If Pinocchio would utter the statement "My nose is going to grow now", it is going to cause a paradox. That's a pretty straight-forward paradox that you have probably heard of. If the statement is true, his nose shouldn't grow, but if his nose wouldn't grow than the statement would be false. If the statement is false, his nose should grow, but if his nose would grow, the statement would be true.
Thinking about such paradoxes is always interesting because it shows us that our understanding of something in the world might actually be questionable. I personally think that a paradox like this shows that there is something wrong with some of the concepts we are using and the way we are applying them. In a way, the absurdity of a paradox comes from our understanding and not neccessarily form a real and grounded contradiction.
Possibility and Impossibility
Let's examine the Pinocchio Paradox in the most trivial of ways first. What we can say to characterize the situation is that it is in fact conceivable, but we can't really say that it is possible. What the paradox postulates is actually a type of system that is most likely physically impossible. Pinocchio's nose must be some real object that has the property of reliably distinguishing between a false and a true statement virtually instantaneously.
Since Pinocchio could utter statements involving the whole universe, this object would require both data about the whole universe and the computing power to determine if a specific statement about it is true or false. An object of that size simply would not have enough particles to allow for neither the computing power, nor the data required from such calculations. This means that by looking at the laws of physics that we are aware of, we can determine that such object is physically impossible.
So if we want to approach this paradox from the standpoint of physical reality, not just statements and logic, this paradox is in fact an impossible scenario. Pinocchio's nose is an object that simply is outside of the realm of possibility. If that is the case, even if we want to apply logic to it, we can't expect the results to make any sense as the premise we are starting with has been demonstrated to contradict reality and to not make sense itself.
This is why possibility and conceivability are not equivalent and not everything we can imagine even in an infinite universe is actually possible. The possibility and impossibility of something both require demonstration and substantiation and cannot simply be postulated. For instance, we can look at the claim that some god exists. Someone can say that it is possible for this statement to be true and I guess that is OK to say. But is it really fair to say "It is possible for a god to exist". I would say that this would be fair to say only after the possibility of a god actually existing is somehow demonstrated. Until then, the idea of a god is conceivable, but it might be just like Pinocchio's nose - something that simply couldn't really exist and is thus impossible.
Of course, while this approach to the Pinocchio Paradox might be yielding interesting topics of discussion, one could say that it is far too trivial to approach a thought experiment (which any paradox basically is) from the point of view of its possibility. After all, of course this is not possible, Pinocchio is a fictional character living in a fictional and magical world. This doesn't mean that asking the question about this impossible situation doesn't make sense or doesn't pose other interesting questions.
Objectivity and Subjectivity
Well, then we come to the question of how do you really establish what is a lie. Pinocchio's nose is supposed to grow only when he lies. But should it grow if Pinocchio unknowingly utters a false statement while believing it was true? Or when he is simply unsure? If his nose would only grow when he lies, then his nose will grow not based on the truthfulness of his statement, but based on his intentions. So the existence of the paradox suddenly hinges on the reason Pinocchio is making the statement in the first place, his internal conscious attitude toward the statement, his beliefs of its truthfulness and his goal and object of communication. This opens the door to a certain level of subjectivity and the need of somewhat conscious assessment. Or at least an impartial and supposedly objective assessment of Pinocchio's internal and possibly subjective state of consciousness.
One could argue that if we are truly in an imaginary world, we could postulate an all-knowing nose that would be aware of his intentions and the paradox will remain a paradox. While objective truthfulness of a statement might be something a supposedly all-knowing nose could determine, intention, lying and trying to deceive or mislead are not necessarily binary and can't always be expected to adhere to the boolean true vs false paradigm. In fact, Pinocchio might be making the statement because he has become aware of the paradox and might be testing his nose with the expectation that his nose would grow, but without really being convinced of the likelihood of either outcome. In this case, is he lying or saying the truth? Is he trying to mislead his nose to reveal additional information to him? Absolute objectivity when dealing with such concepts is simply not attainable.
Additionally, I think we can reasonably come to the conclusion that Pinocchio might make a false statement without an intention to deceive, so it shouldn't really be a paradox for him to say that his nose is going to grow while his nose decides to stay the same length. If he sincerely believed that the paradox was going to make his nose grow for some reason, his statement might not have been a lie. And if his statement was not a lie, his nose was not supposed to grow. So the statement could be false and not a lie at the same time and there wouldn't really be much of a paradox.
Since it's quite reasonable to expect that under different criteria to assess what is a lie and what is simply a false statement that does not constitute a lie, there would be opposing assessments of the same statement. And since there is no way to objectively establish the criteria, the whole assessment becomes quite subjective.
This only highlights further the need for clear definitions and sound concepts that correspond to reality or make sense according to certain hopefully objective criteria. It is in fact essential to distinguish between lying and being incorrect. Determining that something is true requires an assessment of facts and reality. Lying on the other hand requires for us not to just look at the facts, but to pass subjective judgment on the intention of the speaker and often actually includes a moral element.
Truth and Falsehood
But we can in fact easily do away with all this definitional subjectivity by slightly modifying Pinocchio's nose. If his nose grows only when he utters a false statement, regardless of his intentions we are finally faced with the full weight of the paradox and it becomes harder to get away from it by just examining the boundaries and definitions of the concepts it hinges on.
To finally add some history and to give some credit to the person that came up with it, we should mention that the Pinocchio Paradox is not absolutely unique - it's actually a version of the Liar Paradox where a person that always lies would say "I'm lying". It was thought up by Veronique Eldridge-Smith at the tender age of 11 when her father, the logician and philosopher Peter Eldridge-Smith, explained the Liar Paradox to her and her brother and asked them to come up with their own examples of it. The father realized it was actually an interesting proposition and an interesting modification on the original paradox and published a paper about it.
If we want to look at the Pinocchio Paradox not by criticizing its physical or conceptual limitations, but by looking at the underlying logical and mathematical consequences, we will arrive at Gödel's first incompleteness theorem. Gödel actually used a modification of the Liar Paradox to show that there are in fact things that are bound to remain unprovable and thus unknowable. My interpretation of this is that the intrinsic mathematical problem with the Pinocchio Paradox is that it requires an impossible determination. Knowing if the statement is true is impossible much like it's likely impossible that one could determine the final digit of Pi. There are just infinite regressions that cannot be solved and many other things that are both mathematically and physically unknowable.
This has a very important implication about truth in general that is easy to forget. Sometimes we not only don't know what the truth about something is, it is in fact utterly and intrinsically impossible for us to determine what the truth in that case is. And when we encounter such statements, we should accept the apparent paradox. In those cases we should accept the fact that we don't and can't know instead of inventing explanations or declaring the fact that a real determination cannot be made entails some contradiction. In fact, such paradoxes are more likely to contradict our incorrect intuitions about the universe and the laws that govern it than to contradict the laws themselves.
I hope this examination of the Pinocchio Paradox has raised some valid questions about what is possible, real, objective or true and I think examining those concepts through the lens of a paradox like that can actually help us further our understanding of what they actually are.
Image credits: [1] | [2] | [3] | [4] | [5]
