Journal of Logic and Computation Advance Access originally published online on August 21, 2007
Journal of Logic and Computation 2008 18(1):59-76; doi:10.1093/logcom/exm054
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Original Articles |
A Presentation of Quantum Logic Based on an and then Connective
Selim and Rachel Benin School of Computer Science and Engineering, Hebrew University, Jerusalem 91904, Israel.
E-mail: lehmann{at}cs.huji.ac.il
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Abstract |
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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.
Keywords: Generalized Boolean algebras; non-associative Boolean algebras; non-commutative Boolean algebras; quantum measurements; measurement algebras; Quantum Logic; orthomodular lattices; modular lattices