I've been trying to understand Cox's Theorem and the problems surrounding it. There's so much information on this topic that I've become confused as to the exact state of the theorem. I've gathered there are three main issues, but since they cover a broad variety of topics I've split it up into multiple questions. I hope this is sufficiently narrowed down (the original question).
My main reference is K. Van Horn, A Guide to Cox's Theorem, 2003.
The assumption that a plausibility $(A|B)$ (A given B) can always be represented by a real number is often a point of controversy for various reasons (see here for the other two issues with this assumption).
The first is that this assumes universal comparability between propositions, which some find unwarranted. I know Jaynes (Probability Theory as Extended Logic, 2003, Appendix A) argues for universal comparability, though I'm not entirely sure that his argument is sufficient.
The second issue with this assumption is something that I think ties into the first, namely that representing belief in proposition $(A|B)$ as a real number, says nothing about our doubt of $(A|B)$; $(\neg A|B)$. This objection is often raised in the context of belief-function theory (which I'm not really familiar with) (G. Shafer, A Mathematical Theory of Evidence, 1976).
However, further on in his proof Cox postulates that a function $S$ exists such that $(\neg A|B) = S(A|B)$, which seems to show that we can express doubt in terms of belief. For that reason, I don't really understand the objection. Is this just a matter of which axioms we should find reasonable?
Is Jaynes' argument for universal comparability considered sufficient?
What is the problem with one-dimensional theory not saying anything about doubt?