The Pinocchio paradox

You’ll certainly have heard of the Liar paradox: “All Cretans are liars”, Epimenides – himself a Cretan – said. But this utterance contains a contradiction, for if the sentence is true, Epimenides does not lie, while he says he does. And if he lies, it’s just a confirmation of the statement, so not a lie. Since the days of Epimenides (who lived circa 600 BC) philosophers have discussed a lot about the Liar paradox and developed several variants. Do you know the Pinocchio variant?

Once the logician Peter Eldridge-Smith explained the Liar paradox to his children and asked whether they knew a version of their own, so he tells us in a paper with his daughter Veronique as co-author. Veronique replied: “Pinocchio says ‘My nose will be growing’.” I assume that you’ll know the story of Pinocchio, whose nose grows every time he tells a lie. Since the use of the future tense makes the statement a bit complicated, the father changed it into “Pinocchio says ‘My nose is growing’.” And here we have the Pinocchio paradox. As Peter Eldridge-Smith explains: “So, Pinocchio’s nose is growing iff it is not growing. It is clearly a version of the Liar [paradox].” However, there is an important difference between the original Liar paradox and the Pinocchio paradox. The former is semantic: It is about what the speaker (Epimenides in my example) says and the meaning of his (or her) words. But in what way ever we interpret “my nose is growing”, it is not semantic for it is not about the meaning of words. “My nose is growing” is a statement about a fact, which may be the case or may not be the case. Therefore, we could call this paradox pragmatic (in distinction to a semantic paradox). Anyway, it is a real paradox, for if Pinocchio says that his nose grows and he speaks the truth, it will not grow. But if his nose doesn’t grow, when Pinocchio utters this sentence, the sentence is false and so Pinocchio lies and his nose must grow. Voilà.

After a discussion about some logical implications of the Pinocchio paradox, Peter Eldridge-Smith writes against the end of his paper: “The Pinocchio paradox raises a purely logical issue for any metalanguagehierarchy solution, strict or liberal. The Pinocchio scenario is not going to arise in our world, ...” (the italics are mine). Is it true? Without a doubt, there are no Pinocchios in this world: There is nobody whose nose will grow if and because s/he tells a lie. Nevertheless, Pinocchio scenarios do exist. One of the main political issues in present politics is the fact of the global warming. It’s a theme on local levels, national levels, regional levels and globally. All scientific data make clear that it is not a mere opinion that the world gradually becomes warmer and that man is the main cause of this global warming. It is a fact. Therefore, the statement “Man is the main cause of the present global warming” (the “global warming statement” for short) is true. But alas, there are always people who deny what is clearly true, and so there are still many people who deny the global warming statement, and these people happen to be still quite influential. Let me call them “global warming deniers”. If these global warming deniers will determine global policy, they will take no measures to stop the man-caused global warming and the earth will keep warming up. However, if those who endorse the global warming statement will gain the upper hand and will determine global policy, they’ll take all kinds of measures that will stop the global warming and then the earth will not warm up any longer. Briefly: If those who deny the global warming statement win, there’ll be a global warming caused by man; if those who endorse the global warming statement win, there’ll be no such global warming. This is clearly a version of the Pinocchio paradox. However, I don’t doubt that there are more realistic instances of this paradox, for this world is full of lies.

Source

Peter Eldridge-Smith, Veronique Eldridge-Smith, “The Pinocchio paradox”, in: Analysis, Volume 70, Issue 2, April 2010, Pages 212–215.