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Classical negation and game-theoretical semantics. (English) Zbl 1342.03030
Summary: 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 \(L_2\).

MSC:
03B60 Other nonclassical logic
03C80 Logic with extra quantifiers and operators
03B15 Higher-order logic; type theory (MSC2010)
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