Cox wanted his system to satisfy the following conditions: The postulates as stated here are taken from Arnborg and Sjödin.
The postulates as originally stated by Cox were not mathematically rigorous (although more so than the informal description above), as noted by Halpern.
[6][7] However it appears to be possible to augment them with various mathematical assumptions made either implicitly or explicitly by Cox to produce a valid proof.
such that It is important to note that the postulates imply only these general properties.
We may recover the usual laws of probability by setting a new function, conventionally denoted
The measure-theoretic formulation of Kolmogorov assumes that a probability measure is countably additive.
[9] Cox's theorem has come to be used as one of the justifications for the use of Bayesian probability theory.
For example, in Jaynes it is discussed in detail in chapters 1 and 2 and is a cornerstone for the rest of the book.
[8] Probability is interpreted as a formal system of logic, the natural extension of Aristotelian logic (in which every statement is either true or false) into the realm of reasoning in the presence of uncertainty.
It has been debated to what degree the theorem excludes alternative models for reasoning about uncertainty.
For example, if certain "unintuitive" mathematical assumptions were dropped then alternatives could be devised, e.g., an example provided by Halpern.
[6] However Arnborg and Sjödin[3][4][5] suggest additional "common sense" postulates, which would allow the assumptions to be relaxed in some cases while still ruling out the Halpern example.
János Aczél[13] provides a long proof of the "associativity equation" (pages 256-267).
Jaynes[8]: 27 reproduces the shorter proof by Cox in which differentiability is assumed.
A guide to Cox's theorem by Van Horn aims at comprehensively introducing the reader to all these references.
[14] Baoding Liu, the founder of uncertainty theory, criticizes Cox's theorem for presuming that the truth value of conjunction
, which excludes uncertainty theory's "uncertain measure" from its start, because the function