logic · paradox

Liar paradox

This sentence is false.

This sentence is false. What follows if you assume it? The sentence is about itself. Try each label.

This sentence is false.

The sentence is about itself. It says that it is false.

Call it true. Then what it says holds, and it says it is false, so it is false.

It cannot be true and false together, so it cannot be true.

Call it false. Then what it says does not hold. It says it is false, so it is not false, so it is true.

Each label turns into the other one. Nothing else in the sentence can settle it.