Possibility theory (Dubois & Prade 1988) is based on fuzzy set theory (Zadeh 1978). According to the latter theory, an element need not belong to a given set either completely or not at all, but may be a member of the set to a certain degree. For instance, Sophia may belong to the set of politically active people to a degree of .88. This is represented by a membership function [ltr]μ[size=13]A:W→[0,1]μA:W→[0,1][/ltr], where [ltr]
μA(w)μA(w)[/ltr] is the degree of membership to which person [ltr]
w∈Ww∈W[/ltr] belongs to the set of politically active people [ltr]
AA[/ltr].[/size]
Furthermore, the degree [ltr]μ[size=13]W∖AμW∖A[/ltr](Sophia) to which Sophia belongs to the set [ltr]
W∖AW∖A[/ltr] of people who are not politically active equals [ltr]
1−μA1−μA[/ltr](Sophia). Moreover, if [ltr]
μM:W→[0,1]μM:W→[0,1][/ltr] is the membership function for the set of philosophically minded people, then the degree of membership to which Sophia belongs to the set [ltr]
A∪MA∪M[/ltr] of politically active or philosophically minded people is given by[/size]
[ltr]μ[size=13]A∪M(Sophia)=max{μA(Sophia),μM(Sophia)}.μA∪M(Sophia)=max{μA(Sophia),μM(Sophia)}.[/ltr][/size]
Similarly, the degree of membership to which Sophia belongs to the set [ltr]A∩MA∩M[/ltr] of politically active and philosophically minded people is given by[ltr]μ[size=13]A∩M(Sophia)=min{μA(Sophia),μM(Sophia)}.μA∩M(Sophia)=min{μA(Sophia),μM(Sophia)}.[/ltr][/size]
[ltr]m[size=13]A∩M(Sophia)mA∩M(Sophia)[/ltr] is interpreted as the degree to which the vague statement “Sophia is a politically active and philosophically minded person” is true (for vagueness see Égré & Barberousse 2014, Raffman 2014, Williamson 1994 as well as the entry on
vagueness; Field (forthcoming) discusses uncertainty due to vagueness in a probabilistic setting). Degrees of truth belong to the philosophy of language. They do not (yet) have anything to do with degrees of belief, which belong to epistemology. In particular, note that degrees of truth are usually considered to be
truth functional (the truth value of a compound statement such as [ltr]
A∧BA∧B[/ltr] is a function of the truth values of its constituent statements [ltr]
A,BA,B[/ltr]; that is, the truth values of [ltr]
AA[/ltr] and [ltr]
BB[/ltr] determine the truth value of [ltr]
A∧B)A∧B)[/ltr]. Degrees of belief, on the other hand, are hardly ever considered to be truth functional. For instance, probabilities are not truth functional, because the probability of [ltr]
A∩BA∩B[/ltr]is not determined by the probability of [ltr]
AA[/ltr] and the probability of [ltr]
BB[/ltr]. That is, there is no function [ltr]
ff[/ltr]such that for all probability spaces [ltr]
⟨W,A,Pr⟩⟨W,A,Pr⟩[/ltr] and all propositions [ltr]
A,BA,B[/ltr] in [ltr]
AA[/ltr]: [ltr]
Pr(A∩B)=f(Pr(A),Pr(B))Pr(A∩B)=f(Pr(A),Pr(B))[/ltr].[/size]
Suppose someone says that Sophia is tall. How tall is a tall person? Is a person with a height of [ltr]5[size=13]′9′′5′9″[/ltr] tall? Or does a person have to be at least [ltr]
5′10′′5′10″[/ltr] in order to be tall? Although you know that Sophia is tall, your knowledge is incomplete due to the vagueness of the term ‘tall’. Here possibility theory enters by equipping you with a (normalized)
possibility distribution, a function[ltr]
π:W→[0,1]π:W→[0,1][/ltr] with [ltr]
π(w)=1π(w)=1[/ltr] for at least one [ltr]
w∈Ww∈W[/ltr]. The motivation for the latter requirement is that at least (in fact, exactly) one possibility is the actual possibility, and hence at least one possibility must be maximally possible. Such a possibility distribution [ltr]
π:W→[0,1]π:W→[0,1][/ltr]on the set of possibilities [ltr]
WW[/ltr] is extended to a possibility
measure [ltr]
Π:A→RΠ:A→ℜ[/ltr] on the field [ltr]
AA[/ltr]over [ltr]
WW[/ltr] by defining for each [ltr]
AA[/ltr] in [ltr]
AA[/ltr],[/size]
[ltr]Π(∅)=0,Π(A)=sup{π(w):w∈A}.Π(∅)=0,Π(A)=sup{π(w):w∈A}.[/ltr]
This entails that possibility measures [ltr]Π:A→RΠ:A→ℜ[/ltr] are maximitive (and hence sub-additive), i.e., for all [ltr]A,B∈AA,B∈A[/ltr]:[ltr]Π(A∪B)=max{Π(A),Π(B)}.(8)(8)Π(A∪B)=max{Π(A),Π(B)}.[/ltr]
The idea is that, roughly, a proposition is at least as possible as each of the possibilities it comprises, and no more possible than the “most possible” possibility. Sometimes, though, there is no most possible possibility (i.e., the supremum is no maximum). For instance, this is the case when the degrees of possibility are [ltr][size=13]1/2,3/4,7/8,…,2n−1/2n,…1⁄2,3⁄4,7⁄8,…,2n−1⁄2n,…[/ltr] In this case the degree of possibility for the proposition is the smallest number which is at least as great as all the degrees of possibilities of its elements. In our example this is 1. (As will be seen below, this is the main formal difference between possibility measures and non-conditional ranking functions.)[/size]
We can define possibility measures without recourse to an underlying possibility distribution as functions [ltr]Π:A→RΠ:A→ℜ[/ltr] such that for all [ltr]A,B∈AA,B∈A[/ltr]:[ltr]Π(∅)Π(W)Π(A∪B)=0,=1, and =max{Π(A),Π(B)}.Π(∅)=0,Π(W)=1, and Π(A∪B)=max{Π(A),Π(B)}.[/ltr]
It is important to note, though, that the last clause is not well-defined for disjunctions or unions of infinitely many propositions (in this case one would have to use the supremum operation sup instead of the maximum operation max). The dual notion of a necessity measure [ltr]N:A→RN:A→ℜ[/ltr] is defined for all [ltr]AA[/ltr] in [ltr]AA[/ltr] by[ltr]N(A)=1−Π(W∖A).N(A)=1−Π(W∖A).[/ltr]
This implies that[ltr]N(A∩B)=min{N(A),N(B)}.N(A∩B)=min{N(A),N(B)}.[/ltr]
The latter equation can be used to start with necessity measures as primitive. Define them as functions [ltr]N:A→RN:A→ℜ[/ltr] such that for all [ltr]A,B∈AA,B∈A[/ltr]:[ltr]N(∅)N(W)N(A∩B)=0,=1,=min{N(A),N(B)}.N(∅)=0,N(W)=1,N(A∩B)=min{N(A),N(B)}.[/ltr]
Then possibility measures [ltr]Π:A→RΠ:A→ℜ[/ltr] are obtained by the equation:[ltr]Π(A)=1−N(W∖A).[/ltr]
Possibility Theory
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