Cécile, List
I present here, in the most condensed form possible, the merits of a purely algebraic formalization of Peirce's semiotics, entirely indexed to the history of its development. *How do we distinguish the correlates of a triadic sign?* *How do we formalize the triadic sign?* This question arises because the definition of a triad, strictly speaking, implies no a priori distinction between the elements it links together. If you represent them by letters, you're surreptitiously introducing lexicographical order and by numbers, the order of natural integers. This is why I draw attention to an important warning Peirce gives about "First" and "Second" in a footnote to the Syllabus in Part III (EP 2, selection 20): **The conceptions of a First, improperly called an "object," and of a Second should be carefully distinguished from those of Firstness or Secondness, both of which are involved in the conceptions of First and Second. A First is something to which (or, more accurately, to some substitute for which, thus introducing Thirdness) attention may be directed. It thus involves Secondness as well as Firstness; while a Second is a First considered as (here comes Thirdness) a subject of a Secondness. An object in the proper sense is a Second.* ( EP 2: 271) This warning should shed light on the following definition of the Sign (which, in my opinion, is far from the best) on page 272: *A Sign, or Representamen, is a First which stands in such a genuine triadic relation to a Second, called its Object, as to be capable of determining a Third, called its Interpretant, to assume the same triadic relation to its Object in which it stands itself to the same Object.* ( EP 2: 272) On the other hand, the definition given in the fifth version of the Syllabus (EP 2, selection 21), which is much more precise, will avoid confusion: *Representamen **is the First Correlate of a triadic relation, the Second Correlate being termed its Object, and the possible Third Correlate being termed its Interpretant, by which triadic relation the possible Interpretant is determined to be the First Correlate of the same triadic relation to the same Object, and for some possible Interpretant. A Sign is a representamen of which some interpretant is a cognition of a mind*. (EP 2: 290 or CP 2.242 ) This is the version Jon Alan Schmidt has chosen to formalize the sign and semiosis as a spiral. This definition is not that of an ordinary triad since one correlate, the First, has the power to determine another, the Third, to make it the new First of a new triad, conferring on it the same power, and so on. Hence, the image of the spiral, of which Jon Alan Schmidt provides a projection on a plane, requires raising this projection in the reader's mind. The repeated addition of the notation S ïI is intended to capture this "power" and, in fine, "*to capture the idea that the sign mediates between the object and interpretant*." But Peirce himself did much better to achieve this. Once again, I must point out that Peirce modified his definitions of the Sign by introducing, around 1904-1905, the determination of the sign S by the object O. Anybody can consult the list of 76 definitions I published in 1990: it's available on Peirce.org. There's no need to mention Existential Graphs, which require a considerable intellectual investment, especially for non-expert readers. Perhaps the most straightforward is this one (emphasis added): *A sign may be defined as something (not necessarily existent) which is so determined by a second something called its Object that it will tend in its turn to determine a third something called its Interpretant in such a way that in respect to the accomplishment of some end consisting in an effect made upon the interpretant the action of Sign is (more or less) equivalent to what that of the object might have been had the circumstances been different. *(n° 36 - v. 1906 - MS 292. Prolegomena to an Apology for Pragmaticism) The most explicit about the creation of the triadic relation due to the concatenation of the two determinations is the following: *I define a Sign as anything which on the one hand is so determined by an Object and on the other hand so determines an idea in a person's mind, that this latter determination, which I term the Interpretant of the Sign, is thereby mediately determined by that Object. A sign, therefore, has a triadic relation to its Object and to its Interpretant. *(n° 47 bis – 1908 - Letter to Lady Welby in CP 8.343 ). It's clear, then, that the composition of the two determinations gives rise to the triadic relation for Peirce. That's why I've underlined "therefore." Consequently, the formalization is simplified considerably, without any loss of information, by : O à S à I The arrows represent determinations, and this diagram reads: O determines S, which determines I Referring to the Peircean conception of a determination: *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, 1908) We can see that O determines I by transitivity. Peirce verified this in MS 611 (Nov. 1908). This diagram has the considerable advantage of being equivalent to the mathematical object below: Schematic representation of a category with objects *X*, *Y*, *Z* and morphisms *f*, *g*, *g* ∘ *f*. <https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/200px-Commutative_diagram_for_morphism.svg.png> (click) It's an algebraic category, the simplest there is (non-trivial). This one is the archetypal example of a category on the Wikipedia site devoted to this part of mathematics, which emerged in the second half of the 20th century (Category theory - Wikipedia <https://en.wikipedia.org/wiki/Category_theory>). In 1977 (in French) and 1982 (in English), I was able to use it to generate, in just a few pages, not only classes of triadic signs but also, above all, to show that these classes are naturally organized in a lattice structure (which Peirce had intuited in the form of affinities). I've verified that Peirce knew about this type of structure, but limited by set theory, he couldn't obtain it formally. In his classification of the Sciences, this lattice occupies the place of the *Grammatica Speculativa*. It's his ultimate form. Robert Marty Honorary Professor ; PhD Mathematics ; PhD Philosophy fr.wikipedia.org/wiki/Robert_Marty *https://martyrobert.academia.edu/ <https://martyrobert.academia.edu/>*
_ _ _ _ _ _ _ _ _ _ ARISBE: THE PEIRCE GATEWAY is now at https://cspeirce.com and, just as well, at https://www.cspeirce.com . It'll take a while to repair / update all the links! ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the body. More at https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
