I received this question from my mathematics professor as a leisure-time logic quiz, and although I thought I answered it right, he denied. Can someone explain the reasoning behind the correct solution?

Which answer in this list is the correct answer to

thisquestion?

- All of the below.
- None of the below.
- All of the above.
- One of the above.
- None of the above.
- None of the above.

I thought:

- 22 and 33 contradict so 11 cannot be true.
- 22 denies 33 but 33 affirms 2,2, so 33 cannot be true
- 22 denies 4,4, but as 11 and 33 are proven to be false, 44 cannot be true.
- 66 denies 55 but not vice versa, so 55 cannot be true.

at this point only 22 and 66 are left to be considered. I thought choosing 22 would not deny 11 (and it can’t be *all of the below* and *none of the below*) hence I thought the answer is 6.6.

I don’t know the correct answer to the question. Thanks!

Your last point is not correct:

If 6 is true, then 5 is false, which implies that at least one of 1-4 is correct, which is a contradiction. So 6 is false.

On the other hand, 4 is not correct, this implies that 2 is incorrect. Indeed 4 is correct if and only if 2 is correct since 1 and 3 are false. (If 2 is true then 4 is true by the content of 4, but it has been shown that 4 is false.)

Hence the only choice is 5.

Since this is a question on mathemetics, I would argue that the correct answer is that this is not a question at all. Of course it is syntactically valid, but not semantically.

I will try to rephrase it in terms of sets:

This definition is clearly self-referential and therefore not covered in ZFC.

(I think I read something by Bertrand Russel once about sentences that were syntactically valid but semantically invalid. Can’t remember right now. Russel’s Paradox is definitely related.)

All other answers so far really answer a different question:

It doesn’t appear to me that the yet very interesting point brought up by RCT, nor the original post supply a reason for excluding answer number 2. In fact, the original point that it would not deny 1, does not mean it affirms it, still, and therefore, if #3 to #6 are false, #2 is valid. Until we confute #2, we cannot call 5 true either, samewise.

I think an interesting argumentation – while irrelevant after the preceeding answers – is that the answer has to be a not-self-excluding answer (i.e: not #1, #3, #4). In fact, if the answer were to be #4, for instance, it would define the answer to be not #4, excluding the possibility that #4 is indeed the answer (a contradiction).

This, and RCT point leave us with #2 and #5 again.

Answer 4 (“One of the above”) would be true if 2 (“None of the below”) were, but 2 contradicts 4, therefore 2 cannot be true and 5 is the only possible answer.

It’s the same answer, it just appeared to me that I may have missed to see a valid proof in previous statements, but I may be wrong.

Just because there is only one proposition that does not contradict the rest (the 5th one) does not mean that it is the answer. We must consider the question first. And the question does not make any sense. It is a non-starter. You might as well write, “42”, and leave it at that. Thankfully, this does not count towards your grade!