Circumscriptions from What They Cannot Do (preliminary Report)

We show that, in the nite propositional case, traditional circumscrip-tions can be fully described only from the formulas which can never come as a result of the given circumscription (the \inaccessible formulas"). Some work has already been done on the subject. Siegel and Forget have introduced the general notion of X-logic, and they have considered the case of nite proposi-tional circumscriptions. However, their result is restricted to the case where no varying proposition appears, which is known to be a severe restriction in terms of expressivity. We extend this result in the nite propositional case to any cumulative preferential entailment, thus in particular to any \tradi-tional circumscription", that is we allow varying propositions. Moreover, we describe the smallest possible set which can be used for this purpose. We exhibit a spectacular duality between the traditional approach of a given cumulative preferential entailment, and the approach through the inaccessible formulas of this preferential entailment, i.e. the X-logic approach. We extend these results outside the framework of pure propositional preferential entailments.