“Theatre and Embodied Collective Reflexivity” published

The final version of “ Theatre and Embodied Collective Reflexivity” has now been published by the Journal of Dramatic Theory and Criticism. You can download it from the link above (the published version, by permission). The abstract reads:

Theatre is a distinctive type of social reflexivity. The question is how, like other social practices, theatre’s social function shapes its form. By examining theatre in relation to social ontology, the components of agency, and the requirements of an embodied form of collective reflexivity, one finds that theatre is characterized by a double homology with social ontology. Furthermore, conducting embodied collective reflexivity requires the performer/character distinction, which is fundamental to theatrical performance. Social ontology, then, sharpens our concept theatre’s structure and the meaning of its practice.

Book update

The book has undergone significant revisions due to various factors. Some are the same forces that have dogged the project from the start: unexpected topics arose, sometimes required me to invade other fields, others resulting in significant rethinking. But most significantly, an external reader (the infamous Reader 2) for my forthcoming article in the Journal of Dramatic Theatre and Criticism wanted me to discuss postdramatic theater. Doing so in the already tight space I was allowed for the article was out of the question, and I don’t have any personal interest in that genre, but Reader 2 had a point: postdramatic theater, at least in some of its versions, rejects fiction or seeks situations in which it’s unclear whether the activity on stage is fictional or real. Since my argument makes fiction a necessary part of theater, clearly I needed to contend with postdramatic theater’s project. Doing so required more space than I anticipated, ultimately turning into a full chapter plus part of another. With that much attention, postdramatic theater became more or less on a par with online role-playing. The book’s title needed to change in order to reflect the alteration. So the new working title is The Question of Theatre: Online Role-playing, Postdramatic Theater, and Social Ontology. The new title also has the benefit of emphasizing the scope of the project, as the previous one (Online Role-playing and the Question of Theater) risked making the topic seem narrow; the new title will, I hope, attract more readers. But — spoiler alert — they won’t be happy with some of the things I say (or more precisely, say about Lehmann’s theory), so I’ll attract a lot of pissed-off readers.

I write this post as COVID-19 is still in the early stages of ravaging New York City, where I live. It’s conceivable that I might not live to see this project come to fruition. Unfortunately it is nowhere near completion: a chapter, the introduction and the conclusion all have yet to be drafted, and the existing chapter drafts are all very rough: material will move, some topics are but waved at, I need to do a second round of research, various issues need more thorough analysis, the writing needs sharpening — the list goes on. It would be a pity, at least to me, for these years of research, analysis and writing to go to naught. So if I become seriously ill, I may set up a post to publish automatically after a certain time, and make what I’ve written publicly available, warts and all. Unfortunately it’ll only be up as long as the website is paid for, but one takes what one can get.

The Place of Materiality in Peirce’s Thought

Peirce made contradictory statements about the place of materiality in the categories. At one point he declared matter to be a Second (Essential Peirce, vol. 1, 297). Elsewhere he described “the permanence of mass, momentum, and energy” as a type of Thirdness (Essential Peirce, vol. 1, 279), a position that reappears later (Essential Peirce, vol. 2, 186–87). There’s a logic to both stances. But on another occasion, Peirce wrote more extensively:

We not only have an immediate acquaintance with Firstness in the qualities of feelings and sensations, but we attribute it to outward things. We think that a piece of iron has a quality in it that a piece of brass has not, which consists in the steadily continuing possibility of its being attracted by a magnet. In fact, it seems undeniable that there really are such possibilities, and that, though they are not existences [i.e., Seconds], they are not nothing. They are possibilities, and nothing more. But whether this be admitted or not, it is undeniable that such elements are in the objects as we commonly conceive them. (Essential Peirce, vol. 2, 269; Peirce’s italics)

His emphasis on possibility suggests a position similar to critical realism, which identifies three ontological domains: the real, the actual, and the semiosic. The domain of the real consists of causal powers and generative mechanisms as they exist transfactually, whether or not their powers are being exercised; they exist as possibilities, capacities and tendencies. Powers and generative mechanisms are ontologically anterior to their actual interactions—their cause-and-effect relations—which produce the events that constitute the domain of the actual. Finally, we perceive and interpret some portion of the actual: such interpretations compose the semiosic domain.1  Strictly speaking, the domain of the real includes not just material entities, but also ideas, since they too have causal powers, a point that accords with Peirce’s analysis that the sign is “a power . . . to determine some interpretant” (CP 1.542; Peirce’s italics). Thus the domains parallel Peirce’s categories, and might be equivalent to what Peirce called “universes” (Essential Peirce, vol. 2, 478-79; but cf. 435, which seems to drive his universes in an idealist direction). These parallels justify my position that materiality should be considered a First.

