Euler Diagrams, Aristotelian Diagrams and Syllogistics

Euler diagrams and Aristotelian diagrams are two of the most important types of diagrams to visualize (relations between) sets. We have previously shown that Euler diagrams for two sets systematically give rise to various Aristotelian diagrams, such as classical squares of opposition. In this paper, we expand this analysis to Euler diagrams for three sets, and show that they give rise to various kinds of hexagons of opposition as well. This move from two to three sets is philosophically well-motivated and technically non-trivial. On the philosophical side, there is a connection with syllogistics, since syllogisms consist of three terms/sets. On the technical side, moving from two to three sets requires us to take the phenomenon of Boolean subtypes into account.