I'm considering Craig's interpolation theorem in the context of a family of semantically defined logics. I'm somewhat troubled with the duality used in the formulation. As for all logics considered in the literature I've read, there exists a completeness theorem I've seen both a semantic and a syntactic formulation of the theorem.
Now, some members of the family I'm considering can't have an adequate proof calculus(which was proven before). Obviously, I can still work on Craig's theorem in the axiomatizable logics but what about those having a proper semantic definition but no proof calculus. It it worth considering those, is it possible to proof interpolation merely semantically for these?
The statement of the interpolation theorem certainly also makes sense for logics which are defined only via the semantics. Whether it is possible to prove the theorem for a particular such logic of course depends on the logic itself. There are some semantic methods for proving interpolation properties, though, mainly via algebraic semantics if I remember correctly. It might be interesting to have a look at Maksimova's work on interpolation via the amalgamation property in algebraic characterisations of modal and intermediate logics. Apart from a number of papers on this which are out there, the books Interpolation and Definability by Gabbay and Maksimova, or Larisa Maksimova on Implication, Interpolation, and Definability seem like a good place to start.
Edit: The original proof of the Lyndon interpolation theorem, a refinement of Craig interpolation, also seems to have been semantic, i.e., model-theoretic. The reference is this