When epistemologists say that knowledge implies belief (see the entry on epistemology), they use a qualitative notion of belief that does not admit of degrees (except in the trivial sense that there is belief, disbelief, and suspension of judgment). The same is true for philosophers of language when they say that a normal speaker, on reflection, sincerely asserts ‘[ltr]AA[/ltr]’ only if she believes that [ltr]AA[/ltr] (Kripke 1979). This raises the question whether the notion of belief can be reduced to the notion of degree of belief. A simple thesis, known as the Lockean thesis, says that one should believe a proposition [ltr]AA[/ltr] just in case one’s degree of belief for [ltr]AA[/ltr] is sufficiently high (‘should’ takes wide scope over ‘just in case’). Of course, the question is which threshold is sufficiently high. We do not want to require that one only believe those propositions whose truth one assigns subjective probability 1 — especially if we follow Carnap (1962) and Jeffrey (2004) and require every subjective probability to be regular (otherwise we would not be allowed to believe anything except the tautology). We want to take into account our fallibility, the fact that our beliefs often turn out to be false.
Given that degrees of belief are represented as subjective probabilities, this means that the threshold for belief should be sufficiently large, but smaller than 1. In terms of subjective probabilities, the Lockean thesis says that an ideal doxastic agent with subjective probability Pr: [ltr]A→RA→ℜ[/ltr] believes [ltr]A∈AA∈A[/ltr] just in case [ltr]Pr(A)>1−εPr(A)>1−ε[/ltr] for some [ltr]ε∈ε∈[/ltr] (0,1]. [ltr](ε(ε[/ltr] is intended to be a number smaller than 1/2, but the argument to follow holds for any positive number in the unit interval.) This, however, leads to the lottery paradox (Kyburg 1961, and, much clearer, Hempel 1962; a different paradox that does not depend on degrees of belief is Makinson 1965’s preface paradox). For every threshold [ltr]ε∈ε∈[/ltr] (0,1] there is a finite partition [ltr]{A[size=13]1,…,An}{A1,…,An}[/ltr] of [ltr]AA[/ltr] and a reasonable subjective probability Pr: [ltr]A→RA→ℜ[/ltr] such that [ltr]Pr(Ai)>1−εPr(Ai)>1−ε[/ltr] for all [ltr]i=1,…,ni=1,…,n[/ltr], but [ltr]Pr(A1∩…∩An)<1−εPr(A1∩…∩An)<1−ε[/ltr]. For instance, let [ltr]ε=ε=[/ltr] .02 and consider a lottery with 100 tickets that is known for sure to be fair and such that exactly one ticket will win. Then it is reasonable, for every ticket [ltr]i=1,…,100i=1,…,100[/ltr], to assign a subjective probability of 1/100 to the proposition that ticket [ltr]ii[/ltr] will win. We thus believe of each single ticket that it will lose, because [ltr]Pr(W∖Ai)=.99>1−.02Pr(W∖Ai)=.99>1−.02[/ltr]. Yet we also know for sure that exactly one ticket will win. So [ltr]Pr(A1∪…∪An)=1>1−.02Pr(A1∪…∪An)=1>1−.02[/ltr]. We therefore believe both that at least one ticket will win [ltr](A1∪…∪An)(A1∪…∪An)[/ltr] as well as of each individual ticket that it will not win [ltr](W∖A1,…,W∖An)(W∖A1,…,W∖An)[/ltr]. Together these [ltr]n+1n+1[/ltr] beliefs form a belief set that is inconsistent in the sense that its intersection is empty, [ltr]⋂{A1∪…∪An,W∖A1,…,W∖An}=∅⋂{A1∪…∪An,W∖A1,…,W∖An}=∅[/ltr]. Yet consistency (and deductive closure, which is implicit in taking propositions rather than sentences as the objects of belief) have been regarded as the minimal requirements on a belief set ever since Hintikka (1961).[/size]
