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A new deconstructive logic: linear logic

Published online by Cambridge University Press:  12 March 2014


Vincent Danos
Affiliation:
CNRS URA 753, Equipe de Logique Mathématique, Université Paris VII, 2, Place Jussieu, F-75251 Paris Cedex 05, France, E-mail: danos@logique.jussieu.fr
Jean-Baptiste Joinet
Affiliation:
Equipe de Logique Mathématique, Université ParisVII, 2, Place Jussieu, F-75251 Paris Cedex 05, France Université Paris I, (Panthéon-Sorbonne), 17, Rue de la Sorbonne, F-75231 Paris Cedex 05, France, E-mail: joinet@logique.jussieu.fr
Harold Schellinx
Affiliation:
Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, NL-3508 Ta Utrecht, The Netherlands Equipe de Logique Mathématique, Université Paris VII, 2, Place Jussieu, F-75251 Paris Cedex 05, France, E-mail: schellin@math.ruu.nl

Abstract

The main concern of this paper is the design of a noetherian and confluent normalization for LK2 (that is, classical second order predicate logic presented as a sequent calculus).

The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's λμ, FD ([10, 12, 32, 36]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of ‘programming-with-proofs’ ([26, 27]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non-additive proof nets, to be precise) using appropriate embeddings (so-called decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making.

A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these ‘deconstructive purposes’.


Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1997

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