Cox's theorem

1. Belief (degree of plausibility) is represented by a real number
1. Divisibility and comparability - The plausibility of a proposition is a real number and is dependent on information we have related to the proposition.

2. Qualitative correspondence with common sense, and with logic
1. Common sense – Plausibilities should vary sensibly with the assessment of plausibilities in the model.

3. Consistency – If the plausibility of a proposition can be derived in many ways, all the results must be equal.

These

Axioms as

Desiderata can results

• Additivity Rule / Sum rule -
  Marginalization

• Multiplication Rule / Product rule -
  Bayes Theorem

Cox's theorem

Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates.[1][2] This derivation justifies the so-called "logical" interpretation of probability, as the laws of probability derived by Cox's theorem are applicable to any proposition. Logical (also known as objective Bayesian) probability is a type of Bayesian probability. Other forms of Bayesianism, such as the subjective interpretation, are given other justifications.

https://en.wikipedia.org/wiki/Cox's_theorem