A variation of the standard non-associative Lambek calculus with the slightly non-standard yet very traditional semantic interpretation turns out to straightforwardly and uniformly express the instances of non-canonical coordination while maintaining phrase structure constituents. Non-canonical coordination looks just as canonical on our analyses. Gapping, typically problematic in Categorial Grammar–based approaches, is analyzed like the ordinary object coordination. Furthermore, the calculus uniformly treats quantification in any position, quantification ambiguity and islands. It lets us give what seems to be the simplest account for both narrow- and wide-scope quantification into coordinated phrases and of narrow- and wide-scope modal auxiliaries in gapping. The calculus lets us express standard covert movements and anaphoric-like references (analogues of overt movements) in types – as well as describe how the context can block these movements.