Birkhoff and von Neumann [Ann. Math. 37, 823 (1936)] proposed a nonstandard logic to describe quantum mechanics, in which the distributive laws of Boolean logic do not hold. In this paper we develop two algebras of propositions for classical mechanics that, like the quantum logical algebra, are based on a measurement theory. We adopt a simple classical measurement theory that allows the determination of any continuous phase-space function to any finite precision. Surprisingly, the resulting ``classical logics'' are non-Boolean, though the distributive laws hold. It appears that any physical theory with a mathematical space of physical states and an adequate description of measurement naturally yields a logiclike structure of experimental propositions, and that this ``derived logic'' can be non-Boolean even for theories much less radical than quantum theory.
Birkhoff and von Neumann [Ann. Math. 37, 823 (1936)] proposed a nonstandard logic to describe quantum mechanics, in which the distributive laws of Boolean logic do not hold. In this paper we develop two algebras of propositions for classical mechanics that, like the quantum logical algebra, are based on a measurement theory. We adopt a simple classical measurement theory that allows the determination of any continuous phase-space function to any finite precision. Surprisingly, the resulting classical logics'' are non-Boolean, though the distributive laws hold. It appears that any physical theory with a mathematical space of physical states and an adequate description of measurement naturally yields a logiclike structure of experimental propositions, and that this derived logic'' can be non-Boolean even for theories much less radical than quantum theory.
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.
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.
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.
For the first time it is shown that the logic of quantum mechanics can be derived from Classical Physics. An orthomodular lattice of propositions, characteristic of quantum logic, is constructed for manifolds in Einstein’s theory of general relativity. A particle is modelled by a topologically non-trivial 4-manifold with closed timelike curves- a 4-geon, rather than as an evolving 3-manifold. It is then possible for both the state preparation and measurement apparatus to constrain the results of experiments. It is shown that propositions about the results of measurements can satisfy a non-distributive logic rather than the Boolean logic of classical systems. Reasonable assumptions about the role of the measurement apparatus leads to an orthomodular lattice of propositions characteristic of quantum logic. 1 Comment This paper has been published in Foundations of Physics Letters [1]. The work forms the basis of my doctoral thesis [2]. A short, less formal talk about my work is archived in quant-ph/9609021 [3]. 2
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.
For the first time it is shown that the logic of quantum mechanics can be derived from classical physics. An orthomodular lattice of propositions characteristic of quantum logic, is constructed for manifolds in Einstein's theory of general relativity. A particle is modelled by a topologically non-trivial 4-manifold with closed timelike curves--a 4-geon, rather than as an evolving 3-manifold. It is then possible for both the state preparation and measurement apparatus to constrain the results of experiments. It is shown that propositions about the results of measurements can satisfy a non-distributive logic rather than the Boolean logic of classical systems. Reasonable assumptions about the role of the measurement apparatus leads to an orthomodular lattice of propositions characteristic of quantum logic.