Who Watches The Watchmen?

If the first statement is true, it implies both statements are false; in particular, the first statement is false. This is a contradiction. So the first statement is false, so not both statements are false. This means at least one of the statements is true. Since the first statement is false; this gives the only other option of the second statement being true; that is, Superman exists.

A variant of this is Curry's paradox : statements in the form "If this sentence is true, then P" always implies P is true. (To compare with this problem, this problem has the form "(1) Both statements are false, (2) P", which also implies P is true.)

Well, the box as a whole could be false. Just because the box says something, doesn't make it true!

Ok, my solution is to start by saying both statements are false (guess and check). But if they are both false, then 1 would be true. If 1 is true, then 1 can't be false.

So now try making both true. But if both are true, 1 must be false. But this works because 1 is false (because they are not both false) because 2 is true. So then apparently superman is real. Though we really know he isn't because who trusts some random box!

This is something we call a "paradox". I'm surprised there were only two options, out of which neither is correct.

Everyone is giving solutions based on whether the statements are true or false, but the question only asks if they are logical. The first statement is logical as it says both sentences in the box are false. However only one sentence in the box is false, the first one. This would make the first statement false (meaning both the statements in the box aren't false) allowing the second to be true.

A box is a 3 dimensional object...therefore the first statement doesn't count.

The first point says "Both sentences in this box are false." Assuming that sentence is telling the truth, that would then make the sentence false, so we know that this statement is false. So, the truth in this statement becomes "Not both sentences are false," or "At least one sentence is true." Since we know the first sentence is false, that means that the second sentence, "Superman exists," must be true. So, Superman exists.

The first statement can't be true because it says both statements are false, implying that it is false.Since it is false, the other can't be false also because that would make the first statement true. So the second statement is true and Superman exists.

A slight variation: If (1) is true then it must also be false, which is a logical non sequitur. If (1) is false then at least one of the statements in the box must be true, and since (1) is false, statement (2) must be true.

This is somewhat a paradox, but simple to understand. If the first statement is false, which cancels itself out, means the second sentence is still valid.

Example: this is not, not superman. A double negative cancels the first negative, that's how I see the problem anyway.

I made the correct guess of what the asker might have in mind, but this question is really badly formed. What does "if these statements are perfectly logical" mean?

To those who have answered "correctly", is your answer based on statement (1) being false? Mine is, and that's the problem. Is being false considered perfectly logical?

A better way to phrase this question is probably to ask "In a valid valuation, what is the truth value of statement (2)?"

Or more simply, "To make $p \leftrightarrow (\lnot p \land \lnot q)$ true, what truth value should be assigned to $q$ ?"

The sentences cannot have the same logical value, as the first one is stating FALSE both of them, so we know that they have opposite logical values.

Evaluating the first sentence, we have that, if true, both are false, which causes a paradox. For it to be perfectly logical, the first sentence needs to be false, automatically setting the second one to true.

the question says that both the statements in the box are false hence the first statement is itself a false one...... this leads to the conclusion that statement number two is correct

One way you could think about this is that "perfectly logical" doesn't mean true. It could just mean it's a well formed logical expression. The other way is that if the inverse of (1) is that at least one of the sentences are true. Both of the sentences being true results in a paradox, but one of them being true results in an infinite looping "paradox" where Superman always exists.

Superman is real, because if argument 1 is false (both argument 1 and 2 are false) , then for it to be false, only one argument needs to be true. So, if superman does exist, it qualifies as the first argument being wrong.

if the first statement is true, then there is a paradox within the statement ( as it can not be true and false at the same time). That means the first statement is false, and for that to occur, at least one statement must be true, ergo (2) is true

I don't know if this is a valid solution but I thought it like this:

(1) => ¬(¬(2)) ≡ (2)

Because first (1) negates (2) but then it negates itself so it should end up double negating (2)...

Don't try to get confused. There is a hint itself in Statement 1 :- Both statements in this box are false. That means Statement 1 is itself false and so Statement 2 is true.

Well if I were any of u I would first see that is statement 1 is false this means that the possiblity of existaance superman increases also the sentences are completely logical hence the more appropriate option is yes superman exists

This was my logic, which probably isn't logically sound but it got me the correct answer. Nothing says only the first statement can be false, and upon assuming that, in order to keep it false then both statements cannot be false. Therefore the second statement must be true.

The "if" portion of the statement is not true, there for the "then" portion is automatically true. Make a truth table with p, q, p->q. In both cases where p is false, p->q is true.