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.

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Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.

ByHiroakira ONO* § la Introduction Some modal logics based on logics weaker than the classical logic have been studied by Fitch [4] In this paper, we treat modal logics based on the intuitionistic prepositional logic, which we call intuitionistic modal logics (abbreviated as IML's). Our main concern is to compare properties of several IML's of S4-or S5-type 03^ using some model theoretical methods. The study of modal logics based on weak logics seems to reveal to us various properties of classical modal logics, especially of S5, which will be indistinguishable by dealing them only on the classical logic.We will introduce some IML's in the Hilbert-style formalization in § 2. Then we will define IML's in the form of sequent calculi, all of which are given by restricting or modifying the sequent calculi S4 and S5 of . We will show the proper inclusion relationship between these IML's by using a kind of algebraic models. In § §3 and 5, we will introduce two kinds of models for IML's. One of them is a natural extension of Kripke models for the intuitionistic Jogic and the other is for modal logics (see [11], [12]). Then we will prove the completeness theorem with respect to these models. In § 4, the finite model property for some IML's will be shown. We would like to thank M. Sato for his valuable suggestions. § 2 B Intuitionistic Modal LogicsWe will introduce some intuitionistic modal logics. We take A? V?

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