Relevance implication and orthomodular posets

We establish five distinct multiple-valued quantum implications for orthogonal 2-lub-complete posets, encompassing Kalmbach implication, non-tollens implication, Dishkant implication, Sasaki implication, and relevance implication. We demonstrate that if any of these implications adhere to the order property, the corresponding poset transforms into a quantum logic, and vice versa. Additionally, we obtain a corresponding result for the modus ponens rule. Drawing parallels to the quantum case, we introduce a multiple-valued classical implication for orthogonal 2-lub-complete posets. When this classical implication fulfills the order property, the corresponding poset becomes a Boolean algebra, and vice versa. Furthermore, we achieve a parallel conclusion for the modus ponens rule.