Indefinite Reasoning with Definite Rules

In this paper we present a novel explanation of the source of indefinite information in common sense reasoning: Indefinite information arises from reports about the world expressed in terms of concepts that have been defined using only definite rules. Adopting this point of view, we show that first-order logic is insufficiently expressive to handle important examples of common sense reasoning. As a remedy, we propose the use of circumscribed definite rules, and we then investigate the proof theory of this more expressive framework. We consider two approaches: First, prototypical proofs, a special type of proof by induction, which yields a sound proof theory. Second, we describe cases in which there exists a decision procedure for answering queries, a particularly significant result because it shows that it is possible to have decidable query processing in circumscribed theories that are not equivalent to any first order theory.