Re: [PEIRCE-L] Problems in mixing quantifiers with modal logic (was Delta Existential Graphs
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 on behalf of Jon Alan Schmidt Date: Thursday, March 7, 2024 at 7:42 PM To: Peirce-L 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
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
Re: [PEIRCE-L] Problems in mixing quantifiers with modal logic (was Delta Existential Graphs
John, List: JFS: In his letter on Delta graphs, Peirce was breaking new ground. He was proposing a totally new foundation for modal logic, based on metalanguage. There is no "letter on Delta graphs." Peirce wrote only *one sentence* that mentions them in a letter to Risteen (R L376, 1909 Dec 6), and he was *not *"proposing a totally new foundation for modal logic, based on metalanguage." He simply expresses the need to "add a *Delta *part [to EGs] in order to deal with modals," and nothing else in the 19 extant pages is about modal logic or unique to Delta. JFS: The full letter [to James], which mentions the Big Book that Peirce had in mind, is in NEM 3:867-875. That is quite a lot of text to wade through, so it would still be very helpful if you could provide exact quotations of what you consider to be the most relevant portions to support the points that you wish to make. JFS: The outline for the Big Book has a large overlap with L231 and with topics in his last long letter of 1913. That suggests that the Big Book was intended to be the long awaited proof of pragmatism. That also suggests that he intended Delta graphs to be the logic for his proof. Peirce only briefly discusses EGs near the end of the letter to James. In fact, that is where he states, "This ought to be the Logic of the Future" (NEM 3:874, 1909 Dec 28), supplying Pietarinen's title for his five-volume comprehensive collection of Peirce's writings about them. His sole mention of the Big Book comes several sentences later--"I have done a lot of work in Methodeutic that is valuable and very little of it is printed. This will be the most widely useful part of my Big Book" (ibid). Pragmatism falls within methodeutic in Peirce's architectonic classification of the sciences, but he does not say anything about *proving *it in this letter, let alone using EGs to do so, much less a new Delta part that he would not even give that name until two years later. JFS: Peirce's Delta graphs and the IKL logic have very similar goals. Is the *only *goal of the IKL logic "to deal with modals"? That is Peirce's *only *stated goal for Delta EGs. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Wed, Mar 6, 2024 at 1:26 PM John F Sowa wrote: > Gary, > > In his letter on Delta graphs, Peirce was breaking new ground. He was > proposing a totally new foundation for modal logic, based on metalanguage. > The important references are in the future, not the past. For Peirce's > past, the most relevant references were cited and discussed many times: > His 1903 Lowell lectures, his tinctured graphs of 1906, and the references > in R514, L231, and the primary source L376. > > For modern references, I have cited quite a few. The most important one > is to the IKL logic of 2006. For that, I repeatedly pointed to the web > page, which has hundreds of references: https://jfsowa.com/ikl/ . > > The title of that web page is "Semantics for Interoperable Systems" with > four sections. Each section has multiple short paragraphs with several > links for individual systems of that type: > > 1. From the conceptual schema to the semantic web. > 2. The IKRIS project > 3. A foundation for multiple projects > 4. Supporting an open-ended diversity > > Section 1 is historical, but many of the systems mentioned there are still > in use or are the foundation for later developments. Section 2 and the > references there are the basis for IKL and related projects that build on > the IKL base. Section 3 also includes some projects that use IKL. And > Section 4 discusses issues that are still being debated and developed > today. Every section has multiple references. Just look at that article > and note all the phrases in blue. Each one is clickable. > > As for the letter to William James, I was in a hurry, and I didn't have > the time to dig up references to a point that was not directly relevant to > the topic of the letter. For the record, it's the last letter to WJ in > EP2. Unfortunately, the end of the letter was deleted in EP2. The full > letter, which mentions the Big Book that Peirce had in mind, is in NEM > 3:867-875. > > Now that I dug up that reference, I realize that I should have mentioned > something I wrote in an unfinished article. I think I'll include it in my > article on Delta graphs. It shows why Peirce realized the need for a new > approach to modality, and it certainly goes far beyond Alpha graphs. I'll > say more in another note. > > And by the way, that letter was written on 25 Dec 1909. In the part that > was deleted in EP2, Peirce wrote that he was planning to include a section > of the Big Book in an article for Carus. The outline for the Big Book has > a large overlap with L231 and with topics in his last long letter of 1913. > That suggests that the Big Book was intended to be the long awaited proof >
