Jon, List, Since your article has just been accepted for publication, you probably still have time to make a few corrections. Following are some suggestions.
JAS: Indeed, given that Peirce already had a notation for metalanguage in his 1903 Gamma EGs--in fact, five years earlier--how could that be what he had in mind for the new Delta part that he added in 1911? What exactly are you proposing to add that goes beyond the dotted (or thinly drawn) oval and line for asserting propositions about propositions? What must be added is all of L376. More would be better, but we can look to the future to see what is needed. More precisely, Peirce had a notation for metalanguage in 1898, which is simpler and does not need the useless baggage of Gamma graphs. There is no need for them when you adopt the notation for metalanguage. But metalanguage, by itself, is not sufficient for a proof of pragmatism. There is an open-ended variety of ways for using metalanguage. I also copied page 340r from the logic notebook (see attached). Please note that EGs that refer to "circumstances" can be translated to ordinary statements in Peirce's algebraic notion for first-order logic. There is nothing special about circumstances, by themselves, that makes them modal. You should make that correction in your article. Did you read the Wikipedia article about Arthur Cayley and the comments about Risteen in EP2? Those topics would explain why Delta graphs (as described in L376) have much more structure than just a notation for metalanguage. Peirce used that structure to support a phemic sheet that has multiple "papers". And the remainder of L376 describes how they may be used in an "investigation". That is necessary to support a logic of pragmatism. That additional structure is what makes Delta graphs the fourth kind of logic that goes beyond Gamma graphs. I suggest that you include all of L376 in an appendix to your article. That is what I'm doing. In summary, Peirce was years ahead of his time, and it's essential to give him full credit for his accomplishments. It would be insulting to claim that he didn't realize that his logic of 1892, by itself, was sufficient. And by the way, when you make these additions to your article, you can include an acknowledgment to thank me for all this help in explaining what Peirce was doing. John ---------------------------------------- From: "Jon Alan Schmidt" <jonalanschm...@gmail.com> John, List: JFS: If Peirce thought that the notations for his Gamma graphs of 1903 were adequate to represent everything in Delta graphs, why did he claim that the he needed to add a fourth part to his system of EGs? If the Gamma EGs of 1903 were adequate to represent metalanguage (as I have demonstrated), then why would Peirce need to add a new Delta part for that purpose in 1911? JFS: In your diagram below, the row of four EGs on the left represent pure first-order predicate calculus. Any sentences outside of the EGs (whether in English or EGs or some other notation) do nothing to change those sentences from FOL to any version of modal logic. They are pure, unadulterated FOL. No, I have refuted this claim repeatedly. There is a fundamental semiotic difference between describing indefinite individuals (lines of identity) with general concepts (names) and describing possible states of things (lines of compossibility) with propositions (letters). The additional axioms of modal propositional logic do not translate into valid theorems of first-order predicate logic as implemented by Beta EGs, and there are no counterparts in the latter for iterated modalities and propositions that are actually true instead of possibly true (analogous to existentially quantified) or necessarily true (analogous to universally quantified). JFS: In short, that thin line attached to an oval is Peirce's 1898 notation for metalanguage--five years before the Gamma graphs. I agree with you about that. I thanked you for correcting my mistake (and Ketner's) in misreading the thin line in the second EG on RLT 151 as part of the cursive "i" in the rheme "is false." I brought to your attention Peirce's slight revision of that 1898 notation in 1903, only changing from the thinly drawn oval and line to the dotted oval and line, presumably to distinguish them from the thinly drawn lines that still represented cuts at that time. I suppose that it makes little difference once shading replaces cuts, but I prefer the dotted oval and line to minimize the potential for confusion. JFS: The metalanguage of 1898 combined with EGs that contain symbols such as "possibly true" in a verb phrase does represent modality. Exactly! That is why it is very unlikely that Peirce had this in mind when he said in 1911 that he needed to add a new Delta part "in order to deal with modals." JAS: As I have spelled out in a soon-to-be-published paper, "Peirce and Modal Logic: Delta Existential Graphs and Pragmaticism" ... JFS: I suggest that you study the paper I'm writing, since any peer-reviewed paper that makes any claims about modal logic is likely