Conditional Syntax and Semantic Splittings of Strongly and Weakly Consistent Belief Bases

Conditional syntax splitting for inductive inference from conditional belief bases has been proposed as a generalization of syntax splitting, which also covers cases where the conditionals in the subbases may share some atoms. While p-entailment and system Z fail to satisfy conditional syntax splitting, two other inductive inference operators, lexicographic inference and system W, have been shown to satisfy this property for reasoning from strongly consistent belief bases. In this article, we introduce the concept of conditional semantic splitting of a belief base. For both syntax splitting and semantic splitting, we take not only strongly consistent belief bases into account, but also belief bases that are only weakly consistent, enforcing some worlds to be fully infeasible. We show that c-representations satisfy a core postulate relating conditional splittings on the syntax and the semantic level. Based on these findings, we investigate conditional syntax splitting for nonmonotonic inference with c-representations. Regarding single c-representations, we utilize the concept of selection strategies and show that a straightforward property of the selection strategy leads to inference operators satisfying conditional syntax splittings. Furthermore, we show that c-inference, taking all c-representations of a belief base into account fully complies with conditional syntax splitting, and we prove that credulous and weakly skeptical inference based on c-representations also satisfies conditional syntax splitting. In particular, we show that these syntax splitting properties hold for strongly consistent belief bases and also in the case of only weakly consistent belief bases.