The lottery paradox has led some people to reject the notion of qualitative belief altogether (Jeffrey 1970), whereas others have been led to the idea that belief sets need not be deductively closed (Foley 1992; Foley 2009; see also Hawthorne 2004). Still others have turned the analysis on its head and elicit a context-dependent threshold parameter [ltr]εε[/ltr] from the agent’s belief set (Hawthorne and Bovens 1999; Hawthorne 2009). Another view is to take the lottery paradox at face value and postulate two doxastic attitudes towards propositions, viz. beliefs and degrees of beliefs, that are not reducible to each other. Frankish (2004; 2009) defends a particular version of this view (in addition, he distinguishes between a mind, where one unconsciously entertains beliefs, and a supermind, where one consciously entertains beliefs). Kroedel (2012) suggests to avoid the lottery paradox by considering justification a form of permissibility: an agent’s high subjective probability for a given proposition is not sufficient for believing this proposition, but merely for the permissibility of believing this proposition. The paradox is avoided, because being permitted to believe [ltr]AA[/ltr] and being permitted to believe [ltr]BB[/ltr] does not imply that one is permitted to believe the conjunction or intersection [ltr]A∩BA∩B[/ltr]. For further discussion on the relation between qualitative belief and probabilistic degrees of belief see Christensen (2004), Kaplan (1996), and Maher (2006). For a very different approach to combining qualitative notions from traditional epistemology with probabilistic notions see Moss (2013), who defends the thesis that properties of subjective probabilities can constitue knowledge.
Leitgeb (2013) proposes that an agent believes a proposition [ltr]BB[/ltr] if and only if there is a proposition [ltr]CC[/ltr] logically implying [ltr]BB[/ltr] such that the agent’s subjective probabilities for [ltr]CC[/ltr]conditional on any [ltr]AA[/ltr] consistent with [ltr]CC[/ltr] are above a certain threshold not smaller than 1/2. (Leitgeb 2014 relativizes this notion of belief to a question or partition. This makes it easier for an agent to have beliefs she is not certain of, but it has surprising consequences. Suppose an agent believes she has hands relative to the question of whether or not she has hands. This agent cannot lose her belief that she has hands relative to the question if she has hands or if she merely has the delusion of having hands or if she does not have hands in some other way. However, this agent can easily lose her belief that she has hands relative to the question if she has clean hands or if she has dirty hands or if she has no hands.) Leitgeb (2013)’s notion of belief satisfies the AGM axioms of belief revision presented below. However, as Lin & Kelly (2012) show, there is no “sensible” belief revision method that tracks conditionalization and satisfies these AGM axioms. This means that what an agent ends up believing according to Leitgeb (2013) and any other sensible proposal satisfying the AGM axioms if she first conditionalizes her subjective probabilities on evidence [ltr]EE[/ltr] and then extracts her beliefs will in general not coincide with what she ends up believing if she first extracts her beliefs from her subjective probabilities and then revises those beliefs by evidence [ltr]EE[/ltr]. For a different critique see Staffel (2015).
Lin & Kelly (2012) consider a mutually exclusive and jointly exhaustive set of alternative propositions. They suggest that an agent considers such an alternative proposition to be more plausible than another alternative proposition if and only if her subjective probability for the former is sufficiently higher than her subjective probability for the latter. According to them, the agent believes a proposition if and only if this proposition is implied by the disjunction of the most plausible alternative propositions. The agent believes a proposition after revision by evidence [ltr]EE[/ltr] if and only if this proposition is implied by the disjunction of the most plausible alternative propositions compatible with [ltr]EE[/ltr]. This method of belief revision violates the AGM axioms, but it is sensible and tracks conditionalization: what an agent ends up believing if she first conditionalizes her subjective probabilities on evidence [ltr]EE[/ltr] and then extracts her beliefs coincides with what she ends up believing if she first extracts her beliefs from her subjective probabilities and then revises those beliefs by evidence [ltr]EE[/ltr].