The Pinocchio Paradox
The Paradox
Premise 1. Pinocchio's nose grows if and only if he claims any falsehood
Premise 2. Pinocchio claims "my nose grows now"
If his claim is true then his nose will not grow as it only grows if he claims any falsehood – but his claim is that his nose will grow and so if true his nose will grow. This is a contradiction. If his claim is false then his nose will grow as it grows if he claims any falsehood; but his claim is that his nose will grow and so if false his nose will not grow. This is a contradiction.
The Solution
If Pinocchio's nose grows, G, if and only if he claims any falsehood then:
?x: G ? C(x) ? ¬x
If Pinocchio's claim, x, is "my nose grows now" then the application of the above is:
G ? C(G) ? ¬G
Pinocchio's nose grows if and only if his nose doesn't grow (and if he claims that his nose will grow). This is a contradiction. This then shows that there must be either at least one falsehood that if claimed will not cause his nose to grow or at least one truth that if claimed will cause his nose to grow:
?x: (C(x) ? ¬x ? ¬G) ? (C(x) ? x ? G)
The first premise is necessarily false:
?x: ¬(G ? C(x) ? ¬x)
Premise 1. Pinocchio's nose grows if and only if he claims any falsehood
Premise 2. Pinocchio claims "my nose grows now"
If his claim is true then his nose will not grow as it only grows if he claims any falsehood – but his claim is that his nose will grow and so if true his nose will grow. This is a contradiction. If his claim is false then his nose will grow as it grows if he claims any falsehood; but his claim is that his nose will grow and so if false his nose will not grow. This is a contradiction.
The Solution
If Pinocchio's nose grows, G, if and only if he claims any falsehood then:
?x: G ? C(x) ? ¬x
If Pinocchio's claim, x, is "my nose grows now" then the application of the above is:
G ? C(G) ? ¬G
Pinocchio's nose grows if and only if his nose doesn't grow (and if he claims that his nose will grow). This is a contradiction. This then shows that there must be either at least one falsehood that if claimed will not cause his nose to grow or at least one truth that if claimed will cause his nose to grow:
?x: (C(x) ? ¬x ? ¬G) ? (C(x) ? x ? G)
The first premise is necessarily false:
?x: ¬(G ? C(x) ? ¬x)
Comments (31)
There is a contradiction here.
Quoting Sir2uOnly superficially, I think. Pinocchio's claim is "my nose grows now." It is only as a linguistic convention that @Michael is using the future tense to discuss that claim from a perspective outside the event of its utterance. A bit infelicitous perhaps, but nothing that cannot be solved with a dash of charity.
When is "now"? And how soon after Pinocchio tells a lie does his nose grow? Does looking at this on a timeline dissolve the paradox?
"Now" is a place in time though. "My nose grows now" is not self-referential like the liar paradox.
I see it is like this: utterance(now)-growth/not growth(not now). It only seems like he's telling the truth because of the outcome. If the rules of Pinocchio's world determined that it would take a minute after he lies for his nose to grow, it wouldn't seem this way. It is causal, but not because Pinocchio was telling the truth, but because he was lying. It seems like nothing more than a party trick.
I am not sure exactly what you mean here. If you say "now" is the cause of the growth then it is obviously false because he has not said anything before this to cause it to happen. So his nose will grow for lying.
Would there be any difference if instead of using "claims" one used states or says?
Premise 1. Pinocchio's nose grows if and only if he states any falsehood
Premise 2. Pinocchio states "my nose grows now"
Premise 1. Pinocchio's nose grows if and only if he says any falsehood
Premise 2. Pinocchio says "my nose grows now"
This won't do. It is declared a contradiction only because a paradox arises. When you then explain the paradox as arising because there is a contradiction taken to be true, you have rather gone in a circle.
There is good reason to suppose that there is no infallible truth detector possible. But suppose there were, and Pinocchio's nose was such, then in such a world, there would I suggest be no paradox. And it is precisely because of the time element, that you have tried to eliminate. Causes must precede effects, and so there will be a feedback loop created such that the nose will begin to grow, and almost immediately stop, and almost immediately start again, round and round.
It is only in the timeless world of logic that such feedback becomes a contradiction demanding both states at once.
It's even worse than that. The timeless world of logic, by definition, cannot be either the state of Pinocchio lying nor his nose growing. Both those events are states of existence. Each is a moment of time. The argument is trying to talk about Pinocchio does when that's exactly what it says nothing about. Here there is not even a contradiction in the world of logic; the argument is entirely incoherent.
If we pay attention to states the world, what happens is obvious when Pinocchio tells the lie.
Pinocchio (nose not growing) claims "My nose grows now." A falsehood.
Then a moment later, since he lied about his nose growing at the previous point in time, his nose grows.
It's declared a contradiction because where x is G the given rule ?x: G ? C(x) ? ¬x becomes G ? C(G) ? ¬G. There's no paradox here. It's just a contradiction.
