(See BOBIK, J., AQUINAS On Being and Essence, 1970, p.16/9, p.20, p.55).
Now the consideration does not refer anymore to the Genus as Genus, etc., not what it means to be a Genus, etc., but it now refers to that what one or another genus, etc. is, that is, the first intention (prima intentio) that is found to be a genus, etc. (that has as its significatum an extra-mental object, i.e. that refers to an extra-mental thing).
The consideration of Genus, Species, etc. as second intentions refers to the relationships of their extensions in the sense of set theory. The extension of a first intention is its range of significata, for example the first intention ANIMAL refers to all sensible organisms of past, present, and future, while the first intention HOMO refers to all human beings of past, present, and future. The extension of ANIMAL is clearly larger than that of HOMO. And this in such a way, that ANIMAL relates to HOMO as genus to species. So in second intentions it is about their relative extensions (ranges of reference).
Let us characterize the praedicabilia (predicables) in terms of their extensions.
[ Here ext(genus) means : the extension of a genus. The sign > stands for 'larger than', while the sign < stands for 'smaller than' . diff = difference (differentia specifica), prop = proprium (necessary determination [propery s.str.] ), accid = accidens ] :
Again, the extension (ext) of an intention (term) is the total of all objects to which the intention refers (to which the intention intends).