Friday, December 21, 2012

The Pinocchio paradox

Suppose that Pinocchio's nose grows if, and only if, he says something that is not true, and that he says "My nose is growing". Then his nose is growing if and only if it is not growing. (This paradox originated with Veronique Eldridge-Smith.) According to Peter Eldridge-Smith:
The Pinocchio scenario is not going to arise in our world, so it is not a pragmatic issue. It seems though that there could be a logically possible world in which Pinocchio’s nose grows if and only if he is saying something not true. However, there cannot be such a logically possible world wherein he makes the statement ‘My nose is growing’.
In the world in which Pinocchio's nose grows and shrinks in such a way, suppose that he says, of various uniformly coloured objects, that they are blue. What happens if the object is as blue as not? (There must be such colours, because otherwise some colour that is blue is the same colour as some colour that is not blue.) Well, whatever happens, that could also be what happens when he says "My nose is growing". It is, for example, possible that Pinocchio's nose is in a quantum-mechanically entangled state, as much growing as not. That seems to be a logically possible world.

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