to be reviewed by two kinds of people: My paper has already been peer-reviewed, revised in response to comments, and accepted for publication. I suggest that you study it when it comes out, and maybe even consider referencing it in yours, especially if you remain convinced that your conjecture about what Peirce intended for Delta EGs is more plausible than mine. You might also want to think about changing your title to be less similar to mine. JFS: Ordinary readers who will assume that any talk about specifications in a paragraph that immediately follows "I shall now have to add a Delta part in order to deal with modals" is very likely to be a specification of something called Delta. ... How could any reader think that those conventions do not specify Delta graphs? Because what immediately precedes "I shall now have to add a Delta part in order to deal with modals" is "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 ..." Accordingly, ordinary readers will assume that any subsequent talk about specifications is very likely to be about all four parts, especially when everything that follows turns out to be applicable to Alpha, Beta, and Gamma--there is nothing in the extant 19 pages of the letter that deals with modals or would otherwise be unique to the new Delta part. I ask again, can you provide any exact quotations from it to the contrary? CSP: 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. THE CONVENTIONS. The ultimate purpose of contriving this diagrammatic syntax, is to enable one with facility to divide any necessary, or mathematical, reasoning into its ultimate logical steps. JFS: Note especially the words 'now', 'here', and 'conventions' in the ending of the first paragraph, and the title "The conventions" at the beginning of the second paragraph. The word "now" appears only once, referring to the new Delta part; but "here" appears in the first two sentences--as well as "1903" and "division"--in both cases referring to the Alpha, Beta, and Gamma parts. According to the second sentence, what will follow is "a cross division" into "the conventions, the Rules, and the working of the System"--i.e., these three aspects apply equally to all the parts of EGs, just as they did in "the better exposition of 1903." The first sentence of the next paragraph confirms this--the common purpose of all the parts of EGs is to analyze necessary/mathematical/deductive reasoning into its ultimate logical steps. JFS: Finally, if Peirce thought that the notations in his Gamma graphs of 1903 were adequate to represent everything in the Delta graphs, why did he claim that he needed to add a fourth part to his system of EGs? There must be something in those Delta graphs that cannot be represented with just some excerpts from the old Gamma graphs. Indeed, given that Peirce already had a notation for metalanguage in his 1903 Gamma EGs--in fact, five years earlier--how could that be what he had in mind for the new Delta part that he felt the need to add in 1911? What exactly are you proposing to add that goes beyond the dotted (or thinly drawn) oval and line for asserting propositions about propositions? Is there any evidence that Peirce ever used that notation to represent a modal proposition by writing "is possible" or "is necessary" as the attached rheme? On the other hand, consider what he says in the four sentences right before the one where he mentions the need for "a Delta part in order to deal with modals." CSP: In this ["Prolegomena," 1906] I made an attempt to make the syntax cover Modals; but it has not satisfied me. The description was, on the whole, as bad as it well could be, in great contrast to the one Dr. Carus rejected [in 1897]. For although the system itself is marked by extreme simplicity, the description fills 55 pages, and defines over a hundred technical terms applying to it. The necessity for these was chiefly due to the lines called "cuts" which simply appear in the present description as the boundaries of shadings, or shaded parts of the sheet. The broken cuts of 1903 and the tinctures of 1906 were Peirce's previous attempts to "deal with modals," but they were ultimately unsatisfactory--even "nonsensical" in the case of the tinctures, as he called them two years later (R L477, LF 3/2, 1913 Nov 8). The badness of his description of EGs in 1906 was chiefly due to the use of cuts, a deficiency that he had already rectified by replacing them with shading (R 670, LF 3/1, 1911 Jun 11-12). What he needed now was a better graphical notation, compatible with shading, for representing and reasoning about modal propositions. Attaching heavy lines denoting possible states of things to the top of letters denoting propositions that would be true in them fits the bill. Moreover, Peirce himself suggested this solution (R 339:[340r], LF 1:624, 1909 Jan 7), although he evidently never worked out the details. I believe that I have done so in my forthcoming paper. 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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