I've tried to do something similar with odds once, but the assumption about (AB|C) = F[(A|C), (B|AC)] made me give up.

Indeed, one can calculate O(AB|C) given O(A|C) and O(B|AC) but the formula isn't pretty. I've tried to derive that function but failed. It was not until I appealed to the fact that O(A)=P(A)/(1-P(A)) that I managed to infer this unnatural equation about O(AB|C), O(A|C) and O(B|AC).

And this use of classical probabilities, of course, completely defeats the point of getting classical probabilities from the odds via Cox's Theorem!

Did I miss something?

By the way, are there some other interesting natural rules of inference besides odds and log odds which are isomorphic to the rules of probability theory? (Judea Pearl mentioned something about MYCIN certainty factor, but I was unable to find any details)

EDIT: You can view the CF combination rules here, but I find it very difficult to digest. Also, what about initial assignment of certainty?

EDIT2: Nevermind, I found an adequate summary ( http://www.idi.ntnu.no/~ksys/NOTES/CF-model.html ) of the model and pdf ( http://uai.sis.pitt.edu/papers/85/p9-heckerman.pdf ) about probabilistic interpretations of CF. It seems to be an interesting example of not-obviously-Bayesian system of inference, but it's not exactly an example you would give to illustrate the point of Cox's theorem.