Visualizing logical aspects of conceptual structures

Conceptual structures are studied in many different disciplines and represented in a variety of forms including natural language, formal logic, and graphical or spatial representations. This article analyzes the common foundations of such representations across all disciplines, and the relationships between the different forms of representation. Representations of the logical relations in conceptual structures are compared, including logical symbolism, Euler diagrams, semantic networks, conceptual grids and conceptual spaces. It is shown that these representations are formally equivalent and can be inter-translated algorithmically, but provide different and complementary visualizations such that the use of multiple representations may provide greater insight than any alone. It is shown that a wide range of significant conceptual structures can be represented and visualized in a substructural logic having only two logical relations, entailment and contrast/opposition, that is naturally represented in Euler diagrams and in semantic networks with two types of connecting arrow. It is shown that human everyday reasoning not involving definitions and rules but based on abduction over schemata representing traces of past experience can be modeled and visualized in these representations. The extension of the visual representation to include the constructs of a description logic and bridge from the substructural logic to structural mathematical logics is illustrated. It is concluded that interactive, computerbased visualization tools supporting a range of different representation schemes and inference based on a heterogeneous mix of representations can provide significant support for education and research across the many disciplines concerned with conceptual structures.