A not uncommon thought in philosophy is that we should distinguish (in philosophy, anyway) between "sparse" ("real", "serious") and "abundant" ("ideal", "superficial") properties/classes and relations. The abundant ones are the things we get from axioms -- the comprehension scheme in the context of type theory, and set existence principles in the case of set theory. But if you are willing to countenance ideal classes (or relations or whatever), it might seem like just a prejudice to not also countenance ideal elements. Moreover, the history of mathematics is filled with examples of cases where we expanded our ontology by allowing for specific ideal objects like the square root of minus one. (Also, though its connection to my question is hardly clear, forcing is often seen as a means of adding ideal elements.) My general question is: what kinds of systems have been studied that include the idea of postulating ideal elements and/or allow for greater symmetry between the treatment of individuals and classes?
My first impression, having thought about the matter not too long, is that combining comprehension principles with principles allowing the postulation of ideal elements tends to lead to inconsistency. For instance, suppose we just have a two-sorted system, with individuals and sets of individuals, and even suppose that comprehension is restricted in the way it is in ZFC. So we have the scheme
(1) There is a set such that, for every individual $s$, $s$ is in it iff $s$ is in $X$ and $\phi(s)$.
Now add the "inverted" principle
(2) There is an individual such that, for every set $Y$, $Y$ contains it iff $Y$ contains $t$ and $\phi(Y)$.
Together they lead to a contradiction if we assume that some individual $t$ is in two sets, for by (1) we can form singleton $t$, and by (2) we can find a member of singleton $t$ that is in no set but singleton $t$. Here are three things one might be able to do. First, one might be able to symmetrically restrict both comprehension and inverse comprehension in some way that would result in them not being inconsistent with one another. Second, one might try to build a system using some powerful form of inverse comprehension but without any ordinary comprehension or with only some very weak form of ordinary comprehension. If that could be done, it would show that, although we cannot unrestrictedly have both ideal classes and ideal elements, the choice of which to have is somehow arbitrary. Third, one might might try to build up a model iteratively by alternately applying (forms of) comprehension and inverse comprehension to what one has gotten so far. Have any of these avenues been pursued? Also, are there any natural ways of "inverting" any of the other set existence principles used in set theory?