Semantics of Informational Independence

First-order languages of imperfect information (IF rst-order languages) have been introduced by Hintikka and Sandu 2]. These languages contained independent quantiiers and connectives of the form (9x=W); (8x=W); (_=W); (^=W) where W is a set of variables (x = 2 W), the values of which the quan-tiiers and the connectives on the left side of the slash are supposed to be independent of. The notion of a variable being free or bound is the same as in ordinary rst-order logic. In the formula (9x=W)' the variables in W are free. In Hintikka and Sandu 3] only sentences receive a semantic interpretation, and this is done in terms of games G('; A) of imperfect information played with an IF rst-order sentence ', and a model A. The game G('; A) is played by two players, Eloise (9) and Abelard (8): The standard example which goes back to Henkin (1961) is with ' of the form 8x9y8z(9w=fxg)Rxyzw: In this case the game can be pictured like this: 9 : b d=a 8 : a c Here a; b; c; d are individuals from the domain j A j of the model, and 'd=a' means that when choosing the individual d Eloise did not "know" which individual a Abelard chose earlier. The notion of "not knowing" is made explicit in the strategies of the players: they are deened only on earlier "known" choices of the opponent. In other words, a strategy for Eloise in the game consists of two functions f; g :j A j!j A j 1