0. If-then reasoning

A key concept is ``if-then'' reasoning. For example,

For any real number $x$, if $x > 0$ then $x^2 > 0$.

This is true. For short, we usually just say ``if $x > 0$ then $x^2 > 0$''. Another way to write it is $x > 0 \Rightarrow x^2 > 0$, which in words is ``$x > 0$ implies $x^2 > 0$''.

Is this statement true the other way around? In other words, is it true that

For any number $x$, $x^2 > 0 \Rightarrow x > 0$ ??

NO, since $x=-1$ is a counterexample: $(-1)^2 > 0$ is true but $-1 > 0$ is false.