Pinocchio Paradox Solved
It all depends on hod the whole lie detection thing works. There will be no paradox if by lying you mean that Pinocchio says something that he thinks isn’t true.
So if Pinocchio thinks that his nose will grow, and he says “My nose will grow”, then it won’t, and Pinocchio will be surprised. If he thinks his nose won’t grow, it will. Clearly after doing this for a while Pinocchio won’t know what to think; he’ll just know that he’s always somehow surprised by what happens. But there is no paradox.
On the other hand, if the nose growing mechanism (NGM) kicks in simply when Pinocchio doesn’t tell the truth, intentionally or not, then we have a situation. I think in order to answer this question I’ll have to fall back on Godel’s incompleteness theorem and say that such an NGM cannot exist.
For one thing, if Pinocchio’s NGM worked in this way, he’d be totally omniscient. For example, he could say “the stock market will go up tomorrow” and know if his statement is true by whether or not his nose grows. So here you get into a bunch of time travel paradoxes in addition to the original paradox.
Still, let’s assume that this NGM exists. Also, for the sake of argument, let’s assume that nose grows => Pinocchio lied in addition to Pinocchio lies => nose grows. Consider what would happen if Pinocchio made this proclamation:
“Either my nose will grow or I will become a real boy!”
The only way for this statement to not cause a paradox is for Pinocchio to become a real boy. So if somehow the universe does whatever it can to avoid paradoxes, it will turn Pinocchio into a real boy.