Here is a single python statement that solves the problem:
print([
q for q in itertools.product((True, False), repeat=6)
if q == (
all(q[1:]), # 1. All of the below.
not any(q[2:]), # 2. None of the below.
all(q[:2]), # 3. All of the above.
any(q[:3]), # 4. One of the above.
not any(q[:4]), # 5. None of the above.
not any(q[:5]), # 6. None of the above.
)
])
import z3
answers = [z3.Bool(f"answer{i}") for i in range(1,7)]
implications = [
z3.And(answers[1:]), # All of the below
z3.Not(z3.Or(answers[2:])), # None of the below
z3.And(answers[:2]), # All of the above
z3.Or(answers[:3]), # Any of the above
z3.Not(z3.Or(answers[:4])), # None of the above
z3.Not(z3.Or(answers[:5]))] # None of the above
constraints = [z3.Implies(ans, impl) for ans,impl in zip(answers, implications)]
z3.solve(constraints)
Not enough constraints. You must (EDIT: should? see discussion below) also constrain that there is exactly one `answer{i}` which is true, and all the implications should be equalities (else, for example, 6 is a valid answer, even though that would imply 5 is true and thus contradict 6).
It just so happens that the first result Z3 finds is the correct one. But if you exclude that result with an additional constraint, it will find another. (This is general is a good way to check your work.)
(I just did the same exact exercise with Z3 and CVC4, but using SMTLIBv2 syntax.)
I feel like it's implied by the question itself? "Which answer [singular] is the [definite article] correct answer"
But you may be right. I detest word puzzles for exactly this type of ambiguity.
Same goes for implication vs. equality. Why "should" 5 being true contradict 6 being correct? Just because 5 happens to be true doesn't necessarily mean it is "the" "correct" answer. The "real" answer depends on an interpretation of the English-language formulation that most people will apply, but not all.
The puzzle itself contains enough contradictions that a single answer is forced. So it is unclear from the question, but the result is definite: There is exactly one answer. (Of course, that fact is not obvious.)
import z3
answers = [z3.Bool(f"answer{i}") for i in range(1,7)]
implications = [
z3.And(answers[1:]), # All of the below
z3.Not(z3.Or(answers[2:])), # None of the below
z3.And(answers[:2]), # All of the above
z3.Or(answers[:3]), # Any of the above
z3.Not(z3.Or(answers[:4])), # None of the above
z3.Not(z3.Or(answers[:5]))] # None of the above
# An answer should be True if and only if its "implication" is true
constraints = [ans == impl for ans, impl in zip(answers, implications)]
z3.solve(constraints) # Prints the right solution
# Try to find another solution by rejecting the previous one
constraints.append(z3.Or(*answers[:4], z3.Not(answers[4]), answers[5]))
z3.solve(constraints) # no solution
Sorry for the late reply. I don't get notified on replies to my posts. Thanks for the posted solution. It seems to work fine.
I am trying to understand what is happening on line 18 (what's the * operator in front of `answers[:4]` for?) and wondering if it could be re-written more generically based on the output of line 14 (i.e. by saving the result from the first `.solve(constraints)` call on line 14 and automatically appending it as a constraint of something to reject on line 18.
Does that make sense?
is the same as
z3.Or(answers[0], answers[1], answers[2], answers[3])
Of course it can be re-written more generically, but I thought it would be less explicit in this example
import z3
answers = [z3.Bool(f"answer{i}") for i in range(1,7)]
implications = [
z3.And(answers[1:]), # All of the below
z3.Not(z3.Or(answers[2:])), # None of the below
z3.And(answers[:2]), # All of the above
z3.Or(answers[:3]), # Any of the above
z3.Not(z3.Or(answers[:4])), # None of the above
z3.Not(z3.Or(answers[:5]))] # None of the above
# An answer should be True if and only if its "implication" is true
constraints = [ans == impl for ans, impl in zip(answers, implications)]
solver = z3.Solver()
solver.add(constraints)
# Print all solutions
while solver.check() == z3.sat:
solution = solver.model()
print(solution)
solver.add(z3.Or([variable() != solution[variable] for variable in solution]))
The trick with the `while` loop is smart! ;) Exactly what I wanted to achieve. Thanks!
In your first post, you mentioned that the Z3 (which I agree with you is awesome!) solution is less clever than the generic Python one. What additional changes would you implement in order to make it more clever or did you mean something else entirely?
Ah, gotcha. That makes sense.
