Jon S., List,
My strategy for interpreting these passages is to take Peirce at his word when he refers to the triadic relations that are involved. In order to interpret "μ is the surrender by A of B" and "ν is the acquisition by A of D" as triadic and not merely dyadic relations, my hunch is that he is considering these actions as intentional in character. The object surrendered and the agent who surrenders it are existing individuals in the relation of agent and patient, but that existential description of the individuals is part of an intentional action by A. As a general sort of thing, the intention makes the action of surrendering triadic in character--and so too with the action of A acquiring object D. The case that you cite of an object being sold involves a transfer of money and a contract. The simpler case of exchange as barter with no contract is illustrative of how other kinds of relations may be involved when more general things, such as property laws and legal systems, are governing the intentional acts. As a historical point, it is reasonable to suppose that social conventions governing exchanges by barter developed prior to any contracts or legal systems. Consequently, I think that the proper analysis of every genuine triadic relation involves a correlate that, itself, has the character of a general rule. As a correlate, that intention, or property law, or what have you, may involve a general rule that is part of a larger system of rules (such as a legal system). Having said that, the reason the number of triadic relations involved in tetradic, pentadic and higher order relations goes up by a power of 10 is not obvious to me. While it isn't obvious, here is a conjecture: Peirce may be thinking about the operation of general laws and intentions as conforming to a general model that applies to all genuinely triadic relations. One such model is articulated in "The Logic of Mathematics, an attempt to develop my categories from within". In that account of genuinely triadic relations, the law of quality and most general laws of fact each involves three clauses. The first clause governs each correlate considered in itself. The second clause governs the dyadic relations between pairs of correlates. The third clause governs the triadic relations between the three correlates. It is possible that the operation of the three clauses involved in such law might multiply the number of relations that may be involved in tetradic, pentadic, sextadic (etc.) relations by a power of ten in each case. The long explanations that he provides in this essay of the triadic relations that are part of the laws of space and the laws of physics may be illustrative of this general pattern. Generalizing on these points, I think that the principles of logic that govern self-controlled acts of inference are, similarly, parts of larger systems of logical rules. Something as straightforward as the rule governing the first figure of the syllogism (the nota notae) is a rule that is related, as part of a larger logical system, to the principles of identity, non-contradiction, excluded middle, etc. The systematic connections that hold between the underlying laws of logic are probably richer and deeper than anything we are able to express in our little theories of logic (logica utens or logica docens). As a result, the analysis of the triadic relations that are involved in a symbolic argument must take into account the relations that hold between the guiding principle of the inference and the other rules that are essential to inferences of that form. Peirce is offering an outline of how this might work in his explanation of the three clauses that are part of the general law of logic. In order to explain how our conceptions of the laws of logic--conceived as principles that govern self-controlled acts of reasoning--might have evolved in the human species, Peirce considers the simple case of a child learning that hot stoves should not be touched. In "Faculties", "Consequences" and "Further Consequences," Peirce starts with the child forming a more or less deliberate intention to touch the stove. When the child learns that it is not possible for him to hold his hand on the stove, he learns that his parents were correct and his supposition was incorrect. As such, it looks like there is a strategy that may be at work, which is to explain how relatively simpler cases of intentional actions might later give rise to new conceptions--including a conception of the leading principle that governs the type of inferences that were involved in the learning about the stove. Consequently, I think that some clarity could be achieved by applying the analysis of the triadic relations that are involved in progressively more complicated tetradic, pentadic, sextadic, etc. relations to simple examples--such as that of a child learning how to engage more or less self-controlled patterns of logical reasoning. My assumption is that the child was already capable of thinking in a manner that conformed to the laws of logic from an early age. The instinctive patterns of inference were not subject to much self-control at the ages of 1 and 2, but the child was learning how to use a conventional system of symbols (i.e., a natural language) as a matter of habit. In time, what the child learned was how to represent those laws to himself as principles. In turn, the child learned to recognise what those principles, functioning as imperatives, might require of him in terms of the future conduct of his inquiry. How many triadic relations are involved in this process of a young child learning? Well, it appears to grow according to a power law. As such, it grows into a multitude that exceeds any system of numbers that is numerable or even any system that is abnumerable. The upshot of what I am suggesting is that Peirce's observation that there may be a power law involved in richer relations would explain his earlier assertions about the sort of infinity and resulting continuity that is involved in the growth of our cognitions. Yours, Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: Jon Alan Schmidt <jonalanschm...