Jon, Jerry, List, We had discussed this issue many times before. R 669 was an attempt by Peirce to relate all the versions of EGs he had written, published, and toyed with. The result (R 669) was a hodge-podge that had many ad hoc constructions that Peirce was unable to justify by any convincing proof. He knew that it was bad.
In R 670, he began to sketch out a new version, and a few weeks later he produced his clearest, most precise, and most elegant foundation for EGs. And he confirmed that version as his final choice in his last major letter in 2013. Peirce's three primitives are conjunction (AND), negation (NOT), and the existential quantifier (line of identity). These three primitives with Peirce's 1911 rules of inference are so general and powerful, that they unify and simplify Gerhard Gentzen's two systems -- clause form and natural deduction. As a result an unsolved research problem about the relationship between the two systems (stated in the 1970s) was finally solved by a simple proof when translated to Peirce's 1911 notation and rules of inference. That is conclusive evidence beyond any shadow of a doubt that Peirce's 1911 system is one of his most brilliant achievements. I'll send another note with all the references. John ---------------------------------------- From: "Jon Alan Schmidt" <jonalanschm...@gmail.com> Sent: 1/11/24 6:13 PM To: Peirce-L <peirce-l@list.iupui.edu> Subject: Re: [PEIRCE-L] Categorizations of triadic Relationships (Was Re: Graphical Representations of the Sign by Peirce) Jerry, List: JLRC: The classical logic of mathematical reasoning (symbolized by five signs - negation, conjunction, disjunction, material conditional, and bi-conditional. Actually, Peirce points out that only two signs are needed as primitives, with the others being derived from them. CSP: Out of the conceptions of non-relative deductive logic, such as consequence, coexistence or composition, aggregation, incompossibility, negation, etc., it is only necessary to select two, and almost any two at that, to have the material needed for defining the others. What ones are to be selected is a question the decision of which transcends the function of this branch of logic. (CP 2.379, 1902) For example, in the Alpha part of Existential Graphs for propositional logic, the simplest approach is to select the two primitives as juxtaposition for conjunction (coexistence) and shading for negation* such that disjunction is then defined as multiple unshaded areas within a shaded area, material conditional (consequence) as one unshaded area within a shaded area (scroll), and bi-conditional as juxtaposed scrolls with the antecedent and consequent reversed. The Beta part for first-order predicate logic adds one more primitive, the line of identity for existential quantification such that universal quantification is then defined as a line of identity whose outermost part is within a shaded area. *As I have discussed on the List many times before, although this choice is practically more efficient and easier to explain, Peirce suggests on several occasions that it is philosophically more accurate to select the scroll for material implication as the second primitive such that negation is then defined as a scroll with a blackened inner close shrunk to infinitesimal size, signifying that every proposition is true if the antecedent is true (CP 4.454-456, 1903; CP 4.564n, c. 1906; R 300:[47-51], 1908; R 669:[16-18], 1911). Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Thu, Jan 11, 2024 at 12:52 PM Jerry LR Chandler <jerry_lr_chand...@icloud.com> wrote: On Jan 11, 2024, at 11:28 AM, Edwina Taborsky <edwina.tabor...@gmail.com> wrote: But you already know this Edwinia: If I understood the meaning of the “triadic relations”, I would not waste my time attempting to frame precise questions and intensely analyzing the grammatical structures of your and other responses. Mathematical reasoning is grounded in set theory - the relation between ordered pairs. The classical logic of mathematical reasoning (symbolized by five signs - negation, conjunction, disjunction, material conditional, and bi-conditional. These signs are often interpreted in terms of the Aristotelian syllogisms. Which in turn, are related to sentences and sentence grammars. For a discussion of Peircian “tokens and types” from a categorical perspective, see the recent text by Ursula Skadowski, Logic - Language - Ontology. 2022. Or, asserted in similar terms, is the meaning of a triadic relation constrained to multi-valued logics? My interpretation of the posts by the John / Jon / Robert posts is that the classical logic for deduction preserves the truths of propositions of molecular sentences. (Note, it was not necessary to invoke either Robert Rosen’s writings on the philosophy of science or thermodynamics or entropy or dogmas or…. Just seeking a scientifically useful meaning for my research. Cheers Jerry
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