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

TL;DR

This paper investigates the modal completeness of various sublogics of the interpretability logic IL, introducing new sublogics, proving completeness and incompleteness results, and establishing decidability of all studied logics.

Contribution

It introduces the sublogic IL−, proves its completeness, analyzes twelve intermediate logics, and shows the completeness of eight others with respect to generalized structures, advancing understanding of IL sublogics.

Findings

IL− is sound and complete with respect to Veltman prestructures.

Twelve logics between IL− and IL are modally complete.

Eight natural sublogics are modally incomplete but complete with generalized structures.

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.