Category Archives: sixteenfolds

Four qua Four

Here is a diagram for a set of four things standing in for each other, as in their being, character, function, or capacity. Qua is a proper English word (from Latin) meaning “as” or “as being” but also “in the function of” or “in the capacity of”. Often used as “A qua A” rather than “A qua B”.

I suppose the diagram could be altered to be four things qua four other things, with apostrophes or something. It could also be altered for a double duality, as shown below, with some nice symmetries.

It is somewhat similar to my diagram of the Tetrameria, but of course arranged differently. The only other example that I can think of as being “Four qua Four” is the Archic Matrix.

Further Reading:

Carl Jung’s Alchemical Tetrameria



Georgia Guidestones, RIP

Georgia Guidestones
Born: March 22, 1980
Died: July 6, 2022

Let These Be
to an Age
of Reason (*)

Capstone with (*) written in Four Ancient Languages

    • Cuneiform
    • Hieroglyphics
    • Greek
    • Sanskrit

Ten Recommendations (**) about

    • Population
    • Environment
    • Governance
    • Spirituality

Four Upright Stones with (**) written in Eight Modern Languages

    • English and Spanish
    • Swahili and Hindi
    • Hebrew and Arabic
    • Chinese and Russian

One Center Stone supporting Capstone for Three Astronometries

    • Finding the Celestial Pole
    • The Sun’s Annual Travel
    • Sun’s Position at Noon over the Year

Fear no more the heat o’ the sun,
Nor the furious winter’s rages;
Thou thy worldly task hast done,
Home art gone, and ta’en thy wages:
Golden lads and girls all must,
As chimney-sweepers, come to dust.

– William Shakespeare (from Cymbeline)

Further Reading:



Sixteen Emotions

Emotions are also often divided into sixteen different types instead of the eight we saw in The Circumplex Model of Affect. The Geneva Emotion Wheel (1.0) used the same Valence and Arousal axes and produced the following 16 emotions:

(Circular Order) Anger, Contempt, Disgust, Envy, Guilt, Shame, Fear, Sadness, Surprise, Interest, Hope, Relief, Satisfaction, Joy, Elation, Pride

(Alphabetized) Anger, Contempt, Disgust, Elation, Envy, Fear, Guilt, Hope, Interest, Joy, Pride, Relief, Sadness, Satisfaction, Shame, Surprise

(But an updated GEW Version 3.0 has 20 emotions in a circle on Valence and Control axes. See below for details.)

Another study used a Deep Neural Network on a training set of emotional facial expressions. They used the algorithm to track instances of 16 facial expressions one tends to associate with amusement, anger, awe, concentration, confusion, contempt, contentment, desire, disappointment, doubt, elation, interest, pain, sadness, surprise and triumph. (See below for details.)

Further Reading:

Deep Neural Network Study:

GEW (Geneva Emotion Wheel, Version 1.0)

GEW Version 3.0, with 20 emotions on a Valence and Control wheel with 2 options per emotion (in Circular Order):

Irritation, Anger
Contempt, Scorn
Disgust, Repulsion
Envy, Jealousy
Disappointment, Regret

Guilt, Remorse
Embarrassment, Shame
Worry, Fear
Sadness, Despair
Pity, Compassion

Longing, Nostalgia
Astonishment, Surprise
Feeling Disburdened, Relief
Wonderment, Feeling Awe
Tenderness, Feeling Love

Enjoyment, Pleasure
Happiness, Joy
Pride, Elation
Amusement, Laughter
Involvement, Interest

There are also several Classroom Charts available for Teachers and Students, for example (both in Alphabetical order):

Angry, Bored, Confused, Curious, Disappointed, Embarrassed, Excited, Grumpy, Happy, Nervous, Proud, Sad, Scared, Shy, Silly, Surprised

Angry, Afraid, Bored, Embarrassed, Excited, Guilty, Happy, Hopeful, Loved, Jealous, Proud, Sad, Shy, Sorry, Surprised, Tired

Images for Sixteen Emotions:


Kant’s Tables of Judgments and Categories

Even though Immanuel Kant’s tables of Judgments and Categories are each made up of four triples, both are divided into the same four headings: Quantity, Quality, Relation, and Modality. And as to the tripartite structure of their divisions, I can’t say I’m convinced of their coherence and completeness, c.f. Lovejoy’s article below.


  • Quantity: Universal, Particular, Singular
  • Quality: Affirmative, Negative, Infinite
  • Relation: Categorical, Hypothetical, Disjunctive
  • Modality: Problematic, Assertoric, Apodictic


  • Quantity: Unity, Plurality, Totality
  • Quality: Reality, Negation, Limitation
  • Relation: Substance, Cause, Community
  • Modality: Possibility, Existence, Necessity

Further Reading:

Immanuel Kant: Logic

Kant, Immanuel

Arthur O. Lovejoy / Kant’s Classification of the Forms of Judgment, The Philosophical Review, Nov. 1907, Vol. 16, No. 6, pp. 588-603

Click to access 2177294.pdf


Some researchers discussing Kant’s judgments that look interesting:

[*4.136, *7.114]


The Art of the Syllogism

The syllogism is a logical system that was invented by Aristotle which deduces valid inferences from given premises. It is categorical in nature because each of two premises and the conclusion has an internal relationship of belonging or inclusion. Specifically, there is a major premise of a general nature and a minor premise that is usually specific, or of reduced generality. Both are combined deductively to reach or prove the conclusion.

