Pinocchio’s Paradox

My nose will grow

Pinocchio, who is well known for having his nose grow whenever he lies, says, “My nose will grow”. With this utterance, Pinocchio never lies again. And, presumably, his nose never grows again.

So what happens?

If his nose won’t grow, then he must have told a lie. But, if he told a lie, then his nose would grow. But if his nose grows, then he didn’t lie after all. But if he didn’t lie after all, his nose shouldn’t of grown. And so on it will go.

How does this paradox get started? It seems at least logically possible that some magic should make Pinocchio’s nose grow whenever he lies. Yet is this a false assumption? A contradiction is obviously unacceptable logically speaking. But where does the contradiction come from? The purpose of this post is to give a clear exposition of the paradox and the routes that bring about the contradiction.

I think this is a tough paradox to resolve. Part of the difficulty is that the conditions put forth on Pinocchio’s utterance and the world at large need to be fixed and, as we’ll see, this can be done in a few different ways, with different underlying assumptions that need to be made explicit. In beginning to come to terms with the conditions of the paradox, we must be careful not to conflate different conditions when proposing a source of the problems, since this can readily make the proposal appear to be invalidated for having it fail to meet a different condition than the one it is for. All possible conditions of Pinocchio’s Paradox will need a resolution for Pinocchio’s Paradox to be resolved. Truth and reason will stand tall on solid ground once more.

Lying as Necessary and Sufficient For Nose Growth

In the story, whenever Pinocchio lies, his nose grows. For the strongest version of the paradox, we will assume that Pinocchio’s lying is sufficient for nose growth. That is, there will be no exceptions: whenever Pinocchio lies, his nose must grow (if you’ve followed previous posts, we should say that in all circumstances or contexts of his lying there is a necessary connection between his lying and his nose growing). If we rather didn’t think his lying was sufficient for his nose growing, then if we stipulate that his utterance “My nose will grow” is a lie, it won’t follow that his nose has to grow. The magic sometimes fails to work, or can be thwarted. The paradox is then solved, since his utterance can remain a lie since his nose doesn’t have to grow. However, I’m interested only in a solution to the paradox that assumes that the magic always work, since this assumption at least seems logically possible (i.e. non-contradictory), yet the paradox would indicate otherwise.

Not only will we assume that Pinocchio’s lie is sufficient for nose growth, we will assume that it is necessary. That is, we will assume that, in Pinocchio’s case, the only way that his nose can grow is through his lying. Again, if we didn’t make this assumption, there would be an easy solution available that his nose first grow from a cause other than his lying, which proves his utterance to be a lie, and thus results in just more nose-growth. Again, assuming that the magic always works and that the causal laws on Pinnochio dictate that only his lies cause his nose to grow seem like logical possibilities, despite the contradictory results. So how do the contradictory results come about?

With the necessary and sufficient conditions in place, the conditions on Pinocchio are thus:

If Pinocchio tells any lie (L), then Pinocchio’s nose grows (N). Also, if Pinocchio’s nose grows (N), then Pinocchio told a lie (L). That is,
L ← → N

Now, we can gauge this condition from two directions. For one, we can ask: is there nose growth? If we determine that there is nose growth, then we also thereby determine that there was a lie. On the other hand, if we determine that there is a lie, we determine that there will be nose growth. Pinocchio’s utterance, “My nose will grow” is either a lie or it is not a lie. Which is it?

Utterances can be lies only if something is said that is false. That is, if something is said that is true, it cannot be a lie (even if you intended to lie, you lied unsuccessfully). Also, if something is uttered, but it is not false, perhaps for being mere meaningless mouth noises, then it is not a lie. Again, lies are not just untrue, but must be false. Thus, in order for him to lie, Pinocchio must utter a falsehood.

What’s a falsehood? We’ll say that falsehoods say something about how things are that is not actually the way they are. So, if what Pinocchio says is false, his nose won’t grow. Yet is that Pinocchio’s nose won’t grow the way that things are? I think answering this depends on how the future is; or, more specifically, the status of future events as either open or fixed.

