On Bayesian Induction (and Pythagoric Arithmetic)

Colin Howson has called “Hume’s Other Principle” the idea that, in order to make sound inferences from observational data, we need “at least one independent assumption (an inductive assumption) that ... weights some of the possibilities consistent with the evidence more than others” (Howson, this volume, p. 315). I agree with Howson that subjectivist Bayesians are constructive skeptics in the sense that they admit, with Hume, that the circularity of any inductive argument can be broken only by an independent premise, and I would like to show that this premise, in the case of inferences from observed frequencies, is de Finetti’s exchangeability and its generalisation, partial exchangeability.