Logical Diagrams, Visualization Criteria and Boolean Algebras

Abstract

This paper presents logical diagrams as attempts to visualize facts about logical/linguistic/conceptual systems. It introduces four criteria for assessing visualization: 1) completeness, 2) correctness, 3) lack of distortion, and 4) legibility. It then studies presents well-known families of diagrams, based on the geometrical figures of a) the hexagon, and b) the tetrakis hexahedron. These are usually presented as exemplary diagrams. 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.