What are in your opinion some good propositional and first order logic textbooks (undergraduate level) ? I need one that focuses mainly on the aspects of logic related to computer science. thanks in advance.

My favorite introductory book on mathematical logic is Robert S.Wolfe's A TOUR THROUGH MATHEMATICAL LOGIC.Amazingly well written,it covers an extraordinary amount of material in both logic and set theory,complete with biographical vignettes and historical insights.There a deep discussion of first order logic and it's place in metamathematical systems.It also has a wonderful introductory chapter on the theory of computation and it's origins.This is the book I would recommend to any of my students if they asked me about logic. After that,there are several more advanced texts that I like,particularly the ones by Shoenfield and the classic by my old teacher,Elliot Mendelson. You can't go wrong with any of those. 


The question as currently stated is a little vague, but I took a course on logic and theorem proving as it relates to computer science as an undergraduate, and the textbook, Huth and Ryan's "Logic in Computer Science" (which I see now has a second edition) was a reasonable textbook. For a somewhat different value of "aspects of logic related to computer science," Pfenning's notes on from his undergraduate course on Constructive Logic are relatively complete and might be useful. 


A first course in logic by S. Hedman (OUP) is excellent. 


I'm suprised that no one has yet mentioned Boolos 'Computability and Logic'. The first 100 pages are pretty solid computability theoyr, and then follows Metalogic, with an especially good précis on firstorder logic that emphasizes the difference between a Language, an theory, and logical/nonlogical symbols. Worth a gander, espeically if you're interested in CS related logic. Robert Sores 'Recursively Enumerable Sets and Degrees' is another good place to start for recursion theory, as well as general logic. His approach is very clear (if a little assuming that you're following everything at once) and the first 120pp are very informative on recursively enumerable sets (now usually called computably enumerable). But for undergraduates, I'd go with the Boolos... Note of warning  Mendelson's book Introduction to Mathematical Logic I found confusing when starting out  this was confirmed by supervisors/colleagues. I hope this helps! 


It depends on what you mean by "related to computer science". Undergraduate computer science "logic" books tend to focus more on computability, like
Undergraduate mathematical "logic" books tend to focus on propositional logic and firstorder logic but not things like computational complexity. One wellregarded book of that sort is
That book does prove the unique readability (parsing) algorithm for propositional and firstorder formulas. 


As a clear introduction to propositional and first order logic for the mathematically minded, I think Logic and Structure by Van Dalen is in a class of its own. The majority of the book is not particularly CS focused, but the beginnings of recursion theory are covered in the last chapter. 


A very clear (and free) introductory textbook is A Problem Course in Mathematical Logic  "a freeware mathematics text by Stefan Bilaniuk" available as pdf, ps and Latex source on http://euclid.trentu.ca/math/sb/pcml/pcml.html:



Not quite a textbook, but a collection of historical projects about elementary logic and related issues in discrete mathematics and computer science can be found at www.cs.nmsu.edu/historicalprojects Click on the project "Deduction Through the Ages ..." 


Though a little advanced, Logic for Applications by Nerode and Shore fits the bill. (Google books) The book has an extensive treatment of propositional and firstorder logic, with an emphasis on resolution as a proof method. Hilbert style systems are also discussed to some length, but natural deduction is lacking. I found that the Prolog related parts can be successfully skipped if necessary. 


I don't understand what people usually mean when they say "aspects of some mathematical field related to computer science" (how they know what is really related to CS?), but if you really want to study or teach first order logic, then, in my opinion, there is old but still the best book for that  Joseph Shoenfield's Mathematical logic. 


For a discussion of logic textbooks, highly recommended: Teach Yourself Logic 2016:A Study Guide Peter Smith Available online at http://www.logicmatters.net/tyl/ 

