Jon, I'll go into much more detail in the preview article, which I am now working on. I'll just respond to the following point: JAS: Your quotation here omits the crucial first part of the only sentence in R L376 that mentions Delta--"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." I answered that before: Peirce is not saying that he is preserving the details of the 1903 logics. He is saying that he is preserving that DIVISION into Alpha (propositional logic), Beta (predicate logic), and Gamma (something beyond Alpha and Beta). The most significant Gamma graphs are the the ones that represent the second-order version of his 1885 Algebra of Logic. He had reviewed Russell's logic of 1903, and he must have heard about (but didn't have time to study) Whitehead & Russell's logic of 1910. Both of them discussed higher-order logic (second order and higher), but not modal logic. During the years after 1903, Peirce mentioned the modal words in English many, many times. And he experimented with new notations for modality, but he never used or even mentioned his 1903 modal logic for any purpose. In fact, he had only used it for a few examples in 1903. But the most important evidence is to look at the developments in the years after Peirce. C. I. Lewis introduced a new version of propositional modal logic in 1932, which had been inspired by Peirce's 1903 modal logic. It was different from Peirce's version, but equivalent in expressive power to the propositional subset of his modal logic of 1903. During the 30 or 40 years after 1932, many logicians built on that logic. But many others (Quine among them) rejected it. Quine correctly said that modal logic was just a version of metalanguage about logic. Other logicians criticized it or ignored it altogether. Very few did much with it after the 1960s. From the 1970s and later, new versions of logic were developed to handle modal issues, but (a) they did not use the box and diamond operators for modality; (b) they used different words. such as contexts, situations, or domains; and (c) they combined predicate calculus with metalanguage, as Peirce did in L376. In my preview of the Delta graph article, I'll explain these issues in more detail and discuss the directions taken in 1973 and later. Short summary: All the useful applications are based on some version of metalanguage, along the lines of the December 1911 article. Logics that use the two operators for necessary and possible, have no practical applications of any kind. Peirce had good taste and good insights into the kind of logic required for problems in philosophy, science, and engineering. Metalanguage is the foundation for all useful modal reasoning in the 21st C. Textbooks still mention the Lewis-style of modal logic, but there are no applications to any kind of practical applications. Summary: Any version of mathematics and/or logic that has no applications is. literally, useless. There are many such versions in the many years of published tomes. And most of them have few or no citations. On rare occasions, something from the distant past is revived and becomes a big success. Peirce has an unusually large percentage of successful revivals. His Delta graphs are among them. I recognized their importance, because I have used and worked with similar logics from the late 20th and early 21st C. John ---------------------------------------- From: "Jon Alan Schmidt" <jonalanschmidt(a)gmail.com> Sent: 3/11/24 9:07 PM To: Peirce-L <peirce-l(a)list.iupui.edu> Subject: [PEIRCE-L] Peirce's Letter to Risteen (was Higher-Order Logics) John, List: JFS: For convenience, see the attached Delta376.txt. I appreciate the complete transcription, although it would still be very helpful if you could quote specific sentences that you interpret as supporting each of your claims. JFS: I believe that there is no way to interpret that text without acknowledging the fact that it is the beginning of a specification of Delta graphs. Note the ending of the second paragraph: "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." Your quotation here omits the crucial first part of the only sentence in R L376 that mentions Delta--"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." Accordingly, I believe that there is no way to interpret the 19 extant pages of this letter as the beginning of a specification of (only) Delta graphs. On the contrary, Peirce plainly states his intention to describe all four parts of EGs, but he never gets around to explaining their differences, let alone dealing with modals or discussing anything else that is unique to the new Delta part. JFS: Then the paragraph immediately after that begins "The Conventions." And it continues with a specification of the conventions for something. I cannot imagine that the "something" is anything other than Delta graphs. Your failure of imagination is not dispositive. Can you identify even one sentence from the entire section on "The Conventions"--or, for that matter, the rest of the letter--that is about EGs but not applicable to Alpha, Beta, and Gamma? JFS: Note the later discussion about different "parts" of the phemic sheet, which may be asserted and interpreted in different ways. Peirce states, "For they two [utterer and interpreter] may conceive that the 'phemic sheet' embraces many papers, so that