Hello Jon, Lists, Two things:
1. As you prepare to explain in greater detail what Peirce is doing in this 1880 essay on the algebra of logic, let me ask if you are reading the essay in light of C.S. Peirce's reflections on his father's work on linear associative algebra? In particular, in what ways are you reading the essay in light of his work on algebras dealing with higher dimensions (e.g., systems involving quaternions)? I assume that you are thinking in these terms, but I wanted to check to see if we are on the same page here at the outset of the discussion. 2. My hope is that you are taking up these questions about putting values to the correlates in a triad because you were intending to work on them anyways--and not because I've asked a few questions about what is involved in assigning quantitative values to the triplets you are using in your explanation of triadic relations. The point I was trying to make about triadic relations was meant to be fairly straightforward--and I was not trying to saddle you with projects that might take away from your other work. My point was based on the explanations that Peirce gives in his discussion of dyads and triads in the 1898 "The Logic of Mathematics, an attempt to develop my categories from within." He makes similar points in his discussion of the logic of relatives of 1897 (CP, 3.456) and the later essays on the nomenclature and divisions of dyadic and triad relations. The main idea I was trying to draw out about Peirce's account of relatives, relations and relationships is that both dyadic and triadic relations need to have a certain level of richness in order for quantitative values to be applied in a meaningful way. Heck, we can't even put them in a nominal ordering (e.g., like the arbitrary assignment of numbers to the players on a football team) until things are distinguished as individuals. So, to take an example, a dyad of monads is something that Peirce calls an essential dyad. An example of such a dyad is the relationship of contained and container that holds when scarlet is brought into a dyadic relation with red. It is clear that such relations are pretty simple. In order to have something like a correlate that functions as an individual, where a quantity might meaningfully be applied to the individual, the dyadic relation needs to be richer than an essential dyad, and the same is true of a contingent dyad that is inherential. At the very least, the dyadic relations must be accidental, relational dyads of diversity. It is an interesting question of how we should think about dyads of diversity that are qualitative versus those that are dynamical--and what is needed for such dyadic relations to be ordered in one way or another. After all, material and formal order can only be applied, on Peirce's account, to dyads that are dynamical in their relations between the subjects. One might think that the point I making is really about applying arithmetic quantities to the relate and correlate of a dyadic relation, but that is not the case. The same basic point applies to any ordering where we are treating the relate as first in the relation and the correlate as second. What is more, the same kinds of points can be made about triads. Triads that are monadically degenerate are similar in character to essential dyads, because the correlates of such relations are combined in the way that scarlet and red are related to each other. In the essential triad, scarlet and red are related in terms of both being colors. As with an essential dyad, this triad is really just a relation of container and contained. Dyadically degenerate triads may result from the combination of three essential triads, so the points that were made earlier about dyads holds here. All genuinely triadic relations require a correlate that is general and sufficient to determine the relations between the correlates. For now, I'll largely leave to the side the questions that others have been debating about what is determining what in a genuinely triadic relation, and what it is for one thing to be determined by another. I will, however, make a couple of quick remarks off the cuff: I would think that the kind of determination that we're talking about would depend on whether the genuinely triadic relation is governed by a law of quality, a law of fact, or a representation. After all, only the latter kind of triad is thoroughly general. On Peirce's account, these kinds of triads are thoroughly general because triads involving representations are not a mere matter of law and they are not a mere matter of fact. They are not a mere matter of law because such relations are living, and they are not mere matters of fact because such relations are general. So, to restate the point, relations involving representation don't determine the things that are represented in the way that the