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Elliot Mendelson
Introduction to mathematical logic
The first chapter is on the (relatively simple) propositional calculus. The second deals with quantification theory  I would recommend that you look at the start of chapter 3 to start with, so as to have a concrete example. The later parts of chapter 3 get on to more complicated mathematical logic, such as Gödel's incompleteness theorem. Chapter 4 is on set theory. Now you may have come across set theory at school, but I have to tell you that the axiomatic version is a totally different ball game. But with the introduction to axiomatics from the previous chapters, this book would be a good place to start studying it. The fifth chapter is on computability. I wouldn't suggest this book as a starting point for this  it's much easier to approach it via real computers. However, it could be useful for seeing the links with other forms of logic.