The justification of Bayes' rule by cognitive rationality principles is undertaken by extending the propositional axiom systems usually proposed in two contexts of belief change: revising and updating. Probabilistic belief change axioms are introduced, either by direct transcription of the set-theoretic ones, or in a stronger way but nevertheless in the spirit of the underlying propositional principles. Weak revising axioms are shown to be satisfied by a General Conditioning rule, extending Bayes' rule but also compatible with others, and weak updating (...) axioms by a General Imaging rule, extending Lewis' rule. Strong axioms (equivalent to the Miller–Popper axiom system) are necessary to justify Bayes' rule in a revising context, and justify in fact an extended Bayes' rule which applies, even if the message has zero probability. (shrink)

I am indebted to Zwirn and Zwirn [1989] for their extended and careful comments on the arguments of Popper & Miller [1983], [1987], and also for friendly and illuminating conversations. Their judgement seems to be that although Popper and I fail to make a satisfactory case for our conclusion that inductive probability is impossible, that conclusion is nonetheless defensible on quite other grounds. I don’t really agree with this, as I shall explain.