Defeasible Entailment: from Rational Closure to Lexicographic Closure and Beyond

In this paper we present what we believe to be the first systematic approach for extending the framework for de- feasible entailment first presented by Kraus, Lehmann, and Magidor—the so-called KLM approach. Drawing on the properties for KLM, we first propose a class of basic defea- sible entailment relations. We characterise this basic frame- work in three ways: (i) semantically, (ii) in terms of a class of properties, and (iii) in terms of ranks on statements in a knowlege base. We also provide an algorithm for computing the basic framework. These results are proved through vari- ous representation results. We then refine this framework by defining the class of rational defeasible entailment relations. This refined framework is also characterised in thee ways: se- mantically, in terms of a class of properties, and in terms of ranks on statements. We also provide an algorithm for com- puting the refined framework. Again, these results are proved through various representation results.
We argue that the class of rational defeasible entail- ment relations—a strengthening of basic defeasible entail- ment which is itself a strengthening of the original KLM proposal—is worthy of the term rational in the sense that all of them can be viewed as appropriate forms of defeasi- ble entailment. We show that the two well-known forms of defeasible entailment, rational closure and lexicographic clo- sure, fall within our rational defeasible framework. We show that rational closure is the most conservative of the defeasi- ble entailment relations within the framework (with respect to subset inclusion), but that there are forms of defeasible en- tailment within our framework that are more “adventurous” than lexicographic closure.