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A new deconstructive logic: linear logic

Published online by Cambridge University Press:  12 March 2014

Vincent Danos ,
Jean-Baptiste Joinet  and
Harold Schellinx
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
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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
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 755 - 807
Copyright
Copyright © Association for Symbolic Logic 1997

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