A recurrent theme in working with formal models of inference is the often puzzling connection between probabilistic and qualitative representations of information. A particularly salient case concerns the relationship between numerical credences and propositional all-or-nothing belief. In order to bridge the two, Leitgeb (following an earlier proposal by Skyrms) defends an acceptance rule based on the notion of probabilistically stable (or resilient) hypotheses. When applied to discrete probability spaces, the stability rule offers a possible formal solution to the Lottery Paradox and suggests a promising account of the relationship between subjective probabilities and qualitative belief. This talk will first address the problem of applying the stability rule to more realistic probability models: Bayesian statistical models featuring continuous distributions and, more generally, atomless probability spaces. Mierzewski will show that, for a very wide class of probabilistic learning problems, Leitgeb's rule yields a notion of acceptance that either fails to be conjunctive (accepted hypotheses are not closed under (finite) conjunctions) or is trivial (only hypotheses with probability one are accepted). An analogous result also affects refined notions of stability which take into account the evidence structure in the learning problem at hand.

More generally, as will be explained, similar difficulties affect a wide class of acceptance rules. This will be shown by presenting some (old and new) limitative theorems for acceptance, which suggest a fundamental tension between logical constraints on accepted hypotheses and invariance properties of acceptance rules. Roughly put, on probability spaces that are sufficiently rich in symmetries, desirable logical restrictions on acceptance rules conflict with the requirement that the rules respect certain natural transformations of the underlying probability model. These observations raise a number of questions about the question-sensitivity of acceptance, the aggregation of uncertain information in highly symmetric models, and the relationship between acceptance rules and statistical testing procedures.