Reddit mentions of Model Theory: Third Edition (Dover Books on Mathematics)

The best introductory logic text you will ever find: Logic: Techniques of Formal Reasoning, 2nd Edition Donald Kalish, Richard Montague.
This book is especially good if you have done any programming. The structure of main and sub-proofs corresponds to main program and subroutine calls. You can pick up a used copy for around $23 here: https://www.abebooks.com/book-search/author/kalish-montague-mar/ and you can see the table of contents here: https://www.powells.com/book/logic-techniques-of-formal-reasoning-9780195155044 (but, obviously, don't buy it for $133!)

For meta-theory, take a look at: Metalogic: An Introduction to the Metatheory of Standard First Order Logic by Geoffrey Hunter, https://www.amazon.com/Metalogic-Introduction-Metatheory-Standard-First/dp/0520023560. This book explains things in a clear way using ordinary English, before setting out the proofs.
And, if you are interested in model theory, take a look at Model Theory by C.C. Chang and H. Jerome Keisler, https://www.amazon.com/Model-Theory-Third-Dover-Mathematics/dp/0486488217 and you should get a good idea of what additional mathematics you might want to pursue.

> The distinction is that in math, all foundational meta-theories are require to get the right answers on simple object-level questions like "What's 1 + 1?". If your mathematical metatheory answers, "-3.7" rather than "2", then it is not "different", it is simply wrong. We can thus say that Foundations of Mathematics is always done with a realist view.

The natural numbers are an interesting example, which goes back to ADefiniteDescription's point about a privileged model. The basic axioms of arithmetic are categorical, i.e. have only one model, up to isomorphism. Not all theories have this property, though.

If it could be shown that some moral theory similarly has only one correct interpretation - that all alternative interpretations end up being isomorphic - then that could support a kind of realism, at least in the context of that theory. A lot would depend on the nature and scope of the theory in question, and its interpretation.

So perhaps Parfit's position would be better captured by saying that he believes there are unique true answers to moral questions, as there are for questions in categorical mathematical theories.

> What's a good textbook for that field, anyway?

The books I studied are quite outdated now, but a classic modern text is Model Theory by Chang & Keisler. That might be more comprehensive than you're looking for. You could try Model Theory: An Introduction - its first chapter is quite a concise basic intro. There's also A Shorter Model Theory.

Thank you very much for your reply and explanation. It's starting to make more sense now - I think I just need to study Model Theory on its own. (I saw that you praised this book in another thread. Would you recommend it for self study?)