A Science of Reasoning

This paper addresses the question of how we can understand reasoning in general and mathematical proofs in particular. It argues the need for a high-level understanding of proofs to complement the low-level understanding provided by Logic. It proposes a role for computation in providing this high-level understanding, namely by the association of proof plans with proofs. Proof plans are deened and examples are given for two families of proofs. Criteria are given for assessing the association of a proof plan with a proof. 1 Motivation: the understanding of mathematical proofs The understanding of reasoning has interested researchers since, at least, Aristotle. Logic has been proposed by Aristotle, Boole, Frege and others as a way of formalising arguments and understanding their structure. There have also been psychological studies of how people and animals actually do reason. The work on Logic has been especially innuential in the automation of reasoning. For instance, resolution, Robinson 65], the paradigm technique for automatic reasoning, was based on the work of logicians such as Herbrand. Logic has been used for the representation of knowledge in artiicial intelligence, where it has inspired the invention of new kinds of logics 1 , e.g. for non-monotonic reasoning. In this paper we argue that Logic is not enough to understand reasoning. It provides only a low-level, step by step understanding, whereas a high-level, strategic understanding is also required. Robinson has expressed this requirement with the slogan: Proof = Guarantee + Explanation where the \Guarantee" is provided by the logical, low-level, step by step check of soundness, and the \Explanation" is a high-level outline showing how the parts of the proof relate to each other. Many commonly observed phenomena of reasoning cannot be explained without such a high-level understanding. Furthermore, automatic reasoning is impractical without a high-level understanding. We propose a science of reasoning which provides both a low-and a high-level understanding of reasoning. It combines Logic with the concept of proof plans, Bundy 88]. We illustrate this with examples from mathematical reasoning, but it is intended that the science should eventually apply to all kinds of reasoning. am grateful for comments on the rst draft of this paper from two anonymous referees. 1 We adopt the convention of using uncapitalisedìogic' for the various mathematical theories and capitalised`Logic' for the discipline in which these logics are studied. 1 In the rest of this paper we describe how Logic provides a low-level …