Cretan Paradox

The Cretan Paradox is attributed to the Cretan poet Epimenides [http://en.wikipedia.org/wiki/Epimenides_paradox]. Epimenides the Cretan says "All Cretans are liars." This statement, because it was uttered by a Cretan, is true if and only if it is false. (But see below...)

Interestingly, St. Paul makes reference to this paradox, in a way that shows he knew of it but may not have understood it: He says of Cretans that "even one of their own prophets has said, 'Cretans are always liars, evil brutes, lazy gluttons.' This testimony is true." But the context [http://bible.gospelcom.net/cgi-bin/bible?passage=titus+1:12-14&version=NIV] suggests that he took it seriously as a denunciation of Cretans.

On the other hand, [http://www.bismikaallahuma.org/Bible/Text/reply_epimenides.htm] notwithstanding, it could be argued that St. Paul was using a hyperbole to make his point. Paul is exaggerating to express his strong objection to the actions of the Cretans obstructing Titus' mission. Some Bible scholars contend that Paul wrote the words with a wry sense of humor. In any case, from the context of the whole paragraph [http://bible.gospelcom.net/cgi-bin/bible?passage=titus+1:5-16] it's clear that he's angry about what the Cretans are doing.

The CretanParadox, dating back to the 6th century BCE, is the earliest known (attempt at formulating a) mathematical paradox.

The caveat is necessary because under some analyses the statement is not paradoxical. The Wikipedia entry http://www.wikipedia.org/wiki/Epimenides_paradox distinguishes between the "Liar Paradox" ("Everything I tell you is a lie," or, more simply, "This statement is false," which is genuinely paradoxical) and the Epimenides statement.

But this is a quibble that, I feel, misses the point. The Epimenides statement was clearly an attempt, however flawed, to exhibit a logical antinomy. (It was not in any case intended as a sociological observation on Cretan culture.)

Mathematically, it is interesting to analyze why the Epimenides statement fails as an antinomy but the more directly self-referential Liar statement succeeds. (For one analysis, see http://david.tribble.com/text/liar.htm, but note that that page treats the terms "Epimenides Paradox" and "Liar Paradox" as interchangeable.) But in the historical view, it is more interesting to note that twenty-six centuries ago, careful thinkers were already aware of, and exploring, the limits and limitations of their language and their methods of reasoning.

Here's an example of why the Epimenides statement "All Cretans are liars" is not paradoxical, but merely contingent. Suppose there are 10 Cretans in the world, one of them being Epimenides. By saying "All Cretans are liars" Epimenides presumably means that all 10 Cretans are liars, i.e that all 10 of the Cretans that exist in the world always make false statements. Could Epimenides be a liar? Yes, simply if one of the other Cretans doesn't always lie. Then Epimendes statement is simply a false statement, as you would expect from a liar.

[Not quite. One can hardly accept it as validly constructed, since (as noted below) it would be self-contradictory if there was only one Cretan.]

Epimenides can't be a truth-teller, because they can only make true statements. "All Cretans are liars" implies "Epimenides is a liar", so if "All Cretans are liars" is true then so is "Epimenides is a liar", which is a contradiction, hence Epimenides can't be a truth-teller.

However, if Epimenides was the only Cretan, then "All Cretans are liars" is a paradox, since it boils down to "I am a liar".

liar: one that tells lies, not one that tells lies and only lies. You can say "I am a liar", and be telling the truth, because a liar is not prohibited from telling the truth. A paradox would be "Everything Cretans say is a lie".

Mr. Spock once demonstrated that this can be a UsefulLie. He used it to incapacitate Norman the android.

A Cretan saying, "All Cretans are liars," is not a paradox. "All Cretans are liars," does not mean "Every statement uttered by a Cretan is a lie." It implies that Cretans have a greater proclivity to lie than non-Cretans. But it doesn't even come close to implying that it is impossible for a Cretan to make a truthful statement.

"This statement is false," is a genuine paradox...or a half-truth.

Perhaps Epimenides should have said "all Cretans speak only untruths".

It seems what he really wrote (at least, according to http://en.wikipedia.org/wiki/Epimenides_paradox) translates better to "Cretans, always liars".

Even in that form, it is still true that (A) being a liar, even always, doesn't preclude uttering the occasional true statement & (B) the author is implying, especially in the context of the poem, that he himself is excluded. It seems dubious to borrow Epimenides clearly unparadoxical expression to explore this issue. Naming it after him is doubly dubious.

The denial of Absolute Truth is in its self proof of Absolute Truth. Good one for the Religious defenders. - Pat

Aren't we all liars?

See MathematicalParadoxes, and especially the RussellParadox.