Basaldella, Michele and Terui, Kazushige - On the meaning of logical completeness

lmcs:1066 - Logical Methods in Computer Science, December 22, 2010, Volume 6, Issue 4
On the meaning of logical completeness

Authors: Basaldella, Michele and Terui, Kazushige

Goedel's completeness theorem is concerned with provability, while Girard's theorem in ludics (as well as full completeness theorems in game semantics) are concerned with proofs. Our purpose is to look for a connection between these two disciplines. Following a previous work [3], 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 the orthogonal of A which defeats P. Compared with proofs of full completeness in game semantics, ours exhibits a striking similarity with proofs of Goedel's completeness, in that it explicitly constructs a countermodel essentially using Koenig's lemma, proceeds by induction on formulas, and implies an analogue of Loewenheim-Skolem theorem.

Source :
DOI : 10.2168/LMCS-6(4:11)2010
Volume: Volume 6, Issue 4
Published on: December 22, 2010
Submitted on: November 22, 2009
Keywords: Computer Science - Logic in Computer Science,F.3.2, F.4.1


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