Gärdenfors’s Supplementary Postulates for Partial Product Contractions
Franz Baader, Renata Wassermann · Lecture notes in computer science · 2025
In the area of belief change, contraction operations are used to modify a given belief set or belief base such that certain unwanted consequences no longer follow. In previous work we have introduced a framework for constructing contraction operations that generalizes the well-known partial meet contraction approach, called partial product contractions (PPCs). The main idea was to replace the remainders employed by partial meet contractions with optimal repairs, which were first considered in ontology engineering. We were able to characterize PPCs with variants of well-known rationality postulates, and provided a large number of concrete instances of the general framework. In the present work, we start to investigate whether the rather weak conditions imposed by our framework are sufficient to generalize further classical results from belief change to this setting. To this purpose, we consider Gärdenfors’s supplementary postulates for belief contractions. We are able to show that, under two reasonable additional conditions, PPCs induced by maximizingly and transitively relational selection functions indeed satisfy these postulates, similarly to the classical case. However, unlike the classical case, in our general framework these conditions are not strong enough to prove a characterization theorem. In fact, we provide an example that shows that there are partial product contractions satisfying the supplementary postulates that cannot be obtained from a weakly maximizingly and transitively relational selection function. We also introduce a third condition that ensures that already transitively relational selection functions yield PPCs that satisfy the supplementary postulates.