Conditional Splittings of Belief Bases and Nonmonotonic Inference with c-Representations

The concept of 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 share some atoms.

p-Entailment and system Z fail to satisfy conditional syntax

splitting, and up to now, only two inductive inference

operators, lexicographic inference and system W, have been

shown to satisfy this property. In this paper, we introduce

the concept of conditional semantic splitting. 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 also

fully complies with conditional syntax splitting.