For a few hours on Friday, Mitt Romney was the subject of positive press. Why? Because, in an interview on MSNBC’s Morning Joe, the former Republican presidential candidate pushed his party to back a minimum wage increase. “I, for instance, as you know, part company with many of the conservatives in my party on the issue of the minimum wage. I think we ought to raise it,” Romney said. “Because frankly, our party is all about more jobs and better pay.”

Romney wasn’t alone. “There are some basic things that we should be for, and one of them is a reasonable increase from time to time in the minimum wage,” said former Minnesota Gov. Tim Pawlenty, also speaking on Morning Joe. Former Pennsylvania Sen. Rick Santorum followed suit on The Daily Rundown, telling Chuck Todd that Republicans should be OK with an occasional increase in the minimum wage. And on Tuesday, in an address to the American Enterprise Institute, Ohio Sen. Rob Portman endorsed his state’s move to index the minimum wage to inflation.

On the surface, this trend looks like a GOP shift to the center. But these men are outliers. For the large majority of Republican lawmakers, there’s no reason to raise the minimum wage. And when Democrats raise the issue, they push back with the usual reactionary rhetoric.

“[W]hen you raise the cost of something, you get less of it,” said House Speaker John Boehner, after the White House revealed its plan to raise the minimum wage for federal contract workers. He continued: “We know from increases in the minimum wage in the past that hundreds of thousands of low income Americans have lost their jobs.”

Likewise, after voting to block cloture on a minimum wage bill in the Senate—thus killing the proposal—Indiana Sen. Dan Coats, the senior Senate Republican on the Joint Economic Committee, said, “Raising the minimum wage creates winners and losers—it will raise the wages of some but result in job losses for many low-income workers. The true problem plaguing impoverished Americans is not low wage rates but a lack of good job opportunities.”

In both instances, the message is that an increase is either futile or counterproductive—a cure that’s worse than the disease. We know that’s an exaggeration. According to a February report from the Congressional Budget Office, the Democratic proposal to raise the minimum wage to $10.10 would reduce total employment by 500,000 workers over the next three years. At the same time, it would lift more than 900,000 families out of poverty and increase the incomes of 16.5 million low-wage workers. What’s more, it’s reasonable to think we would gain jobs as a result of new economic activity generated by higher wages and new consumer spending.

But none of this mattered to Republicans who read the report. For them, it was vindication. “Raising the minimum wage could destroy as many as one million jobs, a devastating blow to the very people that need help most in this economy,” said Senate Minority Leader Mitch McConnell, all but speaking for most of the Republican Party, and echoing decades of rhetoric against increasing the minimum wage.

All of this doomsaying raises a question: If raising the minimum wage destroys jobs and prevents employment, then lowering it would do the opposite. And if you gain from lowering the minimum wage, then why have one at all?

At least a few Republicans—including two presidential contenders—seem to have wondered as much, and have gone to where that logic leads: Complete opposition to the minimum wage.

Kentucky Sen. Rand Paul, for example, has no tolerance for the minimum wage. “When you set the minimum wage, it may cause unemployment,” he told ABC News during his 2010 Senate campaign. “The least skilled people in our society have more trouble getting work the higher you make the minimum wage.”

Similarly, Florida Sen. Marco Rubio sees the minimum wage as an exercise in futility. “We all support—I certainly do—having more taxpayers, meaning more people who are employed,” said Rubio. “The problem is you can’t do that by mandating it in the minimum wage laws. Minimum wage laws have never worked in terms of having the middle class attain more prosperity.”

The same is true of Tennessee Sen. Lamar Alexander, who—when asked if he does not “believe in the concept of a minimum wage”—responded in the affirmative, and said that he would abolish the standard if he had a chance.

Most recently, one of the Republican candidates for Senate in Georgia, Karen Handel, told ThinkProgress, “The federal government has absolutely no business being involved in mandating salary and wages in the private sector.”

The problem with this position, of course, is that it’s incredibly unpopular. As it stands, huge majorities support a minimum wage increase, which—by definition—includes Republican voters. In a winter poll from CBS News and the New York Times, 65 percent of Americans favored raising the minimum wage, including 53 percent of non–Tea Party Republicans. A Gallup poll conducted in March found similar results. There, 71 percent of adults favored an increase in the minimum wage, including 50 percent of self-described Republicans and 54 percent of self-described conservatives.

Republican politicians might be united on the minimum wage, but Republican voters aren’t, and that difference is why Democrats are laser-focused on raising the minimum wage. Touting an increase doesn’t just motivate the base—a key part of any turnout strategy for Democrats—it potentially wins Republicans, or at the very least sows discontent between the party and its supporters.

For both sides, turnout is everything in this year’s elections. And if Democrats can discourage even a handful of Republicans, then they’ve won themselves an important boost.