Jon, List, You recently wrote:
JAS: I do not know whether anyone has posted a mathematical proof of Peirce's reduction thesis on the Internet. Robert Burch wrote an entire book to present his [. . .] while Sergiy Koshkin purports to demonstrate it even more rigorously in a recent paper. I vividly recall, although it was over a decade ago, a young scholar (either Joachim Hereth or Reinhard Poschel) whom I'd previously not met coming up to me at an outdoor party during an ICCS conference, excitedly announcing that he and another scholar had completed a mathematical proof of Peirce's Reduction Thesis that went beyond Robert Burch's PAL. This is described in the Abstract of the paper below. Peircean Algebraic Logic and Peirce's Reduction Thesis - Joachim Hereth and Reinhard Pöschel *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 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. Keywords:: Peirce <https://www.degruyter.com/search?query=keywordValues%3A%28%22Peirce%22%29%20AND%20journalKey%3A%28%22SEMI%22%29&documentVisibility=all&documentTypeFacet=article> ; Existential Graphs <https://www.degruyter.com/search?query=keywordValues%3A%28%22Existential%20Graphs%22%29%20AND%20journalKey%3A%28%22SEMI%22%29&documentVisibility=all&documentTypeFacet=article> ; Burch <https://www.degruyter.com/search?query=keywordValues%3A%28%22Burch%22%29%20AND%20journalKey%3A%28%22SEMI%22%29&documentVisibility=all&documentTypeFacet=article> ; Peircean Algebraic Logic <https://www.degruyter.com/search?query=keywordValues%3A%28%22Peircean%20Algebraic%20Logic%22%29%20AND%20journalKey%3A%28%22SEMI%22%29&documentVisibility=all&documentTypeFacet=article> ; Relational Graphs <https://www.degruyter.com/search?query=keywordValues%3A%28%22Relational%20Graphs%22%29%20AND%20journalKey%3A%28%22SEMI%22%29&documentVisibility=all&documentTypeFacet=article> Published Online: 2011-08-08 *Peircean Algebraic Logic and Peirce’s Reduction Thesis* https://wwwpub.zih.tu-dresden.de/~poesch-r/poePUBLICATIONSpdf/2011_Hereth_Poe.pdf Published in Print: 2011-August I must admit that the complexities of the math involved in their paper as well as Koshkin's (which I only recently became aware of) has precluded my reading much of either of them. But then, I tend to strongly agree with you in this comment of yours. JAS: I find Peirce's own diagrammatic demonstration to be simple and persuasive enough--relations of any adicity can be built up of triads, but triads cannot be built up of monads or dyads despite involving them (EP 2:364, 1905). In *A Thief of Peirce *Kenneth Ketner calls these diagrams "valental graphs" and discusses them in one or two of the appendices of that book (I remember buying the book for just those appendices even though it seemed a small fortune at the time). [image: image.png] Best, Gary
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