A central theme in the study of human reasoning is the construction of explanations. Abduction is explanatory reasoning. It is a reasoning process to produce possible explanations and select the best ones for an observation. Abduction is not restricted to formal classical logic. In many situations, information is incomplete. From a logical point of view, it is not purely deductive or inductive; sometimes, nonmonotonic, higher-order, or statistical reasoning is applied. It is closely related to proofs and to belief revision in artificial intelligence (AI).

The authors aim to classify abstract properties of abduction and preferred explanations with different structural rules. They restrict their scope to propositional logical languages. They define explanatory relations as binary relations over formulas: relations between an observation and its possible explanation (called a preferred one in the paper). A preference criterion over the explanation of an observation is a partial ordering.

To make a detailed analysis of the relationship between preference criteria and explanatory relations, many structural properties are defined for the two notions, and relationships among them are investigated. Interesting examples demonstrate the introduced properties.

The authors prove some representation theorems. Under fairly general (and different sets of) conditions, the explanatory relations can be derived from suitable preference criteria.

The paper deals with the case of a static background language and theory. A very interesting further direction of investigation would be to allow the introduction of new notions by extending the background theory with their definitions to see how preferred explanations behave.