Updated: 3:57 p.m.

During Wednesday’s debate, Republican nominee Mitt Romney muddied the waters over how he might pay for his 20 percent tax rate cut, even as he clarified that his top principle would be not to add to the deficit.

Romney had floated a specific $17,000 cap on tax deductions earlier in the week as one possible way to pay for the rate cut, but that figure, as President Barack Obama’s campaign quickly pointed out, would potentially hit millions in the middle class who have sizable mortgages, charitable donations, state and local tax deductions and the like, though many others would benefit from the lower rates. A Romney aide even acknowledged to Roll Call before the debate that some middle-income Americans could see their taxes go up under that plan. It’s impossible to know exactly who would pay more without more details, however, and when pressed the aide said it would be “speculation” at this stage to say who might see taxes go up.

A cap on deductions would not prevent the wealthy from getting an overall tax cut under Romney’s rate cuts, according Roberton Williams, an analyst for the Tax Policy Center. But that is something Romney vowed would not happen, arguing during the debate that eliminating other loopholes and deductions would prevent the wealthy from receiving a net tax cut.

“I’m not going to reduce the share of taxes paid by high-income people,” he said.

More than anything else related to taxes, Romney’s previous promises to cut all tax rates by 20 percent are what have put him on the defensive during the general election campaign, and his forceful denials about what his tax plan might do seemed to show his recognition of that. The nonpartisan Tax Policy Center said in an analysis of the Romney tax plan that it would reduce federal revenue by $480 billion by 2015 under a current policy baseline without broadening the base. The TPC said it would be essentially impossible to reduce tax rates by 20 percent without producing a net benefit for the rich, no matter how much tax breaks are reduced if Romney also keeps his promise to not raise taxes on investment income.

As part of the recasting of his tax plan, Romney also dropped his very specific $17,000 figure Wednesday night.

“Make up a number – $25,000, $50,000. Anybody can have deductions up to that amount. And then that number disappears for high-income people. That’s one way one could do it,” Romney said.

The potential problem for Romney is that if you “make up a number” – especially one as high as $50,000 –  it becomes harder to adhere to his stated principles. A higher cap means less effect on the middle class, but it also would result in less revenue to pay for rate cuts and his other tax proposals, which include eliminating the estate tax.

That, in turn, could run into Romney’s vow not to add to the deficit.

“My No. 1 principle is there’ll be no tax cut that adds to the deficit,” Romney said, in a clear break from longtime Republican orthodoxy that has held that tax cuts do not need to be offset.

Romney’s statement appears to be an endorsement of the pay-as-you-go rules most Republicans have resisted for the past decade, a position that would presumably mean he’d have to block the small-business tax cut passed by House Republicans or the tax cuts proposed in vice presidential nominee Paul Ryan’s own budget blueprint, for example. Neither would offset the tax breaks.

Obama has repeatedly violated pay-as-you-go budget rules and has added hundreds of billions of dollars to the deficit by passing various tax cuts, including much of his stimulus package and his payroll tax cuts. He has vowed to block any further extensions of the Bush-era tax cuts for people making more than $250,000, which would raise about $800 billion over the next decade, and he has enacted a number of other tax increases, particularly to help pay for his health care overhaul. Most of those tax increases, however, have not yet taken effect.

Obama has also proposed a much more modest plan to limit tax breaks – although not nearly to the extent Romney has – for only families making more than $250,000.

Romney noted, as he has previously, that you could eliminate various deductions instead of putting in a deduction cap, and he said he would work out the details with Congress.

“There are alternatives to accomplish the objective I have, which is to bring down rates, broaden the base, simplify the code and create incentives for growth,” he said.

Sam Goldfarb contributed to this report.