Compositional Diagrammatic First-Order Logic

Peirce’s \(\beta \) variant of Existential Graphs (EGs) is a diagrammatic formalism, equivalent in expressive power to classical first-order logic. We show that the syntax of EGs can be presented as the arrows of a free symmetric monoidal category. The advantages of this approach are (i) that the associated string diagrams share the visual features of EGs while (ii) enabling a rigorous distinction between “free” and “bound” variables. Indeed, this diagrammatic language leads to a compositional relationship of the syntax with the semantics of logic: we obtain models as structure-preserving monoidal functors to the category of relations.