The Interpolation Theorem for IL and ILP

In this article we establish interpolation for the minimal system of interpretability logic IL. We prove that arrow interpolation holds for IL and that turnstile interpolation and interpolation for the-modality easily follow from this. Furthermore, these properties are extended to the system ILP. The related issue of Beth Deenability is also addressed. As usual, the arrow interpolation property implies the Beth property. From the latter it follows via an argumentation which is standard in provability logic, that IL has the xed point property. Finally we observe that a general result of Maksimova 11] for provability logics can be extended to interpretability logics, implying that all extensions of IL have the Beth property.