Diagrams as Physical Models1

We discuss a variety of roles for diagrams in helping with reasoning, focusing in particular on their role as physical models of states of affairs, much like an architectural model of a building or a 3-D molecular model of a chemical compound. We discuss the concept of a physical model for a logical sentence, and the role played by the causal structure of the physical medium in making the given sentence as well as a set of implied sentences true. This role of a diagram is consistent with a widely-held intuition that diagrams exploit the fact that 2-D space is an analog of the domain of discourse. One line of research in diagrammatic reasoning is that diagrams, rather then being models, are formal representations with specialized rules of inference that generate new diagrams. We reconcile these contrasting views by relating the usefulness of diagrammatic systems as formal representations to the fact that their rewrite rules take advantage of the diagrams’ model-like character. When the physical model is prototypical, it supports the inference of certain other sentences for which it provides a model as well. We also informally discuss a proposal that diagrams and similar physical models help to explicate a certain sense of relevance in inference, an intuition that so-called Relevance Logics attempt to capture. Roles of Diagrams in Reasoning Diagrams give many different types of assistance during problem solving. We identify five roles here: helping extend short term memory, helping organize problem solving by spatial organization of related information, as sentences in a 2-D language with specialized rules of inference, providing a model of the premises so that 1 This paper was prepared through participation in the Advanced Decision Architectures Collaborative Technology Alliance sponsored by the U.S. Army Research Laboratory under Cooperative Agreement DAAD19-01-2-0009, and by federal flow-through by the Department of Defense under contract FA8652-03-3-0005 (as a subcontract from Wright State University and Wright Brothers Institute). I am indebted to Peter Schroeder-Heister and Gerard Allwein for significant assistance in thinking about these ideas, to Neil Tennant and Stewart Shapiro for useful discussions, and to one of the reviewers who made useful suggestions for improvement.