Abstract
Let FL ew be the logic obtained from the intuitionistic propositional logic by deleting contraction rule if we formulate it in a sequent system. Sometimes, this logic is called intuitionistic affine logic. The class of logics over FL ew , i.e. logics stronger than or equal to FL ew , includes many interesting logics, e.g., intermediate logics, Łukasiewicz’s many-valued logics, Grisin’s logic and product logic, etc. (See, e.g., Cignoli et al., 2000; Grisin, 1976; Hajek, 1998.) The study of logics over FL ew , will enable us to discuss these different kinds of logics within a uniform framework (see Ono and Komori, 1985; Ono, 1999 for the detail).
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