Categorical interpretation of Peice's Existential Graphs

Discussing Graphs within graphs nodes on categorytheory.zulipchat.com I was pointed to
a recent paper that presented this from a CT perspective:
"Compositional Diagrammatic First-Order Logic"

https://www.ioc.ee/~pawel/papers/peirce.pdf���
peirce
PDF Document · 3,5 MB

 Abstract:

> Abstract. Peirce’s β 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. In addition to a diagrammatic syntax for formulas, Peirce developed a sound and complete system of diagrammatic reasoning that arose out of his study of the algebra of relations. Translated to string diagrams we show the implied algebraic structure of EGs sans negation is that of cartesian bicategories of relations: for example, lines of identity obey the laws of special Frobenius algebras. We also show how the algebra of negation can be presented, thus capturing Peirce’s full calculus.

This seems to go one stage further than Evan Patterson’s
"Knowledge Representation in Bicategories of Relations"
https://arxiv.org/abs/1706.00526
in that it brings in negative surfaces and so brings together the positive and the negative
Fragments in one diagramatic system. But Evan Patterson’s paper has the
advantage for us in that it directly speaks about the semantic web.

Henry Story