction. The key to our result is the translation of quantum negation as classical negation of observability.
Quantum logic has been introduced by Birkhoff and von Neumann as an attempt to base the logical primitives, the propositions and the relations and operations among them, on quantum theoretical entities, and thus on the related empirical evidence of the quantum world. We give a brief outline of quantum logic, and some of its algebraic properties, such as nondistributivity, whereby emphasis is given to concrete experimental setups related to quantum logical entities. A probability theory based on quantum logic is fundamentally and sometimes even spectacularly different from probabilities based on classical Boolean logic. We give a brief outline of its nonclassical aspects; in particular violations of Boole-Bell type consistency constraints on joint probabilities, as well as the Kochen-Specker theorem, demonstrating in a constructive, finite way the scarcity and even nonexistence of two-valued states interpretable as classical truth assignments. A more complete introduction of the author can be found in the book "Quantum Logic" (Springer, 1998)
Quantum Logic Distributivity And The Algebraic Structure Of Propositions As with many subatomic particles, the question of under what conditions the electron behaves as a particle or as a wave continues to be the subject of research. It was necessary to employ appropriate logic to organise the propositions describing the behaviour of such an object at the subatomic scale into a precise from and to establish coherent relationships between them. Quantum logic, which can be regarded as an interpretation of quantum mechanics, arose from this need. In classical mechanics, the state and properties of a system are described using the propositions of classical logic, which is based on Boolean algebra. In quantum mechanics, however, the fact that the operators representing the measured quantities do not commute has shown that quantum phenomena cannot be described using the propositions of classical logic. It is therefore necessary to have a formal framework suited to the propositions describing quantum phenomena. In our study, we highlight the various differences between classical logic and quantum logic, examine the distributive property and discuss, from the perspective of Birkhoff and von Neumann, the algebraic structure to which quantum logic might correspond. At the conclusion of our research, we find that, in order to assert that all experimental propositions logically equivalent to a given experimental proposition constitute a physical property, the operators corresponding to the measured quantities must commute with one another. Boolean algebra has an algebraic structure that allows classical logic to be formalised. In such algebra, the distributive property holds; thus, distributivity does not pose a problem for classical logic. Consequently, the distributive property does not always hold in quantum logic and quantum systems can be modelled using orthomodular lattices associated with closed subspaces of Hilbert space. We believe that this model must evolve in parallel with the discovery of new quantum phenomena and that it provides a framework conductive to this evolution. Keywords: Quantum logic; distributivity; partially order sets (posets); lattice; orthomodular lattice; Hilbert space; Boolean algebra.
Due to the failure of the classical principles of bivalence and verifunctionality, the logic of experimental propositions relative to quantum systems cannot be interpreted in Boolean algebras. However, we cannot say neither that this logic is captured by orthomodular lattices, as claimed by many authors along the line of Birkhoff's and von Neumann's standard approach. For the alleged violation of distributivity is based on the possibility of combining statements relative to complementary contexts, which does not refer to any experience and, consequently, has no meaning. Indeed, quantum logic should be interpreted in partial, transitive Boolean algebras whose compatibility relation limits the application of the connectives within each of its Boolean sub-algebras, which refer to partial, classical descriptions. Moreover, this approach of quantum logic makes it possible to deal with composite systems, which was not possible to do within the standard approach, and then to deal with the fundamental notion of quantum entanglement. The latter notion can be represented by a series of axioms of the object language that restrict the set of experimental statements bearing on a composite system, while its close link to the notion of complementarity can be expressed in the metalanguage.