 Soon after Henkin’s work, Jaakko Hintikka added that the same idea applies with conjunctions and disjunctions. We can regard a conjunction ‘\(\phi \wedge \psi\)’ as a universally quantified statement expressing ‘Every one of the sentences \(\phi , \psi\) holds’, so it should be for the player \(\forall\) to choose one of the sentences. As Hintikka put it, to play the game \(G(\phi \wedge \psi), \forall\) chooses whether the game should proceed as \(G(\phi)\) or as \(G(\psi)\). Likewise disjunctions become existentially quantified statements about sets of sentences, and they mark moves where the player \(\exists\) chooses how the game should proceed. To bring quantifiers into the same style, he proposed that the game \(G(\forall x \phi(x))\) proceeds thus: player \(\forall\) chooses an object and provides a name \(a\) for it, and the game proceeds as \(G(\phi(a))\). (And likewise with existential quantifiers, except that \(\exists\) chooses.) Hintikka also made an ingenious suggestion for introducing negation. Each game G has a dual game which is the same as G except that the players \(\forall\) and \(\exists\) are transposed in both the rules for playing and the rules for winning. The game \(G(\neg \phi)\) is the dual of \(G(\phi)\).