My preferred intro logic text is Enderton's 'A mathematical introduction to logic.' Smullyan's book on first-order logic is very nice, though it uses proof trees, which is (awesome but) somewhat unusual. I'm also a fan of Peter Smith's 'An Introduction to Godel's Theorems.' Spiderman 1967 Rapidshare Premium more.

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As far as set theory goes, it really depends how far you'd like to go. I imagine that if you're doing graduate research in CS you know the 'basics' that would be covered in Halmos' 'Naive Set Theory.' If you're looking for more, I also like Enderton's 'Set Theory,' and after that a big standard text is Jech's. There's obviously more.

You need a good textbook for modal logic, and I don't know any. Hopefully someone else can help fill in the picture. I second the recommendations for Enderton's texts: Both 'A mathematical introduction to logic' and his text on set theory. If you want something to peruse right now, Paul Teller has a fairly good introduction to logic, available on-line (free, no strings attached) or, more directly,. The first volume is pretty elementary, but the second gets slightly more complex. It's a good start, depending on your background in logic. Another 'starter' book, a bit more rigorous than the usual 'Intro to Logic' text, is the classic by Benson Mates, 'Elementary Logic', 2nd edition.

It's not nearly as 'elementary' as some introductory texts, and would be a great start if you're just getting your 'feet wet'. My first suggestion would be the following article: Samuel R. Buss, Alexander S. Kechris, Anand Pillay & Richard A. Shore,, 2001, Bulletin of Symbolic Logic 7 (2):169-196.

As a second suggestion, I would suggest reading the articles on, the quality is wonderful. Each article is written by an expert on the topic. If you want to know more about a topic, check the list of references at the button of the article. Ps: and also have quite good programs at the intersection of logic and philosophy, and you can find similar programs by Googling.