NONMONOTONIC
Logics individuated by dropping monotony — the rule that adding premises never removes conclusions. Monotony is the consequence-relation form of weakening, as transitivity is the consequence-relation form of cut, so this subdivision stands to weakening as Nontransitive stands to cut: the structural rule is withdrawn at the level of the relation itself, and everything local stays classical.
What replaces monotony is a hierarchy of weaker closure conditions, and the hierarchy is the subdivision's shape: cumulative relations keep cut and cautious monotony, so that adding a derived conclusion to the premises neither adds nor removes consequences; preferential relations add disjunctive reasoning and are represented by smooth preference orders over states; rational relations add rational monotony and are represented by ranked orders. The representation theorems — postulate set on one side, model class on the other — are the subdivision's metatheory, and they are completeness theorems in everything but name.
The division of labour with Applications/Knowledge is fixed by individuation. The systems born of knowledge representation — default logic, circumscription, autoepistemic logic, answer set programming — are individuated by the representation problem and are canonical there; the abstract theory of the consequence relations they induce is individuated by the dropped rule and is canonical here, with Preferential Logic cross-listed as the bridge.