Classical Negation and Game-Theoretical Semantics

Typical applications of Hintikka’s game-theoretical semantics (GTS) give rise to semantic attributes—truth, falsity—expressible in the Σ11-fragment of second-order logic. Actually a much more general notion of semantic attribute is motivated by strategic considerations. When identifying such a generalization, the notion of classical negation plays a crucial role. We study two languages, L1 and L2, in both of which two negation signs are available: ⇁ and ∼. The latter is the usual GTS negation which transposes the players’ roles, while the former will be interpreted via the notion of mode. Logic L1 extends independence-friendly (IF) logic; ⇁ behaves as classical negation in L1. Logic L2 extends L1, and it is shown to capture the Σ12-fragment of third-order logic. Consequently the classical negation remains inexpressible in L2.