Major spelt out the first part of the conundrum in the Commons yesterday. Sinn Fein, he said, had "a huge question mark" hanging over them. "Their leaders have spoken often of their commitment to peace and peaceful methods. But they have always ducked and weaved when they have been questioned about the IRA...."
True. But he then went on: "Sinn Fein must decide whether they are a front for the IRA or a democratic political party committed to the ballot not the bullet." To which all one can say is, oh no, they mustn't. Just now, that would cut the IRA itself out of any peace process. In which case, what's the point?
The link between Sinn Fein and the IRA is murky. Perhaps, as rumour has it, some of the top Sinn-Feiners are on the IRA Army Council. Perhaps, as others whisper, there has been a growing apart. But the hinge between the two is an essential part of any wider negotiation. So is the hinge between some of the small Unionist parties and loyalist paramilitaries. No one demands that they make that relationship explicit, or renounce it - and rightly so.
These are not merely talks between politicians, but an attempt to draw terrorists away from terror. To pretend that the shadow of the gunmen is not present in the process, and necessarily so, is unrealistic. And though the Prime Minister suggested yesterday that, in the absence of Sinn Fein and the IRA, "the peace process goes on", this cannot, strictly speaking, be true. It would be like Lloyd George announcing a general peace conference in 1917 but forgetting to invite the German army.
This isn't to say that the IRA must not return to a ceasefire before talks can start. It must; but that is an entirely different point from trying to divide Sinn Fein finally from the IRA. One day, hopefully, the latter will wither. But until it does, there has to be a way to speak to it.
With that, the British and Irish governments would agree. If Major was really insistent that Sinn Fein smash their hinge with the IRA, he would presumably have demanded yesterday that they condemn the Canary Wharf bombing before there were any further contacts. But he didn't, not quite. And both governments are keeping open official-level links with Sinn Fein - what the Northern Ireland Minister John Wheeler called "sinews of contact".
The most important aspect of yesterday's statement was not the aggressive- sounding sentences designed to make headlines, but the emollient hints carefully inserted between them, and intended for a more knowing audience. It was in fact a careful and largely successful attempt to correct the failures of Major's response on 24 January to the Mitchell report.
It is very easy for commentators to play the hindsight game; so let me admit that, though Major got it wrong in his instant reaction to the Mitchell report, my instant reaction to his instant reaction was just as wrong. Then, he seemed to be making a big concession by retreating from his "no talks before decommissioning" line and going along with the Unionist proposal for elections instead.
What I hadn't properly appreciated was the damage caused by the way Major sprang his proposal. He believed that what he was delivering, however wrapped up, was a concession. Dublin and the nationalist parties in the north thought it was a delaying device. The failure to consult the Irish before making his statement on 24 January now seems a very serious blunder - the first one that Major has made in the whole tortuous process.
It was compounded by two other mistakes which none of us noticed at the time. He said that the elections wouldn't only provide a pool of delegates for all-party talks, but would be "a means to index the strength of the parties' delegations" in the talks. That may have been a fatal phrase. It suggested that these would not be peace talks - a meeting of two sides, whose views needed similar consideration - but would be majoritarian, inevitably dominated by the Unionist parties.
And the final mistake was Major's apparent assent to the Unionist timetable of "elections in April and May", long after the previously agreed deadline for talks. In each case, one can see why Major said what he said. And, as soon as he realised just how angry Dublin was, he got on the phone to John Bruton.
But it was too late. Damage was done. Why? Both Major and the Unionists know that their support for his government is selfish: the Ulstermen are in a better position now, with a small Tory majority, than they would be under a secure Labour government. So he had no need to concede anything to them. Cock-up is likelier than conspiracy - perhaps a failure of briefing at Number 10, from where a key foreign affairs official, Rod Lyne, had recently left.
That is for the historians. But whatever the reason, on 24 January the process had started to roll towards that old abyss, the gulf in comprehension that divides nationalists in Northern Ireland and the British elite. It is centuries old and deepened by accent and class. Both Major and, more particularly, the patrician Sir Patrick Mayhew can sound Anglo-smug about Gerry Adams. They may appreciate his problems but they cut him no slack, not an inch. Coming from a different world, they speak as if he is nothing more than a terrorist Godfather.
On a personal level, one can fully understand why they feel that way. They are targets. But without Adams, or someone like him, no peace is possible. If he is forced out or killed it may be 20 years before an entirely militarised IRA throws up a similar pivotal figure.
And Adams is now in a desperate situation. Some Tory MPs seem to think that by saying he either knew about the bombing, and therefore cannot be spoken to, or didn't, and therefore isn't worth speaking to, they are making a clever point. Maybe; but Westminster debating cleverness has never seemed so futile.
Major seems, at least, to half-accept that. In yesterday's statement, there was a new tone, designed to make things a little easier for the nationalists. He rammed home that any elections would be intended to lead quickly to all-party talks and that such a body would be strictly time-limited - not points he had emphasised three weeks earlier.
In reply to John Hume, who proposed a referendum on the need to renounce violence and set up all-party talks, the Prime Minister was flexible, accepting that there were "a range of options" about how to proceed. That hadn't seemed so on 24 January. His language on multi-sided talks mimicked that of Dublin, after a pre-statement call to Bruton; again, unlike three weeks ago. All this may make elections more plausible to Dublin and Washington.
This was, in short, an attempt to row back a little, to open some more space for dialogue without making any of the overt concessions that would have been unacceptable after a bombing.
And rightly so. And the next step is to admit that Adams is not a normal politician and that Sinn Fein is not a normal party; and that this is precisely why they need to be spoken to. John Major knows it; so do the rest of us - and so he might as well say so. Making peace is a mucky trade; just not as mucky as the other one.