Generative grammar for I Ching divination (Part 6): Linguistics II

(link to part 5)

Feature structure

Generative grammarians love using features to represent linguistic objects. This is a basic methodology in both transformational-generative grammar and other, nontransformational generative frameworks (most typically in head-driven phrase structure grammar). Features are usually put in square brackets in an [attribute: value] format.

 Example   That dog is a singular noun can be represented by the feature [number: singular] and that ran is a past-tense verb by the feature [tense: past]. Features can be grouped into sets, called feature structures, such as

[person: third, number: singular, gender: masculine]

which represents the pronoun he. This way of representing properties shouldn’t be too alien; it’s also used in many other fields. For instance, the CSS code for my  Example  icon is

{font-family: serif; font-variant: small-caps; background-color: GoldenRod;}

which is essentially also a feature structure.

The divinatory properties associated with I Ching lines and line slots can be represented by features. Recall from Parts 3 and 4 that there are four types of lines in the I Ching, which are defined by two properties: yin/yang (i.e., the line’s gender) and old/young (i.e., the developmental stage the line represents).

 Example   The old yang line (⚊◯) can be represented as

[gender: yang, stage: old]

and the young yin line (⚋) as

[gender: yin, stage: young]

A hexagram is just a more complex feature structure—with an additional feature indicating each line’s position—such as

[
  [position: 1, gender: yin, stage: young],
  [position: 2, gender: yang, stage: young],
  [position: 3, gender: yang, stage: old],
  [position: 4, gender: yin, stage: old],
  [position: 5, gender: yang, stage: young],
  [position: 6, gender: yin, stage: old]
]

which represents ䷯ (ching ‘the well’ 井) with three changing lines (1, 3, 4). Since old lines are changing by definition, there’s no need for an extra transformation feature [changing].

Now, suppose there’s a transformation function, which takes a hexagram as input and returns another hexagram as output. It would convert the above feature structure to a new one with

1. the gender values of all old lines toggled, and
2. all stage features deleted.

transform(䷯) =
[
  [position: 1, gender: yin],
  [position: 2, gender: yang],
  [position: 3, gender: yin],
  [position: 4, gender: yang],
  [position: 5, gender: yang],
  [position: 6, gender: yang]
]
= ䷅

This is just the aforementioned sung ‘conflict’ (訟), which by the way isn’t too good a change.😔

Further exploitation of the featural tool

Agreement

An important functionality of features in generative grammar is to model the widespread grammatical phenomenon of agreement. In transformational-generative grammar this is done via a pair of “valued” and “unvalued” features.

 Example   Nominal phrases have valued person and number features while tensed verbs have unvalued ones. By copying over the person/number values from the former to the latter we can derive the subject-verb agreement phenomenon. Thus, in he sings the pronoun he bears

[person: third, number: singular]

and the tensed verb sing bears

[person: [], number: []]

We copy the two values from he onto sing, and the latter gets pronounced as sings, which reflects the agreement process.

There’s no feature-copying or overt pronunciation in I Ching divination, but some sort of feature-based agreement does exist. We’ve already seen this in Part 4, where I mentioned that one of the criteria for an auspicious line was the agreement between its gender and the gender of its slot. This can be modeled by a boolean-valued function

agree(line gender, slot gender) = True | False

Thus, a yin line in a yang slot isn’t good because

agree(yin, yang) = False

We can evaluate the overall degree of auspiciousness of a hexagram by counting the number of Trues returned by the agreement process, though we should probably count in other factors like the cross-line relationships as well if we want to get an accurate evaluation (more on this later).

 Example   Applying the agreement function to the six lines in ䷓ (kuan ‘view’ 觀), we get three Trues and three Falses.

agree(line1, slot1) = [gender(line1) == gender(slot1)]
= [yin == yang] = False
agree(line2, slot2) = [yin == yin] = True
agree(line3, slot3) = [yin == yang] = False
agree(line4, slot4) = [yin == yin] = True
agree(line5, slot5) = [yang == yang] = True
agree(line6, slot6) = [yang == yin] = False

However, since two out of the three Trues occur in central slots (i.e., they are perfectly central), the auspiciousness level of the hexagram is actually much higher than average. Next I turn to represent this state of affairs in featural terms.

Unification

We can let the feature structure of a line and that of its insertion slot go through a unification process similar to that in unification-based grammars like HPSG.

Unification is a conjunction-like operation that integrates information from compatible feature structures. The following examples are adapted from Shieber (2003). Example 1:
[category: noun] ⨆ [agreement: [number: singular]] =
[category: noun, agreement: [number: singular]]
Example 2: [agreement: [number: singular]] ⨆
[agreement: [person: third]] =
[agreement: [number: singular, person: third]]

Now let’s do two things:

1. Suppose the “gender agreement” in I Ching divination replaces the respective gender features of a line and its insertion slot with a new, boolean feature [agree: True | False].
2. Equip the line slot feature structure with an additional boolean feature [central: True | False], which indicates whether a slot has a central position in its ambient trigram.

With the above preparation, if we sequence line/slot unification after line/slot agreement, we’ll get an integrated feature structure that contains both agree and central.

(unify ∘ agree) (line1, slot1) =
[stage: young, agree: False] ⨆
[position: 1, central: False, agree: False] =
[
  position: 1,
  stage: young,
  central: False,
  agree: False
]

And we are now able to represent the central and correct line property in featural terms—by counting how many times the substructure [central: True, agree: True] occurs in a hexagram’s unify ∘ agree values. Let’s still take ䷓ (kuan ‘view’ 觀) for example—and let’s ignore its potential changing lines for now.

