Jon AS, List, As I mentioned in my reply to Jeff, Peirce's ideas were often far ahead of his time, and it's important to see which of them not only stood the test of time, but even improved on later developments. I changed the subject line to emphasize a critical issue that famous logicians (Frege, Whitehead, Russell, Gentzen...) overlooked. For his version of first-order logic, Frege (1879) chose negation, if-then, and the universal quantifier as his three primitives. That was an unfortunate choice, which Whitehead and Russell adopted for their _Principia_. That book had the worst theorem-proving procedure ever inflicted on innocent students. In the 1930s, Gentzen presented two much better procedures: natural deduction and the sequent calculus. But Gentzen also made the mistake of making if-then a primitive. After a bit of digging, I found the passage where Peirce discovered what was essential: Vol 5:107 of _Writings_, MS R506. summer of 1884: CSP: The first chapter of [1880 Algebra of Logic] develops the reasons for choosing the copula of inclusion, exhibits its formulae, and attempts by means of it to consolidate syllogistic with Boolian algebra. But the study of Professor O. H. Mitchell's important paper "On a New Algebra of Logic" has led me to think that the passage from premiss to conclusion ought not to be considered as the essential and elementary type of logical movement. We have rather two elementary modes of modifying assertions and two corresponding modes of transforming them. The two modes of changing assertions are 1st to drop part of what has been asserted and assert less, and 2nd to add to what has been asserted and assert more. Those two "modes of changing assertions" are the basis for his "permissions" (AKA rules of inference) for existential graphs. Each rule inserts or erases an EG or part of an EG in some area. No rule mentions or requires a scroll (if-then statement). Peirce discovered that principle a dozen years before he invented EGs (December 1896). JAS: Peirce explained on multiple occasions--including R 669, written just a couple of weeks before R 670 and L 231--negation is not a primitive. It is derived from the fundamental logical relation of illation or (less archaically) implication. No. There are many different relationships among the 16 binary Boolean operators and the two constants v (verum) and f (falsum). Peirce wrote about them many times from different points of view. But there is no reason to consider any of those operators more fundamental than negation. Children learn the word 'no' long before they learn implication. Even your pet dog or cat learns the word 'no'. The following historical comment occurs only in R669, and Peirce deleted it in the revised version, R670: CSP: In the order of the actual mental evolution of the syntax of existential graphs, the Scroll was first adopted as a sign required before all others because it represented a necessary Reasoning... (R669:18-20[16-18], 1911 May 31) The syntax of the scroll may have influenced Peirce's choice of notation in 1896. But his EG rules of inference refer only to areas -- positive (unshaded) or negative (shaded). They don't refer to scrolls. JAS: As [Ahti] said in his introduction accompanying R 669-670 ... AVP: These last two manuscripts concerning "Assurance through Reasoning" present what may be Peirces most successful attempt to explain the logic of existential graphs, and the philosophy concerning the notation of diagrammatic syntax in particular. The notions of identity, teridentity, composition of graphs, plurality, conditional, scroll, and the derivation of the idea of negation as a consequence of the scroll, all get their fair shares of exposition. In this comment, Ahti did not explain why the scroll, which was considered important in R669, was demoted to "punctuation" in R670, and was not mentioned at all in L231. The most likely reason is that Peirce was not thinking about the EG rules of inference when he wrote R669. But he wrote his best and simplest statement of the rules in L231. Since he wrote R670 in the short time between those two MSS, he was starting to think about those rules. They depend only on the areas of EGs, not on the cuts or the scrolls. Even more important, L231 mentions reasoning about "stereoscopic moving images" . The 2-D areas can be generalized to 3-D or even 4-D regions for space + time. But scrolls are limited to 2-D. For the EG excerpts from L231, see http://jfsowa.com/peirce/eg1911.pdf . JAS: Since my personal interest in EGs is primarily philosophical rather than pedagogical, I am inclined to agree with Pietarinen's assessment. This is a matter of the most profound logical and philosophical issues. In the 1930s, Gerhard Gentzen published a highly respected book on logical deduction. Unfortunately, he was misled to think that the if-then statement (whatever it may be called) was essential. For my 2010 article in _Semiotica_, I made a mistake in the choice of title: "Peirces Tutorial on Existential Graphs". That led many readers to consider it a pedagogical recommendation. But Section 6 "Advanced Topics" proves some fundamental theorems: Peirce's rules of inference are a *generalization* of Gentzen's system of natural deduction. See http://jfsowa.com/pubs/egtut.pdf . There is another important theorem: Peirce's generalization of Gentzen's system solves a research problem that was stated in 1988 and remained unsolved for years. For a statement of that problem and an outline of the proof, see the attached ppe65.png (which is slide 65 of "Peirce, Polya, and Euclid", http://jfsowa.com/talks/ppe.pdf ). I presented those slides in 2015 at an APA conference (session on Peirce) and later at a workshop on EGs in Bogota. That led to a 76-page article in the Journal of Applied Logics (2018) that spells out all the details. For the URL of that article, see slide 2 of ppe.pdf . Summary: The proof in slide 65 is simple for anybody who thinks in terms of Peirce's EG rules. But those who thought that an operator for if-then is essential for deduction were unable to discover the proof. John
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