John and list,
Peirce’s “Improvement on the Gamma Graphs” (CP 4.573-84) in indeed a fascinating read; Frederik Stjernfelt comments on it extensively in Chapter 8 of Natural Propositions. But according to Don Roberts (1973, p.89), it’s from the spring of 1906, and preceded the drafts of the “Prolegomena”, which was published later in that year. You seem to be reversing that chronology by including it with “further developments” in the EGs after the Prolegomena. Anyway, what happened to EGs after 1906 is not at all clear to me, although I’ve worked my way through the relevant papers on your site. Given the central role Peirce wanted them to play in his “apology for pragmaticism,” I’m still trying to understand this, and your list doesn’t give me many clues. Roberts (p.92) describes the “Prolegomena” as “Peirce’s last full scale revision of EG,” and notes that the “tinctures” did not really solve the problems with representing modal logic that Peirce thought he had solved in the spring of 1906. Some of his later comments on the “Prolegomena” (included in http://www.gnusystems.ca/ProlegomPrag.htm) are quite critical of it — one even refers to the “tinctures” and “heraldry” as “nonsensical” — but they don’t really say how these problems can be solved diagrammatically. Are you saying that his later manuscripts did solve these problems, or that Peirce “simplified” his system of EGs by abandoning further development of the Gamma graphs and reverting to a version of the Beta? For me, these questions have large implications for Peirce’s late semiotics, phaneroscopy, Synechism, pragmaticism and metaphysics (as he suggested at the end of his “Improvement on the Gamma Graphs” talk (CP 4.584). I have to confess that for me, the mapping back and forth between EGs and other diagrammatic or algebraic systems doesn’t throw any light on those implications. I’d appreciate any help you (or anyone) can give toward clarifying them. I’m also curious as to what people think of my “Rhematics” post (http://gnusystems.ca/wp/2017/05/rhematics/) and Gary Richmond’s comment on it, as I have a follow-up in mind … Gary f. -----Original Message----- From: John F Sowa [mailto:s...@bestweb.net] Sent: 27-May-17 22:00 To: peirce-l@list.iupui.edu Subject: Re: [PEIRCE-L] Jay Zeman's existentialgraphs.com On 5/26/2017 8:49 AM, <mailto:g...@gnusystems.ca> g...@gnusystems.ca wrote: > my own site, <http://www.gnusystems.ca/ProlegomPrag.htm> > http://www.gnusystems.ca/ProlegomPrag.htm, which I think > improves on Zeman’s version in some respects, even correcting a few > errors. Yes, that looks good. > your contribution to the “Five Questions” collection, > <http://www.jfsowa.com/pubs/5qsigns.htm> > http://www.jfsowa.com/pubs/5qsigns.htm — which i highly recommend Thanks. For the further development of EGs, I recommend Peirce's later MSS and his "Improvement on the Gamma Graphs", which Jay posted on his site. (See below for an excerpt.) The later MSS (around 1909) simplified the foundation of EGs, the rules of inference, and the mapping to and from algebraic notations and natural languages. Basic innovations: 1. Major simplification in the treatment of lines of identity, ligatures, and teridentity. (See the excerpt below.) 2. Elimination of talk about cuts, recto, and verso. Instead, he introduced shaded (negative) and unshaded (positive) areas. 3. Simplification and generalization of the rules of inference to three pairs of rules: each pair has an insertion rule and an erasure rule, each of which is an exact inverse of the other. 4. The same rules apply to both Alpha and Beta: therefore, there is no need to distinguish Alpha and Beta. Any proposition in Alpha may be treated as a medad (0-adic relation) in Beta. 5. The above innovations make Peirce's proof procedure an extension and generalization of *both* Gentzen's natural deduction *and* Alan Robinson's widely used method of resolution theorem proving. 6. Theorem: Every proof by resolution (in any notation for first- order logic) can be converted to a proof by resolution with Peirce's rules. Then by negating each step of the proof and reversing the order, it becomes a proof by Peirce's version of natural deduction. Finally, that proof can be systematically converted to a proof by Gentzen's version of natural deduction. 7. Peirce's rules can be stated in a notation-independent way. With a minor generalization, they can be applied to Peirce- Peano notation, to Kamp's discourse representation structures, and to any statement in English that has an exact translation to and from Kamp's DRS. For the details of points #1 to #6, see <http://www.jfsowa.com/pubs/egtut.pdf> http://www.jfsowa.com/pubs/egtut.pdf For the slides of an introduction to EGs that use Peirce's later rules and notation, see <http://www.jfsowa.com/talks/egintro.pdf> http://www.jfsowa.com/talks/egintro.pdf For an article that discusses all seven points above, see <http://www.jfsowa.com/pubs/eg2cg.pdf> http://www.jfsowa.com/pubs/eg2cg.pdf For these reasons, I believe that Peirce's publications of 1906 should be considered an intermediate stage in the development of existential graphs. The version of 1909 is his preferred version. John ____________________________________________________________________ The last four sentences of CP 4.583 anticipate his later MSS on EGs: <http://www.jfsowa.com/exgraphs/peirceoneg/improvement_on_the_gamma_Graphs.htm> http://www.jfsowa.com/exgraphs/peirceoneg/improvement_on_the_gamma_Graphs.htm Since no perfectly determinate proposition is possible, there is one more reform that needs to be made in the system of existential graphs. Namely, the line of identity must be totally abolished, or rather must be understood quite differently. We must hereafter understand it to be potentially the graph of teridentity by which means there always will virtually be at least one loose end in every graph. In fact, it will not be truly a graph of teridentity but a graph of indefinitely multiple identity. (CP 4.583, 1906) Note by JFS: I interpret the last sentence to imply that a line of (single) identity and a ligature of several lines are both treated as "a graph of indefinitely multiple identity." That would simplify the mapping from an existential graph to other versions of logic, including Peirce-Peano algebra or Kamp's DRS notation.
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