On Different Ways to be (dis)similar to Elements in a Set. Boolean Analysis and Graded Extension

Abstract

We investigate here two questions, first in a Boolean setting and then in a gradual setting: Can we give a formal meaning to “being at odds” (in the sense of being an outlayer) with regard to a subset and, as a dual problem, can we give a meaning to “being even” (in the sense of conforming to a given set of values). Is there a relation between oddness and evenness? Such questions emerge from recent proposals for using oddness or evenness measures in classification problems. This paper is dedicated to a formal study of the oddness and evenness indices in the case of subsets with three or four elements, which are at the basis of the associated measures. Triples are indeed the only subsets such that adding an item that conforms to the triple minority, if any, destroys the majority. It appears that the notions of oddness and evenness are not simple dual of each other; a third notion of being “balanced” interplays with the two others. This is discussed in the setting of squares and hexagons of opposition. The notions of oddness and evenness are related to the study of homogeneous and heterogeneous logical proportions that link four Boolean variables through the conjunction of two equivalences between similarity or dissimilarity indicators pertaining to pairs of these variables. Although elementary, the analysis provides an organized view of new notions that appear to be meaningful when revisiting the old ideas of similarity and dissimilarity in a new perspective. As a side result, it is also mentioned that the logical proportion underlying the idea of being balanced corresponds to the logical encoding of Bongard problems.