Liar Paradox

In philosophy and logic, the classical liar paradox or liar's paradox is the statement of a liar who states that they are lying: for instance, declaring that "I am lying" or "everything I say is false". If they are indeed lying, they are telling the truth, which means they are lying. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar themself. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.

If "this sentence is false" is true, then the sentence is false, but if the sentence states that it is false, and it is false, then it must be true, and so on.

The Epimenides paradox (circa 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The semi-mythical seer Epimenides, a Cretan, reportedly stated that "All Cretans are liars." However, Epimenides' statement that all Cretans are liars can be resolved as false, given that he knows of at least one other Cretan who does not lie. It is precisely in order to avoid uncertainties deriving from the human factor and from fuzzy concepts that modern logicians proposed a "strengthened" liar such as the sentence "this sentence is false".

The paradox's name translates as pseudómenos lógos (ψευδόμενος λόγος) in Ancient Greek. One version of the liar paradox is attributed to the Greek philosopher Eubulides of Miletus who lived in the 4th century BC. Eubulides reportedly asked, "A man says that he is lying. Is what he says true or false?"

The paradox was once discussed by St. Jerome in a sermon:

"'I said in my alarm, Every man is a liar!' Is David telling the truth or is he lying? If it is true that every man is a liar, and David's statement, "Every man is a liar" is true, then David also is lying; he, too, is a man. But if he, too, is lying, his statement: "Every man is a liar," consequently is not true. Whatever way you turn the proposition, the conclusion is a contradiction. Since David himself is a man, it follows that he also is lying; but if he is lying because every man is a liar, his lying is of a different sort."

...
Wikipedia