Epimenides paradox

The Epimenides paradox is a problem in logic. There is no single statement of the problem; a typical variation is given in the book Gödel, Escher, Bach (page 17).

Epimenides was a Cretan who made one immortal statement: "All Cretans are liars."

Is the statement (that "all Cretans are liars," asserted by a Cretan) true or false? This problem is named after the Cretan philosopher Epimenides of Knossos (flourished circa 600 BC), who apparently did state Κρητες αει ψευσται, "Cretans, always liars". We will first consider the logical status of this statement and then review the history of this famous quote.

Logical Analysis

If we define "liar" to mean that every statement made by a liar is false (so that Epimenides' statement amounts to "Anything said by a Cretan is false"), then the statement "All Cretans are liars," if uttered by the Cretan Epimenides, cannot be consistently true. (And, as will be noted below, according to one interpretation it also cannot be consistently false, either.)

The conjunction of "Epimenides said all Cretans are liars" and "Epimenides is a Cretan" would, if true, imply that a Cretan has truthfully asserted that no Cretan has ever spoken the truth; the truth of Epimenides' statement would be a counterexample (some Cretan has told the truth at least once) and would mean that not all Cretans are necessarily always liars, which would contradict Epimenides' statement and thus would render it false.

Several interpretations and analyses are available, if the statement is considered false. It might be contended that the truth-value "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 (not Epimenides in this instance) 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. But if Epimenides' statement is understood as in essence asserting its own falsehood, then the statement cannot consistently be false, either, because its falsehood would imply the truth of its self-asserted falsehood.

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.

Alternatively, if, by "liar", we were to mean someone whose statements are usually but not always false, the logical problem would dissolve: Epimenides might usually lie, but on this occasion it might be that he happened to speak the truth. This interpretation would not lead to an interesting logical problem.

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 Burali-Forti paradox, all of which have self-reference in common with Epimenides. (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.

History

The Epimenides paradox, as a problem in logic, appears to have a relatively recent origin, although the statement "Cretans, always liars" has quite a history itself.

Epimenides was a philosopher and religious prophet who, against the general sentiment of Crete, proposed that Zeus was immortal. As he wrote in his poem Cretica,

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.

Denying the immortality of Zeus, then, was the lie of the Cretans. From the poem's context, it appears that by "Cretans", Epimenides simply intended "Cretans other than myself." The phrase "Cretans, always liars" was quoted by the poet Callimachus in his Hymn to Zeus, evidently with the same theological intent as Epimenides.

The entire second line is quoted in the Epistle to Titus, chapter 1, verse 12, and identified as such by Clement of Alexandria. The apparently paradoxical statement is, "One of them, a prophet of their own, said 'The Cretans are always liars . . .' This testimony is true." In this oblique version of the paradox (as it is viewed by some), the author of the Epistle to Titus (who has traditionally been identifed as St. Paul) quotes the statement by a Cretan (presumably Epimenides) that Cretans always lie, and then goes on to endorse the truth of that logically puzzling utterance.

In the Epistle to Titus, the reference to Epimenides and lying Cretans does not expressly relate to Cretan denial of divine immortality. Some have suggested that the passage in Titus has no theological or logical import but is instead intended simply to defame Cretans. However, others point out that several versions of the pseudomenon (the liar's paradox) were well known in antiquity. Because of this, so the argument goes, the paradoxical dimension of this passage would have been obvious in the ancient world (even if oblique in Titus), so that it is therefore unlikely to have been an accident for St. Paul (or whoever the author was) to make a point of mentioning "truth" in that context. The appended statement in Titus, "This testimony is true," tends to reinforce the logically paradoxical aspect of the underlying statement, by explicitly affirming its truth. Moreover, the opening verses of the Epistle to Titus use unusual phrasing to characterize God as never-lying ("theos apseudes" Titus 1:2) and can be read as establishing "truth and falsehood" as a major theme in Titus. However, there is no extra-textual direct historical evidence that this epistolary text was used as a basis for the study of truth and falsehood in the early Church.

Thus, according to one view, the inconsistency of a Cretan's assertion that all Cretans are liars simply did not occur to Epimenides, nor to Callimachus, nor to the author of Titus, nor to Clement. From that point of view, Epimenides' reference to "Cretans" simply meant "Cretans other than myself." If so, there would be no self-reference and thus no logical problem to speak of. For example, neither the exegesis nor the exposition of Titus in the Interpreter's Bible (1955) mentions the presence of a logical paradox, although the editors do acknowledge Epimenides as the probable source of the quotation in Titus 1:12 and they observe that it would have been "singularly untactful" for a real letter addressed to Cretans, to say such harsh things about all Cretans. From another point of view, however, ancient awareness of the so-called "pseudomenon" logic problem undermines the notion that the inconsistency simply did not occur to the author of Titus. An examples of this class of paradox having been studied within the early Christian Church is found in Saint Augustine, Against the Academicians III.13.29.

The liar paradox was known in antiquity, but it was not associated with Epimenides at that time. Many variations of the liar paradox (called insolubilia) were studied by logicians in the Middle Ages, but none of the extant medieval works on insolubilia refers to Epimenides, either directly or through the Epistle to Titus. The earliest express discussion of Epimenides' statement as logically paradoxical dates only to the nineteenth century. Since that time, the Epimenides paradox has been commonly employed in discussions of logic.

Sources

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 Mero (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 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 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 (Basic Books, 1980), which accords the paradox a prominent place in a discussion of self-reference.

Real-world example

One real world paradox similar to the Epimenides paradox is can be seen, as of 2004, at the web page http://www.icann.com/. On this web page by the Internet Corporation for Assigned Names and Numbers (ICANN), it says:

"There is no web site at the Internet address that you entered."

Just like the Epimenides paradox, this statement at first sight appears to be paradoxical: How can a web page claim that there isn't a web site at the very address it occupies? However this statement, too, is not paradoxical if false. It also assumes that a single web page can be considered a web site.