Gilles Dowek 
Proofs and Algorithms [PDF ebook] 
An Introduction to Logic and Computability

Logic is a branch of philosophy, mathematics and computer science. It studies the required methods to determine whether a statement is true, such as reasoning and computation.Proofs and Algorithms: Introduction to Logic and Computability is an introduction to the fundamental concepts of contemporary logic — those of a proof, a computable function, a model and a set. It presents a series of results, both positive and negative, — Church’s undecidability theorem, Godel’s incompleteness theorem, the theorem asserting the semi-decidability of provability — that have profoundly changed our vision of reasoning, computation, and finally truth itself.Designed for undergraduate students, this book presents all that philosophers, mathematicians and computer scientists should know about logic.