Minds and Machines

, Volume 21, Issue 3, pp 411–430 | Cite as

NO Revision and NO Contraction

  • Gregory Wheeler
  • Marco AlbertiEmail author


One goal of normative multi-agent system theory is to formulate principles for normative system change that maintain the rule-like structure of norms and preserve links between norms and individual agent obligations. A central question raised by this problem is whether there is a framework for norm change that is at once specific enough to capture this rule-like behavior of norms, yet general enough to support a full battery of norm and obligation change operators. In this paper we propose an answer to this question by developing a bimodal logic for norms and obligations called NO. A key to our approach is that norms are treated as propositional formulas, and we provide some independent reasons for adopting this stance. Then we define norm change operations for a wide class of modal systems, including the class of NO systems, by constructing a class of modal revision operators that satisfy all the AGM postulates for revision, and constructing a class of modal contraction operators that satisfy all the AGM postulates for contraction. More generally, our approach yields an easily extendable framework within which to work out principles for a theory of normative system change.


Norms Modal logic Revision Contraction 



This research was supported in part by award LogiCCC/0001/2007, project DiFoS, from the European Science Foundation. Thanks to Erich Rast, Choh Man Teng, and three anonymous referees for very helpful comments on earlier drafts.


  1. Alchourrón, C., Gärdenfors P., & Makinson D. C. (1985). On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50(2), 510–530.zbMATHCrossRefMathSciNetGoogle Scholar
  2. Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal logic. New York: Cambridge University Press.zbMATHGoogle Scholar
  3. Boella, G., Pigozzi, G., van der Torre, L. (2009). Normative framework for normative system change. In D. Sichman, C. Sierra, & C. Castelfranchi (Eds.), Proceedings of the 8th international conference on autonomous agents and multiagent systems (AAMAS 2009), (pp. 169–176).Google Scholar
  4. Chellas, B. (1980). Model logic. Cambridge: Cambridge University Press.Google Scholar
  5. Darwiche, A., & Pearl, J. (1997). On the logic of iterated belief revision. Artificial Intelligence 89, 1–29.zbMATHCrossRefMathSciNetGoogle Scholar
  6. Dixon, S., & Wobcke, W. (1993). The implementation of a first-order logic AGM belief revision system. In Proceedings of the fifth IEEE international conference on tools in artificial intelligence, (pp. 40–47). IEEE Computer Society Press.Google Scholar
  7. Gabbay, D., Rodrigues, O., & Russo, A. (2008). Belief revision in non-classical logics. The Review of Symbolic Logic 1(3), 267–304.zbMATHCrossRefGoogle Scholar
  8. Gärdenfors, P., & Makinson, D. C. (1988). Revisions of knowledge systems using epistemic entrenchment. In The 2nd conference on theoretical aspects of reasoning about knowledge (TARK) (pp. 83–96).Google Scholar
  9. Goranko, V., & Passy, S. (1992). Using the universal modality: Gains and questions. Journal of Logic and Computation 2, 5–30.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Governatori, G., & Rotolo, A. (2010). Changing legal systems: Legal abrogations and annulments in defeasible logic. Logic Journal of the IGPL.Google Scholar
  11. Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic 17, 157–170.zbMATHCrossRefMathSciNetGoogle Scholar
  12. Haenni, R., Romeyn, J. -W., Wheeler, & G., Williamson, J. (2010). Probabilistic logic and probabilistic networks. Synthese library. Dordrecht: Springer.Google Scholar
  13. Hansson, S. O. (1999). A textbook of belief dynamics: Theory change and database updating. Berlin: Kluwer Academic Publishers.zbMATHGoogle Scholar
  14. Jin, Y., & Thielscher, M. (2007). Iterated belief revision, revised. Artificial Intelligence 171(1), 1–18.zbMATHCrossRefMathSciNetGoogle Scholar
  15. Jorgensen, J. (1937). Imperatives and logic. Erkenntnis 7, 288–296.Google Scholar
  16. Katsuno, H., & Mendelzon, A. (1991). On the difference between updating a knowledge base and revising it. In The 2nd international conference on the principles of knowledge representation and reasoning (KR 1991) (pp. 387–394).Google Scholar
  17. Kyburg, H. E. Jr., Teng, C. M., & Wheeler, G. (2007). Conditionals and consequences. Journal of Applied Logic 5(4), 638–650.zbMATHCrossRefMathSciNetGoogle Scholar
  18. Levi, I. (2004). Mild contraction. Oxford: Clarendon Press.CrossRefGoogle Scholar
  19. Makinson, D. C., & van der Torre, L. (2000). Input-output logics. Journal of Philosophical Logic 30(2), 155–185.CrossRefGoogle Scholar
  20. Makinson, D. C., & van der Torre, L. (2001). Constraints for input/output logics. Journal of Philosophical Logic 30, 155–185.zbMATHCrossRefMathSciNetGoogle Scholar
  21. Makinson, D. C., & van der Torre, L. (2007). What is input/output logic? input/output logic, constraints, permissions. In G. Boella, L. van der Torre, & H. Verhagen, (Eds.), Normative multi-agent systems, number 07122 in Dagstuhl seminar proceedings, Dagstuhl, Germany, 2007. Internationales Begegnungs und Forschungszentrum für Informatik (IBFI).Google Scholar
  22. Nute, D. (1994). Defeasible logic. In G. Dov, C. Hogger, J. Robinson (Eds.), Handbook of logic in artificial intelligence and logic programming (Vol. 3). New York: Oxford University Press.Google Scholar
  23. Pagnucco, M., & Rott, H. (1999). Severe withdrawal—and recovery. Journal of Philosophical Logic 28, 501–547. (Re-printed with corrections to publisher’s errors in February 2000).Google Scholar
  24. Poole, D. (1988). A logical framework for default reasoning. Artificial Intelligence 36, 27–47.zbMATHCrossRefMathSciNetGoogle Scholar
  25. Reiter, R. (1980). A logic for default reasoning. Artificial Intelligence 13, 81–132.zbMATHCrossRefMathSciNetGoogle Scholar
  26. Rott, H. (2001). Change choice and inference. Oxford: Oxford University Press.zbMATHGoogle Scholar
  27. Spohn, W. (1987). Ordinal conditional functions: A dynamic theory of epistemic states. In W. L. Harper, & B. Skyrms, (Eds.), Causation in decision, belief change and statistics (Vol. 2, pp. 105–134.). Dordrecht: Reidel.Google Scholar
  28. Stolpe, A. (2010). A theory of permission based on the notion of derogation. Journal of Applied Logic 8, 97–113.zbMATHCrossRefMathSciNetGoogle Scholar
  29. van Ditmarsch, H., van der Hoek, W., B., & Kooi, B. (2008). Dynamic Epistemic Logic. Berlin: Synthese Library, Springer.zbMATHGoogle Scholar
  30. Wheeler, G. (2010). AGM belief revision in monotone modal logics. In: E. D. Clarke, & A. Voronkov, (Eds.), International conference on logic for programming, artificial intelligence, and reasoning (LPAR-16) short paper proceedings, Dakar, Senegal, 2010.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.CENTRIA—Artificial Intelligence Center, Department of Computer ScienceUniversidade Nova de Lisboa, FCT/UNLCaparicaPortugal

Personalised recommendations