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# What would happen if Pinocchio said, "My nose will now grow"?

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after 9 votes the winner is...
socialpinko
 Voting Style: Open Point System: 7 Point Started: 11/2/2011 Category: Philosophy Updated: 6 years ago Status: Post Voting Period Viewed: 25,577 times Debate No: 19094
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36 comments have been posted on this debate. Showing 1 through 10 records.
Posted by Wallstreetatheist 6 years ago
Grape is going to solve this problem, if it's the last f*cking thing he does.
Posted by OberHerr 6 years ago
STOP!! My head HURTS! O.O
Posted by Grape 6 years ago
Upon further reflection, I have changed my mind. Maninorange is correct. I will elaborate tomorrow or Wednesday on why I think so.
Posted by Grape 6 years ago
I'm going over the proofs that you've presented and it's getting very confusing. You're using a sort of implied disjunction elimination rule, but that rule is only is only valid if you keep your assumptions the same and you keep varying them.
Posted by Grape 6 years ago
@ maninorange -

The move from 4) and 6) to 7) in my proof is completely valid. Formally, it is called the rule of negation elimination:

"A sentence A may occur in a proof psi if among the earlier items of psi are both a sentence and its negation." (I stole this definition from my professor, who stole it from his professor. It's the circle of life, or something...)

Here is another way to look at it:

Consider the sentence A --> B (if A, then B). This sentence is true when B is true or A is false (that's the definition of if...then in formal logic). In other words, it just tells us that you can't have A without B. If A is a contradiction, then A --> B is true regardless of the truth value of B because A will always be false and that is sufficient to make the whole thing true. The conjunction of a sentence and its negation is always false (C and not C, for instance). So if we have C in the proof and not C in the proof, we can conjoin them into (C and not C). Then we can infer (C and not C) --> B for any sentence B. because it will always be true. Then we derive B from (C and not C) and (C and not C) --> B by modus ponens.

Here is an example proof.

Proof B from, A, A --> C and A --> not C

1) A (assume)
2) A --> C (assume)
3) A --> not C (assume)
4) C (1, 2, modus ponens)
5) not C (1, 3, modus ponens)
6) (C and not C) (4, 5, conjunction introduction)
7) (C and not C) --> B (6, contradiction implies anything)
8) B (6, 7, modus ponens)

A, B, and C can be any sentences whatsoever.

"As opposed to saying that Pinocchio's nose would not grow, you could just as well say that Godzilla ate the Tooth Fairy." - That's right. That conclusion is at the very core of what it means for logic to be truth functional. I simplified this into one step (going straight from C and not C on separates lines in the main proof to B on a new line) because I did not except it to be controversial. You really lose a lot in terms of what you can prove in a logical syste
Posted by maninorange 6 years ago
You have made the same mistake in the same way three times now. I'll try a different tack.

1) If Pinocchio lies, his nose grows.
=> If Pinocchio's nose does not grow, Pinocchio did not lie.
2) Pinocchio states "My nose will now grow."
3) Assume Pinocchio's nose does not grow.
4) By (1), Pinocchio did not lie.
5) By (2) and (3), Pinocchio did lie.

(4) and (5) are contradictory.

1) If Pinocchio lies, his nose grows.
=> If Pinocchio's nose does not grow, Pinocchio did not lie.
2) Pinocchio states "My nose will now grow."
3) Assume Pinocchio's nose grows.
4) NONE of our assumptions allow us to assume anything further with this information. Yes, he told the truth. However, as stated, this does not disallow his nose from growing. There is no contradiction in saying that Pinocchio's nose would grow.

I conclude that since the assumption that Pinocchio's nose would NOT grow leads to a contradiction and that since the assumption that Pinocchio's nose WOULD grow does not, Pinocchio's nose must grow.

YES there is a paradox, just as there is with all liar's paradoxes. However, for Pinocchio's statement, there is no such paradox, as I have demonstrated.
Note: I state again that I know that paradoxes are not lies. However, a nose growth for a paradox is the only non-contradictory way of handling the situation. Your reasoning to go from 4 and 6 to 7 is ludicrous. If this actually worked, then it would work for all assumptions. As opposed to saying that Pinocchio's nose would not grow, you could just as well say that Godzilla ate the Tooth Fairy. Pinocchio lied and did not lie. Translation: Pinocchio presented a paradox.
Posted by Grape 6 years ago
Let me show the proof of that Pinocchio's nose will grow and not grow under the assumption that he is lying more clearly:

1) If Pinocchio tells a lie, then his nose will then grow. (assumption)
2) A person tells a lie if and only if they say a proposition, and that proposition is false. (definition)
3) Pinocchio says, "My nose will now grow." (assumption)
4) Pinocchio lies. (assumption)
5) Pinocchio's nose grows. (from 1, 4)
6) Pinocchio does not lie (from 2, 3, 5)
7) Pinocchio's nose does not grow (from 4, 6)
8) Pinocchio's nose grows and Pinocchio's nose does not grow (from 5, 7)

Can we at agree that I did not use the premise, "If Pinocchio does not lie, then his nose does not grow"? I use the statement "Pinocchio does not lie" to get to "his nose does not grow" BUT I do not use the premise ""if Pinocchio does not lie, then his nose does not grow" to do so because there is another way.

