  We introduce proper display calculi for intuitionistic, bi-intuitionistic and
classical linear logics with exponentials, which are sound, complete,
conservative, and enjoy cut-elimination and subformula property. Based on the
same design, we introduce a variant of Lambek calculus with exponentials, aimed
at capturing the controlled application of exchange and associativity.
Properness (i.e. closure under uniform substitution of all parametric parts in
rules) is the main interest and added value of the present proposal, and allows
for the smoothest proof of cut-elimination. Our proposal builds on an algebraic
and order-theoretic analysis of linear logic, and applies the guidelines of the
multi-type methodology in the design of display calculi.
