"Similar criticisms seem to us to apply to the original and elegant formal development given by Cox (1946, 1961) and Jaynes (1958), who showed that the probability axioms constitute the only consistent extension of ordinary (Aristotelian) logic in which degrees of belief are represented by real numbers."
Bernardo and Smith, 2000, page 90
"R. T. Cox (1946) published a paper that showed that any set of rules for inference, in which we
represent degrees of plausibility by real numbers, is necessarily either equivalent to the Laplace-
Jeffreys rules, that is (1)-(3), or inconsistent."
Evans (2002)
"Cox’s argument (see Chapter 2) for obtaining a quantitative representation for qualitative or comparative axioms about conditioning, a goal essential to Jaynes’s program, has been re-examined carefully by Halpern (J.Y. Halpern, “Cox’s Theorem Revisited,” Journal of Artificial Intelligence Research, Vol. 11, 1999, pp. 429-435). When we restrict ourselves to finitely many propositions, as advocated by Jaynes, some technicalities must be addressed if we are to reach the conclusion desired by Jaynes; in my view and Halpern’s, the assumptions required are not as compelling as glossing over them makes them appear."
Fine, 2004 (book review)
"The reason for this is clear to one who has studied the theorems of R. T. Cox (1946, 1961). He shows that any method of pausible reasoning in which we represent degrees of plausibility by real numbers, is necessarily either equivalent to Laplace's, or inconsistent."
Jaynes, "Jaynes' reply to Kempthorne's Comments", In: Rosenkrantz (1989), pages 192-193,
"Thus, Cox proved that any method of inference in which we represent degrees of plausibility by real numbers, is necessarily either equivalent to Laplace's, or inconsistent."
Jaynes, "Where do we stand on Maximum Entropy?", In: Rosenkrantz (1989), page 220
"Cox proved that it is the only consistent extension of logic in which degrees of plausibility are represented by real numbers"
Jaynes, "What is the Question?", In: Rosenkrantz (1989), page 382
"It is clear that, not only is the quantitative use of the rules of probability theory as extended logic the only sound way to conduct inference; is is the failure to follow those rules strictly that has for many years been leading to unnecessay errors, paradoxes, and controversies."
Jaynes (2003), page 143
"...the rules of probability theory can be derived as necessary conditions for consistency, as expressed by Cox’s functional equations." Jaynes (2003), page 486
"A third justification for belief as probability (or at least a scaled version of probability) appeared in a paper by R.T. Cox in the American Journal of Physics in 1946 [9]. Cox's proof is not, perhaps, as rigorous as some pedants might prefer and when an attempt is made to fill in all the details some of the attractiveness of the original is lost. Nevertheless his results certainly provide a valuable contribution to our understanding of the nature of belief.
We state here a rigorous version of Cox's main theorem which has aspects which are both stronger and weaker than the original. Slightly stronger versions still can be proved but the increased complications do not seem to justify doing so."
Paris, 1994, page 24
"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. The only freedom is to take some monotonic function instead, such as 100 Pr() (percentage) or Pr()=(1 ?? Pr()) (odds), but such changes are merely cosmetic. It follows that other methods are either equivalent to probability calculus (in which case they are unnecessary), or are wrong."
Skilling, 1998