Formal explanations as logical derivations

Abstract

According to a longstanding philosophical tradition dating back to Aristotle, certain proofs do not only certify the truth of their conclusion but also explain it. Lately, much effort is being devoted to logically characterise the explanatory relation of grounding, especially by proof-theoretical means. Nevertheless, no thorough investigation of the resulting notion of formal explanation exists. We show that formal explanations can be seen as logical derivations of a particular kind and study the interactions between grounding and logical rules, formal explanations and logical derivations. We define a minimal calculus that captures both grounding and logical derivability, and show by a normalisation procedure that grounding rules are proof-theoretically balanced with respect to logical elimination rules. The introduced calculus enables us to combine logical derivations and explanations, to distinguish explanatory parts of derivations from non-explanatory parts, and to compose explanations in order to construct chains of consecutive grounding steps.