Randomized game semantics for semi-fuzzy quantifiers

Abstract

Abstract. Building on a generalization of classical Henkin-Hintikka evaluation games to Giles' game semantics for Lukasiewicz logic, a concept for evaluating (semi-)fuzzy quantifiers by randomizing the choice of witnessing elements is developed. In particular the adequateness of simple dialogue game rules for so-called proportionality quantifiers is shown.

Hintikka Style Game Rules for Semi-Fuzzy Quantifiers

“…As mentioned in the Introduction, game rules for semifuzzy quantifiers based on Giles's game (G-game) [7] for Łukasiewicz logic Ł have been presented in [5] and in [6]. These rules refer to a much richer notion of a game state, given by…”

“…The above mentioned challenge has been addressed in [5], [6] by extending Giles's game for Łukasiewicz logic [7]. This game models the stepwise reduction of logical complex assertions into atomic ones in a rule guided dialogue between two players.…”

“…While the game includes rules for standard existential and universal quantification, Giles did not consider (semi-)fuzzy quantifiers. The main ingredient of the semi-fuzzy quantifier rules presented in [5] and investigated more systematically in [6] are bets for and against certain numbers of instances of the scope formula that result from random choices of domain elements.…”

“…Here we deal with quantification. Like in [5] and [6], random choices of domain elements are essential for obtaining interesting families of truth functions. However, note that Hintikka's principle prevents the reduction to simultaneous bets for and against whole multisets of corresponding instances of the quantified formula, that is central for Giles-style games.…”

On matrices, Nmatrices and games

“…However, with hindsight, it is clear that Giles's game is a species of semantic game, rather than a Lorenzen style dialogue game. For a recent exposition that makes the relation to Hintikka's game transparent and that explores one of the many applications beyond standard Lukasiewicz logic we refer to [10].…”