I really appreciate your help with the above code. I've been meaning to get deeper into the wonderful world of Z3 for a while now and it seems like the time has come :)
> It just so happens that the first result Z3 finds is the correct one. But if you exclude that result with an additional constraint, it will find another.
Having an additional constraint would be a different question, would it not?
import itertools
def below(n):
return slice(n+1, None)
def above(n):
return slice(None, n)
print([
q for q in itertools.product((True, False), repeat=6)
if q == tuple(
f(i) for i, f in enumerate([
lambda n: all(q[below(n)]), # 1. All of the below.
lambda n: not any(q[below(n)]), # 2. None of the below.
lambda n: all(q[above(n)]), # 3. All of the above.
lambda n: any(q[above(n)]), # 4. One of the above.
lambda n: not any(q[above(n)]), # 5. None of the above.
lambda n: not any(q[above(n)]), # 6. None of the above.
])
)
])
Thanks for posting this, because I'm a Python n00b and I had no idea what this line was doing:
itertools.product((True, False), repeat=6)
The site made it easy for me to spend 30 seconds figuring it out. My interpretation is that that line generates what I'll call a truth table (not sure if that's what it formally is) then brute force searches for rows that satisfy all of the listed constraints.
I'd also hazard that the fact that there's only one answer is engineered into the question and not a natural product of this search.
I think you have the general idea down. But as you suspected it's not a truth table. It's all cross-product permutations (hence the name `product`) of six items from the set {True, False}. So that line is creating a iterator that looks like this:
Though not necessarily in this order... I'm unfamiliar with the actual implementation. Since the values are bits, they could have achieved the same effect by incrementing a six-bit integer. Which is a nice proof that the number of permutations here is 2^6. (Generally it's s^n, where s is the set size and n is the count of elements.)
The number of elements of the product of two iterables is the product of the number of elements of each. You can similarly have a product of 3, 4, etc. iterables.
The repeat keyword argument to product specifies that we want not a product of several different iterables, but the product of an iterable with itself N times. So we pass a tuple of all possible boolean values (True, False) and ask that it be raised to the 6th power, giving us all possible 6-tuples of all possible boolean values.
wouldn't any(q[:3]) (the 4th sentence) return true even if 2 or 3 of the above were correct? If I understand correctly, it should be true if exactly one of the above is correct, not at least one.
Although, I'm probably wrong, since the question says "which answer is correct", implying only one can be correct.
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.
I'm not saying that 'this' doesn't resolve to anything.
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 ?
I don't agree with that. The pronoun "this" does not appear in the example question you just provided. There's a difference between "this" and "his" and the difference is how they resolve contextually.
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.
>Because there are still unresolved pronouns in (Q2).
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.
I've updated the answer again with hopefully clearer examples.
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
Well, if you're forced to choose one, then you should minimize your error. So the answer is E), and as 19% is part of "None of the above", you would be only 1% off, much closer than any other answer...
We can probably assume the response selection and establishing the set of correct responses are independent events, as they are being done by different people, and the person going second has no prior knowledge of the results from the first.
That range doesn't overlap with the possible responses. So there's an incorrect assumption in there somewhere. Maybe the one about the question being fair. Backing up:
So the correct answer is 0.25, and the correct multiple choice response is one of either A or D. One's best response is based on knowledge of the psychology of the test-writer. It is not random. The person writing the question consciously chose one or the other, intentionally violating an assumption of the test-taker--that knowing the correct answer to a question would translate to being able to unambiguously select the correct multiple-choice response.
You’re probably right, but this is not a paradox. There is an answer that is logically sound without conflating the semantics beyond their reasonably-assumed meaning.
Normally 25% but since there are 2 answers with 25% this cant be correct. There is only one answer with 50% but if that is the correct one there would be only 25% chance.
That depends on whether we consider A and D the same answer or not. If we can assume that A does not imply D and vice-versa, then we can pick either A or D and be correct (but not both) - in essence our picking either answer collapses the question's wave-function to be one or the other.
I don't think so. A and B are contradictory.
A asserts: (A & !B & !C & !D & !E & !F),
B asserts (B & !C & !D & !E & !F) which would imply B and !B therefore A is false.
Only E (or 5 depending on notation) can be true without contradiction.
Your answer above is incorrect in a very funny way. "At random" implies an uninformed prior, but the question presupposes there's a correct answer, as 0% is missing.
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.
At random has to imply uniform random for the question to make any sense, otherwise you can simply bias your answer choices so the answer is always correct.
