A paradox describes a figure of speech which, although seeming to make no sense and being self-contradictory, does contain an element of truth.

paradox
noun
A paradox is a statement that apparently contradicts itself and yet might be true. Most logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking.
Dictionary.com defines paradox thus:
a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth.
a self-contradictory and false proposition.
any person, thing, or situation exhibiting an apparently contradictory nature.
an opinion or statement contrary to commonly accepted opinion

Synonyms for paradox:  puzzle, anomaly, riddle.

A quote about paradox:
“The most exquisite paradox… as soon as you give it all up, you can have it all. As long as you want power, you can’t have it. The minute you don’t want power, you’ll have more than you ever dreamed possible.” Ram Dass

The Greek word paradoxon, from which paradox derives, means contrary to beliefs. The statement may seem self-contradictory but it contains some fundamental truth. Paradoxes tend to make people think beyond their preconceived opinions and beliefs.

examples of paradox

“You must be cruel to be kind”  Shakespeare. Hamlet
“Some day you will be old enough to start reading fairy tales again.” – C.S. Lewis.  The Lion, The Witch and The Wardrobe

In literature, poetry, and song lyrics paradoxes are used not only because they are an amusing and smart use of words, but because paradoxes more often than not convey a serious sub-text, touching on elemental themes and universal truths.