Randomized game semantics for semi-fuzzy quantifiers

We take up the challenge to extract particular truth functions for fuzzy quantifiers from a game semantic framework. To this aim, we start with a fresh look at Hintikka's evaluation game for classical first-order logic and show that randomizing payoffs in that classical game results in a characterization of so-called weak Aukasiewicz logic. A further step of generalization, considering more than one formula as available for attack at a given state of the game, leads to Giles's game for full Aukasiewicz logic. Finally, we extend this framework to random choices of witnesses for quantified statements. This allows us to characterize two families of extensions of Aukasiewicz logic with different semi-fuzzy proportionality quantifiers that include candidate models for vague natural language quantifiers like about half.