# Iterated Descriptor Revision and the Logic of Ramsey Test Conditionals

- 669 Downloads
- 2 Citations

## Abstract

Two of the major problems in AGM-style belief revision, namely the difficulties in accounting for iterated change and for Ramsey test conditionals, have satisfactory solutions in descriptor revision. In descriptor revision, the input is a (set of) metalinguistic sentence(s) specifying the success condition of the operation. The choice mechanism selects one of the potential outcomes (available belief sets) in which the success condition is satisfied. Iteration of this operation is unproblematic. Ramsey test conditionals can be introduced without giving rise to the paradoxical results that they generate in other systems. In addition to standard (sentential) Ramsey test conditionals, a more general variant of epistemic conditionals is defined, representing statements of the form ”if the belief state is changed to satisfy condition A then it will satisfy condition B”. An axiomatic characterization of such descriptor conditionals is presented. It is related in intricate ways to the KLM postulates for cumulative reasoning.

## Keywords

Descriptor revision Iterated revision Conditionals Darwiche-Pearl postulates Ramsey test Cumulative reasoning## Notes

### Acknowledgment

I would like to thank John Cantwell for a series of conversations on the logic of conditionals that were unusually helpful in the preparation of this paper.

## References

- 1.Alchourrón, C., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: partial meet contraction and revision functions.
*Journal of Symbolic Logic*,*50*, 510–530.CrossRefGoogle Scholar - 2.Alchourrón, C., & Makinson, D. (1982). On the logic of theory change: contraction functions and their associated revision functions.
*Theoria*,*48*, 14–37.CrossRefGoogle Scholar - 3.Bezzazi, H., Makinson, D., & Pino Pérez, R. (1997). Beyond rational monotony: some strong non-Horn conditions for nonmonotonic inference operations.
*Journal of Logic and Computation*,*7*, 605–631.CrossRefGoogle Scholar - 4.Burgess, J.P. (1981). Quick completeness proofs for some logics of conditionals.
*Notre Dame Journal of Formal Logic*,*22*, 76–84.CrossRefGoogle Scholar - 5.Darwiche, A., & Pearl, J. (1997). On the logic of iterated belief revision.
*Journal of Artificial Intelligence*,*89*, 1–29.CrossRefGoogle Scholar - 6.Edgington, D. (2007). On conditionals. In D.M. Gabbay, & F. Guenthner (Eds.),
*Handbook of philosophical logic*. 2nd edn., (Vol. 14 pp. 127–221). Dordrecht: Springer.CrossRefGoogle Scholar - 7.Fermé, E., & Hansson, S.O. (2001). Shielded Contraction. In H. Rott, & M.-A. Williams (Eds.),
*Frontiers of belief revision*(pp. 85–107): Kluwer 2001.Google Scholar - 8.Fermé, E., & Hansson, S.O. (2011). AGM 25 years. Twenty-Five years of research in belief change.
*Journal of Philosophical Logic*,*40*, 295–331.CrossRefGoogle Scholar - 9.Fishburn, P.C. (1970).
*Utility theory for decision making*. New York: Wiley.Google Scholar - 10.Galliers, J.R. (1992). Autonomous belief revision and communication. In P. Gärdenfors (Ed.),
*Belief revision*(pp. 220–246). Cambridge: Cambridge University Press.CrossRefGoogle Scholar - 11.Gärdenfors, P. (1986). Belief revisions and the Ramsey test for conditionals.
*Philosophical Review*,*95*, 81–93.CrossRefGoogle Scholar - 12.Gärdenfors, P., & Makinson, D. (1988). Revisions of knowledge systems using epistemic entrenchment. In M.Y. Vardi (Ed.),
*Proceedings of the 2nd conference on theoretical aspects of reasoning about knowledge*(pp. 83–95). Los Altos: Morgan Kaufmann.Google Scholar - 13.Hansson, S.O. (1991). Belief contraction without recovery.
*Studia Logica*,*50*, 251–260.CrossRefGoogle Scholar - 14.Hansson, S.O. (1992). In defense of the Ramsey test.
*Journal of Philosophy*,*89*, 522–540.CrossRefGoogle Scholar - 15.Hansson, S.O. (1999).
*A textbook of belief dynamics*. Dordrecht: Kluwer.CrossRefGoogle Scholar - 16.Hansson, S.O. (2001). Preference logic. In D. Gabbay, & F. Guenthner (Eds.),
