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An Axiom System for Orthomodular Quantum Logic

Gary M. Hardegree
Studia Logica: An International Journal for Symbolic Logic
Vol. 40, No. 1 (1981), pp. 1-12
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015003
Page Count: 12
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An Axiom System for Orthomodular Quantum Logic
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Abstract

Logical matrices for orthomodular logic are introduced. The underlying algebraic structures are orthomodular lattices, where the conditional connective is the Sasaki arrow. An axiomatic calculus OMC is proposed for the orthomodular-valid formulas. OMC is based on two primitive connectives -- the conditional, and the falsity constant. Of the five axiom schemata and two rules, only one pertains to the falsity constant. Soundness is routine. Completeness is demonstrated using standard algebraic techniques. The Lindenbaum-Tarski algebra of OMC is constructed, and it is shown to be an orthomodular lattice whose unit element is the equivalence class of theses of OMC.

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