Generic Figures: using Category Theory to Model Diagrammatic Logic
In this tutorial we introduce the mathematical notion of generic figures as a way to formalize logical diagrammatic notation and forms of diagrammatic reasoning. Generic figures are constitutive components of presheaf categories, which are themselves standard constructions in the mathematics of category theory. Using presheaves and generic figures, it is possible to capture in an intuitive and natural way the relevant structural properties of certain diagrammatic logical systems, in particular the Existential Graphs developed by C. S. Peirce. Not only are the distinct parts and compositional properties of logical diagrams readily represented with generic figures, but the comparative and transformational processes linking and distinguishing diagrams in such systems with respect to one another, thus serving as the basis for reasoning with them, are also susceptible to formalization using the same mathematical tools. This tutorial aims to make these widely applicable techniques available to a broad audience without presuming any background in category theory or previous familiarity with Peirce’s system. All of the mathematical understanding necessary in order to understand presheaves, generic figures, and their application to Peirce’s Existential Graphs will be presented in a self-contained manner.
In this tutorial we introduce the mathematical notion of generic figures as a way to formalize logical diagrammatic notation and forms of diagrammatic reasoning. Generic figures are constitutive components of presheaf categories, which are themselves standard constructions in the mathematics of category theory. Using presheaves and generic figures, it is possible to capture in an intuitive and natural way the relevant structural properties of certain diagrammatic logical systems, in particular the Existential Graphs developed by C. S. Peirce. Not only are the distinct parts and compositional properties of logical diagrams readily represented with generic figures, but the comparative and transformational processes linking and distinguishing diagrams in such systems with respect to one another, thus serving as the basis for reasoning with them, are also susceptible to formalization using the same mathematical tools. This tutorial aims to make these widely applicable techniques available to a broad audience without presuming any background in category theory or previous familiarity with Peirce’s system. All of the mathematical understanding necessary in order to understand presheaves, generic figures, and their application to Peirce’s Existential Graphs will be presented in a self-contained manner.
Session 1
Session 1
The first session introduces the elementary concepts and structures of category theory, with the aim of showing how categories of presheaves and what in [7] are called generic figures (representable functors) are especially useful for representing diagrammatic structures in general. We focus on two elementary examples: directed graphs and discrete dynamical systems. Our approach follows the lead of [5] and especially [7].
The first session introduces the elementary concepts and structures of category theory, with the aim of showing how categories of presheaves and what in [7] are called generic figures (representable functors) are especially useful for representing diagrammatic structures in general. We focus on two elementary examples: directed graphs and discrete dynamical systems. Our approach follows the lead of [5] and especially [7].
Session 2
Session 2
The second session introduces a particular diagrammatic logical system: C. S. Peirce’s Existential Graphs (EG). We summarize the two levels of EG-alpha and EG-beta which correspond to classical propositional logic and first-order logic with identity respectively. Background material providing details on Peirce’s system can be found in [6], [1] and [2].
The second session introduces a particular diagrammatic logical system: C. S. Peirce’s Existential Graphs (EG). We summarize the two levels of EG-alpha and EG-beta which correspond to classical propositional logic and first-order logic with identity respectively. Background material providing details on Peirce’s system can be found in [6], [1] and [2].
Session 3
Session 3
In the third session, the category theory tools introduced in session 1 are applied to Peirce’s logical system as outlined in session 2, following the basic approach found in [3]. It is shown both how the diagrammatic notation of Peirce’s Existential Graphs finds a straightforward representation in the context of categories of presheaves and how the system’s logical deduction rules can be recast in a perspicuous way in this categorical framework.
In the third session, the category theory tools introduced in session 1 are applied to Peirce’s logical system as outlined in session 2, following the basic approach found in [3]. It is shown both how the diagrammatic notation of Peirce’s Existential Graphs finds a straightforward representation in the context of categories of presheaves and how the system’s logical deduction rules can be recast in a perspicuous way in this categorical framework.
Readings
Readings
[1] Bellucci, F. and A.-V. Pietarinen (2016) “Existential Graphs as an instrument of logical analysis. Part I: alpha,” The Review of Symbolic Logic 9(2): 209-237.
[1] Bellucci, F. and A.-V. Pietarinen (2016) “Existential Graphs as an instrument of logical analysis. Part I: alpha,” The Review of Symbolic Logic 9(2): 209-237.
[2] Bellucci, F. and A.-V. Pietarinen (2021) “An analysis of Existential Graphs – part 2: Beta,” Synthese 199: 7705-7726.
[2] Bellucci, F. and A.-V. Pietarinen (2021) “An analysis of Existential Graphs – part 2: Beta,” Synthese 199: 7705-7726.
[3] R. Gangle, G. Caterina and F. Tohmé (2020) “A Generic Figures Reconstruction of Peirce’s Existential Graphs,” Erkenntnis 85: 1-34.
[3] R. Gangle, G. Caterina and F. Tohmé (2020) “A Generic Figures Reconstruction of Peirce’s Existential Graphs,” Erkenntnis 85: 1-34.
[4] R. Goldblatt. Topoi: The Categorical Analysis of Logic (Dover 2006).
[4] R. Goldblatt. Topoi: The Categorical Analysis of Logic (Dover 2006).
[5] W. Lawvere and S. H. Schanuel, Conceptual Mathematics: A First Introduction to Categories 2nd ed. (Cambridge University Press 2009).
[5] W. Lawvere and S. H. Schanuel, Conceptual Mathematics: A First Introduction to Categories 2nd ed. (Cambridge University Press 2009).
[6] C. S. Peirce, ed. by K. L. Ketner, Reasoning and the Logic of Things: The Cambridge Conferences Lectures of 1898 (Harvard University Press 1993).
[6] C. S. Peirce, ed. by K. L. Ketner, Reasoning and the Logic of Things: The Cambridge Conferences Lectures of 1898 (Harvard University Press 1993).
[7] M. Reyes, G. Reyes and H. Zolfaghari, Generic figures and their glueings: a constructive approach to functor categories (Polimetrica 2004).
[7] M. Reyes, G. Reyes and H. Zolfaghari, Generic figures and their glueings: a constructive approach to functor categories (Polimetrica 2004).