Re: [PEIRCE-L] Problems in mixing quantifiers with modal logic (was Delta Existential Graphs
Gary, In his letter on Delta graphs, Peirce was breaking new ground. He was proposing a totally new foundation for modal logic, based on metalanguage. The important references are in the future, not the past. For Peirce's past, the most relevant references were cited and discussed many times: His 1903 Lowell lectures, his tinctured graphs of 1906, and the references in R514, L231, and the primary source L376. For modern references, I have cited quite a few. The most important one is to the IKL logic of 2006. For that, I repeatedly pointed to the web page, which has hundreds of references: https://jfsowa.com/ikl/ . The title of that web page is "Semantics for Interoperable Systems" with four sections. Each section has multiple short paragraphs with several links for individual systems of that type: 1. From the conceptual schema to the semantic web. 2. The IKRIS project 3. A foundation for multiple projects 4. Supporting an open-ended diversity Section 1 is historical, but many of the systems mentioned there are still in use or are the foundation for later developments. Section 2 and the references there are the basis for IKL and related projects that build on the IKL base. Section 3 also includes some projects that use IKL. And Section 4 discusses issues that are still being debated and developed today. Every section has multiple references. Just look at that article and note all the phrases in blue. Each one is clickable. As for the letter to William James, I was in a hurry, and I didn't have the time to dig up references to a point that was not directly relevant to the topic of the letter. For the record, it's the last letter to WJ in EP2. Unfortunately, the end of the letter was deleted in EP2. The full letter, which mentions the Big Book that Peirce had in mind, is in NEM 3:867-875. Now that I dug up that reference, I realize that I should have mentioned something I wrote in an unfinished article. I think I'll include it in my article on Delta graphs. It shows why Peirce realized the need for a new approach to modality, and it certainly goes far beyond Alpha graphs. I'll say more in another note. And by the way, that letter was written on 25 Dec 1909. In the part that was deleted in EP2, Peirce wrote that he was planning to include a section of the Big Book in an article for Carus. The outline for the Big Book has a large overlap with L231 and with topics in his last long letter of 1913. That suggests that the Big Book was intended to be the long awaited proof of pragmatism. That also suggests that he intended Delta graphs to be the logic for his proof. And by the way, please read that section 2 about the IKRIS project. Its goal was to support interoperability among multiple systems. And the IKL logic is a major part. That shows a definite convergence: a logic of pragmatism would indeed support interoperability among multiple projects in science and engineering. Peirce's Delta graphs and the IKL logic have very similar goals. That's why they are so closely related. I'll mention that in my article on Delta graphs. John From: "Gary Richmond" Sent: 3/5/24 8:44 PM To: John F Sowa Cc: Jon Alan Schmidt , Peirce-L Subject: Re: [PEIRCE-L] Problems in mixing quantifiers with modal logic (was Delta Existential Graphs John, I have been following this exchange between you and Jon Alan with considerable interest. Thank you both for discussing these most interesting -- and I think, important matters relating to modals, Delta graphs, etc. -- in the generally collegial manner in which you have been proceeding. You wrote: And please read what Peirce was writing to William James around that time. He was talking about a Big Book with some rather complex requirements for the logic -- far more than alpha graphs, even with modality. And the content of the Big Book had a large overlap with L231 of June 1911. That letter mentioned his goal of a logic for representing moving motion pictures. That's not possible with Alpha graphs. I would like to suggest that it would be helpful if, instead of suggesting that, for example, Jon (and, I assume, anyone reading this exchange) should "read what Peirce was writing to William James around that time," that you offer exact quotations, something you've not infrequently have suggested is 'best practice' in considering what Peirce actually said, actually had in mind. In truth, I haven't seen many exact quotations in your posts in this exchange (something Jon famously -- or infamously, depending on your perspective -- can't be accused of) and this has often made it difficult to discern exactly what your critique of Jon's position is nor, for that matter, what your's is in certain given cases. I doubt that many, following this recent exchange, have the time or inclination to hunt for quotations that are only very generally pointed to. Again,
