Subjective probability theory is the best developed account of degrees of belief. As a consequence, there is much more material to be presented here than in the case of other accounts. This section is structured into six subsections. The topics of these subsections will also be discussed in the sections on Dempster-Shafer theory, possibility theory, ranking theory, belief revision theory, and nonmonotonic reasoning. However, as there is less (philosophical) literature about these latter accounts, there will not be separate subsections there.

2.1 The Formal Structure

Sophia believes to degree .55 that tomorrow it will be sunny in Vienna. Normally degrees of belief are taken to be real numbers from the interval [0,1], but we will consider an alternative below. If the ideal doxastic agent is certain that a proposition is true, her degree of belief for this proposition is 1. If the ideal doxastic agent is certain that a proposition is false, her degree of belief for this proposition is 0. However, these are extreme cases. Usually we are not certain that a proposition is true. Nor are we usually certain that a proposition is false. That does not mean, though, that we are agnostic with respect to the question whether the proposition we are concerned with is true. Our belief that it is true may well be much stronger than our belief that it is false. Degrees of belief quantify this strength of belief.
The dominant theory of degrees of belief is the theory of subjective probabilities (for an accessible exposition see Easwaran 2011a, 2011b). On this view, degrees of belief simply follow the laws of probability. Here is the standard definition due to Kolmogorov (1956). Let [ltr]AA[/ltr] be a field of propositions over a set [ltr]WW[/ltr] of possibilities. A function Pr: [ltr]A→RA→ℜ[/ltr] from [ltr]AA[/ltr] into the set of real numbers, [ltr]Rℜ[/ltr], is a (finitely additive and non-conditional) probability measure on [ltr]AA[/ltr] if and only if for all propositions [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr]:
The triple [ltr]⟨W,A,Pr⟩⟨W,A,Pr⟩[/ltr] is a (finitely additive) probability space. Suppose [ltr]AA[/ltr] is also closed under countable intersections (and thus a [ltr]σσ[/ltr]-field). Suppose Pr additionally satisfies, for all propositions [ltr]A[size=13]1,…An,…A1,…An,…[/ltr] in [ltr]AA[/ltr],[/size]
Then Pr is a [ltr]σσ[/ltr]- or countably additive probability measure on [ltr]AA[/ltr] (Kolmogorov 1956, ch. 2, actually gives a different but equivalent definition; see e.g. Huber 2007a, sct. 4.1). In this case [ltr]⟨W,A,Pr⟩⟨W,A,Pr⟩[/ltr] is a [ltr]σσ[/ltr]- or countably additive probability space.
A probability measure Pr on [ltr]AA[/ltr] is regular just in case [ltr]Pr(A)>0Pr(A)>0[/ltr] for every non-empty or consistent proposition [ltr]AA[/ltr] in [ltr]AA[/ltr]. Let [ltr]A[size=13]PrAPr[/ltr] be the set of all propositions [ltr]AA[/ltr] in [ltr]AA[/ltr] with [ltr]Pr(A)>0Pr(A)>0[/ltr]. The conditional probability measure [ltr]Pr(⋅∣−):A×APr→RPr(⋅∣−):A×APr→ℜ[/ltr] on [ltr]AA[/ltr] (based on the non-conditional probability measure Pr on [ltr]A)A)[/ltr] is defined for all pairs of propositions [ltr]AA[/ltr] in [ltr]AA[/ltr] and [ltr]BB[/ltr]in [ltr]APrAPr[/ltr] by the ratio[/size]
(Kolmogorov 1956, ch. 1, §4). The domain of the second argument place of [ltr]Pr(⋅∣−)Pr(⋅∣−)[/ltr] is restricted to [ltr]A[size=13]PrAPr[/ltr], since the ratio [ltr]Pr(A∩B)/Pr(B)Pr(A∩B)/Pr(B)[/ltr] is not defined if [ltr]Pr(B)=0Pr(B)=0[/ltr]. Note that [ltr]Pr(⋅∣B)Pr(⋅∣B)[/ltr] is a probability measure on [ltr]AA[/ltr], for every proposition [ltr]BB[/ltr] in [ltr]APrAPr[/ltr]. Some authors take conditional probability measures [ltr]Pr(⋅,given −):A×(A∖{∅})→RPr(⋅,given −):A×(A∖{∅})→ℜ[/ltr] as primitive and define (non-conditional) probability measures in terms of them as [ltr]Pr(A)=Pr(APr(A)=Pr(A[/ltr], given [ltr]W)W)[/ltr]for all propositions [ltr]AA[/ltr] in [ltr]AA[/ltr] (see Hájek 2003). Conditional probabilities are usually assumed to be Popper-Rényi measures (Popper 1955, Rényi 1955, Rényi 1970, Stalnaker 1970, Spohn 1986). Spohn (2012, 202ff) critizices Popper-Rényi measures for their lack of a complete dynamics, a feature already pointed out by Harper (1976), and for their lack of a reasonable notion of independence. Relative probabilities (Heinemann 1997, Other Internet Resources) are claimed not to suffer from these two shortcomings.[/size]

