The readability of proofs in diagrammatic logic

The fact that diagrams can be an expressive and accessible way to communicate is widely recognised and valued, but what makes one diagram more effective than another is seldom measured. Formal diagrams attempt to combine the effectiveness of graphics with mathematical logic, and have the potential to bring the benefits of this expressiveness and accessibility to the field of logical reasoning. Diagrammatic reasoning (or visual logic) consists of using diagrams to represent logical theorems and to create diagrammatic proofs or counter-arguments for those theorems. In this project we will analyse and measure, for the first time, the factors that affect comprehension in diagrammatic reasoning. Our research question is as follows: is it possible to develop a systematic understanding of readability in diagrammatic proofs? (We use the term "readable" to mean easy to understand and use.) To be effective, such understanding would yield strategies to be used to automatically construct readable proofs. Furthermore, these strategies should be general enough to be applied to any visual logic and possess predictive qualities that can inform the design of future visual logics.

Euler diagrams are a simple visual logic which is used to represent data in a great many contexts. As well as being used informally or semi-formally in fields such as education, Euler diagrams have been used as a formal logic since the 1990s. The widespread use of Euler diagrams (for instance, in teaching set theory to school children) testifies to the fact that they are generally considered easy to understand. Logicians, philosophers and cognitive scientists have attempted to describe the origins of this ease of understanding, but have not done so in the specific context of the use of Euler diagrams in proofs, or in a way that yields general strategies for creating Euler diagram proofs which are easily understood.

The aims of this project are to provide a detailed, formal understanding of readability in diagrammatic proofs and to provide ways of automatically generating readable diagrammatic proofs. We will achieve the second of these aims by adapting an existing automated Euler diagram theorem prover. The aims allow us to address what we believe to be two of the most challenging issues for the diagrams community today: the need to explore when diagrammatic reasoning is understandable by people, and the need to produce effective software tools that make the benefits of diagrams available to a wide range of people. The project addresses the need to categorise and to empirically measure the features that make diagrammatic proofs more, or less, easy to understand. Identifying these features and the ways in which they interact will help us to make better use of existing notations and design more effective logics in the future.

The research outcomes will have significant benefits for a wide range of beneficiaries. Some of these benefits will be realised during the lifetime of the project or in the months after it ends, while some benefits will take several years to come to fruition. We anticipate that the project will benefit the following groups:

. Users of diagrammatic modelling systems in industry: examples include NOKIA, who have applied Euler-based logic developed by the Visual Modelling Group in privacy engineering. In the short-term, this group will benefit from increased awareness of and access to the potential role of reasoning (automated and interactive) within modelling processes such as ontology engineering. In the medium and long-term, this group will benefit from access to improved modelling processes through access to readable proofs, leading to better understanding
of aspects of the systems being modelled and to economic gains. Our work will enhance the communication of precise ideas within businesses, enabling non-mathematicians to
participate in reasoning activities via more accessible methods. Our research objectives will allow our industrial collaborators (such as NOKIA) to enrich society by providing software systems
with better designs.

. Developers and users of diagrammatic and/or heterogeneous reasoning: our collaborators at Stanford and Cambridge, the other developers and research teams who contribute to the Openbox and Speedith codebases, and the users of the software. This group will benefit from improved knowledge of integrating diagrammatic reasoning tools with Openbox. Our research objectives have the potential to attract more users and developers to the Openbox platform by offering a more diverse selection of components and, in the medium and long-term, enrich society by making courseware and source code of our tools available.

. Teachers and students of logic: those organisations and individuals who want to find accessible ways to teach the skills of reasoning in a logical system, and those students who may find diagrammatic or heterogeneous logic more accessible than the purely symbolic alternative. Our objectives will have an immediate benefit for this group by enabling the enhancement and extension of current methods of teaching analytical reasoning by improving the state-of-the-art of teaching tools based on diagrams. The research-based software tools we deliver will make analytical reasoning more accessible for those people with cognitive preferences for diagrammatic as opposed to symbolic reasoning. In the medium and long-term this has the potential to enrich society by making diagrammatic reasoning more accessible to wider groups of people, and provide the foundation for enhancing the digital economy by making it possible to improve
education in mathematical logic.

These impacts will be achieved by our commitment to make our results widely and freely available, and by publicising our work through public forums and user groups such as mailing lists relevant to Openbox and mathematical education.