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 Sigma(1)(1)-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, L-1 and L-2, in both of which two negation signs are available: -> and similar to. The latter is the usual GTS negation which transposes the players' roles, while the former will be interpreted via the notion of mode. Logic L-1 extends independence-friendly (IF) logic; -> behaves as classical negation in L-1. Logic L-2 extends L-1, and it is shown to capture the Sigma(2)(1)-fragment of third-order logic. Consequently the classical negation remains inexpressible in L-2.