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 Pöschel (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 Universität Dresden dissertation, 2008).

In this paper, we provide proof of Peirce's Reduction Thesis using a graph notation similar to Peirce's existential graphs.

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The official journal of the International Association for Semiotic Studies, founded in 1969 as one of the first scholarly journals in the field, Semiotica features articles reporting results of research in all branches of semiotic studies, in-depth reviews of selected current literature in the field, and occasional guest editorials and reports. The journal also publishes occasional Special Issues devoted to topics of particular interest.