The Lambek calculus was introduced in [J. Lambek, Am. Math. Mon. 65, 154–170 (1958; Zbl 0080.00702)] under the name syntactic calculus, being called the associative Lambek calculus and denoted by \(\boldsymbol{L}\), while the nonassociative Lambek calculus, denoted by \(\boldsymbol{NL}\), was introduced by J. Lambek [“On the calculus of syntactic types”, in: R. Jakobson (ed.), Structure of language and its mathematical aspects. Providence: AMS. 166–178. (1961)], Residuated groupoids and semigroups are algebraic models of \(\boldsymbol{NL}\) and \(\boldsymbol{L}\), respectively. This paper aims to emphasize the role of the Lambek calculus in the realm of nonclassical logics. A better name for substructural logics could presumably be residuation logics. Products, also called multiplicative or intensional conjunction, tensor, fusion, is always related to the equivalences.
The synopsis of the paper goes as follows.

§ 2

discusses a residuation connection, which occupies a central position in the subject.

§ 3

recalls sequent systems for \(\boldsymbol{L}\) and \(\boldsymbol{NL}\), briefly discussing some properties of these systems (cut-elimination, completeness, decidability, complexity) as well as their role in categorical grammars. As a matter of fact, most results on these two logics were obtained by scholars with interest in linguistic applications.

§ 4

brings a survey of different substructural logics of intuitionistic nature giving up double negation. Since this area has been extensively explored since the 1980s, the author considers only some basic systems and their most fundamental properties. At the end, some related multi-modal logics are described.

§ 5

addresses (classical) linear logics, extending some logics from the previous section by the double negation laws. The presentation is very concise, considering only logics without exponentials with emphasis on noncommutative logics.

§ 6

contains some final comments.

For the entire collection see [Zbl 1470.03008].