Further research

• For the $\Box$-modality we know many readings. We have treated in this paper the interpretation of $\Box$ as the provability predicate. But also other readings are possible. The most prominent (and also the original) reading is that of necessity. Modalities can often have various interpretations. (Think of epistemic logic, temporal logic, etc. ) It is always an interesting venture to try to vary the interpretation of a logic in a certain way to then study the purport of the logic under this new interpretation. One obvious variation is to go back to the original inspiration for the semantics of interpretability logics, namely to read the $\rhd$-modality as a conditional in the setting of entailment logics as in Veltman's dissertation. See [Vel85]. As far as we know this has never been done.

• Understanding the relation between Veltman semantics and the arithmetical properties.
  The Veltman semantics have proved to be a very fruitful tool. However, there is no clear understanding about what is the precise relation between Veltman frames and the arithmetics. Neither is there a clear intuitive way of thinking about frame conditions in terms of the intended arithmetic. It might be interesting to investigate if such a clear connection can be described.

• Understanding why certain principles are not complete w.r.t. their corresponding class of characteristic frames.
  When the principle P₀ was found, it was immediately conjectured to be modally incomplete. The grounds for this conjecture were the similarities with an earlier investigated modally incomplete principle ${\mathit KW4}$, and a general intuition. It might be interesting to try to capture this intuition by a general theorem about incompleteness.

• Performing a schematic enclosure of $\mathit{GIL}$ as proposed.
  In paragraph 6.2 a general approach is proposed for the enclosure of $\mathit{GIL}$. As far as we know, such a systematic approach has not yet been executed. Within this program various other questions fit in well, like for example can one say something about nice principles in the meet of $\mathit{ILP}$ and $\mathit{ILM}$?