Foundational belief change
An operation of contraction for a set \(A\) of sentences is an operation such that for any sentence \(\alpha\), \(A - \alpha\), the outcome of contracting \(A\) by \(\alpha\), is a subset of \(A\) that does not imply \(\alpha\) (unless \(\alpha\) is logically true). This paper is concerned with the operation \(-\) of contraction on a theory \(K\) that can be derived from an operation \(\div\) of contraction on a base \(B\) for \(K\) (i.e., a set \(B\) such that \(\text{Cn}(B) = K\)) according to the principle \(K - a = \text{Cn} (B \div a)\). The paper starts from previous work by \textit{A. Fuhrmann} [J. Philos. Logic 20, No. 2, 175-203 (1991; Zbl 0723.03010)] in which the operation \(\div\) on the base is essentially a variant of save contraction, as defined by \textit{C. Alchourrón} and \textit{D. Makinson} [Stud. Logica 44, 405-422 (1985; Zbl 0605.03002)]. The author attempts to find conditions on \(\div\) that are sufficient for -- to satisfy the so-called supplementary postulates for rational belief contraction that have been introduced by Peter Gärdenfors. After showing that even quite strong restrictions on Fuhrmann's construction are insufficient to yield these properties, the author introduces a modified construction with the desired properties. This construction involves a modification of safe contraction to make it maxichoice, i.e., to ensure that \(B \div \alpha\) is a maximal subset of \(B\) that does not imply \(\alpha\).
scientific article; zbMATH DE number 683098
scientific article