Atila, List: Just to finish the story here, in his third 1903 Lowell Lecture (LF 2/2:237-51), Peirce acknowledges that the advertised topic was "the three Universal Categories" but announces that he will mostly be discussing Existential Graphs (EG) instead. He summarizes what he covered in the previous lecture about the Alpha part and adds that the "true distinction" of the Beta part is that it "takes account of individuals." He then mentions the Gamma part, saying that it "is by far the most important of the three, and which is distinguished by its taking account of *abstractions*" (241-2).
Nevertheless, Peirce begins with phenomenology, defining it as "the science which describes the different kinds of elements that are always present in ... whatever is before the mind in any kind of thought, fancy, or cognition of any kind" (242). He briefly explains 2ns, 1ns, and 3ns in that order, then sketches "a proof that the idea of meaning is irreducible to those of Quality and Reaction" (244). He cites two relevant aspects of EG--first, "a node connecting three lines of identity" is "a triadic idea" (245); and second, "a graph with three tails cannot be made out of graphs each with two or one tail; yet combinations of graphs of three tails each will suffice to build graphs with every higher number of tails" (246). This is what came to be known as his "reduction thesis." Peirce eventually gets back to the Gamma part of EG by invoking Bolzano's analysis that "One collection is more multitudinous than another if, and only if, these can be no relation in which every member of the former collection can stand to a member of the latter collection to which no other member of the former stands in the same relation" (247-8). This is noteworthy because in some of his earlier manuscripts about EG, Peirce explicitly highlights the difficulty of representing one-to-one-relations and collections. He goes so far as to call this "a grave defect in this system of graphs," requiring the "makeshift contrivance" of using the dyadic relation "____ is a character possessed by the individual ____" and the triadic relation "____ is a dyadic relation in which ____ stands to ____" until EG could be "remodelled" to make these "mere formal parts of almost any graph" (LF 1:358-60, 1898; see also LF 1:377-9, c. 1899). Specifically, "a sign for a relation" is needed so that it can serve as the *subject* of a proposition represented by a graph, and none of the signs used in the Alpha and Beta parts qualifies. In particular, "a relation does not possess individuality; and we should be in danger of error if we assumed that it had," so a Beta line of identity cannot denote a relation (LF 2/2:248). Instead, Peirce introduces different colors of heavy lines for qualities and relations, as well as one-letter "selectives" and the special spots that he later calls "potentials" because they denote "logical possibilities"--p or /0\ for a proposition (medad), q or /1\ for a quality or "reference" (monadic relation), r or /2\ for a (dyadic) relation, and s or /3\ for a sign (triadic relation). He also proposes "a dotted enclosure to denote the single character which consists in the logical possibility of the rhema written within it" (249-50; see also 267-9, 361-4, and 370). Again, these innovations likely constitute Peirce's remedy for "a certain fault in the system" that had been bothering him, and he came up with them only upon reexamining EG "from the point of view of the categories" (PPMRT 186, EP 2:176, 1903). Even so, he admits in the Syllabus that he intended to publish in conjunction with the Lowell Lectures, "The Gamma part of the system of Graphs can never be perfected until we have precisely analyzed all the conceptions of logic in terms of the three Categories, together with such other exact conceptions as it may be found necessary to add to those of the Categories. But this is a labor for generations of analysts, not for one" (LF 2/2:365). More than a century later, that additional labor is still needed; but as Ahti-Veikko Pietarinen notes in his introduction to LF 2/2, "it cannot yield a perfect system of rules, given that second-order logic is semantically incomplete" (102), unlike the first-order logic of the Beta part. Regards, Jon On Thu, Oct 16, 2025 at 4:22 PM Jon Alan Schmidt <[email protected]> wrote: > Atila, List: > > Following up on my previous post in this thread, Peirce begins his "slight > sketch" of Existential Graphs (EG) in the entry for "symbolic logic" in > Baldwin's *Dictionary of Philosophy and Psychology* (1902) by describing > what he later distinguishes as the Alpha part for propositional logic. > Rather than thin oval lines as cuts, "to facilitate the printing," he uses > square brackets, parentheses, and braces to enclose different areas; for > example, he represents "if A then B" as [A(B)]. > > > > Upon introducing the line of identity, Peirce does not immediately shift > to the (future) Beta part for first-order predicate logic, where it denotes > an indefinite individual to which general concepts are attributed by > attaching names. Instead, he initially uses a heavy line connecting A and B > to denote a "quasi-instant" at which both propositions are true. This > directly anticipates his Logic Notebook entry of 1909 Jan 7 (R 339:340r > <https://iiif.lib.harvard.edu/manifests/view/drs:15255301$637i>), where > the heavy line represents "circumstances" or "times" when propositions > attached to it are true--a candidate notation for implementing modal logic. > Accordingly, I have suggested in two recent papers (here > <https://revistas.pucsp.br/index.php/cognitiofilosofia/article/view/60449/46975> > and here <https://muse.jhu.edu/article/939654>) that this might be what > he had in mind nearly three years later when he expressed the need "to add > a *Delta* part in order to deal with modals" (R 500, 1911 Dec 6). > > > > Returning to Baldwin's *Dictionary*, Peirce does move on quickly to what > we know today as Beta, although he continues to attach capital letters to > lines of identity instead of names. The only exception is when he briefly > switches to lowercase letters when assigning specific words to them--*l* > for the relation of loving, *m* for man, *w* for woman, etc. He concludes > with the following remarks. > > > CSP: For all considerable steps in ratiocination, the reasoner has to > treat qualities, or collections, (they only differ grammatically), and > especially relations, or systems, as objects of relation about which > propositions are asserted and inferences drawn. It is, therefore, necessary > to make a special study of the logical relatives "____ is a member of the > collection ____," and "____ is in the relation ____ to ____." The key to > all that amounts to much in symbolical logic lies in the symbolization of > these relations. But we cannot enter into this extensive subject in this > article. (CP 4.390) > > > > After some further investigation, I now strongly suspect that this is the > "certain fault in the system" and "vexatious inelegance" that Peirce > mentions in his third 1903 Harvard Lecture (PPMRT 186, EP 2:176)--in the > Beta part of EG that implements first-order predicate logic, heavy lines of > identity *only* denote individuals, such that there is no way to denote > qualities, collections, relations, or abstractions as *subjects* of > propositions. The remedy, which he evidently discovered along with other > "new possibilities of perfectionment" upon reexamining EG "from the point > of view of the categories," was to develop the Gamma part that he > subsequently introduced in his 1903 Lowell Lectures. > > 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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