Epimenides paradox


The Epimenides paradox reveals a problem with self-reference in logic. It is named after the Cretan philosopher Epimenides of Knossos 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.

Logical paradox

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 a 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 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 above is to think that the negation of "all Cretans are liars" is "all Cretans are honest" 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.
In other words, it is not a paradox once one realizes "All Cretans are liars" being untrue only means "Not all Cretans are liars" instead of the assumption that "All Cretans are honest".
Perhaps better put, for "All Cretans are liars" to be a true statement, it does not mean that all Cretans must lie all the time. In fact, Cretans could tell the truth quite often, but still all be liars in the sense that liars are people prone to deception for dishonest gain. Considering that “All Cretans are liars” has been seen as a paradox only since the 19th century, this seems to resolve the alleged paradox. Of course, if ‘all Cretans are continuous liars’ is actually true, then asking a Cretan if they are honest would always elicit the dishonest answer ‘yes’. So arguably the original proposition is not so much paradoxical as invalid.
A contextual reading of the contradiction may also provide an answer to the paradox. The original phrase, "The Cretans, always liars, evil beasts, idle bellies!" asserts not an intrinsic paradox, but rather an opinion of the Cretans from Epimenides. A stereotyping of his people not intended to be an absolute statement about the people as a whole. Rather it is a claim made about their position regarding their religious beliefs and socio-cultural attitudes. Within the context of his poem the phrase is specific to a certain belief, a context that Callimachus repeats in his poem regarding Zeus. Further, a more poignant answer to the paradox is simply that to be a liar is to state falsehoods, nothing in the statement asserts everything said is false, but rather they're "always" lying. This is not an absolute statement of fact and thus we cannot conclude there's a true contradiction made by Epimenides with this statement.

Origin of the phrase

Epimenides was a 6th-century BC philosopher and religious prophet who, against the general sentiment of Crete, proposed that Zeus was immortal, as in the following poem:
Denying the immortality of Zeus, then, was the lie of the Cretans.
The phrase "Cretans, always liars" was quoted by the poet Callimachus in his Hymn to Zeus, with the same theological intent as Epimenides:

Emergence as a logical contradiction

The logical inconsistency of a Cretan asserting all Cretans are always liars may not have occurred to Epimenides, nor to Callimachus, who both used the phrase to emphasize their point, without irony, perhaps meaning that all Cretans lie routinely, but not exclusively.
In the 1st or 2nd century AD, the quote is mentioned by the author of the Epistle to Titus as having been spoken truly by "one of their own prophets."
Clement of Alexandria, in the late 2nd century AD, fails to indicate that the concept of logical paradox is an issue:
During the early 4th century, Saint Augustine restates the closely related liar paradox in Against the Academicians, but without mentioning Epimenides.
In the Middle Ages, many forms of the liar paradox were studied under the heading of insolubilia, but these were not explicitly associated with Epimenides.
Finally, in 1740, the second volume of Pierre Bayle's Dictionnaire Historique et Critique explicitly connects Epimenides with the paradox, though Bayle labels the paradox a "sophisme".