  There are multiple proposed interpretations of probability theory: one such
interpretation is true-false logic under uncertainty. Cox's Theorem is a
representation theorem that states, under a certain set of axioms describing
the meaning of uncertainty, that every true-false logic under uncertainty is
isomorphic to conditional probability theory. This result was used by Jaynes to
develop a philosophical framework in which statistical inference under
uncertainty should be conducted through the use of probability, via Bayes'
Rule. Unfortunately, most existing correct proofs of Cox's Theorem require
restrictive assumptions: for instance, many do not apply even to the simple
example of rolling a pair of fair dice. We offer a new axiomatization by
replacing various technical conditions with an axiom stating that our theory
must be consistent with respect to repeated events. We discuss the implications
of our results, both for the philosophy of probability and for the philosophy
of statistics.