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.
> 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.
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?
Again, not for a question that has a defensible answer present when you're picking at random. (Uninformed prior.)
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.
Of course, a uniform distribution is implied. The fact that you choose to misunderstand this on purpose, doesn't make you smart or right. It makes you a pedant who is still wrong.
I've always thought of these as "Wayside School" problems, because I first encountered them in the children's book, Sideways Arithmetic From Wayside School by Louis Sachar. I highly recommend it for any child who is into puzzles. I really enjoyed how the book deconstructed the format of tests and quizzes that I was so familiar with at the time. My favorite section was the sideways math, where words are added or multiplied and you had to deduce the values of the letters. E.g.,
D O G
x A D
_______
D O G
A D O
_______
A G O G
Maybe children don't learn to do arithmetic like this anymore and this section will be utterly impenetrable to them.
It seems that 5 is a winner, as stated in the accepted answer.
That's a pity, first I thought the goal of the question was to create an instance of Yablo's Paradox. Yablo's paradox is important because you cannot just argue that the paradox arises from an obviously incorrect definition, as people sometimes do for classical semantic paradoxes.
Edit: I just looked up Yablo's paradox. It only works because there are infinite statements. Since there is a finite number of statements here, it could not be an instance of Yablo's paradox. Only a circular paradox.
One interesting aspect of this concrete example is that even though the two statements seem contradictory at first sight, there are in fact two assignments of truth values to statements so that no contradiction arises.
For instance, using Scryer Prolog and its SAT solver to model the situation:
As answer, we get a symbolic expression that compactly captures all concrete solutions:
clpb:sat(S1=\=S2)
This means that as long as the truth value of S1 is different from that of S2, the puzzle is solved. This is intuitively admissible, because if one of the statements is false, then the other is true.
It would be different if for example Statement 1 said “Statement 1 is false”, because then there is no satisfiable assignment at all:
But is it an answer to the question? If the question were something completely different, "How green is interstellar space?", then we would still find 5 as the answer. But "None of the above" is a non-answer. And "What is the answer to this question?" is not well defined.
I wish this wasn't the top comment, and not because it is a bad comment, but because I caught a peak of it before I could try the puzzle without hints.
In my opinion there isn't an answer to the question.
The italicized _this_ is what's important. You need to answer the question itself. As far as I can tell, everyone is caught up reading the answers as
1. All of the below _are true_.
2. None of the below _are true_.
etc.
but the answers don't say that. They don't refer to other answers. The are direct responses to the question. None of those answers make sense in context.
"Which answer in this list is the correct answer to this question? All of the above." That doesn't make sense.
"All of the above _are true_" (etc.) would make sense, but as stated none of the answers provided offer a coherent response to the question.
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?'
I'm a bit confused. It seems that we're all looking for the answer that can hold "true". That's fine, however, does 5 answer _this question_? From my understanding, the question itself is impredicative[1].
"Which answer in this list is the correct answer to this question?"
Being self-referential makes it difficult to decide what correct in that context means. Does the question even allow for a no-answer/inconclusive result?
(declare-const a Bool)
(declare-const b Bool)
(declare-const c Bool)
(declare-const d Bool)
(declare-const e Bool)
(declare-const f Bool)
(define-fun conjecture () Bool
(and
(= a (and b c d e f))
(= b (and (not b) (not c) (not d) (not e) (not f)))
(= c (and a b))
(= d (or a b c))
(= e (and (not a) (not b) (not c) (not d)))
(= f (and (not a) (not b) (not c) (not d) (not e)))
(xor a b c d e f)
)
)
(assert conjecture)
(check-sat)
(get-model)
result :
sat
(model
(define-fun f () Bool
false)
(define-fun b () Bool
false)
(define-fun a () Bool
false)
(define-fun c () Bool
false)
(define-fun d () Bool
false)
(define-fun e () Bool
true)
)
Implication is not strong enough. That permits e.g. f to be true and e to be false.
Also I don't know the specifics of the language that tool uses, but typically variadic XOR simply constrains that an odd number of its operands are true. (It so happens that there are no solutions to this problem with 3 or 5 true answers.)
The model is satisfiable where only a5 is true, anyone can run it themselves and play with it [1].
Additionally forcing a5 to be false by adding the following to the list of assertions:
(assert (not a5))
And the model becomes unsatisfiable.
