I'm not sure I can absolutely distinguish between absence and non-existence. Absence might be just "not here", but absent everywhere is non-existence.Seems to me that you're dealing above with absence rather than non existence. For instance, My arm , if it has not been amputated, cannot be anywhere that I am not nor can I be somewhere that my arm is not. Therefore if my arm is not in my house I am not in my house. The absence of me in my house is not proof of my non existence.
This is not my argument but:
P1: If God exists, prayers are answered.
P2: Prayers are not answered.
C: God doesn't exist.
In this case, there isn't a difference between absence and non-existence since P1 assumes that God's location is irrelevant.
Now of course we can spend hours debating whether P1 is true then a few more hours debating P2. Then we can argue about what 'answered' means. But, the point is that if we were to agree on all the terms and on the truth of the propositions, then we would have proven the non-existence of the god presumed in the argument.
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