Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct?
A) 25%
B) 50%
C) 60%
D) 25%If they would accept both correct answers, then the answer would be 50%, but then if it's 50% those two correct answers would be incorrect.
The question displays a lack of understanding of language, especially the relation between a word and its meaning. More specifically, the relation between a noun and its pronoun.
Suppose Question 1 was:
> "What day is it today?"
And suppose Question 2 was:
> "Which answer in this list is the correct answer to this question? > Friday, Saturday, Sunday"
If someone asks, 'What question is the pronoun "this" in Question 2 referring to?'
Then you can reply - 'Question 1'
And then they can proceed to answer Question 2 as 'Saturday'
Suppose I just ask you out of the blue : 'what's his height?'
You will immediately respond : 'who are you talking about?'
A pronoun must come AFTER a noun has been established. If it comes before, the speaker must clarify the noun after.
In other words, you must be able to do a Ctrl-H Find & Replace in the original sentence, and change "this" to whatever it's referring to, and the new sentence should still make sense. That is the point of pronouns.
If I ask 'what is the weather in this city?'
Does that question make sense without mentioning the city's name either before or after? It might make sense grammatically, but practically, it is not answerable.
If you now come back and say - Just replace the pronoun "this" with the text of the question, then it becomes :
"Which answer in this list is the correct answer to "Which answer in this list is the correct answer to this question" question?
Is the question answerable now ? Still NO. Because there is still one unresolved "this". Ad-infinitum.
So, to answer your question:
> 'which question ?'
The question, as is, is not answerable. Because it does not make sense until you resolve what 'this' refers to. Until then, it is just a bunch of words without a corresponding meaning. It's not a paradox or a contradiction. Barber's 'paradox', 'This sentence is false' etc. all are basically just a poor understanding of language/pronouns.
You might as well ask :
> 'Which answer in this list is the correct answer to oogabooga question?'
>>If I ask 'what is the weather in this city?' >>Does that question make sense without mentioning the city's name either before or after?
Yes, it does, if we're standing in a city when the question is asked. "This" would obviously refer to the city we're in. Likewise, in this instance, "this" obviously refers to the question that is currently being asked.
Of course it does. It's like pointing to something and asking 'what's this?'. Of course the act of pointing, or the act of standing in a city resolves 'this' to something else.
But the point is where 'this' resolves to something which has meaning.
A question makes sense, if it is understandable and answerable.
Q1: //TODO
Q2: what's the population of this city ?
Q3: what's the location of this city ?
...
Q22: what's the altitude of this city?
Q23: what's the weather in this city ?
Suppose I ask you "What is the answer to Question 23?".
You'll respond "what is Question 23?" (trying to resolve/concrete 'Q 23').
I'll reply "Question 23 is - 'What's the weather in this city?' ".
You'll respond "well, what does 'this' refer to in Question 23?" (trying to resolve/concrete pronoun 'this')
I'll reply "this refers to the city in Question 22"
You'll respond "well, what is Question 22?' (again trying to resolve/concrete 'Q 23')
I'll reply "Question 22 is - 'What's the weather in this city?' "
You'll respond "well, what does 'this' refer to in Question 22?" (again trying to resolve/concrete pronoun 'this')
So-on, so-forth.. until we reach Question 2, referring to Question 1.
Now, If I define question 1 to be - 'What is the capital of USA?'. Then, by chaining, you can answer Question 23.
But if it do not define question 1, you can't answer Question 23, because I have not given you enough information to resolve pronoun 'this' in 'this city'.
That is the case with OP's question. In order to answer a question, the question must be answerable. In order for a question to be answerable, it must be understandable. In order for a question to be understandable, all pronouns in it must be resolvable to a valid question.
Do you agree that the question 'what is his height?' - is unanswerable IF the asker is not pointing to anyone, or there is no one else in the room, or there is no prior context - because the pronoun 'this' is unresolved ?
And in the OP's question, it's obvious that "this" resolves to the current question that it appears in. It seems like you want to say that this is not obviously the case simply because the question is unanswerable. Yet I'm not convinced that you didn't invent this requirement yourself. Anyone reading the OP's question would clearly understand to what the "this" is referring.
Basically, OP has not provided a base case for his recursive question. As a result, it goes on infinitely, ending up in stackoverflow.
When something is self-referential and resolves to itself, that's the end of the recursive process. There's no need to produce a Q2 that is Q layered inside itself. You go and substitute the entire sentence in place of "this" and form some unresolveable chain of recursion but you do it for no apparent reason. If these were statements in a programming language your expansion might make sense, but not in the context of spoken english language.
