# Cox's Theorem: controversy surrounding the proposition domain size

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 references are K. Van Horn, A Guide to Cox's Theorem, 2003 and A. Terenin & D. Draper, Rigorizing and Extending the Cox-Jaynes Derivation of Probability, 2015.

Halpern (A Counterexample to Theorems of Cox and Fine, 1999; Revisiting Cox's Theorem, 1999) has claimed to have constructed a counterexample to Cox's Theorem. Snow (The Reasonableness of Possibility from the Perspective of Cox, 2001) disagrees, saying that Cox implicitly made an assumption that negates the counterexample. Paris (The Uncertain Reasoner's Companion, 1994) seems to formalise this assumption, but the consequence is that the Cox-Jaynes probability function cannot have a finite domain.

Halpern considers this problematic, since it may not be natural to postulate an infinite proposition space. Snow and Van Horn each have their own arguments against Halpern's objection. Van Horn (A Guide to Cox's Theorem, 2003, p.12-13) remarks that even if we restrict ourselves to a finite number of propositions $(A|B)$, there can still be an infinite number of plausibility values if we don't restrict ourselves to a finite number of information states for that domain. Snow (On the Correctness and Reasonableness of Cox's Theorem for Finite Domains, 1998, section 4) gives examples of situations where an infinite domain is required to give sensible answers.

Terenin & Draper (Rigorizing and Extending the Cox-Jaynes Derivation of Probability, 2015, p.8) claim Van Horn's approach is difficult to evaluate formally since it lacks rigor. They do however, remark that:

Jaynes (2003) makes an important distinction between expressions of information that are ontological (e.g., ''there is noise in the room") describing the world as it is and those that are epistemological (e.g., ''the room is noisy") describing Your information about the world. There is no contradiction in assuming a finite number of world states (ontology), which is certainly true in some problems, and using an uncountably infinite number of propositions to describe Your uncertainty about those world states (epistemology); the latter is a modeling choice that we (and many other Bayesian statisticians) and to be extremely useful.

Which seems to in spirit agree with Van Horn, if I'm interpreting everything correctly.

I haven't seen anyone claim fault in Snow's position, though Jaynes (2003) notes:

It is very important to note that our Consistency Theorems have been established only for probabilities assigned on finite sets of propositions. In principle, every problem must start with such finite-set probabilities; extension to [countably] infinite sets is permitted only when this is the result of a well-defined and well-behaved limiting process from a finite set.

Which I'm not sure contradicts Snow, but at least questions the naturalness of infinite domains yet again.

S. Arnborg and G. Sjödin (Bayes Rules in Finite Models, 2000) show that Paris' assumption is not necessary to justify the theorem. They replace his assumption with weaker ones, and prove Cox' Theorem for finite domains. However, their method does not work for infinite domains, which seems somewhat problematic considering Snow's claim that infiniteness of the domain is natural.

Finally, Terenin and Draper claim to formalise Cox-Jaynes for uncountably infinite domains (2015, p.7), though I'm not sure whether this means infinite propositions, or just infinite plausibilities (in the information state sense of Van Horn, the epistemological sense of Jaynes, and/or the reasonable results sense of Snow).

In short, there seems to be a lot of disagreement here. It might be that I am misinterpreting statements though, so I would like for someone more knowledgeable to confirm.

In summary:

1. Is there a concensus regarding the finite-/(un)countably-infinite-ness of the domains in the Cox-Jaynes approach?

2. If not, what are the exact problems?