Testability and Ockham’s Razor: How Formal and Statistical Learning Theory Converge in the New Riddle of Induction

Abstract

Nelson Goodman’s new riddle of induction forcefully illustrates a challenge that must be confronted by any adequate theory of inductive inference: provide some basis for choosing among alternative hypotheses that fit past data but make divergent predictions. One response to this challenge is to distinguish among alternatives by means of some epistemically significant characteristic beyond fit with the data. Statistical learning theory takes this approach by showing how a concept similar to Popper’s notion of degrees of testability is linked to minimizing expected predictive error. In contrast, formal learning theory appeals to Ockham’s razor, which it justifies by reference to the goal of enhancing efficient convergence to the truth. In this essay, I show that, despite their differences, statistical and formal learning theory yield precisely the same result for a class of inductive problems that I call strongly VC ordered, of which Goodman’s riddle is just one example.