  We study modal completeness and incompleteness of several sublogics of the
interpretability logic $\mathbf{IL}$. We introduce the sublogic
$\mathbf{IL}^-$, and prove that $\mathbf{IL}^-$ is sound and complete with
respect to Veltman prestructures which are introduced by Visser. Moreover, we
prove the modal completeness of twelve logics between $\mathbf{IL}^-$ and
$\mathbf{IL}$ with respect to Veltman prestructures. On the other hand, we
prove that eight natural sublogics of $\mathbf{IL}$ are modally incomplete.
Finally, we prove that these incomplete logics are complete with respect to
generalized Veltman prestructures. As a consequence of these investigations, we
obtain that the twenty logics studied in this paper are all decidable.
