Is there a non-monotonic logic where these two definitions of redundance are not equivalent?

Question

Let $S$ be a set of axioms, meaning, just any old set of sentences in some logical system, and let $A$ be an element of $S$. There are two definitions of the property "$A$ is redundant with respect to $S$". The first definition is, $A$ is in the deductive closure of $S - \{A\}$. The second definition is, the deductive closure of $S$ is equal to the deductive closure of $S - \{A\}$. These two definitions are equivalent for any monotonic logic, but I wonder, is there a non-monotonic logic and some $S$ and some $A \in S$ where these two definitions are not equivalent?