Jon AS, JAS: I continue to agree that NEM 3:162-169 is Peirce's simplest and clearest explanation of existential graphs. That'a good. But the reason why it appears so clear and simple is that Peirce discovered a simple, universal, notation-independent mathematical structure that underlies all reasoning. That includes formal (mathematcal) reasoning and the informal common sense in ordinary thought and language. The version summarized in L231 has the expressive power of first-order logic, but it can be extended to the same subset of higher-order logic as Peirce's algebraic logic of 1885: 1. Three primitives: existence, negation, and predication (first, second, or third). 2. Two operations: insertion and deletion of any primitive or combination of primitives in any area. 3. Three symmetric pairs of permissions (AKA rules of inference) that state the conditions for performing either of the two operations. Each member of any pair is the exact inverse of the other. For this logic, Peirce continued his practice of writing diagrams on a two-dimensional sheet. But he stated the rules with such generality that they could be applied to any notation in a space of any number of dimensions -- linear, 2-D sheet, 3-D region, or 4-D region of space + time. With this flexibility, eg1911 becomes a universal, notation-independent system of deduction. The followig point from 1908 is just a special case: CSP: [The] purpose of the System of Existential Graphs ... [is] to afford a method (1) as simple as possible (that is to say, with as small a number of arbitrary conventions as possible), for representing propositions (2) as iconically, or diagrammatically and (3) as analytically as possible. (CP 4.561n, 1908) Peirce knew that graphs on a two-dimensional sheet would need ad hoc methods (selectives or bridges) for lines that cross. But 2-D shaded areas could be generalized to 3-D shaded regions to represent graphs without crossovers. For evidence that Peirce was thinking about 3-D or even 4-D logic in 1911, see L231 (or NEM 3.191), where he mentioned reasoning with "stereoscopic moving images". Peirce also knew that Euclid's diagrams were far more iconic than EGs for reasoning about geometry. NEM 3.191 is a hint that he may have considered some way of applying his 1911 rules directly to 2-D or 3-D Euclidean diagrams. In fact, it's quite easy to apply Peirce's 1911 rules of inference to Euclid's diagrams and proofs. For an example, see the slides http://jfsowa.com/talks/ppe.pdf . Slide 6 shows a diagram for Euclid's Proposition 1 and Euclid's proof, as translated to English by Heath. For a proof of Proposition 1 by Peirce's rules, see slides 30 to 43. In that proof, Euclid's diagrams are used to state propositions; Euclid's letters are used as selectives to show lines of identity; Euclid's diagrams may be enclosed in shaded areas for negations and implications; and diagrams or parts of diagrams may be inserted or erased according to Peirce's rules. Since Peirce stated his rules near the beginning of L231 and he talked about reasoning with images later in the same MS, it's quite likely that he may have been thinking of something along these lines. He may have written something about that possibility in some MS that was lost. It's even possible that some such MS might someday be found. JAS: Technically a scroll is not "just a way of drawing two cuts." These are indeed equivalent in classical logic, such that "if A then B" entails "not (A and not-B)" and vice-versa. No. In mathematics and mathematical logic, two equivalent notations (one-to-one mappings and identical entailments) have identical semantics. A choice between one or the other is just a matter of convenience in reading or writing. In R670 (1911), Peirce showed semantically identical EGs with cuts in Fig. 9, scrolls in Fig. 10, and shaded areas in Fig 11. Immediately after Fig. 11, he wrote "the cuts in themselves have no meaning whatever, but are, at most, pure punctuation-marks". JAS: However, Peirce recognized that "there is but one primary and fundamental logical relation, that of illation" (CP 3.440, 1896), which is what the scroll signifies when drawn as a single continuous cut that crosses itself so that the inner close touches the outer close at a node (CP 4.435-436, 1903; CP 4.564, 1906; R 669:16, 1911). The word 'illation' is an archaic synonym for 'inference', and 1896 was the time of Peirce's first thoughts about EGs. After he defined his more elegant system of 1911, he explicitly wrote that the cuts are "pure punctuation marks". As Figures 9 to 11 of R670 show, the scrolls are convenient punctuation marks for indicating the more fundamental shading. JAS: when adapting existential graphs to intuitionistic logic as Arnold Oostra has outlined. here a scroll is not equivalent to nested cuts; "if A then B" still entails "not (A and not-B)," but "not (A and not-B)" does not entail "if A then B." Intuitionistic logic was developed after Peirce's time, and Oostra chose the scroll notation for a kind of implication that was different from anything that Peirce represented. That's a good reason for *not* using scrolls as a synonym for a shaded area with a nested unshaded area. JAS: from a strictly philosophical standpoint, I suggest that the scroll is more analytical than shading. The word 'analytical' in philosophy or any other subject means "pertaining to analysis". Unless you specify what kind of analysis is intended, the word is more confusing than insightful. For further discussion of these issues, I recommend the slides http://jfsowa.com/talks/ppe.pdf which I presented for a session on Peirce at an APA meeting in April 2005. Later in 2015, Fernando Zalamea organized a workshop on existential graphs in Bogota, where I presented an extended version of those slides. In 2018, some presentations from the workshop were published in a special issue of the Journal of Applied Logics. When all the formal details were spelled out, my article took 76 pages. For the issue with all the talks, see http://www.collegepublications.co.uk/downloads/ifcolog00025.pdf . John
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