David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Logic Journal of the IGPL 10 (3):299-337 (2002)
The system of natural deduction that originated with Gentzen , and for which Prawitz proved a normalization theorem, is re-cast so that all elimination rules are in parallel form. This enables one to prove a very exigent normalization theorem. The normal forms that it provides have all disjunction-eliminations as low as possible, and have no major premisses for eliminations standing as conclusions of any rules. Normal natural deductions are isomorphic to cut-free, weakening-free sequent proofs. This form of normalization theorem renders unnecessary Gentzen's resort to sequent calculi in order to establish the desired metalogical properties of his logical system.Ultimate normal forms are well-adapted to the needs of the computational logician, affording valuable constraints on proof-search. They also provide an analysis of deductive relevance. There is a deep isomorphism between natural deductions and sequent proofs in the relevantized system
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Nissim Francez (2015). On the Notion of Canonical Derivations From Open Assumptions and its Role in Proof-Theoretic Semantics. Review of Symbolic Logic 8 (2):296-305.
Peter Schroeder-Heister (2014). The Calculus of Higher-Level Rules, Propositional Quantification, and the Foundational Approach to Proof-Theoretic Harmony. Studia Logica 102 (6):1185-1216.
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