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 form
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 , in connection with the study of BCK-algebras (for information on BCK-algebras, see ). 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., , , ).
In this paper, we will study these logics more comprehensively. We notice here that the distributive law
does 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.