Chapter 6 Quantifiers
The authors of this text have aimed to get you started proving things as quickly as possible. This meant that we had to skip over several important topics and return to them later. That is why the text has bounced between logic and proof and logic and proof, and now we return one last time to logic.
The first result we really proved in this text (way back at Result 3.2.7) was
\begin{equation*}
(n \text{ is even}) \implies (n^2 \text{ is even}).
\end{equation*}
We approached the proof by thinking about how the implication could possibly be false. That, in turn, led us to assume the hypothesis to be true, and to show that the conclusion could not possibly be false. In so doing, we have hidden something from you, the reader. Sorry, but the authors felt this was a necessary but well-intentioned untruth 46 to achieve their aim of getting you to start proving things as quickly as possible.
Consider the truth-values of hypothesis and conclusion of the above implication carefully. “\(n\) is even” and “\(n^2\) is even” are not a statement, they are both open sentences 47 whose truth values depend on the variable \(n\text{.}\) We have hidden from you, our reader, is the implicit scope on the variable \(n\text{.}\) We implied that we want this result to be true every possible integer \(n\). To make this implicit explicit:
\begin{equation*}
\text{For every integer } n, (n \text{ is even}) \implies (n^2 \text{ is even}).
\end{equation*}
The effect of that extra bit of text is to provide scope to the variable \(n\text{,}\) and so turns the open sentence into a statement with a well-defined truth value. And once we have a statement we can try to prove it.
A teenie-tiny one.
The reader who has momentarily forgotten the difference between statement and open sentence should quickly jump back to Chapter 2
