The Democrats held the year’s first “official” mayoral debate Tuesday evening. The event was official because the city’s Campaign Finance Board sponsored it, with participation a prerequisite for obtaining public matching funds.

The “Gang of Four” had sparred previously, indeed frequently – so frequently that the forums made hardly a blip on the public radar screen. But Tuesday’s affair seemed to receive a bit more attention, not so much because of its official status but because the calendar tells us there will be just four more weeks until the Democrats either choose an opponent to Mayor Bloomberg or at least narrow the field.

Conventional wisdom says that only the front-runner, Fernando Ferrer, has a chance of achieving the 40% plurality in the September 13 primary that is needed to avoid a runoff. I agree.

The same wisdom holds that avoiding a runoff is a good thing for the Democrats. I disagree.

A runoff is the only chance the Democrats have to clearly define a rationale for themselves as they attempt to challenge the well-heeled and popular incumbent. Two weeks of intense back-and-forth between the top finishers would help define that rationale, if there is one. That will be difficult, however, as three of the four candidates Tuesday evening helped Mr. Bloomberg, by asserting that he was a better mayor than his predecessor, Rudolph Giuliani.

Since the easily re-elected Mr. Giuliani reduced crime to a degree unimaginable when he took office, halved the number of recipients of welfare benefits, reversed the decades-long perception that the quality of life here was declining, and provided memorable leadership in the dark hours of the September 11, 2001, terrorist attack, rating Mr. Bloomberg as a better mayor would seem to make a good case for his retention.

Why did Mr. Ferrer, C. Virginia Fields, and Gifford Miller say Mr. Bloomberg is a better mayor than Mr. Giuliani? It is because Mr. Bloomberg, the liberal former Democrat, is much closer to them philosophically. Mr. Giuliani was more resistant to raising taxes, was dismissive of race-based initiatives, and shunned the Reverend Al Sharpton.

Only Anthony Weiner avoided this trap. Indeed, the debate clearly defined the congressman as marching to a slightly different drummer than his Democratic opponents. Mr. Weiner implied that Mr. Bloomberg was not up the standard set by Mr. Giuliani. As his opponents run madly to the left, Mr. Weiner strolls calmly to the center.

He won the debate Tuesday, a perception shared by most of those who gave instant commentary on NY1 immediately following the forum, as well as by the callers and e-mailers who weighed in on the cable station’s “The Call” program a half-hour after the debate ended.

Whether that means anything is still anyone’s guess. It is the common perception of those, including myself, who have attended previous forums that Mr. Weiner almost always wins. Yet he languishes in the polls. If, however, the Tuesday debate is the beginning of an electoral blitzkrieg – as opposed to the sitzkrieg we have witnessed so far – then perhaps there will finally be some movement in Mr. Weiner’s direction.

Much discussed has been and will be Mr. Miller’s response to moderator Dominic Carter’s “lightning round” question whether the candidates would send their children to public school. On a certain level, the question was unfair. It personalizes public policy in an uncomfortable way, and in this field, it affects only Mr. Miller.

Both Ms. Fields and Mr. Weiner are unmarried and childless, although Mr. Weiner cleverly quipped that he would send his children to public school but needed to find a wife first. Mr. Ferrer’s daughter is grown, though he inexplicably gave a misleading answer, something instantaneously picked up on by reporters: He stated that she “graduated from” public school, which was true only through the eighth grade, but Cardinal Spellman High School, a parochial school, was the school from which she graduated.

It was Mr. Miller who was put on the spot. His children are 3 and 4 years old. I don’t think that there was anyone in the room last night who believed, as Mr. Miller angrily asserted, that a school choice hadn’t already been made, at least for the older child. Nor would I expect that the choice was anything other than the same type of elite private school that Mr. Miller himself attended. The point was very forcefully made by El Diario’s Gerson Borrero on the post-debate analysis show on NY1.

When you make education the central part of your public agenda, as Mr. Miller has tried to do with his initiative on class size, private school is not the answer your supporters want to hear. Nor is evasion a popular option, the reason Mr. Miller received a chorus of boos when he tried to dodge Mr. Carter’s bullet.

Mr. Ferrer, as front-runner, had the most to lose on Tuesday, and he tried mightily to offend no one and avoid mistakes. It seemed that every answer was carefully scripted, making the former president of the Bronx seem insincere. Competing for much the same constituency as Mr. Ferrer, Ms. Fields, the Manhattan president, tried, weakly, to confront him on the “flip-flop” controversy by raising abortion and the death penalty, issues he has visited from both sides. Mr. Ferrer emerged unscathed.

The question Ms. Fields couldn’t bring herself to ask him, the one that could put her back in the race, was: “Do you believe that the four officers indicted in the fatal shooting of Amadou Diallo should have been convicted of the crime with which they were charged?”

Ms. Fields did poorly, which is too bad. One on one, she is sharper and better-informed than she appears to be in these forums. She did score with a cogent response on the technical question regarding emergency response, so perhaps, if she can confront Mr. Ferrer more effectively next time, she can, after all, win the runoff spot she covets.