Epimenides paradox

The Epimenides paradox reveals a problem with self-reference in logic.

It is named after the Cretan philosopher Epimenides of Knossos (alive circa 600 BC) who is credited with the original statement.

A typical description of the problem is given in the book Gödel, Escher, Bach, by Douglas Hofstadter:

A paradox of self-reference arises when one considers whether it is possible for Epimenides to have spoken the truth.

Thomas Fowler (1869) states the paradox as follows: "Epimenides the Cretan says, 'that all the Cretans are liars,' but Epimenides is himself a Cretan; therefore he is himself a liar. But if he be a liar, what he says is untrue, and consequently the Cretans are veracious; but Epimenides is a Cretan, and therefore what he says is true; saying the Cretans are liars, Epimenides is himself a liar, and what he says is untrue. Thus we may go on alternately proving that Epimenides and the Cretans are truthful and untruthful."

The Epimenides paradox in this form can however be solved. A paradox only results when the statement is assumed to be true. Namely, if the statement "all Cretans are liars" (stated by Epimenides, himself a Cretan) is true, then Epimenides, being a Cretan, would be a liar; making the assumption that liars only make false statements, the statement should be false. So assuming the statement is true leads us to conclude that the statement is false and cannot be accepted, therefore it must be true, continuing in a self-referential paradox.

However, if we assume the statement is false and that Epimenides is lying about all Cretans being liars, then there must exist at least one Cretan who is honest. This does not lead to contradiction, since it is not required that this Cretan be Epimenides, meaning that Epimenides can say the false statement that all Cretans are Liars while knowing at least one honest Cretan and lying about this particular Cretan. Hence, from the assumption that the statement is false it does not follow that the statement is true. So we can avoid a paradox as seeing the statement "all Cretans are liars" as a false statement, which is made by a lying Cretan, Epimenides. The mistake made by Thomas Fowler (and many other people) above is to think that the negation of "all Cretans are liars" is "all Cretans are honest" (a paradox) when in fact the negation is "there exists a Cretan who is honest", or "not all Cretans are liars". The Epimenides paradox can be slightly modified as to not allow the kind of solution described above, as it was in the first paradox of Eubulides but instead leading to a non-avoidable self-contradiction. Paradoxical versions of the Epimenides problem are closely related to a class of more difficult logical problems, including the liar paradox, Socratic paradox, and the Burali-Forti paradox, all of which have self-reference in common with Epimenides. Indeed, the Epimenides paradox is usually classified as a variation on the liar paradox, and sometimes the two are not distinguished. The study of self-reference led to important developments in logic and mathematics in the twentieth century.

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