President Obama continued to push for a “big deal” on the debt limit Friday, sharpening his push for Republicans to accept tax increases as a part of that deal.

The takeaway from the press conference – Obama’s third in less than three weeks – was essentially this: the American people recognize the need for tax increases (a.k.a. “revenues”) in addition to spending cuts, and the Republican Party should too.

“The bottom line is, this is not an issue of salesmanship to the American people. The American people are sold,” Obama said. “You have 80 percent of the American people who support a balanced approach. Eighty percent of the American people support an approach that includes revenues and cuts. ... The problem is, members of Congress are dug in ideologically.”

Some polls do indeed show that even many Republicans say tax increases should be a part of any deal, but it’s not quite so simple as the president would like it to be.

First, though, a look at the poll the president was referring to (and be sure to check out our polling wrap-up).

A Gallup poll this week showed that just 20 percent of Americans – including 26 percent of Republicans – say the deficit should be reduced solely through spending cuts. This leaves 80 percent of all Americans who don’t rule out tax increases. (Important note: Some were undecided and 4 percent said only to use tax increases, so the actual number who support using both spending cuts and tax increases is actually less than 70 percent.)

Quinnipiac University put those ruling out tax increases at 25 percent overall, including 48 percent of Republicans — much worse numbers for Obama’s argument. But for our argument’s sake, we’ll use the numbers more favorable to Obama.

A look at that 80 percent number is really a mixed bag.

Nearly one-third of people (30 percent) say they want a deal that is mostly spending cuts with some tax increases, while another 32 percent say the plan should be balanced equally between tax increases and spending cuts. Only 11 percent say the deal should be mostly or solely tax increases.

So saying that 80 percent of Americans support a “balanced approach” may be over-stating things. The Gallup poll actually shows 50 percent of Americans say either no tax increases or that tax increases should be the smaller part of the package.

Drilling down on exactly what kind of tax increases are fine with that 30 percent is the next problem. While some Republicans may be open to repealing the Bush tax cuts for the wealthy – which would result in a major change – others may be referring to tax loopholes that would have a much smaller effect.

Getting all Republicans to agree to specific tax hikes is a tall order, and the polling can’t tell the complete story about where Republican voters stand on the issue.

But even that misses the point a little bit.

The stated reason that Republicans are resisting tax increases is not that they are broadly unpopular with the American people; it’s because they say the tax increases would harm the economy. Many of these Republicans have also signed pledges promising not to raise taxes — a fact that Obama lamented in the press conference.

This leads to the problem of primary campaigns. Many Republicans faced significant primary opposition in the 2010 election, and some even lost. Regardless of what the polls saying, voting for anything that could be construed as a “tax increase” risks earning the ire of the most active Republican primary voters. And if you look at Quinnipiac’s polling, it’s actually a plurality of all Republicans who say tax increases should be off the table.

Even if only 26 percent of Republicans will completely rule out tax increases, though, that 26 percent is arguably the most active group of voters and has shown a willingness to send a message on primary day.

In the end, getting Republicans to get out of their default no-new-taxes mode is an extremely tall task. Members would be putting their careers on the line, and the GOP would risk a chasm even bigger than the one that has already resulted from the debt limit debates.

If Obama still wants a “big deal,” he’ll have to prove that tax increases aren’t a big deal to Republican voters. And that’s a tough sell.