statement that, despite apparently valid reasoning from true premises, leads to an apparently-self-contradictory conclusion

A paradox is a sentence in logic that cannot be true but also cannot be false. It is self-contradictory. Many famous problems of this kind exist.

Robert Boyle's self-flowing flask fills itself in this picture, but perpetual motion machines cannot exist.

A famous paradox is called the liar's paradox. It is the simple sentence "This sentence is a lie", or equivalently, "This statement is false."[1]

If the sentence is true, then it is a lie as it says. But if it is a lie, it cannot be true. A lie cannot also be a truth. So the sentence being true makes it a lie.

On the other hand, if the sentence is a lie, then it is not as it says: it is true. But that is just what the sentence says, which makes the content of the sentence true. So the sentence being a lie makes it true.

This paradox is not just in English, but in any language. It is true of mathematics as well. Paradox can never be removed from any symbol system that makes claims about itself.

## Other examples

Another example is the statement that "there is no cabal". Only a cabal can know if there is no cabal, so this is either a guess, or, it is a cabal trying to pretend it does not exist.

Not all paradoxes are true logical paradoxes, since they can also be common-sense-defying statements that appear true.[2] Some famous examples of this kind of paradox include: