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.

Mythology of lying Cretans

According to Ptolemaeus Chennus, Thetis and Medea had once argued in Thessaly over which was the most beautiful; they appointed the Cretan Idomeneus as the judge, who gave the victory to Thetis.
In her anger, Medea called all Cretans liars, and cursed them to never say the truth.

Logical paradox

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."

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 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'' (III.13.29), 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". '' Dictionnaire Historique et Critique'' at Wikipedia.

References by other authors

All of the works of Epimenides are now lost, and known only through quotations by other authors. The quotation from the ''Cretica'' of Epimenides is given by R.N. Longenecker, "Acts of the Apostles", in volume 9 of ''The Expositor's Bible Commentary'', Frank E. Gaebelein, editor (Grand Rapids, Michigan: Zondervan Corporation, 1976–1984), page 476. Longenecker in turn cites M.D. Gibson, ''Horae Semiticae X'' (Cambridge: Cambridge University Press, 1913), page 40, "in Syriac". Longenecker states the following in a footnote:

An oblique reference to Epimenides in the context of logic appears in "The Logical Calculus" by W. E. Johnson, ''Mind'' (New Series), volume 1, number 2 (April, 1892), pages 235–250. Johnson writes in a footnote,

The Epimenides paradox appears explicitly in "Mathematical Logic as Based on the Theory of Types", by Bertrand Russell, in the ''American Journal of Mathematics'', volume 30, number 3 (July, 1908), pages 222–262, which opens with the following:

In that article, Russell uses the Epimenides paradox as the point of departure for discussions of other problems, including the Burali-Forti paradox and the paradox now called Russell's paradox. Since Russell, the Epimenides paradox has been referenced repeatedly in logic. Typical of these references is ''Gödel, Escher, Bach'' by Douglas Hofstadter, which accords the paradox a prominent place in a discussion of self-reference.

It is also believed that the "Cretan tales" told by Odysseus in ''The Odyssey'' by Homer are a reference to this paradox.

Notes

References

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{{Paradoxes

Self-referential paradoxes
Paul the Apostle
New Testament words and phrases
Ancient Crete

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