During last week’s presidential debate, Mitt Romney repeatedly promised to “lower taxes on middle-income families” without reducing “the share paid by high-income individuals.” But this combination will prove difficult, if not impossible, for the Republican candidate to deliver given the other elements of his tax reform plan – especially his illogical definition of “middle-income families.”

Romney’s basic idea, which in broad outline has bipartisan support, is to “lower tax rates” and “broaden the base” by reducing deductions, credits and exemptions. He proposes cutting individual income tax rates by 20 percent, so that the top rate would be 28 percent rather than the current 35 percent and the bottom rate would drop from 10 percent to 8 percent. He also wants to abolish the estate tax, repeal the alternative minimum tax, and eliminate taxes on interest, dividends and capital gains for taxpayers earning less than $200,000.

Since Romney insists “there’ll be no tax cut that adds to the deficit,” he needs to make up for the lost revenue by cutting back on tax breaks, and he is committed to doing so without increasing the burden on “middle-income” households, shrinking the share of taxes paid by “high-income” households or reducing the tax code’s incentives for savings and investment. According to a widely cited August report from the Tax Policy Center (TPC), a joint project of the Urban Institute and the Brookings Institution, this task “is not mathematically possible,” a point that President Obama emphasized during the debate.

The Romney campaign dismissed the TPC’s “biased study,” saying it failed to take into account “the positive benefits to economic growth” from his deficit reduction plan and his proposed cut in the corporate income tax. If those changes boost economic output, tax revenue will increase, reducing the amount that needs to be raised by closing loopholes.

As usual, however, Romney did not show his math. He did not even estimate the magnitude of those “positive benefits” (as opposed to negative benefits?) so people could judge whether he was being realistic. More crucially, he has never specified which deductions he would scale back or abolish, and his plan to restrain spending is mostly a mix of penny-ante items (e.g., the $146 million National Endowment for the Arts, which Romney would not even eliminate) and wishful thinking (e.g., $60 billion a year saved by controlling “waste and fraud”).

Sympathetic economists such as Harvey S. Rosen of Princeton and Martin Feldstein of Harvard provided much more substantive responses to the TPC report, arguing that a tax reform plan similar to Romney’s could indeed work. Their analyses differed from the TPC’s in various ways, including the tax data they used (historical vs. projected), their assumptions about how tax reform would affect economic growth and the deductions they deemed to be “on the table.”

But the most important factor in all these studies seems to be the definition of “high-income” vs. “middle-income” households. The TPC used a cutoff of $200,000, while Rosen and Feldstein preferred $100,000. (Rosen also ran the numbers based on the higher threshold, with results considerably less favorable to Romney.)

As The Washington Post’s Dylan Matthews pointed out, the broader definition of middle-class households is counterintuitive, to say the least, since it applies to 96 percent of Americans. The narrower definition, by contrast, makes sense if the middle class corresponds to the middle fifth of household incomes. The problem for Romney is that he (like Obama) uses the broader definition, which is politically useful but fiscally inconvenient.

Similarly, Romney’s lack of specificity avoids the risks of identifying which popular deductions he would target, but at the cost of letting his critics fill in the details. More important, his plan is backward as well as vague. Serious reform would start by eliminating the arbitrary, meddlesome and economically distorting complications that have made the tax code such a headache-inducing mess and only then ask how much rates should be lowered to keep revenue about the same.

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