2007/01/16 by Daniel Lehmann
Computer Science · Mathematics · Medicine · Physics and Astronomy · #Advanced Algebra and Logic #Classics #Computer science #Hebrew #History #Library science #Logic, Reasoning, and Knowledge #Logic, programming, and type systems #Medicine #Physics #Presentation (obstetrics) #Quantum #Quantum computer #Quantum logic #Quantum mechanics #cs.LO #math.LO #quant-ph
paper · pdf · doi:10.1093/logcom/exm054
published as Journal of Logic and Computation 18 (1): 59-76 Feb. 2008 · 28 pages. Submitted
arxiv created 2007/01/16 · openalex publication_date 2007/08/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
When a physicist performs a quantic measurement, new information about the system at hand is gathered. This article studies the logical properties of how this new information is combined with previous information. It presents Quantum Logic as a propositional logic under two connectives: negation and the and then operation that combines old and new information. The and then connective is neither commutative nor associative. Many properties of this logic are exhibited, and some small elegant subset is shown to imply all the properties considered. No independence or completeness result is claimed. Classical physical systems are exactly characterized by the commutativity, the associativity, or the monotonicity of the and then connective. Entailment is defined in this logic and can be proved to be a partial order. In orthomodular lattices, the operation proposed by Finch in [3] satisfies all the properties studied in this article. All properties satisfied by Finch's; operation in modular lattices are valid in Quantum Logic. It is not known whether all properties of Quantum Logic are satisfied by Finch's; operation in modular lattices. Non-commutative, non-associative algebraic structures generalizing Boolean algebras are defined, ideals are characterized and a homomorphism theorem is proved.