International Conference on Theory and Application of Diagrams

Diagrams 2014: Diagrammatic Representation and Inference pp 246-260

# Logical and Geometrical Complementarities between Aristotelian Diagrams

• Hans Smessaert
• Lorenz Demey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8578)

## Abstract

This paper concerns the Aristotelian relations of contradiction, contrariety, subcontrariety and subalternation between 14 contingent formulae, which can get a 2D or 3D visual representation by means of Aristotelian diagrams. The overall 3D diagram representing these Aristotelian relations is the rhombic dodecahedron (RDH), a polyhedron consisting of 14 vertices and 12 rhombic faces (Section 2). The ultimate aim is to study the various complementarities between Aristotelian diagrams inside the RDH. The crucial notions are therefore those of subdiagram and of nesting or embedding smaller diagrams into bigger ones. Three types of Aristotelian squares are characterised in terms of which types of contradictory diagonals they contain (Section 3). Secondly, any Aristotelian hexagon contains 3 squares (Section 4), and any Aristotelian octagon contains 4 hexagons (Section 5), so that different types of bigger diagrams can be distinguished in terms of which types of subdiagrams they contain. In a final part, the logical complementarities between 6 and 8 formulae are related to the geometrical complementarities between the 3D embeddings of hexagons and octagons inside the RDH (Section 6).

## Keywords

Aristotelian relations square of oppositions hexagon of oppositions logical geometry 3D visualisation subdiagrams complementarity embedding

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