Finite contractions on infinite belief sets
In classical studies of belief change, a belief set is a set of sentences closed under logical consequence. Thus, a belief set is an infinite entity. However, it is possible to define a special class where the belief set can be represented by a finite set of formulae. A belief set \(K\) is finite-based if and only if it has a finite representation, i.e. if and only if there is a finite set \(A\) of sentences such that \(K = \mathrm{Cn}(A)\). A partial meet contraction is a function that selects the ``best'' elements of the set \(K \bot p\) (maximal subsets of \(K\) that do not imply the sentence \(p\)). These subsets, if the language is not finite, are not finite-based, from which it follows that partial meet contraction does not in general preserving this property. Even if \(K\) is finite-based, the contraction outcome \(K \sim \gamma\;p\) may nevertheless lack a finite representation. The desired property is: \[ \text{If } K \text{ is finite-based, then so is } K \div p \text{ (finite-based outcome).} \] In previous works, the author proved that the AGM contractions that satisfy finite-based outcome exactly coincide with those AGM contractions that can alternatively be constructed as instances of specified meet contraction (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)]). In this paper, Hansson extends these results to belief sets that are not finite-based.
scientific article; zbMATH DE number 6121542
scientific article