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Presidential candidate Herman Cain on Sunday defended his opposition to a new mosque in Tennessee, expressing concern about Shariah law and declaring Americans "have the right" to ban mosques in their communities.

Cain, who stirred controversy this year by saying he would be uncomfortable appointing a Muslim to his Cabinet if elected, first expressed concern Thursday about the controversial mosque in Murfreesboro, Tenn. That mosque has been the subject of demonstrations and legal challenges in the wake of the controversy over the so-called "Ground Zero mosque" in New York City.

Speaking on "Fox News Sunday," Cain said he came out against the Tennessee mosque after talking to members of that community. He said the site is "hallowed ground" to Murfreesboro residents and that they're concerned about "the intentions of trying to get Shariah law" -- the code governing conduct in Islamic societies.

"It's not just a mosque for religious purposes. This is what the people are objecting to," he said.

Asked whether any community should be able to prohibit a mosque, Cain said they should.

"They have the right to do that. That's not discriminating ... against that particular religion. That is an aspect of them building that mosque that doesn't get talked about," he said.

Cain again argued that residents were objecting to "the fact that Islam is both a religion and a set of laws, Shariah law. That's the difference between any one of our other traditional religions."

But while Cain said he expects the case to come before the Supreme Court, a local judge has allowed the project to go forward.

Cain is taking heat for his comments about Muslims. The Council on American-Islamic Relations, which accused him of using "bigoted" language with his Cabinet comments, said Sunday that he should "apologize" for his latest remarks.

CAIR spokesman Ibrahim Hooper, describing Cain's stance on the Tennessee mosque as a possible "sign of desperation," said other Republican candidates and leaders should also distance themselves from that kind of rhetoric.

"It's incumbent on reasonable people within the Republican Party to come out strongly and repudiate these kinds of un-American unconstitutional views," he said. "It's just so bizarre."

Cain, the former CEO of Godfather's Pizza, is struggling to build on earlier momentum and break into the top tier of candidates in the 2012 Republican primary race. He expressed confidence Sunday that he could have a strong showing in an upcoming Iowa straw poll.

FoxNews.com's Judson Berger contributed to this report.