A survey of multiple contractions
The authors extend the AGM account of belief contraction [as set out by \textit{C. E. Alchourrón}, \textit{P. Gärdenfors} and \textit{D. Makinson} in ``On the logic of theory change: partial meet contraction and revision functions'', J. Symb. Logic 50, 510-530 (1985; Zbl 0578.03011)] to multiple contraction, i.e. contraction by a set \(A\) of sentences rather than by a single sentence. This should not be confused with iterated contraction, which is another possible extension of the AGM theory, where one considers the result of making one contraction after another. Two principal kinds of multiple contraction are distinguished -- package contraction in which all the sentences in \(A\) are removed from the belief set, and choice contraction in which at least one of them is removed. The authors provide semantic definitions of both kinds using natural generalizations of the AGM ``partial meet'' operations, and also give them both syntactic characterizations in terms of natural generalizations of the ``basic postulates'' of AGM. They observe that in general, neither package nor choice contraction can be reduced to contraction by a single sentence. Even when \(A\) is finite, package contraction still resists such reduction, whilst in that case choice contraction reduces trivially to contraction of the conjunction of all the sentences in \(A\). The presentation is extremely clear, with the authors taking the time to discuss adequately the intuitive background to each formal operation. As they remark, their paper is only a beginning of a study of the subject, and they list several respects in which existing theory for the singleton case, as already developed by Alchourrón, Gärdenfors and Makinson, still awaits extension to the multiple case. These include relationally constrained partial meet contraction and its characterization by ``supplementary postulates'', revision and its relation to contraction via the Levi and Harper identities, and the Makinson/Gärdenfors translation between the logic of belief change and nonmonotonic inference. As observed in correspondence by the reviewer, and noted in passing by the authors, consideration of the second of these questions reveals the existence of yet a third kind of multiple contraction, which one might call radical contraction, which is eminently natural when contraction is seen as a step in constructing, via the Levi identity, a revision of a belief set by consistently adding all sentences in a set \(A\).
scientific article; zbMATH DE number 502990