Paradox

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.
Liar's paradox
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."

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. Some famous examples of this kind of paradox include:
 Zeno's paradoxes of motion
 Simpson's paradox in statistics
 Grandfather paradox
Banach–Tarski paradox

 Quine's classification 
Willard Van Orman Quine did a classification of paradoxes. He found three different types:

 A veridical paradox  produces a result that looks absurd, but that can be proved to be true. A well-known example is the birthday paradox, which states that if just 23 random people were put in a group, the chances of two people having the same birthday is more than 50%. Other examples are  Arrow's impossibility theorem that demonstrates difficulties in mapping voting results to the will of the people. Monty Hall paradox (or equivalently three prisoners problem) that demonstrates that a decision that has an intuitive fifty–fifty chance is in fact heavily biased towards making a decision that, given the intuitive conclusion, the player would be unlikely to make. In 20th-century science, Hilbert's paradox of the Grand Hotel, Schrödinger's cat, Wigner's friend or the Ugly duckling theorem are famously vivid examples of a theory being taken to a logical but paradoxical end.
 A falsidical paradox gives a result that is false. This is because there was a fallacy in the demonstration. Examples are All horses are the same color, or Zeno's paradoxes.
 A third class are those paradoxes that reach a contradictory result; they are true and false at the same time. An example of this is the Grelling–Nelson paradox.

 Informal uses of "paradox" 
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 an ethical dilemma but not a logical paradox.

Related pages
 Contradiction
 Dilemma
 Irony
 Oxymoron
 Buttered cat paradox
 Thought experiment

 References 

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