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“If I were in Congress in 1941, I would have signed a declaration of war against Japan,” he said. “I’m not going to attack Japan today.

“The bottom line is that we can’t have sequestration, we can’t go off the fiscal cliff,” Mr. King said.

Sen. Lindsey Graham, South Carolina Republican, said flatly that he is willing to violate Mr. Norquist’s pledge if Democrats agree to cuts in entitlement programs. Mr. Graham said he still opposes increasing tax rates but that he supports capping deductions to help the federal government balance its books.

“I agree with Grover we shouldn’t raise rates, but I think Grover is wrong when it comes to [not capping] deductions,” Mr. Graham said Sunday on ABC’s “This Week.”

On Friday, two days after Sen. Saxby Chambliss of Georgia said, “I care more about my country than I do about a 20-year-old pledge,” Mr. Norquist fired back with a warning that lawmakers who break the pledge will face consequences from their own constituents, not him.

If Mr. Chambliss “wants to change his mind and become a tax increaser so we don’t have to reform government, he needs to have that conversation with the people of Georgia,” Mr. Norquist said on CNN.