The importance of open and recursive circumscription

摘要

Circumscription is known to result in an inconsistency when applied to certain consistent theories. To counter this problem, closed nonrecursive circumscription, a restricted form of circumscription that has been proved not to affect the consistency of the theory over which circumscription is applied, has been proposed. We show that closed nonrecursive circumscription involves an excessive weakening of standard circumscription by establishing that closed nonrecursive circumscription is incomplete for some crucial theories over which standard circumscription is consistent and complete. First, we prove that closed circumscription cannot yield the desired uniqueness formula for the simplest of existential theories. Second, we prove that nonrecursive circumscription fails to be as strong as predicate completion for Horn clause theories. Third, we prove that the natural way to strengthen circumscription, that is, adding more variable predicates, may weaken nonrecursive circumscription.