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Studies in logic and the foundations of mathematics

The Relevance of Quantum Logic in the Domain of Non-Classical Logics

María Luisa Dalla Chiara

Published Jan 1, 1986• 2 citations
Publisher
DOI
10.1016/s0049-237x(09)70695-0

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We consider the issue of computability at the most fundamental level of physical reality: the Planck scale. To this aim, we consider the theoretical model of a quantum computer on a non commutative space background, which is a computational model for quantum gravity. In this domain, all computable functions are the laws of physics in their most primordial form, and non computable mathematics finds no room in the physical world. Moreover, we show that a theorem that classically was considered true but non computable, at the Planck scale becomes computable but non decidable. This fact is due to the change of logic for observers in a quantum-computing universe: from standard quantum logic and classical logic, to paraconsistent 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.