# Cox's Theorem: the necessity of (un)countably additivity

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).

This one issue seems kind of obscure. There appears to be a problem with countable additivity in Cox's Theorem, which only A. Terenin & D. Draper (Rigorizing and Extending the Cox-Jaynes Derivation of Probability, 2015) mention in the sources I could find. Indeed, it seems most of their motivation for writing the paper is to add countable additivity to the Cox-Jaynes approach. My guess is that this is necessary when discussing domains of propositions that are countably infinite in size, which other authors don't explicitly deal with. K. Van Horn (A Guide to Cox's Theorem, 2003) mentions the controversy surrounding the domain size in Cox's Theorem, but leaves it as an unsolved problem (for that issue see here).

From what I understand there's also arguments in favor of uncountably infinite domains. For example, Snow (On the Correctness and Reasonableness of Cox's Theorem for Finite Domains, 1998, section 4) gives examples of situations where uncountably infinite domains are required to give sensible answers. Terenin and Draper claim to address the uncountable case as well, but I can only find a proof of countable additivity.

I'm not especially familiar with these concepts, and I wonder whether countable additivity a sufficient condition to reason about uncountably infinite sets of propositions. If not, then what am I missing?

EDIT: At request, a little more specific: the Cox-Jaynes approach involves a plausibility function $\mathbb{P}_{CJ} : (\mathscr{C} \times \mathscr{C}) \rightarrow [0,1]$ that maps a proposition $(A|B) \in (\mathscr{C} \times \mathscr{C})$ to a real number in $[0,1]$. $\mathscr{C}$ is the set of propositions such as $A$ or $B$. There's some controversy concerning the cardinality of the domain of $\mathbb{P}_{CJ}$ (here), which is the set of conditionals we're reasoning about. The most general case would be that the domain has no cardinality restrictions, which is also the case that Terenin and Draper argue. Consequently, they prove both finite and countable additivity for their edited version of Cox's axioms. I don't have much experience with this field, and I'm left wondering whether countable additivity is sufficient when the domain has no cardinality restrictions.

• This question appears to be confounding three distinct and different things: countable additivity, a countable domain, and countable sets of propositions. Could you clarify the sense(s) in which you think they are all related in this context? – whuber Jan 11 '16 at 20:28
• Of course! I've added a bit that hopefully explains the connection as I see it. – Timsey Jan 11 '16 at 21:32