Both premises and the conclusion deal with three categories two at a time, a subject term (S), a middle term (M), and a predicate term (P), joined by one of four binary inclusion relations. The major premise deals with M and P, the minor premise deals with S and M, and the conclusion with S and P. The four types of relations are denoted by the letters A, E, I, O (also a, e, i, o) and are described below. The premises may have M first or second, but the conclusion always has the S first and the P second.

S = Subject
M = Middle
P = Predicate

A = a = XaY = All X are Y
E = e = XeY = All X are not Y
I = i = XiY = Some X are Y
O = o = XoY = Some X are not Y

Major premise: MxP or PxM, x = a, e, i, or o
Minor premise: SxM or MxS
Conclusion: SxP

The distinction between the four Figures concerns the placement of the middle term M in each of the premises. In order to highlight this order, I’ve written them with ( and ) on the side of the relation where the M is.

Figure 1: MxP, SyM, SzP: (xy)z
Figure 2: PxM, SyM, SzP: x(y)z
Figure 3: MxP, MyS, SzP: (x)yz
Figure 4: PxM, MyS, SzP: x()yz

There are only 24 valid inferences out of all possible combinations, six for each of the four Figures (and some of these may be erroneous sometimes due to the existential fallacy). In addition, they were given mnemonic names in the Middle Ages by adding consonants around the vowels of the relations. And so the valid inferences and their names (or something close to it) are as follows (by my notation and in no special order):

(aa)a, B(arba)ra
(ea)e, C(ela)rent
e(a)e, Ce(sa)re
a(e)e, Ca(me)stres
a()ee, Ca(l)emes
(ai)i, D(ari)i
(a)ii, D(at)isi
(i)ai, D(is)amis
i()ai, Di(m)atis
(ei)o, F(eri)o
e(i)o, Fe(sti)no
(e)io, F(er)ison
e()io, Fre(s)ison
a(o)o, Ba(ro)co
(o)ao, B(oc)ardo
(aa)i, B(arba)ri
a()ai, Ba(m)alip
(ea)o, C(ela)ront
e(a)o, Ce(sa)ro
a(e)o, Ca(me)stros
a()eo, Ca(l)emos
(e)ao, F(el)apton
e()ao, Fe(s)apo
(a)ai, D(ar)apti

For example, (aa)a, or Barbara, is a syllogism of the form: All Y are Z; All X are Y; thus All X are Z.

Further Reading:


Vaughan Pratt / Aristotle, Boole, and Categories (PDF, October 12, 2015)

Click to access PrattParikh.pdf

Vaughan Pratt / Aristotle, Boole, and Chu: Duality since 350 BC (Slides, August 12, 2015)

Click to access PrattABCD.pdf



The Genetic Code

There are many ways to show the genetic code, the map between triplets of nucleotides and the amino acids of proteins. Here is one that may be a bit awkward to understand, but other more standard ones are easily found.

 First, here are the codes for the four nucleotides:

  • U = Uracil
  • C = Cytosine
  • A = Adenine
  • G = Guanine

As well, let

  • $ = U or C
  • % = A or G
  • & = U or C or A
  • * = U or C or A or G

And so, here are the amino acids and their nucleotide codes

A = Ala = Alanine = GC*
C = Cys = Cysteine = UG$
D = Asp = Aspartic Acid = GA$
E = Glu = Glutamic Acid = GA%
F = Phe = Phenylalanine = UU$
G = Gly = Glycine = GG*
H = His = Histidine = CA$
I = Ile = Isoleucine = AU&
K = Lys = Lysine = AA%
L = Leu = Leucine = UU% + CU*
M = Met = Methionine = AUG
N = Asn = Asparagine = AA$
P = Pro = Proline = CC*
Q = Gln = Glutamine = CA%
R = Arg = Arginine = CG* + AG%
S = Ser = Serine = UC* + AG$
T = Thr = Threonine = AC*
V = Val = Valine = GU*
W = Typ = Tryptophan = UGG
Y = Tyr = Tyrosine = UA$
# = Stop = UA% + UGA

Note that some letters encode both nucleotides as well as amino acids, which might be confusing.

Further Reading:

[*10.146, *10.147]


Carl Jung’s Alchemical Tetrameria

Jung’s diagram of his alchemical tetrameria is supposed to represent the evolving self, and suggests movement, succession, and change and yet stillness, consistency, and renewal. His own diagram is quite different from mine, but I do think that mine has some merit.

What are those elements A, B, C, D, and a, b, c, d, and the subscripts 1, 2, 3, 4, indicating the modification of them? I’m not quite sure that it matters, except that for the relationships between the two, and the relationships between the four squares, and the relationships between the four parts of the four squares.

