Game-theoretical semantics: insights and prospects

The paradigm problem for game-theoretical semantics (GTS) is the treatment of quantifiers, primarily logicians’ existential and universal quantifiers. As far as the uses of quantifiers in logic and mathematics are concerned, the basic ideas codified in GTS have long been an integral part of logicians’ and mathematicians’ folklore. Everybody who has taken a serious course in calculus remembers the definition of what it means for a function y = f(x) to be continuous at x0. It means that, given a number w, however small, we can find ɛ such that |f(x) - F(x0) | < δ given any x such that | x – x0| < ɛ.1 The most natural way of making this jargon explicit is to envision each choice of the value of an existentially bound variable to be my own move in a game, and each choice of the value of a universally bound variable a move in the same game by an imaginary opponent. The former is what is covered by such locutions as “we can find”, whereas the latter is what is intended by references to what is “given” to us. This is indeed what is involved in the continuity example.