Willi Hennig (“the father of cladistics”) invented the notion of “paraphyletic groups”, which he then set out as a goal to get rid of.
However, “paraphyletic groups” is just an illusion that emerges in inconsistent logical reasoning, that is, logical reasoning that rests on the fundamental assumption that kinds are real.
The problem with this fundamental assumption is that it leads into backward logical reasoning only acknowledging sets that include itself as member (called “clades”), when such sets actually are contradictory, and this set (ie, the set of all sets that include itself as member) actually is a paradox (see Russell’s paradox). It thus denies all consistent sets, ie, sets that do not include itself as member, including the so-called “paraphyletic groups”. It only acknowledges inconsistency.
Such “paraphyletic groups”, and in a broader sense consistent sets, are thus actually impossible to get rid of, because the opposite (ie, the set of all sets that include itself as member) is impossible to find (as being a paradox). Instead, such “paraphyletic groups” is nothing but an illusion that emerges if one only acknowledges inconsistency, like Hennig did and his followers do. Their struggle is thus like that of The Ingenious Gentleman Don Quixote of La Mancha.