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Revista Colombiana de Matemáticas

Print version ISSN 0034-7426

Rev.colomb.mat. vol.53 no.2 Bogotá July/Dec. 2019  Epub Mar 20, 2020

 

Artículos originales

Existential Graphs on nonplanar surfaces

Gráficos existenciales sobre superficies no planas

ARNOLD OOSTRA1 

1Departamento de Matemáticas y Estadística Universidad del Tolima Facultad de Ciencias Barrio Santa Helena Ibagué, Colombia e-mail: noostra@ut.edu.co


ABSTRACT.

Existential graphs on the plane constitute a two-dimensional representation of classical logic, in which a Jordan curve stands for the negation of its inside. In this paper we propose a program to develop existential Alpha graphs, which correspond to propositional logic, on various surfaces. The geometry of each manifold determines the possible Jordan curves on it, leading to diverse interpretations of negation. This may open a way for appointing a "natural" logic to any surface.

Key words and phrases. Existential graphs; 2-dimensional topological manifolds; Jordan's Curve Theorem

RESUMEN.

Los gráficos existenciales sobre el plano constituyen una representación bidimensional de la lógica clásica, en la cual una curva de Jordan indica la négation de su interior. En este artículo se propone un programa para desarrollar los gráficos existenciales Alfa, que corresponden a la lógica preposicional, sobre diferentes superficies. La geometría de cada variedad determina las posibles curvas de Jordan sobre ella, lo cual conduce a interpretaciones diversas de la negación. Esto puede abrir el camino para asignar una lógica "natural" a cualquier superficie.

Palabras y frases clave. Gráficos existenciales; 2-variedades topolágicas; teorema de la curva de Jordan

Full text available only in PDF format.

Aknowledgements

Parts of this paper were presented in talks at the Universidad del Tolima (Ibague) and at the Universidad Nacional de Colombia (Bogota). Special thanks to Fernando Zalamea, Leonardo Solanilla, Andres Villaveces, and Franciso Vargas for very enlightening discussions on this topic, and also for their substantial comments on earlier versions of this paper. The author wishes to thank the anonymous referee for his valuable suggestions, which improved the submitted draft. The drawings in this paper were created with pst-solides3d.

References

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[2] F. Bellucci, D. Chiffi, and A.-V. Pietarinen, Assertive Graphs, Journal of Applied Non-Classical Logics 28 (2018), no. 1, 72-91. [ Links ]

[3] T. tom Dieck, Algebraic Topology, European Mathematical Society, Ziirich, 2008. [ Links ]

[4] C. Fuentes, Calculo de secuentes y graficos existenciales Alfa: Dos estructuras equivalentes para la lógica proposicional, undergraduate thesis, Universidad del Tolima, Ibague, Colombia, 2014. [ Links ]

[5] T. C. Hales, Jordan's Proof of the Jordan Curve Theorem, Studies in Logic, Grammar and Rhetoric 10 (2007), no. 23, 45-60. [ Links ]

[6] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. [ Links ]

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[9] Y. Martínez, Un modelo real para los graficos Alfa, undergraduate thesis, Universidad del Tolima, Ibague, Colombia, 2014. [ Links ]

[10] A. Oostra and D. Diaz, Algebras booleanas libres en algebra, topología y logica, Boletin de Matematicas 23 (2016), no. 2, 143-163. [ Links ]

[11] D. D. Roberts, The Existential Graphs of Charles S. Peirce, Mouton, The Hague, 1973. [ Links ]

[12] J. Taboada and D. Rodriguez, Una demostracion de la equivalencia entre los graficos Alfa y la logica proposicional, undergraduate thesis, Universidad del Tolima, Ibague, Colombia, 2010. [ Links ]

[13] F. Zalamea, Los graficos existenciales peirceanos, Universidad Nacional de Colombia, Bogotá, 2010. [ Links ]

[14] J. J. Zeman, The Graphical Logic of C. S. Peirce, Ph.D. dissertation, University of Chicago, Chicago, 1964. [ Links ]

Revised: April 2019; Accepted: August 2019

2010 Mathematics Subject Classification. 03B99, 14H99.

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