Leave aside for a moment the unintentionally funny spectacle of a member of the current Republican Party declaring that religion should be kept out of public life, and that Mr. Carson, as an African-American, is a member of a much belittled minority. The freedom of religion embedded in the First Amendment rules out the very idea of a religious test for public office, as John F. Kennedy so eloquently argued and then proved by becoming the first Catholic president.

As for Shariah law, Catholicism has canon law and Judaism has the Halakha and nobody is painting them as threats to the republic — at least not this year.

Following Mr. Carson’s comments, some Republicans tried to suggest that anti-Muslim feelings existed only in parts of the party’s grass-roots base. Some argued that former President George W. Bush drew a line between the country’s antiterrorism efforts after 9/11 and a broader campaign against Muslims. “We’ve worked so hard to try to make it clear that that isn’t the case,” said Tony Fratto, who was a press aide to Mr. Bush. “But each time somebody does this kind of thing, it makes it harder.”

But Mr. Carson is hardly alone in his demagogy. Donald Trump has accused President Obama of being foreign born, and is continuing to stoke suspicions that Mr. Obama is a Muslim — a myth that a disturbing number of Republicans believe. Last week, Mr. Trump welcomed a question from a man who asserted “we have a problem in this country, it’s called Muslims,” that “our current president is one,” and the man asked “when can we get rid of them?” Instead of telling the truth, that Mr. Obama is an American-born Christian, Mr. Trump said, “We’re going to be looking at that and plenty of other things.”

Senators Ted Cruz of Texas and Lindsey Graham of South Carolina publicly disagreed with Mr. Carson, but neither truly matters in the campaign or the party. Other Republican hopefuls — Governors Chris Christie of New Jersey and John Kasich of Ohio, and Jeb Bush at least deigned to acknowledge that Mr. Obama is American and is not Muslim. But these efforts have been tepid, at best, because the Republicans are playing to polls that show support for Muslims ranks far below other religious groups among Republicans and Republican-leaning voters.