Analytic Calculi for Logics of Ordinal Multiples of Standard t-Norms

For two propositional fuzzy logics, we present analytic proof calculi, based on relational hypersequents. The logic considered first, called MŁ, is based on the finite ordinal sums of Łukasiewicz t-norms. In addition to the usual connectives—the conjunction ⊙, the implication → and the constant 0—we use a further unary connective interpreted by the function associating with each truth value a the greatest ⊙-idempotent below a. MŁ is a conservative extension of Basic Logic. The second logic, called MΠ, is based on the finite ordinal sums of the product t-norm on (0, 1]. Our connectives are in this case just the conjunction and the implication.