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What criterion do modern taxonomists use to classify organisms?

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The prevailing theory is called cladistics. This relies on unique homologies, or structures which members of the clade possess to the exclusion of all other clades. For example, all mammals give milk (the females, anyway) and no other animal gives milk, so mammals are a distinct clade, in this case, a "class." Smaller distinctions determine that certain creatures belong in smaller clades, called "orders," "families," "genera" and "species."