2.2 Interpretations

What does it mean to say that Sophia’s subjective probability for the proposition that tomorrow it will be sunny in Vienna equals .55? This is a difficult question. Let us first answer a different one. How do we measure Sophia’s subjective probabilities? On one account Sophia’s subjective probability for [ltr]AA[/ltr] is measured by her betting ratio for [ltr]AA[/ltr], i.e., the highest price she is willing to pay for a bet that returns $1 if [ltr]AA[/ltr], and $0 otherwise. On a slightly different account Sophia’s subjective probability for [ltr]AA[/ltr] is measured by her fair betting ratio for [ltr]AA[/ltr], i.e., that number [ltr]r=b/(a+b)r=b/(a+b)[/ltr] such that she considers the following bet to be fair: $[ltr]aa[/ltr] if [ltr]AA[/ltr], and $[ltr]−b−b[/ltr] otherwise [ltr](a,b≥0(a,b≥0[/ltr] with inequality for at least one). As we may say it: Sophia considers it to be fair to bet you [ltr]bb[/ltr] to [ltr]aa[/ltr] that [ltr]AA[/ltr].
It need not be irrational for Sophia to be willing to bet you $5.5 to $4.5 that tomorrow it will be sunny in Vienna, but not be willing to bet you $550,000 to $450,000 that this proposition is true. This reveals one assumption of the measurement of probabilistic degrees of belief in terms of (fair) betting ratios: the ideal doxastic agent is assumed to be neither risk averse nor risk prone. Gamblers in the casino are risk prone: they pay more for playing roulette than the fair monetary value according to reasonable subjective probabilities (this may be perfectly rational if the additional cost is what the gambler is willing to spend on the thrill she gets out of playing roulette). Sophia, on the other hand, is risk averse when she refuses to bet you $100,000 to $900,000 that it will be sunny in Vienna tomorrow, while she is happy to bet you $5 to $5 that this proposition is true. This may be perfectly rational as well: as a moderately wealthy philosopher, she might lose her standard of living along with this bet. Note that it does not help to say that Sophia’s fair betting ratio for [ltr]AA[/ltr] is that number [ltr]r=b/(a+b)r=b/(a+b)[/ltr] such that she considers the following bet to be fair: $[ltr]1−r=a/(a+b)1−r=a/(a+b)[/ltr] if [ltr]AA[/ltr], and $[ltr]−r=−b/(a+b)−r=−b/(a+b)[/ltr] otherwise [ltr](a,b≥0(a,b≥0[/ltr] with inequality for at least one). Just as stakes of $1,000,000 may be too high for the measurement to work, stakes of $1 may be too low.
Another assumption is that the agent’s (fair) betting ratio for a proposition is independent of the truth value of this proposition. Obviously we cannot measure Sophia’s subjective probability for the proposition that she will be a billionaire by the end of the week by offering her a bet that returns $1 if she will, and $0 otherwise. Sophia’s subjective probability for being a billionaire by the end of the week will be fairly low. However, assuming her to be rational and that being abillionaire is something she desires, her betting ratio for this proposition will be fairly high.
Ramsey (1926) avoids the first assumption by using utilities instead of money. He avoids the second assumption by presupposing the existence of an “ethically neutral” proposition (a proposition whose truth or falsity does not affect the agent’s utilities) which the agent takes to be just as likely to be true as she takes it to be false. See Section 3.5 of the entry on interpretations of probability.

Let us return to our question of what it means for Sophia to assign a certain subjective probability to a given proposition. It is one thing for Sophia to be willing to bet at particular odds or to consider particular odds as fair. It is another thing for Sophia to have a subjective probability of .55 that tomorrow it will be sunny in Vienna. Sophia’s subjective probabilities are measured by, but not identical to her (fair) betting ratios. The latter are operationally defined and observable. The former are unobservable, theoretical entities that, following Eriksson & Hájek (2007), we should take as primitive.