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Towards a General Theory of Classifications pp 209–228Cite as

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For an Axiomatic Theory of Classifications

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

Abstract

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).

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Notes

  1. 1.

    Recall that the axiom of separation is the axiom saying that, given any set A, there is a set B such that, given any set x, x is a member of B iff x is a member of A and f holds for x (see [518–520]).

  2. 2.

    The axiom of comprehension is the axiom saying that there exists a set B whose members are precisely those objects that satisfy a predicate P without any kind of restriction on P. This axiom schema was tacitly used in the early days of naive set theory, i.e. before a strict axiomatization had been adopted. Unfortunately, it leads directly, as we have seen in Sect. 3.7.1, to Russell’s paradox, by defining P as x∉x.

  3. 3.

    Isaac Malitz (see [330]) originally introduced this designation in his 1976 PHD thesis at UCLA (Church was the chairman of the committee of supervisors). A more recent paper on the consistency of positive theory of sets is: [156], 105–116.

  4. 4.

    A formula f is said to be stratified if there exists a function f from the parts of syntax of the natural numbers, such that, for any atomic subformula of f, we have f(y)=f(x)+1, while, for any atomic subformula x=y of f, we have f(x)=f(y).

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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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  2. Pierre Neuville
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Parrochia, D., Neuville, P. (2013). For an Axiomatic Theory of Classifications. In: Towards a General Theory of Classifications. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0609-1_8

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