ACTA issues

An implicational logic for orthomodular lattices

Ivan Chajda, Jānis Cirulis

Acta Sci. Math. (Szeged) 82:3-4(2016), 383-394
63/2015

 Abstract. Orthomodular lattices were introduced to get an algebraic description of the propositional logic of quantum mechanics. In this paper, we set up axiomatization of this logic as a Hilbert style implicational logical system $\LOM$, i.e., we present a set of axioms and derivation rules formulated in the signature $\{\to,0\}$. The other logical operations $\vee, \wedge, \neg$ are expressed in terms of implication (which is the so-called Dishkant implication) and falsum. We further show that the system $\LOM$ is algebraizable in the sense of Blok and Pigozzi, and that orthomodular lattices provide an equivalent algebraic semantics for it. DOI: 10.14232/actasm-015-813-6 AMS Subject Classification (1991): 06C15, 03G12 Keyword(s): algebraizable logic, axiom system, derivation rule, Dishkant implication, logic of quantum mechanics, orthomodular implication algebra, orthomodular lattice, semi-orthomodular lattice, weak BCK-algebra Received August 16, 2015, and in final form January 16, 2016. (Registered under 63/2015.)