Preferential Logics are X‐logics

This paper shows how to define nonmonotonic logics from any classical logics ℒ and any set X of formulas of L. In this context, the nonmonotonic inference relation ⊢X is defined 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 verified for any X. Moreover, we prove that cumulativity is verified 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 finite case every preferential nonmonotonic logic is an X ‐logic. We also study a particular form of the set X for general propositional circumscription.