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Logics without the contraction rule. (English) Zbl 0583.03018
Syntactical and semantical properties of propositional logics weaker than the intuitionistic, in which the contraction rule does not hold, are studied. Main tools of this paper are Gentzen-type formulation on the one hand, and Kripke-type semantics using partially ordered monoids on the other hand. These tools give us a uniform way of treating a lot of nonclassical logics, e.g. $${\L}ukasiewicz's$$ many-valued logics, relevant logics, the intuitionistic logic and logics related to BCK-algebras, as shown in § 6 and § 7. As an application, some algebraic results on BCK and related algebras are obtained in § 8.
On this occasion, we notice that Problem 1 in § 9 was solved affirmatively by Wroński and Idziak, and that Idziak solved also Problem 7, by giving the necessary and sufficient condition for the class of D-BCC-algebras to form a variety, after the publication of this paper.

##### MSC:
 03B60 Other nonclassical logic 03B20 Subsystems of classical logic (including intuitionistic logic)
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