Structures of Opposition and Comparisons: Boolean and Gradual Cases

Didier Dubois (1, 2) , Henri Prade (1, 2) , Agnès Rico (3)

1 IRIT-ADRIA - Argumentation, Décision, Raisonnement, Incertitude et Apprentissage
2 CNRS - Centre National de la Recherche Scientifique
3 ERIC - Entrepôts, Représentation et Ingénierie des Connaissances

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.

Mots clés: Mathematics Subject Classification (2010). Primary 68T30; Secondary 03A05; 03B05; 68T37 Square of opposition; hexagon of opposition; cube of opposition; partition; difference; similarity; analogy; ordered weighted min; comparison; preferences

Domaines: Logique en informatique [cs.LO]

Didier Dubois, Henri Prade, Agnès Rico. Structures of Opposition and Comparisons: Boolean and Gradual Cases. Logica Universalis, 2020, 14 (1), pp.115-149. ⟨10.1007/s11787-020-00241-6⟩. ⟨hal-03121904⟩