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Structures of Opposition and Comparisons: Boolean and Gradual Cases

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Abstract

This paper first investigates logical characterizations of different structures of opposition that extend the square of opposition in a way or in another. Blanché’s hexagon of opposition is based on three disjoint sets. There are at least two meaningful cubes of opposition, proposed respectively by two of the authors and by Moretti, and pioneered by philosophers such as J. N. Keynes, W. E. Johnson, for the former, and H. Reichenbach for the latter. These cubes exhibit four and six squares of opposition respectively. We clarify the differences between these two cubes, and discuss their gradual extensions, as well as the one of the hexagon when vertices are no longer two-valued. The second part of the paper is dedicated to the use of these structures of opposition (hexagon, cubes) for discussing the comparison of two items. Comparing two items (objects, images) usually involves a set of relevant attributes whose values are compared, and may be expressed in terms of different modalities such as identity, similarity, difference, opposition, analogy. Recently, J.-Y. Béziau has proposed an “analogical hexagon” that organizes the relations linking these modalities. Elementary comparisons may be a matter of degree, attributes may not have the same importance. The paper studies in which ways the structure of the hexagon may be preserved in such gradual extensions. As another illustration of the graded hexagon, we start with the hexagon of equality and inequality due to R. Blanché and extend it with fuzzy equality and fuzzy inequality. Besides, the cube induced by a tetra-partition can account for the comparison of two items in terms of preference, reversed preference, indifference and non-comparability even if these notions are a matter of degree. The other cube, which organizes the relations between the different weighted qualitative aggregation modes, is more relevant for the attribute-based comparison of items in terms of similarity.

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Notes

  1. In the crisp case it means \(\overline{A}\cap (A\cup B) = B\) if A and B are disjoint.

  2. In previous papers [16, 17, 32], we wrongly credited Reichenbach [49] for inventing this cube in the setting of syllogisms, because we omitted to consider additional constraints of non-equality of involved predicates introduced by him. This point is discussed later on in the next subsection.

  3. The actual octagon of Johnson and Keynes does not materialize the bold lines expressing mutual exclusiveness, while other lines relating A to a and E to e appear and are labeled “complementaries”; likewise, lines relating I to i and O to o appear and are labeled “sub-complementaries”. For instance, A= all P’s are Q and a = all Q’s are P complement each other in the sense that if they both hold, P and Q are identical. Interestingly, in the Reichenbach cube, both A and a, and E and e, are mutually exclusive (and called “opposite”), since P is supposed not to be equal to Q. Moreover yet other lines relating A to o, E to i, O to a and I to e are labeled “contra-complementaries”. Observe that these different forms of complementarity do not appear in the traditional square.

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Acknowledgements

The second author thanks Lorenz Demey (Center for Logic and Philosophy of Science, Leuven, Belgium) for interesting discussions on cubes of opposition in Kolymbari (Crete) in early November 2018 at the 6th world congress on the Square of Opposition. The authors are indebted towards an anonymous reviewer for pointing out references to Johnson–Keynes octagon as the proper ancestor of their cube.

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  1. Institut de Recherche en Informatique de Toulouse (IRIT), Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Didier Dubois & Henri Prade

  2. Equipe de Recherche en Ingénierie des Connaissances - ERIC, Université Claude Bernard Lyon 1, Lyon, France

    Agnès Rico

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  1. Didier Dubois
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Correspondence to Henri Prade.

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Dubois, D., Prade, H. & Rico, A. Structures of Opposition and Comparisons: Boolean and Gradual Cases. Log. Univers. 14, 115–149 (2020). https://doi.org/10.1007/s11787-020-00241-6

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