Uniform Rules and Dialogue Games for Fuzzy Logics ⋆ A. Ciabattoni, C.G. Ferm¨ uller, and G. Metcalfe Technische Universit¨ at Wien, A-1040 Vienna, Austria {agata,chrisf,metcalfe}@logic.at Abstract. We provide uniform and invertible logical rules in a framework of re- lational hypersequents for the three fundamental t-norm based fuzzy logics i.e., Lukasiewicz logic, G¨ odel logic, and Product logic. Relational hypersequents gen- eralize 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 valid- ity 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. 1 Introduction Fuzzy logics based on t-norms and their residua are formal systems providing a founda- tion for reasoning under vagueness. Following e.g., [10], conjunction and implication are interpreted on the real unit interval [0, 1] by a continuous t-norm and its residuum, respectively. The most important of these logics are Lukasiewicz logic L, G¨ odel logic G, and Product logic Π. These three are viewed as fundamental since all continuous t-norms can be constructed from their respective t-norms. A variety of proof methods have been proposed for L, G, and Π. In particular, calculi for many fuzzy logics have been presented in a framework of hypersequents,a generalization of Gentzen sequents to multisets of sequents (see e.g., [2]). A very attrac- tive calculus has been defined for G in [2] by embedding Gentzen’s LJ for intuitionistic logic into a hypersequent calculus without modifying the rules for connectives. Elegant hypersequent calculi have also been defined for L [16] and Π [14], but using different rules for connectives. A further calculus for G, which unlike the respective hyperse- quent calculus has invertible rules, has been introduced in a framework of sequents- of-relations [5]. More proof search oriented calculi include a tableaux calculus for L [9], decomposition proof systems for G [3], and goal-directed systems for L [15] and G [13]. Finally, a general approach is presented in [1] where a calculus for any logic based on a continuous t-norm is obtained via reductions to suitable finite-valued logics. In this paper we introduce a generalization of both hypersequents and sequents-of- relations, that we call relational hypersequents. A relational hypersequent, or, for short, r-hypersequent, is a multiset of two different types of sequents, where Gentzen’s se- quent arrow is replaced in one by < and in the other by ≤. Intuitively we may think ⋆ Research supported by C. B¨ uhler-Habilitations-Stipendium H191-N04, FWF Project Nr. P16539-N04, and Marie Curie Fellowship 501043.