An axiom system for orthomodular quantum logic

Studia Logica 40 (1):1 - 12 (1981)
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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 calculusOMC 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 ofOMC is constructed, and it is shown to be an orthomodular lattice whose unit element is the equivalence class of theses ofOMC.

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Gary Hardegree
University of Massachusetts, Amherst

Citations of this work

The Birth of quantum logic.Miklós Rédei - 2007 - History and Philosophy of Logic 28 (2):107-122.
Unified quantum logic.Mladen Pavičić - 1989 - Foundations of Physics 19 (8):999-1016.

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References found in this work

Semantic analysis of orthologic.R. I. Goldblatt - 1974 - Journal of Philosophical Logic 3 (1/2):19 - 35.
An axiom system for the modular logic.Jerzy Kotas - 1967 - Studia Logica 21 (1):17 - 38.
Lattice Theory.Garrett Birkhoff - 1940 - Journal of Symbolic Logic 5 (4):155-157.
The conditional in quantum logic.Gary M. Hardegree - 1974 - Synthese 29 (1-4):63 - 80.
Implication connectives in orthomodular lattices.L. Herman, E. L. Marsden & R. Piziak - 1975 - Notre Dame Journal of Formal Logic 16 (3):305-328.

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