David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jonathan Jenkins Ichikawa
Jack Alan Reynolds
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Bulletin of Symbolic Logic 9 (4):477-503 (2003)
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
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References found in this work BETA
Dag Prawitz & J. E. Fenstad (1975). Ideas and Results in Proof Theory. Journal of Symbolic Logic 40 (2):232-234.
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Citations of this work BETA
Willem Heijltjes (2010). Classical Proof Forestry. Annals of Pure and Applied Logic 161 (11):1346-1366.
Alberto Naibo & Mattia Petrolo (2015). Are Uniqueness and Deducibility of Identicals the Same? Theoria 81 (2):143-181.
L. Mehats & Sergei Soloviev (2007). Coherence in SMCCs and Equivalences on Derivations in IMLL with Unit. Annals of Pure and Applied Logic 147 (3):127-179.
Kosta Dosen & Zoran Petric (2012). Isomorphic Formulae in Classical Propositional Logic. Mathematical Logic Quarterly 58 (1):5-17.
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