The truth is that I do not know how to negate a logical proposition, for example the following:
$$(\forall x \in \mathbb R) (x^2 > x)$$
I do not want to determine its truth value since I know it is false. I want to negate the affirmation. I would greatly appreciate your help.
In general, we have that:
$$\neg (\forall x) \phi(x) \Leftrightarrow (\exists x) \neg \phi(x)$$
Applied to your formula:
$$\neg (\forall x \in \mathbb{R}) (x^2 >x) \Leftrightarrow$$
$$(\exists x \in \mathbb{R}) \neg (x^2 > x)$$
but of course the claim $\neg (x^2 > x)$ is equivalent to the claim $x^2 \le x$, and so we get:
$$(\exists x \in \mathbb{R}) (x^2 \le x)$$