  The present paper investigates consequence relations that are both
non-monotonic and paraconsistent. More precisely, we put the focus on
preferential consequence relations, i.e. those relations that can be defined by
a binary preference relation on states labelled by valuations. We worked with a
general notion of valuation that covers e.g. the classical valuations as well
as certain kinds of many-valued valuations. In the many-valued cases,
preferential consequence relations are paraconsistant (in addition to be
non-monotonic), i.e. they are capable of drawing reasonable conclusions which
contain contradictions. The first purpose of this paper is to provide in our
general framework syntactic characterizations of several families of
preferential relations. The second and main purpose is to provide, again in our
general framework, characterizations of several families of preferential
discriminative consequence relations. They are defined exactly as the plain
version, but any conclusion such that its negation is also a conclusion is
rejected (these relations bring something new essentially in the many-valued
cases).
