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There Is No Cube of Opposition

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The Square of Opposition: A Cornerstone of Thought

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

Abstract

The theory of opposition has been famously crystallized in a square. One of the most common generalizations of the square is a cube of opposition. We show here that there is no cube such that each of its faces is a square of opposition. We discuss the question of generalization and present two other generalizations of the theory of opposition to the third dimension: one based on Blanché’s hexagon of opposition, the other on the square of contrariety.

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Notes

  1. 1.

    We introduced this colouring of the square in 2003, cf. [2].

  2. 2.

    We have elaborated this distinction in our paper “La puissance du symbole” [11] published in the book La pointure du Symbole [10] which is the result of the interdisciplinary workshop we organized at the University of Neuchâtel in 2005. Saussure gives as an example of symbol the balance but he does not specify the double aspect of symbolization.

  3. 3.

    Our choice for the colours of the square as in Fig. 1 was more or less intuitive: red for contradiction, because it is the strongest opposition, black for subalternation, because it is not an opposition. The choice of blue and green was more intuitive, we didn’t know at this time the RBG theory which was later on formalized by Dany Jaspers using the theory of opposition, see [35].

  4. 4.

    The square of opposition is an interesting way to classify propositions and it can be seen as a tool for classification, which is at once more complex yet more compact than the most famous classificatory structure, the tree—about the theory of classification see [47].

  5. 5.

    Let us point out here that there is a difference between generalization reached by induction and generalization reached by abstraction from a single example. There can be some mixed cases. In the case of the square it looks more like pure abstraction than induction.

  6. 6.

    About recent advances on visual reasoning see e.g. [43].

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Acknowledgements

Thanks to Catherine Chantilly and Robert Purdy for discussion and comments.

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Correspondence to Jean-Yves Béziau .

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Béziau, JY. (2017). There Is No Cube of Opposition. In: Béziau, JY., Basti, G. (eds) The Square of Opposition: A Cornerstone of Thought. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-45062-9_11

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