Robert, List: I sincerely appreciate your response, which confirms what I suspected--your mathematical lattice approach *can *be extended to any number of trichotomies, but only *after *they are arranged in the proper order for applying the "rule of determination" to classify signs. That remains controversial, especially for the three interpretants, and indeed must ultimately be established by community consensus. Fortunately, we can set it aside in *this *thread, which is focused on the two objects, since we agree that the proper order of the four associated trichotomies is Od → Oi → S → Od-S. As I said before, Peirce himself clearly affirms this in his December 1908 draft letters to Lady Welby, but some of his examples seem inconsistent with it.
By the way, we could replace S with R in the last two trichotomies--as I have acknowledged previously, Peirce indeed describes a sign as a *species *of representamen in his 1903 writings about speculative grammar. However, in the 1908 letters, he consistently uses "sign" and *never *mentions "representamen," having decided by then that the two terms are effectively synonymous after all. "I use 'sign' in the widest sense of the definition. It is a wonderful case of an almost popular use of a very broad word in almost the exact sense of the scientific definition. ... I formerly preferred the word *representamen*. But there was no need of this horrid long word" (SS 193, 1905 July). RM: In the case of "beauty," I agree with him that the first occurrence is a Qualisign and the second is a Rhematic Symbol, like all words in the language. Just to clarify, agree with whom? Again, if the *quality *of beauty serves as a sign of some *other *possible quality, then it is indeed a tone (qualisign). However, Peirce's problematic example is the *word *"beauty," which *cannot *be a tone--like every word, it is a symbolic seme (rheme) and thus must be a type (legisign), whose instances (replicas) are tokens (sinsigns). That is why I remain puzzled by his claim that it is an abstractive, which can only be a descriptive, a tone, and an icon. Are there any passages in Peirce's writings where he *explicitly *classifies an ordinary word as a tone (or qualisign or potisign)? Where you say, "the first occurrence is a Qualisign ... like all words in the language," are you perhaps suggesting that *every *word is a tone the very first time that someone hears or reads a token of it? If so, then is the word "beauty" also an abstractive upon that initial encounter, with a quality as its dynamical object? If so, then what exactly is that quality, and how does the word *iconically *represent it? Thinking along those lines is reminiscent of Peirce's explanation that a "proper name, when one meets with it for the first time, ... is *then*, and then only, a genuine Index," becoming "an Icon of that Index" the next time and "a Symbol" once someone has "habitual acquaintance with it" (CP 2.329, EP 2:286, 1903). However, he offers no such qualification when he classifies the word "beauty" as an abstractive and a descriptive. RM: The answer would therefore be that "beauty," as a representation of a Quality incarnate, is a replica of Iconic Legisign and that, as a word, it is a replica of Rhematic Symbol. There is nothing intrinsically beautiful about the *word *"beauty," so it does not *iconically *represent the quality of beauty, even as embodied in something. An instance of the word "beauty" *does *iconically represent the corresponding type (NEM 3:887, 1908 Dec 5), but as a token that embodies "a definitely significant Form," not as a tone that is *itself* an "indefinite significant character such as a tone of voice" (CP 4.537, 1906). RM: In the example "All S is P," which is a Dicisign, i.e., either a Dicent Indexical Legisign or a Dicent Indexical Sinsign, there are therefore two possibilities. Actually, the proposition "All S is P" is a *symbolic *pheme (dicent) type--although its instances are indexical pheme tokens--and Peirce straightforwardly classifies it as a copulant accordingly. The problem is that he does not say the same about "Some S is P," instead claiming (twice) that it is a descriptive, which is not feasible for *any *pheme or indexical sign because *every *descriptive is an iconic seme. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Mon, Sep 29, 2025 at 3:49 AM robert marty <[email protected]> wrote: > Atila, Jon, Edwina, Jack, List, > > It is easier to reach an agreement when we can rely on mathematical > structures that are indisputable because they are well-defined and > well-documented. That is why I continue with other, more developed > structures. Jon recalls the well-known divisions of the object O into Od > and Oi and of the Interpreter I into Ii, Id, and If, as with the relations > of determination that they interrelate. In this regard, I am not going to > resurrect the recurring debates we have had in the past about the order of > the three interpretants, since the structural extensions I propose do not > depend on their respective names, because, *a priori*, they are > considered only as letters of algebra. I have not changed my mind, and I > assume the same is true for you. These are questions that are settled at > the level of communities of scholars, who, according to Peirce, define all > science, which plunges us into the social sciences, historically dated to > the precise moment when we are discussing them. > > The only way, it seems to me, to clarify the