Are we heading for another election? It remains the Labour leadership’s official position that the best way out of the Brexit mess is a fresh electoral contest, and a growing number of Conservative MPs believe that it may take another election to fix the mess.

It’s become customary to say that an election “wouldn’t change anything”, sometimes, but not exclusively, from supporters of a second referendum, who also believe that the only way to end the deadlock is to get the British public to weigh in.

But this isn’t true: an election could change the Brexit crisis in a number of ways. Most obviously, every Labour gain makes it slightly easier to pass some kind of Brexit deal – the number of Labour MPs who are willing to contemplate a no deal exit is two, and there is, to my knowledge, no prospective parliamentary candidate in any of the 99 seats Labour has already selected its standard-bearer in who supports no deal. The number of Conservative MPs who are already in Parliament who are willing to contemplate no deal is considerably higher than two. There are no seats which Labour could plausibly gain from the Conservative Party that don’t make it easier to reach a Brexit deal.

In Northern Ireland, every other electorally viable political party supports the backstop, so any losses by the DUP also make it easier to pass a Brexit deal, whether through directly adding to the pile of MPs willing to back an accord, or, if the DUP lose seats to Sinn Féin, by reducing the number of votes needed to pass anything, as Sinn Féin do not take their seats.

But the risk with an election is that while those outcomes make reaching an accord more likely, almost every other outcome makes the Brexit deadlock worse not better. Even in an election with no net gains, it is possible – perhaps even likely – that not every Conservative MP currently backing some form of Brexit deal will be re-adopted by their local parties. Added to that, in the event of an early election, all three of the former Conservative MPs who have joined the Independent Group would certainly lose their seats, and in two cases (Sarah Wollaston and Heidi Allen) they are certain to be replaced by Conservative MPs, who may well be supporters of no deal, whether through genuine ideological commitment or because of pressure from their local parties.

Any gains made by the SNP will be on a explicitly anti-Brexit platform and many of their party’s MPs are of the opinion that it would be “impossible” for them to back any sort of Brexit as their voters are strongly opposed and it would defang their main strategic argument that Brexit represents a sufficient change that there must be another referendum on Scotland’s relationship with the European Union.

As for the Liberal Democrats, likewise, any gains by that party will come on an explicitly anti-Brexit platform, which makes it hard for them to back any flavour of Brexit even with a deal. But the problem is that, while replacing a Tory MP with a Labour one makes it easier to pass a Brexit deal, it doesn’t mean that it gets easier to pass a second referendum, as just as in the existing parliamentary Labour party, many Labour candidates are agnostic or outright opposed to a fresh referendum.

But those problems are as nothing to the Conservative side. For Labour, the path to resolving the Brexit crisis is easy enough in theory – they just need to win a parliamentary majority of 20 or above. In practice this may be quite difficult but it is at least theoretically achievable. The difficulty for the Conservatives is that in the country as a whole, their parliamentary candidates have a similar set of views on Brexit as their current MPs – that is to say, they are badly split, perhaps irreconcilably so. Increasing the number of Tory MPs doesn’t really fix that essential problem.

So it’s not really accurate to say that an election wouldn’t change things. Increasing the number of Labour MPs and/or decreasing the number of DUP MPs makes it easier to pass a Brexit deal but still makes it very difficult to stop Brexit. Increasing the number of SNP, Plaid Cymru or Liberal Democrat MPs makes it easier to stop Brexit but harder to pass a Brexit deal. And increasing the number of Conservative MPs increases the chances of continuing deadlock and no deal at all.