Donald Trump and his allies at the Republican National Convention in Cleveland peddled two falsehoods about America’s immigration problem. One was the vision presented by speaker after speaker of a nation overrun with foreigners crossing American borders and infiltrating communities to rob and kill. Another was the notion that most Americans are desperate for the kind of tough-guy response — including massive deportation and building a wall — that Mr. Trump offers as his solution.

A careful examination of the facts undermines both claims. It is true that this is a complex issue inspiring strong passions. But its resolution, or at least progress toward a resolution, requires clear thinking. It benefits not at all from wild and poisonous assertions. People on all sides of this issue, including Republicans of good will who might be seduced by Mr. Trump’s hyperbole, would do well to take a moment or more to reflect on a few simple truths.

One is that the country is divided over immigration, but not nearly as much as Mr. Trump claims. Americans by wide margins have long supported the principles of sensible immigration reform: modernized laws to better match workers and jobs, strong border security (though not the equivalent of a 2,000-mile wall), better workplace standards and wages, and an opportunity for the 11 million immigrants living outside the law to earn inclusion and citizenship.

Recent polling from the Pew Research Center is instructive: Fifty-nine percent of the public said immigrants “strengthen the country,” while only 33 percent said they were a burden. And 75 percent said immigrants should be allowed to stay legally, if they meet certain conditions. These principles — recognizing the good that immigrants do for the country and rejecting the folly of high walls and mass expulsion — are the basis of the reform legislation that Mr. Trump so deplores as “amnesty” and that many Republicans themselves support. As recently as 2013, bipartisan reform legislation passed the Senate by a vote of 68 to 32.