Games and Logic

The idea behind these games is to obtain an alternative characterization of logical notions cherished by logicians such as truth in a model, or provability (in a formal system). We offer a quick survey of Hintikka’s evaluation games, which offer an alternative notion of truth in a model for first-order langauges. These are win-lose, extensive games of perfect information. We then consider a variation of these games, IF games, which are win-lose extensive games of imperfect information. Both games presuppose that the meaning of the basic vocabulary of the language is given. To give an account of the linguistic conventions which settle the meaning of the basic vocabulary, we consider signaling games, inspired by Lewis’ work. We close with IF probabilistic games, a strategic variant of IF games which combines semantical games with von Neumann’s minimax theorem. 1. EVALUATION GAMES: GAME-THEORETICAL SEMANTICS Evaluation games for first-order logic have arisen from the work of Henkin in the 50’s and the work of Hintikka in the 70’. For Hintikka the purpose is to give a game-theoretical characterization of truth and falsity for first-order sentences. The driving force here are the standard quantifiers, the universal “for every x”, and the existential “for some Games and Logic 2 x”. Their scopal dependencies and independencies in a first-order sentence are recast in terms of the strategic interaction of two players in a game of perfect information. For instance, the first-order sentence ∀x∃yB(x , y) is analyzed in terms of a game between the universal player (Abelard) and the existential player (Eloise). They choose individuals from the underlying universe of discourse to be the values of the variables x and y , respectively. The order of the choices and the information sets of the players are indicated by the syntax of the sentence. Relative to the universe, the sentence is true if and only if there is a method (a strategy) for Eloise that produces a choice b for every choice a by Abelard so that B(a, b) holds. And the sentence is false if there is a way for Abelard to choose an individual a so that for every possible choice b by Eloise, B(a, b) does not hold. When the dependencies and independencies between quantifiers become more complex, the interaction between the two players becomes less trivial. For instance, consider the game associated with the first-order sentence ∀x∃y∀x ∃y Q(x , x , y, y ). A strategy for Eloise will have to give her, first, a choice b for every individual a chosen by her opponent, and, second, another choice b for each individual a and a, chosen by Abelard. But for the second choice, the strategy may use information encoded in her first choice for b. The game-theoretical view connects in a nice way the dependencies and independencies of quantifiers with the choice functions which codify the strategies of the two players in semantical games. This view of quantification is at “the heart of how classical logicians in the twenties viewed the nature of quantification. " (Goldfarb 1979). 1.1. Semantical games: extensive games of perfect information Let us make the above informal notions more precise. A semantical game is played by two players, ∃ (Eloise) and ∀ (Abelard). They consider a first-order formula φ in negation normal form, model M, which interprets the vocabulary of φ, and an assignment s in M which includes the free variables of φ. ∃ tries to show that φ is true in M (relative to the assignment s), and ∀ tries to show that φ is false in M. The game starts from the initial position (φ, s), and after each move Vol. 8: Games, Game Theory and Game Semantics