Edit: Finally, to make sure there's not some possible solution where a5 is true as well as another assertion you could alternatively add this assertion:
The second answer is a C program. It specifies the constraints and loops over all the possible solutions. That's the most naive constraint solver possible, but adequate for this task.
Once you put in the effort of translating the problem into constraints, the Prolog solution is trivial, because the Prolog compiler is a constraint solver.
I feel the question is unnecessarily convoluted. To really answer the question, you have to recursively refer to the question itself. So to parse the question, I'm calling a function recursively without a stop condition. They could have just asked, which statement below is true. And for all intents and purposes the answer would have been the same.
I too am of the opinion the question recurses indefinitely, which would mean it's unanswerable and perhaps not even a valid question technically speaking.
LinkedIn's skill tests are hilariously poor. They ask unanswerable questions, and mistype the answers. The more familiar you are with the subject matter, the more uncertain you become.
I noticed those randomly one day and did a few of them. I failed the one for the primary programming language I have worked in for the past decade and passed ones for two different languages I have hardly used and AWS, which I barely know anything about outside of EC2 and R53.
I felt like there was a big difference in the quality of the questions and answers for each quiz. For the AWS one, for example, the phrasing used for most of the questions and answers made the correct answer basically obvious without actually knowing it to be correct.
We can easily narrow down the correct answer to "none of the above", which leaves 5 or 6. But since 5 is above 6 and is a correct answer, 6 cannot be correct. So it must be 5.
I could be wrong, but the question is ill-stated; should be something like 'which of the following statements are true', I think, or maybe 'what is the only true statement in the following list.' It's been awhile since logic classes though, maybe those are essentially the same as the original question, I can't quite figure out what I think is wrong with the original question, it just doesn't seem quite right to me.
Also, 4 is ill-stated -- "One of the above is true" -- does that mean exactly one of the above is true, or at least one of the above is true. Although in that case I don't think it matters because 1/2 are mutually exclusive and 2/3 are mutually exclusive, and the combination of those two means that 1/3 are mutually exclusive., so only exactly one of 1-3 can possibly be true anyway.
6 can not be true because if it where then 1,2,3,4 & 5 would be false, but that would make 5 true as well which is a contradiction, therefore 6 can only be false
1 Can not be true because we just determined 6 as false so 1 can only be false
3 can not be true because we just determined 1 as false so 3 can only be false
With 1 and 3 now proven false, 2 can only be true if 4 is false, but if 2 is true then 4 would be true as well. This is a contradiction so 2 can not be true.
With 1,2 and 3 now proven false, 4 is false.
5 is true since we just proved 1,2,3 and 4 to be false
Everyone in the comment thread seems to be focused on whether each proposition can be true. This is looking for your keys under the lamppost despite the fact that you dropped them in the dark. It's easy to determine whether a proposition can be true.
But the question doesn't ask about that. It just asks "which of these is the correct answer to this question?" There are no stated criteria for being the correct answer, and there is no reason to assume that the correct answer must be true or even capable of being true.
Consider this other question:
Which of the following is the correct answer to this question?
(A) This one.
(B) This one.
That's arguing about semantics. It seems obvious from context that the intended interpretation of "the correct answer" is "the only true answer" — since "true" and "correct" are nearly synonymous, and the use of definite article singular implies that there exists exactly one answer.
To be fair, making the argument that something is obvious from context is also arguing about semantics. The other poster makes a defensible point (as do you!).
This seems like a crucial problem with the initial question. As with so many puzzles like this, it assumes that the reader will make similar (unargued, unstated) assumptions as the writer.
But what justifies that methodological assumption? Nothing, as thaumasiotes is pointing out.
Any communication relies by necessity on a mutual implicit frame of reference, and while there most certainly are cases where question tread the line of unambiguous interpretability (I have suffered these many times, especially in education tests), I'd say in this case the demands are rather benign.
I didn't see a logical explanation that explains why an option is correct or not, so here's one:
Q Which answer in this list is the correct answer to this question?
1. All of the below.
2. None of the below.
3. All of the above.
4. One of the above.
5. None of the above.
6. None of the above.
I'll assume "is the answer" to be "True" and "is not the answer" to be "False" in my explanations to make it more readable.
Lets say 1 is True. This implies - 2, 3, 4, 5, 6 are also True. Now, if 2 is True, then it means 3, 4, 5, 6 are False. This contradicts what 1 says. So, 1 cannot be True.