In Jung’s diagram, A equals a cycle of a, b, c, and d, and likewise B a cycle of a1, b1, c1, and d1, etc., and so we can instead say A is a cycle of Aa, Ab, Ac, and Ad, and likewise B is a cycle of Ba, Bb, Bc, and Bd, etc. In that sense my diagram denotes much the same as Jung’s.

Nevertheless, I’m going to have to cycle through some more thoughts about why one should spend too much time contemplating this diagram.

Further Reading:

Murray Stein / Jung’s Map of the Soul: an introduction

Leslie Stein / Becoming Whole: Jung’s equation for realizing God

Carl Jung / Aion

[*12.10, *12.11]


The Arcane Arts of Ramon Llull : the Dignities

Oh, Ramon Llull, where have you been all my life? I’m sure he’s been there all along, death now over seven hundred years in the past, just like always. His legacy seems at first glance to be quite the essence of medieval religion and scholastic philosophy, but still significantly and obscurely different to be enticing to this one. And on further examination, much more.

My schema above has little to do with his grand elaborate figures, except for listing the sixteen attributes he called “dignities”. Llull’s diagrams are full of clock-like wheels within wheels, complicated tableau, and combinatorial patterns. He wished to create a universal model to understand reality, and who wouldn’t want to discover the same? It is said that his methods are akin to an early computer science, and I’m just now starting to understand why.

The magister based the substance of his methods on his Christian faith, although he converted in midlife from Islam. Living in Barcelona, it was probably a good place to make such a change, but felt his calling was to convert others as well, so traveling he went. The methods he developed to convince others of their errors in belief were quite remarkable, as were the volume of his writing.

Like Gottfried Wilhelm Leibniz, who lived four hundred years later and was influenced by him, Llull wished to automate reasoning. But instead of building mechanical devices, Llull built computers from paper and ink, rulers and drawing compasses, scissors and glue. And instead of numbers as the smallest tokens of his computer, he used abstractions (i.e. words) that he felt would be understood by everyone in exactly the same way.

For example, he enumerated these sixteen dignities or aspects of his Christian diety, although sometimes he used the first nine. His constructions allowed one to pose questions and then obtain answers mechanistically that would be convincing to all observers of the correctness of the result. Too bad he was ultimately stoned to death while on his missionary work, although he lived to be eighty two.

Llull’s devices remind me of some of my pitiful charts and diagrams, and make me wonder if I may either adapt some of his techniques to my own use, or be inspired to develop others. I suspect I have locked myself into limitations by my approach, or are these constraints to my advantage? It might be hard to have spinning elements, but I can envision sliding elements like Napier’s Bones, origami-style folding and pleating, and even physical constructions like linkages and abacuses.

Now a martyr within the Franciscan Order, Llull’s feast day is June 30, which I’ve now missed. I hope to remember him to repost or improve on this by next year.

Further Reading:

The memory wheel



Richard McKeon’s Aspects of Knowing, Part 3

Previously on this blog:

Each method can be associated with a discursive process: operational with debate, dialectical with dialogue, logistic with proof, and problematic with inquiry. Each method is also associated with a mode of thought which in turn has two moments and one dependency or assumption: the operational method is debate by discrimination and postulation dependent on chosen theses, the dialectical method is dialogue by assimilation and exemplification dependent on changeless models, the logistic method is proof by construction and decomposition dependent on indivisible constituents, and the problematic method is inquiry by resolution and question dependent on discoverable causes.

For this diagram, the four dependencies or assumptions are in the center, and the associated methods are adjacent to them. Filling out the outer edge are the four pairs of moments. Listed, these facets are:

  • Methods (associated discursive process): Operational (Debate), Dialectical (Dialogue), Logistic (Proof), Problematic (Inquiry)
  • Assumptions: Chosen Theses, Changeless Models, Indivisible Constituents, Discoverable Causes
  • Modes of Thought: discrimination and postulation, assimilation and exemplification, construction and decomposition, resolution and question

The second diagram comes from a chart in McKeon’s “Philosophic Semantics and Philosophic Inquiry”. Here, the four methods are in the upper left corner (Universal) and lower right corner (Particular), and four principles are in the lower left corner (Meroscopic) and upper right corner (Holoscopic). Four interpretations are in the center (the vertical pair is Ontic, and the horizontal pair is Phenomenal), and four selections are adjacent to them. Listed, these facets are:

  • Methods: Operational, Dialectical, Logistic, Problematic
  • Principles: Simple, Actional, Comprehensive, Reflexive
  • Interpretations: Existentialist, Entitative, Ontological, Essentialist
  • Selections: Knower (Types), Knowable (Matters), Knowledge (Hierarchies), Known (Kinds)

Note that the Archic Matrix of Watson and Dilworth is essentially derived from this, and even has many of the same terms. However and obviously, the sixteen-fold arrangements of the two diagrams are different.

Further Reading:

[*5.184, *5.185, *6.20, *11.102, *11.106]




The Mouse’s Tale

No apologies to Lewis Carroll.

Further Reading:

[* 11.100]