Here is a link to the Wikipedia article on the subject: https://en.wikipedia.org/wiki/Negative_and_positive_atheism.
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What I have to say about it at the moment:
For any logical statement X, the default, prior to any analysis, is that it is unknown whether X is true. The same applies to the negation of X. Prior to analysis, it is also unknown whether the negation of X is true.
When X is the statement that "a god exists," and we have not met the burden to prove X, then we are negative atheists. For the same X, if we judge that the burden has been met to prove the negation of X, then we are positive atheists. "Positive" here means that we are asserting a statement; "negative" means that we are not asserting any statement. An alternate standard for positive atheism would be to prove or establish that the negation of X is more likely to be true than X.
In the predicate calculus, "a god exists," would be encoded by applying the existential quantifier. I expect that a reasonable encoding of it would read something along the lines of, "there exist in experience phenomena that conform to the definition of a deity." The negation of that statement is a universally quantified assertion. So, I expect a reasonable encoding of the negation would read something along the lines of, "for all phenomena in experience, none conform to the definition of a deity." [edit - A more literal translation, though perhaps one more awkward in English, would be, "every phenomenon in experience does not conform to the definition of a deity."] This would be a falsifiable assertion, that could be proven false by observing a deity. As a falsifiable assertion, it could not be scientifically proven, except within the context of a particular theory of physical reality that itself has evidential support. This situation is not a bug, it is a feature in science that, generally speaking, opens the door to progress.
Nevertheless, because of this situation, in my view, it is more rational to be a negative atheist than a positive atheist.
And, it is more than rational to take one's negative atheism and run with it, holding no expectation that it will turn out that a god, in fact, exists.
Of course, a critical issue here is that we have not actually formalized a definition of a deity in a way that people can agree on, or at all really, I'd argue.
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Thanks!
