Jeff BD, Terry R, Jon AS, List I endorse Jeff's comments about the need to relate any author's work to his or her predecessors, contemporaries, and successors. I copied an excerpt from his note after my signature below. A major reason why Peirce's logic and semiotic were so advanced is that he had mastered all the major works from antiquity to the end of the 19th c -- especially the Scholastic logic, which was far more advanced than the typical textbooks of the 19th c. TR> [human]logic is essentially sensate and intuitive -- based on imminent pre-analytic experiential iconicity but equally vulnerable to error, mistake, and dissonance: e.g., it looks exactly like guacamole... I agree. The metaphor I use is "knowledge soup". William James called it the "blooming buzzing confusion". That's the content of the phaneron. The iconic structure of EGs makes them ideal for selecting lumps" from the soup and reasoning about them. TR> As far as possible-world semantics and models beyond FOL are concerned, two of the most helpful seminal works... _Essential Formal Semantics_ and _Topics in Conditional Logic_ [by Donald Nute]. I checked the Amazon previews of those two books. I agree that they provide good background for relating Peirce's writings to developments in the 20th c. Since Peirce was thinking "ahead of his time", texts from his future can help us understand some cryptic comments he had not fully developed. TR> Not convinced this is true: "JFS: And the syntax and semantics of any other versions of logic can be specified by mathematical theories expressed in FOL." Please note: I am *NOT* claiming that it's possible to translate other versions of logic to FOL. But every known version of mathematics can be specified by axioms stated in FOL. For examples, look at tools such as Mathematica or MathLab. The semantics of modal logic (every version, including Peirce's) can be specfied by a purely first-order theory of possible worlds. You don't have to believe me, you can look at references I cite (in http://jfsowa.com/pubs/5qelogic.pdf ) or at the references cited by Donald Nute or anybody else. JAS> RL 231 (June 1911) includes Peirce's simplest and clearest explanation of Beta EGs (NEM 3:162-169), which are equivalent to classical first-order logic. It is also his last thorough explanation of EGs in the extant manuscripts other than RL 378 (September 1911 in French), which is fully consistent with it. With this point of agreement, we can end that debate. I'll add the following observations, which I believe are not controversial: 1. The explanation of semantics (endoporeutic) is clearer than Peirce's earlier comments, and it is consistent with Risto Hilpinen's observation that endoporeutic is a version of Hintikka's game-theoretical semantics (GTS). 2. The rules of inference (permissions) are stated as three symmetric pairs instead of five separate rules. This symmetry is a major advance over other proof procedures, such as Frege's or Gentzen's. Among other important results, it enables a simple solution to an unsolved research problem from 1988. 3. The formalization in terms of shaded/unshaded areas permits a direct generalization to regions in higher-dimenions. Peirce had previously said that a limitation to a 2-D sheet required selectives or bridges that would not be required in 3-D graphs. And later in L231, he discussed reasoning in "stereoscopic moving images". That thought was probably in the back of his mind while he was writing the earlier pages. 4. Finally, the smaller number of technical terms reduces the time and effort for teaching and learning EGs. The absence of the word 'scroll' also avoids any questions about a possible difference in meaning. JAS> Peirce's multiple derivations of negation from the primitive logical relation of consequence are not "horribly contorted" at all. There is no need to derive negation from anything else. Affirmation and negation are the foundation for every logic from Aristotle to the present. For Peirce (L376, December 1911), "A denial is logically the simpler, because it implies merely that the utterer recognizes, however vaguely, some discrepancy between the fact and the speech, while an affirmation implies that he has examined all the implications of the latter and finds no discrepancy with the fact." The so-called "derivation" of negation from consequence is just a simple theorem of Boolean algebra: "not p" is equivalent to "if p then 0", where 0 represents Falsum. Boole's original algebra had 'not' as a primitive, but it didn't have 'if' until Peirce introduced his "claw" symbol. The exaggerated claim that Peirce made for "illation" results from one of his rare blunders: "For [the reader] cannot reason at all without a monstrative sign of illation" (CP 4.76, 1893). That claim is refuted by Peirce's 1884 discovery (R506, W 5:107): "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... 