PaL Diagrams: A Linear Diagram-Based Visual Language

Linear diagrams have recently been shown to be

more effective than Euler diagrams when used

for set-based reasoning. However, unlike the

growing corpus of knowledge about formal aspects

of Euler and Venn diagrams, there has been no

formalisation of linear diagrams. To fill this

knowledge gap, we present and formalise Point

and Line (PaL) diagrams, an extension of simple

linear diagrams containing points, thus providing

a formal foundation for an effective visual

language.We prove that PaL diagrams are exactly

as expressive as monadic first-order logic with

equality, gaining, as a corollary, an equivalence

with the Euler diagram extension called spider

diagrams. The method of proof provides translations

between PaL diagrams and sentences of monadic

first-order logic.