Belief models: An order-theoretic investigation

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.

[1]  Mary-Anne Williams,et al.  Transmutations of Knowledge Systems , 1994, KR.

[2]  Didier Dubois,et al.  Mathematical models for handling partial knowledge in artificial intelligence , 1995 .

[3]  Dov M. Gabbay,et al.  Handbook of logic in artificial intelligence and logic programming (vol. 1) , 1993 .

[4]  Joseph B. Kadane,et al.  Rethinking the Foundations of Statistics: Subject Index , 1999 .

[5]  Robert Nau.  The Shape of Incomplete Preferences , 2003, ISIPTA.

[6]  Adam J. Grove,et al.  Two modellings for theory change , 1988, J. Philos. Log..

[7]  D. Nute.  Topics in Conditional Logic , 1980 .

[8]  Didier Dubois,et al.  Possibility Theory: Qualitative and Quantitative Aspects , 1998 .

[9]  Gert de Cooman,et al.  POSSIBILITY THEORY II: CONDITIONAL POSSIBILITY , 1997 .

[10]  Didier Dubois,et al.  Possibilistic and Standard Probabilistic Semantics of Conditional Knowledge Bases , 1997, J. Log. Comput..

[11]  Dov M. Gabbay,et al.  Handbook of defeasible reasoning and uncertainty management systems: volume 2: reasoning with actual and potential contradictions , 1998 .

[12]  Nic Wilson,et al.  Revision rules for convex sets of probabilities , 1995 .

[13]  Gabriele Kern-Isberner.  The Principle of Conditional Preservation in Belief Revision , 2002, FoIKS.

[14]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[15]  Gert de Cooman.  Confidence Relations and Ordinal Information , 1998, Inf. Sci..

[16]  Peter Walley,et al.  A possibilistic hierarchical model for behaviour under uncertainty , 2002 .

[17]  David Makinson,et al.  General patterns in nonmonotonic reasoning , 1994 .

[18]  J. Pearl,et al.  On the Logic of Iterated Belief Revision , 1994, Artif. Intell..

[19]  Isaac Levi,et al.  The Enterprise Of Knowledge , 1980 .

[20]  P G rdenfors,et al.  Knowledge in flux: modeling the dynamics of epistemic states , 1988 .

[21]  Gert de Cooman,et al.  On the extension of P-consistent mappings , 1995 .

[22]  Donald Nute,et al.  Counterfactuals , 1975, Notre Dame J. Formal Log..

[23]  Francisco Javier Girón González-Torre,et al.  Quasi-Bayesian behaviour: a more realistic approach to decision making? , 1980 .

[24]  F. J. Girón,et al.  Quasi-Bayesian Behaviour: A more realistic approach to decision making? , 1980 .

[25]  R. Aumann.  UTILITY THEORY WITHOUT THE COMPLETENESS AXIOM , 1962 .

[26]  M. Schervish,et al.  A Representation of Partially Ordered Preferences , 1995 .

[27]  Wolfgang Spohn,et al.  Ordinal Conditional Functions: A Dynamic Theory of Epistemic States , 1988 .

[28]  Robert J. Aumann,et al.  UTILITY THEORY WITHOUT THE COMPLETENESS AXIOM: A CORRECTION , 1964 .

[29]  Eric Schechter,et al.  Handbook of Analysis and Its Foundations , 1996 .

[30]  G. Cooman.  POSSIBILITY THEORY I: THE MEASURE- AND INTEGRAL-THEORETIC GROUNDWORK , 1997 .

[31]  Gert de Cooman,et al.  Precision–imprecision equivalence in a broad class of imprecise hierarchical uncertainty models , 2002 .

[32]  Didier Dubois,et al.  A survey of belief revision and updating rules in various uncertainty models , 1994, Int. J. Intell. Syst..

[33]  H. Jeffreys,et al.  Theory of probability , 1896 .

[34]  David Lewis.  Counterfactuals and Comparative Possibility , 1973 .

[35]  M. Gupta,et al.  FUZZY INFORMATION AND DECISION PROCESSES , 1981 .

[36]  Gert de Cooman,et al.  POSSIBILITY THEORY III: POSSIBILISTIC INDEPENDENCE , 1997 .

[37]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .