Interpretation of Cox's Theorem I am breaking down Cox's Theorem in an attempt to understand it and its implications. This is my first time really getting into the details of probability theory, its history, the different schools of thought, and how these schools of thought relate to each other and the overall idea of "probability."
I'm reading in several different places that Cox's Theorem was an attempt to prove that there are systems "like" probability theory (e.g. here, Van Horn), namely (I suppose) classical propositional calculus.
Moreover, Skilling (1997) stated:

Probability calculus is unique: it is the only calculus within which uncertainties about propositions are manipulated consistently with the logical
  (TRUE/FALSE) status of the propositions... Kolmogorov
  (1950) is widely quoted as the author of the axiomatic basis of probability calculus, but it was R.T. Cox (1946, 1961) who showed that no other
  calculus is admissible... It follows that other methods are either
  equivalent to probability calculus (in which case they are unnecessary), or
  are wrong

I'm failing to find this result in Cox's 1946 paper, though. From my perspective, I see Cox lay out several fascinating relationships between symbolic logic and the rules of probability theory (e.g. the sum rule, product rule, negation). I see how he arrived at an alternative derivation of the rules of probability from propositional calculus. 
But, how has this proven that probability theory is unique and that "no other calculus is admissible"? How did the proofs he laid forth result in the conclusion: "abstract reasoning under uncertainty is isomorphic to finitely additive probability theory" (Terinin and Draper). Perhaps, what's another way of wording the results of Cox (1946) to illuminate these conclusions (pointing me to a reference is all I'm asking)?
 A: Probably for my own good, I'll attempt to answer the question. Hopefully someone who knows more will be able to correct me where I'm wrong.
I asked, "How did the proofs [Cox] laid forth result in the conclusion: 'abstract reasoning under uncertainty is isomorphic to finitely additive probability theory'". Moreover, why does the Theorem claim that probability theory is unique?
To my understanding, Cox's Theorem provides two explicit, major requirements for a system of plausible reasoning. Should any system of reasoning meet these two requirements (plus satisfy some assumptions along the way), that system must then be equivalent to a system of probability. These have been expounded by many since his publication, as is found in Van Horn.
Cox seems inspired by propositional calculus (for anyone new to this: nothing related to differentiation and integrals), which is a system of deductive logic. Propositional logic is based upon propositions, such as A) "If it's raining outside then it's cloudy" and B) "It's raining." From these two premises, one can conclude if B and A, then "It is cloudy." This is a logical deduction the calculus allows.
Using these ideas of propositional calculus, Cox reasons about how they might build up a system of probability under the same rules that were derived by Kolmogorov. For example, Cox puts together proofs for $p(c \cdot b|a) = p(c|b\cdot a) \cdot p(b|a)$ (i.e. the product rule), $p(c \cup b | a) = p(c|a) + p(b|a) - p(c \cdot b | a)$ (i.e. sum of non-mutually exclusive propositions), and $p(a) + p(\neg a) = 1$ (i.e. negation).
Despite these results already being formulated by Kolmogorov, Cox derived them from a different philosophy. Namely, logic. Kolmogorov had come to them by assumptions about what happens given infinite events/samples. Consequently, Kolmogorov's probability theory was know as frequentist because it dealt with frequencies of events over many trials. Cox's was built from propositional calculus and was tied less to events and more to propositions with degrees of belief.
Yet, propositional calculus does not handle degrees of belief. In fact, it is strictly about truth values. True or False. Importantly, Cox "extends" this to create a calculus which accounted for degrees of beliefs in propositions. Hence, Cox's theorem was important because it provided an alternative interpretation of probability less bound to infinitely sampled spaces and instead relied on deductive logic and the idea that "belief" can be quantified and represented as probability.
Now, my question: how is it that no other calculus is admissible because reasoning with uncertainty is isomorphic to probability theory? I think, instead of interpreting this as "no other calculus is admissible" because nothing else is possible, one should interpret it as, no other probability calculus is unique because by Cox's Theorem they are all one-in-the-same. Reasoning under uncertainty, i.e. performing logical deductions outside of the True/False paradigm, is thus proven to actually be an interpretation of probability.
