By basing Bayesian probability theory on five axioms, we can give a trivial proof of Cox's Theorem on the product rule and sum rule for conditional plausibility without assuming continuity or differentiablity of plausibility. Instead, we extend the notion of plausibility to apply to unknowns, giving them plausible values. Thus, we combine the best aspects of two approaches to Bayesian probability theory, namely the Cox-Jaynes theory and the de Finetti theory.
"New axioms for rigorous Bayesian probability." Bayesian Anal. 4 (3) 599 - 606, 2009. https://doi.org/10.1214/09-BA422