Why would someone even consider investigating whether or not the square root of 47 is 25 if no one claimed it's true?
Well, if you were a math teacher I´m sure you´d come across this claim eventually.
There has to be evidence in the form of a claim in order for anyone to investigate the truth of the claim.
Indeed it´s not very reasonable to even start investigating a claim for which no evidence is presented. We can simply lean back, say "
I don´t believe you" and await evidence be presented.
(And where asking you for evidence leads, we know from previous discussions with you.)
It's logical to investigate a truth claim in order to determine if it's actually true or not.
No. It´s logical to say "present your evidence, just so I can see whether it´s even worth putting effort in investigating it."
It's illogical to investigate a truth claim that has never been made, such as the claim that the square root of 47 is 25.
Yet there are people who don´t believe that the square root of 47 is 25. Nobody needs to seriously make this claim for there to be lack of belief in the claim. Which blows your previous idea out of the window: namely that a lack of belief in X depends on the belief that X exists.
Anyway, it´s completely irrelevant whether the truth claim has seriously been made. That the truth claim has been made doesn´t make your argument any more valid. People make false and mistaken truth claims all the time. Your argument (that a truth claim is more likely to be accurate than its rejection because the rejection depends on the truth claim) is therefore patently absurd. It´s just nonsense.
But just because it´s you, I´ll give you another analogous argument, this time with a belief actually held by people:
As long as there are people who don´t believe that Nazism is the best societal form, their lack of belief depends on the claim that Nazism is the best societal form. This should point a rational thinking person to "Nazism is the best societal form" over "I don´t believe that Nazism is the best societal form", if they value evidence and truth.