Organizing families of aggregation operators into a cube of opposition

Abstract

The cube of opposition is a structure that extends the traditional square of opposition originally introduced by Ancient Greek logicians in relation with the study of syllogisms. This structure, which relates formal expressions, has been recently generalized to non Boolean, graded statements. In this paper, it is shown that the cube of opposition applies to well-known families of idempotent, monotonically increasing aggregation operations, used in multiple criteria decision making, which qualitatively or quantitatively provide evaluations between the minimum and themaximum of the aggregated quantities. This covers weighted minimum and maximum, and more generally Sugeno integrals on the qualitative side, and Choquet integrals, with the important particular case of Ordered Weighted Averages, on the quantitative side. The main appeal of the cube of opposition is its capability to display the various possible aggregation attitudes in a given setting and to show their complementarity.