Opposition and Implication in Aristotelian Diagrams

In logical geometry, Aristotelian diagrams are studied in a systematic fashion. Recent developments in this field have shown that the square of opposition generalizes in two ways, which correspond precisely to the theory of opposition (leading to α-structures) and the theory of implication (leading to ladders) it exhibits. These two kinds of Aristotelian diagrams are dual to each other, in the sense that they are the oppositional and implicative counterpart of the same construction. This paper formalizes this duality as OI-companionship, explores its properties, and applies it to various σ-diagrams. This investigation shows that OI-companionship has some interesting, but unusual behaviors. While it is symmetric, and works well on the level of Aristotelian families, it lacks (ir)reflexivity, transitivity, functionality, and seriality. However, we show that all important Aristotelian families from the literature do have a unique OI-companion. These findings explore the limits that arise when extending the duality between opposition and implication beyond the limits of α-structures and ladders.