We study modal completeness and incompleteness of several sublogics of the interpretability logic $\mathsf{IL}$. We introduce the sublogic ${\mathsf{IL}}^{-}$, and prove that ${\mathsf{IL}}^{-}$ is sound and complete with respect to Veltman prestructures which are introduced by Visser. Moreover, we prove the modal completeness of twelve logics between ${\mathsf{IL}}^{-}$ and $\mathsf{IL}$ with respect to Veltman prestructures. On the other hand, we prove that eight natural sublogics of $\mathsf{IL}$ 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.

中文翻译：

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可解释性逻辑 IL 的子逻辑的模态完备性

我们研究可解释性逻辑的几个子逻辑的模态完备性和不完备性 $\mathsf{伊利诺伊州}$. 我们引入子逻辑${\mathsf{伊利诺伊州}}^{-}$，并证明 ${\mathsf{伊利诺伊州}}^{-}$相对于 Visser 引入的 Veltman 预结构而言，它是健全和完整的。此外，我们证明了 12 个逻辑之间的模态完备性${\mathsf{伊利诺伊州}}^{-}$ 和 $\mathsf{伊利诺伊州}$关于 Veltman 预结构。另一方面，我们证明了 8 个自然子逻辑$\mathsf{伊利诺伊州}$模态不完整。最后，我们证明这些不完全逻辑对于广义 Veltman 预结构是完备的。作为这些调查的结果，我们得到本文研究的 20 种逻辑都是可判定的。