Agnostic and gnostic atheism aren't real positions. And literally every position has the burden of proof. There is no position which lacks the burden of proof.
Furthermore, while agnosticism and atheism are separate, they are not mutually exclusive. Agnosticism is the claim that divinity is unknown or unknowable, whereas atheism is a rejection of theism which may be characterized as a lack of belief and thus justified by agnosticism.
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Atheism is the belief that god does not exist. That's incompatible with agnosticism.
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Atheism is a rejection of theism and not necessarily the positive belief that god does not exist. Atheism refers to a lack of belief. Agnosticism says nothing of belief.
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Atheism is the belief that god does not exist. Agnosticism was a position carved out specifically in rejection of both atheism and theism, which only makes sense of atheism is a claim about the existence of god (Huxley was an interesting figure in this regard)
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and the term is used in the relevant literature as a positive position about the belief in god.
End of conversation
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