Pinocchio is Incompatible with Free-will

Pinocchio’s Paradox (original)

The story of Pinocchio is an exercise of assuming the following mechanics.

Pinocchio’s nose grows if and only if he tells lies

You’ve heard of the liar paradox, right? Or, as I like the call it, the Pinocchio paradox, which is a fun little thought-experiment: what happens when Pinocchio says “My nose will grow now”?

You guessed it, there’s no way to classify that sentence as true or false. If he is telling a lie, his nose will not grow. But, his nose grows if and only if he tells lies. “A contradiction!” shouts Kurt Gödel.

Pinocchio’s Paradox (extended)

I’m not the first to extrapolate the paradox to the future-tense. The modified Pinocchio paradox rephrases the statement and introduces the concept of time.

At time t=0, Pinocchio makes this strange claim.

NoseGrows and TellsLie are now a time-varying function (otherwise known as a fluents). Formally, Pinocchio has made the following claim at t=0.

Equivalently, Pinocchio is admitting that he will tell a lie in the future.

Here’s some interesting observations to conclude:

If his nose grows in the future, he told the truth at t=0.

If his nose does not grow in the future, he told a lie at t=0.

Free-will

If a person has free-will, they have the choice to either tell the truth or tell a lie at any given time in the future. A little more formally, free will means no simulation of a person can predict their behavior.

How does this relate to Pinocchio? First, let’s consider the two possible cases:

Case 1: For t>0, Pinocchio chooses to never lie again. Therefore his statement at t=0, “I will lie in the future” is false! Consequently, his nose will grow at t=0.

Case 2: For t>0, Pinocchio chooses to lie. His statement at t=0, “I will lie in the future” was true! Therefore his nose will remain normal and not grow at t=0.

Here’s the exciting part: Regardless of whether Pinocchio’s nose grows or doesn’t, there is no such thing as free will!

What? Yes, you heard right!

Assume free-will exists. Therefore, the future must be unpredictable. But, knowing the state of Pinocchio’s nose at t=0 reveals with certainty whether he will lie in the future. A contradiction! Therefore, free-will does not exist.

Even stranger argument:

Assume free-will exists. If Pinocchio chooses never to lie, then his past-self’s nose at t=0 will grow. Thus his current nose at t>0 will also grow, even if he was not lying. A contradiction! Therefore, free-will does not exist.

Poor Pinocchio.