tom: would you briefly explain what, in your opinion, is the difference between a 'negative' and 'positive' claim? what does that distinction mean to you?
Negative claims are an absence, not a reworded positive claim. It is not "0". It is "-".
Negative claims hold a special distinction. If I claim that the window is NOT open, it does not mean I am claiming that the window is closed. I am claiming that the window is NOT open.
It is possible that the window does not even exist, making the claims that the window is both 'not open' and 'not closed' simultaneously true.
It is also possible that the window is positively open or closed, but due lack of evidence of its positive state, and lack of evidence of its existence altogether, we are compelled to define it as it is, a NOT. Barring some sort of evidence otherwise, the window does not exist and it is neither open or closed. It is the burden of the person with the positive claim -- that the window exists, or that the window is open or closed -- to provide evidence for that claim.
We must believe in nothing because there is no evidence. Once there is evidence we can start believing in things.