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Belief Contraction in the Context of the General Theory of Rational Choice

  • Hans RottEmail author
Part of the Springer Graduate Texts in Philosophy book series (SGTP, volume 1)

Abstract

This paper reorganizes and further develops the theory of partial meet contraction which was introduced in a classic paper by Alchourrón, Gärdenfors and Makinson. Our purpose is threefold. First, we put it in a broader perspective by decomposing it into two layers which can respectively be treated by the general theory of choice and preference and elementary model theory. Second, we reprove the two main representation theorems of AGM and present two more representation results for the finite case that “lie between” the former, thereby partially answering an open question of AGM. Our method of proof is uniform insofar as it uses only one form of “revealed preference”, and it explains where and why the finiteness assumption is needed. Third, as an application, we explore the logic characterizing theory contractions in the finite case which are governed by the structure of simple and prioritized belief bases.

Keywords

Preference Relation Selection Function Rational Choice Contraction Function Belief Revision 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

I gratefully acknowledge numerous comments and suggestions by David Makinson which have once again been extremely helpful.

References

  1. Alchourrón, C., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet contraction functions and their associated revision functions. Journal of Symbolic Logic, 50, 510–530.CrossRefGoogle Scholar
  2. Alchourrón, C., & Makinson, D. (1986). Maps between some different kinds of contraction function: The finite case. Studia Logica, 45, 187–198.CrossRefGoogle Scholar
  3. Arrow, K. J. (1959). Rational choice functions and orderings. Economica, N.S., 26, 121–127.Google Scholar
  4. Chernoff, H. (1954). Rational selection of decision functions. Econometrica, 22, 422–443.CrossRefGoogle Scholar
  5. de Condorcet, N. (1785). Essai sur l’application de l’analyse á la probabilité des décisions rendues á la pluralité des voix. Paris: Imprimerie Royale. Reprint Cambridge: Cambridge University Press (2014).CrossRefGoogle Scholar
  6. Fuhrmann, A., & Morreau, M. (Eds.) (1991). The logic of theory change (Lecture notes in computer science, Vol. 465). Berlin: Springer.Google Scholar
  7. Gärdenfors, P. (1988). Knowledge in flux: Modeling the dynamics of epistemic states. Cambridge: Bradford Books/MIT.Google Scholar
  8. Gärdenfors, P. (Ed.) (1992). Belief revision. Cambridge: Cambridge University Press.Google Scholar
  9. Gärdenfors, P., & Makinson, D. (1988). Revisions of knowledge systems using epistemic entrenchment. In M. Vardi (Ed.), Theoretical aspects of reasoning about knowledge (pp. 83–95). Los Altos: Morgan Kaufmann.Google Scholar
  10. Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17, 157–170.CrossRefGoogle Scholar
  11. Hansson, S. O. (1992). Similarity semantics and minimal changes of belief. Erkenntnis, 37, 401–429.CrossRefGoogle Scholar
  12. Herzberger, H. G. (1973). Ordinal preference and rational choice. Econometrica, 41, 187–237.CrossRefGoogle Scholar
  13. Katsuno, H., & Mendelzon, A. O. (1991). Propositional knowledge base revision and minimal change. Artificial Intelligence, 52, 263–294.CrossRefGoogle Scholar
  14. Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44, 167–207.CrossRefGoogle Scholar
  15. Lehmann, D., & Magidor, M. (1992). What does a conditional knowledge base entail? Artificial Intelligence, 55, 1–60.CrossRefGoogle Scholar
  16. Lewis, D. (1973). Counterfactuals. Oxford: Blackwell.Google Scholar
  17. Lewis, D. (1981). Ordering semantics and premise semantics for counterfactuals. Journal of Philosophical Logic, 10, 217–234.CrossRefGoogle Scholar
  18. Lindström, S. (1991). A semantic approach to nonmonotonic reasoning: Inference and choice. University of Uppsala, April 1991 (manuscript).Google Scholar
  19. Makinson, D., & Gärdenfors, P. (1991). Relations between the logic of theory change and nonmonotonic logic. In Fuhrmann & Morreau (1991) (pp. 185–205).Google Scholar
  20. Nebel, B. (1989). A knowledge level analysis of belief revision. In R. Brachman, H. Levesque, & R. Reiter (Eds.), Proceedings of the 1st International Conference on Principles of Knowledge Representation and Reasoning (pp. 301–311). San Mateo: Morgan Kaufmann.Google Scholar
  21. Nebel, B. (1992). Syntax-based approaches to belief revision. In Gärdenfors (1992) (pp. 52–88).Google Scholar
  22. Rott, H. (1991). Two methods of constructing contractions and revisions of knowledge systems. Journal of Philosophical Logic, 20, 149–173.CrossRefGoogle Scholar
  23. Rott, H. (1992a). On the logic of theory change: More maps between different kinds of contraction function. In Gärdenfors (1992) (pp. 122–141).Google Scholar
  24. Rott, H. (1992b). Preferential belief change using generalized epistemic entrenchment. Journal of Logic, Language and Information, 1, 45–78.Google Scholar
  25. Rott, H. (2003). Basic entrenchment. Studia Logica, 73, 257–280.CrossRefGoogle Scholar
  26. Samuelson, P. A. (1950). The problem of integrability in utility theory. Economica, N.S., 17, 355–381.Google Scholar
  27. Sen, A. K. (1982) Choice, welfare and measurement. Oxford: Blackwell.Google Scholar
  28. Suzumura, K. (1983) Rational choice, collective decisions, and social welfare. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  29. Uzawa, H. (1956). Note on preference and axioms of choice. Annals of the Institute of Statistical Mathematics, 8, 35–40.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of RegensburgRegensburgGermany

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