The Modal Logic of Aristotelian Diagrams

Abstract

In this paper, we introduce and study AD-logic, i.e., a system of (hybrid) modal logic that can be used to reason about Aristotelian diagrams. The language of AD-logic, LAD, is interpreted on a kind of birelational Kripke frames, which we call “AD-frames”. We establish a sound and strongly complete axiomatization for AD-logic, and prove that there exists a bijection between finite Aristotelian diagrams (up to Aristotelian isomorphism) and finite AD-frames (up to modal isomorphism). We then show how AD-logic can express several major insights about Aristotelian diagrams; for example, for every well-known Aristotelian family A, we exhibit a formula χA∈LAD and show that an Aristotelian diagram D belongs to the family A iff χA is validated by D (when the latter is viewed as an AD-frame). Finally, we show that AD-logic itself gives rise to new and interesting Aristotelian diagrams, and we reflect on their profoundly peculiar status.

Boolean Subtypes of the U4 Hexagon of Opposition

“…The contemporary research program of logical geometry studies Aristotelian diagrams as objects of independent mathematical and philosophical interest [36][37][38][39][40]. A major (and still ongoing) research effort in this area is the development of a comprehensive typology of Aristotelian diagrams, which allows us to systematically classify these diagrams into various families and subfamilies [41][42][43]. For example, the diagram in Figure 1a belongs to the family of 'classical squares of opposition', while those in Figure 1b,c both belong to the family of 'Jacoby-Sesmat-Blanché (JSB) hexagons' [8,9,11].…”