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Substructural logics and residuated lattices – an introduction. (English) Zbl 1048.03018
Hendricks, Vincent F. (ed.) et al., Trends in logic. 50 years of Studia Logica. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1601-8/hbk). Trends Log. Stud. Log. Libr. 21, 193-228 (2003).
This is an introductory survey of substructural logics and their algebraic semantics by residuated lattices. Beginning with basic substructural logics introduced in sequential stile, the author explains through several proof-theoretical results on them, particularly on their cut-elimination property, how the structural characteristics are closely related to such fundamental properties of logical system as decidability, disjunction property, and so on.
The algebraic semantics by residuated lattices is discussed in the latter half, where the notion of residuation is featured as essential for the structural characteristics of logical systems and the author concludes that substructural logics are logics of residuated structure and this is the reason why sequent systems are suitable for formalizing substrutural logics. With this point of view an approach to the algebraic study of substructural logics is proposed under a global perspective.
For the entire collection see [Zbl 1029.00014].

03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03G10 Logical aspects of lattices and related structures
03F05 Cut-elimination and normal-form theorems
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations