On the Meaning of Logical Completeness

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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.