The Lemon Fool

Posted: **May 25th, 2022, 6:51 am**

Which answer in this list is the correct answer to this question?

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

Posted: **May 25th, 2022, 6:54 am**

second from the bottom?

Posted: **May 25th, 2022, 11:39 am**

Agreed.

Posted: **May 27th, 2022, 7:32 pm**

servodude wrote:second from the bottom?

I agree. Lettering the statements A-F:

E asserts that A-D are all false. F asserts that A-E are all false, and substituting in the value of E means that F is "ABCD are all false, and (ABCD are all false) is false". So F requires X and not-X, and so is itself false.
Since A asserts that all of B-F are true, and we know F is false, A is also false.
C asserts that both A and B are true, but A is false, so C is false
D asserts that exactly one of A, B, C are true. Since we know A and C are false, D asserts that B is true, so D and B are the same. And B asserts that all of C-F are false - which is equivalent to C, B, E and F being false (and since we know C and F are false, equivalent to B and E being false). If E is false, then B is true if B is false, and B is false if B is true. So to avoid a paradox, E must be true - which makes B false, which in turn makes D false, which means that all of A-D are false, which makes E true!

So the only true statement is E, the last-but-one "none of the above".

Posted: **May 28th, 2022, 9:28 am**

Since servodude got in before I saw it, and steelcamel IMHO overcomplicated a very simple problem, let's see if we can add £0.02 or thereabouts ....

There's an alternative logic here, based on the formulation of the question. The wording, unambiguously in the singular, implies there is exactly one correct answer, which in itself reduces the candidates to three. I won't insult anyone's intelligence by going further.

Posted: **May 28th, 2022, 10:49 am**

UncleEbenezer wrote:Since servodude got in before I saw it, and steelcamel IMHO overcomplicated a very simple problem, let's see if we can add £0.02 or thereabouts ....

There's an alternative logic here, based on the formulation of the question. The wording, unambiguously in the singular, implies there is exactly one correct answer, which in itself reduces the candidates to three. I won't insult anyone's intelligence by going further.

So, that handles the "correct" case
Which would be most "vintage"

?

Posted: **May 28th, 2022, 4:18 pm**

UncleEbenezer wrote:There's an alternative logic here, based on the formulation of the question. The wording, unambiguously in the singular, implies there is exactly one correct answer, ...

While I occasionally use similar arguments to solve problems like Sudokus, that is in a context where I can easily verify the results by other means. It's very shaky ground in cases like this. Verifying the result is exactly the same as steelcamel's solution so you have gained nothing.

Moreover in a formal logic system* many, in some sense most, statements are ungrammatical, i.e. nonsense. You can reorder redsturgeon's answers to get such a statement. Various paradoxes are other examples of nonsense. Assuming consistency is simply not on.

You may argue that redsturgeon is master of his puzzle and he can create a system where whatever he wants to be true is. But that is not correct. Even if he could if he sets up a universe where, say, one of the multiple definitions of π gives a different value, then at least one of the rules we accept as true is not true. We don't know which one, so we can't say much about his universe.

*that is powerful enough to contain arithmetic. And note that English is powerful enough to contain formal logical systems that contain arithmetic, so there is no escape that way.

Posted: **May 28th, 2022, 4:59 pm**

9873210 wrote:
UncleEbenezer wrote:There's an alternative logic here, based on the formulation of the question. The wording, unambiguously in the singular, implies there is exactly one correct answer, ...

While I occasionally use similar arguments to solve problems like Sudokus, that is in a context where I can easily verify the results by other means.

As you can in this case. I didn't need to think about it, and neither did servodude (as witness his solution's posting time compared to the OP). Nor, I would guess, any of the regulars here. And my first-reaction solution didn't rely on the correct language of the problem. But to have stated that would have added nothing.

Moreover in a formal logic system* many, in some sense most, statements are ungrammatical, i.e. nonsense. You can reorder redsturgeon's answers to get such a statement. Various paradoxes are other examples of nonsense. Assuming consistency is simply not on.

Indeed. Speaking as a pedant who doesn't always understand what is meant when a problem is misstated (or inconsistently formulated), I could easily be one to point that out. If you look you can find examples on this very board.

In this instance, the question was correctly stated, but would arguably have made more sense stated in an equally correct manner that didn't rule out any of the candidate solutions. The solution is of course unchanged by generalising to

Which (if any) statement or statements in this list are true?

Posted: **May 31st, 2022, 12:40 pm**

UncleEbenezer wrote:Since servodude got in before I saw it, and steelcamel IMHO overcomplicated a very simple problem, let's see if we can add £0.02 or thereabouts ....

There's an alternative logic here, based on the formulation of the question. The wording, unambiguously in the singular, implies there is exactly one correct answer, which in itself reduces the candidates to three. I won't insult anyone's intelligence by going further.

I did consider that. The question clearly says "which statement", not "which statements", or even "which statement(s)", which would imply only one. But some of the possible answers clearly imply the possibility of multiple answers - even "none of..." implies it was possible that some were. And the question is written in informal language, so I thought it a bit doubtful to use the grammar to infer the answer unless necessary - and it fact it's possible to resolve the question without this, so I didn't see any advantage in incorporating a slightly doubtful point.

Also, the solution may be over-complex, but as this was stated as a "logic problem" I wanted to make sure I'd covered every possibility. I didn't actually go through all those steps to find the correct answer, but wrote it as a proof that it was the only possible answer once I'd found which one it was.

The Lemon Fool

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