debates with a view to > facilitating their convergence towards a possible community agreement, is > to begin by distinguishing between what belongs to the structure and what > belongs to the experience from which it is derived by hypostatic > abstraction. This is what is known, as everyone knows—even if they have > their own ideas about it—as modeling. This is the rule in the physical > sciences, where we can see that debates most often boil down to conflicts > between competing mathematical models in the same field of experience. It > is the scientific community that, more or less provisionally, decides in > favor of one model or another. There is no reason to proceed differently in > other sciences whenever the possibility exists, and this is the case for > Peirce's semiotics, since I propose a mathematical model directly inspired > by his MS. This is what he expresses in the opening lines of his 1903 > Syllabus, 5th Lecture, by distinguishing between *“a priori” *and *“a > posteriori”*: > > The principles and analogies of Phenomenology enable us to describe, in a > distant way, what the divisions of triadic relations must be. But until we > hav met with the different kinds *a posteriori, *and have in that way > been led to recognize their importance, the *a priori *descriptions mean > little;—not nothing at all, but little. Even after we seem to identify the > varieties called for *a priori *with varieties which the experience of > reflection leads us to think important, no slight labor is required to make > sure that the divisions we have found *a posteriori *are precisely those > that have been predicted *a priori. *In most cases, we find that they are > not precisely identical, owing to the narrowness of our reflectional > experience. It is only after much further arduous analysis that we are able > finally to place in the system the conceptions to which experience has led > us. In the case of triadic relations, no part of this work has, as yet, > been satisfactorily performed, except in some measure for the most > important class of triadic relations, those of signs, or representamens, to > their objects and interpretants. (CP 2.233) > > As early as 1902, he wrote that: > > One might equally argue *a priori *in favor of the Truth. For suppose > there is not any proposition which is correct independently of what is > thought about it. Then, if there be any proposition which nobody ever > thinks incorrect, it is as correct as possible and has all the truth there > is. Consider, then, the proposition: "This proposition is thought by > somebody to be incorrect." Now, if it is, in fact, thought by somebody to > be incorrect, then it is true. For that is precisely the statement. But if > it is not thought by anybody to be incorrect, it has all the truth > possible, if there is no truth independent of opinion. Here, then, is a > proposition which is correct whether it is thought to be so or not. > Therefore, there is such a thing as a proposition correct whatever may be > opinions about it. But when we come to study logic, we shall find that all > such *a priori *arguments, whether *pro *or *con, *about positive fact > are rubbish. This question is a question of fact, and experience alone can > settle it. (CP 2.137, 1902) > > Everything Peirce writes, up to and including 2.242, falls within “*the a > priori*” and concerns only triadic relations. 2.242 contains the > definitions of Representamen and also of Sign: “A *Sign *is a > Representamen of which some Interpretant is a cognition of a mind. Signs > are the only representamens that have been much studied.” A Sign is > therefore a “specified” Representamen, i.e., “a species distinguished > within a more general genus.” This means that everything that can be said > about triadic relations can be said about Signs. In this regard, I would > like to issue a warning concerning the wording of the definition that > Peirce gave in the second lecture (CP 2.274, EP2: 272, 3rd section of the > Syllabus, MS 478), which begins with “*A Sign, or Representamen*,” as it > may suggest that the two terms are synonymous, since the distinction “A *Sign > *is a Representamen with a mental Interpretant” appears 15 lines later. > Furthermore, “mental Interpretant” is a specification of “cognition of a > mind” that leaves room for quasi-minds and disciplines such as > zoosemiotics, phytosemiotics, LLM, and many others. All this to say that I > continue in *a priori* by working solely with triadic relations in order > to argue that the lattice of ten classes of abstract triadic relations that > I obtain *a priori*, as a theorem of relational algebra, is applicable, *“a > posteriori”*, to the signs of social life. > > So I remain *a priori* by writing Od → Oi→ R → Ii→ Id → If, where the > letters are just letters and the arrows are abstract concatenable morphisms > of a transitive relation (Representamens), while Od → Oi → S→ Ii→ Id → If > are species of these Representamens, still *a priori*. On the other hand, *a > posteriori*, the letters are the names of elements of the real world > (exterior world and interior world, see CP 5.474) as you yourself have > specified them, and the arrows represent relations of determination between > these elements, which are concatenated because the verb “to determine” is a > transitive verb. It remains to clarify what Peirce means by “determine.” He > does so in the item you cite: “We thus learn that the Object determines > *(i.e., > renders