Michael Moortgat Symmetric categorial grammar Fifty years ago, Jim Lambek published the seminal "Mathematics of sentence structure". In that paper, the familiar parts of speech (nouns, verbs, ...) take the form of formulas of a substructural logic; determining whether a phrase is well-formed, and assigning it an interpretation, i.e. parsing, can then be seen as a process of deduction in the grammar logic. The original Syntactic Calculus has turned out to be strictly context free, and it has an NP-complete decision problem. The goal of recent extensions of Lambek-style categorial type logics is to find a balance between expressivity and computational tractability: can one combine the ability to recognize patterns beyond context-free with polynomial parsing algorithms? In the talk, I show how symmetric categorial grammar answers that challenge. In addition to the Lambek connectives (product, for phrasal composition, and residual left and right division) one considers a dual family: coproduct with residual left and right difference operations. The two families interact via structure-preserving distributivity principles originally studied by Grishin in 1983. I discuss modeltheoretic and prooftheoretic properties of the resulting Lambek-Grishin calculus. I show how its derivations can be given a proofs-as-programs interpretation in the continuation passing style.