A Fixed Vs Open Future

“My nose will grow”

Let’s first assume that the future is fixed. If the future is fixed and there is nose growth in the future, then Pinocchio’s utterance is true after all. However, this is a problem. Recall that nose growth comes only as the result of Pinocchio’s lies. This is fine if Pinocchio also lies in addition to the utterance we’re investigating. However, suppose Pinocchio never utters anything beyond “My nose will grow” that is a lie (this seems logically possible). Yet the source of the contradiction is revealed: lies are the only way that nose growth results. If we suppose the utterance is true, this must be because there is nose growth. Yet, without a lie, the nose growth results from something else, or perhaps from nothing at all, which directly contradicts our condition that Pinocchio’s nose growth results only from Pinocchio’s lie. Here, what is revealed is that Pinocchio’s never uttering anything beyond “My nose will grow” that is a lie is not logically consistent with the conditions we are supposing. Keeping track of the conditions, of a fixed future and that his nose growth comes only from his lies, yet his nose growth would result from something other than a lie once the nose growth makes the utterance true and not false (I think the form in this case is similar to that of the Barber’s Paradox).

If the future is fixed and there is, on the other hand, no nose growth in the future, then Pinocchio’s utterance will achieve the status of a lie. However, lies are causally efficacious for nose growth (even for nose growth in the future), and this contradicts there being no nose growth in the future.

Where does the contradiction lie exactly? I would be highly skeptical of any conclusion that says that our accounts of lies, falsehoods, or truth are somehow in jeopardy. The contradictory result has little bearing on such accounts. Rather, we simply have contradictory conditions. The way to see the contradiction is that we are assuming that there is no nose growth in the future. By this fact, Pinocchio’s utterance is a lie. Since we suppose that lies are causally and necessarily efficacious for nose growth in the future, the nose grows. Yet we assumed there is no nose growth in the future. One step further than this, if the nose grows despite what we initially supposed, then it is again for some other cause than a lie (since there is no lie in this case). Our assumption therefore contradicts a necessary causal law we’re assuming.

I think the conditions of Pinocchio’s Paradox seem logically possible because, at least from our point of view, the future is open. Let’s now assume that the future is open. That is, let’s assume that future events are not yet determined by present events. Future events are to be determined.

In the case that the future is open, then the nose could grow sometime in the future since the conditions for the nose to grow sometime in the future could be. If the future is open and there is nose growth, then Pinocchio’s utterance becomes true. If, on the other hand, there is no nose growth, then it would still be inappropriate say that there will be no nose growth, but only that there hasn’t yet been nose growth. This is because we suppose that the future is open: the conditions for nose growth could be in place sometime in the future. And why not?–The future’s forever. This sort of open future is also consistent with the fact that Pinocchio’s nose growth comes only from lies. The event that is open is whether or not Pinocchio will lie.

If this is right, then Pinocchio’s utterance is never a lie, since it cannot ever be resolved as being false. It rather has a truth condition that is as open as the future we’re supposing. It is forever in limbo for being attached to a never-ending open time, but it is not false, since it could always turn out to be that the conditions arise for the nose to grow. Another way to put this point is that the truth conditions for ‘My nose will grow’ depend on the necessary and sufficient conditions for nose growth (telling a lie), and in this case the condition yet could be, sometime later.

For completeness, suppose that the future is a hybrid between fixed and open events. Suppose that Pinocchio dies after his utterance in a way that there is no way to bring him back such that he could lie, even if while he lives he could lie in the future and thus cause his nose to grow. While Pinocchio lives, the future event of his lying is open. If Pinocchio dies, however, it becomes fixed that Pinocchio will not lie. Since his lying is the necessary and sufficient condition for his nose growth, the truth of Pinocchio’s utterance would also seem to become fixed.

With this we run into another contradictory wall, firstly concerning whether or not the future is fixed or open. We suppose that it is fixed that Pinocchio will not lie. However, this is self-contradictory in one reading of it, for if we suppose that Pinocchio will not lie, then this suffices for there to be nose growth, and thus for Pinocchio’s utterance to become a lie, against our supposition that future lying was determined in the negative.

