On the Meaning of Logical Completeness

  • Michele Basaldella
  • Kazushige Terui
Conference paper

DOI: 10.1007/978-3-642-02273-9_6

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5608)
Cite this paper as:
Basaldella M., Terui K. (2009) On the Meaning of Logical Completeness. In: Curien PL. (eds) Typed Lambda Calculi and Applications. TLCA 2009. Lecture Notes in Computer Science, vol 5608. Springer, Berlin, Heidelberg

Abstract

Gödel’s completeness theorem is concerned with provability, while Girard’s theorem in ludics (as well as full completeness theorems in game semantics) is concerned with proofs. Our purpose is to look for a connection between these two disciplines. Following a previous work [1], we consider an extension of the original ludics with contraction and universal nondeterminism, which play dual roles, in order to capture a polarized fragment of linear logic and thus a constructive variant of classical propositional logic.

We then prove a completeness theorem for proofs in this extended setting: for any behaviour (formula) A and any design (proof attempt) P, either P is a proof of A or there is a model M of \({\mathbf{A}}^{\bot}\) which beats P. Compared with proofs of full completeness in game semantics, ours exhibits a striking similarity with proofs of Gödel’s completeness, in that it explicitly constructs a countermodel essentially using König’s lemma, proceeds by induction on formulas, and implies an analogue of Löwenheim-Skolem theorem.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Michele Basaldella
    • 1
  • Kazushige Terui
    • 1
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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