Sorry, I don't see the relevance of the odd and even analogy - we are discussing a simple polarity, whether a specific entity exists or not. There are only two possible states; 1. The belief that God exists and 2. The lack of that belief.
Correct. But option 2 - lacking belief - is not the same as holding a positive belief in the non-existence of God. The odd/even analogy is used to illustrate that saying "not X" is not equal to saying "therefore Y", even if X and Y are mutually exclusive positions (just as a quantity of something is either even or odd and cannot be both or neither, God either exists or doesn't). By saying "I don't believe the number is even" I am not saying "the number is odd", even though that is the only other option. Belief is being withheld. In this analogy, not believing the quantity is even is analogous to atheism, as being an atheist only requires not believing in God, not the positive belief that God does not exist. However, both lack of belief and belief in God's nonexistence come under the heading of atheism. They aren't the same position, but they are both atheistic positions.
Sorry, that does not make sense to me - if you don't believe God exists, you therefore believe he does not exist. There is no third position.
Again, if you can understand why saying "I don't believe the number of grains of salt in the ocean is even" does not mean you must necessarily say "the number of grains of salt in the ocean is odd" - despite the fact that you would agree they are the only two possible options - then you should be able to understand how a person is capable of saying "I don't believe God exists, but I don't believe God does not exist".
To be honest, I think that the point people seem to be missing is that weak and strong atheists are not different groups, they are not exclusive categories. They are responses to different claims.
This is something I agree with and have addressed. Both positions are atheistic in that they both come under the heading of lacking a belief that God exists, and both address different claims.
Strong and weak atheism are frankly pretty silly and arbitrary distinctions, invented by Christian apologists in order to try to paint atheism into some kind of corner. The reality of course is that weak atheism is the stronger and more sustainable position, and strong atheism essentially a strawman. An atheist who says that there is no God is in reality no more making a knowledge claim than is the weak atheist - and the burden of proof, the rationale remains the same.
There is an extent to which I agree with you, and it's one of the reasons I argue so strongly for the broad definition of atheism in the majority of atheism/theism debates. There is a tendency in kinds of debates to frame the discussion in terms of a battle between two extremes, with a vast "neutral space" in between that people can escape to in order to evade the actual debate entirely. It's useful for some theists, because they can acknowledge their own position as being at one end of the scale while criticising atheists for doing the same (you see it all the time in many debates: "You have faith too", etc.), while people who are self-identified "non-theist-non-atheist-agnostics" can simply cast a wry eye on both sides of the debate and smirk at the apparent stupidity of these two "totally identical" sides. I see the debate merely as those who accept the proposition a God exists vs. those who do not, and I think it is more conductive to frame the debate in those terms not just for purposes of the null hypothesis, but also to close up that middle ground entirely.
However, I do also think there is room for discussion about positive claims of God's non-existence, which is a claim that should be addressed separately but is no less worthy of discussion. The claim that God does not exist still carries a burden, but it is a burden that should be met in any discussion in which that is the singular claim being addressed - not any discussion about God's existence rather than non-existence.
