We will study syntactical and semantical properties of propositional logics weaker than the intuitionistic, in which the contraction rule (or, the exchange rule or the weakening rule, in some cases) does not hold. Here, the contraction rule means the rule of inference of the formif 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 [11], 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 lawdoes not hold necessarily in these logics. By adding some axioms (or initial sequents) and rules of inference to these basic logics, we can obtain a lot of interesting nonclassical logics such as Łukasiewicz's many-valued logics, relevant logics, the intuitionistic logic and logics related to BCK-algebras, which have been studied separately until now. Thus, our approach will give a uniform way of dealing with these logics. One of our two main tools in doing so is Gentzen-type formulation of logics in syntax, and the other is semantics defined by using partially ordered monoids.

In [8] (1920), Lukasiewicz introduced a 3-valued propositional calculus with one designated truth-value and later in [9], Lukasiewicz and Tarski generalized it to an m-valued propositional calculus (where m is a natural number or ^0) with one designated truth-value. For the original 3-valued propositional calculus, an axiomatization was given by Wajsberg [16] (1931). In a case of m Φ ^0> Rosser and Turquette gave an axiomatization of the m-valued propositional calculus with an arbitrary number of designated truth-values in [13] (1945). In [9], Lukasiewicz conjectured that the ^o-valued propositional calculus is axiomatizable by a system with modus ponens and substitution as inference rules and the following five axioms:Here we use P V Q as the abbreviation of (P 3 Q) Z) Q. We associate to the right and use the convention that 3 binds less strongly than V. In [15] p. 51, it is stated as follows: "This conjecture has proved to be correct; see Wajsberg [17] (1935) On the other hand, Rose [11] (1953) showed that the cardinality of the set of all super-Lukasiewicz propositional logics is ^0. Surprisingly it was before Rose and Rosser's completeness theorem [12]. The proof in Rose [11] was also due to McNaughton's theorem. Some of our theorems in this paper have already been obtained by Rose [11], But our proofs are completely algebraic.In our former paper [5], we gave a complete description of super-

In the traditional study of Łukasiewicz propositional logic, the finite-valued or infinite-valued linearly ordered model exists at the start, and then the axiomatization of the set of all formulas valid in its model are studied. On the other hand, we are in a point of view such that the set of provable formulas is important and models are no more than means to characterize the set.

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