Epistemology is the study of knowledge and justified belief. Belief is thus central to epistemology. It comes in a qualitative form, as when Sophia believes that Vienna is the capital of Austria, and a quantitative form, as when Sophia’s degree of belief that Vienna is the capital of Austria is at least twice her degree of belief that tomorrow it will be sunny in Vienna. Formal epistemology, as opposed to mainstream epistemology (Hendricks 2006), is epistemology done in a formal way, that is, by employing tools from logic and mathematics. The goal of this entry is to give the reader an overview of the formal tools available to epistemologists for the representation of belief. A particular focus will be on the relation between formal representations of qualitative belief and formal representations of quantitative degrees of belief.2. Subjective Probability Theory
2.1 The Formal Structure
2.2 Interpretations
2.3 Justifications
2.4 Update Rules
2.5 Ignorance
2.6 Qualitative Belief
3. Other Accounts
3.1 Dempster-Shafer Theory
3.2 Possibility Theory
3.3 Ranking Theory
3.4 Belief Revision Theory
3.5 Nonmonotonic Reasoning
Bibliography
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[size=30]1. Preliminaries
1.1 Formal Epistemology versus Mainstream Epistemology
One can ask many questions about belief and the relation between belief and degrees of belief. Many of them can be asked and answered informally as well as formally. None of them can be asked or answered only informally in the sense that it would be
logically impossible to ask or answer them formally — although it is, of course, often impossible
for us to do so. Just think of how you would come up with a counterexample to the claim that some questions can be asked or answered only informally. You would list objects, and properties of these, and maybe even relations between them. But then you already have your formal model of the situation you are talking about. On the other hand, some epistemological questions can only be answered formally, as is illustrated by the following example (modeled after one given by Sven Ove Hansson in Hendricks & Simons 2005).
Consider the following two proposals for a link between the qualitative notion of belief and the quantitative notion of degree of belief. According to the first proposal an agent should believe a proposition if and only if her degree of belief that the proposition is true is higher than her degree of belief that the proposition is false (Weatherson 2005). According to the second proposal (known as the
Lockean thesis and discussed in section 2.6) an agent should believe a proposition if and only if her degree of belief for this proposition is higher than a certain threshold. We can ask formally as well as, maybe, informally under which conditions these two proposals are equivalent. However, we can answer this question only formally.
This provides one reason why we should care about
formal representations of belief, and
formalepistemology in general. For different reasons why formal epistemology is important see Hájek (2006).
1.2 The Objects of Belief
Before we can investigate the relation between various beliefs and degrees of belief, we have to get clear about the relata of the (degree of) belief relation. Belief is commonly assumed to be a relation between a doxastic agent at a particular time to an object of belief. Degree of belief is commonly assumed to be a relation between a number, a doxastic agent at a particular time, and an object of belief. For the purposes of this entry we may focus on ideal doxastic agents who do not suffer from the computational and other physical limitations of ordinary doxastic agents such as people and computer programs. These ideal doxastic agents get to voluntarily decide what to believe (and to what degree of numerical precision); they never forget any of their (degrees of) beliefs; and they always believe believe all logical and conceptual truths (to a maximal degree). We may define an agent to be
ideal just in case any action that is physically possible is an action that is possible for her. Such ideal agents ought to do exactly that which they ought to do if they could, where the ‘can’ in ‘could’ expresses possibility for the agent, not metaphysical possibility.
It is difficult to state what the objects of belief are. Are they sentences, or propositions expressed by sentences, or possible worlds (whatever these are: see Stalnaker 2003), or something altogether different? The received view is that the objects of belief are
propositions, i.e., sets of
possible worlds or truth conditions. A more refined view is that the possible worlds comprised by those propositions are
centered at an (ideal doxastic) agent at a given time (for an overview see Ninan 2010). Whereas a(n) (uncentered) possible world completely specifies a way the world might be, a centered possible world additionally specifies who one is when in a given (uncentered) possible world. In the latter case propositions are often called
properties. Most epistemologists stay very general and assume only that there is a non-empty set
[ltr]WW[/ltr] of possibilities such that exactly one element of
[ltr]WW[/ltr] corresponds to the actual world. If the possibilities in
[ltr]WW[/ltr] are centered, the assumption is that there is exactly one element of
[ltr]WW[/ltr] that corresponds to your current time slice in the actual world (Lewis 1986 holds that this element not merely corresponds to, but is identical with, your current time slice in the actual world).
