Logics Without the Contraction Rule

if we formulate our logics in a Gentzen-type formal system. Some syntactical properties of these logics have been studied firstly by the second author in [I I], in connection with the study of BCK-algebras (for information on BCK-algebras, see [9]). There, it turned out that such a syntactical method is a powerful and promising tool in studying BCK-algebras. Using this method, considerable progress has been made since then (see, e.g., [8], [18], [27]). In this paper, we will study these logics more comprehensively. We notice here that the distributive law