Classical Negation and Game-Theoretical Semantics

Abstract

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: $\rightharpoondown$ and $\sim$. 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; $\rightharpoondown$ behaves as classical negation in $L_1$. Logic $L_2$ extends $L_1$, and it is shown to capture the ${\Sigma }_1^2$-fragment of third-order logic. Consequently the classical negation remains inexpressible in $L_2$.