 One can prove that for any first-order sentence \(\phi\), interpreted in a fixed structure \(A\), player \(\exists\) has a winning strategy for Hintikka’s game \(G(\phi)\) if and only if \(\phi\) is true in \(A\) in the sense of Tarski. Two features of this proof are interesting. First, if \(\phi\) is any first-order sentence then the game \(G(\phi)\) has finite length, and so the Gale-Stewart theorem tells us that it is determined. We infer that \(\exists\) has a winning strategy in exactly one of \(G(\phi)\) and its dual; so she has a winning strategy in \(G(\neg \phi)\) if and only if she doesn’t have one in \(G(\phi)\). This takes care of negation. And second, if \(\exists\) has a winning strategy for each game \(G(\phi(a))\), then after choosing one such strategy \(f_a\) for each \(a\), she can string them together into a single winning strategy for \(G(\forall x \phi(x))\) (namely, ‘Wait and see what element \(a \forall\) chooses, then play \(f_a\)’). This takes care of the clause for universal quantifiers; but the argument uses the axiom of choice, and in fact it is not hard to see that the statement that Hintikka’s and Tarski’s definitions of truth are equivalent is itself equivalent to the axiom of choice (given the other axioms of Zermelo-Fraenkel set theory).