I show that there is a common order-theoretic structure underlying many of the models for representing beliefs in the literature. After identifying this structure, and studying it in some detail, I argue that it is useful. On the one hand, it can be used to study the relationships between several models for representing beliefs, and I show in particular that the model based on classical propositional logic can be embedded in that based on the theory of coherent lower previsions. On the other hand, it can be used to generalise the coherentist study of belief dynamics (belief expansion and revision) by using an abstract order-theoretic definition of the belief spaces where the dynamics of expansion and revision take place. Interestingly, many of the existing results for expansion and revision in the context of classical propositional logic can still be proven in this much more abstract setting, and therefore remain valid for many other belief models, such as those based on imprecise probabilities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.J. Aumann, Utility theory without the completeness axiom, Econometrica 30 (1962) 445–462.
R.J. Aumann, Utility theory without the completeness axiom: A correction, Econometrica 32 (1964) 210–212.
S. Benferhat, D. Dubois and H. Prade, Possibilistic and standard probabilistic semantics of conditional knowledge bases, Journal of Logic and Computation 9 (1999) 873–895.
L. Boyen, G. de Cooman and E.E. Kerre, On the extension of P-consistent mappings, in: Foundations and Applications of Possibility Theory – Proceedings of FAPT '95, eds. G. de Cooman, D. Ruan and E.E. Kerre (World Scientific, Singapore, 1995) pp. 88–98.
A. Darwiche and J. Pearl, On the logic of iterated belief revision, Artificial Intelligence 89 (1997) 1–29.
B.A. Davey and H.A. Priestley, Introduction to Lattices and Order (Cambridge University Press, Cambridge, 1990).
G. de Cooman, Confidence relations and ordinal information, Information Sciences 104 (1997) 241–278.
G. de Cooman, Possibility theory I–III, International Journal of General Systems 25 (1997) 291–371.
G. de Cooman, Precision–imprecision equivalence in a broad class of imprecise hierarchical uncertainty models, Journal of Statistical Planning and Inference 105 (2002) 175–198.
G. de Cooman and P. Walley, A possibilistic hierarchical model for behaviour under uncertainty, Theory and Decision 52 (2002) 327–374.
B. de Finetti, Teoria delle Probabilità (Einaudi, Turin, 1970).
B. de Finetti, Theory of Probability, Vol. 1 (Wiley, Chichester, 1974). English Translation of [11].
D. Dubois and H. Prade, A survey of belief revision and updating rules in various uncertainty models, International Journal of Intelligent Systems 9 (1994) 61–100.
D. Dubois and H. Prade, Possibility theory: qualitative and quantitative aspects, in: Handbook on Defeasible Reasoning and Uncertainty Management Systems, Vol. 1: Quantified Representation of Uncertainty and Imprecision, ed. Ph. Smets (Kluwer, Dordrecht, 1998) pp. 169–226.
P. Gärdenfors, Knowledge in Flux – Modeling the Dynamics of Epistemic States (MIT, Cambridge, MA, 1988).
R. Giles, Foundations for a theory of possibility, in: Fuzzy Information and Decision Processes, eds. M.M. Gupta and E. Sanchez (North-Holland, Amsterdam, 1982) pp. 183–195.
F.J. Giron and S. Rios, Quasi-Bayesian behaviour: A more realistic approach to decision making? in: Bayesian Statistics, eds. J.M. Bernardo, M.H. DeGroot, D.V. Lindley and A.F.M. Smith (Valencia University Press, Valencia, 1980) pp. 17–38.
A.J. Grove, Two modellings for theory change, Journal of Philosophical Logic (1988) 157–170.
G. Kern-Isberner, The principle of conditional preservation in belief revision, in: Proceedings of the Second International Symposium on Foundations of Information and Knowledge Systems (FoIKS 2002), Lecture Notes in Computer Science 2284 (Springer, 2002), pp. 105–129.
I. Levi, The Enterprise of Knowledge (MIT, London, 1980).
D.K. Lewis, Counterfactuals (Harvard University Press, Cambridge, MA, 1973).
D.K. Lewis, Counterfactuals and comparative possibility, Journal of Philosophical Logic (1973) 2.
D. Makinson, General patterns in non-monotonic reasoning, in: Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 3, eds. D.M. Gabbay, C.H. Hogger and J.A. Robinson (Oxford University Press, Oxford, 1994) pp. 35–110.
S. Moral and N. Wilson, Revision rules for convex sets of probabilities, in: Mathematical Models for Handling Partial Knowledge in Artificial Intelligence, eds. G. Coletti, D. Dubois and R. Scozzafava (Plenum, New York, 1995) pp. 113–128.
R. Nau, The shape of incomplete preferences, in: ISIPTA '03 – Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications, eds. J.-M. Bernard, T. Seidenfeld and M. Zaffalon (Carleton Scientific, 2003) pp. 408–422.
R. Nau, The shape of incomplete preferences, Technical report, Decision Analysis Working Paper WP030009, September 2003. Extended version of [25].
D. Nute, Topics in Conditional Logic (Reidel Company, Dordrecht, Holland, 1980).
E. Schechter, Handbook of Analysis and Its Foundations (Academic Press, San Diego, California, 1997).
T. Seidenfeld, M.J. Schervish and J.B. Kadane, A representation of partially ordered preferences, The Annals of Statistics 23 (1995) 2168–2217. Reprinted in [30] pp. 69–129.
T. Seidenfeld, M.J. Schervish and J.B. Kadane, Rethinking the Foundations of Statistics (Cambridge University Press, Cambridge, 1999).
W. Spohn, Ordinal conditional functions: A dynamic theory of epistemic states, in: Causation in Decision, Belief Change and Statistics, Vol 2, eds. W.L. Harper and B. Skyrms (Reidel, Dordrecht, 1987) pp. 105–134.
P. Walley, Statistical Reasoning with Imprecise Probabilities (Chapman and Hall, London, 1991).
M.A. Williams, Transmutation of knowledge systems, in: Proceedings of the Fourth International Conference on Principles of Knowledge Representation and Reasoning (Morgan Kaufmann, 1994) pp. 619–629.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
de Cooman, G. Belief models: An order-theoretic investigation. Ann Math Artif Intell 45, 5–34 (2005). https://doi.org/10.1007/s10472-005-9006-x
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10472-005-9006-x
Keywords
- belief model
- belief revision
- classical propositional logic
- imprecise probability
- order theory
- possibility measure
- system of spheres