Posted 23 October 2007 - 04:08 AM

davidsparkman was the first to make the point that "all Cretans are liars" is not equivalent to "all Cretans lie all the time." So many of you keyed on this same point. You are right of course, and this observation resolves the issue very easily. So easily, in fact, that you have to ask why this would be handed down over the generations as a paradox.

I believe the answer is that the original paradox meant the latter (agreeing strongly with oranfry and haxxor), and translations/semantics have added this extra discrepancy. Let's give Chrysippos the benefit and allow that "liar" means "lies all the time." We accept that in this situation each Cretan is either a full liar all the time or a full truth-teller all the time. Now we have something to really chew on.

While I would love to address sexsidexy's questions, in particular the issue about whether thick-crusted pizza is in fact Odysseus' favorite (I think he might say, "too much bread, needs more meat, give me the thin crust, good Esophitimenes, and pass the mead"), I'll stick to the paradox at hand.

Consider two possibilities: (1) "All Cretans are liars (lie all the time) or (2) "All Cretans are not liars (tell the truth all the time)."

If (1) is true, then the familiar Cretan would have been required to say that Cretans are truth-tellers. Since he didn't say this, then by contrapositive reasoning we can deduce that statement (1) is not true.

If (2) is true, then the familiar Cretan AGAIN would have been required to say that Cretans are truth-tellers. Since he didn't say this, then similarly we can deduce that statement (2) is not true.

The essence of this type of paradox is to get you to make the assumption that (1) and (2) are the only two possibilities. Then, seeing that both statements are false, you would be stuck with a paradox that has an irreconciliable contradiction (definition 2 in dictionary.com for paradox).

Of course, a paradox may have a resolution (definition 1 in dictionary.com; American Heritage also has both definitions). In this case, eliminating (1) and (2), you must accept the third possibility, that some Cretans are liars and some are truth-tellers. cpotting and Adeori first got at this and several others followed suit.

So, since we know that some Cretans are liars and some are not, the first statement made by the Cretan is a lie (or perhaps he was just innocently mistaken ... I'm kind of forgiving in this way...). Note that the narrator did not say that the Greeks were confused by this statement, probably because they were good at resolving paradoxes.

The second statement made by the familiar Cretan is an inherent contradiction. Given that he is truly a Cretan (a fact we did not question because it was provided by the narrator, whose integrity is not in question), his first clause applies to himself, so he must be a liar, but then in his second clause he states that he is a truth-teller. Such an inherent contradiction cannot be true regardless of whether Cretans are liars or not, so this is a paradox with no resolution.

Notice that it seems the familiar Cretan did not sail there a third time. I guess the Greeks made sure to get rid of that wacko.

Cheers!