Chapter 1 Logic and Proofs
Logic is the study of consequence. Given a few mathematical statements or facts, we would like to be able to draw some conclusions. For example, if I told you that a particular real-valued function was continuous on the interval \([0,1]\text{,}\) and \(f(0) = -1\) and \(f(1) = 5\text{,}\) can we conclude that there is some point between \([0,1]\) where the graph of the function crosses the \(x\)-axis? Yes, we can, thanks to the Intermediate Value Theorem from calculus. Can we conclude that there is exactly one point? No. Whenever we find an “answer” in math, we really have a (perhaps hidden) argument.
Mathematics is really about establishing general statements (like the Intermediate Value Theorem). This is done via an argument called a proof. We start with some given conditions, the premises of our argument, and from these, we find a consequence of interest, our conclusion.
The problem is, as you no doubt know from arguing with friends, not all arguments are good arguments. A bad argument is one in which the conclusion does not follow from the premises, i.e., the conclusion is not a consequence of the premises. Logic is the study of what makes an argument good or bad. In other words, logic aims to determine in which cases a conclusion is, or is not, a consequence of a set of premises.
We will start in Section 1.1 by considering statements, the building blocks of arguments. Understanding what counts as a statement and what form statements can take is the first step in understanding arguments. We will take a closer look at how statements can be combined in Section 1.2. Then we will see what mathematical tools we can develop to better analyze these statements and how they interact in Section 1.3. Finally, we will put all of this together in Section 1.4 and Section 1.5 to see how we can use these tools to construct arguments and prove statements.