But why is the inconsistent among Peirce’s statements? One reason could be that his views evolved. But another is that context matters, so it could be he has different aspects of materiality in mind during the various instances he addressed the matter. From this perspective, consider his case for the concept of “real possibility.” He gives the example of a diamond that had formed but then had been destroyed before ever being pressed by anything hard: was it hard or not? Clearly, a diamond is a diamond whether it’s been tested for hardness or not, because a diamond has a certain set of properties, so if it had instead been tested, it would have been hard (Essential Peirce, vol. 2, 354-57). He concludes,

what does that behavior [of a material substance] consist in, except that if a substance of a certain kind should be exposed to an agency of a certain kind, a certain kind of sensible result would ensue, according to our experiences hitherto. As for the pragmaticist, it is precisely his position that nothing else than this can be so much as meant by saying that an object possesses a character. He is therefore obliged to subscribe to the doctrine of a real Modality, including real Necessity and real Possibility. (Essential Peirce, vol. 2, 357; Peirce’s italics)

The diamond’s hardness is transfactual—testing it for hardness is contingent. In other words, whether or not the diamond is actually tested, as a real entity, if it were tested it would exhibit its powers and tendencies, including the hardness it had when it wasn’t tested (see also CP 1.422). So as something bearing certain possibilities which may or may not be tested, it is a First. Otherwise his use of the word “possibility” would be meaningless or misleading. But should a particular condition or circumstance arise, it would behave in such-and-such a way, as a natural necessity. It follows a law, which is a Third.

I’m uncertain how to parse this, but my but guess at the moment (I’m sure Peirce scholars have explored it thoroughly) stems from Peirce’s distinction between two “generals” (generalities): qualities and laws. I’m inclined to view the qualities qua Firsts as static, whereas law-like behavior is the tendency (Thirdness) that regulate its actions or behaviors (Secondness). There are also probabilities that are rooted not in material qualities but in other ones: if you repeatedly flip a coin, the probability is that you’ll increasingly approach an equal number of heads and tails, which I suppose is the result of the coin’s formal properties (having two flat sides makes it far more likely the coin will land on a side rather than the slim edge). But material’s aspects as both First and Third would seem to leave Secondness out of the picture, except insofar as it’s a degenerate Third, perhaps insofar as one doesn’t examine its behavioral tendencies.

Peirce, Trichotomies, and Emergence

In this post, I’m going to amplify my comment in the previous post that Peirce’s trichotomies are a type of stratified emergence.

In “A Guess at the Riddle” and “The Categories Defended,” Peirce sets forth a number of arguments defending his use of three categories: (1) Firstness, Secondness, and Thirdness are distinctive, such that Secondness isn’t simply a matter of two monads, and Thirdness isn’t just a combination of dyads; (2) anything higher (Fourthness, Fifthness, etc.) can be reduced to a combination of Thirdness; (3) looking the other direction, that Firstness and Secondness can’t be collapsed into Thirdness; yet at the same time, (4) Secondness contains an element of Firstness, and Thirdness contains an element of both Secondness and Firstness, and in this relation of containment, the lower categories are not present as mere parts, but as necessary parts.

I will not review Peirce’s categories and the arguments above; in this post I’m focusing instead in this conjunction of qualitative distinctions, irreducibility, and necessary containment. It is remarkably similar to the modern concept of emergence, according to which a new entity (property, structure, etc) develops out of preexisting, more basic ones. The emergent entity has causal powers that could not be predicted from its constituents and cannot be reduced to them. It depends on its constituents for its existence, but it’s also capable of “acting back” on them, such as by setting boundary conditions or controlling their activity (downward causation).1 New powers mean new “levels” of entities: emergence leads to the stratification of reality. The most basic is the physical stratum, which has its own strata, e.g., sub-atomic particles, atoms, molecules, and four fundamental forces. From the physical stratum emerged the biological stratum. Biology depends on the workings of the physical stratum, but it also does new things, like reproduce itself and evolve. From the realm of biology emerged creatures able to interpret signs in their environment and produced among themselves; humans are the most sophisticated of these species, but it is scarcely alone in possessing some form of semiosic capacity. But the semiotic relationship—meaning per se—isn’t reducible to biology, it’s a different kind of power. (Obviously reductionists disagree.)

That example is diachronic, but once the entity has emerged, its stratification continues synchronically: semiosis still depends on and contains a biological substrate and biology still depends on and contains physical processes as long as that creature lives. Emergence doesn’t only apply to entities (ontically); as I’ve argued previously, it also characterizes ontology, as conceptualized in critical realism.2 The domain of the real is that of powers and generative mechanisms existing as substantive but (so to speak) static possibilities or potentialities. When they interact, concrete events and manifestations arise, constituting the domain of the actual. Of these, a subset are perceived, experienced or detected, which for Bhaskar constitute the empirical domain. He adjusted that later to the subjective domain, but I argue that it should be conceptualized more broadly (and less anthropocentrically) as the semiosic domain.

The three domains clearly parallel Peirce’s three categories (see e.g. CP 1.23-24, 1.325, 8.216). More, Peirce conceptualized the three categories as having a line of dependency that is fully consonant with emergence:

Thirdness as such . . . can have no concrete being without action, as a separate object on which to work its government, just as action cannot exist without the immediate being of feeling on which to act. (Essential Peirce, vol. 2, 345)

The categories, then, describe an emergent relationship.

