Identity of Proofs Based on Normalization and Generality

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 form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz's, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in terms of 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. ?1. General proof theory. When a question is not in the main stream of mathematical investigations, and is also of a certain general, conceptual, kind, it runs the danger of being dismissed as "philosophical". Before the advent of recursion theory, many mathematicians would presumably have dismissed the question "What is a computable function?" as a philosophical question (or perhaps even as a psychological, empirical, question). It required something like the enthusiasm of a young discipline on the rise, which logic was in the first half of the twentieth century, for such a question to be embraced as legitimate, and seriously treated by mathematical means-with excellent results. An outsider might suppose that the question "What is a proof?" should be important for a field called proof theory, and then he would be surprised to find that this and related questions, one of which will occupy us here, Received on May 27, 2003; revised July 4, 2003; revised August 26, 2003. 2000 Mathematics Subject Classification. 03F03, 03F07, 03A05, 03-03.