A taxon (pl. taxa) is any group of organisms that is given a formal taxonomic name. Traditionally, a monophyletic taxon is understood to be one that includes a group of organisms descended from a single ancestor [as in (a)], whereas a polyphyletic taxon is composed of unrelated organisms descended from more than one ancestor [as in (b)]. One type of monophyletic taxon is a paraphyletic taxon, which includes an ancestor and a group of organisms descended from it [as in (c)].
These loose definitions fail to recognize the fact that all organisms are related, therefore any conceivable taxon is logically "monophyletic". In modern usage, a monophyletic taxon is defined as one that includes the most recent common ancestor of a group of organisms, and all of its descendents; such groups are sometimes called holophyletic. A paraphyletic taxon includes the most recent common ancestor, but not all of its descendents. A polyphyletic taxon is one that does not include the common ancestor of all members of the taxon.
Taxonomists tend to fall into two schools, "Evolutionary" or "traditional" systematics versus "Phylogenetic" or "cladistic" systematics. Since the 1970s, "phylogenetic systematics" has been gradually replacing "traditional systematics" Because the older literature and textbooks often use "evolutionary" classifications, the student must understand both systems.
Another unfortunate circumstance for the student is that the two schools use the same terms, but in different ways, and often refuse to recognize the alternative usage. Evolutionary taxonomists claim to recognize only "monophyletic" taxa, but use the traditional definition to include both holophyletic and paraphyletic taxa. Phylogenetic taxonomists also claim to recognize only "monophyletic" taxa, but limit the term to what is defined above as "holophyletic," although most reject that particular term. Both schools reject the use of polyphyletic taxa, although most phylogenetic taxonomists would use that term to included paraphyletic taxa.
Homework: The diagram above recognizes Taxa 1, 2, & 3 as examples of "monophyletic", "polyphyletic", and "paraphyletic" groups. (1) In the diagram, identify two additional examples each of groups that would be considered "holophyletic," "polyphyletic," and "paraphyletic". (2) Identify all possible "holophyletic" taxa.