The above description hits on (at least) two different logical topics:
The first is paradoxes:
“This statement is false” creates a paradox (in systems that consider such a statement meaningful) – if the statement is true, what it says is true, and it is therefore false. If, on the other hand, it is false, than the statement must be true. Any way you look at this sentence, it won’t work.
The second topic addressed is self evident falsehood:
“Everything is relative” creates no paradox. Even if we assume that something that is relative is not true, and that saying “this statement is relative” creates a paradox, than all we know is that the statement cannot be true. Its negation – “It is not true that everything is relative” only means that there is some statement or thing that is not relative. It could be this statement or any other, and neither option causes trouble.