John, List: No one is disputing that there are two operations (erasure and insertion) and three symmetric pairs of permissions for performing them. I have simply pointed out that according to Peirce--not just once, but in several different passages--negation is not a *primitive* in logic, it is *derived* from illation/implication. Pietarinen confirms this in his introduction to volume 1 of *Logic of the Future: Writings on Existential Graphs* (here <https://books.google.com/books?id=IUnSDwAAQBAJ>).
AVP: R 481 seems to be the very first paper in which the scroll is indeed recognised as the primitive logical sign to be used, thus capturing the relation of illation better than any other sign would. Also realised for the very first time is the fact that negation, on the other hand, is not a primitive sign of logic, and that in the course of the logical development of reason, one in fact derives the idea of the negation from the idea of the scroll. (LoF 1:85) As Peirce states in the referenced manuscript itself, "It will be observed that we did not assume that any sign represented negation, but we have proved that certain signs having certain significations, otherwise defined, must express negation. In other words, we have virtually analyzed the conception of negation" (R 481:8, LoF 1:288, 1896). Every rule for EGs that mentions or implies a distinction between evenly and oddly enclosed areas requires a scroll, although its inner close may be blackened and shrunk to infinitesimal size to make it a cut, and its outer close may be shaded to make it more iconic. JFS: Children learn the word 'no' long before they learn implication. Even your pet dog or cat learns the word 'no'. There is obviously more to negation in logic than the word "no." As Peirce explains in a quote from R 669 (1911) that I provided previously, it requires the concept of *falsity*, which we do not yet grasp during "the state of paradisaical logic" through which "[p]robably every human being passes." Besides, *how *does a child (or a pet) learn the word "no"--or anything else, for that matter--while possessing only a rudimentary *logica utens*? Presumably by drawing simple inferences from experience. For example, *if* an adult says "no" and I do it anyway, *then* something unpleasant happens to me. Again, recognizing that implication is primitive while negation is derived is essential for adapting EGs to intuitionistic logic, which is the logic of continuity as Fernando Zalamea has observed (here <https://uberty.org/wp-content/uploads/2015/07/Zalamea-Peirces-Continuum.pdf>, p. 20). It is likewise the logic of synthetic differential geometry (here <https://pages.physics.ua.edu/staff/fabi/InvitationSDG.pdf>, pp. 8-9) and smooth infinitesimal analysis (here <http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf>, p. 5), which is widely regarded among Peirce scholars as a better candidate for rigorous mathematical treatment of his conception of continuity than Robinson's non-standard analysis (e.g., Herron <https://www.jstor.org/stable/40320632>, pp. 621-623; Moore <https://www.jstor.org/stable/40321199>, p. 468 n. 45; Havenel <http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.473.9336&rep=rep1&type=pdf>, pp. 99-100 & 111-112). Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Mon, Jul 27, 2020 at 11:20 PM John F. Sowa <[email protected]> wrote: > 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 Peirce’s 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: "Peirce’s 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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