@gmail.com> Sent: Friday, May 10, 2019 6:40 PM To: peirce-l@list.iupui.edu Subject: Re: [PEIRCE-L] Triadic and Tetradic relations Jeff, List: That passage by Peirce is quite a head-scratcher. For one thing, the relations of surrendering and acquiring are clearly dyadic, rather than triadic. For another, it seems obvious that just as any triadic relation involves exactly three dyadic relations, likewise any tetradic relation involves exactly four triadic relations. In the case of "A gives up B to C in exchange for D," they would be "A gives B to C," "C gives D to A," "A exchanges B for D," and "C exchanges D for B." The difference is that a tetradic relation is always reducible to the combination of its constituent triadic relations, while a genuine triadic relation is irreducible. Of course, "A gives B to C" is a paradigmatic example of a genuine triadic relation; so it is irreducible to the combination of its constituent dyadic relations--"A surrenders B," "C acquires B," and "A benefits C" (cf. CP 6.323; 1908). I wonder if what Peirce had in mind as the result of analyzing a tetradic relation were not ten triadic relations, but ten relations of any lower adicity. Besides the four triadic relations, there are six dyadic relations involved in any tetradic relation--e.g., "A surrenders B," "C surrenders D," "A acquires D," "C acquires B," "A trades with C," and "B is traded for D." However, such an approach would still not translate to the alleged "power law" for relations of increasing adicity--a pentadic relation involves five tetradic, ten triadic, and ten dyadic relations for 25 total relations, rather than 100; a hexadic relation involves 6+15+20+15=56 relations, rather than 1,000; and an enneadic relation involves 9+36+84+126+126+84+36=501 relations, rather than 1,000,000. By the way, Peirce elsewhere gave a different but (in my view) equally puzzling analysis of what amounts to the very same tetradic relation. CSP: Suppose a seller, S, sells a thing, T, to a buyer, B, for a sum of money, M. This sale is a tetradic relation. But if we define precisely what it consists in, we shall find it to be a compound of six triadic relations, as follows: 1st, S is the subject of a certain receipt of money, R, in return for the performance of a certain act As; 2nd, This performance of the act As effects a certain delivery, D, according to a certain contract, or agreement, C; 3rd, B is the subject of a certain acquisition of good, G, in return for the performance of a certain act, Ab; 4th, This performance of the act Ab effects a certain payment, P, according to the aforesaid contract C; 5th, The delivery, D, renders T the object of the acquisition of good G; 6th, The payment, P, renders M the object of the receipt of money, R. (CP 7.537; no date) Why introduce so many additional subjects, rather than sticking with the four in the initial proposition? Is an essential element somehow omitted if we simply analyze selling as a combination of giving and exchanging? Regards, Jon Alan Schmidt - Olathe, Kansas, USA Professional Engineer, Amateur Philosopher, Lutheran Layman www.LinkedIn.com/in/JonAlanSchmidt<http://www.LinkedIn.com/in/JonAlanSchmidt> - twitter.com/JonAlanSchmidt<http://twitter.com/JonAlanSchmidt> On Fri, May 10, 2019 at 4:05 PM Jeffrey Brian Downard <jeffrey.down...@nau.edu<mailto:jeffrey.down...@nau.edu>> wrote: List, In a draft of a 1905 letter to Lady Welby, Peirce analyzes the tetradic relationship of A gives up B to C in exchange for D (Semiotic and Significs, 190). I am interested in his remarks about the exchange of goods for the sake of better understanding his account of the relations that hold between signs, objects and interpretants. Peirce argues that any tetradic or higher order relationship (i.e., valency >3) is complex and can be analyzed into elementary monadic, dyadic and triadic relations. Here is the upshot of his analysis of exchanging goods: The tetradic relationship is reducible to, at most, 10 elementary triadic relations. Here they are: a. λ is an exchange of property (B and D) between A and C b. ι is a transposition of ownership of B and D c. L is an accomplishment of λ through ι d. μ is the surrender by A of B e. m is the surrender by C of D f. Μ<https://en.wikipedia.org/wiki/%CE%9C> is the performance of μ in reciprocal consideration of m (Note the error in the transcription, which has M instead of m at the end.) g. ν is the acquisition by A of D h. η is the acquisition by C of B i. Ν<https://en.wikipedia.org/wiki/%CE%9D> is the performance of μ in reciprocal consideration of η (Note the error in the transcription, which has N instead of η at the end.) j. L is carried out by the union of M and N. On the basis of this type of analysis, Peirce generalizes to relationships of higher adicity. He claims there is a power law that holds for relations of adicity four or greater. 1. Tetradic relations such as exchanging appear to be reducible to, at most, 10 triadic relations. 2. Pentadic relations are reducible to, at most, 100 triadic relations. 3. Hexadic relations are reducible to, at most, 1,000 triadic relations. 4. Enneadic relations are reducible to, at most, 1,000,000 triadic relations. I'd like to ask two questions. First, what is the basis of this claim concerning the power law that seems to govern higher-order relations? Second, how does this analysis apply to Peirce's paradigmatic case of giving--offered as an example of a genuinely triadic relation. Following the analysis of exchange offered above, giving is reducible to, at most, three triadic relations. A gives B to C: a. μ is the surrender by A of B b. η is the acquisition by C of B c. Μ<https://en.wikipedia.org/wiki/%CE%9C> is the performance of μ with the intent of bringing about η I'd like to flag two points that can be made about this analysis. First, I'd like to note that the performance of μ with the intent of bringing about η has the implied condition that, if C does not accept B as a gift, then A may reassert ownership. Second, each of the relations might be understood as having the following condition: _____ in accordance with property laws R in legal system S. My hunch is that Peirce is, in this analysis, ignoring the relevance of the property laws that would, typically, govern exchanges and gifts. Let me end by restating the question posed above: is this a fair analysis of the more elementary triadic relations that are involved in a genuinely triadic relation, such as giving? Yours, Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354
----------------------------- 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 .
