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We want now to take a glance at the logical form a theory of classifications (whatever these may be) could get in general. We must remark, first, that a logical approach to classifications (especially from an intensional viewpoint) never won unanimous support among the ancient philosophers and logicians (as proven by the numerous discussions between Aristotle and Plato, Ramus and Pascal, Jevons and Joseph).

We have seen that classic representations of classifications give hierarchical structures, which are the chains of partitions of a partition lattice. But as we have seen also, such structures may change with time, either because of the internal modifications in the description of objects’ properties (intrinsic instability), or because of the external increasing or decreasing of the number of objects to be classified (extrinsic instability). For these two reasons, it is clear that it would be pleasant to get an algebra of all the possible transformations of classifications, and particularly explicit rules to explain how we can extend them, without breaking their initial structures.

We have presented in the previous chapter, a theory able to generate the infinite set of classifications as the continuum, each classification being in a one-to-one correspondence with a real number. But, as we know, there are, in the mathematics of the infinite, since the works of Cantor, Suslin and others, a lot of possible views of the continuum. Moreover, since the undecidability results of Cohen, there exist also a lot of possible set theories. So, it may be useful to ask some questions about what happens concerning the existence of classifications in those alternative theories, in particular, when they admit higher forms of the infinite. Though the risk is obvious, there, to end up at some undecidability results, several arguments speak for such an extension.

As we have seen (see Chap. 5), classifications cannot be reduced to the classic representations of Barbut and Monjardet (Ordre et classification, algebre et combinatoire, vol. 1, Librairie Hachette, Paris, 1970) or of Lerman (La classification automatique, Paris, 1970; Classification automatique et analyse ordinale des donnees, Dunod, Paris, 1981). Note, however, that Lerman (La classification automatique, Paris, 1970, 73) has proposed to extend partitions search’s algorithms to covering problems. A more recent reference on the subject is (Alpert and Kahng in Integr. VLSI J. 19(1–2):1–81, 1995. It is obvious that these representations are not sufficient for describing k-classifications proposed by Jardine and Sibson (Math. Biosci. 2:465–482, 1968). But as soon as we admit overlapping classes, classifications are no more chains of partitions. They are made of covers. So we define first this notion (Sect. 6.3), count the number of covers on a set (Sect. 6.4), study the set of all covers on a set (Sect. 6.5) and the lattice of all minimal covers (Sect. 6.6). It appears that, when the generalized classifications still keep a hierarchic form, they are in fact chains of covers (Sect. 6.7). Now, as we shall see, if those covers are not complete ones, classifications become chains of parts of covers. And as a part of a cover is just the same a part of a partition, i.e. a class, such classifications are just chains of classes (Sect. 6.8). As the lattice of minimal covers does not contain all possible generalized classifications, and as the set of all covers over a set is not a lattice, we are necessarily led to introduce topological notions (Sect. 6.9), because topological relations are the only relations that subsist on this kind of classifications. We can then apply to classifications some basic theorems of topology: first, it is a useful way to get information on them (Sect. 6.10); but, more generally, it is possible to develop a topological representation of information (Haouas et al. in J. Appl. Sci. 8(20):3743–3747, 2008). Finally, those systems of classes happen to be a kind of hypergraphs, so, it appears that the language of relational structure (Sect. 6.11) is the more convenient to describe those structures.

As most of the classifications man has constructed during a long period going from Plato and Aristotle to Linnaeus (and even to the XIXth century) were hierarchical classifications, we devote this complete chapter to the exposition of mathematical models of hierarchies. After some historical views on the subject (Sect. 3.2), we introduce (Sect. 3.3) the basic notions of partition, partition lattice, and chain of partitions, this last one being the exact mathematical model of a hierarchical classification, classically represented by a tree diagram. In Sect. 3.4, we give the structure of the set of all hierarchical classifications on a finite set (which allows us to know exhaustively all the possible hierarchies we can make). Then, in Sect. 3.5, we present the exact correspondence between tree diagram, hierarchical structure, and the distance we can define on it, which is an ultrametric. These ideal models and their algebraic representations (Sect. 3.6) are those that the taxonomists want generally to obtain, but the appearance of the real world, in general, is quite far from such nice orders. So, we explain (Sect. 3.7) how we can replace the empirical quasi-chaotic data with due mathematical taxonomies. Though, in many cases, we are able to do that and can easily adapt our numerical models to empirical data, some problems arise in this operation (Sect. 3.8): either because of the existence of mathematical limits within the models (intrinsic instability) or because of the presence of changes in human perception of the world in the course of time (extrinsic instability). However (Sect. 3.9), we list finally some possible answers to these important questions.