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

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 $\textit{probabilistically stable propositions}$: propositions to which the agent assigns resiliently high credence. The stability rule generates a class of $\textit{probabilistically stable belief revision}$ 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.