Hintikka-Style Semantic Games for Fuzzy Logics

Abstract

Various types of semantics games for deductive fuzzy logics, most prominently for Łukasiewicz logic, have been proposed in the literature. These games deviate from Hintikka’s original game for evaluating classical first-order formulas by either introducing an explicit reference to a truth value from the unit interval at each game state (as in [4]) or by generalizing to multisets of formulas to be considered at any state (as, e.g., in [12,9,7,10]). We explore to which extent Hintikka’s game theoretical semantics for classical logic can be generalized to a many-valued setting without sacrificing the simple structure of Hintikka’s original game. We show that rules that instantiate a certain scheme abstracted from Hintikka’s game do not lead to logics beyond the rather inexpressive, but widely applied Kleene-Zadeh logic, also known as ‘weak Łukasiewicz logic’ or even simply as ‘fuzzy logic’ [27]. To obtain stronger logics we consider propositional as well as quantifier rules that allow for random choices. We show how not only various extensions of Kleene-Zadeh logic, but also proper extensions Łukasiewicz logic arise in this manner.