Centered propositions are needed to adequately represent self-locating beliefs such as Sophia’s belief that she lives in Vienna, which may well be different from her belief that Sophia lives in Vienna (these two beliefs differ if Sophia does not believe that she is Sophia). Self-locating beliefs have important epistemological consequences (Elga 2000, Lewis 2001, Bostrom 2007, Meacham 2008, Bradley 2012, Titelbaum 2013, Halpern 2015), and centered propositions are ably argued by Egan (2006) to correspond to what philosophers have traditionally called
secondary qualities (Locke 1690/1975). Lewis’ (1979, 133ff) claim that the difference between centered and uncentered propositions plays little role in how belief and other attitudes are formally represented, and postulated to behave in a rational way, can only be upheld for synchronic constraints on the statics of belief. For diachronic constraints on the dynamics of belief this claim is false, because the actual centered world (your current time slice in the actual uncentered world) is continually changing as time goes by. We will bracket these complications, though, and assume that, unless noted otherwise, the difference between centered and uncentered possibilities and propositions has no effect on the topic at issue.
1.3 The Structure of the Objects of Belief
Propositions have a certain set-theoretic structure. The set of all possibilities,
[ltr]WW[/ltr], is a proposition. Furthermore, if
[ltr]AA[/ltr] and
[ltr]BB[/ltr] are propositions, then so are the complement of
[ltr]AA[/ltr] with respect to
[ltr]W,W∖AW,W∖A[/ltr], as well as the intersection of
[ltr]AA[/ltr] and
[ltr]B,A∩BB,A∩B[/ltr]. In other words, the set of propositions is a (finitary)
field or
algebra [ltr]AA[/ltr] over a non-empty set
[ltr]WW[/ltr] of possibilities: a set that contains
[ltr]WW[/ltr] and is closed under complementation and finite intersection. Sometimes the field
[ltr]AA[/ltr]of propositions is assumed to be closed not only under finite, but also under countable intersection. This means that
[ltr]A[size=13]1∩…An∩…A1∩…An∩…[/ltr][/size] is a proposition (an element of
[ltr]A)A)[/ltr], if each of
[ltr]A[size=13]1,…,An,…A1,…,An,…[/ltr][/size] is. Such a field
[ltr]AA[/ltr] is called a
[ltr]σσ[/ltr]-field. Finally, a field
[ltr]AA[/ltr] is
complete just in case the intersection
[ltr]⋂B=⋂[size=13]A∈BA⋂B=⋂A∈BA[/ltr][/size] is in
[ltr]AA[/ltr], for each subset
[ltr]BB[/ltr] of
[ltr]AA[/ltr].
If Sophia believes today (to degree .55) that tomorrow it will be sunny in Vienna, but she does not believe today (to degree .55) that tomorrow it will not be not sunny in Vienna, propositions cannot be the objects of Sophia’s (degrees of) belief(s) today. That tomorrow it will be sunny in Vienna and that tomorrow it will not be not sunny in Vienna is one and the same proposition (if stated by the same agent at the same time). It is merely expressed by two different, but logically equivalent sentences. (Some philosophers think that propositions are too coarse-grained as objects of belief, while sentences are too fine-grained. They take the objects of belief to be
structured propositions. These are usually taken to be more fine-grained than ordinary propositions, but less fine-grained than sentences. For an overview see the entry on
structured propositions. Other philosophers think that ordinary propositions are just fine, but that they should be viewed as sets of
epistemic or
doxastic rather than
metaphysical or
logicalpossibilities.)
