Studia Logica

, Volume 71, Issue 3, pp 415–442 | Cite as

Proof Nets for the Multimodal Lambek Calculus

  • Richard Moot
  • Quintijn Puite
Article

Abstract

We present a novel way of using proof nets for the multimodal Lambek calculus, which provides a general treatment of both the unary and binary connectives. We also introduce a correctness criterion which is valid for a large class of structural rules and prove basic soundness, completeness and cut elimination results. Finally, we will present a correctness criterion for the original Lambek calculus Las an instance of our general correctness criterion.

Cut Elimination Lambek Calculus Linear Logic Multimodal Lambek Calculus Proof Nets Proof Theory 

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References

  1. Danos, V.: 1990, ‘La Logique Linéaire Appliquée à l'étude de Divers Processus de Normalisation (Principalement du λ-Calcul)'. Ph.D. thesis, University of Paris VII.Google Scholar
  2. Danos, V. and L. Regnier: 1989, ‘The Structure of Multiplicatives'. Archive for Mathematical Logic 28, 181-203.Google Scholar
  3. de Groote, P.: 1999, ‘An Algebraic Correctness Criterion for Intuitionistic Proof-nets'. Theoretical Computer Science 224(1–2), 115-134.Google Scholar
  4. de Groote, P. and F. Lamarche: 2002, ‘Classical non-associative Lambek calculus'. Studia Logica, in this issue, 245-278.Google Scholar
  5. Girard, J.-Y.: 1987, ‘Linear Logic'. Theoretical Computer Science 50, 1-102.Google Scholar
  6. Girard, J.-Y.: 1996, ‘Proof-Nets: The Parallel Syntax for Proof-Theory'. In: P. Agliana and A. Ursini (eds.): Logic and Algebra. New York: Marcel Dekker.Google Scholar
  7. Lafont, Y.: 1995, ‘From Proof Nets to Interaction Nets'. In: J.-Y. Girard, Y. Lafont, and L. Regnier (eds.): Advances in Linear Logic. Cambridge University Press, pp. 225-247.Google Scholar
  8. Lambek, J.: 1958, ‘The Mathematics of Sentence Structure'. American Mathematical Monthly 65, 154-170.Google Scholar
  9. Lambek, J.: 1961, ‘On the Calculus of Syntactic Types'. In: R. Jacobson (ed.): Structure of Language and its Mathematical Aspects, Proceedings of the Symposia in Applied Mathematics, Vol. XII. pp. 166-178, American Mathematical Society.Google Scholar
  10. Moortgat, M.: 1997, ‘Categorial Type Logics'. In: J. van Benthem and A. ter Meulen (eds.): Handbook of Logic and Language. Elsevier/MIT Press, Chapt. 2.Google Scholar
  11. Puite, Q.: 1998, ‘Proof Nets with Explicit Negation for Multiplicative Linear Logic'. Technical report, Department of Mathematics, Utrecht University. Preprint 1079.Google Scholar
  12. Puite, Q.: 2001, ‘Sequents and Link Graphs: Contraction Criteria for Refinements of Multiplicative Linear Logic’. Ph.D. thesis, Department of Mathematics, Utrecht University.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Richard Moot
    • 1
  • Quintijn Puite
    • 2
  1. 1.Utrecht Institute of Linguistics OTSUtrecht UniversityNetherlands
  2. 2.Department of MathematicsUtrecht UniversityNetherlands

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