The Logic of Quantum Mechanics

Birkhoff, Garrett; Neumann, John von

doi:10.1007/978-94-010-1795-4_1

Cited by 484 publications

(759 citation statements)

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“…According to Birkhoff and Neumann [69], the closed subspace of a Hilbert space formulation of quantum mechanics forms an orthomodular lattice. Although Mackey attempted to provide an axiomatic system of quantum logic as an orthocomplemented lattice, it is has not been satisfactorily demonstrated [13].…”

Section: -4 Orthomodular Lattice Derived From Ambiguous Representationmentioning

confidence: 99%

Quantum cognition based on an ambiguous representation derived from a rough set approximation

2016

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Over the last years, in a series papers by Arecchi and others, a model for the cognitive processes involved in decision making has been proposed and investigated. The key element of this model is the expression of apprehension and judgment, basic cognitive process of decision making, as an inverse Bayes inference classifying the information content of neuron spike trains. It has been shown that for successive plural stimuli this inference, equipped with basic non-algorithmic jumps, is affected by quantum-like characteristics. We show here that such a decision making process is related consistently with an ambiguous representation by an observer within a universe of discourse. In our work the ambiguous representation of an object or a stimuli is defined as a pair of maps from objects of a set to their representations, where these two maps are interrelated in a particular structure. The a priori and a posteriori hypotheses in Bayes inference are replaced by the upper and lower approximations, correspondingly, for the initial data sets that are derived with respect to each map. Upper and lower approximations herein are defined in the context of "rough set" analysis. The inverse Bayes inference is implemented by the lower approximations with respect to the one map and for the upper approximation with respect to the other map for a given data set. We show further that, due to the particular structural relation between the two maps, the logical structure of such combined approximations can only be expressed as an orthomodular lattice and therefore can be represented by a quantum rather than a Boolean logic. To our knowledge, this is the first investigation aiming to reveal the concrete logic structure of inverse Bayes inference in cognitive processes.

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Section: -4 Orthomodular Lattice Derived From Ambiguous Representationmentioning

confidence: 99%

Quantum cognition based on an ambiguous representation derived from a rough set approximation

2016

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“…He argued that it is the relation between these physical properties and the projection operators definable on a Hilbert space that should make it possible to obtain some sort of logical calculus. The details of this calculus were worked out later in the paper that von Neumann co-authored with Birkhoff in (1936). Together their aim was to discover the logical structure of quantum mechanics.…”

Section: Origin Of Quantum Logicmentioning

confidence: 99%

“…Together their aim was to discover the logical structure of quantum mechanics. According to them, the logic of quantumexperimental propositions is different from classical logic but nonetheless "resembles the usual calculus of propositions with respect to and, or and not" (Birkhoff and von Neumann 1936). The difference does not concern properties of negation (as for instance in intuitionistic logic), but rather the properties linking conjunction and disjunction.…”

Section: Origin Of Quantum Logicmentioning

confidence: 99%

“…P (Q ∩ R) = (P Q) ∩ (P R) and P ∩ (Q R) = (P ∩ Q) (P ∩ R)). As stressed in Birkhoff and von Neumann (1936): "Distributivity is a law in classical, not quantum mechanics". The Orthomodular Law is satisfied by the lattice of closed linear subspaces of any Hilbert space (while the usual Distributivity of meet over join is not satisfied).…”

Section: Orthomodular Latticesmentioning

confidence: 99%

“…The work of von Neumann in (1932) and the paper he co-authored with Birkhoff in (1936) are generally considered to form the birth of Quantum Logic. In their seminal paper, Birkhoff and von Neumann argue that the logic underlying the formalism of Quantum Mechanics is not classical logic.…”

Section: Introductionmentioning

confidence: 99%

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The dynamic turn in quantum logic

Baltag

Smets

2011

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In this paper we show how ideas coming from two areas of research in logic can reinforce each other. The first such line of inquiry concerns the "dynamic turn" in logic and especially the formalisms inspired by Propositional Dynamic Logic (PDL); while the second line concerns research into the logical foundations of Quantum Physics, and in particular the area known as Operational Quantum Logic, as developed by Jauch and Piron (Helve Phys Acta 42:842-848, 1969), Piron (Foundations of Quantum Physics, 1976). By bringing these areas together we explain the basic ingredients of Dynamic Quantum Logic, a new direction of research in the logical foundations of physics.

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