List: In light of the following statements by John Sowa last month, I decided to take a fresh look at R 669-670, including both the online digital images ( https://rs.cms.hu-berlin.de/peircearchive/pages/home.php) and the transcriptions published by Ahti-Veikko Pietarinen in 2014 ( https://www.researchgate.net/publication/271419583_Two_Papers_on_Existential_Graphs_by_Charles_Peirce) as well as in the first volume of *Logic of the Future: Writings on Existential Graphs* (LoF).
JFS: R669 is based on the 1906 notation, but it ends abruptly just after Peirce wrote the first two rules of inference. He did not write the third rule, even though there was enough room on the page. Five days later, he began R670 with the same title, but with a summary of the new version. ( https://list.iupui.edu/sympa/arc/peirce-l/2020-12/msg00075/eg1911x.png) JFS: While Peirce was writing the three EG rules of inference around 8 pm on 2 June 1911, he suddenly realized that the rules depend *only* on whether an area is positive or negative. ( https://list.iupui.edu/sympa/arc/peirce-l/2020-12/msg00087.html) It turns out that Peirce had a very good reason for not writing a third rule at the end of R 669, and it was *not *because "he suddenly realized" something at that moment in time and "abruptly" abandoned his previous train of thought. It was simply because he had *already *stated the third rule a few paragraphs earlier, and had explicitly pointed out that it is not an *illative *permission; i.e., it is not a rule of *inference*. CSP: It now only remains to formulate those general permissions to modify what has already been scribed which express the logicality of those several forms of elementary deductive inference, out of which all other deductions can be built up. There are but two of these general illative permissions; but before stating them there is one other thing that has to be said. Namely, it is to be imagined that every graph-instance anywhere on the sheet can be freely moved about upon the sheet; and since a scroll both of whose closes are empty asserts nothing, it is to be imagined that there is an abundant store of empty scrolls on a part of the sheet that is out of sight, whence one of them can be brought into view whenever desired. What is here said ought to be reckoned as a permission, but it is not an illative permission, i.e. a permission authorizing a species of inference. (R 669:21-22[19-20], LoF 1:583) Pietarinen comments in a footnote, "This corresponds to the double-cut rule," which is the "Third Permission" in "Prolegomena to an Apology for Pragmaticism" (CP 4.567, 1906). After proceeding to present deletion/insertion as the "First Illative Permission" and iteration/deiteration as the "Second Illative Permission," Peirce concludes, "These two permissions will suffice to enable any valid deduction to be performed" (R 669:23[21], LoF 1:584). This echoes his remark in "Prolegomena" that "These two Rules (of Deletion and Insertion, and of Iteration and Deiteration) are substantially all the undeduced Permissions needed; the others being either Consequences or Explanations of these" (CP 4.567). Hence the exposition of existential graphs in R 669 is in fact complete, such that only "The few examples that shall forthwith be given" are missing. The final sentences note the inadequacy of automated reasoning to apply "the two illative permissions," since they require "a living intelligence" (R 669:23-24[21-22], LoF 1:584). Peirce does not discuss *any *of the transformation rules in R 670, whose first page is dated five days after the last page of R 669. However, in RL 231 a couple of weeks later he presents deletion/insertion as the "1st Permission," iteration/deiteration as the "2nd Permission," and double-cut as the "3rd Permission" (NEM 3:166). His descriptions of the first two are very similar to those in R 669, except that he replaces "evenly enclosed" with "unshaded" and "oddly enclosed" with "shaded" as an alternative way of distinguishing the different areas. For the third, he similarly refers to "a vacant ring-shaped area" rather than "empty scrolls." Unlike "Prolegomena" (CP 4.569), none of these manuscripts includes a "4th Permission" expressing "the strange rule" that Peirce deemed to be inconsistent with "the reality of some possibilities" as affirmed by his pragmatism (CP 4.580-581, 1906), such that he was ultimately "sceptical as to the universal validity of" it (RL 477:33[13], 1913). Again, I readily acknowledge that shading is *more iconic* than thin oval lines for signifying that oddly enclosed areas are a different surface from evenly enclosed areas, representing a universe of possibilities rather than the universe of actuality. I also concede that for classical/dyadic logic with excluded middle, which "is not absolutely false" (R 339:515[344r], 1909), treating shading as a primitive for negation is *simpler *than properly recognizing that the scroll is a primitive for consequence and then deriving negation from it as a scroll with a blackened inner close (CP 4.564n, c. 1906; R 669:18-20[16-18]). Nevertheless, I continue to maintain that the latter is *more analytical* because it preserves the fundamental asymmetry of reasoning and can thus be easily adapted for intuitionistic/triadic logic without excluded middle, which "is universally true" (R 339:515[344r]). Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt
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