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Logic can exist within the materialist world view. Materialism just requires that it arise from the material nature of reality.
No, you have to explain how logic is universal in your world view. We all operate under the law of non-contradiction, it's part of the basis of rational thought.
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Atheism tells you nothing about a persons belief system other than whether or not it includes gods. That's it.
Yes, that's a belief system.
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Yes, I acknowledge that all beliefs (even mine) are built on certain assumptions that might not be provable. I could be a brain in a jar hallucinating all of this. I can't prove otherwise, but I assume that isn't the case. We assume that the natural laws we observe locally are consistent elsewhere, but without verifying we can't know that to be the case. We'll never be able to escape the fact that our beliefs have some amount of assumptions baked in. So we're left to argue which ones are the most reasonable.
You assume uniformity of natural laws? Why? Don't you see the amazing amount of tension in assuming uniformity in a world where there is nothing but the random process of atoms and molecules as if they could be driven to create order instead of disorder? It's not a question of what is most reasonable, it's the question of what is even possible. This is the point very point of the OP.
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There's a bit of misdirection going in your post when you talk about "truth". The debate about objective and subjective truth is typically used when discussing morality with the atheists usually on the side of rejecting objective moral truth. It doesn't mean that atheists reject the idea of a shared reality in which we all coexist as actually existing and being observable.
There is no misdirection, you are making the point. Yes, for one, this is why ethics are not tangential, they are a fundamental basis for belief. Belief entails knowledge, knowledge entails what we think is true. Your epistemology is intimately tied with ethics. That's why, for any rational world to exist, these categories have to have a transcendent basis because for them to arise from within the material world would only make them relative. In reality, we don't operate on relative particulars, we "assume" (to use your word) objective universals.
But once again, an atheistic world view cannot provide the very fabric of what you need to make your very own argument.
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Fundamentally what I see when I read your post is a massive misunderstanding of what atheists actually believe and why.
Fundamentally, there are only a handful of ways of viewing metaphysics. You can talk about nuance, but they all boil down to a few different frameworks.
Naturalism vs supernaturalism, impersonal supernaturalism vs personal supernaturalism. Even Platonic dualism boils down into these categories.