The Epimenides paradox is a problem in logic. It is named after the Cretan philosopher Epimenides of Knossos (alive circa 600 BC), There is no single statement of the problem; a typical variation is given in the book Gödel, Escher, Bach, by Douglas Hofstadter:
The Epistle to Titus makes reference to Epimenides: the author says of Cretans that "they are all liars, as one of their own has said." (Titus 1:12)
A paradox of self-reference is commonly supposed to arise when one considers whether Epimenides spoke the truth. However, if Epimenides knew of at least one Cretan (other than himself) who was not a liar, then his statement is a non-paradoxical lie in that it does not lead to a logical contradiction. (The negation of the statement, "All Cretans are liars" is the statement, "Some Cretans are not liars," which might be true at the same time as the statement, "Some Cretans are liars.")
History of the phrase
Epimenides was a philosopher and religious prophet who, against the general sentiment of Crete, proposed that Zeus was immortal, as in the following poem:
They fashioned a tomb for thee, O holy and high one
The Cretans, always liars, evil beasts, idle bellies!
But thou art not dead: thou livest and abidest forever,
For in thee we live and move and have our being.
– Epimenides, Cretica
Denying the immortality of Zeus, then, is the lie of the Cretans. It appears that by "Cretans", Epimenides intended "Cretans other than myself". The phrase "Cretans, always liars" was quoted by the poet Callimachus in his Hymn to Zeus, with the same theological intent as Epimenides. The entire second line is quoted in the Epistle to Titus:
One of Crete's own prophets has said it: 'Cretans are always liars, evil brutes, lazy gluttons'.
He has surely told the truth. For this reason correct them sternly, that they may be sound in faith instead of paying attention to Jewish fables and to commandments of people who turn their backs on the truth.
– Epistle to Titus, 1:12-13
The logical inconsistency of a Cretan asserting all Cretans are always liars may not have occurred to Epimenides, nor to Callimachus. In the original context, Epimenides necessarily meant "Cretans other than myself", so there is no self-reference and thus no logical problem to speak of; accusing Cretans (other than himself) denying the immortality of Zeus while he did not deny it himself. It is also quite natural to understand the Cretan poet as possibly having employed the figure of speech known as hyperbole (deliberate exaggeration) rather than having advanced a vigorous logical claim. It is not clear when Epimenides became associated with the Epimenides paradox. Epimenides himself does not appear to have intended any irony or paradox in his statement, "Cretans, always liars", nor did Callimachus, nor the author of Titus, nor Clement. The logical contradiction exists on Saint Paul's epistle rather than the poem of Epimenides:
Saint Augustine restates the liar paradox, without mentioning Epimenides or Titus, in Against the Academicians (III.13.29). 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. The second volume of Pierre Bayle's Dictionnaire Historique et Critique of 1740 explicitly connects Epimenides with the paradox, though Bayle labels the paradox a "sophisme." 
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