Multi-sorted model theory

Question

I've started reading Model Theory: An Introduction by Marker. I'm not sure whether it covers multi-sorted languages. It seems that it doesn't. So, the question is whether there is something I could read on the topic and whether multi-sorted model theories make sense.

Answer 1

As Mark Kamsma says in the comments, the traditional approach is to learn the basics of model theory in a single-sorted context, and then at some point observe that "everything generalizes" to the multi-sorted context. This is basically correct (see this related question and its answers), but it may seem a bit unsatisfying.

I once taught a class on basic model theory in which we used multi-sorted logic (and allowed empty structures and empty sorts) from the beginning. This included setting up a proof system for multi-sorted first-order logic (with empty sorts allowed) and proving the completeness theorem. My lecture notes are available here. I'm not sure this is the most pedagogically sound approach - I mostly did it this way because I wanted to work through the details for myself at least once.

Since you ask about Marker's book: it (very briefly) introduces multi-sorted logic on p. 28, in the section "Multi-sorted Structures and $\mathcal{M}^{\mathrm{eq}}$". Most of the book implicitly works in the single-sorted context, but occasionally (especially in Chapters 6-8 on stability theory), it is useful to work in the multi-sorted expansion $\mathcal{M}^{\mathrm{eq}}$, which eliminates imaginaries.