On the meaning of logical completeness

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.

💡 Research Summary

The paper investigates a deep connection between Gödel’s classical completeness theorem and the full‑completeness results that arise in ludics and game semantics. While Gödel’s theorem links provability to semantic truth, full‑completeness identifies each semantic object (a strategy or model) with a syntactic proof. The authors aim to bridge these perspectives by extending the original ludics framework with two dual operators: contraction, which re‑introduces the ability to duplicate resources and thus captures the classical structural rule, and universal nondeterminism, which simultaneously explores all possible choices and serves as the dual of contraction. This enriched system faithfully represents a polarized fragment of linear logic and, consequently, a constructive version of classical propositional logic.

The central result is a “completeness for proofs” theorem. For any behaviour A (interpreted as a formula) and any design P (interpreted as a proof attempt), exactly one of the following holds: (i) P is a genuine proof of A, or (ii) there exists a model M belonging to the orthogonal A⊥ that defeats P. The proof proceeds in two main phases. First, a structural induction on the shape of A handles atomic, additive, multiplicative, and negation cases, taking into account the new contraction and nondeterminism rules. Second, the authors invoke König’s Lemma to construct a counter‑model when P fails. They view the interaction between P and a potential opponent as an infinite branching tree; König’s Lemma guarantees an infinite branch, which is then read as a concrete model in A⊥ that refutes P. This construction mirrors Gödel’s original method of building a term model from a maximally consistent set, but it is carried out entirely within the ludics interaction paradigm.

A further contribution is an analogue of the Löwenheim‑Skolem theorem: the counter‑model can be chosen of minimal (often finite) size, showing that the completeness theorem does not rely on large or exotic structures. This mirrors the classical observation that any satisfiable first‑order theory has a countable model, and it strengthens the computational relevance of the result.

The paper also situates its theorem alongside full‑completeness results in game semantics. Traditional game‑theoretic full‑completeness often provides existence proofs without explicit construction of the opponent strategy. By contrast, the ludics‑based approach yields an explicit, algorithmic construction of the defeating model, thanks to the interplay of contraction and universal nondeterminism. This makes the result particularly attractive for applications such as proof search, automated theorem proving, and the design of verification tools where one needs not only to know that a counter‑example exists but also to generate it effectively.

In conclusion, the authors demonstrate that by enriching ludics with dual operators, one can obtain a completeness theorem that is structurally similar to Gödel’s, yet operates at the level of proofs rather than mere provability. The work opens several avenues for future research, including extensions to richer logical fragments, deeper analysis of the duality between contraction and nondeterminism, and practical implementations in proof assistants and model‑checking frameworks.