What Are Progressive Taxes?

There has been quite a bit of discussion of the term progressive with reference to taxation and how that term should be interpreted. I pulled out a really old Public Finance textbook (it's the one I used as an undergraduate) to see what it says about defining progressivity. There are two concepts of progressive, the technical definition and a definition in terms of distributional equity. This is from Modern public finance by Bernard P. Herber:

In a "technical" sense, it was noted that that the base of a tax is that object to which a tax is applied. This object may be income, wealth, transactions, or people. Although these technical definitions of progressive, proportional, and regressive taxes are necessary for the purpose of understanding the detailed discussion of the different types of taxes throughout Part Two of this book, "distributional equity" considerations nonetheless require a somewhat different definition of these terms. Moreover, it should be recognized that the latter distributional definitional approach serves today as the more popular usage in both academic and nonacademic circles. The significant equity interpretation of the term derives, of course, from the overall acceptance by society of (1) the ability-to-pay principle of tax equity, and (2) intertemporal comparisons of utility along with diminishing marginal utility of income concept. The acceptance of these concepts by society has led to income being selected as the best indicator of taxpaying ability and, thus, to its general use as the "tax base" for reference to progressivity, proportionality, or regressivity in distributional equity terms.

Thus, in a distributional equity sense, a tax (or tax system) is said to be progressive if the tax paid as a percentage of income increases as income increases. This is true regardless of the type of tax in question, that is, whether it be revenues derived from any form of income, wealth, transactions, or personal tax which increases as a proportion of income. However, if the tax paid as a percentage of income remains unchanged as income increases, the tax is termed proportional. Or, if the tax paid as a percentage of income diminishes as income increases, the tax is said to be regressive...

Only a tax such as a progressive personal income tax is "progressive" in both distributional equity and technical terms. This occurs because the "income" which is the indicator of taxpaying ability as the equity benchmark, is also said to be the technical base of the tax unlike, for example, a general retail sales tax whose technical base is stated in terms of the value of transactions or a property tax whose technical base is stated in terms of wealth. Taxes such as the latter are usually "proportional" in their technical structure, but regressive in the sense of tax paid as a percentage of income...

Public Finance is not my main area, and it may be that definitions and conventions have changed since 1975 when the Modern public finance text was last updated, but the distinction between the technical and equity base of a tax is a useful distinction, as is having a specific conventional definition of what is meant by progressive in the literature ("the tax rate increases as the tax base grows larger" in a technical sense, and "tax paid as a percentage of income increases as income increases" for equity comparisons using income as the equity base).

It is also helpful to understand the assumptions underlying the use of income as the equity base (i.e. (1) and (2) above) because we do not always define equity in this way. For example, for inheritance taxes, the equity base is often wealth rather than income. Thus, as we aggregate federal, state, and local taxes to assess equity, we should be careful that the taxes that are summed up (e.g. income plus capital gains plus sales plus dividend plus etc.)  are appropriate for the particular equity base being used to assess fairness. What is the appropriate equity base for, say, a capital gains tax? [Update: More in comments.]