  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.
