An implicational logic for orthomodular lattices

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 L, i.e., we present a set of axioms and derivation rules formulated in the signature {→, 0}. The other logical operations ∨, ∧, ¬ are expressed in terms of implication (which is the so-called Dishkant implication) and falsum. We further show that the system L is algebraizable in the sense of Blok and Pigozzi, and that orthomodular lattices provide an equivalent algebraic semantics for it. © 2016 Bolyai Institute, University of Szeged.