On the selective lambek calculus

An intermediate logical system in comparison with the non-associative (NL) and the associative Lambek Calculus(L) is obtained by extending their language by means of a so-called “combining permission”, which is used to regulate the introduction of a product formula in the scope of a derivation and to impose a control on applications of the association structural rules. A cut elimination theorem for such a system, so-called Selective Lambek Calculus, is presented.