Quantum implications in orthomodular posets

Kadir Emir; Jan Paseka

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arXiv · 2311.12191

Quantum implications in orthomodular posets

Kadir Emir·Jan Paseka

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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.

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Kadir Emir, Jan Paseka. 2023-11-20. Quantum implications in orthomodular posets. https://arxiv.org/abs/2311.12191

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