The logic of orthomodular posets of finite height

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.