On the Meaning of Logical Completeness

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.