You're getting stuck because you're used to only 1-level resolution/replacements for pronouns, because that's what we're used to in daily speak.
But I've given a concrete 2-level example, where you will question your own stance of "that's the end of the recursive process", and help you expand the logic to 3-levels deep, 4-levels, 5... to infinity.
Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct?
A) 20%
B) 40%
C) 60%
D) 20%
E) None of the above P(A) = 0.25
P(B) = 0.25
P(C) = 0.25
P(D) = 0.25
P({A,D}) + P({B}) + P({C}) = 1
P(x) = P(A & {A,D}) + P(B & {B}) + P(C & {C}) + P(D & {A,D})
P(x|{A,D}) = 0.25
P(x|{B}) = 0.5
P(x|{C}) = 0.6
P(x) = (P(A)+P(D)) * P({A,D}) + P(B) * P({B}) + P(C) * P({C})
P(x) = 0.5 * P({A,D}) + 0.25 * P({B}) + 0.25 * P({C})
P(x) = 0.25 * P({A,D}) + 0.5 * P({B}) + 0.6 * P({C})
P(x) = P({A,D})/4 + 1/4
P({B}) = 7/2 - 6 * P({A,D})
P({C}) = 5 * P({A,D}) - 5/2
1/2 <= P({A,D}) <= 7/12
0 <= P({B}) <= 1/2
0 <= P({C}) <= 5/12
3/8 <= P(x) <= 19/48
P({A}) + P({B}) + P({C}) + P({D}) = 1
P(x) = P(A & {A}) + P(B & {B}) + P(C & {C}) + P(D & {D})
P(x|{A}) = 0.25
P(x|{B}) = 0.5
P(x|{C}) = 0.6
P(x|{D}) = 0.25
P(x) = P(A) * P({A}) + P(B) * P({B}) + P(C) * P({C}) + P(D) * P({D})
P(x) = 0.25 * P({A}) + 0.25 * P({B}) + 0.25 * P({C}) + 0.25 * P({D})
P(x) = 0.25 * P({A}) + 0.5 * P({B}) + 0.6 * P({C}) + 0.25 * P({D})
P(x) = 0.25
0 <= P({A}) <= 1
0 <= P({D}) <= 1
P({A}) = 1 - P({D})
"Is the teacher a Cretan?" :-P
Therefore it must be C) 60%
Now, what would be the answer to the same question with the answer options:
A) 20%
B) 40%
C) 60%
D) 20%
E) 0%
0% and 100% are indefensible for any uninformed prior.
There are two answers giving the correct probability for uniform prior. One of the 25% is a wrong answer despite being indistinguishable. Such a thing can happen because of a typo or in any myriad of ways. It's not a paradox.
If it asked the question "knowing that probability of success for random single choice question out of 4 answers is 25%, what is the probability of selecting the correct answer at random from one of the below?" That formulation implies an informed prior.
That'd have the answer of prior% with 50% for uniform prior, but it's not duplicated, so the actual answer is 0% making it a paradox. If it were duplicated, (answers only 25% and 50% available) the either answer of 50% would hold.
But that was not the question asked in yours or former question.
If none of the answers are correct, the chance of picking the correct answer is 0%.
If there is no defensible answer present, it's still not 0%, it's a paradox.
The answer is dependent on base axioms accepted. (Axiom of choice and axiom of regularity are a few.) The answer set is Dedekind-finite and countable, whereupon probability is 1/logically consistent answers given the axioms chosen.
Maybe I'll go for a walk.
Given uniform random none of the answers is correct:
25%) There's a 50% chance of choosing this answer
50%) There's a 25% change of choosing this answer
60%) There's a 25% chance of choosing this answer
If you're allowed to pick your own random distribution then any answer is equally valid. Say we pick one of the 25% answers 25% of the time, and the other ones uniformly:
25%) There's a 25% chance of picking this, as defined by our random distribution.
50%) 37.5% chance
60%) 37.5% chance
But if we instead pick our distribution so we pick the 60% answer 60% of the time and the others uniformly we've suddenly got this answer:
25%) ±27% chance
50%) ±13% chance
60%) 60% chance
So depending on if you interpret random as "uniform random" or as "whichever random distribution you like" either none of the answers are right or you can pick any answer to be right.
Sorry but that seems disingenuous... You are ready to accept that there's a mistake in one of the answers but not that none of the answers are correct?