laws of fact determine the relations between existing facts, and neither kind of determination is a matter of mere accidental relations of dynamical and productive difference. So, once again, I'd like to point out that we're not just talking about applying arithmetic quantities to the correlates in genuinely triadic relations under a law of fact or to a thoroughly genuine triad under a representation. The question crops up whenever we ask about the ordering of correlates as first, second in any triadic relation. We need to get to level of richness in the subjects that are being brought together, and we need certain kinds of relations to obtain between those subjects in order for one correlate to be ordered as first in the triad, and another to be second and another to be third. So, I'm more than happy to encourage you, Jon, to continue down the path of explaining what is involved in applying quantitative values in an algebraic logic of relations--especially where we are working with a logic that has triadic relatives and where the system has the expressive power of a logic of second intention. It will certainly be of great benefit for me to think more about how conceptions drawn from the mathematics of calculus are being put to work in such logical systems. Having said that, I do want to point out that, on Peirce's classification of mathematics, the calculus belongs to the middle branch that lies between truly continuous systems and those that are finite and discrete. As such, the typical conception of a limit does not, on Peirce's account find its natural home in a truly continuous system of mathematical relations. Systems involving the relations that are embodied in quaternions appear to be different in character. On Peirce's account, these are algebraic representations of spaces that are truly geometric (i.e., genuinely continuous) in character. As such, I'd also be especially interested in hearing more about how you think Peirce is drawing from the mathematics of quaternions and other similarly rich systems of hypotheses involving true continuity in the development of his algebraic logic of relations. --Jeff Jeff Downard Associate Professor Department of Philosophy NAU (o) 523-8354 ________________________________________ From: Jon Awbrey [jawb...@att.net] Sent: Sunday, February 01, 2015 7:32 AM To: John Collier Cc: Peirce List Subject: [PEIRCE-L] Re: Triadic Relations John, List, To follow up on the question of the representamen vs. sign distinction, as far as I can recall Peirce uses that only to distinguish the maximally abstract and general concept of a sign, which he calls a "representamen", from the more special concept of a sign with a mental interpretant, which he calls a "sign", as in the following oft-quoted passage: http://stderr.org/pipermail/inquiry/2004-February/001182.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SOP. Note 1. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 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. | | The triadic relation is 'genuine', that is, its three members are | bound together by it in a way that does not consist in any complexus | of dyadic relations. That is the reason the Interpretant, or Third, | cannot stand in a mere dyadic relation to the Object, but must stand | in such a relation to it as the Representamen itself does. | | Nor can the triadic relation in which the Third stands be merely similar | to that in which the First stands, for this would make the relation of the | Third to the First a degenerate Secondness merely. The Third must indeed | stand in such a relation, and thus must be capable of determining a Third | of its own; but besides that, it must have a second triadic relation in | which the Representamen, or rather the relation thereof to its Object, | shall be its own (the Third's) Object, and must be capable of determining | a Third to this relation. All this must equally be true of the Third's | Third and so on endlessly; and this, and more, is involved in the familiar | idea of a Sign; and as the term Representamen is here used, nothing more | is implied. | | A 'Sign' is a Representamen with a mental Interpretant. | | Possibly there may be Representamens that are not Signs. | | Thus, if a sunflower, in turning towards the sun, becomes by that very act | fully capable, without further condition, of reproducing a sunflower which | turns in precisely corresponding ways toward the sun, and of doing so with | the same reproductive power, the sunflower would become a Representamen of | the sun. | | But 'thought' is the chief, if not the only, mode of representation. | | C.S. Peirce, "Syllabus" (c. 1902), 'Collected Papers', CP 2.274 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o But I have never found this degree of subtlety to serve any purpose in justifying the ways of Peirce to ordinary mortals, so I think it a far better thing we do simply to use the word "sign" in the maximal general sense, and to say "sign with a mental interpretant" when we intend that. Regards, Jon On 2/1/2015 4:00 AM, Jon Awbrey wrote: > John, List, > > The whole sign relation, say L subset of O x S x I, is called a sign > relation. For relations in general, Peirce wrote a k-tuple as x1 : x2 : ... : > xk and called it an elementary relative or an individual relative. Strictly > speaking, relatives are terms denoting, not the objects denoted, so we could > call a triple (o, s, i) in L an elementary sign relation. > > Regards, > > Jon > > http://inquiryintoinquiry.com > >> On Feb 1, 2015, at 3:09 AM, John Collier <colli...