A New Conjecture About Identity of Proofs

TL;DR

This paper proposes a novel criterion for proof identity based on natural transformations, unifying various rule-permutation concepts across multiple logical systems.

Contribution

It introduces a new approach to proof identity using natural transformations, offering a unified perspective on rule permutations in propositional and higher-order logics.

Findings

Natural transformations effectively characterize proof identity.

The approach unifies existing criteria based on normalization and graph structures.

It applies to propositional, first- and second-order logic systems.

Abstract

A central problem in proof-theory is that of finding criteria for identity of proofs, that is, for when two distinct formal derivations can be taken as denoting the same logical argument. In the literature one finds criteria which are either based on proof normalization (two derivations denote the same proofs when they have the same normal form) or on the association of formal derivations with graph-theoretic structures (two derivations denote they same proof when they are associated with the same graph). In this paper we argue for a new criterion for identity of proofs which arises from the interpretation of formal rules and derivations as natural transformations of a suitable kind. We show that the naturality conditions arising from this interpretation capture in a uniform and elegant ways several forms of "rule-permutations" which are found in proof-systems for propositional, first- and second-order logic.