BACKGROUND: For too many years the practice of systematics has been impeded by profound disagreements about the very foundations of this discipline, that is to say the type of information that should or should not be incorporated into a proper classification of life. Two main schools of systematics, both recognizing evolution, oppose each other: cladism states that the classification should only reflect the branching order of the lineages on the tree of life whereas evolutionism states that the length of the branches, that is the degree of modification, should also be taken into account so as to reflect macroevolutionary leaps. The first one forbids the exclusion of any descendant from a group that contain its ancestors, while the second one explicitly requires that the descendants too much different from their ancestors must be classified separately. Moreover, both schools often use the same words, such as " monophyly, " to designate different ideas. This prevents proper communication between the proponents of either side. Consequently, the research in phylogenetics is globally erratic and the taxonomic classification is highly unstable. RESULTS: I rigorously define the terms which designate the phyletic relationships and explore their properties through use of graph theory. I criticize a similar work (Kwok 2011) that was unable to properly catch these notions. This leads me to provide three independent arguments –– one historical, one utilitarian, and one morphosemantic –– in order to retain the original Haeckelian meaning of the term " monophyly " rather than the redefined Hennigian one. I identify some polysemy regarding the term " clade, " and that is why I define two new words, " holoclady " and " heteroclady, " to contrast respectively with " holophyly " and " heterophyly. " I also show that a strictly holocladic or holophyletic classification advocated by cladists is formally impossible. I therefore review and criticize the philosophical postulates subtending such an illogical paradigm. I show that cladism is part of a more general philosophical movement named structuralism, which is mainly characterized by anti-realism and a metaphysical way of thinking. I identify the biologically unrealistic assumptions on which cladism is based and argue that they have been empirically falsified. I therefore defend the use of paraphyletic groups in the scientific classification of life and review the main arguments that have been opposed to this solution. Some of them, such as anthropocentrism or the lack of an objective manner to determine paraphyletic groups, are grossly outdated, while others simply rest upon the difficulty in conceptualizing emergent phenomena. CONCLUSION: Since clades are still useful for methodological reasons, I offer a compromise that should make possible the coexistence of the two main opposing schools of systematics by eliminating competition between clades and taxa for the same names. I propose therefore that in a future revision, the BioCode should approve a dual system by recognizing both a " phyletic arrangement " made of clades and a " phylogenetic classification " made of taxa.