Re: [PEIRCE-L] Problems in mixing quantifiers with modal logic (was Delta Existential Graphs
John, I have been following this exchange between you and Jon Alan with considerable interest. Thank you both for discussing these most interesting -- and I think, important matters relating to modals, Delta graphs, etc. -- in the generally collegial manner in which you have been proceeding. You wrote: And please read what Peirce was writing to William James around that time. He was talking about a Big Book with some rather complex requirements for the logic -- far more than alpha graphs, even with modality. And the content of the Big Book had a large overlap with L231 of June 1911. That letter mentioned his goal of a logic for representing moving motion pictures. That's not possible with Alpha graphs. I would like to suggest that it would be helpful if, instead of suggesting that, for example, Jon (and, I assume, anyone reading this exchange) should "read what Peirce was writing to William James around that time," that you offer exact quotations, something you've not infrequently have suggested is 'best practice' in considering what Peirce actually said, actually had in mind. In truth, I haven't seen many exact quotations in your posts in this exchange (something Jon famously -- or infamously, depending on your perspective -- can't be accused of) and this has often made it difficult to discern exactly what your critique of Jon's position is nor, for that matter, what your's is in certain given cases. I doubt that many, following this recent exchange, have the time or inclination to hunt for quotations that are only *very generally* pointed to. Again, I have found your exchange most interesting and valuable. But on a listserv such as Peirce-L, it would be more than helpful to have exact quotations provided in posts to the List, especially in consideration of the topics which you and Jon have been discussing. Best, Gary On Tue, Mar 5, 2024 at 6:24 PM John F Sowa wrote: > Jon, > > The first point to emphasize is that Peirce's primary goal in the last > decade of his life was to provide a proof of pragmatism. That would > require a system of logic that could express and analyze rather > sophisticated texts about science. The metalanguage of the IKL logic in > 2006 is very close to Peirce's EGs supplemented with the operator that he > specified in R514 (June 1911), which seems to be very similar to what he > was specifying in L376 (December 1911). > > Alpha graphs for Boolean logic are a trivial subset of EGs. Peirce made > an important contribution to Boolean logic by adding the symbol -< for > implication. He also made a few other important modifications. But that > was a very early project. In 1903, he presented his version of modal > logic, which included EGs with lines of identity. As far as I know, there > was never a reason for him to say anything further about Alpha graphs other > than the fact that they were a simple subset that could be freely mixed > with Beta graphs. > > JAS> I am still wondering exactly how your candidate would represent the > five modal propositions that Peirce wrote in his Logic Notebook, if not > exactly as he scribed them on that page (R 339:[340r], 1909 Jan 7. > > Did you read my response? I showed that the five EGs on the excerpt you > included were *not *modal. They were simple first-order (Beta) EGs. And > I included a translation of all five to English sentences that did not > require a single occurrence of the words 'possible' or 'necessary'. > > As far as I know, Peirce never used the modal logic of 1903 for any > purpose in any MSS after 1903. If you can find any examples, please send > us a copy. But he did write quite a bit about modality, including his > tinctured graphs of 1906. He did criticize them in L376 for their > *notation*, but not their goal of representing rather sophisticated modal > content -- much more than modal alpha graphs. > > And please read what Peirce was writing to William James around that > time. He was talking about a Big Book with some rather complex > requirements for the logic -- far more than alpha graphs, even with > modality. And the content of the Big Book had a large overlap with L231 of > June 1911. That letter mentioned his goal of a logic for representing > moving motion pictures. That's not possible with Alpha graphs. > > Since Risteen had considerable experience with Cayley's trees, that is an > excellent reason for his visit, and for Peirce to be constructing a tree of > "papers". It's inconceivable that he would have invited Risteen (a former > collaborator who had an excellent understanding of his 1885 logic of first > order and higher order logic) to discuss some trivial work with a subset of > the modal logic of 1903. > > Four points: (1) there is evidence of metalanguage (postulates in the > margin about nested graphs) in R514 and L376: (2) there is no evidence that > Peirce intended to adopt a subset of his 1903 modal notation, which he had > not used in any MSS after 1903; Risteen's
