AbstractThe triplet consisting of two monophyletic taxa and one paraphyletic taxon as constructive element of the phylogenetic systemEvolution has produced very many novelties (apomorphies). Most of them are small and relatively inconstant, these are more or less indicative of the phylogenetic relationships between closely related species. They cannot be the constitutive character of a supraspecific taxon that exists since a long time and comprises many diversified species. Such a taxon of higher rank can only be characterized by an improbable, rare novelty that has developed only once and has been preserved in all descendent species. Two consecutive apomorphies of this persistent type (‘fixed apomorphies’) characterize three supraspecific taxa, the triplet “A”, “B” and “A minus B” (Fig. 1). The group “A minus B” is rejected in Hennig's theory because it is ‘paraphyletic’, but it is not an artefact created by the systematicist. It is an inevitable mathematical consequence of the differentiatison of the group “B” within the group “A”. Being the result of a subtraction, it is necessarily associated with the two monophyletic partners in the triplet, as it is delimited on one side by the synapomorphy of the group “A”, of which it is a part, and on the other side by the autapomorphy of the separate group “B”.Traditional classifications often include paraphyletic groupings that are inconsistent with phylogenetics, e. g. the Reptilia and the Apterygota. The fault in such cases is that these groups are extended beyond the limits of a triplet and cover more than a single interval between consecutive monophyletic taxa. Paraphyletic groups are admitted in the phylogenetic system only for bridging the gaps in our cladistic information.According to HENNIG'S theory, all supraspecific taxa should be arranged two by two as sister‐groups originating from one ancestral species and comprising all descendents of that species. The fixed evolutionary novelties which characterize higher supraspecific taxa are, however, rare and scattered. It is highly improbable that they have developed in sister species, therefore the taxa marked by them cannot be sister‐groups (except in very rare cases). In HENNIG'S earlier papers, e. g. in his system of Lepidoptera (1953: 46–49), the alleged ‘sister‐groups' are, in reality, the groups “B” and “A minus B” of a triplet (see Fig. 2). In his revised concept (1957 and later), two autapomorphic groups which are most closely related in the recent fauna (“B” and “C” in Fig. 3) are called ‘sister‐groups’. But these have originated independently from different ancestors in a plesiomorphic complex of extinct species and are more closely related to parts of this complex than to each other. True sister‐groups (“Bx” and “Cx” in Fig. 4) would be formed if these related plesiomorphic species were included, but this extension of the ’backward‘ border of the taxon is not justified by synapomorphy (in the terms of logic, it is a ’metabasis‘), and it would make the classification of fossil species impossible, unless these show at least one synapomorphy with either “B” or “C”. In the system of the recent fauna the sister‐groups are identical with the autapomorphic groups, because the plesiomorphic species are extinct.The natural system based on synapomorphies and autapomorphies is the triplet‐system as outlined in Figure 6. It is not a new type of classification, but its theoretical foundation was missing, and precise instructions were needed for its use in phylogenetics. The information obtained by HENNIG'S method is entirely preserved in this system and can be retrieved from it, and both recent and extinct species can be classified together. The disadvantage of the triplet‐system is that it contains twice as many taxa as HENNIG'S classification. This complexity will limit its use in the practice of taxonomy, but it may be simpli