[Request] Is it possible to answer correctly?

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image showing [Request] Is it possible to answer correctly?

Djorgal on July 11st, 2020 at 16:09 UTC »

No it is not possible to provide a correct answer to that question as the question itself is incoherent.

This is a variant of the self-reference paradox. Other famous variants would be:

"This statement is false"

"The barber is the one who shaves all those, and those only, who do not shave themselves. Who shaves the barber?"

"The judge convicts a prisoner to be hanged at noon on one weekday in the following week such that the day of the execution will be a surprise to the prisoner."

In every case, the rules stated are making references to themselves.

Edit4: I know the edits are not in order, but I think this one is the most relevant to the question. I get far too many answers telling me that the Who Wants to Be a Millionaire question is not a paradox and that there is a definitive answer and that it is [A/B/C]. No one picks D. So let's show that this is indeed a paradox.

Let's assume that A is correct. That means the likelihood to get the correct answer if you picked at random is 25%. Since that percentage is what is being asked, then A and D are correct answers. Two correct answers out of four possibilities. Thus, you are 50% to pick a correct answer if you choose at random. This contradicts A.

When we assumed that A was correct, it led us to a contradiction. This is a proof by contradiction that A is not correct. There is no circular reasoning. This proof ended here and it showed that A is not correct.

(Some people might say that there is a problem in the previous proof when I say that A and D are correct answers because the rules of WWM do not allow that, but it just means we've reached a contradiction one step sooner. The fact that we have two answers contradicts the rules already.)

Now let's do another, independent, proof and this time we will assume that A is incorrect. That means that the likelihood is not 25%. So, what can it be?

0%? That would mean B is the only correct answer. You would be 25% likely to pick it at random. It contradicts our assumption.

50%? That would mean C is the only correct answer. You would be 25% likely to pick it at random. It contradicts our assumption.

x% where x is something different from 0, 25 or 50? Then this does not correspond to any of the answers. So you would be 0% likely to pick the correct answer at random. Which contradicts the fact that x is not 0.

No matter what, if you assume that A is incorrect, you reach a contradiction. It proves that A must be correct.

You can do the same for all other possible answer. For every single answer A, B, C and D you can prove that they are correct and you can prove that they are incorrect. So stop telling me that the answer is B, because it may be true, but it's also false.

Edit: It seems the judge and the prisoner's example requires a bit of explaining. Especially since in some cases it may be possible to still respect all the rules stated.

The paradox comes from the reasoning that the surprise hanging can't be on Sunday, as if the prisoner hasn't been hanged by Saturday, there is only one day left - and so it won't be a surprise if he's hanged on Sunday.

But then, the surprise hanging cannot be on Saturday either, because Sunday has already been eliminated and if he hasn't been hanged by Friday evening, the hanging must occur the next day, making a Saturday hanging not a surprise either. By similar reasoning, the hanging cannot occur on any other day of the week.

The next week, the executioner knocks on the prisoner's door at noon on Tuesday to the prisoner's greatest surprise.

So it turns out that the rules ended up being respected. He was indeed executed the next week and the day of the execution was indeed a surprise. The problem with self-referential rules isn't that they are wrong, it is that they are incoherent. In other words, the rules are bullshit.

Let's say that I guess a random day and tell you that you are going to die that day. When I do that, I am full of shit, there is no way I could possibly know and I did guess at random. But I might nonetheless happen to be correct by coincidence. Just because I am full of shit doesn't mean I am necessarily incorrect.

The same goes for incoherent rules. They might still happen to be respected, but that would be coincidently so, not as a result of the rules themselves. In the case of the prisoner, he was not surprised on Tuesday because of a necessity caused by the rules. He was surprised because he was rather oblivious to the Judge's tricks. For a paranoid prisoner who always believe he's going to be executed as early as possible, no matter the rules, his execution cannot take him by surprise.

Edit2: To the smart asses thinking that "the barber is a woman", it doesn't solve the paradox the way I wrote it.

Edit3: No, a second barber doesn't solve the problem either. My statement was a definition of what a barber is. If there is a second barber who shave the first one. Does that mean the first barber doesn't shave himself? But, he is a barber, he must shave all those who don't shave themselves, thus he must shave himself.

There is actually a solution to the barber paradox as I have written it: "The town is empty, no one in it, no barber, nothing." If there is no one to whom the rules apply, they don't lead to a contradiction.

If we add as a second axiom that "there is a barber". That ought to do the trick.

acoollobster on July 11st, 2020 at 16:55 UTC »

you have two 25% so the change of getting one of those is 50%, making it incorrect, 25% chance of getting the 0, so its incorrect if you get it too, 50% has a 25% chance too so its incorrect, its in fact impossible to get the correct answer

bbman225 on July 11st, 2020 at 17:40 UTC »

No, it is not. We can prove this by contradiction. If we suppose B to be the correct answer, we find that we would have a 25% chance, which contradicts our assumption that the probability would be 0%. If we suppose C to be correct, we again find a 25% chance, contradicting our assumption of 50% again. If we suppose either A or D is correct, then they both must be correct, so we find the probability to be 50% which again contradicts our 25%.

Since we've proven none of the answers can be true by themselves, we can conclude there is no correct answer to the question.

Edit: Well it might seem that because I end up with 0 correct answers that would make B the correct answer, this is not true because if B was correct, the probability must be at least 25%, which contradicts B being correct.

Edit 2: Okay, some of the replies have me doubting this explanation. They point out that when we eliminate all the other possible answers, the probability becomes 0%, making B correct. I believe what's really happening here is a paradox where B can't be resolved to either true or false. So, to say all of the answers are incorrect is wrong.