Craig's trick and a non-sequential system for the Lambek calculus and its fragments

We show that Craig's trick is not valid for the Lambek calculus, i.e. there exists such a recursively enumerable theory (set of sequents) over the Lambek calculus, which does not have a decidable axiomatization. We show that Lambek's non-emptiness restriction (the constraint that left-hand sides of all sequents should be non-empty) and an infinite set of variables are crucial for the failure of Craig's trick. We also present a non-sequential formulation of the product-free fragment of the Lambek calculus and show its equivalence to the sequential one.