Negation of propositions

by Jeremy Perret Mar 26 2016

The proposition that is false if another one is true and vice-versa.

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.