On the foundations of Bayesianism

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.

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