Hello John, Jon, List, Peirce examines both first and second intentional logics. The distinction appears to be similar, in some respects, to the contemporary distinction between first and second order logics. Here, for instance, is an SEP entry on higher order logics: https://seop.illc.uva.nl/entries/logic-higher-order/#HighOrdeLogiVisVisTypeTheo
Does Peirce’s explorations in the Gamma system of the EG, and his contemplation of a possible Delta system, bear some similarities to contemporary discussions of higher order logics, such as third order, or fourth order, etc.? --Jeff D From: peirce-l-requ...@list.iupui.edu <peirce-l-requ...@list.iupui.edu> on behalf of Jon Alan Schmidt <jonalanschm...@gmail.com> Date: Thursday, March 7, 2024 at 7:42 PM To: Peirce-L <peirce-l@list.iupui.edu> Subject: Re: [PEIRCE-L] Problems in mixing quantifiers with modal logic (was Delta Existential Graphs John, List: It looks like you sent the message quoted below only to me, but I assume that you intended it for the entire List, so I am replying accordingly. JFS: In the copy of your note, included below, please note that the five EGs are BETA graphs. The lines of identity refer to things that Peirce called circumstances. A circumstance is a THING that is indistinguishable from a context by McCarthy or a situation by Barwise. In fact, a possible world can also be called a THING. If this were true, then there would be no need for all the different formal systems of modal logic--S1-S5, T, B, P, etc.--because we could just use first-order predicate logic (FOPL). There would also be no need for "a Delta part [of EGs] in order to deal with modals," because we could just use Beta EGs. On the contrary, circumstances as possible states of things (PSTs) are not themselves "things" that can be denoted by ordinary lines of identity (LoIs), and propositions are not "properties" that can be attributed to such "things" by attaching letters to those heavy lines. As I acknowledged when I corrected your mistranslations of Peirce's modal EGs on R 339:[340r] the first time, there is an analogy between quantifying predicates (general concepts) over subjects (indefinite individuals) and quantifying propositions over PSTs, but they still require different formal systems. In fact, the usual permissions for transforming LoIs in Beta EGs served as my starting point for working out the new permissions for transforming lines of compossibility (LoCs) in my candidate for Delta EGs, but then I had to adjust them to account for the peculiarities of PSTs vs. individuals and propositions vs. concepts. There are at least two obvious notational differences--in Beta, a name must always be attached to at least one LoI, and it can only be attached to more than one LoI if it denotes a dyadic or triadic relation; while in Delta, a letter can be unattached to any LoCs when it denotes a proposition that is true in the actual state of things (AST), and it can be attached to multiple LoCs for iterated modalities unless system P is being implemented to preclude them. JFS: I don't have the page number of R514 in front of me, but I remember that the following sentence ended in the middle with [end]. Again, why the rush? This is Peirce-L, not Sowa-L nor Schmidt-L, so we should always take the time and make the effort to look up any passages in Peirce's writings that we are planning to cite in a post, make sure that they actually say what we remember them saying, and then include the relevant exact quotations to support our points. In this case, you are presumably referring to the following. CSP: One of my possibly slight improvements, is that I begin by drawing (preferably with a red pencil), a line all round my sheet at a little distance from the edge; and in the margin outside the red line, whatever is scribed is merely asserted to be possible. Thus, if the subject were geometry, I could write in that margin in the postulates, and any pertinent problems stated in the form of postulates such as, that “if, on a plane, there be circle with a ray cutting it, and two points be marked [end] (R 514:[18-19], 1909) As I keep emphasizing, this notational innovation has nothing to do with modal logic nor metalanguage. It simply converts the entire sheet--there is only one sheet here--into a scroll for material implication with the antecedent (e.g., postulates) in the margin (outer close) and the consequent (e.g., theorems) within the red line (inner close). Peirce does not say anything about this in R L376, and he does not say anything about the "many papers" concept in R 514, so I am still not seeing any explicit connection between those two manuscripts. As I have said before, a further improvement is shading the margin instead of drawing a red line as its inner boundary. This is a more iconic way of conveying that it is a different surface from the unshaded interior, representing a universe of possibility--"in the margin ... whatever is scribed is merely asserted to be possible." JFS: As for McCarthy's circumstances, please note his article titled "Modality Si! Modal Logic No!" I read the article (http://jmc.stanford.edu/articles/modality/modality.pdf), and it does not say much about circumstances/contexts/situations. McCarthy even admits in the abstract, "I have no proof that modal logic is inadequate." Moreover, Heinrich Wansing subsequently wrote a rebuttal entitled "Modality, of Course! Modal Logic, Si!" (https://www.jstor.org/stable/40180140). He states, "I think that it would be a bad move to avoid modal logic, in particular because modal logic is alive and thriving, perhaps more so than ever. It has become a mature field that can be of great benefit to many areas--including AI [artificial intelligence] and KR [knowledge representation]." JFS: The many different axioms for modal logic that had been defined over the years were constrains on the domain of quantification. But those constraints were just useless baggage. You could keep them, reject them, or replace them with any kind of constraints that were relevant to the problem(s) being considered. That is what Peirce was specifying in L376. Please provide an exact quotation from R L376 where Peirce specifies what you describe here, because I am not seeing it. As I have explained previously, the different modal axioms added to classical propositional logic are hardly "useless baggage." They correspond to different properties of the binary alternativeness/accessibility relation (AR) between the AST and PSTs--serial for D, reflexive for T, symmetric for B, transitive for 4, euclidean for 5--thus formalizing different criteria for a state of things to be possible in the first place. None of this was worked out during Peirce's lifetime, and various logicians contributed pieces to the puzzle decades after his death. How would such complexity be handled in your candidate for Delta EGs? So far, you have not even been willing/able to show/tell us how you would represent several very simple modal propositions and then derive others from them. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt<http://www.LinkedIn.com/in/JonAlanSchmidt> / twitter.com/JonAlanSchmidt<http://twitter.com/JonAlanSchmidt> On Wed, Mar 6, 2024 at 5:47 PM John F Sowa <s...@bestweb.net<mailto:s...@bestweb.net>> wrote: Jon, In every one of these notes, I have urged you to read the references to the IKL logic of 2006, which is based on earlier developments from 1973 and later by Dunn, Hintikka, and others. The fundamental principle involves metalanguage and quantifiers that range over possible worlds -- or subsets of worlds called contexts (John McCarthy) or situations (Barwise & Perry). Hintikka was also developing ideas along similar lines with his game theoretical semantics. He also allowed quantifiers to range over possible worlds or subsets of them. In the copy of your note, included below, please note that the five EGs are BETA graphs. The lines of identity refer to things that Peirce called circumstances. A circumstance is a THING that is indistinguishable from a context by McCarthy or a situation by Barwise. In fact, a possible world can also be called a THING. Peirce's tinctured graphs of 1906 had a notation that he later rejected -- not because they referred multiple possible worlds, but because the notation was bad. He improved the notation in R514 and L231. I don't have the page number of R514 in front of me, but I remember that the following sentence ended in the middle with [end]. As for McCarthy's circumstances, please note his article titled "Modality Si! Modal Logic No!" I don't have that article in front of me at the moment, but you can google it by typing "McCarthy" and "modality si". And by the way, both McCarthy and Barwise had invited me to give talks in their seminars at Stanford. That suggests that they did not reject my interpretation. Saul Kripke showed how to define a semantics for any version of modal logic by allowing quantifiers to range over possible worlds. Dunn (1973) showed that it was possible to simplify and generalize the method by using metalanguage and quantifiers to range over worlds, contexts, circumstances, situations or anything you wanted to call them. The many different axioms for modal logic that had been defined over the years were constrains on the domain of quantification. But those constraints were just useless baggage. You could keep them, reject them, or replace them with any kind of constraints that were relevant to the problem(s) being considered. That is what Peirce was specifying in L376. In December 1911, Peirce did not have any knowledge of the following century of logic. But he had a deep insight into the issues, which led him to a version that is very similar to the logics cited above. I recommend that you read the references. John
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