Probabilistically stable revision and comparative probability: a representation theorem and applications

2025-09-02

AI Summary

This paper fully characterizes belief revision operators satisfying stability constraints within a probabilistic stability framework and provides a qualitative choice-function semantics for probabilistic stable belief revision logic. To model the dynamics of all-or-nothing beliefs, it integrates Bayesian updating with stability conditions; the revision framework is constructed over finite probability spaces using comparative probability theory, measure theory, and ratio-comparison logic. Key contributions are: (1) a necessary and sufficient characterization theorem for probabilistic stable belief revision operators; (2) a qualitative characterization of the strongest stable set operator, revealing its strong monotonicity in nonmonotonic reasoning; and (3) a representation theorem for the logic and necessary and sufficient conditions for joint representability by comparative probability orders—applied to voting games and revealed preference theory.

Technology Category

Knowledge Representation and Reasoning: Reasoning with BeliefsReasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyGame Theory and Economic Paradigms: Social Choice / Voting

Application Category

Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactionsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsUser Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

The stability rule for belief, advocated by Leitgeb [Annals of Pure and Applied Logic 164, 2013], is a rule for rational acceptance that captures categorical belief in terms of $ extit{probabilistically stable propositions}$: propositions to which the agent assigns resiliently high credence. The stability rule generates a class of $ extit{probabilistically stable belief revision}$ operators, which capture the dynamics of belief that result from an agent updating their credences through Bayesian conditioning while complying with the stability rule for their all-or-nothing beliefs. In this paper, we prove a representation theorem that yields a complete characterisation of such probabilistically stable revision operators and provides a `qualitative' selection function semantics for the (non-monotonic) logic of probabilistically stable belief revision. Drawing on the theory of comparative probability orders, this result gives necessary and sufficient conditions for a selection function to be representable as a strongest-stable-set operator on a finite probability space. The resulting logic of probabilistically stable belief revision exhibits strong monotonicity properties while failing the AGM belief revision postulates and satisfying only very weak forms of case reasoning. In showing the main theorem, we prove two results of independent interest to the theory of comparative probability: the first provides necessary and sufficient conditions for the joint representation of a pair of (respectively, strict and non-strict) comparative probability orders. The second result provides a method for axiomatising the logic of ratio comparisons of the form ``event $A$ is at least $k$ times more likely than event $B$''. In addition to these measurement-theoretic applications, we point out two applications of our main result to the theory of simple voting games and to revealed preference theory.

Problem

• Characterizing probabilistically stable belief revision operators
• Providing qualitative semantics for non-monotonic belief logic
• Establishing representation conditions for comparative probability orders

Innovation

• Representation theorem for probabilistically stable belief revision
• Qualitative selection function semantics for non-monotonic logic
• Comparative probability orders with joint representation conditions