  We study nonlinear connectives (exponentials) in the context of Type Logical
Grammar (TLG). We devise four conservative extensions of the displacement
calculus with brackets, DbC, DbCM, DbCb and DbCbMr which contain the universal
and existential exponential modalities of linear logic (LL). These modalities
do not exhibit the same structural properties as in LL, which in TLG are
especially adapted for linguistic purposes. The universal modality ! for TLG
allows only the commutative and contraction rules, but not weakening, whereas
the existential modality ? allows the so-called (intuitionistic) Mingle rule,
which derives a restricted version of weakening. We provide a Curry-Howard
labelling for both exponential connectives. As it turns out, controlled
contraction by ! gives a way to account for the so-called parasitic gaps, and
controlled Mingle ? iteration, in particular iterated coordination. Finally,
the four calculi are proved to be Cut-Free, and decidability is proved for a
linguistically sufficient special case of DbCbMr (and hence DbCb).