However, Peirce’s categories more emphatically incorporate action and process than critical realism (and the latter’s is far from weak). I say this because Peirce constantly underscores the active, processual, evolutionary character of semiosis. This is undoubtedly best known through his argument that every sign becomes—or better, has the potential to become—the object of a new sign, which in principle means it could spawn an infinite process of semiosis, growing in complexity, specificity, vividness, generality, or in some other manner:

Symbols grow. They come into being by development out of other signs, particularly from likenesses or from mixed signs partaking of the nature of likenesses and symbols. We think only in signs. These mental signs are of mixed nature; the symbol-parts of them are called concepts. If a man makes a new symbol, it is by thoughts involving concepts. So it is only out of symbols that a new symbol can grow. (Essential Peirce, vol. 2, 10)

Another processual element resides in what Peirce called the logical interpretant: a habit-change, consisting of “a modification of a person’s tendencies toward action,” and also such things as a habitual association of ideas or a way of thinking, which we might describe as actions of thought (CP 5.476-78). This is one more instance of dynamism and transformative elements in his thought, one in which a rule itself may changes and semiosis can ultimately reenter the body. Nor is all of it necessarily conscious:

The action of thought is all the time going on, not merely in that part of consciousness which thrusts itself upon attention, but also in those parts that are deeply shaded and of which we are too little conscious to be much affected by what takes place there. (Essential Peirce, vol. 2, 23)

Today we’d reject the idea that what happens in the subconscious doesn’t affect us, but my point is the depth to which semiotic activity goes.

But Peirce’s signs (particularly symbols) don’t only grow through their activity: they also ramify. Saussure’s sign-structure, the simple signifier + signified combination, is static. The only differences among them that one might point to are as parts of speech (nouns, verbs, etc), but these involve no structural distinctions in the Sr/Sd relationship: the relationship was solely an arbitrary, conventional one (Peirce’s symbols). In contrast, from the very first Peirce had three distinct sign-types (icon, index, symbol), arising from different types of relationship between the sign and object, relationships that were themselves defined by his three categories. Over the years Peirce realized that this taxonomy, although satisfactory for many purposes, was in logical or scientific terms too unrefined, and he identified ten categories, which in his later years he expanded to 66.

As I explained in my previous post, the methodology behind these distinctions is governed by the trichotomies. They operate as an engine of ramification, and they do so because the relationship between them is emergent and stratified. In the two paragraphs following his argument that symbols grow, he writes:

we may liken the indices we use in reasoning to the hard parts of the body, and the likenesses we use to the blood: the one holds us stiffly up to the realities, the other with its swift changes supplies the nutriment for the main body of thought. . . . The reasoner makes some sort of mental diagram by which he sees that [a] conclusion must be true, if the premise is so; and this diagram is an icon or likeness. (Essential Peirce, vol. 2, 10)

Thus in the reasoning process, icons have a special weight or function, a point that melds well with cognitive science’s findings concerning image schemas. (See also CP 2.170.) The symbol produced at one point in thought changes the quality of its sign-structure (or the emphasis of its quality) from conventional to iconic—and, moreover, to a potential, as it is not inevitable that a new step in reasoning will occur.

In addition, when he developed the ten sign categories, Peirce realized that the icon-index-symbol distinction was already too concrete and actual, in fact a Second, and he would have to back up to identify consisting of a triad of semiotic Firstness, a set of thoroughly qualitative potentialities. Thus in his derivation of the ten categories, the icon-index-symbol triad came second, and the next was determined by Thirdness (consisting of rhemes, dicents and arguments). But note that because the three triads are distinguished by the trichotomy of Firstness, Secondness, and Thirdness, the structure of ten sign-types is itself stratified and emergent: they have a necessary order, a line of dependency, and because they are qualitatively different they cannot be reduced to any of the others.

For these reasons, I hold that Peirce’s system of trichotomy provides a method for analyzing emergent entities, and perhaps of emergence as such.

However, two additional points. First, the fact that Fourthness, Fifthness etc can be resolved into combinations of Thirdness implies that the growth of symbolic thought will remain at the level of symbols, and not produce a novel type of sign. This is particularly significant when considering reflexivity as recursion: purely discursive (especially linguistic) reflexivity loops within semiosis, although it may reach a final interpretant in the form of a habit-change. (There’s an ethical point about reflexivity there!) Of course, nevertheless within the level of Thirdness, there can be (and is) increasing degrees of complexity.

Second, Peirce didn’t explore the fact that because signs are triune—a form of Thirdness—it’s possible to back up further to find the forms of Firstness and Secondness preceding the semiotic relationship. This is what I am doing in analyzing agency, which is rooted in our material being. But agency itself is triune (embodied, causal, and intentional), and as agency, it is a Second. The Firstness preceding it, I’ll hypothesize, is the natural one I discussed above: physics, biology, semiosis (biosemiosis, including non-human semiosis). Perhaps there is a triad preceding even that, but that notion might lead us into “triads all the way down,” an infinite regression of who knows what sort.

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.