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, Grišin's logic and product logic, etc. (See, e.g., Cignoli et al., 2000; Grišin, 1976; Hájek, 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).