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/>* Le sam. 27 sept. 2025 à 17:34, Jon Alan Schmidt <[email protected]> a écrit : > Helmut, List: > > If the *quality *of beauty (or whiteness or mass) *itself *serves as a > sign of some *other *possible quality, then it is obviously a > qualisign/potisign/tone, so it can *only *be an abstractive and a > descriptive--there is nothing problematic about this. However, Peirce > states that the *word *"beauty" is an abstractive, even though (like all > words) it is a legisign/famisign/type, which--according to what he says > elsewhere in the very same text--can *only *be a collective and a > copulant. > > Another puzzling example is when he says (twice) that while the universal > proposition "Any S is P" is a copulant, the particular proposition "Some > S is P" is a descriptive (CP 8.357&361, EP 2:486&488). The only difference > between them is the *quantification *of the subject, and Peirce sometimes > even labels the trichotomy for the immediate object as > vague/singular/general accordingly. However, this is plainly inconsistent > with *every *proposition being either indexical or symbolic as firmly > established in his 1903 taxonomy, and thus either a sinsign/actisign/token > or a legisign/famisign/type, such that it *cannot *be a descriptive, only > a designative or a copulant (CP 8.361&367, EP 2:488-9). > > Further working out these kinds of later developments in Peirce's > speculative grammar is not merely a matter of terminology, unless every > sign having two objects (and three interpretants) has no relevance to "the > actuality of semiosis." I wonder, can Robert's mathematical lattice > approach be extended to the resulting additional trichotomies and sign > classes? Might it shed some helpful light on these apparent inconsistencies > in Peirce's initial classifications of signs according to "the Mode of > Being of the Dynamical Object," "the Mode of Presentation of the Immediate > Object," "the Mode of Apprehension of the Sign itself," and "the Relation > of the Sign to its Dynamical Object" (CP 8.344, EP 2:482)? > > Regards, > > > Jon Alan Schmidt - Olathe, Kansas, USA > Structural Engineer, Synechist Philosopher, Lutheran Christian > www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt > > On Sat, Sep 27, 2025 at 8:47 AM Helmut Raulien <[email protected]> wrote: > >> Supplement: Fur colours/ whiteness might count, that e.g. a white >> orchid´s colour for an insect is of a different colour (ultraviolet ith >> black stripes?). For mass, I don´t know. Was Peirce wrong by subsuming it >> under abstractives, or has it something to do with as well being effete >> mind? >> 26. September 2025 um 21:19 >> "Helmut Raulien" <[email protected]> wrote: >> Jon, List, >> >> I think, there´s on one hand the possible meaning of a token of a >> necessitant type, its immediate interpretant, and on the other hand an >> abstractive, whose type is not a necessitant, but a possible. "Beauty" on >> first sight seems like of a necessitant type, like a copulant, because >> beauty commonsensely exists. But beauty does not exist in any explicit >> place, because it always may be, that what one person regards for >> beautiful, another person does not. So there is a chance, that there is >> nothing everybody would find beautiful. Meaning: It is not clear, that >> beauty exists, it merely is possible. That´s why Peirce says, beauty is an >> abstractive, I´d say. >> >> Best, Helmut >> > _ _ _ _ _ _ _ _ _ _ > ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. 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