We discuss precise assumptions entailing Bayesianism in the line of investigations started by Cox, and relate them to a recent critique by Halpern. We show that every finite model which cannot be rescaled to probability violates a natural and simple refinability principle. A new condition, separability, was found sufficient and necessary for rescalability of infinite models.We finally characterise the acceptable ways to handle uncertainty in infinite models based on Cox's assumptions. Certain closure properties must be assumed before all the axioms of ordered fields are satisfied. Once this is done, a proper plausibility model can be embedded in an ordered field containing the reals, namely either standard probability (field of reals) for a real valued plausibility model, or extended probability (field of reals and infinitesimals) for an ordered plausibility model.

We discuss the justifications of Bayesianism by Cox and Jaynes, and relate them to a recent critique by Halpern(JAIR, vol 10(1999), pp 67–85). We show that a problem with Halperns example is that a finite and natural refinement of the model leads to inconsistencies, and that the same is the case with every model in which rescalability to probability cannot be done. We also discuss other problems with the justifications and assumptions usually made on the function F describing plausibility of conjunction. We note that the commonly postulated monotonicity condition should be strengthened to strict monotonicity before Cox justification becomes convincing. On the other hand, we note that the commonly assumed regularity requirements on F (like continuity) or its domain (like denseness) are unnecessary.

Of the many justifications of Bayesianism, most imply some assumption that is not very compelling, like the differentiability or continuity of some auxiliary function. We show how such assumptions can be replaced by weaker assumptions for finite domains. The new assumptions are a non-informative refinement principle and a concept of information independence. These assumptions are weaker than those used in alternative justifications, which is shown by their inadequacy for infinite domains. They are also more compelling.