 Hodges (1997a,b) showed, however, that \(\mathbf{IFL}\) can be given a compositional  semantics.[54]  The semantics is given by recursively defining the satisfaction relation ‘\(M \vDash_X \phi\)’ (read: \(\phi\) is satisfied in \(M\) by \(X)\), where \(X\) is a set of variable assignments. While the Tarskian semantics for \(\mathbf{FO}\) is in terms of single variable assignments, Hodges’s semantics employs sets of variable assignments. The game-theoretical semantics of \(\mathbf{IFL}\) is captured by this compositional semantics: for every formula \(\phi\) of \(\mathbf{IFL}\), player 2 has a winning strategy in \(G(\phi , M, g)\) iff the condition \(M \vDash_{\{g\}} \phi\) holds. Due to Hodges’s semantic clauses for existential quantifiers and disjunctions involving independence indications, the evaluation of \(\phi\) relative to the singleton set \(\{g\}\) generally leads to evaluating syntactic components of \(\phi\) relative to sets of (possibly infinitely) many variable assignments. Hintikka remarks (2006a: 65) that if one is sufficiently ruthless, one can always save compositionality by building the laws of semantic interaction of different expressions into the respective meanings of those  expressions.[55]