Languages for imperfect information Gabriel Sandu University of Helsinki, Finland sandu@mappi.helsinki.fi Abstract. This chapter gives a self-contained introduction to game- theoretical semantics (GTS) both for classical first-order logic and for one of its extensions, Independence-Friendly Logic (IF logic). The games used for the interpretation of IF logic are 2-player win-lose extensive games of imperfect information. Several game-theoretical phenomena will be discussed in this context, including signaling and indeterminacy. To overcome indeterminacy, we introduce mixed strategies and apply Von Neumann’s Minimax Theorem. This results in a probabilistic interpre- tation of IF sentences (equilibrium semantics). We shall use IF logic and its equilibrium semantics to model some well known examples which involve games with imperfect information: Lewis’ signaling games and Monty Hall. 1 Introduction Evaluation games for first-order logic have arisen from the work of Hintikka in the 1970s. They led to various applications to natural language phenomena (e.g., pronominal anaphora) in a single framework which is now known as game- theoretical semantics (GTS). Van Benthem [6] is right in emphasizing that these games analyze the ‘logical skeleton’ of sentence construction: connectives, quan- tifiers, and anaphoric referential relationships, with logic being the driver of the analysis here. Hintikka and Kulas [29,30] and Hintikka and Sandu [31–34] syn- thesize some of the work done in this area. This paradigm was amplified in the years that followed but since the late 1980s and during the 1990s there was a switch of interest: logic was still the driving force but the emphasis was on the role of evaluation games for the foundations of mathematics. Drawing on some earlier work by Henkin [26], Hintikka [27] observed that the connection between quantifier dependence and choice functions could be naturally extended to go beyond the patterns allowed by traditional first-order logic. Hintikka and Sandu [31] sketch the general lines of the programme and Hintikka [28] explores the con- nections between quantifier-independence, choice functions, games of imperfect information and expressive power (descriptive completeness) in the foundations of mathematics. Since the very beginning, Hintikka conceived evaluation games as a challenge to the more traditional compositional paradigm which arose from the work of Tarski and Montague and constituted the underlying methodology in the model-theoretical tradition. The seminal paper by Hodges [35] showed, nevertheless, that compositional and game-theoretical methods can go hand in