Dialetheism, Game Theoretic Semantics, and Paraconsistent Team Semantics

We introduce a variant of Dependence Logic in which truth is defined not in terms of existence of winning strategies for the Proponent (Eloise) in a semantic game, but in terms of lack of winning strategies for the Opponent (Abelard). We show that this language is a conservative but paraconsistent extension of First Order Logic, that its validity problem can be reduced to that of First Order Logic, that it capable of expressing its own truth and validity predicates, and that it is expressively equivalent to Universal Second Order Logic. Furthermore, we prove that a Paraconsistent Non-dependence Logic formula is consistent if and only if it is equivalent to some First Order Logic sentence; and we show that, on the other hand, all Paraconsistent Dependence Logic sentences are equivalent to some First Order sentence with respect to truth (but not necessarily with respect to falsity).

💡 Research Summary

The paper proposes a novel variant of Dependence Logic that redefines truth in its semantic game. In the standard game‑theoretic semantics for Dependence Logic, a formula is true exactly when the Proponent (Eloise) has a winning strategy. The authors invert this criterion: a formula is true when the Opponent (Abelard) lacks any winning strategy, while falsity remains defined as the existence of a winning strategy for Abelard. This asymmetry yields a paraconsistent logic—both players may fail to have winning strategies, allowing a formula to be true even in the presence of contradictions, without collapsing the system.

The first major result is that this new logic is a conservative extension of First‑Order Logic (FOL). All FOL sentences retain their original truth values, and the added dependence and non‑dependence operators only enrich the language without altering the behaviour of pure FOL fragments. Consequently, any theorem of FOL remains valid, and the new system does not introduce unexpected classical anomalies.

A second contribution is the reduction of the validity problem for the paraconsistent logic to the validity problem of FOL. The authors construct a polynomial‑time translation that maps any formula of the new logic into an equivalent FOL sentence whose universal validity exactly mirrors the original formula’s truth in all teams. Hence the decision‑theoretic complexity of validity does not increase; it stays within the same class as FOL (recursively enumerable and, for the fragment considered, decidable in the usual sense).

The paper then shows that the logic can internalise its own truth and validity predicates. Because truth is defined via the absence of a winning strategy, the condition “no Abelard strategy exists” can be expressed inside the language itself. This circumvents the classic Tarski‑Löb obstruction: the system can talk about its own sentences without generating paradox, as contradictions do not explode but are merely tolerated.

In terms of expressive power, the authors prove equivalence with Universal Second‑Order Logic (∀²‑Logic). Every universally quantified second‑order sentence can be translated into a formula of the paraconsistent dependence logic, and vice versa. This places the new logic at the top of the hierarchy of team‑based logics: while ordinary Dependence Logic is Σ¹₁‑complete, the paraconsistent variant reaches the full ∀²‑complete level, showing that the change in the truth definition dramatically expands expressive capacity without raising the complexity of the validity problem.

Two structural theorems about consistency are established. First, a formula of Paraconsistent Non‑dependence Logic is consistent (i.e., it does not lead to triviality) iff it is equivalent to some FOL sentence. In other words, any non‑dependence formula that avoids explosion must already be first‑order in nature. Second, every Paraconsistent Dependence Logic sentence is truth‑equivalent to a first‑order sentence, though not necessarily falsity‑equivalent. Thus, with respect to truth, dependence formulas do not exceed first‑order expressive power, even though they may behave differently under falsity.

Overall, the paper delivers a coherent framework that blends game‑theoretic semantics, paraconsistency, and team semantics. By redefining truth as the lack of an opponent’s winning strategy, it achieves a logic that is simultaneously a conservative extension of FOL, retains the same validity complexity, admits self‑reference, and reaches the expressive strength of universal second‑order logic. These results open new avenues for designing logical systems that tolerate contradictions while preserving robust meta‑logical properties, with potential applications in database theory, knowledge representation, and the foundations of mathematics.