By "desktop reference" I mean a book that aims to be comprehensive and self-contained, rather than didactic1.
The best example I can think of of the sort of book I am looking for is Jech's Set Theory.
Also, by "self-contained" is mean that it includes all non-trivial proofs.
The book should cover, at least, propositional and first-order logic. I am agnostic as to what other areas of logic it should cover, though, of course, the more comprehensive the book the better.
I would prefer a recent book (> 2000) over a classic. Also, it is important to me that the book uses standard notation and terminology. (In fact, the reason I am passing up Hinman's Fundamentals of Mathematical Logic, is it's idiosyncratic "overdot" notation.)
1 Hence, this question differs from other similar [reference-request]-tagged questions (such as Reference request: Standard textbook for first-order predicate logic in english), where the OP is asking for a book to learn logic from. Also, and likely for this reason, this question appears to be outside of the scope of the Teach Yourself Logic page.
Best Answer
For many years the standard "advanced student level" reference for mathematical logic has been Handbook of Mathematical Logic edited by Jon Barwise (1977).
Reviews of Handbook of Mathematical Logic: [1] John Lane Bell, British Journal for the Philosophy of Science 30 #3 (September 1979), pp. 306-309. [2] Daniel Lascar, Journal of Symbolic Logic 49 #3 (September 1984), pp. 968-971. [3] Akihiro Kanamori, ibid., pp.971-975. [4] Sy-David Friedman, ibid., pp. 975-980. [5] Alasdair Urquhart, Canadian Journal of Philosophy 14 #4 (December 1984), pp. 675-682.
For other types of logic (less focused on topics such as set theory foundations and recursion theory), see A Companion to Philosophical Logic edited by Dale Jacquette (2002; relatively cheap and has several very readable essays on non-standard logic) and the incredibly comprehensive and encyclopediac 18-volume (current number) Handbook of Philosophical Logic edited by Gabbay/Guenthner (1983-present).
Regarding textbooks, besides those already mentioned, one of the best known (and also not very didactic) is Shoenfield's Mathematical Logic (1967), but since that's a classic, maybe look at An Introduction to Mathematical Logic by Richard E. Hodel (2013), which is somewhat of a more student-friendly and detailed version of Shoenfield's book.