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 www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Mon, Mar 4, 2024 at 2:14 PM John F Sowa <s...@bestweb.net> wrote: > 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 >
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