Jon, Jerry, List, My interpretation of L376 depends on two ways of interpreting Peirce's L376. The first way is the one followed by most scholars: Comparing the content to an MS to everything written by Peirce and his sources prior to the date of the MS and to everything written later by him. Since the content of L376 is very different from his sources and from his own writings before and after, that provides very little guidance. That's why nobody was able to interpret L376 to determine what Peirce wrote and how he intended to use what he was specifying.
JFS> The single most important innovation of Delta graphs is an operator for metalanguage or metalogic. JAS> That is not what Peirce says about Delta EGs in the letter to Risteen. He simply states, "I shall now have to add a Delta part in order to deal with modals..." That is true. The second way of interpreting Peirce is to look backwards from the developments in logic in the century or more after Peirce and interpreting what he wrote in comparison to ALL developments in the same or similar subjects. The words 'metalanguage' and 'metalogic' were coined by Tarski and Carnap a few decades after Peirce died. But the that-operator in RLT (1898) can support the methods they used for metalanguage. It is logically identical to writing postulates in the margin of a paper in R514 (June 1911) and to the "papers" of a phemic sheet in L376 (December 1911). It is also identical to methods used by Hintikka and others from the 1970s and later. It's not possible to interpret what Peirce intended in L376 with just the vocabulary he used. It's likely that he would have coined more terminology if he had been able to finish that MS. But his accident and the six months of morphine by the "quack" who treated him prevented him from finishing it and explaining his intentions and applications in detail. JLRC> First, the question of modern modal symbolic logic is remote from probability theory and even remoter from the Peircian notion of “qualisign, sinsign, legisign” That is true of Peirce's modal logic of 1903, which was the mainstream of modal logic for most of the 20th C and which is still taught in introductory courses. But Peirce became very interested in probability theory, as shown in his writings in the Logic Notebook. The that-operator from 1898 and the "papers" of June and December 1911 can support the kind of metalanguage that is widely used today for computational and theoretical methods for either or both possibilities and probabilities. JLRC> Is not the distinction between logic of syntax and the logic of semantics? Is not the semantic gap in the meanings of signs was probably a constitutive factor in the categorization of signs, would you agree? I agree that those distinctions are important. But any operators for metalanguage, including Peirce's three versions, can be and are used to represent, reason about, and compute with representations for syntax and/or semantics of any notation of any kind. See the many references in https://jfsowa.com/ikl . That text, by me, is very short. I wrote it as a guide to a wide range of documents from the 1980 to 2010. I haven't added anything since then because the amount of publication is huge. But it is still a useful guide to 30 years of developments, many of which take advantage of various methods of metalanguage. And Peirce's three notations for metalanguage are logically equivalent to methods that have been reinvented in several versions since the 1970s. The second way is to look backwards from the developments in logic in the century or more after Peirce and interpreting what he wrote in comparison to ALL developments in the same or similar subjects. From the perspective of the late 20th and 21st C, the specifications in RLT (1898), R514 (June 1911), and L376 (December 1911) define the that-operator of IKL. That operator specifies 20th and 21st C operations for metalanguage and metalogic. That single operator, when added to first-order logic, supports a very powerful version of logic. JAS> in the letter to Risteen. He simply states, "I shall now have to add a Delta part in order to deal with modals," and we do not have to guess at what he means by "modals" since he provides a straightforward definition elsewhere. "A modal proposition takes account of a whole range of possibility. According as it asserts something to be true or false throughout the whole range of possibility, it is necessary or impossible. According as it asserts something to be true or false within the range of possibility (not expressly including or excluding the existent state of things), it is possible or contingent" (CP 2.323, EP 2:283, 1903). What he wrote about modals in 1903 represents his views about modals in 1903. But 1903 was the end of the line for earlier projects, especially lexicography (Century & Baldwin dictionaries) and Minute Logic (rejection). It was also the beginning of three major projects: publications for Carus, proof of pragmaticism, and correspondence with Lady Welby. His eight years of correspondence with Lady Welby broadened his views enormously. It led to his transition from phenomenology to phaneroscopy. It's important to compare his correspondence with her to what he was writing in the LNB and in other letters, MSS, and publications at the same dates. His views, his style, and his topics developed in new ways on new topics during that last decade. When in doubt compare what he wrote technical subjects to his letters at the same dates to his correspondents, especially Lady Welby and William James. John ---------------------------------------- From: "Jon Alan Schmidt" <jonalanschm...@gmail.com> John, List: I fully agree with your comment last week that "Peirce List is a collaboration, not a competition," and I hope that you will receive this response in that spirit. My questions are genuinely intended to help me (and others) better understand your position, and I would appreciate direct answers. JFS: The single most important innovation of Delta graphs is an operator for metalangage or metalogic. That is not what Peirce says about Delta EGs in the letter to Risteen. He simply states, "I shall now have to add a Delta part in order to deal with modals," and we do not have to guess at what he means by "modals" since he provides a straightforward definition elsewhere. "A modal proposition takes account of a whole range of possibility. According as it asserts something to be true or false throughout the whole range of possibility, it is necessary or impossible. According as it asserts something to be true or false within the range of possibility (not expressly including or excluding the existent state of things), it is possible or contingent" (CP 2.323, EP 2:283, 1903). Hence, the 1898 example--"That you are