dorsal/arxiv
View SchemaA presentation of Quantum Logic based on an "and then" connective
| Authors | Daniel Lehmann |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0701113 |
| URL | https://arxiv.org/abs/quant-ph/0701113 |
| DOI | 10.1093/logcom/exm054 |
| Journal | Journal of Logic and Computation 18 (1): 59-76 Feb. 2008 |
Abstract
When a physicist performs a quantic measurement, new information about the system at hand is gathered. This paper 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 (1969) satisfies all the properties studied in this paper. All properties satisfied by Finch's operation in modular lattices are valid in Hilbert Space Quantum Logic. It is not known whether all properties of Hilbert Space 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.
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"abstract": "When a physicist performs a quantic measurement, new information about the\nsystem at hand is gathered. This paper studies the logical properties of how\nthis new information is combined with previous information. It presents Quantum\nLogic as a propositional logic under two connectives: negation and the \"and\nthen\" operation that combines old and new information. The \"and then\"\nconnective is neither commutative nor associative. Many properties of this\nlogic are exhibited, and some small elegant subset is shown to imply all the\nproperties considered. No independence or completeness result is claimed.\nClassical physical systems are exactly characterized by the commutativity, the\nassociativity, or the monotonicity of the \"and then\" connective. Entailment is\ndefined in this logic and can be proved to be a partial order. In orthomodular\nlattices, the operation proposed by Finch (1969) satisfies all the properties\nstudied in this paper. All properties satisfied by Finch\u0027s operation in modular\nlattices are valid in Hilbert Space Quantum Logic. It is not known whether all\nproperties of Hilbert Space Quantum Logic are satisfied by Finch\u0027s operation in\nmodular lattices. Non-commutative, non-associative algebraic structures\ngeneralizing Boolean algebras are defined, ideals are characterized and a\nhomomorphism theorem is proved.",
"arxiv_id": "quant-ph/0701113",
"authors": [
"Daniel Lehmann"
],
"categories": [
"quant-ph",
"cs.LO",
"math.LO"
],
"doi": "10.1093/logcom/exm054",
"journal_ref": "Journal of Logic and Computation 18 (1): 59-76 Feb. 2008",
"title": "A presentation of Quantum Logic based on an \"and then\" connective",
"url": "https://arxiv.org/abs/quant-ph/0701113"
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