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Quantum Implications in Orthomodular Posets
Kadir Emir and Jan Paseka
We show that, for every orthogonal lub-complete poset 𝐏, we can introduce multiple-valued implications sharing properties with quantum implications presented by Kalmbach for orthomodular lattices. We call them classical implication, Kalmbach implication, non-tollens 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-tollens 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.
Keywords: Multiple-valued implication, classical implication, Kalmbach implication, non-tollens implication, Dishkant implication, Sasaki implication, orthomodular poset, modus ponens rule