a good girl is much to be wished"--is not what Peirce considered to be a modal proposition; only something like "That you are a good girl is possible" would qualify. Where exactly do you see anything about "an operator for metalanguage or metalogic" in the letter to Risteen? Again, what does Peirce say in that text that would not be fully applicable to Alpha, Beta, and Gamma EGs as he had described them previously? Please provide exact quotations. JFS: Although Peirce never developed it further (as far as I know), the option of attaching a line of identity to an oval is exactly the same operation as taking a sheet of paper, drawing a line around the nested text (You are a good girl), and stating postulates in the margin (as in R514 and L376). It is not the same operation at all since "--is much to be wished" is not a postulate from which "you are a good girl" follows necessarily. As I explained before, Peirce's "red pencil" operation in R 514 effectively turns each individual sheet of paper on which EGs are scribed into a conditional proposition. Its physical edges and the red line drawn just inside them are cuts, the latter nested within the former, so that the margin is the outer close (antecedent) and the area within the red line is the inner close (consequent). Any propositions in the margin (postulates) are "merely asserted to be possible," and if they are all true, then all the propositions within the red line (theorems) are also true. There is no "line of identity" connecting the red line to the postulates in the margin. Where exactly do you see anything about "stating postulates in the margin" in R L376? Please provide exact quotations. JFS: As for the five EGs from 1909, quoted below, none of them express modal logic. All five of them can be translated to statements in first-order logic: Those translations are incorrect. It is unambiguous from Peirce's own handwritten translations that the EGs scribed on that Logic Notebook page are not Beta graphs with heavy lines for indefinite individuals attached to lowercase letters for general concepts being attributed to them. Instead, the heavy lines are for "circumstances," and they are attached to lowercase letters for propositions (as in Alpha) that would be true in them. There is an analogy between quantifying predicates over subjects in first-order predicate logic and quantifying propositions over possible states of things in propositional modal logic--in Peirce's words, "The distinction between the Indefinite, the Singular, and the General ls obviously only another application of the distinction between the Possible, the Actual, and the Necessary, for which the Germans have invented the convenient name Modality" (NEM 3:814, 1905)--but they still require different formal systems. In modern standard notation, Peirce's five modal propositions are (1) ◇p, (2) ¬◇¬p = □p, (3) ◇p ∧ ◇q, (4) ◇(p ∧ q), and (5) ◇p ∧ ◇q ∧ ¬◇(p ∧ q); in each case, p and q are atomic non-modal propositions. How would you represent them in your candidate for Delta EGs? For example, would ◇p simply be p inside an oval with a heavy line attached to the verb phrase "--is possible," and would □p simply be p inside an oval with a heavy line attached to the verb phrase "--is necessary"? If so, then that seems much more cumbersome--much less iconic--than my candidate for Delta EGs. Instead of formulating new graphical transformation rules, would you just stipulate the usual modal axioms--for example, "necessary" may always be changed to "possible" (D), "actual" (T), or "necessarily necessary" (4)? Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Thu, Feb 22, 2024 at 10:18 PM John F Sowa <s...@bestweb.net> wrote: Jon, The single most important innovation of Delta graphs is an operator for metalangage or metalogic. Just that one operator, when added to ordinary first-order logic, makes it possible to define a wide range of modal logics and logics for probability. In fact, Peirce's modal logic of 1903 and his tinctured modal logic of 1906 (as well as may other kinds of modalities and probabilities) can all be defined in terms of Delta graphs (which I assume to be first-order EGs with the operator summarized below). The reason why I make that claim is that I was on the committee of 9 logicians and computer scientists that defined the IKL logic of 2006. And as exercises, we showed how to define all those options by extending FOL with just one operator, which is equivalent to what Peirce defined in RLT (1898), in R514 (June 1911), and in L376 (Dec. 1911). See below. Peirce introduced an operator for metalanguage in RLT (1898). The example he used was the sentence "That you are a good girl is much to be wished." The notation he adopted was a plain white oval with a line of identity attached to the oval. Inside the oval was the sentence "You are a good girl". The line of identity attached to the oval may be read "There exists a proposition p, which is stated by the nested graph for 'You are a good girl'." Outside the oval, he attached the verb phrase "--is much to be wished" to the same line of identity. Although Peirce never developed it further (as far as I know), the option of attaching a line of identity to an oval is exactly the same operation as taking a sheet of paper, drawing a line around the nested text (You are a good girl), and stating postulates in the margin (as in R514 and L376). That is identical the IKL extension to the base logic (called Common Logic). See the cited references about IKL. In IKL, the operator for stating postulates outside the nested statements is named 'that' -- which happens to be the first word in Peirce's example of 1898. When the nine of us defined the IKL logic, I was the only person who had read RLT, but I was not the first person who suggested the word 'that' for the operator. (As they say, great minds run in the same rut.) But as an exercise, we showed that first-order logic plus the that-operator can be used to define all the operators that Peirce defined for his 1903 version of modal logic. So if you like Peirce's 1903 version of modal logic, you can have it. Just use the 'that' operator of 1898 or the Delta papers of 1911 to define the 1903 modal graphs. In short, adopting the Delta graphs of 1911 does not reject the modal logic of 1903, because every option of 1903 can be defined in terms of Delta graphs. As for the five EGs from 1909, quoted below, none of them express modal logic. All five of them can be translated to statements in first-order logic: There exists x such that p(x). If there exists x, then p(x). There exist x and y, such that p(x) and q(y). There exists x, such that p(x) and q(x). There exist x and y, such that p(x) and q(y) and x is not equal to y. John
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