Abstract
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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Notes
- 1.
See [5] for an alternative method for generating countermodels.
- 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.).
References
Alchourrón, C.: Philosophical foundations of deontic logic and the logic of defeasible conditionals. In: Meyer, J.-J., Wieringa, R. (eds.) Deontic Logic in Computer Science, pp. 43–84. John Wiley & Sons Inc, New York (1993)
Åqvist, L.: Deontic logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. II, pp. 605–714. Springer, Dordrecht (1984). https://doi.org/10.1007/978-94-007-6730-0_1002-1
Asher, N., Bonevac, D.: Common sense obligation. In: Nute [20], pp. 159–203
Avron, A.: The method of hypersequents in the proof theory of propositional non-classical logics. In Logic: from Foundations to Applications, pp. 1–32. OUP, New York (1996)
Benzmüller, C., Farjami, A., Parent, X.: Åqvist’s dyadic deontic logic E in HOL. IfCoLog 6, 715–732 (2019)
Danielsson, S.: Preference and Obligation. Filosofiska Färeningen, Uppsala (1968)
Forrester, J.: Gentle murder, or the adverbial samaritan. J. Phil. 81, 193–197 (1984)
Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: Analytic tableaux calculi for KLM logics of nonmonotonic reasoning. ACM Trans. Comput. Log. 10(3), 1–47 (2009)
Girlando, M., Lellmann, B., Olivetti, N., Pozzato, G.L.: Standard sequent calculi for Lewis’ logics of counterfactuals. In Proceedings JELIA, pp. 272–287 (2016)
Goble, L.: Axioms for Hansson’s dyadic deontic logics. Filosofiska Notiser 6(1), 13–61 (2019)
Hansson, B.: An analysis of some deontic logics. No\(\hat{\rm {u}}\)s, 3(4), 373–398 (1969)
Hilpinen, R. (ed.): Deontic Logic. Reidel, Dordrecht (1971). https://doi.org/10.1007/978-94-010-3146-2
Horty, J.: Deontic modals: why abandon the classical semantics? Pac. Philos. Q. 95(4), 424–460 (2014)
Kurokawa, H.: Hypersequent calculi for modal logics extending S4. In: Nakano, Y., Satoh, K., Bekki, D. (eds.) JSAI-isAI 2013. LNCS (LNAI), vol. 8417, pp. 51–68. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10061-6_4
Kuznets, R., Lellmann, B.: Grafting hypersequents onto nested sequents. Log. J. IGPL 24(3), 375–423 (2016)
Lewis, D.: Counterfactuals. Blackwell, Oxford (1973)
Loewer, B., Belzer, M.: Dyadic deontic detachment. Synthese 54, 295–318 (1983)
Makinson, D.: Five faces of minimality. Stud. Logica. 52(3), 339–379 (1993)
Minc, G.: Some calculi of modal logic. Trudy Mat. Inst. Steklov 98, 88–111 (1968)
Nute, D. (ed.): Defeasible Deontic Logic. Kluwer, Dordrecht (1997)
Parent, X.: Completeness of Åqvist’s systems E and F. Rev. Symb. Log. 8(1), 164–177 (2015)
Parent, X.: Preference semantics for Hansson-type dyadic deontic logic: a survey of results. In: Handbook of Deontic Logic and Normative Systems. vol. 2, pp. 7–70. College Publications, London (2021)
Prakken, H., Sergot, M.: Dyadic deontic logic and contrary-to-duty obligations. In Nute [20], pp. 223–262
Shoham, Y.: Reasoning About Change. MIT Press, Cambridge, MA, USA (1988)
Spohn, W.: An analysis of Hansson’s dyadic deontic logic. J. Phil. Logic 4(2), 237–252 (1975)
J. Tomberlin. Contrary-to-duty imperatives and conditional obligation. No\(\hat{\rm {u}}\)s, pp. 357–375 (1981)
van Benthem, J., Girard, P., Roy, O.: Everything else being equal: a modal logic for ceteris paribus preferences. J. Phil. Logic 38(1), 83–125 (2009)
van der Torre, L., Tan, Y.-H.: The many faces of defeasibility in defeasible deontic logic. In: Nute [20], pp. 79–121
van Fraassen, B.: The logic of conditional obligation. J. Phil. Logic 1(3/4), 417–438 (1972)
Acknowledgements
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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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. https://doi.org/10.1007/978-3-031-21203-1_4
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