The latest cunning strategy for repairing the reputation of the Prime Minister is to keep him off the television screen. On the very first day of Gordon Brown’s holiday in the Lake District, Lord Mandelson blew that strategy off course by suggesting that the Prime Minister was now ready for a TV debate with David Cameron during the next general election campaign. Until now, Mr Brown has fended off the request for a debate with the excuse that served Tony Blair — that Prime Minister’s Questions allows voters a weekly assessment of the party leaders in combat. Downing Street insists that Mr Brown’s reluctance still stands.

But Lord Mandelson rarely speaks without prior calculation and there is a simple reason why the Prime Minister’s opposition to a debate could be softening. That reason is the state of the opinion polls. Lord Mandelson has conceded that, in the electoral battle to come, the Labour Party is now the underdog. Suddenly a TV debate — which, it is usually thought, helps the struggling candidate — looks a more attractive proposition. The hope, probably forlorn, is that a debate allows the candidate who is being out-boxed a puncher’s chance of victory.

This changes the logic for Mr Cameron, for the inverse reason. When he was a challenger a debate seemed to be in his interest. Now that the Conservative Party is consistently polling numbers that would give it an overall majority in the next Parliament, Mr Cameron could be less keen. He can hardly admit that he has changed his mind, having demanded a debate so often. No doubt Mr Cameron genuinely feels confident that he would perform well. But, with every successive opinion poll, a live debate is beginning to look like an unwarranted risk.

Beyond all this political calculation, there is another way of looking at the question. That is to ask: is it in the national interest for the main pretenders to Downing Street to enter a TV debate? Lay aside all the negotiations about the style of moderation, the venue, the presence of Nick Clegg, who speaks first and whether they can question one another. Whatever the format, would it improve our democratic process if we had a TV debate during the electoral campaign?

Although Britain’s polity is not presidential, the media age magnifies the importance of the leader. A live debate would generate a lot of interest in the arguments that, at the moment, are poorly carried by choreographed events, vague manifestos and clunky photo opportunities. It would give the candidates a great opportunity to crystallise their message to the nation, to offer a critique of their opponent and, each in turn, to rebut that critique. It would help to define the terms of the election and it may help to expose the strengths and weaknesses of the respective arguments on which the parties choose to stand.

It is unlikely that a TV debate would alter the course of the election. Even when presidential debates have had an impact it has usually been to give a visual metaphor — think of Richard Nixon’s five o’clock shadow — to an impression already held. But an open debate, on primetime television, is long overdue. Chesterton once said that “people generally quarrel because they cannot argue”. The quality of the political conversation in Britain is not good. A live debate will not solve that problem but it will help.