Reddit mentions of Type-Logical Semantics (Language, Speech, and Communication)

Introduction to Logic, Patrick Suppes --- This is the best introductory text (understandable and complete) to logic I've run across.

Language Proof and Logic , J. Barwise and J. Echtemendy --- This one comes with with a CD-ROM that allows you to work out logical relationships inside a geometric world, a fun way to reinforce the concepts.

The best text I know of that uses logic to analyze natural language is:

Type-Logical Semantics It has a good foundational section in logic and the rest of the book is a thorough treatment of current methods in using logic to analyze the meaning of natural language.

You might also want to look at Logic, Language and Meaning, volumes 1 and 2 for something similar but with a larger emphasis on logic and less on natural language.

Here's a list of linguistics books that would be good to read that I produced for another question similar to yours. I recommend especially the Seuren; Sapir; Cann, Kempson and Lutz books. If you want to learn Chomsky thoroughly from first principles, read the two books on that list. Chomsky's other big books are just "here are the latest revisions to notation and assumptions I've come up with". His early book, "Syntactic Structures" is an unsatisfying and thin abridgment of "The Logical Structure of Linguistic Theory", which is in my list. LSLT is a full justification of his approach with lots of supporting argument. His best justifications of certain methodological points, such as introspection, are in this book. "On Nature and Language" is a recent revision of those arguments in light of what he is trying to do with the "Minimalist Program".

If you are interested in doing formal linguistics of any type, Chomskian or otherwise, you owe it to yourself to acquire some familiarity with the abstract tool you will be using, read this book:

Introduction to Formal Languages

You also should read Saussure's Course in General Linguistics