Levels of syntax for Euler diagram logics

Euler diagrams have been used for centuries to convey logical information and have been formalized for this purpose. Related notations, such as Venn-II, Euler/Venn, spider diagrams, and constraint diagrams, have been developed and fully formalized as logics in their own right. Issues regarding their abstract (type) syntax and concrete (token) syntax have been discussed in the literature but there are more fine grained levels of syntax that have not, to the authors’ knowledge, been given serious consideration. We discuss different levels of Euler diagram syntax. These include the drawn level, where the images of the diagrams live, for example, on a computer screen, a piece of paper, or a whiteboard; mathematical models of the drawn diagrams; the dual graph of an Euler diagram; and the abstract level of syntax that captures the semantic information present in the diagram and forgets all geometric and topological properties that have no impact on the semantics.