Quantum implications in orthomodular posets

Abstract

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