Jeff, I’ll take a crack at it, inserting my answers after your questions.
Gary f. -----Original Message----- From: Jeffrey Brian Downard [mailto:jeffrey.down...@nau.edu] Sent: 4-Jan-16 19:37 Hello, I'd like to follow up on the post that Gary F. made some weeks back about the first two pages in NDTR. Let me focus on two paragraph that are found on the second page of the essay in the EP: Triadic relations are in three ways divisible by trichotomy, according as the First, the Second, or the Third Correlate, respectively, is a mere possibility, an actual existent, or a law. These three trichotomies, taken together, divide all triadic relations into ten classes. These ten classes will have certain subdivisions according as the existent correlates are individual subjects or individual facts, and according as the correlates that are laws are general subjects, general modes of fact, or general modes of law. (CP 2.238) There will be besides a second similar division of triadic relations into ten classes, according as the dyadic relations which they constitute between either the First and Second Correlates, or the First and Third, or the Second and Third are of the nature of possibilities, facts, or laws; and these ten classes will be subdivided in different ways. (CP 2.239) How do these claims fit together? For the time being, let's set aside the two systems of classification, and let's focus on the relations themselves. In CP 2.238, Peirce describes triadic relations between the first, second and third correlates. GF: Well, that’s what he does in the entire essay, based on the principle that the correlates are numbered first, second and third in order from simplest to most complex. Specifically in 238, he first considers each of the correlates monadically. For instance, in the case of a sign, considering it monadically (i.e. regardless of its relations to object or interpretant) gives us one trichotomy (later designated as qualisign/sinsign/legisign). Considering the object (second correlate) monadically would give us another trichotomy, and considering the interpretant (third correlate) monadically would give us a third trichotomy. (I say “would” because Peirce does not, in his later focus on sign relations, actually consider either the object or the interpretant monadically.) JD: In 2.239, he describes three dyadic relations between the correlates. How are the three dyadic relations connected to the triadic relation or relations? GF: As Peirce says in 239, every triadic relation constitutes three dyadic relations. In the semiotic case these would be sign-object, object-interpretant and sign-interpretant. In other words, each of these can be considered dyadically, and that would give us another set of three trichotomies, since dyadic relations (like monads) can be of the nature of possibilities, facts, or laws. (Only one of those trichotomies, icon/index/symbol, is actually given in NDTR.) JD: In the opening remarks in the discussion of triadic relations in "The Logic of Mathematics; an attempt to develop my categories from within," he says the following about the connections between the dyadic and the triadic relations: Each of the three subjects introduces a dyad into the triad, and so does each pair of subjects. (CP 1.472) How should we understand his claims about the manner in which the dyads are being introduced into the triad? GF: It should be clear enough that each pair of subjects “introduces” a dyad into the triad (following the order of involution or analysis) because the triad includes three pairs, each of which can be considered dyadically (as per CP 2.239). I think the reason that each subject introduces a dyad into the triad is that each of its three subjects must have a dyadic relation to the dyad which is the other two considered as a “unit”. CP 1.471 gives some context to this, including a couple of examples which may be helpful: 471. We come now to the triad. What is a triad? It is a three. But three what? If we say it is three subjects, we take at the outset an incomplete view of it. Let us see where we are, remembering that logic is to be our guide in this inquiry. The monad has no features but its suchness, which in logic is embodied in the signification of the verb. As such it is developed in the lowest of the three chief forms of which logic treats, the term, the proposition, and the syllogism. The dyad introduced a radically new sort of element, the subject, which first shows itself in the proposition. The dyad is the metaphysical correlative of the proposition, as the monad is of the term. Propositions are not all strictly and merely dyadic, although dyadism is their prominent feature. But strictly dyadic propositions have two subjects. One of these is active, or existentially prior, in its relation to the dyad, while the other is passive, or existentially posterior. A gambler stakes his whole fortune at an even game. What is the probability that he will gain the first risk? One half. What is the probability that he will gain the second risk? One fourth; for if he loses the first play, there will be no second. It is one alternative of the prior event which divides into two in the posterior event. So if A kills B, A first does something calculated to kill B, and then this subdivides into the case in which he does kill B and the case in which he does not. It is not B that does something calculated to make A kill him; or if he does, then he is an active agent and the dyad is a different one. Thus, there are in the dyad two subjects of different character, though in special cases the difference may disappear. These two subjects are the units of the dyad. Each is a one, though a dyadic one. Now the triad in like manner has not for its principal element merely a certain unanalyzable quality sui generis. It makes [to be sure] a certain feeling in us. [But] the formal rule governing the triad is that it remains equally true for all six permutations of A, B, C; and further, if D is in the same relation at once to A and B and to A and C, it is in the same relation to B and C; etc. This leads directly to your quoted sentence: “Each of the three subjects introduces a dyad into the triad, and so does each pair of subjects.” (CP 1.472) Anyway, that’s how I understand it!
----------------------------- 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 the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