Premise 1 is necessarily false: ?x: ¬(G ? C(x) ? ¬x)
We don't need to talk about Pinocchio or time or tense to discuss the Pinocchio paradox. The logic of the rule which governs the growing of Pinocchio's nose can be examined on its own terms.
I understand the language implies duration or a particular moment, but the point of the paradox is not to fixate on the time interval between the lie and the growth, but merely to point out the causal connection between a self-referential statement and the outcome.
Quoting Sir2u
The word "now" was only meant to indicate that the statement is an indexical and that the cause of lying (i.e., Pinocchio's nose growing) is tied to the utterance (i.e., self-referential). The word is indexical insofar as the truth condition of the statement is dependent on the statement itself, but not the time of the utterance.
Quoting Michael
There is a fictional world of Pinocchio whereby his nose grows if and only if he claims any falsehood. In replying that it is impossible by virtue of a contradiction is to import metaphysical baggage into the Pinocchio world. What justification is there for the import of that baggage? Are "true contradictions" impossible? If so, why? The Pinocchio paradox does not shed light on how to resolve the contradiction.
The dialetheists would contend that the contradiction will be simultaneous, not oscillating. Our cognitive faculties want to resolve the paradox by collapsing it into a definite state (i.e., by oscillating truth values), but there doesn't appear to be a reasonable justification for that collapse. Pinocchio's nose will grow and not grow simultaneously.
If helpful, it can be likened by analogy to Schrödinger's cat.
To the extent that fictional worlds can contain logical impossibilities, perhaps. But then I might as well say that there is a fictional world where there are four-sided triangles or married bachelors.
It has nothing to do with metaphysics. It just points out that it is logically impossible for the rule G ? C(x) to hold for any x.
Why do we need to resolve the contradiction? Do we need to resolve the contradictions "an X is a four-sided triangle" and "I am a married bachelor"? No. We just point out that they're contradictions and so necessarily false. The same with the rule "Pinocchio's nose grows if and only if he claims any falsehood".
Then let's keep it simple and avoid any peculiarities of quantum mechanics; the rule describes what happens after observation/collapse in a single world and what not. Or, as we're discussing a fictional world, we might as well say that this world behaves according to classical mechanics rather than quantum mechanics.
Other contradictions (e.g., square circle) are semantic contradictions. We can dismiss them as a peculiarity of language. Redefine the terms and move on. The Pinocchio paradox posits a "true contradiction" insofar as it has a consequence of a simultaneous contradictory state of affairs.
Quoting Michael
The metaphysics is introduced when the contradiction is recognized. It is logically impossible for the rule to hold without contradiction, but what does that mean for poor old Pinocchio? Does logically impossible entail ontological impossibility? How does logical impossibility impose itself on ontological possibility?
Isn't G ? ¬G a semantic contradiction?
How can "X exist" be true if "X" is logically-impossible? It's less a case of logical impossibility imposing itself on ontological impossibility and more a case of ontological claims being false if they're illogical.
Indeed. But where is x G? First, x wasn't G and everything was fine, and then becomes G.
Quoting Michael
We need to resolve it because we want to use expressions of the form ?x: G ? C(x) ? ¬x all the time.
Where the claim refers to the consequence of the rule which governs the growth of his nose.
That we want to is irrelevant. It's necessarily false. Where x is G a contradiction arises. Therefore the rule fails where x is G.
Consider the simplification ?x: G ? ¬x. Let x be any activity and G be me jumping. So, for any activity I will jump if and only if I don't do this activity. But what if x is me jumping? I jump if and only if I don't jump. The rule can't be followed. There is an x, G, where ¬(G ? ¬x).
There are some that want to reduce it to a semantic contradiction. The dialetheists do not, and it's not clear to me where they've gone wrong, if at all.
Quoting Michael
Because "true contradictions" can exist. Logical impossibility is distinct from ontological impossibility.
You are importing the metaphysical baggage without justification again. Why can't the rule be followed? Because it results in a logical contradiction? Why does that preclude the ontological possibility of following the rule?
[quote=Eldridge-Smith]The Pinocchio paradox is, in a way, a counter–example to solutions to the Liar that would exclude semantic predicates from an object–language, because "is growing" is not a semantic predicate.[/quote]
It was the follow up article Pinocchio against the dialetheists that addressed dialetheism – and as a critic rather than an advocate:
[quote=Eldridge-Smith]If it is a true contradiction that Pinocchio's nose grows and does not grow, then such a world is metaphysically impossible, not merely semantically impossible.[/quote]
But I'm interested in the paradox as a variation of the Liar paradox. I want to show that Eldridge-Smith is mistaken in saying that "there could be a logically possible world in which Pinocchio's nose grows if and only if he is saying something not true".
The paradox as initially presented by Eldridge-Smith anticipates the response he uses against dialetheism, in particular,
Eldridge-Smith used the paradox to allegedly undermine dialetheism, but he doesn't address the issues I raised here (i.e., the justification for claiming logical impossibility entails ontological impossibility).