(unify ∘ agree) (line1, slot1) =
[position: 1, central: False, agree: False]
(unify ∘ agree) (line2, slot2) =
[position: 2, central: True, agree: True]
(unify ∘ agree) (line3, slot3) =
[position: 3, central: False, agree: False]
(unify ∘ agree) (line4, slot4) =
[position: 4, central: False, agree: True]
(unify ∘ agree) (line5, slot5) =
[position: 5, central: True, agree: True]
(unify ∘ agree) (line6, slot6) =
[position: 6, central: False, agree: False]

There are two [central: True, agree: True] occurrences in kuan’s unify ∘ agree values, which makes it a perfectly-central hexagram—namely, it’s awesome.

Next, let’s turn to a hexagram with changing lines. Take the above-mentioned ䷯ (ching ‘the well’ 井) for example, whose first, third, and fourth lines are changing. I’ve also filled in the slot-quality features mentioned in Part 4.

(unify ∘ agree) (line1, slot1) =
[
  position: 1,
  quality: n/a,
  stage: old,
  central: False,
  agree: False
]
(unify ∘ agree) (line2, slot2) =
[
  position: 2,
  quality: meritorious,
  stage: young,
  central: True,
  agree: False
]
(unify ∘ agree) (line3, slot3) =
[
  position: 3,
  quality: dangerous,
  stage: old,
  central: False,
  agree: True
]
(unify ∘ agree) (line4, slot4) =
[
  position: 4,
  quality: disquieting,
  stage: old,
  central: False,
  agree: True
]
(unify ∘ agree) (line5, slot5) =
[
  position: 5,
  quality: best,
  stage: young,
  central: True,
  agree: True
]
(unify ∘ agree) (line6, slot6) =
[
  position: 6,
  quality: n/a,
  stage: young,
  central: False,
  agree: True
]

There’s only one double-True line in this hexagram (i.e., the hexagram has a central and correct line), but it occupies the best (i.e., fifth) slot in the hexagram. As such, the hexagram ching is not as auspicious as kuan but still quite okay, especially considering it in addition has four single-True lines.

The two examples above well illustrate how the feature tool in generative grammar can provide a new perspective on I Ching divination. There may still be other aspects of the divination process that can be given a feature-based modeling, but I’ll stop here before the post becomes too long-winded. Anyway, I think I’ve made my point.

Final thoughts

Generative grammar is a theoretical school in linguistics, but its ideas and methods can potentially have a wider impact. In this article I demonstrated how it can be used to examine the symbolic system of the I Ching, and I remember once seeing a Nobel Prize lecture entitled “The Generative Grammar of the Immune System” (Niels K. Jerne, 1984). Perhaps someday there will eventually be a field of study named “applied generative grammar.” That would be super cool.

I appreciate the profoundness and neatness of the I Ching—both as a divination manual and as a philosophical classic—and also admire its intricate yet highly organized symbolic system, which essentially builds the entire universe on just two notions: yin and yang (at least that’s the idea). This system is so remarkable—especially considering how old it is—that Gottfried Leibniz actually referenced it in the title of one of his most influential papers: “Explanation of the binary arithmetic, which uses only the characters 1 and 0, with some remarks on its usefulness, and on the light it throws on the ancient Chinese figures of Fu Xi.”

Binary code, powered by modern computers, has an amazing capacity to represent reality, which Gottfried Wilhelm von Leibniz could barely have conceived. The ancient authors of the I-Ching might have understood its potential—and its pitfalls—even better than we do.
— The Guardian

Leibniz wasn’t the only famous scholar in history who had publicly expressed their interest in the I Ching—Niels Bohr had even sewed the yin-yang symbol (☯) on his coat of arms when he received the Order of the Elephant from the King of Denmark.

Leibniz’s copy of I-Ching hexagrams (left) and Bohr’s coat of arms (right)

In our own era, the advent of new technology has only increased the public’s interest in this ancient “book of wisdom” and its implications for various domains of knowledge. For instance, I have seen repeated attempts to draw parallelisms between the I Ching and DNA (see this book, this paper, and this blogpost). I don’t have the expertise to evaluate the scientific validity of these attempts, though I find this blogger’s cold water–throwing not without reason:

On the internet, whole armies of crazies advanced their theories about the book.

My personal opinion on this renewed I Ching craze is that any serious result in “applied I Ching studies”—if that’s a well-defined area of study at all—must be based on scholarly team effort rather than merely enthusiastic surmise. Former President of Peking University Hu Shih (胡適) has a famous epithet regarding the methodology of learning: Daring in putting forward hypotheses; careful in searching for proofs (大膽假設小心求證). I find this a great criterion to follow, especially as we step into a zone as mysterious as the I Ching.

In the last years of his life, Confucius told his disciples that if he could live longer he’d devote his remaining time to the I Ching.

The Master said, “If some years were added to my life, I would give five or ten to the study of the Yi, and then I might come to be without great faults.”
(子曰：「加我數年，五十以學易，可以無大過矣。」)
— Shu Er, The Analects (論語・述而)

Since the I Ching has appealed to such great minds as Confucius, Leibniz, and Bohr, I imagine having a real understanding of it must be a highly illuminating experience. People tend to assume that such an old and mystical book must be all kinds of obscure, but that’s not necessarily true. With the help of modern tools, either from mathematics (as in this paper, this website, and this book) or from linguistics (as in the current article), I believe a systematic rethinking of the old book of wisdom should be not only feasible but also fruitful.🙏

-The End-