Constructive modelings for theory change
Two themes in AGM-style belief revision are discussed. First, the authors introduce a definition of update operators in terms of revision operators, calling it the Winslett identity: \[ T\diamondsuit \alpha= \begin{cases} \bigcap_{K\in [T]} K* \alpha\text{ if } T\text{ is consistent}\\ T\text{ if } T\text{ is inconsistent}\end{cases} \] Here \(\diamondsuit\) represents updating of a theory and \(*\) represents belief revision. \([T]\) is the set of possible worlds that include \(T\). Updating is intended to represent modification of a theory in response to a change in the world, and revision represents modification in response to new information. A nice correspondence is shown to hold between update operators satisfying a set of axioms (that extend the Katsuno-Mendelzon postulates) and revision operators satisfying the standard AGM postulates: (1) If \(*\) satisfies the revision postulates, then \(\diamondsuit\) (defined via the Winslett identity) satisfies the update postulates. (2) If \(\diamondsuit\) satisfies the update postulates, then there exists an operator \(*\) satisfying the revision postulates, from which \(\diamondsuit\) can be derived via the Winslett identity. This result is a major achievement in the study of update operations. Secondly, the authors provide a new construction for revision, namely a nice preorder on models. This construction is fairly closely related to Adam Grove's system of spheres. They also give conditions under which a nice preorder on models, an entrenchment ordering and a system of spheres represent the same revision function. These results are clarifying although not very surprising. The authors do not seem to be aware that their Theorem 6.1 follows directly from results already in the literature [e.g., \textit{A. Grove}, J. Philos. Logic 17, No. 2, 157-170 (1988; Zbl 0639.03025), and \textit{P. Gärdenfors}, Knowledge in flux (1988)].
scientific article; zbMATH DE number 824595