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From Sets and Types to Topology and AnalysisTowards practicable foundations for constructive mathematics$
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Laura Crosilla and Peter Schuster

Print publication date: 2005

Print ISBN-13: 9780198566519

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198566519.001.0001

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THE DUALITY OF CLASSICAL AND CONSTRUCTIVE NOTIONS AND PROOFS

THE DUALITY OF CLASSICAL AND CONSTRUCTIVE NOTIONS AND PROOFS

Chapter:
(p.149) 9 THE DUALITY OF CLASSICAL AND CONSTRUCTIVE NOTIONS AND PROOFS
Source:
From Sets and Types to Topology and Analysis
Author(s):

Sara Negri

Jan Von Plato

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566519.003.0009

The method of converting mathematical axioms into rules of sequent calculus reveals a perfect duality between classical and constructive basic notions, such as equality and apartness, and between the respective rules. Derivations with the mathematical rules of a constructive theory are duals of corresponding classical derivations. The class of geometric theories is among those convertible into rules, and the duality defines a new class of ‘co-geometric’ theories. Examples of such theories are projective and affine geometry. The logical rules of classical sequent calculus are invertible. For quantifier-free theories, this has the effect that logical rules can be permuted to apply after the mathematical rules. For mathematical rules involving variable conditions, this separation of logic does not always hold because quantifier rules may fail to permute down. A sufficient condition for the permutability of mathematical rules is determined and applied to give an extension of Herbrand theorem from universal to geometric and co-geometric theories.

Keywords:   sequent calculus, axioms, rules, classical, constructive, geometric theories, Herbrand theorem

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