Lets say 2 is True. This implies - 3, 4, 5, 6 are False. If 3 is False, it means both 1 and 2 cannot be True. Since, we've assumed 2 to be True, 1 can indeed be False, so this is consistent. If 4 is False, then it means either less than one or more than one out of 1, 2, 3 are True. Zero out of 1, 2, 3 cannot be True since it contradicts our assumption of 2 being True. So either two or three out of 1, 2, 3 have to be True. Since, we've already established than 1 is False, that leaves 3 to be True along with our assumption of 2 being True. But 3 cannot be True since it contradicts 2. So 2 cannot be True.
Lets say 3 is True. This implies 1, 2 are True. We've already established that 1, 2 are False. Moreover, 2 being True will contradict 3 being True. So 3 cannot be True.
Lets say 4 is True. This implies that exactly one of 1, 2, 3 is True. We've already established that they are False. So 4 cannot be True.
Lets say 5 is True. We now know that 1, 2, 3, 4 are not True. So, this is consistent. We'll come back to this.
Lets say 6 is True. This implies that 5 is False. If 5 is False, it means all of the above 5 - 1, 2 , 3, 4 are True which we've found not to be True. So 6 cannot be True.
Can't edit. Explanation for 6 is incorrect as pointed above/below. Correct explanation is: Lets say 6 is True. This implies that 5 is False. If 5 is False, it means at least one of the above 5 - 1, 2, 3, 4 is True. But we've established above that all of them are False. So, 5 cannot be False. Hence, 6 cannot be True.
Maybe I made a mistake, but I think this can be solved really simply just by taking each option in turn, assuming it is correct and looking for an inconsistency.
1 can't be right because it both affirms and contradicts 2.
2 can't be right because it both denies and fulfils 4 (even if 'one' means 'one and only one', because we already know that 1 is false, and looking ahead we can see that 3 is also false).
3 affirms 1, which we already know to be false.
4 affirms one of 1-3, which we have determined are all false.
5 is correct, as demonstrated by our finding that 1-4 are all false.
6 is false; we would know this even if we hadn't already worked out that 5 was true, because 6 implies that 1-4 are all false, which implies that 5 is true.
Let's assume 6 is T so we have xxxxxT. In that case 5 need to be F. But if 5 is F, then any from [1,2,3,4] need to be T.
But if any from [1,2,3,4] is T then 6 cannot be T.
Thus 6 cannot be T, so it is F if correct answer exists.
Now we have: xxxxxF
Next, we try to set 5 to T: xxxxTF. 1 cannot be T, 2 cannot be T, since their last elements are different (TT!=TF, FF!=TF), 3 cannot be T because it requires 1 and 2 be both T , 4 cannot be T because (1 or 2 or 3) are F.
It seems all of the answers assume "You cannot assume there is only one correct answer." However in "...answer in this list is the correct answer..." to my mind the word "the" requires only one correct answer. If it was worded '...a correct answer...' or '...one of the correct answers...' ok sure, there can be more than one. But as it stands there can only be one correct answer so the "correct" answers so far, are based on a false premise.
Regardless of the interpretation it's not a false premise. The uniqueness of the answer is just an assumption that we don't need.
Suppose the question was instead: "What is the unique real number x for which x^3 = 8?" We can guess and verify that x = 2 is a solution. And if we're allowed to assume that this equation has a unique solution, then that's enough. But if we work a little harder, then we don't actually need that assumption: we can prove that x = 2 is the unique solution.
Similarly, in the original problem we could guess and verify that FFFFTF is a solution. And if we're allowed to assume that the solution is unique, then we're done. But again, if we work a little harder, then we don't need that assumption: we can prove that FFFFTF really is the only solution.
Do you agree it can be rephrased as "Which answers in this list are the correct answers to this question?", so as to make it more difficult, while keeping the same answer?
It could be rephrased that way without changing the answer, but I don't think it would be any harder that way. You can still start with question 1 and continue downward to logically deduce the answer.
The question is phrased to mislead as this is a constraint satisfaction problem over the boolean space 2^6. There may be 0, 1, or many correct answers.
Although h53::noneabove.4 |- ( th <-> ( ph \/ ps \/ ch ) ) is not a correct interpretation of "[Exactly] One of the above", it is rather "[At least] One of the above", it doesn't matter because 'th' is not true anyway.
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?'
5 and 6 are the only possible. But if 5 is true then 6 is false
None of the above can be true only with "more than one of the above (but different than all, none or one)", but then we would not be talking about the correct answer. It is assumed than only an answer can be correct.