I don't know much about his attack of dialetheism, but I assume he's not saying that logical impossibility entails ontological impossibility; I assume he's saying that it's ontologically impossible for someone's nose to both grow and not grow, and so the dialetheist cannot resolve the Pinocchio paradox by saying that Pinocchio's claim "my nose grows now" is both true and false (which is how the dialetheist resolves the traditional liar paradox).
This is where the justification is missing. The leap from logical impossibility (e.g., that the Pinocchio paradox demonstrates a logical contradiction in causing someone's nose to both grow and not grow) to the ontological impossibility of a world with simultaneous contradictory truth-values.
The liar paradox can be resolved without dialetheism because the contradiction can be resolved by excluding semantic predicates, however
1. Pinocchio's nose grows within ten seconds of completing an utterance if and only if he claims any falsehood as part of that utterance.
2. Pinocchio claims, "my nose will grow within ten seconds of completing this utterance."
Quoting unenlightenedI don't think this is correct. That liar sentences (and variations on them) contain contradictions is a diagnosis. In that sense, it's true that such sentences are declared contradictions only because an apparent paradox arises: if they didn't present any sort of problem, there would be nothing to diagnose and resolve. But plenty of paradoxes are resolved without any declarations of contradiction, and plenty of sentences are declared contradictions even when no paradox arises. It's just that in this case, the problem is—according to this solution—that we have a sentence that is a contradiction without appearing as such. Only on analysis can we see what the sentence is really saying. So again, it is true that the diagnosis says the paradox only arises in this case because someone tries to understand a necessarily false sentence as true, but that's not circular. It's just one diagnosis: liar sentences are false because they are contradictions, but they give rise to an apparent paradox because some people are fooled by their syntax and don't realize that they are contradictions (and thus think that anyone who says they are false is also committed to saying they are true, which appears paradoxical).
Quoting SoylentThis isn't a point that you will see defended very often because it is generally thought that logical possibility is—by definition—a stricter criterion than metaphysical possibility, which is a stricter criterion than physical possibility. Therefore, that something is physically impossible does not entail that it is metaphysically impossible, and that it is metaphysically impossible does not entail that it is logically impossible. But anything that is logically impossible is metaphysically impossible, and anything that is metaphysically impossible is physically impossible. One can take issue with this, of course, but many would argue that this is just how the terms are used.
My own view is that this point cannot be used against the dialetheist, who after all agrees with the Priorian (contra all other interested parties) that liar sentences are disguised contradictions. The difference between them is just how to respond to this fact. The dialetheist says there are true contradictions, while the Priorian says contradictions are necessarily false so there never was a real paradox to begin with (liar sentences are just unproblematically false sentences that people incorrectly take to be problematically false because they have misanalyzed them). But nothing prevents the dialetheist from claiming that true contradictions go "all the way to the top," so to speak. That is, the dialetheist is not restricted to saying that true contradictions are only physically or metaphysically possible. And if that's right, then Eldridge-Smith should have stuck to the original line that the Pinocchio paradox showed why the Priorian tradition is better than other classical logic solutions and found a different way of going after dialetheism.
(Indeed, he could even rest on the argument that just having a successful classical logic solution is enough insofar as the only real motivation for dialetheism is the lack of such a solution. Such an argument only goes so far, of course, but it is at least dialectically useful. A committed dialetheist can remain a dialetheist but is left without reasons why anyone else should be a dialetheist if they are not already attracted to the view.)
He could just simply be mistaken and not actually lying.
There's another angle that can be taken on this, that is of interest to mathematicians, although perhaps to nobody else. That is that Premise 2 is a statement about the velocity of the end of the nose. Velocity is the derivative of position with respect to time.
We can generalise the notion of ordinary derivative to define a left and a right derivative at a point (in time, say). The former is the rate of change (of position of the end of the nose) in the infinitesimally short period immediately before 'now', and the latter is the rate in the infinitesimally short period immediately after 'now' (I know that sounds very woolly, but it can be made precise if one has access to mathematical symbols).
The ordinary derivative only exists if the left and right derivative both exist and equal one another. In the real world, that always happens, because Newton's laws say nothing can instantaneously change its velocity.
But in this magical Pinocchio world we are considering, Newton's laws might not apply. So it is possible that the nose is growing at a constant rate up to time t='now', but stationary after time t. In that case the nose has no velocity at time t.
So the statement 'my nose is growing now' can be seen as ambiguous, as it can refer to either the left or right derivative of the nose-end's position.
Or it can be seen as meaningless, as the nose-end will have no velocity 'now' if it suddenly stops growing at time t.
If we take it to be ambiguous then when Pinocchio makes the statement, the nose will instantaneously stop growing if it interprets the reference as being to the left derivative, and there will be no contradiction.
But if it interprets it as a reference to the right derivative, it will suffer the same contradiction that occurs in the simpler analysis. However, it's easier to see there why it's a contradiction, because in that case it is a statement about the future. It's equivalent to saying 'my nose will now stop growing if it does not now stop growing', which is as bare a contradiction as 'I am Pinocchio and I am not Pinocchio'.