(CBS News) Republican presidential candidate Herman Cain said Tuesday that he would be willing to take a lie detector test to rebut multiple claims that he engaged in sexual harassment in the 1990s, though he seemed to stop short of promising to do so.
"Yes. I absolutely would," Cain said when asked about taking a test. "But I'm not going to do that unless I have a good reason to do that. Of course I would be willing to do a lie detector test."
Sharon Bialek has accused Cain of putting his hand under her skirt and pushing her head toward his crotch after a dinner meeting in 1997, after she had been let go from the National Restaurant Association. She was the first woman to make the claim publicly, though three other women had done so anonymously.
During a press conference in Scottsdale to address the scandal threatening to engulf his campaign, Cain repeated his claim that he did not recognize Bialek when he saw her at her press conference Monday and said he did not remember her name.
"I tried to remember if I recognized her, and I didn't. I tried to remember if I remembered that name, and I didn't," he said.
As he has repeatedly, Cain categorically denied having all of the allegations leveled at him, saying, "I have never acted inappropriately with anyone. Period."
As for Bialek's claims, Cain said, "They simply didn't happen. They simply did not happen." He said he would not allow "false allegations" reflect negatively on his character.
Cain said that his withdrawal from the presidential race "ain't gonna happen" and vowed "we will get through this. We will get through this."
"The Democrat machine in America has brought forth a troubled woman to make false accusations," said Cain, who did not offer support for the claim that "the Democrat machine" was behind Bialek's decision to go public. Bialek says she is a Republican and member of the Tea Party.
Cain acknowledged that he'd had a "tough" time since the "media started to beat me up" when the story broke last Sunday. He said that "when we allow deceit and false accusations to rule the day and distract us, that part [of the electoral process] is broken."
Cain relayed comments from his wife upon hearing the allegations, quoting her as saying Bialek's allegation "doesn't even sound like anything you would do to anyone." He also called on the media to "not drag my family into this."
Cain said he considers sexual harassment to be a "serious" issue and said respect for women is a "top priority" for him. He said he had seen sexual harassment in the workplace personally, both with men harassing woman and "where women have attempted to sexually harass men."
He also repeated his claim, first made earlier in the day, that the charges were surfacing because there is "an element in this country, in our politics, that does not want to see a businessman succeed at getting the nomination from the Republican Party and does not want me to succeed at becoming president of the United States of America."
Cain's campaign has aggressively attacked Bialek, emailing reporters a note Tuesday laying out what it cast as her "long and troubled history, from the courts to personal finances." Asked if financial issues were a motivation in her coming forward, Cain said Tuesday, "From a common sense standpoint, one would have to ask if that may have been a motivation."
On CBS' "Early Show" Tuesday, Bialek said she came forward because she "wanted to help" Cain come clean about his personal history.
CBS News has confirmed Bialek filed for bankruptcy in 1991 and again ten years later. She also has been sued at least six times -- for amounts under $5000 over such issues as nonpayment of rent. She was evicted once. She also won and got child support in a paternity suit.
Bialek says she is not being paid for having come forward. According to the Chicago Tribune, she lives with her fiance, an executive in the medical equipment industry, in an "upscale home" in the suburbs of Chicago.
Bialek is the fourth woman to claim that Cain has acted inappropriately with her -- but the first to do so publicly. The identity of another of Cain's accusers, 55-year-old federal employee and registered Republican Karen Kraushaar, was revealed by iPad newspaper The Daily on Tuesday. A number of media outlets, including CBS News, had previously withheld Kraushaar's name at her request.
Karen Kraushaar, Herman Cain sexual harassment accuser, goes public
Kraushaar's attorney said last week that Kraushaar "stands by the complaint that she made" of "inappropriate behaviors and unwanted advances" by Cain, though he did not get into specifics.
After her name was revealed Tuesday, Kraushaar told CBS News that she is interested in holding a press conference with all of women who claim Cain sexually harassed them.
She said "the whole situation is so volatile right now I just don't want to say anymore." Speaking to the New York Times, Kraushaar said, "When you are being sexually harassed in the workplace, you are extremely vulnerable. You do whatever you can to quickly get yourself into a job some place safe, and that is what I thought I had achieved when I left."
Asked about Kraushaar Tuesday, Cain said "that is the one [accuser] that I recall that filed a complaint but it was found to be baseless."
"When she made her accusations, they were found to be baseless and she could not find anyone to corroborate her story," he added. Cain said of Kraushaar that he "didn't have regular interaction [with her], periodically I would see her."
Asked to discuss what specifically went on with Kraushaar, Cain repeated a story in which he said he had told Kraushaar that she is the same height as his wife, "because my wife also comes up to my chin."
Cain was introduced at his press conference by attorney L. Lin Wood, a prominent Atlanta-based Libel and Defamation attorney.
Wood represented Richard Jewell, the accused 1996 Olympic bomber, John and Patsy Ramesy and many other high-profile clients including Gary Condit, the victim in the Kobe Bryant case, and the mother of Anna Nicole Smith.
Wood said Cain is "on trial in the court of public opinion, falsely accused."
His client "has to respond not to admissible evidence, he has to respond to hearsay, he has to respond to rumors and speculation," Wood said.