Sometimes sentences of a formal language
[ltr]LL[/ltr] are taken to be the objects of belief. In this case the above mentioned set-theoretic structure translates into the following requirements: the tautological sentence
[ltr]⊤⊤[/ltr] is a sentence of the language
[ltr]LL[/ltr]; and whenever
[ltr]αα[/ltr] and
[ltr]ββ[/ltr] are sentences of
[ltr]LL[/ltr], then so are the negation of
[ltr]αα[/ltr],
[ltr]¬α¬α[/ltr], as well as the conjunction of
[ltr]αα[/ltr] and
[ltr]ββ[/ltr],
[ltr]alpha∧βalpha∧β[/ltr]. However, as long as logically equivalent sentences are required to be assigned the same degree of belief — and all accounts considered here require this — the difference between taking the objects of beliefs to be sentences of a formal language
[ltr]LL[/ltr] and taking them to be propositions from a finitary field
[ltr]AA[/ltr] is mainly cosmetic. The reason is that each language
[ltr]LL[/ltr] induces a finitary field
[ltr]AA[/ltr] over the set of all models or classical truth value assignments for
[ltr]LL[/ltr],
[ltr]Mod[size=13]LModL[/ltr][/size]:
[ltr]AA[/ltr] is the set of propositions over
[ltr]Mod[size=13]LModL[/ltr][/size] that are expressed by the sentences in
[ltr]LL[/ltr].
[ltr]AA[/ltr] in turn induces a unique
[ltr]σσ[/ltr]-field, viz. the smallest
[ltr]σσ[/ltr]-field
[ltr]σ(A)σ(A)[/ltr] that contains
[ltr]AA[/ltr] (
[ltr]σ(A)σ(A)[/ltr] is the intersection of all
[ltr]σσ[/ltr]-fields that contain
[ltr]AA[/ltr] as a subset).
[ltr]AA[/ltr] also induces a unique complete field, viz. the smallest complete field
[ltr]γ(A)γ(A)[/ltr] that contains
[ltr]AA[/ltr] (
[ltr]γ(A)γ(A)[/ltr] is the intersection of all complete fields that contain
[ltr]AA[/ltr] as a subset). In the present case where
[ltr]AA[/ltr] is generated by
[ltr]Mod[size=13]LModL[/ltr][/size],
[ltr]γ(A)γ(A)[/ltr] is the powerset of
[ltr]Mod[size=13]LModL[/ltr][/size],
[ltr]℘(Mod[size=13]L)℘(ModL)[/ltr][/size].
[ltr]σ(A)σ(A)[/ltr], and hence
[ltr]γ(A)γ(A)[/ltr], will often contain propositions that are not expressed by a sentence of
[ltr]LL[/ltr]. For instance, let
[ltr]α[size=13]iαi[/ltr][/size] be the sentence “You should donate at least
[ltr]ii[/ltr] dollars to the Society for Exact Philosophy (SEP)”, for each natural number
[ltr]ii[/ltr]. Assume our language
[ltr]LL[/ltr] contains each
[ltr]α[size=13]iαi[/ltr][/size]and whatever else it needs to contain to be a language (e.g. the negation of each
[ltr]α[size=13]iαi[/ltr][/size],
[ltr]¬α[size=13]i¬αi[/ltr][/size], as well as the conjunction of any two
[ltr]α[size=13]iαi[/ltr][/size] and
[ltr]α[size=13]jαj[/ltr][/size],
[ltr]α[size=13]i∧αj).Lαi∧αj).L[/ltr][/size] generates the following finitary field
[ltr]AA[/ltr] of propositions:
[ltr]A={Mod(α)⊆Mod[size=13]L:α∈L}A={Mod(α)⊆ModL:α∈L}[/ltr][/size], where
[ltr]Mod(α)Mod(α)[/ltr] is the set of models in which
[ltr]αα[/ltr] is true.
[ltr]AA[/ltr] in turn induces
[ltr]σ(A[/ltr][/size]
Formal Representations of Belief
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