[1] Most studied formal logics have a monotonic entailment relation, meaning that adding a formula to the hypotheses never produces a pruning of its set of conclusions.
The belief revision approach is alternative to paraconsistent logics, which tolerate inconsistency rather than attempting to remove it.
[2] Model-theoretic formalization of a non-monotonic logic begins with restriction of the semantics of a suitable monotonic logic to some special models, for instance, to minimal models,[3][4] and then derives a set of non-monotonic rules of inference, possibly with some restrictions on which contexts these rules may be applied in, so that the resulting deductive system is sound and complete with respect to the restricted semantics.
[5] Unlike some proof-theoretic formalizations that suffered from well-known paradoxes and were often hard to evaluate with respect of their consistency with the intuitions they were supposed to capture, model-theoretic formalizations were paradox-free and left little, if any, room for confusion about what non-monotonic patterns of reasoning they covered.
Examples of proof-theoretic formalizations of non-monotonic reasoning, which revealed some undesirable or paradoxical properties or did not capture the desired intuitive comprehensions, that have been successfully (consistent with respective intuitive comprehensions and with no paradoxical properties, that is) formalized by model-theoretic means include first-order circumscription, closed-world assumption,[5] and autoepistemic logic.