Logics without the contraction rule and residuated lattices

In this paper, we will develop an algebraic study of substructural propositional logics over FLew, i.e. the logic which is obtained from the intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCK-logics, Łukasiewicz’s many-valued logics and fuzzy logics, within a uniform framework.  (2009): The draft of the present paper was originally completed in 1999 and then revised slightly in 2001, which I intended to dedicate to R. K. Meyer on the occasion of his 65th birthday. From the middle of 90s, I had been trying to develop an algebraic study of substructural logics over the logic without the contraction rule FLew, and had announced results in several conferences, e.g. the 6th Asian Logic Conference in 1996, Dagstuhl Seminar on Multiple-Valued Logic in 1997 and S. Jaśkowski Memorial Symposium on Parainconsistent Logic, Logical Philosophy, Mathematics & Informatics in 1998. So I planned to make the present paper a comprehensive survey of the state of affairs of the study. While the paper has not been published for many years, the draft has been referred in considerably many papers of substructural logics published in the last decade. On the occasion of the publication of the paper, I have been wondering in which way I should publish the paper and how much I should revise it. For, there have been a remakable progress in the direction of this research within these 10 years, and moreover my joint book [25] on substructural logics was already published. It would be no use to make an entire update of it. Hiroakira Ono, “Logics without the contraction rule”, Australasian Journal of Logic (8) 2010, 50–81 http://www.philosophy.unimelb.edu.au/ajl/2010 51 My decision is to minimize changes, keeping the original form, and to add only necessary information on recent progress. (I put the mark (†) to footnotes which are essentially added in the present revision.) I hope that the paper will be still of interest and also informative, especially in its references, beyond its historical meaning.