Is the word ‘mammal’ more abstract than ‘rodent’? Is ‘rodent’ more abstract than ‘mouse’? Is abstractness thus merely a question of inclusiveness? Surely not. Surely ‘apple’ is not more abstract than ‘winesap’, nor ‘sugar’ more abstract than ‘levulose’. Inclusiveness is one thing, abstractness another.

Sometimes what is conjured up by talk of abstraction is rather the hierarchy of naming. At the bottom there are things; next above them there are names of things; next there are names of those names; and so on up. Lewis Carroll touched on this.

A more venerable example is the tetragrammaton. This fourteen-letter word ‘tetragrammaton’ was the name of a four-letter word, and its utility came of the circumstance that the four-letter word was taboo. The four-letter word⁴ was spelled yodh, he, vav⁵, he, reading from right to left in the Hebrew fashion, and it was pronounced Yahweh (soll mir nicht schuldigen). Here then we have three levels, if for the sake of the example you will grant me the existence of God. At the bottom level we have the Deity Himself. (I do hope I can get through this part without being struck down by a bolt of lightning⁶.) At the next level we have his four-letter name, ‘Yahweh’. At the third level we have its fourteen-letter name, ‘tetragrammaton’. And we can ascend to further levels, uninterestingly, by applying quotation marks and more quotation marks.

This form of hierarchy – the thing, the name, the name of the name – is generally sterile after the first two levels. The initial distinction⁷, between things and their names, is important. But when we talk about the names we ordinarily form names of names by simple quotation, and rise no higher. The case of ‘tetragrammaton’ was a rare case, and due only to a strange taboo.

This hierarchy becomes somewhat richer if instead of limiting ourselves to designation, by singular names, we look also to denotation by general terms, that is, common nouns and adjectives. Three levels then stand vividly forth. At the zero level as always there are the things, God bless them. At the next level there are now not only the names of things, names like ‘Boston’ and ‘Washington Monument’ and ‘Bernard J. Ortcutt’⁸, but also there are general terms like ‘rat’, ‘rodent’, ‘mammal’, ‘city’, ‘monument’, etc. that apply to things. At the level above these there are not only names of names, like ‘tetragrammaton’, but also general terms that apply to names and to general terms. There are such general terms as the word ‘noun’, which applies to the words ‘rat’ and ‘city’ and all the rest; also the word ‘trochee’, which applies to the words ‘Boston’, ‘rodent’, ‘city’, and ‘mammal’; also the word ‘dactyl’, which applies to the word ‘monument’.

This hierarchy thus rejoices in three lively levels, and lapses into dullness above these. It turns out, moreover, that these levels must not be confused, on pain of paradox. The paradox was propounded by Kurt Grelling 65 years ago. It hinges on the general terms ‘autological’⁹ and ‘heterological’. These are terms of level two, if we count things as of level zero. These terms are at the level of the words ‘noun’ and ‘trochee’ and ‘dactyl’: words of level two, applicable to words¹⁰, of level one. Here is what they mean: a word is autological if it is truly applicable to itself. Thus the word ‘short’ is autological, being a short word. The word ‘English’ is autological, being an English word. The word ‘word’ is autological, being a word. The word ‘trochee’ is autological, being a trochee. Other words are called heterological; thus the words ‘long’ and ‘German’ are heterological, not being long or German. And now here comes Grelling’s paradox: is the word ‘heterological’ autological or heterological? If the word ‘heterological’ is itself heterological, and thus true of itself that makes it autological; and vice versa. By respecting the levels of our hierarchy, however, we dodge the paradox. The second-level word ‘heterological’ is applicable only to first-level words, and we confuse levels when we ask whether it is itself heterological.

So we see that these linguistic levels of denotation do matter; the words that apply to words occupy a significantly higher level than the words that apply to things. Still this hierarchy also, however important, fails to capture what one wants to call the levels of abstraction. Abstractness is not a matter of inclusiveness, we saw, and it is also not a matter of loftiness in the hierarchy of names of names of names. It may be better identified, surely, with yet a third kind of thing: it is a question of classes and classes of classes, or of properties and properties of properties. Thus consider again the mice and the rodents. The mice constitute a zoological family; the rodents a zoological order. Each mouse, indeed each rodent, is a thing of abstraction level zero. The mouse family, taken as a class or property, belongs to the next level of abstraction; and so does the order Rodentia. The order Rodentia is more inclusive than the mouse family, but they are both at the same level of abstraction, namely level one. And then at level two we have the class of all zoological families, or the property of being a family; likewise the class of all orders; the class of all species; and also the union of all these classes, hence the class of all taxa, as taxonomists call their taxonomical categories. Thus each mouse, each chipmunk, each rodent, each individual, belongs to the zero level. The mouse family taken as a class, or property, belongs to level one; and so does the order Rodentia, and so does the class Mammalia, and the phylum Chordata. Each of these, some more inclusive and some less, stands at the first level of abstraction. Then at the second level we have classes of classes, or properties of properties. One of them is the class of all families, one is the class of all species, one is the class of all orders, one is the class of all taxa of all sorts. At the third level of abstraction we have classes of such classes of classes; and so on up.

These levels of abstraction are what Russell called types. They are levels of abstraction in a serious sense. Significantly enough, this hierarchy also goes rather dim after a few levels. This is a commentary on our own limitations in relation to levels of abstraction. Mathematicians concerned with abstract set theory blithely encompass an infinite hierarchy of such levels, and even emerge into transfinite levels. Well, they are intelligent men, but let us not overestimate them. They gain their swift ascents by formulating the general principle of ascent and not fretting over specific applications. Conversely our own adherence to a few bottom levels of this hierarchy can be accounted for in terms of utility and diminishing returns.

The two hierarchies that I have last described are sometimes conflated, and we can see why. One is a hierarchy of words and phrases, in which those of higher level are true of those of lower level. The other hierarchy, the types, is a hierarchy rather of classes or properties in which those of higher level are classes or properties of those of lower level. Practical men of an amiably nominalist bent are apt to view the classes or properties as mere words or phrases, and thus to identify the hierarchy of classes or properties with the hierarchy of words and phrases. However, there are differences. There are logical differences of a technical kind that I shall not pause over, and there are common-sense differences. It would be awkward to identify the mouse family with the word ‘mouse’, or the order Rodentia with the word ‘rodent’, even apart from technical troubles. We want to say of the word ‘mouse’ that it is short and of the word ‘rodent’ that it is a trochee; we want to say quite other things of the mouse family or the order of Rodentia – e.g., that it is numerous.

The nominalist urge to reduce abstract objects to mere abstract words is both amiable and understandable. For how, one may ask, can people learn to talk about abstract objects – classes, properties – when only concrete objects are present to the senses? This is a good question and I think it admits of a good answer, though not a brief one. We can reconstruct plausible steps whereby people can have learned to talk not only of observable concrete objects but also of abstract ones. Some of the steps proceed by conspicuous analogy and unconscious extrapolation. Some of them depend on confusions. Confusi...