Fuzzy Extensions of Conceptual Structures of Comparison
Résumé
Comparing two items (objects, images) involves a set of relevant attributes whose values are compared. Such a comparison 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. The hexagon structure extends the logical square of opposition invented in Aristotle time (in relation with the theory of syllogisms). The interest of these structures has been recently advocated in logic and in artificial intelligence. When non-Boolean attributes are involved, elementary comparisons may be a matter of degree. Moreover, attributes may not have the same importance. One might only consider most attributes rather than all of them, using operators such as ordered weighted min and max. The paper studies in which ways the logical hexagon structure may be preserved in such gradual extensions. As an illustration, we start with the hexagon of equality and inequality due to Blanché and extend it with fuzzy equality and fuzzy inequality.