Randomized semantic games for fuzzy logics

Recent work in the field of game theoretic modeling of natural language quantifiers introduces randomizing game rules within the setting of Giles's game for Łukasiewicz logic [3], [15], [16]. The randomization comes about in the form of a non-strategic player, called Nature. This can, taken as a mere selection principle, lead to an immense increase of expressivity, compared to the usual two player zero sum games of perfect information, like Hintikka's game or Giles's game [3]. We will define several games with randomizing game rules and investigate their expressivity, ending with a game which we call the NRG-game, with corresponding logic Łα(Π). We give representation theorems for the class of semi-fuzzy deliberate choice quantifiers, and, as the main result of the paper, show how one can define the operations of Gödel and Product logic, and furthermore, those of all fuzzy logics, that are based on a continuous t-norm that is representable as a finite ordinal sum of the t-norms corresponding to Łukasiewicz, Gödel and Product logic [4], within Łα(Π), as long as the domains of the interpretations are finite.