However, on a different and more explicit reading, let’s suppose that what is fixed is rather that there is no future utterance that is lie from Pinocchio. If so, then this sort of future could be fixed, even if it turns out that Pinocchio’s past utterance can achieve the status of a lie. Let’s examine this possibility in detail.

If we say that there is no future utterance that is a lie from Pinocchio, then we must suppose that there will be no nose growth, since no utterance from Pinocchio achieves the status of a lie. Yet once we assume that, then this firstly gives Pinocchio’s utterance the status of a lie, against what we presumed; this suffices for there to be nose growth, against what we supposed; and it has the added update on the status of Pinocchio’s past utterance as being true. This in turn contradicts our assumption about the necessary and sufficient causal conditions for nose growth, since the nose grew but not for there being a lie.

There is clearly a web of contradictions present in this. What I briefly want to make clear, however, is that there is nothing contradictory to be revealed in the above case about the following fact: an utterance can fix truth conditions about the way things are in the world, and that, once fixed, the truth value of the utterance depends entirely for its truth value on how things actually stand in the world.

Rather, the contradiction is one of the causal processes involved. The utterance fixes a truth condition of the world, that then makes it causally efficacious on the world, necessarily changing the world in a way that also changes its status, contradicting what it is permitted to do causally. Again, in the case just above we are assuming that there is no nose growth in the future, since no utterance from Pinocchio achieves the status of a lie. By this fact, Pinocchio’s past utterance is a lie. Since we suppose that lies are causally and necessarily efficacious for nose growth in the future, the nose grows. Yet we assumed there is no nose growth in the future. Also, Pinocchio’s past utterance cannot be a lie, but is rather true. And again, if the nose grows despite what we initially supposed, then it is for some other cause than a lie, contradicting our causal assumption. I propose that the above conditions are to be rejected for assuming a causal structure that is contradictory. And same goes for the previously examined cases of Pinocchio’s Paradox further above where we assumed a future of fixed events.

One interesting feature of Pinocchio’s utterances is that they are endowed with causal powers that are at once intrinsic to themselves as utterances of Pinocchio, and yet also intrinsic to how things are in the rest of the world (or on whatever is fixed by the truth conditions of the utterances). That is, Pinocchio can fix an utterance how he likes and so can cause his nose to grow or not, but such an effect is also just as much caused by how the world is, once the truth conditions of the utterance are fixed. This is again because once the truth conditions are fixed, it’s up to how things stand that they are either true or false.

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2 thoughts on “Pinocchio’s Paradox

  1. I don’t think it would give him the ability to create future truths by making pronouncements. But it would definitely come in handy for picking the right horse at the race track. 🙂

    But all of this would defeat the purpose of the curse that causes his nose to grow when he tells a lie. The point is to prevent him from benefitting from lying to others. So it would be reasonable to assume that he would not be allowed benefit from this. So the curse would only apply to what he believes to be true or false right now. In other words, his nose only grows when he intentionally tells a lie.

    I suspect that at the root of all paradoxes is a subtle fraud or deception, a little trick that pulls you into a way of thinking there is a problem where there is none. In this case you’re creating a new paradox that has nothing to do with Pinocchio.

    If statement A is true then event B happens.
    If statement A is not true then event B does not happen.
    In what real life situations do those two statement hold true?
    I suspect that the only case would be when statement A is “event B will happen”.
    And that’s what we have when Pinocchio says “My nose will grow.”

    If the nose is a paranormal fortune teller, then it will grow now if in the future he will ever tell a lie.

    If the nose is simply a lie detector, then there is a small fraud in the statement “My nose will grow”. Does he mean to say “My nose will not grow right now, but my nose will grow in the future, because I sincerely believe I will lie again in the future” or does he mean to say “My nose will grow right now because I sincerely believe I will not lie in the future”?

    Thus a puzzle is constructed by unclear language that can be interpreted two different ways.

    Like

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