one part of it is before the common attention at one time and another part at another, and that actual conventions between them equivalent to scribed graphs make some of those pieces relate to one subject and part to another." As I have noted before, the different parts relate to different subjects to which both parties pay attention at different times--equivalent what Peirce describes in previous texts such as R 280 (c. 1905) and CP 4.561n (1908), both of which I quoted last week (https://list.iupui.edu/sympa/arc/peirce-l/2024-03/msg00004.html). As such, this concept is fully applicable to Alpha, Beta, and Gamma EGs--it does not exclusively "deal with modals," and thus is not unique to Delta EGs. JFS: That is why metalanguage must be used to state the many kinds of modality that Peirce discusses in the attached text. Peirce indeed briefly discusses modality in R L376, but he does not address how to represent and reason about modal propositions using EGs, which is his only stated reason for needing a Delta part. JFS: But the original MS, a copy of which you included in your note, had a thin line that connected the oval to the word 'is'. I suspect that who drew that diagram thought that the thin line between the oval and the word 'is' was just part of the word 'is'. But in his handwiriting, Peirce never drew a line in front of an initial letter 'i'. Therefore, that graph was mistakenly drawn. Thank you for correcting my mistake. I noticed that line, drawn even more lightly than the one in the first EG on RLT 151, but assumed that it was part of Peirce's cursive "i"--just as Ketner evidently did. However, after looking at a few other manuscripts, I agree that Peirce generally did not include such a line when handwriting "i" as the first letter of a word, so there is indeed a lightly drawn line connecting the oval (containing a proposition) to the rheme (whose blank that proposition fills). Of course, I already brought to your attention his similar notation in a later manuscript--R 492 (1903), erroneously reproduced in CP 4.471 but corrected by both Roberts and Pietarinen--where the oval and line are dotted instead of lightly drawn (https://list.iupui.edu/sympa/arc/peirce-l/2024-02/msg00141.html). This notation in Gamma EGs asserts a proposition about a proposition, but there is no hint of anything like it in R L376 (nor R 514). Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Mon, Mar 11, 2024 at 3:56 PM John F Sowa <sowa(a)bestweb.net> wrote: Jon, Jeff, Gary, List, I am now writing the article on Delta graphs. In a few days, I'll send a preview. For convenience, see the attached Delta376.txt. (Since Peirce's paragraphs tend to be very long, I added some additional paragraph breaks,) I believe that there is no way to interpret that text without acknowledging the fact that it is the beginning of a specification of Delta graphs. Note the ending of the second paragraph: "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." Then the paragraph immediately after that begins "The Conventions." And it continues with a specification of he conventions for something. I cannot imagine that the "something" is anything other than Delta graphs. (That paragraph break, by the way, is Peirce's.) Note the later discussion about different "parts" of the phemic sheet, which may be asserted and interpreted in different ways. That is why metalanguage must be used to state the many kinds of modality that Peirce discusses in the attached text. John I'll also mention that three people misinterpreted the two diagrams on p. 151 of RLT -- you, me, and Ken Ketner. I misinterpreted the first diagram as having a line of identity between an oval that encloses the sentence "You are a good girl". With that interpretation, it would assert "There exists a proposition that you are a good girl, and that proposition is much to be wished." But you correctly noticed that the line is so thin that it cannot be interpreted as a line of identity. Peirce did not state any reading for that complete EG. Therefore, I read it as asserting a complete grammatical sentence "That you are a good girl is much to be wished. That assertion is correct. It is logically equivalent to the above reading, but it is not syntactically equivalent to it. Then Ken Ketner (or somebody else who drew the second EG) did not show an attached line between the oval and the verb phrase "is false." But the original MS, a copy of which you included in your note, had a thin line that connected the oval to the word 'is'. I suspect that who drew that diagram thought that the thin line between the oval and the word 'is' was just part of the word 'is'. But in his handwiriting, Peirce never drew a line in front of an initial letter 'i'. Therefore, that graph was mistakenly drawn. Neither you nor Ken noticed that error. You did mention that Peirce had not introduced the convention of using an oval for negation until the next example. That is true, but it does not excuse the mistake of not noticing the thin line that connects the previous oval to the word 'is. There is much more to say, and I'll include it in the preview, which I plan to send in the next few days. John In that case, I believe that the thin line implies that the proposition in the oval is a THING that is the subject of the verb phrase "is much to be wished."