Negotiations on Britain’s withdrawal from the EU have now begun in earnest. They are required, according to article 50, to “take account of the framework” for Britain’s “future relationship with the union”. But what is that future relationship to be?

Economically, the EU comprises three elements: a free trade area; a customs union (an area with a common trade policy and a common tariff); and an internal market in which non-tariff barriers to trade (regulations, standards and the like) are harmonised and, indeed, reduced.

Theresa May has declared that the UK will not seek to remain in the single market or the customs union. Campaigning for remain in 2016, she said of the single market: “We would have to make concessions in order to access it; these concessions could well be about accepting EU regulations over which we would have no say, making financial contributions just as we do now, accepting free movement rules just as we do now, or quite possibly all three combined.”

Both membership of the single market and a customs agreement would require us to accept much EU legislation without being able to help formulate it – and not just existing legislation, but any legislation the EU chooses to enact in future. We would become, in effect, a satellite of the EU, relying on the European commission or other member states to defend our interests. Such an outcome – regulation without representation – proved unacceptable to Americans in the 18th century. It would probably prove equally unacceptable to the British people in the 21st.

So, when May said that Brexit means Brexit, she was merely drawing out the constitutional logic of the referendum decision. For there is no logic to a “soft” Brexit – a form of withdrawal that mimics EU membership, but without the influence that comes from membership. The ultimate choice we face is either “hard” Brexit or remain.

Britain will be negotiating, therefore, for a free trade agreement in a “hard” Brexit. But, sadly, our negotiating position may not be very powerful. If one leaves a tennis club because one does not wish to pay the subscription and does not like the rules yet still wishes to play tennis, one’s leverage is not strong. One is a supplicant.

In addition, a trade agreement would probably have to be ratified unanimously by the European council, by a majority in the European parliament, and 27 national and 11 regional parliaments – and we are up against a two-year time limit. There is, apparently, a Japanese saying to the effect that the shorter the time limit, the deeper your wallet needs to be.

Some British politicians suffer from an imperial reflex, however. For them, Britain lies at the centre of the world. We only have to state our aims and other countries will be generous enough to help us achieve them. Last year Brexiteers argued that Britain should leave an EU composed of ill-intentioned foreigners whose interests were in conflict with its own. This year it has been magically transformed into a charitable institution that can be relied on to safeguard our interests.

But now we have had the general election. May called it to resolve the European question and strengthen her negotiating hand. Had she gained the landslide she hoped for, the referendum result would have been confirmed and Brexit would be assured. But the election re-opens the issue of Europe – for four reasons.

First, there is probably no Commons majority for May’s version of Brexit. Indeed, there is probably a stronger representation of remain MPs in parliament today than before the election.

Second, Labour’s electoral gains raise the question of whether the decision in the 2016 referendum is final: for, although Labour was not a remain party this year, the British Election Study found that the party’s “soft Brexit” policy played a large part in its substantial gain in votes. In constituencies where over 55% voted remain, the party achieved a swing of around 7%. The election was the revenge of the remainers.

‘When he thought he would lose, Nigel Farage [above] said a further referendum would be needed.’ Photograph: Neil Hall/Reuters

Third, the election intensifies internal divisions in both major parties. If the eventual deal is too “hard”, Conservative remainers may join with their opposition counterparts to defeat it; if too “soft”, Tory Eurosceptics could ensure its rejection. There may be no majority for any of the forms of Brexit on offer, with the Commons deadlocked.

Fourth, the House of Lords – in which the pro-remain Liberal Democrats and crossbenchers hold the balance of power, and the proportion of remainers is probably even higher than in the Commons – will feel emboldened to reject a hard Brexit, arguing that a minority government has no mandate for it.

With a deadlocked parliament, the possibility of an unfavourable deal and both parties so deeply divided on Europe, it may start to appear that the only way out of the impasse is a second referendum in which the government’s deal is put it to the people for legitimation.

That appears unlikely at the moment. Yet a referendum on Europe appeared even more unlikely when, in 1971, Tony Benn proposed it to Labour’s national executive but failed to find a seconder. James Callaghan presciently declared that for a divided party, the referendum might well prove a “rubber life raft into which the whole party may one day have to climb”. The Conservatives too may come eventually need that life raft.

Why are those who based arguments for Brexit on the will of the people now opposed to consulting the people Lord Butler

In the House of Lords on 21 February, the former cabinet secretary, Lord Butler, inquired why it was that those “who base arguments for Brexit on the will of the people are now opposed to consulting the people on the outcome of the negotiations”. When he thought he was going to lose in 2016, Nigel Farage said that a further referendum would be needed: there is no doubt that Brexiteers would have continued their campaign to take Britain out of the EU and they would have had every democratic right to do so. But so equally do those who have doubts about the decision.

Brexit after all raises fundamental, indeed existential, issues for the future of the country. That is why the final deal needs the consent not only of parliament, but of a sovereign people.

• Vernon Bogdanor is professor of government at King’s College London. This article is based on his Gresham lecture delivered in June.