*Handbook of philosophical logic*. 2nd edn. (pp. 319–393).Google Scholar - 17.
- 18.Hansson, S.O. (2013). Cognitive realism in belief revision. In E.Fermé, & G. Simari (Eds.),
*Trends in belief revision and argumentation dynamics*(pp. 57–74). London: College Publications.Google Scholar - 19.Hansson, S.O. (2013). Blockage contraction.
*Journal of Philosophical Logic*,*42*, 415–442.CrossRefGoogle Scholar - 20.Hansson, S.O. (2013). Outcome level analysis of belief contraction.
*Review of Symbolic Logic*,*6*, 183–204.CrossRefGoogle Scholar - 21.
- 22.Hansson, S.O. (2014). Relations of epistemic proximity for belief change.
*Artificial Intelligence*,*217*, 76–91.CrossRefGoogle Scholar - 23.Hansson, S.O., Fermé, E., Cantwell, J., & Falappa, M. (2001). Credibility-limited revision.
*Journal of Symbolic Logic*,*66*, 1581–1596.CrossRefGoogle Scholar - 24.Jech, T.J. ([1973] 2008).
*The axiom of choice*. Mineola, N.Y.: Dover Publications.Google Scholar - 25.Jin, Y., & Thielscher, M. (2007). Iterated belief revision, revised.
*Artificial Intelligence*,*171*, 1–18.CrossRefGoogle Scholar - 26.Konieczny, S., & Pino Pérez, R. (2000). A framework for iterated revision.
*Journal of Applied Non-classical logics*,*10*, 339–367.CrossRefGoogle Scholar - 27.Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics.
*Artificial Intelligence*,*44*, 167–207.CrossRefGoogle Scholar - 28.Lehmann, D. (1995). Belief revision, revised. In
*Proceedings of the 14th international joint conference on artificial intelligence (IJCAI-95)*(pp. 1534–1540).Google Scholar - 29.Lehmann, D., Magidor, M., & Schlechta, K. (2001). Distance semantics for belief revision.
*Journal of Symbolic Logic*,*66*, 295–317.CrossRefGoogle Scholar - 30.Lindström, S., & Rabinowicz, W. (1991). Epistemic entrenchment with incomparabilities and relational belief revision. In A. Fuhrmann, & M. Morreau (Eds.),
*The logic of theory change*(p. 93126). New York: Springer.Google Scholar - 31.Lindström, S., & Rabinowicz, W. (1992). Belief revision, epistemic conditionals and the Ramsey test.
*Synthese*,*91*, 195–237.CrossRefGoogle Scholar - 32.Makinson, D. (1994). General patterns in nonmonotonic reasoning. In D.M. Gabbay, C.J. Hogger, & J.A. Robinson (Eds.),
*Handbook of logic in artificial intelligence and logic programming*, Nonmonotonic reasoning and uncertain reasoning, (Vol. 3 pp. 35–110). Oxford: Clarendon.Google Scholar - 33.
- 34.Peppas, P. (2014). A panorama of iterated revision. In S.O. Hansson (Ed.),
*David Makinson on classical methods for non-classical problems*(pp. 71–94). Dordrecht: Springer.CrossRefGoogle Scholar - 35.Ramsey, F. (1931).
*Foundations of mathematics and other logical essays*. New York: Routledge.Google Scholar - 36.Roberts, F.S. (1979).
*Measurement theory with applications to decisionmaking, utility and the social sciences*. Reading: Addison-Wesley.Google Scholar - 37.Rott, H. (1999). Moody conditionals: Hamburgers, switches, and the tragic death of an American president. In J. Gerbrandy, M. Marx, M. de Rijke, & Y. Venema (Eds.),
*Essays dedicated to Johan van Benthem on the occasion of his 50th birthday*. http://www.illc.uva.nl/j50/contribs/rott (pp. 98–112). Amsterdam: Amsterdam University Press.Google Scholar - 38.Rott, H. (2001). Change, choice and inference: a study of belief revision and nonmonotonic reasoning. Oxford: Oxford University Press.Google Scholar
- 39.Rott, H. (2009). Shifting priorities: simple representations for twenty-seven iterated theory change operators. In D. Makinson, J. Malinowski, & H. Wansing (Eds.),
*Towards Mathematical Philosophy. Papers from the Studia Logica conference Trends in Logic IV*. Springer (pp. 269–296).Google Scholar - 40.Stalnaker, R. (1968). A theory of conditionals. In N. Rescher (Ed.),
*Studies in logical theory, american philosophical quarterly monograph series, II*(pp. 98–112). Cambridge: Blackwell.Google Scholar - 41.
- 42.Ullmann-Margalit, E., & Morgenbesser, S. (1977). Picking and choosing.
*Social Research*,*44*, 757–78.Google Scholar