O'Connor's opinion on Church and State

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The two words mean two different things. One is the Muslim god, the other is either an all-encompasing term for deities, or the Christian god.

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Actually this is not correct. Islam recognizes god just the same as the christian god. Both are monotheist religions (Judaism and Sikhism are the only other two that I know of). The Koran explicitly recognizes and offers protection to the people of the Book (which includes Christians and Jews) and asks for the conversions of those who dont believe in a single all encompassing God. Historically, this resulted in masses of pagans in the middle east being converted from their ways into a monotheist religion with roots in the old testament.

So, the God that Christians refer to is the same as Allah. You can argue that these are two paths to the same goal and you would be correct.