Kalmbach implication in orthomodular posets

Abstract

We show that for every orthogonal lub-complete poset P = (P,≤,′, 0, 1), we can introduce multiple-valued implications sharing properties with quantum implications presented for orthomodular lattices by Kalmbach. We call them classical implication, Kalmbach implication, and non-tolens implication.If the classical implication satisfies the order property, then the corresponding orthologic becomes classical and vice versa. If the Kalmbach or non-tolens implication meets the order property, then the corresponding orthologic becomes quantum and vice versa. A related result for the modus ponens rule is obtained.