Peirce’s Recursive Trichotomies

In some of my work I’ve written that Peirce’s concept of trichotomies designates a recursive system or methodology.1 Since I haven’t seen trochotomies described this way elsewhere (although it sometimes seems implied), my claim needs to be supported and explained more fully.

Peirce’s 1903 Harvard lecture, “The Categories Defended,” is the key document for my argument.2 He begins by briefly defining his three foundational categories: Firstness, Secondness, and Thirdness. He writes,

Category the First . . . is a Quality of Feeling.

Category the Second . . . is Reaction as an element of the Phenomenon.

Category the Third . . . is Representation as an element of the Phenomenon. (EP 2.161)

Firstness, he continues, is so rudimentary that is has no weakened or “degenerate” form. (His term “degenerate,” I gather, is from conics; a better term today might be “subsidiary.” Nevertheless I’ll use his terminology.) However, Secondness does have a degenerate form (in this text his distinction is merely between strong and weak Secondness; he has better discussions elsewhere). And Thirdness, for its part, has two degenerate forms. “The most degenerate Thirdness is where we conceive a mere Quality of Feeling, or Firstness, to represent itself to itself as Representation. Such, for example, would be Pure Self-Consciousness” (EP 2.161).

In what evidently is an effort to explain the phrase “represent itself to itself,” Peirce takes a curious turn of thought. He imagines a map laid on the soil of a country. The map is so perfectly and infinitely detailed that

the map itself will be portrayed in the map, and in this map of the map everything on the soil of the country can be discerned, including the map itself with the map of the map within its boundary. Thus there will be within the map a map of the map, and within that a map of the map of the map and so on ad infinitum. (EP 2.162)

He goes on to call this image “the precise analogue of pure self-consciousness.” The image, a mise en abyme, is an example of recursion, the application of a process or structure upon itself. One of its outcomes is self-similarity, in which a part is similar to the whole, and so it often embeds other instances of itself (e.g., a sentence can contain another sentence). In mathematics recursion is essential to numerous concepts and formulas. In geometry recursion produces fractals—shapes that are self-similar at every level of scale—which have garnered considerable attention since the 1970s partly because they accurately model various natural phenomena, such as the growth patterns of certain algae and the branching patterns of fern leaves and various trees. In human culture, recursion has figured for centuries and even millennia, expressed especially in images of self-containment (such as microcosms, the matryoshka doll and the mise en abyme), and also in spirals and other designs.

Branching is the most common type of natural and geometrical recursion (i.e., fractal). Simple branching rules apply to every branch in the same way: for instance, ordinary dichotomous recursion has the rule, “each branch generates two new branches.” One can also have a “single branch” rule: for instance, a spiral is a curve that branches with a slightly larger curve of the same shape, and the mise en abyme likewise recurses with an image of itself in itself. A recursive method could also be “the right-hand branch generates two branches, but the left only generates one.” Although this would look rather lopsided, it’s completely valid.

Peirce’s self-representing map prefaces his further discussion of Thirdness:

The relatively degenerate forms of the Third category do not fall into a catena, like those of the Second. What we find is this. Taking any class in whose essential idea the predominant element is Thirdness, or Representation, the self-development of that essential idea . . . results in a trichotomy giving rise to three subclasses, or genera, involving respectively a relatively genuine thirdness, a relatively reactional thirdness or thirdness of the lesser degree of degeneracy, and a relatively qualitative thirdness or thirdness of the last degeneracy. This last may subdivide, and its species may even be governed by the three categories, but it will not subdivide in the manner which we are considering by the essential determinations of its conception. The genus corresponding to the lesser degree of degeneracy, the reactionally degenerate genus, will subdivide after the manner of the second category forming a catena, while the genus of relatively genuine Thirdness will subdivide by trichotomy just like that from which it resulted. Only as the division proceeds, the subdivisions become harder and harder to discern. (EP 2.162-63)

In trichotomous recursion, then, Firstness is too rudimentary to have a degenerate form, so it cannot subdivide; Secondness subdivides in two, a genuine and a degenerate form; and Thirdness subdivides into three forms, one genuine, one “reactionally degenerate” (a type of Secondness), and the other, he soon says, is “qualitatively degenerate” (a type of Firstness). Since “the division proceeds,” these are branching rules, which I depict below. The dashes between numbers indicates whether a result is degenerate or genuine; for instance, 3-1 means the degenerate Firstness of a Third.

