IDentity Of Proofs Based On Normalization And Generality

Abstract

Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal formin natural deduction. Lambek worked on a normalization proposal analogous to PrawitzÕs, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in termsof an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to a renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to low-dimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.