Beyond rational monotony: Some strong non-Horn rules for nonmonotonic inference relations

Lehmann, Magidor and others have investigated the effects of adding the non-Horn rule of rational monotony to the rules for preferential inference in nonmonotonic reasoning. In particular, they have shown that every inference relation satisfying those rules is generated by some ranked preferential model. We explore the effects of adding a number of other non-Hem rules that are stronger than or incomparable with rational monotony, but which are still weaker than plain monotony. Distinguished among these is a rule of determinacy preservation, equivalent to one of rational transitivity, for which we establish a representation theorem in terms of quasi-linear preferential models. An important tool in the proof of the representation theorem is the following purely semantic result, implicit in work of Freund, but here established by a more direct argument: every ranked preferential model generates the same inference relation as some ranked preferential model that is collapsed, in the sense of being both injective and such that each of its states is minimal for some formula. We also consider certain other non-Horn rules which are incomparable with monotony but are implied by conditional excluded middle, and establish a representation result for a central one among them, which we call fragmented disjunction, equivalent to fragmented conjunction, in terms of almost linear preferential models. Finally, we consider briefly some curious Horn rules beyond the preferential ones but weaker than monotony, notably those which we call conjunctive insistence and n-monotony.