Peter Fishburn's analysis of ambiguity

Peter Fishburn's analysis of ambiguity

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Article ID: iaor20163725
Volume: 81
Issue: 2
Start Page Number: 153
End Page Number: 165
Publication Date: Aug 2016
Journal: Theory and Decision
Authors:
Keywords: decision theory, behaviour
Abstract:

In ordinary discourse the term ambiguity typically refers to vagueness or imprecision in a natural language. Among decision theorists, however, this term usually refers to imprecision in an individual’s probabilistic judgments, in the sense that the available evidence is consistent with more than one probability distribution over possible states of the world. Avoiding a prior commitment to either of these interpretations, Fishburn has explored ambiguity as a primitive concept, in terms of what he calls an ambiguity measure a on the power set 2 Ω equ1 of a finite set Ω equ2 , characterized by five axioms. We prove, in purely set‐theoretic terms, that if λ equ3 is a so‐called necessity measure on 2 Ω equ4 and υ equ5 is its associated possibility measure, then a = υ λ equ6 is an ambiguity measure. When Ω equ7 is construed as a set of possible exemplars of a vague predicate ϕ equ8 , then λ equ9 and υ equ10 may be regarded as arising from a fuzzy membership function f on Ω equ11 , where f ( ω ) equ12 designates the degree to which ϕ equ13 is applicable to ω equ14 . In this case a(A) represents the degree to which the partition { A , A c } equ15 differentiates members of Ω equ16 with respect to the predicate ϕ equ17 . When Ω equ18 is construed as a set of possible states of the world, a necessity measure may be regarded as a very special type of lower probability known as a consonant belief function, and a possibility measure as its associated upper probability, whence a(A) represents the degree of imprecision in the pair ( λ , υ ) equ19 with respect to the event A. Fishburn’s axioms are thus consistent with an interpretation of ambiguity as linguistic vagueness, as well as (a very special sort of) probabilistic imprecision.

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