Theresa May gathered her cabinet together for a brainstorming session at Chequers yesterday. About time. Two months on from the EU referendum, and while we all know that “Brexit means Brexit”, we’re still none the wiser about what “Brexit” itself means.

International Trade Secretary Liam Fox seems to think it means he should take over large parts of the Foreign Office, and anything to do with trade and Europe. Foreign Secretary Boris Johnson is equally certain it does not. Nemo me impune lacessit, as the self-styled classics scholar might put it – don't mess with me.

Brexit Secretary David Davis says it is “ten times more difficult than the Schleswig Holstein Question”. If so, answers may not be forthcoming for some time. Lord Palmerston said of Schleswig-Holstein in the 19th Century: ”Only three people have ever really understood it: the Prince Consort, who is dead, a German professor, who has gone mad, and I who have forgotten all about it." There are now three UK cabinet ministers in charge of Brexit and they may be going the same way.

We’ve heard practically nothing from the voluble former Mayor of London except those cryptic remarks in Brussels in July about Brexit not meaning Britain leaves Europe “in any sense”. We've heard a lot about musical chairs in Whitehall as ministers bicker about who should get the biggest offices and the most civil servants. But there have been no significant policy statements from any of the lead ministers on the most important constitutional and diplomatic issue facing Britain since we joined the EU in 1973.

As for the Prime Minister, saying she is all over the place would be an insult to people lacking a sense of direction. We were initially led to believe she was going to bulldoze Brexit using her powers under Royal Prerogative. “Like a Tudor monarch”, as the Labour MP Barry Gardiner put it. This didn't go down too well with the opinion-forming classes because we are supposed to be a democracy. Referendums are advisory and MPs are meant to take these crucial decisions in our sovereign parliament.

But it would, of course, be an abomination for parliament to simply reject the referendum. MPs don't have the authority any more to ignore such a direct expression of the people's will. However, nor is it acceptable for them to have no say at all, as Mrs May initially implied. So that was changed, and her official spokesman has now said that, of course, MPs would “have a say” on Brexit. Which only raises further questions about what having a say would mean. Would they have a vote? Would that vote mean anything? Would the House of Lords be able to veto a Brexit motion? And can it be amended by either chamber of parliament?

What about the Scottish Parliament? Does it have a say too? The PM has suggested it would. After her Bute House meeting with Nicola Sturgeon in June she said she wouldn’t begin the Article 50 negotiations on leaving the EU until there was an “agreed UK approach”. No one seriously believes this gives Ms Sturgeon a veto, but it did suggest there could be significant leverage.

But to have a deal, with sidebars and concessions, you have to have a negotiation, and until we know what is being negotiated, there can't be any meaningful talks about it. Which brings us back to yesterday's Chequers brainstorm. It would have been fascinating to be a fly on the wall, and no doubt the many flies around the table were taking mental notes. Forget the motor industry, the prospects for British publishing have never looked more rosy as ministers and civil servants plan their memoirs. Mr Johnson is probably onto his second draft.

Mrs May insisted we are definitely leaving the EU and there should be no loose talk of a second referendum, as advocated by Labour leadership challenger Owen Smith. This doesn't mean the Brexit process couldn't be stalled or that another general election couldn’t intercede before negotiations are over or even begun. The PM says Article 50, won’t be invoked until January 2017 at the earliest. She is likely to wait until the German and French elections are out of the way later in the year. This means negotiations with Brussels, which take a minimum of two years, may not be completed until the very eve of the 2020 general election, which could itself become a de facto referendum on the Brexit settlement.

If by some miracle, Labour is elected in 2020 on a Remain ticket, Britain could theoretically find itself negotiating re-entry to the EU even before we’ve actually left it. What would Brussels say about that? Probably: you'll have had your Brexit, live with it. The EU may anyway have assumed a rather different shape by then, with a deepening of financial and military integration, and a renewed determination to impose the single currency on all its members. It seems most unlikely Britain would vote to adopt the euro.

But we’re getting ahead of ourselves. Loosely speaking, Brexit currently has a range of meanings in the UK cabinet. Chancellor Philip Hammond and, we understand, Mr Johnson, are arguing for a soft Brexit in which Britain remains in the European Single Market (ESM). They're expressing the interests of what might be called the Whitehall-industrial complex – the UK civil service and the big corporate interests that lobby Whitehall and overlap with it. Richard Branson, the banks, car manufacturers and above all the City of London want as little disruption as possible and believe remaining in the ESM, while out of the EU, provides the best bet for continuity. They envisage joining Norway in the European Economic Area. (Actually, Norway is not terribly keen on big Britain being in this small nation club, but that's another issue).

At the other end of the Brexit spectrum are the true anti-Europeans like the former Chancellor, Lord Lawson, and Tory backbenchers like Jacob Rees Mogg who said yesterday that: “staying in the single market is code for rejecting the referendum result itself”. He says the ESM is the heart of the EU and source of the vexatious regulations and of free movement, neither of which the British people will tolerate. These Brexit hard-liners want nothing to do with Europe and hope to renegotiate Britain’s trading links with the rest of the world from scratch through the World Trade Organisation. Brexit Secretary Mr Davis is thought to be with the fundamentalist faction.

This division is about as deep as it gets and you wonder how these two views can co-exist in one government. If Labour could only pause their civil war for one moment, they would realise the Tories are probably more divided than they are. Tomorrow, Nicola Sturgeon will be announcing her third relaunch of the next independence campaign. But it's difficult to know what she can say about the nature of the Union Scotland would be leaving, given the lack of clarity about the EU the UK is leaving. It's a Brexit imbroglio. Westminster has simply stopped making sense.