Re: [PEIRCE-L] Problems in mixing quantifiers with modal logic (was Delta Existential Graphs
Jon, The first point to emphasize is that Peirce's primary goal in the last decade of his life was to provide a proof of pragmatism. That would require a system of logic that could express and analyze rather sophisticated texts about science. The metalanguage of the IKL logic in 2006 is very close to Peirce's EGs supplemented with the operator that he specified in R514 (June 1911), which seems to be very similar to what he was specifying in L376 (December 1911). Alpha graphs for Boolean logic are a trivial subset of EGs. Peirce made an important contribution to Boolean logic by adding the symbol -< for implication. He also made a few other important modifications. But that was a very early project. In 1903, he presented his version of modal logic, which included EGs with lines of identity. As far as I know, there was never a reason for him to say anything further about Alpha graphs other than the fact that they were a simple subset that could be freely mixed with Beta graphs. JAS> I am still wondering exactly how your candidate would represent the five modal propositions that Peirce wrote in his Logic Notebook, if not exactly as he scribed them on that page (R 339:[340r], 1909 Jan 7. Did you read my response? I showed that the five EGs on the excerpt you included were not modal. They were simple first-order (Beta) EGs. And I included a translation of all five to English sentences that did not require a single occurrence of the words 'possible' or 'necessary'. As far as I know, Peirce never used the modal logic of 1903 for any purpose in any MSS after 1903. If you can find any examples, please send us a copy. But he did write quite a bit about modality, including his tinctured graphs of 1906. He did criticize them in L376 for their notation, but not their goal of representing rather sophisticated modal content -- much more than modal alpha graphs. And please read what Peirce was writing to William James around that time. He was talking about a Big Book with some rather complex requirements for the logic -- far more than alpha graphs, even with modality. And the content of the Big Book had a large overlap with L231 of June 1911. That letter mentioned his goal of a logic for representing moving motion pictures. That's not possible with Alpha graphs. Since Risteen had considerable experience with Cayley's trees, that is an excellent reason for his visit, and for Peirce to be constructing a tree of "papers". It's inconceivable that he would have invited Risteen (a former collaborator who had an excellent understanding of his 1885 logic of first order and higher order logic) to discuss some trivial work with a subset of the modal logic of 1903. Four points: (1) there is evidence of metalanguage (postulates in the margin about nested graphs) in R514 and L376: (2) there is no evidence that Peirce intended to adopt a subset of his 1903 modal notation, which he had not used in any MSS after 1903; Risteen's expertise suggests that trees of "papers" are very likely to be involved in the representation and reasoning with and about Delta graphs; and (4) IKL or some version of metalanguage for representing trees of papers can represent a significant amount of computer science and AI today. That is the topic of the article about Delta graphs, which I am writing. John From: "Jon Alan Schmidt" John, List: JFS: One reason why I did not respond in detail to your previous note (copied below) is that your citations to the writings by Dunn and Goble only apply to PROPOSITIONAL modal logic (no quantifiers). I have acknowledged this all along--my candidate for Delta EGs is an extension of Alpha EGs, not Beta EGs. After all, the various modal axioms are formulated as extensions of classical propositional logic, not first-order predicate logic (FOPL). The heavy lines of compossibility (LoCs) in my Delta EGs represent possible states of things (PSTs) in which propositions denoted by attached letters would be true, while the heavy lines of identity (LoIs) in Beta EGs represent indefinite individuals to which general concepts denoted by attached names are attributed. Roberts suggests that these two notations could be combined, with LoCs attached to the top of names that are also attached to LoIs (1973, pp. 99-100); but as you rightly observe later in your post, this "opens up a huge can of worms." I have mentioned previously an exception to this cautionary note, which is implementing system P with no iterated modalities. LoCs are then attached to letters for propositions on the one sheet for the actual state of things (AST), which are keyed to different Beta graphs with LoIs on the various sheets for PSTs. This demonstrates the sense in which formal propositional logic as implemented by Alpha EGs is a simple metalanguage for reasoning about propositions, each of which can then be more informatively
