what is logic?

What is Mathematical Logic?, J.N. Crossley et.al. [Crossley72] This book has pace: historical survey, completeness, models, turing machines, Gödel and set theory.. all in 80 pp.

first-order logic

The Language of First Order Logic, Jon Barwise and John Etchemendy [Barwise92] A practical approach to learning logic. The book was designed for a first course in logic using the Tarki's World 4.0 software (Logic Software from CLSI), which comes with the book. for PC: for MAC:

Methods of Logic, Willard Van Ormon Quine [Quine62] A lucid introductory text from one of the best.

Logic and Philosophy

Philosophy of Logics, Susan Haack [Haack78] A readable introduction with a slightly broader interpretation of "logic" than the average philosophy text. Philosophy of Logic, Willard Van Orman Quine [Quine70] An excellent short (109pp) introduction with the emphasis on the philosophy. Metalogic - An Introduction to the Metatheory of Standard First-Order Logic, Geoffrey Hunter [Hunter71] An excellent second course for philosophy students who want a good technical understanding of classical first order logic.

Philosophical Logic - An Introduction, Sybil Wolfram [Wolfram89] A worthwhile fairly recent introduction to the kind of problems raised by philosophical logic. Possible Worlds - an introduction to Logic and its Philosophy, Raymond Bradley and Norman Swartz [Bradley79] A substantial (391pp) introduction with the emphasis on propositional and modal logics.

Logic and Mathematics

Principia Mathematica to *56, Bertrand Russell and Alfred North Whitehead [Russell62] The logical and set theoretic1 parts of this great masterpiece, the first consistent logical formalisation of mathematics. Of it, Quine said: "This is the book that has meant most to me.". To understand it today, read [Quine69] first.
Foundations of Constructive Mathematics, Michael Beeson [Beeson80] A wide ranging study of the surprisingly diverse constructive approaches to the logical foundations of mathematics.	Varieties of Constructive Mathematics, Douglas Bridges and Fred Richman [Bridges87] An introduction to and survey of the constructive approaches to pure mathematics.

Logic and Computing

Computability and Logic, George Boolos and Richard Jeffrey [Boolos74] A computationally oriented text for graduate or advanced undergraduate philosophy or mathematics students.

Logic and Set Theory

Notes on Logic and Set Theory, P.T. Johnstone [Johnstone72] A starter for those who want to understand how logic and set theory provide a foundation for mathematics.
Set Theory and its Logic, Willard Van Ormon Quine [Quine69] In the spirit of Principia Mathematica, Quine cleans up the story on how to formalise mathematics using logic (via set theory).	Sets, Functions, and Logic, Keith Devlin [Devlin92] Provides a solid foundation in the basic logical concepts for university/college mathematics.
Set Theory - an Introduction to Independence Proofs, Kenneth Kunen [Kunen80] an introduction to relative consistency proofs in axiomatic set theory, intended as a text for graduate courses

Combinatory Logic and The Lambda Calculus

Introduction to Combinators and the Lambda Calculus, Roger Hindley and Jonathan Seldin [Hindley86] A clear and thorough introduction to a topic fundamental to logic and computer science.

The Lambda Calculus - Its Syntax and Semantics, Henk Barendregt [Barenderegt84] A comprehensive reference on lambda-calculi and combinatory algebras. more info

See also: John Harrison's bibliography on formalised mathematics.