@ukzn.ac.za> wrote: >> >> Hi Jon, >> >> What would you call the whole triadic relation in that case? >> >> I have assumed that Peirce introduced 'representamen' to avoid the potential >> confusion, but he isn't consistent by any means. (His care about terminology >> was not always manifested.) I suppose we could use 'sign triplet', being the >> irreducible triplet containing the sign. >> >> What do you think is best? >> >> John >> >> -----Original Message----- >> From: Jon Awbrey [mailto:jawb...@att.net] >> Sent: February 1, 2015 5:48 AM >> To: Peirce List >> Subject: [PEIRCE-L] Re: Triadic Relations >> >> Sung, List, >> >> I think it best to use the word "sign" in a way that relates as naturally as >> possible to its ordinary use. Of course we expect a technical formalization >> of an informal concept to sharpen up the root idea and cast new light on its >> meaning, but we do that all the better to serve the original purpose of >> using that word. >> >> So I can but recommend using "sign" to mean a thing 's' that has an object >> 'o' and an interpretant sign 'i' in an ordered triple of the form (o, s, i) >> that is an element of a sign relation L that is a subset of a cartesian >> product O x S x I, for a given object domain O, sign domain S, and >> interpretant sign domain I. >> >> If you try your suggestion on any other sort of relation, say, the dyadic >> relations indicated by "brother", "father", "mother", I think you will see >> the sort of confusion that would be caused. >> >> Regards, >> >> Jon >> >>> An excellent post, Jon. >>> >>> So, it may be useful to distinguish between two 'signs' (or >>> designations) of the sign -- (i) "the Sign" (capital letter S, as >>> adopted by Edwina) defined as the irreducible set of three elements, >>> object, representamen, and interpretant, and (ii) "the sign" (small >>> letter s) defined as synonymous with the representamen. >>> >>> If we adopt this convention, the following statement would hold: >>> >>> "The Sign is to the sign what a set is to one of its elements." >>> (013015-10) >>> >>> A corollary to Statement (013115-11) would be >>> >>> "Conflating the Sign and the sign is akin to conflating a set and its >>> elements." (013015-11) >>> >>> All the best. >>> >>> Sung >>> >>> >>> On Sat, Jan 31, 2015 at 10:30 AM, Jon Awbrey <jawb...@att.net> wrote: >>> Re: John Collier >>> JC: http://permalink.gmane.org/gmane.science.philosophy.peirce/15541 >>> JC: http://permalink.gmane.org/gmane.science.philosophy.peirce/15549 >>> JC: http://permalink.gmane.org/gmane.science.philosophy.peirce/15557 >>> JC: http://permalink.gmane.org/gmane.science.philosophy.peirce/15565 >>> >>> John, List, >>> >>> Peirce's concept of determination is apt enough if understood in all >>> its implications and ramifications, but it does get some interpreters >>> locked into absolutist, behaviorist, causalist, determinist, >>> dyadicist, essentialist ways of thinking, especially if they are bent >>> that way to begin with. >>> >>> A less narrow path to understanding is through the concept of constraint, >>> especially as used in classical cybernetics and mathematical systems theory. >>> Constraint is present in a system in measure as the set of likely >>> occurrences subsets the set of conceivable occurrences. >>> >>> Constraint, determination, information, and relation are all affairs >>> of sets and systems of elements, not single elements taken out of context. >>> >>> Sets and systems of elements have properties that their member elements do >>> not. >>> That is why it is important to understand a sign relation as a set of >>> triples, not a single triple. Irreducibility, whether compositional >>> or projective, is a property of the set, not of individual triples. >>> >>> Regards, >>> >>> Jon >> -- academia: http://independent.academia.edu/JonAwbrey my word press blog: http://inquiryintoinquiry.com/ inquiry list: http://stderr.org/pipermail/inquiry/ isw: http://intersci.ss.uci.edu/wiki/index.php/JLA oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey facebook page: https://www.facebook.com/JonnyCache
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