Reddit reviews Logics and Falsifications: A New Perspective on Constructivist Semantics (Trends in Logic (40))

The four-valued one that that I had in mind was Belnap's relevance logic which is outlined in this Stanford article here.

I've never studied 4-valued logics, but I have studied the Łukasiewicz 3-valued logic (just above Belnap's on that page) and I guess it's similar in principle - we just accept that statements can be True, False, or Neither. "Neither" could be cashed out as "Undefined Truth Value". Ł_3 is used by Kripke in his Outline of a Theory of Truth to ascribe a truth-value to those "ungrounded" statements in a construction which aims to circumvent the Liar paradox (and kinda does, at a cost). The four-valued one is similar, we just have a fourth truth-value. I believe both of these logics have been applied in computer science.

The important thing is that these are perfectly reasonable mathematical/logical constructions - whether you think the Law of Excluded Middle holds in some "true logic of the universe" sense, and these are just bastardisations of classical logic, is another thing.

If you want a book, I'd suggest Greg Restall's Introduction to Substructural Logics.

For a fairly high-level modern primer on intuitionistic logic, I'd try Kapsner's book.

And if you're interested in the current research on homotopy type theory (which has "built-in" intuitionistic logic), the main resource is the Univalent Foundations book which is available as a regularly updated PDF, completely gratis. Feel free to donate though :).

Edit: I misremembered, Kripke actually uses Strong-Kleene 3-valued logic, which is almost the same as Ł_3 but has a slightly different IF-THEN rule.