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I saw someone pointing out that the primary point of "separation of church and state" is to make it clear that the President of the US is not also the arch-bishop of the country. That the bishop of Virginia doesn't have an automatic Senate seat. Etc. We're not even close to that level of entanglement.
Very true. However, the law was deliberately written vaguely enough to be applicable beyond the 18th century, and the courts (Supreme included) have ruled time and again that the principle of separation of church and state applies at all levels, including local establishments such as public schools.

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Lower-level stuff, like nativity scenes on public property, I tend to think should be handled democratically and locally.
Using my tax dollars to pay for the nativity scene in front of the local post office? I don't think so.

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I think as a generalization, "Christian country" works pretty well. In the same general way that Pakistan is Muslim, India is Hindu, France is secular, etc.
America is a secular country with Judeo-Christian leanings. It was not ever declared a Christian state. Pakistan, on the other hand, was founded with the intention of being a Muslim country. And if France (a country of many Catholics) can be classed as secular, so can the U.S.

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including the concept of separation of church and state, btw -- Jesus was specific about his followers also being tax-paying, law-abiding Roman subjects/citizens, regardless of the official Roman theology. There's nothing like that in Islam
I'm sorry, but the bolded portion of your statement is wholly inaccurate, as well as a tad offensive.