Fuzzy Extensions of Conceptual Structures of Comparison

Abstract

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.

Structures of Opposition and Comparisons: Boolean and Gradual Cases

“…This section considers different settings for comparing objects where graded hexagons or cubes of opposition are present. Several results in this section were already proposed in [32].…”

“…This subsection deals with a gradual extension of the strong hexagon of opposition [19,32]. A graded hexagon of opposition is obtained by first assigning variables ν = d(α, ) and γ = c(ι, o) to new vertices U and Y.…”

“…similarity) between items. A further extension of the analogical hexagon involving statements with weakened quantifiers, such as "x and y are (highly) similar on at least k over q attributes" has been outlined and discussed in [32]. Since the treatment of such quantifiers is a topic in itself, and would be relevant as well for the other structures of opposition (square, cubes), we leave it aside in this paper.…”

“…Such gradual extensions take into account approximate equality, and attribute importance. A preliminary version of that latter part of the section appeared in [32]. We finally consider the two cube structures and indicate their relevance for the comparison of items.…”