Intuition behind Vacuous proofs My book says we can quickly prove the conditional statement $P \implies Q $ when we know $P $ is false. This much I'm fine with as I can show it with a truth table. But then I'm asked to, using vacuous proofs, prove $P (0)$ is true, given if $n > 1$, then $n^{2}$ > $n $. How can $P (0)$ be true? 0 is not greater than 1. Or is there some other piece to vacuous proofs that I'm overlooking?

It looks like $P(n)$ is supposed to be the statement:

> $n>1\implies n^2>n$.

When $n\le1,$ we have that $n>1$ is _false_. So, for example, $P(0)$ is true.