A Lexicographic Inference for Partially Preordered Belief Bases

Coherence-based approaches are quite popular to reason under inconsistency. Most of them are defined with respect to totally preordered belief bases such as the lexicographic inference which is known to have desirable properties from theoretical, practical and psychological points of view. However, partially preordered belief bases offer much more flexibility to represent efficiently incomplete knowledge and to avoid comparing unrelated pieces of information. In this paper, we propose a lexicographic inference for partially preordered belief bases that extends the classical one. On one hand, we define a natural inference relation which consists in applying classical lexicographic inference from all compatible totally preordered belief bases. On the other hand, we propose a novel cardinality-based preorder between consistent subbases. This cardinality-based preorder can be indifferently applied on partially or totally preordered belief bases. Then, applying classical inference on the preferred consistent subbases, according to this preorder, provides another lexicographic inference relation for partially preordered belief bases. Interestingly enough, we show that these two inference relations are equivalent. Lastly, a semantic characterization of these two equivalent definitions is provided.