With only first-order quantifiers, you can express things like:
(1) There is nothing in the fridge. ¬∃x( In(x, f) )
If you have higher-order quantifiers, you can also express things like:
(2) I like nothing about Hegel. ¬∃P( P(h) ∧ ¬LikesAbout(i, P, h) )
But there are situations where you want to explain 'nothing' or 'nothingness' away, e.g.:
(3) Ruby is nothing special.
(4) Nothingness is characteristic of the contents of my bank account.
In (3) you can say that Ruby is not special, and in (4), that your bank account is empty.
The general concept of 'nothingness' is unlike concepts like whiteness and friendship, which following Frege we can call first-order concepts, that is, functions from individual things to truth-values. The concept of 'nothingness' seems to me to hold of things that contain other things: collections, classes, sets, and so on. Nothingness holds of such set-like objects just in case their contents are empty.
e.g. (1) really means: the fridge as a set of products is empty
e.g. (2) really means: the set of properties of Hegel that I like about him is empty
In other words: nothingness-in-general is a higher-order concept, and it holds of objects that can have members but for one reason or another do not.