| A History of Numbers
| Propositional Logic
| Logical Completeness
| The Liar's Paradox
Logical Consistency | Basic Methods of Mathematical Proof | Integers and Natural Numbers
Rational Numbers | Irrational Numbers | Imaginary Numbers | The Euler Equation
Imagine if you will that you are back in 6 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 citizens and exclaims 'All Cretans are liars!'. This is an interesting statement, coming from a Cretan because:
If Epimenides' statement is true, then he is a liar and hence his statement is false. A contradiction.
If Epimenides' statement is false, then it would be possible to pop your 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 had said 'I am a liar':
If his statement is true, then he is lying and hence his statement is false. A contradiction.
If his statement is false, then he isn't a liar and sometimes tells the truth. This merely happens to be 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 4 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, ie 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. It was completely generalised in 14 AD by a French philosopher named Jean Buridan who wrote;
All statements on this page are false.
... on an otherwise blank page. Buridan also owned a donkey which starved to death while standing in between two identical bales of hay unable to choose one over the other.
In the 20th Century 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, for example.