Reconsideration of circumscriptive induction with pointwise circumscription

Explanatory induction and descriptive induction are two main frameworks for induction in logic. Both frameworks, however, have some serious drawbacks: explanatory induction often exhibits an inductive leap problem, and descriptive induction sometimes fails to explain given observations. Circumscriptive induction is a new framework intended to overcome these difficulties by unifying explanatory induction and descriptive induction. In this paper, we study and improve several aspects of circumscriptive induction. First, we reformulate the concepts of inductive leaps and conservativeness. The reformulated conservativeness becomes a partial generalization of the original conservativeness. We give a simple sufficient condition for the reformulated conservativeness and clarify a relationship between correct solutions and conservativeness. Furthermore, we propose a new tractable induction framework, called pointwise circumscriptive induction, which just uses first-order logic with equality in the formulation, and does not demand any second-order computation. Pointwise circumscriptive induction enables us to derive some interesting hypotheses through ordinary resolution performed in a mechanical way.