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Peirce claims that carefully studying his system of Existential Graphs (EG) "should show what nature is truly common to all significations of concepts," and that comparing this with how his pragmatism defines it would "furnish a test of the truth or falsity of" the latter (R 298:2, 1906). However, he eventually abandons that project, and in a recent paper ( https://www.jstor.org/stable/10.2979/trancharpeirsoc.55.4.01), Dave Beisecker cites two other authors in offering a potential reason why. DB: Hookway (https://books.google.com/books?id=_4KOTIyzMfwC, chapter 11) and Pietarinen ( https://www.researchgate.net/publication/271417080_Moving_pictures_of_thought_II_Graphs_games_and_pragmaticism's_proof) suggest that this is likely due to the graphs’ inability to represent certain features of modal reasoning, which Peirce stresses would need to be captured in order to accommodate subjunctive readings of the pragmat[ist]ic maxim. (p. 378) Peirce unsuccessfully attempts "to represent certain features of modal reasoning" in EG by introducing what he later disparages as "nonsensical 'tinctures' and heraldry" (RL 477:33, 1913 Nov 8). In addition to "Possibility, when Color will be used" and "Actuality shown by Metal," he identifies "Intention, indicated by Fur" as the third "Mode of Being which is to be affirmatively or negatively attributed to the state of things described" (CP 4.554, 1906). This is curious since he states just a few paragraphs earlier, "in addition to actuality and possibility, a third mode of reality must be recognized in that which ... is *destined*" (CP 4.547); i.e., necessity. The third mode in my proposed system of Synechistic Graphs (SG) is *conditional* necessity, corresponding to the inner close of a scroll where its outer close contains a real possibility. I have admitted that the associated restrictions on insertion in shaded areas likely limit the usefulness of SG for propositional logic. The situation is presumably the same with entire propositions in first-order predicate logic since the issue is ensuring the proper relationship between antecedent and consequent. However, unlike in Alpha, in Beta there is often an explicit connection between what is scribed in the two areas. In EG, where a line of identity joins a name "A" in the outer close with a name "B" in the inner close, it asserts a consequence *de inesse*--"if anything *actually* possesses the character of A, then it also *actually* possesses the character of B." In SG, the same graph-instance asserts a subjunctive implication--"if anything *were* to possess the character of A, then it also *would* possess the character of B." Here the insertion rule allows the attachment of any number of other names to the line in the shaded outer close, thus predicating additional concepts of the same object, since this has the effect of *reducing* the range of states of things signified by the entire graph-instance. Consider this series of propositions accordingly. 1. If anything (is / were) a dog, then it also (is / would be) a mammal. 2. If anything (is / were) a brown dog, then it also (is / would be) a mammal. 3. If anything (is / were) a small brown dog, then it also (is / would be) a mammal. In both classical and synechistic logic, and therefore in both EG and SG, #2-3 are true because #1 is true--no matter what other characters any dog may be supposed to possess, it remains (or would remain) a mammal. Even replacing "brown" with "purple" and "small" with "microscopic," such that their antecedents presumably no longer describe actualities or real possibilities at all, #2-3 remain true because #1 is true. However, there are important differences between EG and SG. For example, double negation elimination in classical logic makes "it is not the case that if anything *is* a unicorn, then it *is* hornless" equivalent to "something is a unicorn and is not hornless." Nevertheless, in the *actual* universe as opposed to a fictional one, the former is at least arguably true while the latter is definitely false. By contrast, in synechistic logic, "it is not the case that if anything *were* a unicorn, then it *would* be hornless" can be derived from "something is a unicorn and is not hornless," but not the other way around. This is consistent with the modern rejection of existential import ( http://cstl-cla.semo.edu/hhill/pl120/notes/existential%20import.htm). Consider another example from Peirce himself (R 490:22-25, 1906). The following propositions signify different states of things, and thus are expressed by different graph-instances in EG (see attached R490eg.jpg), yet they are equivalent in classical logic because they are only *false* under exactly the same circumstances. 4. A man exists such that if he *goes* bankrupt, then he *commits* suicide. 5. A man exists such that if he *goes* bankrupt, then a man exists who *commits* suicide. #5 can be derived from #4, but not vice-versa because the identity of the two men is additional information as reflected by the extra unenclosed line segment in the corresponding graph-instance, which can be erased but not inserted. "It seems monstrous to say that these two come to the same thing," but neither proposition is *actually* falsified unless "*every* *man* goes bankrupt, and *no* *man* suicides" (R 490:23-24). Now here are the translations in synechistic logic, where the graph-instances in SG have scrolls rather than nested cuts/ovals. 6. A man exists such that if he *were* to go bankrupt, then he *would* commit suicide. 7. A man exists such that if he *were* to go bankrupt, then a man exists who *would* commit suicide. Again, #7 can be derived from #6, but not vice-versa, yet #7 could be true while #6 is false. On the other hand, a man's own bankruptcy is *prima facie* a more plausible reason *why* he would commit suicide than someone else's bankruptcy. To sum up, it seems feasible in SG to interpret a scroll in Beta graphs as a subjunctive implication, suggesting that SG might indeed be more useful for proving Peirce's pragmatism than EG turned out to be. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt >
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