 The models of Commonsense Entailment differ significantly from those of preferential logic and the logic of extreme probabilities. Instead of having structures that contain sets of models of a standard, default-free language, a model of the language of Commonsense Entailment includes a set of possible worlds, together with a function that assigns standard interpretation (a model of the default-free language) to each world. In addition, to each pair consisting of a world \(w\) and a set of worlds (proposition) \(A\), there is a function \(*\) that assigns a set of worlds \({*}(w,A)\) to the pair. The set \({*}(w,A)\) is the set of most normal \(A\)-worlds, from the perspective of \(w\). A default conditional \((p \Rightarrow q)\) is true in a world \(w\) (in such a model) just in case all of the most normal \(p\) worlds (from \(w\)’s perspective) are worlds in which \(q\) is also true. Since we can assign truth-conditions to each such conditional, we can define the truth of nested conditionals, whether the conditionals are nested within Boolean operators or within other conditionals. Moreover, we can define both a classical, monotonic consequence relation for this class of models and a defeasible, nonmonotonic relation (in fact, the nonmonotonic consequence relation can be defined in a variety of ways). We can then distinguish between a default conditional’s following with logical necessity from a default theory and its following defeasibly from that same theory. Contraposition, for example — inferring \((\neg q \Rightarrow \neg p)\) from \((p \Rightarrow q)\) — is not logically valid for default conditionals, but it might be a defeasibly correct  inference.[8]