Epimenides paradox
Channel: 宇宙のすべての知識 プリンシピア
Date: 2019-02-04

Epimenides paradox, by Wikipedia https://en.wikipedia.org/wiki?curid=9638 / CC BY SA 3.0

#Self-referential_paradoxes
#Paul_the_Apostle
#New_Testament_words_and_phrases
#Ancient_Crete
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 is 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, however, can be solved. There are two options: it is either true or false. First, assume that it is true, but then Epimenides, being a Cretan, would be a liar, and making the assumption that liars only make false statements, the statement is false. So, assuming the statement is true leads us to conclude that the statement is false. This is a contradiction, so the option of the statement being true is not possible. This leaves the second option: that it is false.

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. This means that Epimenides can say the false statement that all Cretans are liars while knowin...