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Peircean Algebraic Logic and Peirce's Reduction Thesis

Peircean Algebraic Logic and Peirce's Reduction Thesis Robert Burch describes Peircean Algebraic Logic (PAL) as a language to express Peirce's ““unitary logical vision”” (1991: 3), which Peirce tried to formulate using different logical systems. A ““correct”” formulation of Peirce's vision then should allow a mathematical proof of Peirce's Reduction Thesis, that all relations can be generated from the ensemble of unary, binary, and ternary relations, but that at least some ternary relations cannot be reduced to relations of lower arity. Based on Burch's algebraization, the authors further simplify the mathematical structure of PAL and remove a restriction imposed by Burch, making the resulting system in its expressiveness more similar to Peirce's system of existential graphs. The drawback, however, is that the proof of the Reduction Thesis from Burch (A Peircean reduction thesis: The foundations of topological logic, Texas Tech University Press, 1991) no longer holds. A new proof was introduced in Hereth Correia, and Pööschel (The teridentity and Peircean algebraic logic: 230––247, Springer, 2006) and was published in full detail in Hereth (Relation graphs and contextual logic: Towards mathematical foundations of concept-oriented databases, Technische Universitäät Dresden dissertation, 2008). In this paper, we provide proof of Peirce's Reduction Thesis using a graph notation similar to Peirce's existential graphs. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Semiotica - Journal of the International Association for Semiotic Studies / Revue de l'Association Internationale de Sémiotique de Gruyter

Peircean Algebraic Logic and Peirce's Reduction Thesis

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Publisher
de Gruyter
Copyright
©© 2011 Walter de Gruyter GmbH & Co. KG, Berlin/Boston
ISSN
0037-1998
eISSN
1613-3692
DOI
10.1515/semi.2011.050
Publisher site
See Article on Publisher Site

Abstract

Robert Burch describes Peircean Algebraic Logic (PAL) as a language to express Peirce's ““unitary logical vision”” (1991: 3), which Peirce tried to formulate using different logical systems. A ““correct”” formulation of Peirce's vision then should allow a mathematical proof of Peirce's Reduction Thesis, that all relations can be generated from the ensemble of unary, binary, and ternary relations, but that at least some ternary relations cannot be reduced to relations of lower arity. Based on Burch's algebraization, the authors further simplify the mathematical structure of PAL and remove a restriction imposed by Burch, making the resulting system in its expressiveness more similar to Peirce's system of existential graphs. The drawback, however, is that the proof of the Reduction Thesis from Burch (A Peircean reduction thesis: The foundations of topological logic, Texas Tech University Press, 1991) no longer holds. A new proof was introduced in Hereth Correia, and Pööschel (The teridentity and Peircean algebraic logic: 230––247, Springer, 2006) and was published in full detail in Hereth (Relation graphs and contextual logic: Towards mathematical foundations of concept-oriented databases, Technische Universitäät Dresden dissertation, 2008). In this paper, we provide proof of Peirce's Reduction Thesis using a graph notation similar to Peirce's existential graphs.

Journal

Semiotica - Journal of the International Association for Semiotic Studies / Revue de l'Association Internationale de Sémiotique – de Gruyter

Published: Aug 1, 2011

Keywords: Peirce ; Existential Graphs ; Burch ; Peircean Algebraic Logic ; Relational Graphs

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