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." When Peirce began R669, he was still thinking along the lines of his 1906 Monist article. On 27 May 1911, he discussed cuts and the Recto/Verso distinction. On May 31, he derived "not A" from "A implies the pseudograph". But his proof took two MS pages and four EGs (Figures 23 to 26). Instead of calling that "horribly contorted", I would be satisfied with Peirce's own words, which he applied to that 1906 text: "very badly and at an intolerable length" (September 1911) or "as bad as it well could be" (December 1911). But at the end of R669, you can see the moment when the metaphorical "lightbulb" flashed on in Peirce's mind. On 1 June 1911, he wrote two "Illative Permissions". On June 2, he added a short paragraph about them. And then he stopped. He suddenly realized that the illative permissions depend only on positive or negative areas. They do not require "a monstrative sign of illation". That was the end of R669. On 7 June 1911, Peirce made a fresh start with R670. He decided to use white for positive areas, and black for negative areas. On p. 17 (12 June 1911), he wrote "the cuts... are, at most, mere punctuation separating particular affirmations from general negations." Illations are gone; reasoning involves affirmations and negations. That suggests a game between Graphist and Grapheus. In L231 (June 22), Peirce dropped the word 'illative' in front of 'permissions'. JAS> Moreover, negation is definitely not a primitive in Gamma EGs, where shading ("blue tint") represents a universe of possibility rather than just the denial of actuality, No. In every version of modal logic from Aristotle to the present, the symbol (word or diagram) for 'not' is just as fundamental as the symbols for 'possible' and 'necessary". In fact, one modal symbol plus 'not' is sufficient to define the others. For any proposition p, "possible p" is equivalent to "not necessarily not p". And "necessary p" is equivalent to "not possibly not p". In his 1903 version of modality, Peirce used a "broken cut" as the sign that the nested graph is contingent: "contingent p" is equivalent to "not necessarily p" and to "possibly not p". Instead of going through all the MSS, just look at the examples in Roberts' Chapter 5. For the use of negations in tinctured EGs, look at the examples in Chapter 6. JAS> in RL 376 (December 1911) ... [Peirce] states that Delta EGs would be necessary "in order to deal with modals," but no evidence has turned up so far that he ever actually developed them. No examples of Delta graphs have been found, but there is some evidence about the directions Peirce was pursuing. In R670 (12 June 1911), he wrote Figure 14, with the word "possible" as a meded for "Something is possible." On June 13th, he wrote "The nature of the universe or universes of discourse (for several may be referred to in a single assertion) ... is denoted either by using modifications of the heraldic tinctures or by scribing the graphs in colored inks". After writing those comments, Peirce continued writing R670 until June 17th. During that time, he continued to draw more EGs that used shading to express ordinary FOL examples. That is the first evidence that Peirce was thinking about combining shaded negations with some version of tinctures. Less than a week later (L231), Peirce wrote his new version of first-order EGs. In that same MS (NEM 3:191), he mentioned reasoning in "stereoscopic moving images". The 2-D shaded areas for negation could be generalized to shaded 3-D or 4-D regions. But neither the scroll nor the recto/verso metaphor could be generalized beyond 2-D. In December 1911 (L376), Peirce wrote "The better exposition of 1903 divided the system into three parts, distinguished as the Alpha, the Beta, and the Gamma, parts; a division I shall here adhere to, although I shall now have to add a Delta part in order to deal with modals. A cross division of the description which here, as in that of 1903, is given precedence over the other is into the Conventions, the Rules, and the working of the System." For Delta graphs, it would a short step to combine the shading of L231 with the 1903 broken cuts for modality. The result would be consistent with modern notations for modal logic. The next question is whether Peirce intended to add the tinctures, which he mentioned in R670. Since L376 ends abruptly just after Peirce began to discuss the phemic sheet, it seems likely that any further discussion would consider tinctures to indicate the "nature of the universe or universes of discourse" (R370). Conclusion: For years, I have considered the version of EGs in L231 to be the best available presentation of first-order EGs. This exercise of "looking over Peirce's shoulder" as he was writing did more than confirm that opinion. It showed that Peirce reviewed, rethought, and revised every aspect of EGs in the six months from M670 to L376. All earlier writings about EGs should be evaluated in terms of those MSS. John ________________________________________________________________________ >From a note by Jeff BD, 10 August 2020: When I'm trying to make sense of Peirce's writings, I find it is essential to draw on the secondary literature and to sort out what seems more and less helpful. At the same time, I'm trying to understand what Peirce is saying by reading what he is reading. That, I think, is necessary to understand what he's saying. John Sowa suggests that a richer understanding of Peirce's inquiries can be gained by seeing where they have taken later reachers who have followed in his wake. As such, there are five sources that seem important to reading Peirce: the texts themselves; the secondary literature on Peirce; the inquiries of philosophers, scientists, mathematicians (etc.) Peirce was reading--especially those he was drawing on in a sustained manner; the inquiries of those following in Peirce's wake (self-consciously or not). In addition to asking how Peirce used this or that term in a given text (as in 1, above), I think that it is essential that we (5) try to reconstruct his arguments and, at the same time, engage in the inquiries ourselves. After all, Peirce's writings were not written for armchair philosophers. Rather, they were written for inquirers willing to engage in philosophy as an experimental science.
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