Abstract: Monophyly, paraphyly and polyphyly are three pivotal terminologies for reconstructing natural system, but their definitions have not been generally accepted as yet. Advantages and drawbacks of every set of definitions of monophyly, paraphyly and polyphyly are discussed with the help of set theory and graph theory of discrete mathematics and transformed cladism of systematic biology. The reason that previous paraphyly couldn’t be differentiated from polyphyly is first clarified based on set theory and graph theory. In addition, every species could be monophyletic based on its unique direct origin which could not be replaced by all other origins, based on comparison of the relation between individuals of the same species with the relation between this species and any other species, or based on its unique evolutionary history. As a result, their exact definitions should ensure that any species could be monophyletic, and ensure that they three are quite distinct from one another.

Key words: graph theory, monophyly, paraphyly, polyphyly, set theory, transformed cladism