General Preferential Entailments as Circumscriptions

A (general) preferential entailment is defined by a “preference relation” among “states”. States can be either interpretations or sets of interpretations, or “copies” of interpretations or of sets of interpretations, although it is known that the second kind and the fourth one produce the same notion. Circumscription is a special case of the simplest kind, where the states are interpretations. It is already known that a large class of preferential entailments where the states are copies of interpretations, namely the “cumulative” ones, can be expressed as circumscriptions in a greater vocabulary. We extend this result to the most general kind of general preferential entailment, the additional property requested here is “loop”, a strong kind of “cumulativity”. The greater vocabulary needed here is large, but only a very simple and small set of formulas in this large vocabulary is necessary, which should make the method practically useful.