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Analogical Proportions and Binary Trees

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

Abstract

Analogical reasoning has been thought for a longtime as something aside, away from logical reasoning. However, it is not exactly so. This chapter in its first part mainly surveys works of the last decade, which have proposed a logical modeling of analogical proportions (i.e., statements of the form “a is to b as c is to d”) among other logically expressible proportions and have shown their use in analogical inference. It also emphasizes the pervasiveness of analogical proportions as soon as we compare situations. The second part of the chapter shows that dichotomous trees built from pairs of mutually exclusive properties have also a reading in terms of Boolean analogical proportions, thus providing another clue of the links existing between analogy and logically expressed taxonomies. This also gives birth to noticeable opposition structures and can be related to formal concept analysis.

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Mathematics Subject Classification (2000)

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Notes

  1. 1.

    Other patterns can be obtained by combination. This in particular the case of the Joint method of agreement and difference:

    A B C occur together with x y z,

    A D E occur together with x v w also B C occur with y z

    —————————————————————————————

    Therefore A is the cause, or the effect, or a part of the cause of x.

  2. 2.

    This cube is distinct from the cube of opposition obtained as an extension of the traditional square of opposition; see [7, 12, 42].

  3. 3.

    The third conjunct is unfortunately missing in [1].

  4. 4.

    This subsection follows from an idea suggested by Bernhard Ganter to the second author.

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Barbot, N., Miclet, L., Prade, H., Richard, G. (2022). Analogical Proportions and Binary Trees. In: Béziau, JY., Desclés, JP., Moktefi, A., Pascu, A.C. (eds) Logic in Question. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-94452-0_22

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