I am currently reading Robert Causey's Logic, Sets and Recursion, which is intended to bridge the gap between introductory freshman/sophomore "introductory discrete math" courses and senior level mathematical logic. I am only about a third of the way through it right now, but I have already learned several useful things, I especially found the discussion of mathematical definition of "relations" enlightening. The biggest complaint I have about it, is that it uses different terminology from the previous logic I have read, "sentential logic" rather than "propositional logic" for example, but at least it mentions the alternative terms when it introduces its own.