The logic of orthomodular posets of finite height. (22nd December 2020)

• Record Type: Journal Article
• Authors: Chajda, Ivan; Länger, Helmut
• Is Part Of: Logic journal of the IGPL. Volume 30:Number 1(2022)
• Page Start: 143
• Page End: 154
• Publication Date: 2020-12-22
• Subjects: orthomodular poset -- connective implication -- implication orthomodular poset -- axiom system -- derivation rules -- Gentzen-style system; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1093/jigpal/jzaa067
• Languages: English
• ISSNs: 1367-0751

Abstract

Abstract: Orthomodular posets form an algebraic formalization of the logic of quantum mechanics. A central question is how to introduce implication in such a logic. We give a positive answer whenever the orthomodular poset in question is of finite height. The crucial advantage of our solution is that the corresponding algebra, called implication orthomodular poset, i.e. a poset equipped with a binary operator of implication, corresponds to the original orthomodular poset and that its implication operator is everywhere defined. We present here a complete list of axioms for implication orthomodular posets. This enables us to derive an axiomatization in Gentzen style for the algebraizable logic of orthomodular posets of finite height.