Euler vs Hasse Diagrams for Reasoning About Sets: A Cognitive Approach

Abstract

AbstractThe literature on diagrammatic reasoning includes theoretical and experimental work on the effectiveness of diagrams for conveying information. One influential theoretical contribution to this field proposes that a notation that is more effective than another would have an observational advantage over it; that is, it would make certain pieces of information observable—by means of some visual, meaning-carrying relationships—that were not observable in the other. Although the notion of observational advantage captures a relevant aspect of the benefit of using one notation over another, we present here an example where this notion is not sufficient to distinguish between a more and a less effective diagram. We suggest to take the theory of observational advantage one step further by linking it to cognitive theories of human conceptualisation and reasoning. Following our previous work, we propose that the act of observing facts about set theory from the geometry of a diagram can be modeled as a conceptual blend of image schemas with parts of the geometric configuration of a diagram. Image schemas are elementary mental structures that crystallize early embodied experiences, allowing agents to make sense of what they perceive by conceptualising it in terms of these structures (e.g., container, link, scale etc.). With our approach, we can extend the theory of observational advantage to take into account the cognitive complexity of the act of observation. Concretely, we present an example of an Euler and a Hasse diagram, and we posit that, while their observational advantage is equivalent, the Hasse diagram requires a much more complex network of conceptual blends to model certain observations made from it. Thus, to reason about certain set-theoretic claims, a Hasse diagram is less cognitively effective than an Euler diagram with equivalent observational advantage. We believe our approach contributes to the theoretical discussion on what factors affect the effectiveness of a diagram, and provides new avenues for the exploration of how the embodied experiences of the users contribute to the way they reason with diagrams.