Logical Diagrams, Visualization Criteria, and Boolean Algebras

Abstract

AbstractThis paper considers logical diagrams as a method for visualizing information concerning logical/linguistic/conceptual systems. I introduce four criteria for assessing visualization: (1) completeness, (2) correctness, (3) lack of distortion, and (4) legibility. Next, I present well-known families of diagrams, based on the geometrical figures of (a) the hexagon, and (b) the tetrakis hexahedron. These two families of diagrams are generally regarded as exemplars in the logical geometry literature. To understand better why they succeed so well at visualizing logical information, they are presented as visualizations of complete (finite) Boolean algebras. This also establishes the connection between the combinatorial concept of partition and the logical concept of opposition (i.e., contradiction, contrariness, and subcontrariness). Finally, the paper suggests that the two geometrical figures in question are part of a larger family of polytopes with deep connections to Boolean algebras.KeywordsOppositional geometryLogical geometryLogical hexagonLogical tetrakis hexahedronOppositionPartitionBoolean algebraVisualizationDiagramMathematics Subject Classification03G05 · 03E02 · 03B05 · 52B11