Re: [PEIRCE-L] Problems in mixing quantifiers with modal logic (was Delta Existential Graphs
John, List: JFS: One reason why I did not respond in detail to your previous note (copied below) is that your citations to the writings by Dunn and Goble only apply to PROPOSITIONAL modal logic (no quantifiers). I have acknowledged this all along--my candidate for Delta EGs is an extension of *Alpha *EGs, not *Beta *EGs. After all, the various modal axioms are formulated as extensions of classical *propositional *logic, not first-order *predicate *logic (FOPL). The heavy lines of compossibility (LoCs) in my Delta EGs represent possible states of things (PSTs) in which propositions denoted by attached letters would be true, while the heavy lines of identity (LoIs) in Beta EGs represent indefinite individuals to which general concepts denoted by attached names are attributed. Roberts suggests that these two notations *could *be combined, with LoCs attached to the top of names that are also attached to LoIs (1973, pp. 99-100); but as you rightly observe later in your post, this "opens up a huge can of worms." I have mentioned previously an exception to this cautionary note, which is implementing system *P* with no iterated modalities. LoCs are then attached to letters for propositions on the one sheet for the *actual *state of things (AST), which are keyed to different Beta graphs with LoIs on the various sheets for PSTs. This demonstrates the sense in which formal propositional logic as implemented by Alpha EGs is a simple metalanguage for reasoning *about *propositions, each of which can then be more informatively represented in FOPL as implemented by Beta EGs. JFS: And the E of EG refers to the existential quantifier, which corresponds to a universal quantifier in a negated area. Not exactly, since Alpha EGs are (obviously) *existential *graphs despite not having *any *quantifiers at all. Moreover, although Peirce invented/discovered quantifiers independently of Frege, and his algebraic notation for them was the basis for what subsequently became the standard one, he apparently never uses the term "existential quantifier" in any of his writings. In fact, I have found only one place where he gives it a name at all, calling it "the *particular *quantifier" (CP 2.339, c. 1895). Regardless, we do not need to guess at Peirce's reason for giving EGs their name--he tells us plainly, "I call it the system of Existential Graphs, because its fundamental symbol expresses the relation of existence" (R 485, LF 1:312, c. 1898). Also, "In order to draw such a graph, the first step is to assign some sheet of paper or enclosure upon a sheet, marked out by a bounding line, to represent so much as we know or recollect of the universe. If on that sheet or in that enclosure we draw a picture, or write a general description, or a letter which is an abbreviation for a general description, the effect is to be understood and agreed upon as being that we *assert* that to something in the universe that picture or description applies. We aver that such a thing *exists*. Hence, I call this the system of *existential* graphs" (R 513, LF 1:316-317, 1898). JFS: Solution: Add metalanguage to a conventional (non-modal) logic. ... That happens to be the solution adopted for the IKL logic of 1906, which appears to be a superset of Peirce's Delta graphs. Again, there is no evidence in R L436 nor elsewhere in Peirce's writings that what he has in mind for Delta EGs is adding metalanguage to classical logic, other than what Alpha EGs already provide for reasoning *about *propositions and an additional sign for asserting them as possibly (or necessarily) true/false instead of actually true/false. All we can say with certainty is that he recognizes the need for "a *Delta *part in order to deal with modals," and my candidate achieves this purpose by implementing various formal systems of modal logic that have been introduced over the last century. As I said yesterday, I am still wondering exactly how your candidate would represent the five modal propositions that Peirce wrote in his Logic Notebook, if not exactly as he scribed them on that page (R 339:[340r], 1909 Jan 7). JFS: As for the two quotations by Peirce below, (1) they're irrelevant to the issues about Delta graphs, and (2) they are not consistent with modern developments in physics and astronomy. My only point in presenting those two quotations was to show a potential *application *of formal modal logic--system *S4*, with a reflexive and transitive alternativeness relation (AR) such that the model set of law-propositions never shrinks but can grow with each iteration of PSTs, is reminiscent of Peirce's hyperbolic cosmology. This is a *metaphysical *hypothesis grounded in synechism, tychism, and objective idealism--not a finding of the *special *sciences, which have largely adopted the opposite assumptions of reductionism, determinism, and materialism. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian
