The Relationship between Aristotelian and Hasse Diagrams

Abstract

The aim of this paper is to study the relationship between two important families of diagrams that are used in logic, viz. Aristotelian diagrams (such as the well-known ‘square of oppositions’) and Hasse diagrams. We discuss some obvious similarities and dissimilarities between both types of diagrams, and argue that they are in line with general cognitive principles of diagram design. Next, we show that a much deeper connection can be established for Aristotelian/Hasse diagrams that are closed under the Boolean operators. We consider the Boolean algebra $\mathbb{B}_n$ with 2ⁿ elements, whose Hasse diagram can be drawn as an n-dimensional hypercube. Both the Aristotelian and the Hasse diagram for $\mathbb{B}_n$ can be seen as (n − 1)-dimensional vertex-first projections of this hypercube; whether the diagram is Aristotelian or Hasse depends on the projection axis. We show how this account provides a unified explanation of the (dis)similarities between both types of diagrams, and illustrate it with some well-known Aristotelian/Hasse diagrams for $\mathbb{B}_3$ and $\mathbb{B}_4$.