Recently an online interlocutor made the point to me that a "flat" tax is regressive rather than fair because minimum living expenses are largely fixed, thus creating a "proportionally" higher tax rate on those of lower income.
In the course of countering his attempted argument, I realized that the definition of "flat" and "progressive" have been shifted one echelon over, creating a distortion of terms in favor of the preferred Marxist/Progressive position.
A truly flat tax resembles a fee. If the government spends $2B on a new aircraft carrier and there are 400 million taxpayers, then a "flat" tax-- a uniform distribution of tax burden-- would simply divide that $2B among the 400M taxpayers and we'd each get a $50 bill to pay from Uncle Sam.
The ingenuity in the verbal gymnastics is in describing the rate as "flat." But the rate is not the tax. It is the first derivative, the rate of change.
A brief physics analogy illustrates this point. If you are traveling at a "flat" rate of speed, then your position is still changing, perhaps at a rate of 55 miles per hour. Likewise, if you are accelerating at a "flat" rate of 200 miles per hour per hour, your speed is changing and you will likely crash or get a ticket if you sustain that.
The key point here is this: tax BURDEN is the tax, not the tax RATE. The obvious corollary follows immediately behind: any taxation scheme in which some pay more than others is in fact progressive.
It may be progressive with respect to income-- those who earn more pay more. It might be progressive with respect to wealth-- like a property tax-- in that those with more taxable wealth pay more tax. Regardless of the basis for the progressivity, it is still "progressive" in taxation.
A proportional tax of a fixed 15% or 20% then isn't "flat" in tax, only in rate. As a tax, it is progressive. Returning to our car example, the flat rate of speed was still producing a change in position (predictable over time).
When we add a progressive rate, then the system becomes doubly progressive, like the way a change in position is affected by a "fixed" rate of acceleration. It's like taxing your income squared or income cubed by some fixed rate. The resulting distribution follows an exponential function and is progressively progressive.
This is why a so-called "flat" income tax is progressive, not regressive.