A Generic Figures Reconstruction of Peirce’s Existential Graphs (Alpha)

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic (Peirce’s Existential Graphs $$\alpha $$). The straightforward construction of a presheaf category $${{\mathcal {E}}}{{\mathcal {G}}}_{\alpha ^{*}}$$ of cuts-only Existential Graphs (equivalent to the well-studied category of finite forests) provides a basis for the further construction of the category $${{\mathcal {E}}}{{\mathcal {G}}}_\alpha $$ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the formal investigation of relations between syntax and semantics in such diagrammatic systems.