In my previous post introducing Phylogenetic Systematics (PS), I explained that its criterion for the recognition of supra-specific taxa (groups of species in the formal, scientific classification of life) is monophyly. A monophyletic group is one that includes all descendant species of its common ancestral species, as opposed to only some of them.
Unfortunately, the way many people memorize this, and the way they are even sometimes taught it as students, is as follows: "All taxa have to be monophyletic". This makes no mention of grouping species into supra-specific taxa, and so they may come to assume that PS demands that species also be monophyletic.
There are three very different groups of people who make that mistake: those who oppose PS and its monophyly criterion and fail to realize that they don't actually understand what they criticize; those who uncritically accept PS but fail to read up on the details or think its implications through; and a few well-informed proponents of PS who should know better but let themselves get carried away by the alluring simplicity of applying the same approach to everything, even where it cannot work. (As the saying goes, "if all you have is a hammer, everything starts to look like a nail.")
So what are the problems with the claim that species must be monophyletic?
For starters, it is incoherent if we follow Hennig's definition of a monophyletic group as "a group of species including all descendant species of its common ancestral species." It makes about as much sense as saying that because libraries arrange their books following the Dewey Decimal Classification, the pages of each book should also be arranged after it. The concept simply does not apply, and the idea of trying to make it apply is complete bonkers.
To make sense of the claim in question, we thus have to expand or generalize our concept of monophyly to be not only about groups of species anymore, but about groups of any type of items. So in the case of species, we would have to clarify the claim to read: "species must be monophyletic groups of individuals".
Does the claim make sense now? Of course not, if we think back to the previous post. Within sexually reproducing species we do not find phylogenetic structure but instead tokogenetic structure. Individuals of such a species have a tokogenetic relationship, not a phylogenetic one, and thus they cannot be whatever-phyletic, be it mono-, para- or poly-.
And this is then the fundamental error committed by all three groups claiming that PS demands species be monophyletic: The failure to understand that phylogenetic categories only make sense when discussing items that show phylogenetic structure, or in other words the failure to appreciate the distinction between tokogeny and phylogeny.
Phylogenetic Systematics is ultimately not concerned with the monophyly of sequence copies within species (although that is at least, in contrast to "monophyly of a species", a coherent concept, but more on that another time), nor with the relationships of individuals from the same species, nor with occasional events of introgression or lateral gene transfer between species, but only with the evolutionary relationships of entire species with each other. All the other issues in this list - the ones connected with the nor's - are relevant mostly insofar as they make it harder to infer the species phylogeny.