Interest in logic at American universities appears to be ebbing at the same time there is a flow tide of applied mathematics. Yet, if applied mathematics is the use of mathematical ideas to study a scientific discipline, logic which is the application of mathematical ideas to the study of mathematics itself must be classified as applied mathematics. We logicians must confront fundamental questions about the role of logic in the mathematics curriculum. Is it of any use at all? I think the answer is definitely yes and I have written this book in an attempt to show the reasons why.
Formal logic is a rigid and pedantic codification of certain logical principles. But just because no real proofs are done formally it does not follow either that these logical principles are not used throughout mathematics or that the study of formal logic does not add to our understanding of them. One does not often see formal logical formulas in written mathematics, yet the ability to analyze a real mathematical statement in terms of its logical connectives is crucial to any understanding of how it is to be proved.
Logical rules are used in every mathematical proof, but most of time they are just used without comment and without naming them. When one insists that all steps be shown and explicitly named, one has formal logic. Students often have difficulty making the jump from calculus and differential equations to more abstract courses involving proof. It would appear that they sometimes do not grasp that even though proofs are all words and seemingly free form, there are very definite and precise rules they must follow. These rules when stated in formal logic take on a calculational nature no less detailed than the laws for differentiation or integration.
Chapter 1 deals with using and understanding logical formulas.
Before one gets anywhere in logic one should be able to understand
and translate formulas such as 
. Chapter 2
then analyzes some real
proofs in algebra and analysis, showing how an understanding of
the logical connectives used in a statement leads to precise rules
for proving it. It may appear strange that a three line proof needs
five pages of logical analysis, but don't forget that naming
and explaining these matters for the first time requires space.
Chapter 
presents a slick system of formal proof which is
used to give completely detailed proofs of the theorems from
Chapter 2. Chapter
presents the mathematical theory of
logic, meta theorems about proving itself.
Many subjects may claim to be capable of providing a bridge to higher math, but logic is the bridge to higher math. The only debate is whether it is better to learn this implicitly by studying some other subject which presents logic by a process of osmosis, or take the bull by the horns and study logic explicitly. It is sometimes said that calculus is college algebra + limits. In the same spirit, we can say that analysis is calculus + logic and that abstract algebra is college algebra + logic.