"I am lying!"
"I am lying!"
An apparently simple statement, but if you think about it, what is the truth value of this statement? If I told the truth, then I lied. On the other hand, if I lied, then I told the truth.
The earliest version of this paradox originated about 2,600 years ago in Greek. Epimenides, a Cretan, said, :"All Cretans are liars." We see an immediate logical paradox, did Epimenides himself tell a lie?
A modern version of this paradox took place in 1947. The first computer that was designed to solve problems in truth value logic was asked to determine the truth value of "This sentence is false.". As you can imagine, the computer went crazy and went into a oscillating phase between true and false.
If you pay attention, paradoxes like this happen all the time. A graffiti saying "NO GRAFFITI!", someone saying to the crowd "Stop talking." and the statement "The only golden rule is there is no golden rule." are all variations of this paradox. My favorite is "This sentence has six words." An obviously false sentence. If we try to correct it, say "This sentence doesn't have six words.", we are still in the same trap. And I couldn't help mentioning two logic puzzle books by Raymond Smullyan, one is called What Is the Name of This Book? and the other, This Book Needs No Title.
All these paradoxes seem to arise from self-reference, i.e. a Cretan talks about Cretans, a graffiti bans graffiti, etc. However, it is not that simple. How about the following.
Plato: "Socrates lies."
Socrates: "Plato was right."
Neither Plato nor Socrates talks about himself, yet we are still trapped between true and false when we go from one to another. We fall into a loop paradox. This loop paradox reminds me of a advertisement I saw recently. On the bill board, all it says is
"The answer is 'Yes!' Now what is your question?"
Can you think of a question that make the above statement self contradictory? (After all, that is what a mathematician does all day long.) The question is "Can you say 'No'?"
This paradox originally involves a computer that can only answer "yes" or "no". The computer was asked "Is your next response a 'no'?". Of course, the computer went crazy again. Note that the question "Is your next response a 'yes'?" does not cause any contradiction.
So much said, this bring us to yet another paradox. It appeared in the book Don Quixote. The king of an island sets up this rule that any visitor to his island must answer the question "Why are you coming here?" If the visitor answers falsely, he will be hanged. Otherwise, he will be set free. Of course, it all depends on the king's decision whether the answer is true or false...until one day a visitor answers "I am here to be hanged!" The king is at lost now. If he hangs the visitor, his answer is true and therefore should be set free. However, if the king doesn't hang the visitor, his answer is false and he should be hanged.
A similar story is about a crocodile and a baby's mother. The crocodile snatched the baby from the mother and asked "Will I eat your baby? If you answer correctly, I will return your baby. Otherwise, I will eat your baby." And the wise mother's answer? "You will eat my baby."