From Euler Diagrams in Schopenhauer to Aristotelian Diagrams in Logical Geometry

In this paper I explore the connection between Schopenhauer’s Euler diagrams and the Aristotelian diagrams that are studied in contemporary logical geometry. One can define the Aristotelian relations in a very general fashion (relative to arbitrary Boolean algebras), which allows us to define not only Aristotelian diagrams for statements, but also for sets. I show that, once this generalization has been made, each of Schopenhauer’s concrete Euler diagrams can be transformed into a well-defined Aristotelian diagram. More importantly, I also argue that Schopenhauer had several more general, systematic insights about Euler diagrams, which anticipate general insights and theorems about Aristotelian diagrams in logical geometry. Typical examples include the correspondence between n-partitions and α-structures (a particular class of Aristotelian diagrams), and the fact that many families of Aristotelian diagrams have distinct Boolean subtypes. Because of his various concrete Euler diagrams and, especially, his more systematic observations about them, Schopenhauer can rightly be considered a distant forerunner of contemporary logical geometry.