Modal completeness of sublogics of the interpretability logic $\mathbf{IL}$
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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2020
- DOI:
- 10.48550/arXiv.2004.03813
- arXiv:
- arXiv:2004.03813
- Bibcode:
- 2020arXiv200403813K
- Keywords:
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- Mathematics - Logic
- E-Print:
- 34 pages