I get 5) by modus ponens. You also have that in your proof.

I get to 6) because, per 2), someone lies if and only if they say something and it is false. The "if and only if" means that "x lies" and "x says y and y is false" must agree in truth value. Pinocchio says his nose grows and Pinocchio's nose grows, then "Pinocchio says his nose will grow and his nose does not grow" is false (because the latter conjunct is false), so it is also false that Pinocchio lies.

I now have 4) "Pinocchio lies" and 6) "Pinocchio does not lie." That is a contradiction, so I can derive anything I want. I choose to derive 7) "Pinocchio's nose does not grow." I could have derived anything. The premise, "If Pinocchio's does not lie, then his nose will not grow" was never involved. I did not have to contradict those two statements, I could have contradicted anything. It is also clearly possible contradict 2) or 3).

And obviously I get 8) by conjunction introduction from 5) and 7).
Posted by Grape 6 years ago
@ maninorange

(In response to my statement "4) Pinocchio's statement was not a lie (he said his nose would grow, and it did)")

"This is irrelevant to whether or not Pinocchio's nose would grow as there is no rule given overning his no requiring that it not grow when he tells the truth. This is not carrying through to the logical conclusion; this is simply the result of seeing his nose as a truth detector as well as a lie detector. Paradoxes are never addressed, and, therefore, the conclusion, as I have shown, must be that his nose would grow. Rules are only broken when we assume the nose would not grow."

I never said that Pinocchio's nose must not grow if he tells the true. I am aware the Pinocchio's nose is not a truth detector. That is not at all what this sentence says. Just to be total clear, we have:

"If Pinocchio tells a lie, his nose will grow."

From which we can get:

"If Pinocchio does not grow, he did not tell a lie."

by the contrapositive, but we cannot get:

'If Pinocchio's does not lie, then his nose will not grow."

Because there are cases in which that is true but the original statement is false.

I am not arguing for that. I am specifically trying to show you that there is a way to derive a contradiction without that premise. All that 4) says is that if Pinocchio said something would happen and it did, so he was not lying. That just follows tautologically from the definition of lying.

By itself, that is indeed not enough to conclude that Pinocchio's nose would not grow. I got 5) from 1) and 4), not 4) by itself.

"However, this [the paradox] is only because Pro specifically stated that Pinocchio's nose would not grow if he told the truth. We are not dealing with the same thing here. Don't let yourself be confused."

No, we are dealing with the same thing. I explicitly telling you that we do not need that premise to show that his nose should grow and not grow. There is another way to derive it, which I just demonstrate
Posted by maninorange 6 years ago
@Grape:

You continue from my first scenario:

"4) Pinocchio's statement was not a lie (he said his nose would grow, and it did)"

This is irrelevant to whether or not Pinocchio's nose would grow as there is no rule given overning his no requiring that it not grow when he tells the truth. This is not carrying through to the logical conclusion; this is simply the result of seeing his nose as a truth detector as well as a lie detector. Paradoxes are never addressed, and, therefore, the conclusion, as I have shown, must be that his nose would grow. Rules are only broken when we assume the nose would not grow.

"Which is why it's a paradox: his nose should grow and not grow."

I understand what the actual paradox is. "This is a lie." This is irreconcilable as a true or false statement. Based on Pro's premises, it is also irreconcilable. However, this is only because Pro specifically stated that Pinocchio's nose would not grow if he told the truth. We are not dealing with the same thing here. Don't let yourself be confused.
Posted by Grape 6 years ago
maninorange:

You have not taken your proofs through to their logical conclusions. Let's start with the first one:

1) Assume Pinocchio's statement is a lie.
2) Pinocchio's nose would grow.
3) As there are no other rules to prevent the growing of the nose given that he told the truth, this is where it ends.

But it does not end there. From 2, we can deduce:

4) Pinocchio's statement was not a lie (he said his nose would grow, and it did)
5) Pinocchio's nose would not not grow (from 1 and 4, because a contradiction implies anything)

So we get

6) Pinocchio's nose will grow and Pinocchio's nose will not grow (from 2 and 5)

Which is why it's a paradox: his nose should grow and not grow.

Looking at the second proof:

1) Assume Pinnochio's statement is true.
2) Pinocchio's nose has no reason to grow, so we can assume it does not.
3) Pinocchio's statement is then a lie.
4) Punocchio's nose would then grow.
5) As in the above case, there is reason to continue.

This contains two contradictions already, which are somewhat hidden in the word usage. 2) says that Pinocchio's nose would not grow and 4) says that it would, so we already have the paradox without having to do any real work. We can just derive

6) Pinocchio's nose will grow and Pinocchio's nose will not grow (from 2 and 4)

Which is the same paradoxical conclusion that we reached before.
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