The Cretan philosopher Epimenides is credited with the assertion, "All Cretans are liars". The liar paradox assumes that liars always lie, and non-liars ("truth-tellers") always tell the truth. Saint Paul refers to the paradox in Titus 1:12-13, though it's unclear whether he's aware of the paradoxical implications.
The cleanest variation: "This sentence is false." The Everything variation: This node is deceptive.
Bertrand Russell claimed that George Edward Moore lied only once in his life: when asked if he always told truth, Moore reflected a moment and replied, "No".
Lord Dunsany's short story, Told Under Oath, exploits this paradox.

A Blather of Paradoxes
Of course, when Epimenides of Crete says "all Cretans are liars", there is no paradox. Epimenides is lying; this means that not all Cretans are liars, i.e. that there exists at least one Cretan who tells the truth. This is entirely plausible. Indeed, suppose Crete were populated by just Epimenides and his brother Epidermis. It could be that Epimenides is a liar, while his brother tells the truth.

So the historically accurate version doesn't work. But having Epimenides say "I am a liar" does.

Ok then, I accept your challenge.

I'll begin with an explanation.

The paradox arises from considering the sentence as possessing a truth value (ie. being either true or false). You see, if the sentence (call it P from now on) is true, then the person saying P must be lying, making P false. Conversely, considering P as being false results in the truth being that the person saying P is telling the truth, meaning that they are actually lying, which means they're telling the truth, and so on.
In other words, whichever truth value you assign the sentence, ananlysis of it in this light produces the opposite truth value. Problem.

What does it all mean? Fucked if I know, but let's see what I can make up as I go along. What do we know about it?

  1. P is in English. English is a human language that, idioms aside, contains the meaning of its sentence in grammatical relations of its particles.
  2. P, when considered as false, turns out to be true. When considered as true, P turns out to be false, as we have just seen.
Prima facie, it would appear that the problem occurs as a result of considering a sentence constructed in English to conform to logical rules. However, the problem can be restated in formal logical language without the paradox cheerfully going away.
P possesses the follwoing logical form:
sentence P:"P is false"

From this, it can be seen that I have no idea what the hell is going on. The best I can say is that considering things like this to have a truth value is what causes the problem in the first place, and so the most obvious way out of this one is simply to make sure it never comes up. Consider P as being neither true nor false.

All sorts of problems go here but I'm sleepy and Wittgenstein or someone has probably beaten me to it, so this is Pseudomancer signing off for now.

This paradox is mentioned in the bible, where the apostle Paul writes "Epimenides the Cretan asserts that all Cretans are liars" and goes on to prove himself one of the abovementioned idiots by adding "and he is right."

The paradox results from the ability of language, or meaning if you wish, to refer to itself. This is what the word lying does.

A more sophisticated example of this paradox applies to formal logic: "I cannot prove everything that is true". This is known as the incompleteness of mathematical logic. Any form of logic or computation suffers from it that is sufficiently powerful to model its own operation and argue about its validity. Such systems can be surprisingly small: lambda calculus and Turing machines are examples.

Even if Epimenides were to say "I am a liar", it still would not be a paradox. I'd say there are two different meanings of the word "liar": Someone who speaks falsely all the time, or someone who speaks falsely sometimes.
  • Lies always.
    Obviously, if he were telling the truth, then he would be a liar, which contradicts his ability to tell the truth. So, the statement is false, meaning that he lies sometimes, but not all the time (which is not a liar by our current definition). The statement is false in this case.
  • Lies sometimes.
    If the statement he made is false, then he does not lie sometimes, which means he tells the truth all the time. Except this contradicts the fact that he made a false statement. So, we are left with truth. This means that he lies sometimes, but not always (a liar by our current definition). The statement is true in this case.
So, he is declaring in both cases that he lies sometimes, but not always. However, whether the statement itself is true or not depends on what "liar" means.

The invention of the Liar Paradox (in the 4th century BC) is credited to Eubulides of Miletus, a student of Euclid of Megara.

Eubulides phrased the paradox something like this (although probably in ancient greek):
"A man says that he is lying. Is what he says true or false?"

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