 Contradictory negation and GTS. The truth-conditions that GTS yields are of the form ‘there are strategy functions \(f_1 ,\ldots ,f_n\) such that —,’ i.e., it gives rise to truth-conditions expressible in \(\mathbf{ESO}\). By the strong inexpressibility of contradictory negation, there is no single \(\mathbf{IFL}\) sentence, not translatable into \(\mathbf{FO}\), whose contradictory negation has a truth-condition of that form. This fact makes it understandable why contradictory negation should not be expected to admit of a game-theoretical interpretation along the same lines in which the other logical operators are interpreted. Different ways of assigning a game-theoretical interpretation to contradictory negation can, however, be developed. To this end, in the context of fully extended IF first-order logic (FEIFL, cf. Subsect. 3.4), Hintikka has proposed to use semantic games with subgames. (See Hintikka 2002c, 2006b; for subgames, see Carlson &amp; Hintikka 1979, Hintikka &amp; Kulas 1983.) This approach leads to mixing the play level with the strategy level: choices of individuals made in a subgame may depend on strategy functions chosen in earlier subgames. In Tulenheimo (2014), a game-theoretical semantics is formulated for the fragment of FEIFL that consists of formulas in prenex form. Plays of the correlated games do not involve choosing any second-order objects, such as strategy functions. Contradictory negation \((\neg)\) is interpreted by introducing an additional component in game positions: a mode. At the play level, the negation \(\neg\) triggers not only a role switch (like the dual negation \({\sim})\), but it also involves changing the mode from positive to negative or vice versa. The semantic effect of modes becomes visible at the strategy level: modes regulate the way in which the truth condition of a sentence involves existentially or universally quantifying over strategy functions. Like independence indications, also occurrences of \(\neg\) are, then, interpreted in terms of conditions that act on the strategy level. For related research, see Figueira et al. (2011, 2014).