Some Ideas of Logic

The dependence of mathematics on logic is obvious. We use reasoning processes in mathematics to prove results, and logic is concerned with reasoning. However, when someone studies mathematics, he does not first study logic in order to learn to think correctly. Rather, he jumps into mathematics, perhaps with high school geometry, and learns to prove things by actually doing proofs. Logic comes into his education only when there seems to be something doubtful or obscure that needs clarification. A person who is working with some parts of mathematics, like foundational studies, where common sense does not provide enough precision, needs the more finely-tuned results of logic. However, in most of the undergraduate mathematics courses you can get by with informal reasoning and common sense. There are a few exceptions. It is necessary to have a clear understanding of some of the vocabulary of logic as used in mathematics and of the logic underlying the idea of a mathematical proof. I will try to clarify some of these points in this chapter. You should read it over quickly and refer back to it when the need arises. At the end of the chapter I will give you an opportunity to use what you have learned by asking you to construct some simple proofs.