He provides his own diagram of the application of this trichotomous subdivision through three recursions, which I approximate:

Looking left to right, a line representing Firstness branches off with another line of Firstness. Next, a line representing Secondness splits in two; the left side is its degenerate aspect of Firstness, and its right side is its genuine Secondness, which follows the rule of recursive branching by splitting in two again. The right-hand branch represents Thirdness, which divides into three. Its left side is its degenerate Firstness, which has no further subdivision. Its middle division is its degenerate Secondness, which in its next recursion divides in two. And the line on the right, representing genuine Thirdness, divides in three. As Peirce says, this method of division can be applied again and again: trichotomy designates recursive branching.

Rotating Peirce’s diagram and using the node marking I employed to describe the rules, we get the following diagram:

Let’s pause for a moment. In ordinary dichotomous recursion, each branch generates two branches, with no special difference between them, and everything looks very balanced. Not so with Peirce’s trichotomies—the recursion is complex and “imbalanced.” Conceivably the rule for trichotomous recursion could have been like the one for dichotomous recursion: each branch sprouts three new branches, end of story. Instead, one branch spawns three branches, another produces two, and the last has just one. The reason is that each branch has a different nature, and the three branches are related and have a line of dependency. While Secondness cannot be reduced to Firstness, there must be Firstness for Secondness to exist, and so Secondness has to have Firstness within it. Similarly for Thirdness: Peirce is keen to argue that triads cannot be reduced to dyads, but it is also clear that dyads must appear within triads; and thus Thirdness possesses an aspect of Secondness and so Firstness as well. This is an instance of stratified emergence: something arises from some level of reality, possessing new characteristics that cannot be reduced to the level from whence it came, and yet they also depend on that lower level. (Which is why I’m interested in it: Peirce’s trichotomy is highly analogous to Bhaskar’s three domains, the three aspects of agency, and more.)

Peirce next applies the concepts of the categories and their degenerate forms to semiotics, tying Firstness to icons, Secondness to indexes, and Thirdness to symbols, and then elaborating on the iconic element of indexes and the iconic and indexical elements of symbols. He adds, significantly, that “The Symbol, or relatively genuine form of Representamen, divides by trichotomy into the Term, the Proposition, and the Argument” (EP 2.164).

I don’t know for certain, but these three concepts appear to have been an early foray into developing new trichotomies. Later that same year, in “Nomenclature and Divisions of Triadic Relations, as Far as They Are Determined,” he presented the theory that there are three trichotomies: (1) Qualisigns, Sinsigns, and Legisigns; (2) Icons, Indexes, and Symbols; and (3) Rhemes, Dicents, and Arguments. He then explains that these three trichotomies produce ten classes of signs, in the following order:

I: Qualisigns
II: Iconic sinsigns
III: Rhematic indexical sinsigns
IV: Dicent indexical sinsigns
V: Iconic legisigns
VI: Rhematic indexical legisigns
VII: Dicent indexical legisigns
VIII: Rhematic symbols
IX: Dicent symbols
X: Arguments

But how did he produce them? Why not nine classes? Or for that matter, twelve? Peirce gives us logical explanations (e.g., EP 2.272-86, 291-97), but they’re rather awkward (well, much of his writing is awkward). However, the classes are quite straightforwardly generated if the trichotomies are applied recursively and diagrammatically, as demonstrated by the following figure—which, again, follows what Peirce himself drew:

The Roman numerals in the diagram correspond to the ten classes above. Peirce drops superfluous terms when he discusses the classes (for example, instead of “Rhematic iconic qualisigns,” he refers to “qualisigns” since the rest is implicit); he identifies them with all three terms using a triangle (2.294-96). Notably, he discusses them in the exact sequence that the recursive methodology would suggest, from bottom to top.

