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Dyadic Obligations: Proofs and Countermodels via Hypersequents

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Part of the Lecture Notes in Computer Science book series (LNAI,volume 13753)


The basic system \(\textbf{E}\) of dyadic deontic logic proposed by Åqvist offers a simple solution to contrary-to-duty paradoxes and allows to represent norms with exceptions. We investigate \(\textbf{E}\) from a proof-theoretical viewpoint. We propose a hypersequent calculus with good properties, the most important of which is cut-elimination, and the consequent subformula property. The calculus is refined to obtain a decision procedure for \(\textbf{E}\) and an effective countermodel computation in case of failure of proof search. By means of the refined calculus, we prove that validity in \(\textbf{E}\) is Co-NP and countermodels have polynomial size.

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  1. 1.

    See [5] for an alternative method for generating countermodels.

  2. 2.

    Put \(x\succ ' y\) iff \(x\succ y\) and \(y\not \succ x\). We can easily verify that an arbitrarily chosen world satisfies exactly the same formulas in both models, viz. for all worlds x, \(M,x\models A\) iff \(M',x\models A\). (The sole purpose of this construction is to extend the result in [21] to the current setting.).


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Work funded by the projects FWF M-3240-N and WWTF MA16-028. We thank the anonymous reviewers for their valuable comments.

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Correspondence to Agata Ciabattoni , Nicola Olivetti or Xavier Parent .

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Ciabattoni, A., Olivetti, N., Parent, X. (2023). Dyadic Obligations: Proofs and Countermodels via Hypersequents. In: Aydoğan, R., Criado, N., Lang, J., Sanchez-Anguix, V., Serramia, M. (eds) PRIMA 2022: Principles and Practice of Multi-Agent Systems. PRIMA 2022. Lecture Notes in Computer Science(), vol 13753. Springer, Cham.

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