  We study the general problem of strengthening the logic of a given (partial)
(non-deterministic) matrix with a set of axioms, using the idea of rexpansion.
We obtain two characterization methods: a very general but not very effective
one, and then an effective method which only applies under certain restrictions
on the given semantics and the shape of the axioms. We show that this second
method covers a myriad of examples in the literature. Finally, we illustrate
how to obtain analytic multiple-conclusion calculi for the resulting logics.
