Research On Axiomatizable Classes Of Injective Models

In nonmonotonic logic, injective preferential models play an important role. Although fruitful representation results induced by some kinds of injective models have been established in the literature, it is still an open problem to characterize the family of all injective inference relations in terms of proof theoretical properties. The types of postulates appeared in the literature seem not to be competent to characterize this family. This brings up an interesting theoretical problem: What kinds of injective inference relations may be characterized by existent types of postulates? This paper aims to obtain some characteristics of axiomatizable classes. The main work is as follows:1. A monadic second-order frame language is presented. The relationship between ?0 -axiomatizability and second-order definability is explored. The influence of the four-node structure destroying the injective preservation under reductions is studied. And then we obtain the characteristics of axiomatizable classes of injective preferential models in finite case.2. A notion of an admissible set is introduced. Based on this notion, we show that any preferential model, which does not contain any four-node substructure, must be a reduct of some injective model.3. A notion of closure under reductions of posets is brought forward. Based on this notion, we use the knowledge of model theory to handle the infinite case, and furnish a necessary and sufficient condition for the axiomatizability of classes of injective preferential models using general rules.