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“She signed a bill she could not defend in court and it’s led to boycotts, to jeopardizing our tourism industry and polarizing our state,” said Mr. Goddard.

The Goddard campaign has also attacked Mrs. Brewer on the jobs lost during her tenure as governor and the state’s budget woes. Her refusal to participate in a second debate “is a disservice to Arizona,” said Goddard spokeswoman Janey Pearl.

Mrs. Brewer’s numbers did dip slightly from last month’s Rasmussen survey, which showed her leading by 60 percent to 38 percent. Unfortunately for Mr. Goddard, his numbers remained virtually unchanged, meaning that most of those who abandoned Mrs. Brewer joined the “undecided” camp, not his.

Brewer spokesman Doug Cole said that Arizona voters are already more than familiar with Mr. Goddard, who served previously as mayor of Phoenix and unsuccessfully sought the Democratic gubernatorial nomination in 1990 and 1994.

“The people of Arizona know who he is. Why should we give him a platform to try to remake himself?” said Mr. Cole.

He also dismissed the impact of her previous debate gaffe. “The people of Arizona appreciate a politician who tries to collect their thoughts,” Mr. Cole said, “as opposed to a politician who rambles on.”