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A Presentation of Quantum Logic Based on an and then Connective

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  • Daniel Lehmann

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 in [3] satisfies all the properties studied in this paper. 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.

Suggested Citation

  • Daniel Lehmann, 2007. "A Presentation of Quantum Logic Based on an and then Connective," Discussion Paper Series dp442, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    • Handle: RePEc:huj:dispap:dp442
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    Cited by:

    1. Daniel Lehmann, 2008. "Similarity-Projection structures: The Logical Geometry of Quantum Physics," Discussion Paper Series dp482, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.

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