James, list, Ben Udell briefly discusses this matter in a *Wikipedia* article on the Categories. He writes:
Part of the justification for Peirce's claim that three categories are both necessary and sufficient appears to arise from mathematical ideas about the reducibility <https://en.wikipedia.org/w/index.php?title=Relation_reduction&action=edit&redlink=1> of *n*-adic relations. According to Peirce's Reduction Thesis, (a) triads are necessary because genuinely triadic relations cannot be completely analyzed in terms or monadic and dyadic predicates, and (b) triads are sufficient because there are no genuinely tetradic or larger polyadic relations—all higher-arity <https://en.wikipedia.org/wiki/Arity> *n*-adic relations can be analyzed in terms of triadic and lower-arity relations. Others, notably Robert Burch (1991) and Joachim Hereth Correia and Reinhard Pöschel (2006), have offered proofs of the Reduction Thesis https://en.wikipedia.org/wiki/Categories_(Peirce) There are several appendices in *A Thief of Peirce: The Letters of Kenneth Laine Ketner and Walker Percy* in which Ketner discusses the valental aspects of Peirce's mathematics and logic; he provides a number of diagrams illustrating the Reduction Thesis. Best, Gary R [image: Gary Richmond] *Gary Richmond* *Philosophy and Critical Thinking* *Communication Studies* *LaGuardia College of the City University of New York* *C 745* *718 482-5690* On Thu, Feb 16, 2017 at 4:55 PM, James Crombie <[email protected] > wrote: > Hello Jerry, Stephen, list : > > > > Somewhere Peirce points out that with combinations of triads one can > produce relations of all other valencies – n-ads for all values of n. This > is not possible with dyads or tetrads. But I think, off hand, that with > pentads one can make a triad… > > > > See the attached image. > > > > Best, > > > > James Crombie > > > > *De :* Jerry Rhee [mailto:[email protected]] > *Envoyé :* 16 février 2017 16:36 > *À :* Jerry LR Chandler <[email protected]> > *Cc :* Stephen C. Rose <[email protected]>; Edwina Taborsky < > [email protected]>; Peirce List <[email protected]> > *Objet :* Re: [PEIRCE-L] Possible Article of Interest - CSP's "Mindset" > from AI perspective > > > > Jerry C, Stephen, list: > > > > When you ask such questions: > > “If triadic thinking is infinitely superior to binary thinking. > > should tetradic thinking be infinitely superior to “triadic" thinking. > > and should pentadic thinking be infinitely superior to tetradic thinking?” > > > > in response to Stephen’s assertion that: > > “more imperative than ever that a way be found to make the triadic mode > more understandable and to say why it is infinitely superior to binary > thinking. Without it we perish. This is NOT an academic matter,” > > > > you are taking the problem to the clowns. That is, the preamble for why > three and not two or four or five or myriad creatures have been addressed > many times over. That is, you are rubbing out the work of great men who > have urged a specific course of action. You do this, in spite of your > clear awareness of Peirce’s work and why three (cf., Letter to Lady > Welby). > > > > Therefore, I must think that you do this not out of immorality but because > you want to bring attention to a particular matter. > > > > So, why three? Why not because there are three parts to the soul? Why > not because a syllogism requires three and only three terms? Why not > because an enthymeme utilizes the structure of syllogism and its quality is > judged by the artfulness by which an audience is engaged to fill in the > incompleteness? > > > > Or, rather simply, why not CP 5.189? > > > > “The comparison should be with famous men; that will strengthen your > case; it is a noble thing to surpass men who are themselves great. > > > > So are the things that continue even after death; those which are always > attended by honour; those which are exceptional; and those which are > possessed by one person alone-these last are more readily remembered than > others.” ~Aristotle, *Rhetoric* > > > > “But I seem to myself to be the sole depository at present of the > completely developed system, which all hangs together and cannot receive > any proper presentation in fragments.” ~ Peirce, *Letter to William James* > > > > Best, > Jerry Rhee > > > > On Thu, Feb 16, 2017 at 1:36 PM, Jerry LR Chandler < > [email protected]> wrote: > > Steven, List : > > > > On Feb 10, 2017, at 8:04 AM, Stephen C. Rose <[email protected]> wrote: > > > > Which makes it more imperative than ever that a way be found to make the > triadic mode more understandable and to say why it is infinitely superior > to binary thinking. Without it we perish. This is NOT an academic matter. > > > > I would contradict your conclusion. > > > > My logic is relatively simple: > > > > Transitivity of meaning is intrinsic to the ordering of the terms. > > If triadic thinking is > > > > infinitely superior to binary thinking. > > > > should tetradic thinking be > > infinitely superior to “triadic" thinking. > > and should pentadic thinking be > > infinitely superior to tetradic thinking?. > > > > Ben has explored the tetradic pathways of thinking. > > I have found it enormously useful to explore the pentadic pathways of > thinking. > > And, occacionally septiadic (7) pathways of speculation. > > > > The deeper question is, > > When is the fullness of the thought expressible in logic terms such that > inference can be made? > > > > From your perspectives, > > do any thoughts exist that can not be expressed in three terms? > > > > The question is, did CSP basically argue that any logical conclusion > required three sentences? > > Or, did CSP basically argue that any logical conclusion required three > connected terms? > > > > Cheers > > > > Jerry > > > > > > ----------------------------- > PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to > [email protected] . To UNSUBSCRIBE, send a message not to PEIRCE-L > but to [email protected] with the line "UNSubscribe PEIRCE-L" in the > BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm > . > > > > > > > > ----------------------------- > PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to > [email protected] . To UNSUBSCRIBE, send a message not to PEIRCE-L > but to [email protected] with the line "UNSubscribe PEIRCE-L" in the > BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm > . > > > > > >
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