A Representation Theorem for Preferential Logics

This paper shows how to deene nonmonotonic logics from any classical logics L and any set X of formulas of L. In this context, the nonmonotonic inference relation`X is deened by A ` X B if every classical theorem of A B which is in X is a theorem of A. The properties of the relation`X are studied. We show, in particular, that the elementary properties (supraclassicity, or, left logical equivalence, cut, etc.) are veriied for any X. Moreover, we prove that cumulativity is veriied if the set of formulas of the language, which are not in X, is deductively closed. Then we prove a representation theorem, i.e., in the nite case every preferential nonmonotonic logic is an X-logic. We also study a particular form of the set X for propositional circumscription.