From Euler Diagrams to Aristotelian Diagrams

Euler and Aristotelian diagrams are both among the most well-studied kinds of logical diagrams today. Despite their central status, very little research has been done on relating these two types of diagrams. This is probably due to the fact that Euler diagrams typically visualize relations between sets, whereas Aristotelian diagrams typically visualize relations between propositions. However, recent work has shown that Aristotelian diagrams can also perfectly be understood as visualizing relations between sets, and hence it becomes natural to ask whether there is any kind of systematic relation between Euler and Aristotelian diagrams. In this paper we provide an affirmative answer: we show that every Euler diagram for two non-trivial sets gives rise to a well-defined Aristotelian diagram. Furthermore, depending on the specific relation between the two sets visualized by the Euler diagram, the resulting Aristotelian diagram will also be fundamentally different. We will also link this with well-known notions from logical geometry, such as the information ordering on the seven logical relations between non-trivial sets, and the notion of Boolean complexity of Aristotelian diagrams.