This article argues that one reason for the Brexit crisis is the traditional binary voting system which counts votes of different meanings as if they were the same. In the 2016 referendum, for example, “remain” had more or less a clear meaning but different voters understood “leave” in different ways. Similarly in the parliamentary vote of January 15, the 432 MPs who voted against Theresa May’s deal meant different things by their “no” with, it’s estimated, a third wanting a “second referendum”, a third a “better deal”, and a third “no deal”. As researchers of the electoral process, we have a simple solution to the impasse. We propose that a second referendum – or a new vote in the House of Commons – be based on a new method of voting, one we’ve called majority judgement. In majority judgement, voters (or MPs) would vote on a number of well-defined “leave” and “remain” alternatives and would be asked to grade each alternative according to a scale of grades from “Terrible” to “Excellent.” The alternative with the highest majority-grade wins. Why would this method fix the Brexit crisis? To prove our claim, first we explain some notions from social choice theory (such as the Condorcet and Arrow paradoxes). Then, based on a plausible estimate of current British public opinion, we contrast how the current binary voting system has gone wrong with how majority judgement could provide a winning solution. Finally, we use existing data from Pew Research surveys to show how majority judgement would have worked in the 2016 US Presidential Elections.