[CG] Four branches of existential graphs: Alpha, Beta, Gamma, Delta

To refresh my memory, I reread Peirce's Lowell Lectures about Gamma graphs. And the following passage from Lecture V (NEM 3, p. 365) explains what he meant in L376 when he said that he would keep the Gamma division: "I must begin by a few words concerning gamma graphs; because it is by means of gamma graphs that I have been enabled to understand these subjects... In particular, it is absolutely necessary to representing the reasoning about these subjects that we should be able to reason with graphs about graphs and thus that we should have graphs of graphs." That explains the issues we have been debating recently. Peirce had recognized the importance of graphs of graphs when he wrote "The better exposition of 1903 divided the system into three parts, distinguished as the Alpha, the Beta, and the Gamma, parts; a DIVISION I shall here adhere to, although I shall now have to add a Delta part in order to deal with modals", That division would require some version of metalanguage for specifying modality and higher-order logic. But it does NOT imply all (or any) details that he happened to specify in 1903. Since he had earlier specified a version of metalanguage in 1898 (RLT), he had previously recognized the importance of metalanguage. The examples in the Lowell lectures are similar to his 1898 version. Since he never again used the details he specified in 1903 in any further MSS, it's unlikely that he would revive them in 1911. The only feature he was reviving was the use of metalanguage. The 1898 version was just as good as anything he specified in 1903. Since it was simpler than the Gamma graphs, that would make it better. In fact, Peirce mentioned another version of metalanguage in R514 (June 1911) that was logically equivalent and syntactically similar to what he was writing in L376 (December 1911). The novel features of L376 are sufficiently advanced to qualify as a fourth branch of EGs. But they require a bit more explanation. As I said before, they depend critically on the expertise of Allan Risteen. For that information, see the references to Risteen that are listed in the index to EP2. And the applications discussed in L376 have strong resemblances to the applications of the very similar IKL logic in 2006. For those, see the brief discussion and detailed references in https://jfsowa.com/ikl . I'll write more about these topics in another note later this week. John