The most famous is the liar's paradox. It is the simple sentence "This sentence is a lie."
If the sentence is true, then it is a lie, as it says. But if it is a lie, how can it be true? A lie cannot also be the truth. So the sentence being true makes it a lie.
If the sentence is a lie, then it is not as it says, it is true. But that is just what the sentence says. So that makes it true. So the sentence being a lie makes it true.
This paradox is not just true in English but in any language powerful enough for a sentence to make a claim about itself. This is true of mathematics as well. Paradox can never be removed from any symbol system that makes claims about itself.
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 doesn't exist.
Other famous examples:
- Xeno's paradox of motion
- Simpson's paradox in statistics
Because a paradox forces us to think "out of the box", about possibilities other than true or false in logic, right or wrong in morality, it is considered very important in education. People who do not see a paradox where others do, are likely to be too certain they are right.
See also irony, Oxymoron.