Yegor Shɨshkin-Skarð 

Mathematically, it is possible to be a subset (⊂) of a set and not to be its member (∈)—e.g. the empty set. Here, I highlight that a similar principle applies to undiscovered objects in sets of real objects. The current definitions of holophyly (monophyly sensu stricto) and paraphyly suggest a direct membership (∈) of ancestors in taxa. These ancestors are almost always unknown (undiscovered) in phylogenetics. Undiscovered entities are hard to be members of a taxonomical system in organismic biology. There are still several questions, which are problematic not to phylogeny but to the current conceptual framework. Are Pisces sensu Linnaeus, 1756 (including cetaceans and sirenians along with fishes) paraphyletic or they are polyphyletic? If a group is composed of all discovered members of some lineage but based on their homoplasy, is this group holophyletic or polyphyletic? Are choanomonads holophyletic, if some ancestor they share with animals was a collar-bearing (choano-) monad? Can something with a true nucleus be called an “ancestor of eukaryotes”? Accompanying the problematic membership of undiscovered ancestors in groups of discovered organisms, the list of the problems includes further facts. (1) Any organisms of now-known life have a common ancestor (deniable for Haeckel who started this conceptual framework). Therefore, the division principle by the inclusiveness of groups, which gives us the dichotomy “polyphyly”–(holophyly + paraphyly) requires clarification. (2) The definition “the ancestor plus all its descendants” does not allow any clade to be divided into subclades completely—at least the ancestor from the definition will remain. (3) The definitions “the ancestor plus all its descendants” and “the ancestor plus not all its descendants” left no place for the sole-member groups. At least two are needed (one ancestor and one its descendant) to be holophyletic or paraphyletic. I wish to propose and discuss a redesigned system of concepts to solve logical issues. Here, the direct membership/non-membership (∈/∉) of undiscovered ancestors was replaced with the ability/disability of the definition of the systematic group to contain them as members. At the same time, the inclusion (⊂) of the sets corresponding to undiscovered objects remains untouched. The proposed system of the types of systematic groups does not ignore ancestors and deals with their existence more carefully. Simultaneously, undiscovered ancestors do not require membership in systematic groups. Other logical problems also could be solved. To provide a concise definition of holophyly some concepts behind this were named and reduced to one word (phylon, startestor, intrendon, jugiphyly). Holophyly was finally defined using them near the end of the paper.