[PEIRCE-L] Problems in mixing quantifiers with modal logic (was Delta Existential Graphs
Jon, One reason why I did not respond in detail to your previous note (copied below) is that your citations to the writings by Dunn and Goble only apply to PROPOSITIONAL modal logic (no quantifiers). Every version of modal logic that Peirce developed included existential graphs as the base logic. And the E of EG refers to the existential quantifier, which corresponds to a universal quantifier in a negated area. Therefore Peirce's modal logic of 1903 included both quantifiers. Any mixture of quantifiers with modal logic opens up a huge can of worms, which Peirce did not address. For a brief summary see the article on the Stanford site: https://plato.stanford.edu/archIves/spr2010/entries/logic-modal/#QuaModLog That article has 13 sections. The first 12 cover the many variations of propositional modal logic. Section 13, the shortest one, summarizes the complexities introduced by mixing the modal operators with quantifiers. Short summary: "Here be dragons." Longer summary: For anybody who considers 13 unlucky, here's more evidence. Solution: Add metalanguage to a conventional (non-modal) logic. That is the solution that is used in nearly all versions of modal reasoning used in computer science, artificial intelligence, etc, That happens to be the solution adopted for the IKL logic of 1906, which appears to be a superset of Peirce's Delta graphs. Furthermore, J. Michael Dunn, whom you cited below, developed a foundation that justifies metalanguage (as in IKL) for combining modality with quantifiers. See the references in the many documents I cited. By the way, Dunn thanked me for all the references in which I cited his work. And he invited me to give a talk on those applications (including IKL) at his university. For a brief (6 page) summary of the issues, see "Modality Si!, Modal Logic No!" by John McCarthy: http://jmc.stanford.edu/articles/modality/modality.pdf . McCarthy hosted the founding meeting in 1956 that adopted the term "Artificial Intelligence", and he has been a strong advocate for using logic in AI and other branches of computer science. Although he died before the 2006 project that developed IKL, many of the people who participated in that project were his students and colleagues. As for the two quotations by Peirce below, (1) they're irrelevant to the issues about Delta graphs, and (2) they are not consistent with modern developments in physics and astronomy. John From: "Jon Alan Schmidt" List: As I continue contemplating my updated candidate for Delta EGs (see earlier posts below), I am finding that, in conjunction with the laws and facts semantics (LFS) developed by Dunn and Goble, it is very helpful for explicating the effects of adding various modal axioms to classical logic. For example, the distribution axiom K = □(p → q) → (□p → □q) that is included in all so-called "normal" modal logics is illustrated by the fact that if p → q is on every sheet for a possible state of things (PST) and p is also on every PST sheet, then q is likewise on every PST sheet or can be derived on any PST sheet where it is initially missing. As I have mentioned before, other axioms assign different properties of the binary alternativeness/accessibility relation (AR) between the actual state of things (AST) and any PSTs, as well as the latter and their higher-order PSTs when there are iterated modalities. - Serial, axiom D = □p → ◇p, or ◇⊤; every law-graph on the AST sheet is a fact-graph on at least one PST sheet, and any graph that can be derived from the blank on the AST sheet can also be derived from the blank on at least one PST sheet. - Reflexive, axiom T = □p → p, or p → ◇p; every law-graph on the AST sheet is also a fact-graph on the AST sheet, and every fact-graph on the AST sheet is a fact-graph on at least one PST sheet. - Symmetric, axiom B = ◇□p → p, or p → □◇p; every law-graph on any PST sheet is a fact-graph on the AST sheet, and every fact-graph on the AST sheet is a fact-graph on at least one second-order PST sheet for every first-order PST sheet. - Transitive, axiom 4 = □p → □□p, or ◇◇p → ◇p; every law-graph on the AST sheet is a law-graph on every PST sheet, and every fact-graph on a second-order PST sheet is a fact-graph on at least one first-order PST sheet. - Euclidean, axiom 5 = ◇□p → □p, or ◇p → □◇p; every law-graph on a PST sheet is a law-graph on the AST sheet, and every fact-graph on a PST sheet is a fact-graph on at least one second-order PST sheet for every first-order PST sheet. LFS effectively stipulates that the AR is serial because every law-graph on the AST sheet is a fact-graph on every PST sheet--its basic principle is that possibility is defined as consistency with the laws of the AST--and any classical tautology can be derived from the blank on every sheet. The AR properties and their corresponding axioms are then combined in different ways for
