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arXiv · 2004.03813

Modal completeness of sublogics of the interpretability logic $\mathbf{IL}$

Taishi Kurahashi·Yuya Okawa
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Abstract

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.

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Taishi Kurahashi, Yuya Okawa. 2020-04-08. Modal completeness of sublogics of the interpretability logic $\mathbf{IL}$. https://arxiv.org/abs/2004.03813

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