 Compared with first-order logic, IF logic is missing a component that the game semantics won’t supply. The game semantics tells us when a sentence is true in a structure. But if we take a formula with \(n\) free variables, what does the formula express about the ordered \(n\)-tuples of elements of the structure? In first-order logic it would define a set of them, i.e. an \(n\)-ary relation on the structure; the Tarski truth definition explains how. Is there a similar definition for arbitrary formulas of IF logic? It turns out that there is one for the slightly different logic introduced by Hodges 1997, and it leads to a Tarski-style truth definition for the language of that logic. With a little adjustment this truth definition can be made to fit IF logic too. But for both of these new logics there is a catch: instead of saying when an assignment of elements to free variables makes a formula true, we say when a set of assignments of elements to free variables makes the formula true. Väänänen 2007 made this idea the basis for a range of new logics for studying the notion of dependence (see entry on  dependence logic).  In these logics the semantics is defined without games, although the original inspiration comes from the work of Hintikka and Sandu. 