Uniform Rules and Dialogue Games for Fuzzy Logics

We provide uniform and invertible logical rules in a framework of relational hypersequents for the three fundamental t-norm based fuzzy logics i.e., Łukasiewicz logic, Gödel logic, and Product logic. Relational hypersequents generalize both hypersequents and sequents-of-relations. Such a framework can be interpreted via a particular class of dialogue games combined with bets, where the rules reflect possible moves in the game. The problem of determining the validity of atomic relational hypersequents is shown to be polynomial for each logic, allowing us to develop Co-NP calculi. We also present calculi with very simple initial relational hypersequents that vary only in the structural rules for the logics.