A statement that negates itself in some way. These are an annoying problem logically, and indicate that something is wrong with our logic system. Some of them are obvious, some of them are subtle. It also turns out that the core of the proofs of both Godel's Theorem and the Halting Problem is self negating statements.

Here's a list of all the self negating statements that I know of.

• This statement is false.
• All generalizations are false.
• Everything is relative.
• Russell's Paradox, i.e. the (male) barber in a town shaves everyone who doesn't shave themselves. This one's kind of subtle. Think about the barber. Who shaves him?
• The next statement is very important. The previous statement is irrelevant.
• Never end a sentence with but.

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.