Note about cardinality-based circumscription

In a very interesting paper, Liberatore and Schaerf showed how to express the leading proposals of belief revision using a circumscription. In order to deal with one of the belief revision theories, they had to introduce a new circumscription, based on the cardinality instead of the classical set inclusion. We give a few properties of this new circumscription. In doing so, we prove that in the finite case, there is inter-definability between ordinary circumscription and cardinality-based circumscription. We show why a technical result published in the initial paper cannot be true, and we give a related result which is true instead, where the respective roles of the varying or fixed propositions are reversed. This result could help automatic computation. We use the logical properties of circumscriptions and cardinality-based circumscriptions in order to highlight the behavior of cardinality-based circumscription and the importance of the fixed propositions. This study allows a full description of the expressive power of cardinality-based circumscription in the finite case, and it provides some limitations in the infinite case.