definitely to be such as it will be**) the Sign in a particular > manner.” *(CP 8.361). It is customary to call these signs “hexadic.” > > In an article published in Semiotica and available online[1] > <https://www.deepl.com/fr/translator#_ftn1>, I showed (using Category > Theory) that the 28 classes of hexadic signs and the possible 66 classes of > decadic signs were also structured by lattice structures (which can be > generated automatically by an application designed by Patrick Benazet[2] > <https://www.deepl.com/fr/translator#_ftn2>). However, like all my work > using this theory, it received very little attention from the community. > That is why, in my current work, I have adopted a method modeled on the one > Peirce uses in the Syllabus, which I am finalizing for triadic signs (Part > 1 and Part 2) and then generalizing to hexadic signs and possible decadic > signs, a generalization that could have been made in 1903, incidentally. > > [1] <https://www.deepl.com/fr/translator#_ftnref1> Marty, Robert. “The > trichotomic machine” Semiotica, vol. 2019, no. 228, 2019, pp. 173-192. > https://doi.org/10.1515/sem-2018-0084 > > [2] <https://www.deepl.com/fr/translator#_ftnref2> > http://patrick-benazet.chez-alice.fr/treillis_en_ligne/lattices/ > > · Here are the steps for triadic signs: > - Obtaining the ten classes of signs (Part 1) > - Normalized definition of affinities between classes (without using > adjacencies in a diagram): > - Study of the properties of the affinity relation, which leads to the > construction of a lattice structure (Part 2). > Here is a diagram of this structure in a more modern presentation: > > [image: Une image contenant texte, capture d’écran, nombre, Police Le > contenu généré par l’IA peut être incorrect.] > > I note that in this diagram, the concepts of normalized affinity and > adjacency of rectangles coincide perfectly, rendering the use of Peirce's > triangle diagram in CP 2.264 obsolete. > > · For hexadic signs: > · Generalization of the concept of normalized affinity between classes. > · Verification of properties and completion (with the timely assistance of > AI), leading to the lattice structure of the following 28 classes: > > [image: Une image contenant texte, capture d’écran, Rectangle, carré Le > contenu généré par l’IA peut être incorrect.] > > > > I maintain that, as with triadic signs, this lattice is the Grammatica > Speculativa of these signs. > > > > · For any decadic signs, we can do the same, but is it really > necessary? Peirce writes the following about them: > > On these considerations, I base a recognition of ten respects in which > Signs may be divided. I do not say that these divisions are enough. But > since every one of them turns out to be a trichotomy, it follows that in > order to decide what classes of Signs result from them, I have 310, or > 59,049, difficult questions to carefully consider*; and therefore I will > not undertake to carry my systematical division of Signs any farther, but > will leave that for future explorers*".(EP: 482) [highlighted by me] > > We are all explorers of Peirce's future. Personally, in this capacity, I > would say that two conditions are necessary to reduce the 59,049 questions > to just 66: > · there must be community agreement on the order of the trichotomies, > · this agreement must extend to nine concatenated relations between the > elements of these trichotomies. > > To date, none of this is a given; my opinion is that it is better to leave > this to future explorers and that it is better to focus on exploiting the > previous results (which I have begun to do). In any case, if agreement is > ever reached, the grammar of these signs will be available! > > Do the lattices of the 10 or 28 classes provide the right answers to the > latest questions posed by JAS? > > In the case of “beauty,” I agree with him that the first occurrence is a > Qualisign and the second is a Rhematic Symbol, like all words in the > language. Where can they meet? In the inner world, they are both rhematic; > in the outer world, the first is incarnated in objects, the second governs > replicas. We therefore look at the possibilities for each of them in the > lattice: > > · Qualigns are incarnated in Iconic Sinsigns, which Iconic Legisigns can > govern, but > > · on the one hand, Rhematic Symbols embody Rhematic Indexical Legisigns, > which govern Rhematic Indexical Sinsigns, which embody Iconic Sinsigns; on > the other hand, they embody Iconic Legisigns, which govern Iconic Sinsigns > (this can be read in CP 2.265). > > · The answer would therefore be that “beauty,” as a representation of a > Quality incarnate, is a replica of Iconic Legisign and that, as a word, it > is a replica of Rhematic Symbol. > > In the example “All S is P,” which is a Dicisign, i.e., either a Dicent > Indexical Legisign or a Dicent Indexical Sinsign, there are therefore two > possibilities. Indeed, we see in the lattice that Dicent Indexical > Legisigns govern certain Dicent Indexical Sinsigns. Is this observation > sufficient to answer your question? > > As for your question about the possibility of extending the concept of > Speculative Grammar, I believe I have answered it in the affirmative above. > > Best regards, > > Robert Marty > Honorary Professor > PhD Mathematics ; PhD Philosophy > fr.wikipedia.org/wiki/Robert_Marty > *https://martyrobert.academia.edu/ <https://martyrobert.academia.edu/>* >
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