THE BOOING BEGAN before CNBC moderator Maria Bartiromo could finish the question. “You know that shareholders are reluctant to hire a CEO where there are character issues,” she said, addressing presidential candidate Herman Cain during Wednesday’s GOP debate. “Why should the American people hire a president if they feel there are character issues?”

It was a legitimate question and one that Mr. Cain must address.

Four women have independently asserted that they were sexually harassed by Mr. Cain. Two of them worked at the National Restaurant Association (NRA) during the 1990s, when Mr. Cain headed that organization; the women secured settlements from the association.

Mr. Cain, who leads in polls for the GOP nomination, has denied the allegations, saying during a Nov. 8 news conference that he has “never acted inappropriately with anyone. Period.” But his shifting accounts do not instill confidence.

Mr. Cain first denied, then acknowledged, knowing about the NRA complaints or settlements. Without a shred of evidence, he accused the rival campaign of Texas Gov. Rick Perry (R) of leaking the existence of settlements; he later asserted that Democratic operatives were responsible. Most recently, his campaign claimed falsely that the son of one of the women is a reporter with Politico, which wrote the first story on the matter. It did not help that Mr. Cain’s lawyer, L. Lin Wood, warned accusers to “think twice” before coming forward. Mr. Wood says he was cautioning that there could be legal consequences for those who lodge false accusations.

The women, too, have contributed to the uncertainty. The restaurant association waived a confidentiality agreement at the behest of Karen Krau­shaar, one of the women who settled with the NRA, but neither woman has provided a full accounting.

Through her lawyer, Ms. Kraushaar claimed that Mr. Cain engaged in a “series of inappropriate behaviors and unwanted advances.” But Ms. Kraushaar would have no further comment “unless and until the other women come forward,” according to a statement from her lawyer.

If she wishes to speak, Ms. Kraushaar should come forward and tell the truth regardless of what others decide. At the least, she should ask the restaurant association to make public her original complaint.

On Nov. 7, Sharon Bialek described an incident she says occurred some 14 years ago after having dinner with Mr. Cain. While in his car, Ms. Bialek said he slid his hand under her skirt and grabbed her “head and brought it toward his crotch.” When she rebuffed him, Ms. Bialek claims he said, “You want a job, right?” Ms. Kraushaar’s lawyer told the New York Times that Ms. Bialek’s account was “very similar” to his client’s and that it “corroborates the claim.” He has declined to provide specifics.

Sexual harassment is immoral, not to mention illegal. If the accusations are true, they depict a man who attempted to use his position of power to coerce sexual favors from subordinates or vulnerable women. The truth may be impossible to discern, given the time that has lapsed. But the public has a right to as much information as possible to weigh the competing accounts and to make a determination about Mr. Cain’s fitness for office.