 It is worth noting that many mathematicians and philosophers proposed diagrammatic systems, often with a didactic motivation. Some systems, like Lewis Carroll’s in “The Game of Logic”  (1896)  are variants on the proposals of Euler and Venn. Others, like Frege  (1879),  used lines rather than plane regions. (For a description of Frege’s notation, see the section on Complex Statements and Generality in the entry on  Gottlob Frege.  See also Englebretsen  1992.)  Carroll’s system supersedes Venn’s in that the complements of sets are explicitly represented as regions of the diagram, rather than being left as the background region against which the circles appear. This means that Carroll’s system is able to draw inferences about relations between complements of properties, at the expense of representing some properties as disjoint (i.e., non-connected) regions. This shift closely mirrors the shift in logic from subject-predicate argumentation to a function-argument representation (Stenning  1999).