Abstract
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.
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Notes
A mathematical category consists of a class of objects together with morphisms or arrows between objects subject to axioms of identity (every object is equipped with an identity morphism that composes inertly), composition (head-to-tail morphisms compose to a unique morphism) and associativity (paths of morphisms compose uniquely). For a comprehensive introduction and details filling out this rough characterization, see Mac Lane (1998) and Spivak (2014).
The text of Reyes et al. (2004) is at times ambiguous between referring to the objects of the base category as the generic figures and using the term to refer to the representable sets that represent those objects. Since these are in one-to-one correspondence and context usually determines which is meant, the ambiguity remains basically innocuous.
The terminology of “representable set” is somewhat unfortunate, as a representable set is, on the one hand, not a set but rather a presheaf and, on the other hand, more characteristically representative than representable. Nevertheless, we maintain the standard usage.
FinSet is the category of finite sets and functions between them (a full subcategory of Set). We take presheaves into FinSet rather than Set in order to exclude the possibility of infinite conjunctions of cuts at any given depth.
Note, however, that the converse does not hold in general. It is nonetheless straightforward, if somewhat tedious, to define a function from morphisms in \({{\mathcal {E}}}{{\mathcal {G}}}_{\alpha ^{*}}\) into \(G^{+}\) that replaces branching extensions of more than one node (cases in which \({\hat{g}}\) is too large) with single nodes. Details are left to the reader.
We represent these lines in the third dimension perpendicular to the plane of the sheet of assertion in order to preclude any possible ambiguity with the “lines of identity” used in Peirce’s EG-\(\beta \) system.
Note that the fact that \(f_{{{\mathcal {V}}}{{\mathcal {T}}}_{g^{+}h^{+}}}\) is a functor between categories (specifically groupoids) guarantees that the variable typing is respected across the mapping. Two variable-tokens that are identified as being of a common type in \(g^{+}_{var}\) must be mapped to variable-tokens that are of the same type in \(h^{+}_{var}\) (the existence of an arrow \(X\longrightarrow Y\) in \({{\mathcal {V}}}{{\mathcal {T}}}_{g^{+}}\) implies the existence of an arrow \(f_{{{\mathcal {V}}}{{\mathcal {T}}}_{g^{+}h^{+}}}(X)\longrightarrow f_{{{\mathcal {V}}}{{\mathcal {T}}}_{g^{+}h^{+}}}(Y)\) in \({{\mathcal {V}}}{{\mathcal {T}}}_{h^{+}}\)).
Note that any discrete category is, trivially, a groupoid.
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Gangle, R., Caterina, G. & Tohme, F. A Generic Figures Reconstruction of Peirce’s Existential Graphs (Alpha). Erkenn 87, 623–656 (2022). https://doi.org/10.1007/s10670-019-00211-5
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DOI: https://doi.org/10.1007/s10670-019-00211-5