Sunday, October 13, 2013

The Pinocchio's paradox is not a paradox.

What would happen if Pinocchio said "My nose grows now"?
Would the nose grow or not?

According to this Wikipedia's article, it is a paradox.

But I think it is not a paradox.
When Pinocchio says "My nose grows now", he is either lying or telling the truth.
If he is lying and his nose is not growing, his nose will start growing.
If he is telling  the truth, his nose will stop growing.

Because the nose only grows AFTER  he tells a lie. If Pinocchio says "My nose grows now" and he is telling the truth, it is because of a lie he must have said before. The sentence itself "My nose grows now" doesn't make his nose grows.

But if his nose was not growing at the moment he says "My nose grows now" then it will start growing AFTER  he says it.
The sentence is a lie because it refers to that particular moment in the past when the nose was not growing.

Can the nose start to grow while he is telling the lie?

I don't think so, because there is no way to know if it is true or false if the sentence is not completed yet. Incomplete sentences cannot be assigned a value of true or false.
That is why the nose can only grow after he tells a lie, never during the sentence. The nose cannot grow while he is still in mid-sentence.

What if he says "My nose will grow as soon as I say this"?

That is an interesting question.
He is making a statement about the future. Is that statement true or false?
But it is also a statement about an instant in time.
At that precise instant, is the nose growing or not?
If it is not, then the nose will start growing.
When will the nose start growing? One minute after he tells a lie? Thirty seconds after he tells a lie? One second after he tells a lie? One millisecond after he tells a lie?

The statement is about an instant in time.
If the statement is a lie, the nose will start growing an instant LATER, not at the same instant.
Even though those 2 instants may be very close together, they are not the same instant.

But that is also a statement about the future, which I'd like to discuss when talking about the crocodile's paradox.

Let's talk about the crocodile's paradox.

Suppose a crocodile who has stolen a child, promises the father that his son will be returned if and only if he can correctly predict whether or not the crocodile will return the child.
What happens if the father correctly predicts the son will not be returned? Should the crocodile return the son or not?

Let's see the premises:

Premise 1: if the father predicts correctly then the crocodile will return the son.

Premise 2: The crocodile will not return the son.

Premise 3: The father predicts correctly.

From premise 1 and 3 we conclude that the crocodile will return the son, which contradicts premise 2, thus the paradox.
The paradox emerges because two contradictory premises cannot both be true at the same time.

Are all paradoxes the result of contradictory premises?

The paradox disappears if one of the contradictory premises is eliminated.
If we eliminate premise 3, we conclude that that father will fail to predict correctly, by saying  the son will be returned.
If we eliminate premise 2, we conclude that the father will correctly predict the son will returned.

Can all paradoxes be solved by just eliminating the contradictory premise?

There is also another aspect: the statement about the future.
Is it possible to make a statement about the future that is absolutely true (or false)?
In the crocodile paradox a statement like "The crocodile will not return the son" assumes an immutable and unchangeable future. Once we assume that the crocodile will not return the son, there is nothing we can do to change that.

But that is not how we talk about the future.
Predictions usually assume the form: "a certain event will happen if certain conditions are met".

For example if the doctor predicts that "the patient will die", is the doctor telling the truth or lying? Actually he is neither.
He is talking about a POSSIBLE future that may happen if certain conditions are met.

Is it possible to predict the future?
To predict the future we have to assume that the future is immutable and predictable.
But if the future is not immutable, if the future is changeable, then it becomes unpredictable.

We assume the future is changeable, because we are trying to change the future all the time.
When we try to predict the future, we are talking about a future that may happen IF WE DO NOTHING. So the question becomes: what should we do to change the future?

All statements about the future need to be conditional.
It seems impossible to make unconditional predictions.
How can anyone know for certain what is going to happen?
A statement like "the patient will die" can neither be true or false. It needs to be rephrased to "the patient will die IF the diagnosis is correct and there is no available treatment".

Is it possible to solve the time travel paradox?
Is it possible to go back in time and prevent your own parents from meeting, thus preventing your own birth? If you were never born, where did you come from?
If a paradox can be solved by just removing the contradictory premise, which premise should be eliminated?

Is the past unchangeable?
If time travel is possible then the past must be changeable because the future is changeable and when we travel back in time THE PAST BECOMES THE FUTURE.

How to solve the liar's paradox?

If someone says "I am lying", is he really lying or telling the truth?

We have to know where the contradiction is.
For any logical argument, we have a set of premises and then draw a conclusion from the premises.
All the premises are assumed to be true and the conclusion obtained from the rules of logic, must also be true.
For any logical argument, all statements need to be true, or at least assumed to be true. It is impossible to deal with a false statement.

Let's have the following premise:
Premise: this sentence is false.

We have to assume that all premises are true. But in this case, the premise declares itself to be false. Since it cannot be both, and it is impossible to deal with a false statement, we have to invalidate the sentence.
In other words the sentence is not a truth-bearer.

That is the same conclusion shown in the link below.
"We get out of this bind by refusing to accord it the privileges of statementhood. In other words, it is not a statement. "

There are many statements which cannot be assigned a value of true or false.
For example, incomplete statements are also not truth-bearers.
Statements about the future are also not truth-bearers.

Logic and mathematics can only deal with truth-bearers.
But since not all statements are truth-bearers, it begs the question: is it possible to construct mathematical models that deal with non-truth-bearers? Sentences that are neither true or false?
Because there is something called three-valued logic.
Such a logic should be able to deal with the liar's sentence, without having a paradox.

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