Monetary bets and degrees of belief (de Finetti (1937/1964)
Scoring rules and degrees of belief (de Finetti (1963, 1964), Savage (1971) and Lindley (1982))
Axiomatic approaches to degrees of belief (Bayes (1763), Cox (1946, 1961) and Jaynes (1958))
Bernardo and Smith (2000), page 86
"Why is probability a reasonable way of quantifying uncertainty? The following reasons are often advanced.
By analogy: physical randomness induces uncertainty, so it seems reasonable to describe uncertainty in the language of random events. Common speech uses many terms such as ‘probably’ and ‘unlikely,’ and it appears consistent with such usage to extend a more formal probability calculus to problems of scientific inference.
Axiomatic or normative approach: related to decision theory, this approach places all statistical inference in the context of decision-making with gains and losses. Then reasonable axioms (ordering, transitivity, and so on) imply that uncertainty must be represented in terms of probability. We view this normative rationale as suggestive but not compelling.
Coherence of bets. [...]
Gelman, et al. (2004), page 13
Degrees of belief and the probability calculus - fair betting quotients (their own, uses Dutch Book theorem)
The Standard Dutch Book Argument
Scoring Rules
Using a Standard (Lindley (1985, pp. 17-20))
The Cox-Good-Lucas Argument (Cox (1961), Good (1950, Appendix III), and Lucas (1970))
Introducing Utilities
Howson and Urbach (1993), chapter 5 in second edition
Abstract: "Arguments are adduced to support the claim that the only satisfactory description of uncertainty is probability. Probability is described both mathematically and interpretatively as a degree of belief. The axiomatic basis and the use of scoring rules in developing coherence are discussed. A challenge is made that anything that can be done by alternative methods for handling uncertainty can be done better by probability. This is demonstrated by some examples using fuzzy logic and belief functions. The paper concludes with a forensic example illustrating the power of probability ideas."
page 17: "Our thesis is simply stated: the only satisfactory description of uncertainty is probability. By this is meant that every uncertainty statement must be in the form of a probability; that several uncertainties must be combined using the rules of probability; and that the calculus of probabilities is adequate to handle all situations involving uncertainty. In particular, alternative descriptions of uncertainty are unnecessary. These include the procedures of classical statistics; rules of combination such as Jeffrey’s (1965); possibility statements in fuzzy logic, Zadeh (1983); use of upper and lower probabilities, Smith (1961), Fine (1973); and belief functions, Shafer (1976). We speak of “the inevitability of probability.”"
Conclusion: "Our argument may be summarized by saying that probability is the only sensible description of uncertainty and is adequate for all problems involving uncertainty. All other methods are inadequate. The justification for the position rests on the formal, axiomatic argument that leads to the inevitability of probability as a theorem and also on the pragmatic verification that probability does work. My challenge that anything that can be done with fuzzy logic, belief functions, upper and lower probabilities, or any other alternative to probability, can better be done with probability, remains."
Lindley (1987)
"The “inevitability of probability” is strengthened by the fact that three distinct axiom systems lead to that result. The first is firmly based on decision theory and is due to Ramsey (1931) and independently to Savage (1954). The second uses scoring rules, originating with de Finetti (1974/75) who also, in common with others, used a method based on betting. A third follows the usual mensuration procedure of comparison with a standard (Pratt, Raiffa and Schlaifer, 1964). There is an admirable survey by Shafer (1986; with discussion) that concentrates on Savage's method."
Lindley (1990), page 51
urn model
"A second justification for belief as probability emerges from some early work of Ramsey, de Finetti, Kemeny and Shimony (see [61], [19], [20], [37], [65]). The idea is to identify an expert’s belief [...] with his willingness to bet..."
"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)
Summary and Conclusion
Various distinct axiom systems developed to deal with uncertainty (the axiomatic approach, Dutch book argument, scoring rules and using a standard) all lead to probability.
The only satisfactory description of uncertainty is probability;
probability is adequate for all problems involving uncertainty; and
alternative descriptions of uncertainty are unnecessary and inadequate.