An important characteristic of many logics for Arti cial Intelligence is their nonmonotonicity. This means that adding a formula to the premises can invalidate some of the consequences. There may, however, exist formulae that can always be safely added to the premises without destroying any of the consequences: we say they respect monotonicity. Also, there may be formulae that, when they are a consequence, can not be invalidated when adding any formula to the premises: we call them conservative. We study these two classes of formulae for preferential logics, and show that they are closely linked to the formulae whose truth-value is preserved along the (preferential) ordering. We will consider some preferential logics for illustration, and prove syntactic characterization results for them. The results in this paper may improve the e ciency of theorem provers for preferential logics.De nition 2 (Persistence) Given a preferential logic (L; Mod; j=; ), a formula 2 L is called downward persistent in this logic, if 8m; n 2 Mod : (m j= and n m) ) n j= ; and it is called upward persistent if 8m; n 2 Mod : (n j= and n m) ) m j= :