Abstract
In local change, the operation \(\circ \) is specific for the original belief set K. Formally it is a function that takes us from a descriptor \(\Psi \) to an element \(K\circ \Psi \) of the outcome set \(\mathbb {X}\) (the set of belief sets that are potential outcomes of belief change). It only represents changes that have K as their starting-point. In this chapter the framework of descriptor revision is widened to global (iterated) belief change. This means that the operation \(\circ \) can be applied to any potential belief set. Formally, it is a function that takes us from a pair consisting of a belief set K and a descriptor \(\Psi \) to a new belief set \(K\circ \Psi \). This makes it possible to cover successive changes, such as \(K\circ \mathfrak {B}p\circ \lnot \mathfrak {B} p\). Several constructions of global descriptor revision are presented and axiomatically characterized. The most orderly of these constructions is based on pseudodistances (distance measures that allow the distance from X to Y to differ from the distance from Y to X). For any elements X and Y of the outcome set, i.e. the set of belief sets that are eligible as outcomes, there is a number \(\delta (X, Y)\) denoting how far away Y is from X. When revising a belief set K by some descriptor \(\Psi \), the outcome \(K\circ \Psi \) is the belief set satisfying \(\Psi \) that is closest to K, as measured with \(\delta \). If we revise \(K\circ \Psi \) by \(\Xi \), then the outcome \(K\circ \Psi \circ \Xi \) is the belief set \(\delta \)-closest to \(K\circ \Psi \) that satisfies \(\Xi \), etc. The chapter also provides a generalization of blockage revision to global operations.
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Notes
- 1.
This will be the case if \(X\circ \Pi _X=X\) holds, cf. Section 4.4.
- 2.
The term “pseudodistance” has been used since the 19th century for various weakenings of, and alternatives to, standard Euclidean distance. See [155, p. 300] for a useful general definition of pseudodistances in this sense.
- 3.
We can achieve the same effect with the postulate \(\delta (X,X)<\delta (X, Y)\) if \(X\ne Y\), but the self-closeness property is easier to work with.
- 4.
Cf. Section 5.3.
- 5.
Let \(\delta \) be a measure that satisfies self-closeness and non-negativity. Let l be a limit function with \(l(X)>0\) for all \(X\in \mathbb {X}\). Furthermore, let a be an accessibility function such that \(Y\in a(X)\) if and only if \(\delta (X, Y)<l(X)\). We can then define \(\delta '\) as the measure such that \(\delta '(X,Y)=\delta (X, Y)/l(X)\) for all \(X, Y\in \mathbb {X}\). Then \(Y\in a(X)\) if and only of \(\delta '(X, Y)<1\).
- 6.
Properties of binary relations are transferred to ternary relations by keeping the middle term constant. Hence, a ternary relation \(\rightharpoondown \) satisfies asymmetry if and only if \(\rightharpoondown _X\) satisfies asymmetry for all X.
- 7.
Negative transmission is also closely related with an axiom introduced under the name “loop” in [155, p. 306]. To see the connection with “loop”, note that Lehmann et al. refer to distances between sets of objects. In their notation, \(X\!\!\mid \!\! Y\) is the set of elements y of Y such that \(min_{x\in X}\delta (x, y)\) is at least as small as is \(min_{x\in X}\delta (x, y')\) for any other element \(y'\) of Y. Therefore their formula \((X_1\mid (X_0\cup X_2))\cap X_0 \ne \varnothing \) can be interpreted in the singleton case (\(X_0=\{x_0\}\), \(X_1=\{x_1\}\), and \(X_2=\{x_2\}\)) as saying that \(x_1\) is at least as close to \(x_0\) as it is to \(x_2\).
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Hansson, S.O. (2017). Global Descriptor Revision. In: Descriptor Revision. Trends in Logic, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-53061-1_6
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DOI: https://doi.org/10.1007/978-3-319-53061-1_6
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