The Frame Problem and the Semantics of Classical Proofs

We outline the logic of current approaches to the so-called “frame problem” (that is, the problem of predicting change in the physical world by using logical inference), and we show that these approaches are not completely extensional since none of them is closed under uniform substitution. The underlying difficulty is something known, in the philosophical community, as Goodman’s “new riddle of induction” or the “Grue paradox”. Although it seems, from the philosophical discussion, that this paradox cannot be solved in purely a priori terms and that a solution will require some form of real-world data, it nevertheless remains obscure both what the logical form of this real-world data might be, and also how such data actually interacts with logical deduction. We show, using work of McCain and Turner, that this data can be captured using the semantics of classical proofs developed by Bellin, Hyland and Robinson, and, consequently, that the appropriate arena for solutions of the frame problem lies in proof theory. We also give a very explicit model for the categorical semantics of classical proof theory using techniques derived from work on the frame problem.