INCLUSION CRITERIA
An entry belongs in this subdivision if and only if it is individuated by the restriction or removal of monotony — the closure of consequence under additional premises — or by the abstract study of consequence relations so restricted.
Required: At least one of the following:
- A consequence relation that fails monotony, with the weaker closure it keeps stated as postulates (cumulativity, cautious monotony, rational monotony)
- A representation theorem tying a postulate set to a model class (cumulative models, smooth preference orders, ranked orders)
- An abstract result about the family of nonmonotonic consequence relations — their lattice, their translations, their relation to conditional logics
Not sufficient: A knowledge-representation formalism individuated by its representation problem — default rules, minimized predicates, stable models — which belongs in Applications/Knowledge however nonmonotonic its consequence relation is. A belief-change operation (Modal/Epistemic and Applications/Knowledge hold revision). A conditional logic individuated by its object-language conditional (Modal/Alethic), even though the flat fragment of such logics and preferential consequence coincide. Paraconsistency or paracompleteness, which drop different things.
Boundary: The seam with Applications/Knowledge runs through individuation, not through subject: default logic and circumscription induce nonmonotonic consequence relations and stay in Knowledge, because the representation problem is what they are; the postulate-and-representation theory of the relations themselves lives here, and Preferential Logic is cross-listed in both as the bridge. The seam with Modal/Alethic's Conditional Logic is the flat-fragment correspondence: a preferential relation is the flat fragment of a conditional logic, and which entry states which half of that fact is recorded in both. The division-level criterion already admits this subdivision's move: transitivity of consequence is a structural rule and so is monotony, and a logic individuated by dropping either belongs in Structural however classical it is otherwise.