Interpolation, Definability and Fixed Points in Interpretability Logics

In this article we study interpolation properties 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 result. Furthermore, these properties are extended to the system ILP. Failure of arrow interpolation for ILW is established by providing an explicit counterexample. The related issues of Beth definability and fixed points are also addressed. It will be shown that for a general class of logics the Beth property and the fixed point property are interderivable. This in particular yields alternative proofs for the fixed point theorem for IL (cf. de Jongh and Visser 1991) and the Beth theorem for all provability logics (cf. Maksimova 1989). Moreover, it entails that all extensions of IL have the Beth property.