  We provide a general and syntactically-defined family of sequent calculi,
called \emph{semi-analytic}, to formalize the informal notion of a "nice"
sequent calculus. We show that any sufficiently strong (multimodal)
substructural logic with a semi-analytic sequent calculus enjoys the Craig
Interpolation Property, CIP. As a positive application, our theorem provides a
uniform and modular method to prove the CIP for several multimodal
substructural logics, including many fragments and variants of linear logic.
More interestingly, on the negative side, it employs the lack of the CIP in
almost all substructural, superintuitionistic and modal logics to provide a
formal proof for the well-known intuition that almost all logics do not have a
"nice" sequent calculus. More precisely, we show that many substructural logics
including $\mathsf{UL^-}$, $\mathsf{MTL}$, $\mathsf{R}$, $\mathsf{L}_n$ (for $n
\geq 3$), $\mathsf{G}_n$ (for $n \geq 4$), and almost all extensions of
$\mathsf{IMTL}$, $\mathsf{L}$, $\mathsf{BL}$, $\mathsf{RM^e}$, $\mathsf{IPC}$,
$\mathsf{S4}$, and $\mathsf{Grz}$, (except for at most 1, 1, 3, 8, 7, 37, and 6
of them, respectively) do not have a semi-analytic calculus.
  Keywords. Craig interpolation, sequent calculi, substructural logics, linear
logics, subexponential modalities
