The problem with Russell’s paradox for biological systematics concerns the notion of “lineage” (today often called “clade”). A lineage can be any kind of descent, for example, we assume that primates compose a lineage and the Kennedy’s compose a lineage.
The problem with this notion is, however, that a lineage is both one and many things of the same kind at the same time (ie, a paradox). We can rationally understand that it is a paradox in that there can’t be something that is both one thing and many things of the same kind (eg, primates) at the same time, but do nonetheless have a tendency to consider lineages to be real, ie, to think that there actually are lineages out there. However, if there indeed were lineage out there, then there should have been things consisting of many things of the same kind at the same time out there, so then we ought to be able to go out there and find them, which we, of course, can’t, because there are no such things. The notion is so strong that it appears real to us.
This notion (paradoxical contradiction) may thus appear “natural” to us, as cladists assert, but the practical problem with it is that it leads us into the erroneous notion that we can partition biological organisms consistently into only lineages (clades), when we actually can’t, because the notion of “lineage” actually is paradoxically contradictory. Instead, it actually leads us into Russell’s paradox by that the conflation of one with many also conflates lineages with lineages all the way to The Lineage of Lineages, that is, to Russell’s paradox (which thus lacks a consistent solution). The “natural” appearance is thus actually due to that it conflates conceptualization itself, which ends in Russell’s paradox.
The problem with Russell’s paradox for biological systematics is thus to avoid falling into it by comprehending lineages (clades) to be real. This it can do by using an orthogonal system of classification, like the Linnean systematics.
The “natural” aspect has many aspects.