Helmut, List, I think the maximum benefit possible at this point is to be gained from studying those simple examples of triadic relations and sign relations that I gave. Once we get used to dealing with small examples like that we can move on to tackling more complex examples on the order of those we might encounter in realistic applications.
The sort of sign relation we normally encounter in practice will be a subset L of a cartesian product O × S × I, where the object, sign, and interpretant-sign domains all have infinitely many members in principle, though of course we tend to get by with finite samples at any given moment and it may even be possible to start small and build capacity over time. All the objects we need to signify in a given application will go into the object domain O and all the signs and interpretant-signs we need to signify these objects will go into the sign domain S and the interpretant-sign domain I. There are such things as monadic projections: proj_O : O × S × I → O proj_S : O × S × I → S proj_I : O × S × I → I For example, proj_O (L) would give all the elements of O that actually occur as first correlates in L, sometimes called the O-range. But I don't get the sense you are talking about that with your idea of projections with two S's and so on. There is however an interesting class of relations that take place internal to the sign domain. These are the syntactic relations or parsing relations that relate a complex sign to its component signs. But again, that's a topic a little ways down the road ... Regards, Jon On 5/8/2017 1:55 PM, Helmut Raulien wrote:
Jon, List, I guess, that the answer to the question, whether or not information is lost due to the projection (of a triadic relation towards three dyadic ones), depends on whether the triadic relation is intransparent, and thus casts three shadow-like pictures, or is sort of transparent, and thus casts three X-ray-like pictures. Another problem is, that, as a relation may be upon one set, meaning between a set and a copy of the same set, then there are not only three, but six dimensions. But I guess, that only the representamen/sign can have a reasonable relation with itself, so perhaps there are only four dimensions: S,S,O,I. Or: S,s,O,I, or sign, representamen, object, interpretant, with sign and representamen being both identical and different in some mysterious (Re-entry?), yet to be elaborated, way... Or has Peirce already solved this mystery, but did not explain it, assuming that it is self-explaining? This hunch is based on the fact, that he used the term "sign" for both the whole and a part. With which matter we have problems, but maybe he did not, because for him, the solution to what for us feels like a paradoxon, felt somehow for granted? Best, Helmut 08. Mai 2017 um 16:40 Uhr *Von:* "Jon Awbrey" <[email protected]> Peircers, Here's the revised, extended edition of my last two posts on this thread: https://inquiryintoinquiry.com/2017/05/06/icon-index-symbol-%e2%80%a2-16/ Icon Index Symbol • 16 ====================== Questions Concerning Certain Faculties Claimed For Signs Re: Peirce List Discussion JAS:https://list.iupui.edu/sympa/arc/peirce-l/2017-04/msg00196.html HR:https://list.iupui.edu/sympa/arc/peirce-l/2017-04/msg00200.html Having lost my train of thought due to a week on the road, I would like to go back and pick up the thread at the following exchange: JAS: “To be honest, given that the Sign relation is genuinely triadic, I have never fully understood why Peirce initially classified Signs on the basis of one correlate and two dyadic relations.” HR: “I have a guess about that: I remember from a thread with Jon Awbrey about relation reduction something like the following: A triadic relation is called irreducible, because it cannot compositionally be reduced to three dyadic relations. Compositional reduction is the real kind of reduction. But there is another kind of reduction, called projective (or projectional?) reduction, which is a kind of consolation prize for people who want to reduce. It is possible for some triadic relations.” The course of discussion after that point left a great many of the original questions about icons, indices, and symbols unanswered, so I’d like to make another try at addressing them. The relevant facts about triadic relations and relational reduction can be found at the following locations: • Triadic Relations : Examples from Mathematics http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation#Examples_from_mathematics • Triadic Relations : Examples from Semiotics http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation#Examples_from_semiotics • Relation Reduction : Projective Reducibility of Triadic Relations http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction#Projective_reducibility_of_triadic_relations I introduced two examples of triadic relations from mathematics, two examples of sign relations from semiotics, and used them to illustrate the question of projective reducibility, in another way of putting it, whether the structure of a triadic relation can be reconstructed from the structures of three dyadic relations derived or “projected” from it. • In the geometric picture of triadic relations, a dyadic projection is the shadow that a 3-dimensional body casts on one of the three coordinate planes. • In terms of relational data tables, a dyadic projection is the result of deleting one of the three columns of the table and merging any duplicate rows. • In Peircean terms, a projection is a type of “prescision” operation, abstracting a portion the structure from the original relation and ignoring the rest. The question then is whether we keep or lose information in passing from a triadic relation to the collection of its dyadic projections. If there is no loss of information then the triadic relation is said to be reducible to and reconstructible from its dyadic projections. Otherwise it is said to be irreducible and irreconstructible in the same vein. Regards, Jon
-- inquiry into inquiry: https://inquiryintoinquiry.com/ academia: https://independent.academia.edu/JonAwbrey oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey isw: http://intersci.ss.uci.edu/wiki/index.php/JLA facebook page: https://www.facebook.com/JonnyCache
----------------------------- 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 .
