A general theory of deduction, induction, and learning

Abstract

Deduction, induction, learning, are various aspects of a more general scientific activity: the discovery of truth. We propose to embed them in a common, logical framework. First, we define a generalized notion of “logical consequence.” Alternating compact and “weakly compact” consequences, we stratify the set of generalized logical consequences of a given theory in a hierarchy. Classical first-order logic is a particular case of this framework; the fact that it is all about deduction is due to the compactness theorem, and this is reflected by the collapsing of the corresponding hierarchy to the first level. Classical learning paradigms in the inductive inference literature provide other particular cases. Finite learning corresponds exactly to the first level (or level Σ1) of the hierarchy, whereas learning in the limit corresponds to another level (namely Σ2). More generally, strong and natural connections exist between our hierarchy of generalized logical consequences, the Borel hierarchy, and the hierarchy which measures the complexity of a formula in terms of alternations of quantifiers. It is hoped that this framework provides the foundation of a unified logic of deduction and induction, and highlights the inductive nature of learning. An essential motivation for our work is to apply the theory presented here to the design of “Inductive Prolog”, a system with both deductive and inductive capabilities, based on a natural extension of the resolution principle.