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Information Data Structures

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Towards a General Theory of Classifications

Part of the book series: Studies in Universal Logic ((SUL))

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Abstract

Data structures, orderings and applied classifications are generally defined on finite sets or sets of relations, which supposes that we know what are such entities (Sect. 2.3). But a part of classification research deals with data mining and the constitution of structured domains of concepts or objects. The fact that a mathematical structure, the Galois connection, contains a quasi-exhaustive information about the correspondence of two sets (Sect. 2.4) has suggested to use this structure in association with an order relation (Sect. 2.5) to initiate formal conceptual analysis (Sect. 2.6). But formal concepts are not real concepts. The exploration of concrete structures of objects has then led to the construction of formal (Sect. 2.7) and regional (Sect. 2.8) ontologies, using sometimes, as Barry Smith does, non-classical logics (the mereology of Lesniewski). In all this chapter, we study these models in relation to the main problems of classification and, finally, discuss (Sect. 2.9) the theories and results that have been introduced.

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Notes

  1. 1.

    The notion of a Galois connection has its roots in Galois theory. By a fundamental theorem of Galois theory, there is a one-to-one correspondence between the intermediate fields between a field L and its subfield F (with appropriate conditions imposed on the extension L/F), and the subgroups of the Galois group Gal(L/F) such that the bijection is inclusion-reversing:

    $$\mathit{Gal}(L/F)\supseteq H\supseteq\langle e \rangle\quad\mbox{iff}\quad F\subseteq L^H\subseteq L, \quad \mbox{and} $$
    $$F\subseteq K\subseteq L\quad\mbox{iff}\quad \mathit{Gal}(L/F)\supseteq \mathit{Gal}(L/K) \supseteq\langle e \rangle $$
  2. 2.

    Recall that a complete lattice is a lattice in which every subset admits a greatest lower bound and a least upper bound. The lattice itself being one of its own subsets, it follows that it has a (unique) infimum and a (unique) supremum.

  3. 3.

    This database comes from the UCI KDD Archive (http://kdd.ics.uci.edu/).

  4. 4.

    The existence, in the language, of such oppositions, and particularly the difference between contradiction and contrariety, has suggested to Sommers (see [468], VII–VIII), a staunch proponent of a traditionalist view of logic, to develop a new “Calculus of Terms”, significantly different—in his eyes—from the predicate logic, but closer to some of the Leibniz’s proposals. In particular, in [468], Chap. 13, the author indicates how traditional logic’s way with contrariety leads to the conception of categories that is at the basis of Ryle’s seminal work in the forties (see [438]) and his own more formal treatment of categories in the early sixties (see [467]). More precisely, Sommers recognizes the need for a notion of contrariety that would allow for saying, for example, that “Saturday is neither fed nor unfed” (which renders both “Saturday is fed” and “Saturday is unfed” category mistakes in the sense of [438]). This kind of problems prompts him to re-examine traditional Aristotelian logic and its characteristic distinction between contrary terms or predicates and contradictory propositions.

  5. 5.

    Open Biological and Biomedical Ontologies.

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Authors and Affiliations

  1. Department of Philosophy, Université Jean Moulin – Lyon III, Lyon, France

    Daniel Parrochia

  2. Ecole Nationale Supérieure des Sciences de l’Information et de la Bibliothèque, Villeurbanne, France

    Pierre Neuville

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  1. Daniel Parrochia
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Parrochia, D., Neuville, P. (2013). Information Data Structures. In: Towards a General Theory of Classifications. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0609-1_2

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