In applications of logic, it is rare to find a unique domain of discourse. I.e. it is customary to have to speak of different entities, as Hao Wang's puts it, for instance in geometry you need to talk about points, lines and planes.
I'm surprised that there seems to be no resources like say the book "The Classical Decision Problem" addressing the decidability of fragments of this logic. The papers I could find have a limited scope. For instance, "Decidable fragments of many-sorted logic" by Abadi et alii. are severely restricted. In my particular application, I cannot have a function $f: A \times I \times C \to A$ since the fragment enforces that the two $A$'s should have different "levels".
Please correct me if I'm wrong regarding this gap in the literature. Moreover, is there a reason for this status of things? I.e. is there a reason that makes many-sorted logic decidability much more complex?