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CUT ELIMINATION AND NORMALIZATION FOR GENERALIZED SINGLE AND MULTI-CONCLUSION SEQUENT AND NATURAL DEDUCTION CALCULI

Part of: Proof theory and constructive mathematics

Published online by Cambridge University Press:  29 June 2020

RICHARD ZACH
RICHARD ZACH*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF CALGARY 2500 UNIVERSITY DRIVE NW CALGARYABT2N 1N4, CANADAE-mail:rzach@ucalgary.caURL: https://richardzach.org/
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Abstract

Any set of truth-functional connectives has sequent calculus rules that can be generated systematically from the truth tables of the connectives. Such a sequent calculus gives rise to a multi-conclusion natural deduction system and to a version of Parigot’s free deduction. The elimination rules are “general,” but can be systematically simplified. Cut-elimination and normalization hold. Restriction to a single formula in the succedent yields intuitionistic versions of these systems. The rules also yield generalized lambda calculi providing proof terms for natural deduction proofs as in the Curry–Howard isomorphism. Addition of an indirect proof rule yields classical single-conclusion versions of these systems. Gentzen’s standard systems arise as special cases.

Keywords

sequent calculuscut-eliminationnatural deductionnormalizationproof termsCurry-Howard isomorphism

MSC classification

Primary: 03F03: Proof theory, general
Secondary: 03F05: Cut-elimination and normal-form theorems 03F07: Structure of proofs
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 3 , September 2021 , pp. 645 - 686
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Copyright
© Association for Symbolic Logic, 2020

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