Severe withdrawal (and recovery)
The authors undertake a comprehensive study of a motivated variant of AGM contraction. The motivation is that if we reduce a belief set \(K\) by eliminating one of its elements \(x\), then we should not retain any elements \(y\) of \(K\) whose credibility, from the point of view of \(K\), is less than or equal to that of \(x\). With this motivation for the concept of `severe withdrawal', the most direct definition is in terms of the Gärdenfors-Makinson apparatus of `epistemic entrenchment', as was already suggested by the first of the authors under review in his paper ``Two methods of constructing contractions and revisions of knowledge systems'' [J. Philos. Log. 20, No. 2, 149-173 (1991; Zbl 0723.03009)]. Indeed, such a definition of severe withdrawal \(K-x\) as \(\{y\in K:x<y\}\) where \(<\) is an epistemic entrenchment relation (in the principal case that \(x\in K\) and \(x\notin Cn(\emptyset))\) is formally simpler and also easier to motivate than the corresponding definition of AGM contraction as \(\{y\in K:x<x\vee y\}\). In the paper under review, the authors examine the operation of severe withdrawal from several angles familiar from the literature on AGM belief revision operations. As well as the definition in terms of epistemic entrenchment, they consider that of nested spheres of possible worlds (following a 1991 suggestion of S. Lindström and W. Rabinowicz), and also that of the syntactic principles or `postulates' satisfied by the operation (on which some work had already been carried out by \textit{E. L. Fermé} and \textit{R. O. Rodriguez} [``A brief note about Rott contraction'', Log. J. IGPL 6, No. 6, 835-842 (1998; Zbl 0923.03041)]). In the context of nested spheres, \(K-x\) is determined by adding to the \(K\)-worlds not only the closest \(\neg x\)-worlds (as for AGM contraction) but also all those \(x\)-worlds that are at least as close to \(K\) as are the closest \(\neg x\)-worlds. On the level of syntactic properties satisfied, the controversial AGM principle of recovery fails (although its limiting case \(K-x=K\) when \(x\in Cn(\emptyset)\) still holds). But an antitony principle, not valid for AGM contraction, is gained: \(K-x\subseteq K-x\wedge y\) whenever \(x\notin Cn(\emptyset)\). The authors also show that for any belief set \(K\), the class of all revision- equivalent withdrawal operations on \(K\) contains exactly one severe withdrawal operation, which is included in the unique AGM contraction operation shown by Makinson to exist in the same class. Reviewer's comments: This very perceptive, thorough and clearly written treatment of severe withdrawal will be a valuable reference point for all those working in the area. Nevertheless, the reviewer does not find the motivation to be stronger than that for AGM contraction. When we eliminate a proposition \(x\) from a belief set \(K\), it is not evident that we should also drop all the \(y\in K\) which, from the point of view of \(K\), were regarded as less credible than \(x\). For the decision to withdraw \(x\) corresponds to a change in our credibility ratings: the relative positions of \(x\) and \(y\) in the new rating may well be reversed, with \(x\) becoming less credible than \(y\), so that the latter may stay in the reduced set. [From the publisher's note (Vol. 29, No. 1, 121 (2000)): ``Due to a technical error at the printing stage, certain characters, important to a proper understanding of the text, \dots failed to appear in the printed version. The correct version has been reprinted on the following pages, and the original pagination has been retained, to enable a replacement of the wrong version in Vol. 28, No. 5'' ].
scientific article; zbMATH DE number 1383887
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