Negation of propositions

We use again our statement $S$, "Socrates is a man", and we add another statement $Q$, "Socrates is not a man".

Clearly, the two cannot be both true or false. The law of excluded middle says that either $P$ is true and $Q$ is false, or the opposite. We call $Q$ the negation of $P$, and write it:

$Q\equiv \neg P$

If $P$ is true, then $\neg P$ is false; if $P$ is false, then $\neg P$ is true.