Perhaps Jaynes is trying here to prevent the reader from even thinking about these questions, but if so his strategy is more bold and unconventional than I can fathom.

His strategy is to make them look like trivial details, things that can be safely assumed, things that only a pedantic mathematician could care about, things that don't matter.

As for P(AB|C) = F[P(A|C),P(B|A&C)] that is equation 2.1. Jaynes considers an alternative in equ 2.2 and then discusses how to organize an exhaustive case split....

This part, in particular is what struck me as the most absolutely, monumentally awful part of the book. The other cases jaynes considers in his "exhaustive case split" are only a tiny, minuscule, arbitrary set of the things that P(AB|C) might depend on. Why should P(AB|C) not depend on the specific structure of the propositions themselves?

What bothers me so much about this part of the book isn't so much that the argument is incomplete, but that Jaynes is downright deceptive in his attempts to convince the reader that it is a complete rigorous justification for the Bayesian approach. Jaynes (and Eliezer) make it sound like Cox proved a generic Dutch book argument against anyone who doesn't use the Bayesian approach. There may indeed be such a theorem, but Cox's theorem just isn't it.