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Disclaimer: I've just skimmed through Van Horn's post and paper, but fully read Clayton & Waddington's paper (which inspired Van Horn's).

I'm not sure about identifying states of information with compound propositions defined on a base of atomic propositions. C&W use such states to provide a family of examples that constrain probability rules allowing to lift some inelegant technical assumptions in Cox's theorem, while I understand that in VH states of information must be of this kind in general. Is this too restricting? Can't you have a consistent assignment of probabilities where the generating atomic propositions are not equally probable, or independent? But maybe you can always "rotate" to that basis.

Also, countable additivity is still missing. Jaynes would say this is not a real problem, but...

I should refrain from commenting further before I've carefully read the whole article.

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What do you think about Kevin S. Van Horn’s formulation of probability as extended logic? It looks interesting because it doesn’t seem to rely on the not-so-intuitive Universality axiom, removing which leads to Halpern’s counter-example, or conventions, axioms/desiderata which are not required by common sense, like representing plausibility using real numbers. Even David Chapman, who is highly critical of probability theory as extended logic, seems to be okay with it. Is it widely accepted or does it have conceptual issues like the original formulation?