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Why aren't they correct?
And why isn't 2 wrong?
I'm saying that providing evidence is better than not providing evidence, if the objective is to verify/confirm/support a claim.
This is universally accepted and applied to just about every aspect of life. It's how you make daily decisions, too. I'm sure you've based 100 decisions on this method just in the last day.
Sorry, but you don't have irrefutable proof that the sky is purple, but you can say that the sky appears purple from inside that room. You haven't been able to explain why it's purple, you've only made an observation.
Science has already explained why the real sky appears in colours, and it was done through more than believing the lie of a single person.
From everything you said, it would be just as right to believe (the lie) without any further investigation. Or even worse, you'd make up a story about the gods being upset with you, and they turned the sky purple.
That makes no sense. Going outside to get a different perspective, realize that the sky does not appear purple, and enter a line of further inquiry and investigation is exactly how you'd get answers.
The more evidence you gather, the closer you get to the truth. And when you have enough evidence, you'll be able to prove and test your claim with mathematical precision.
With evidence to support that hypothesis, you would be as close to right as you can be.
It would surely be better than blindly believing the liar, no?