From the cladistic perspective, many conventional Linnean taxa actually turn out to be paraphyletic (i.e. they include descendants that do not belong within those taxa).  As  T. Mike Keesey pointed out in an email, such traditional taxa can be shown as nested lists, e.g.:

Class Reptilia
  Subclass Synapsida --> Class Mammalia
  Subclass Anapsida
  Subclass Diapsida --> Class Aves
Class Aves <-- Order Diapsida
Class Mammalia <-- Order Synapsida

--o Amniota
  |--Sauropsida
  |  |--+ Mesosauridae
  |  `--o Reptilia
  |     |--Anapsida
  |     `--Diapsida (including Aves)
  `--Synapsida (including Mammalia)

A more serious objection to synthesising these two schemas by doing away with paraphyletic taxa has been raised by  R. K. Brummitt who points out that:

"Linnaean classification without paraphyletic taxa is a logical impossibility. Every monophyletic genus in a Linnaean classification must be descended from something (probably a species) in a different genus, which must be paraphyletic. Similarly every monotypic family must be descended from a species in a genus in a different family.  If one denies paraphyletic taxa, where do genera and families come from? Ultimately, one would end up sinking everything into its ancestral taxon, and the whole classification would telescope into its original taxon....

The theory of a Linnaean classification without paraphyletic taxa is nonsensical. Hennig's proposal to eliminate paraphyletic taxa was based on a failure to see the difference between the Linnaean hierarchy in which all taxa are nested in the next higher taxon, and a phylogenetic hierarchy which is not so nested, the lower levels of the hierarchy being not equivalent to the higher levels. Put another way, all the species of a genus together equal the genus but all the offspring of a parent do not equal the parent."

note:

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