Classical Negation and Game-Theoretical Semantics (original) (raw)

Typical applications of Hintikka's game-theoretical semantics (GTS) give rise to semantic attributes-truth, falsity-expressible in the † 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. 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 † 2 1-fragment of third-order logic. Consequently the classical negation remains inexpressible in L 2 .