Incoporating negation into visual logics: a case study using Euler diagrams

Abstract

Many diagrammatic logics based on Euler diagrams have been defined with the aim of making precise reasoning easier for people. In such logics it is unusual to find the negation operator (¬) included. This is likely to impact the usability of the logics when users try to make statements that are naturally phrased using the ¬ operator. Furthermore, if one wishes to use semantic tableaux methods for the purposes of establishing entailment then including ¬ is essential. Thus, there are good reasons for extending existing diagrammatic logics to include ¬ explicitly. In this paper, we take Euler diagrams and extend the notation to include the ¬ operator, as well as 'or' and 'and'. Various expressiveness results for the logic are established. We present a sound and complete set of reasoning rules for the logic, drawing parallels with existing completeness proof strategies and highlighting differences that arise due to including negation.