Peirce used three trichotomous recursions (i.e., three trichotomies) to identify ten classes of signs. But why only three recursions? Why not three recursions of three recursions? This seems in fact to have been his line of thought just a few years later, when he wrote that there are “ten main trichotomies of signs” (EP 2.486). If three recursions leads to ten classes of signs, then three recursions of three recursions, for a total of ten recursions (hence ten trichotomies) would mean there are 66 types of signs—which was his claim (EP 2.481).

In the end, it doesn’t matter whether Peirce used such diagrams, worked out the trichotomous relationships in his head, or developed some other method of proceeding: the use of recursion accurately explains his results and exposition sequence. For that reason, it can be a valuable method for analyzing the implications of trichotomies in other areas where they apply.

Addendum (1 Nov 2018)

Further evidence supporting my theory of Peirce’s use of trichotomous recursion appears in a passage I came across in one of Peirce’s letters to Lady Welby:

the six trichotomies, instead of determining 729 classes of signs, as they would if they were independent, only yield 28 classes; and if, as I strongly opine (not to say almost prove) there are four other trichotomies of signs of the same order of importance, instead of making 59,049 classes, these will only come to 66. (EP 2.481)

Where did these numbers come from? The letter itself leaps into speaking of six trichotomies. We do however know that three trichotomies produce ten classes of signs. We’ll have to try some math. Since Peirce always thinks in terms of three, if we repeatedly divide 729 by 3, we find that it’s equal to 36 (3 to the power of 6). He says that in addition to the six classes, there are four more, and so a total of 10 classes: 310 (3 to the power of 10) equals 59,049. Evidently, the number of classes that the trichotomies would yield “if they were independent” equals 3 to the power of the number of classes. Backtracking, then, a single trichotomy would be 31 = 3, two trichotomies would be 32 = 9, three trichotomies would be 33 = 27 … if they are independent. But they aren’t. Is there a quick way to calculate the correct numbers?

There is, of course. The hint appears in the triangle in which he places his original ten classes of signs (2.294-96). There is something called a triangle number, which is the sum of the numbers from 1 to n. It’s easy to visualize as a dot, with two dots under it, three dots under that, etc., however far you want to go (your n), an image that looks like a triangle. If you write that as a mathematical function called T(n), T(1) = 1, T(2) = 3, T(3) = 6, T(4) = 10, and so forth. The formula is pretty simple: T(n) = (n*(n+1))/2. For example:

T(4)
= (4*(4+1))/2
= (4*5)/2
= 20/2
= 10

But there’s a problem: three trichotomies make ten classes, but T(3) = 6: to get 10, you need T(4). Why?

Consider it this way: a trichotomy isn’t three things, but one thing with three aspects. In particular, a sign is a single thing with three parts (sign/representamen, object, interpretant). That’s before you start to determine any classes. Each round of determination is a trichotomous recursion. The sign, then, is one thing with zero recursions represented by one dot, and its first recursion is the row of two dots underneath, which yields three dots — the three original sign types. So n recursions is equal to the triangle number n+1. As a formula, if R(n) is the result of n recursions, then R(n) = T(n+1) = ((n+1)*(n+2))/2. For three recursions:

R(3)
= ((3+1)*(3+2))/2
= (4*5)/2
= 20/2
= 10

Using this modified triangle number formula, we find that three trichotomies is the same as three recursions, resulting in the ten types of signs, as described above.

So if there are six trichotomies, then, instead of 36 (=729), we use R(6):

= ((6+1)*(6+2))/2
= (7*8)/2
= 42/2
= 28

Through recursion, six trichotomies yields 28 classes, just as Peirce says; and if you plug in the numbers, ten trichotomies yields 66 classes. Moreover, the reason he goes out to ten trichotomies is that the original three trichotomies recursively yields 10 larger classes.

Endnotes

^1 I make this assertion in “Embodied Collective Reflexivity: Peircean Performatives,” Journal of Critical Realism 16.1 (2017): 43-69; and in “Theater and Embodied Collective Reflexivity,” currently under review.

^2 In Charles S. Peirce, The Essential Peirce: Selected Philosophical Writings, ed. Nathan Houser and Christian J. W. Kloesel (Bloomington: Indiana University Press, 1992), hereafter cited as EP. Italics in quotations are Peirce’s.