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Robert Boyle's self-flowing flask fills itself in this picture, but perpetual motion machines cannot exist.
The Pinocchio paradox is a variant of the Liar paradox

A paradox is a seemingly absurd or self-contradictory statement in logic that, superficially, cannot be true but also cannot be false. Further analysis of the statement or proposition may reveal a fallacious axiom or some obscure underlying truth. Not all paradoxes are fundamentally incongruous, as some may only appear so. Many famous problems of this kind exist.

A famous paradoxes is called 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 does not exist.

Other famous examples:

A paradox can also arise in ethics. Assuming power over others may sometimes be required to protect them while diminishing their right to autonomy. This is defined as an ethical dilemma which means "a paradox arising in ethics". Similarly, an ethical dilemma may be resolved by re-framing of the problem to reveal the false contradiction.

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.

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