Cast your mind back to the 6th century BC when it was fashionable to wear your bedclothes out for a day. A philosopher from Crete named Epimenides is completely and utterly not at all contented with his fellow citzens and exclaims "All Cretans are liars!". This is an interesting statement, coming from a Cretan:

• If Epimenides' statement is true, then he is a liar and hence his statement is false. A contradiction
• If Epimenides' statement is false, then someone living in the 6th century BC (us, for example) could pop their sandals on and sail over to Crete to find a Cretan who sometimes tells the truth.

Because Epimenides didn't contradict himself in the second case (when he was lying), the statement "All Cretans are liars" coming from a Cretan implies that there is at least one Cretan who isn't a liar (note that, "not all apples are green" is the same as "there exists an apple that isn't green").

Ok. So lets refine his statement a little. Suppose Epimenides said "I am a liar":

• If his statement is true, then he is lying and hence his statement is false. Therefore he isn't a liar and sometimes tells the truth. A contradiction.
• If his statement is false, then he isn't a liar and sometimes tells the truth. Here we have an example where he is lying. No contradiction.

So, if someone proclaims that they are a liar, it means that they sometimes tell the truth.

Now, ancient Greeks being ancient Greeks, took this one step further. Eubulides living in the 4th century BC decided to say "I am lying". Since we have a little tradition going:

• If Eubulides' statement was true, then he is lying when he says "I am lying" and so he isn't, i.e. his statement is false.
• If his statement is false, then he isn't lying when he tells us he is, and so his statement is true.

This is known as the Liar's paradox. The paradox was completely generalised in the 14th century AD by a French philosopher named Jean Buridan who wrote

All statements on this page are false

on an otherwise blank page.

In the 20th century AD, a baby was born in the Czech Republic (then Austria-Hungary) and grew up to be a very clever (and paranoid) man called Kurt Gödel who used the Liar's paradox to prove that there are some things you can't prove, his Incompleteness theorem.

Links

Here are a few pages about the Liar's paradox, Kurt Gödel, and, if you're really up to it, the Incompleteness theorems:

• The Liar's Paradox
• Kurt Gödel
• The Incompleteness theorem