Multiple and iterated contraction reduced to single-step single-sentence contraction
Operations of `specified meet contraction' were introduced and studied by the author in his papers ``Contraction based on sentential selection'' [J. Log. Comput. 17, No.~3, 479--498 (2007; Zbl 1135.03008)] and ``Specified meet contraction'' [Erkenntnis 69, No.~1, 31--54 (2008; Zbl 1155.03010)]. The underlying idea is to use a function taking propositions to propositions to do some of the work of the selection operation in partial meet contraction: the specified meet contraction of proposition \(x\) from belief set \(K\) is defined to be the full meet contractions of \(f(x)\) from \(K\). In the paper under review, the author extends the scope of this notion to cover also multiple contraction, that is, the joint contraction of a collection (possibly infinite) of propositions from a belief set. Results obtained include sound and complete postulates for specified meet contraction from an arbitrary belief set closed under classical consequence, and also from finitely based such belief sets; the reduction of multiple specified meet contraction by a finite set to such contraction by a singleton; and the irreducibility in general of multiple specified meet contraction by an infinite set to the same by a finite set. Certain particular kinds of multiple specified meet contraction are also studied. The paper ends with some interesting reflections on why it may sometimes be better, in practice, to refrain from carrying out iterated contractions by single propositions, waiting instead for a suitable occasion to effect a multiple contraction covering a bundle of them.
scientific article; zbMATH DE number 5770279
scientific article