 Epimenides paradox

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:
 Epimenides was a Cretan who made one immortal statement: "All Cretans are liars."
The Epistle to Titus makes apparent reference to Epimenides: the author says of Cretans that "they are always liars, as one of their own has said." (Titus 1:12)
A paradox of selfreference 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 nonparadoxical lie in that it does not lead to a logical contradiction. (The contradictory 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.")
Contents
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 selfreference 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 of Alexandria:
In his epistle to Titus, Apostle Paul wants to warn Titus that Cretans don't believe in the one truth of Christianity, because "Cretans are always liars". To justify his claim, Apostle Paul cites Epimenides.
— Stromata 1.14
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".^{[1]}
Logical analysis
Define a "liar" as someone who is never truthful. Then the statement, "All Cretans are liars", if uttered by a Cretan, in this case Epimenides, implies that the speaker's statement is not true—that is, some Cretans are not liars. Some logicians have treated the Epimenides paradox as identical to the Liar paradox.
For example, 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; hence 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."^{[2]} However, the inference from "not all Cretans are liars", to "the Cretans are veracious" is not valid.
Several interpretations and analyses are available, if the statement is considered false. It might be contended that the truthvalue "false" can be consistently assigned to the simple proposition that "All Cretans are liars," so that this statement by itself, when deemed false, is not, strictly speaking, paradoxical. Thus, if there ever existed a Cretan who even once spoke the truth, the categorical statement "All Cretans are (always) liars", would be false, and Epimenides might be simply regarded as having made a false statement himself.
An interesting asymmetry is possible under one interpretation: the statement's truth clearly implies its falsehood, but, unless the statement is interpreted to refer specifically to itself (rather than referring categorically to all statements by Cretans), the statement could be contingently false without implying its own truth.
Naturally, any truly logical idea of a paradox with the statement falls flat if one understands that while "all Cretans" may be "liars," such a statement in realistic terms does not necessarily mean that all Cretans lie all the time or that they lie only. Even if it is said that "Cretans are always liars," this does not produce a paradox if one understands the various meanings of the term always—as in "John always says No!" does not mean that "No" is all—or the only thing—John ever says. Certainly even the most prolific liars in history told the truth at least some of the time, so an idea that anyone lies in every single sentence they speak is merely simpleminded at best. The word "always" can also be interpreted to mean "in all instances", as in, "when you find a Cretan, you will always have found a liar," where "liar," again, need not mean more than a person who is known to lie.
Paradoxical versions of the Epimenides problem are closely related to a class of more difficult logical problems, including the liar paradox, Russell's paradox, and the BuraliForti paradox, all of which have selfreference 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 selfreference led to important developments in logic and mathematics in the twentieth century.
Notes
 ^ Bayle, Pierre (1740). Dictionnaire Historique et Critique. 2 (5th ed.). p. 414. http://books.google.com/books?id=FKsAAAAcAAJ&dq=epimenides%20logic&pg=PA414#v=onepage&q=qu'Epimenide&f=false. Retrieved 1 April 2011. Dictionnaire Historique et Critique at Wikipedia.
 ^ Fowler, Thomas (1869). The Elements of Deductive Logic (3rd ed.). Oxford: Clarendon Press. p. 163. http://books.google.com/books?id=FSYCAAAAQAAJ&dq=epimenides&pg=PA163#v=onepage&q=epimenides&f=false. Retrieved 1 April 2011.
References
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:
 The Syr. version of the quatrain comes to us from the Syr. church father Isho'dad of Merv (probably based on the work of Theodore of Mopsuestia), which J.R. Harris translated back into Gr. in Exp ["The Expositor"] 7 (1907), p 336.
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,
 Compare, for example, such occasions for fallacy as are supplied by "Epimenides is a liar" or "That surface is red," which may be resolved into "All or some statements of Epimenides are false," "All or some of the surface is red."
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:
 The oldest contradiction of the kind in question is the Epimenides. Epimenides the Cretan said that all Cretans were liars, and all other statements made by Cretans were certainly lies. Was this a lie?
In that article, Russell uses the Epimenides paradox as the point of departure for discussions of other problems, including the BuraliForti 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 selfreference.
Categories: Paradoxes
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