Category Archives: sixteenfolds

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:



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]




The Tangram

Further Reading:



The Chromatic Scale of Music

All art constantly aspires towards the condition of music.

— Walter Pater

I really don’t know much about music theory, but I know I like this chromatic “circle” with solmization center.

Also, if you count out seven triangles clockwise around the rim starting from C you get G, D, A, E, B, F#, etc., which is the Circle of Fifths for chords. (But you get that with any chromatic circle.)

And I imagine if you go counter-clockwise you get the Circle of Fourths, which is important for Jazz chords!

Further Reading:

[*11.68, *11.69]


Combogenesis: a constructive, emergent cosmos

I just finished reading Tyler Volk’s “Quarks to Culture: how we came to be”. In this book Volk outlines an interesting model for what he calls combo-genesis, a “great chain of being” leading from basic physical law up through the highest organizational structures that we know of, human societies. He traces a path through human knowledge: physics, chemistry, biology, zoology, sociology, etc., and I am reminded of E. O. Wilson’s “Consilience: the unity of knowledge” that argued for some of the same things.

But Volk’s work has some good new ideas. He details twelve hierarchical levels, where each level is constructed on a “lower” previous level, and the new “higher” level has new things and different abilities than its predecessor. These levels range from the level of fundamental quanta (the quarks of the title), to geo-political states (the culture of the title).

  • QUA: fundamental quanta
  • PRO: nucleons, which are protons and neutrons
  • NCL: atomic nuclei
  • ATM: atoms
  • MOL: molecules
  • PCL: prokaryotic cells
  • ECL: eukaryotic cells
  • ANI: multicellular organisms, including animals
  • ASG: animal social groups
  • HUM: human tribal meta-groups
  • AGR: agro-villages
  • STA: geo-political states

These levels are within three dynamical realms, the first realm being of physical laws and then those realms of biological and cultural evolution. Each of these realms has a base level that has a capability for great constructive and emergent potential via an “Alpha-kit”. An alpha-kit has two facets, an element set and a cornucopia set, that operate like an alphabet and the myriad combinations that that alphabet can produce.

Dynamical realms:

  • Realm of physical laws: QUA -> MOL
  • Realm of biological evolution PCL -> ASG
  • Realm of cultural evolution HUM -> STA

Base levels and their Alpha-kits:

  • QUA, means for chemistry and molecules (atomic)
  • PCL, mechanisms for biology and its evolution (genetic)
  • HUM, faculty for culture and its evolution (linguistic)

As Volk’s model has each higher level based on or constructed from the previous lower one, I make the following suggestion utilizing my four-fold Structure-Function. The structures of each lower level serve as the parts of the next higher level, and the functions of each lower level serve as the actions of the next higher level. In this way a chain of actions and parts, structures and functions are built giving different entities and capabilities to different operational domains.

In the diagram shown, the sets of structures (S) and functions (F) of level i are used for the sets of parts (P) and actions (A) of level i+1, so S(i)=>P(i+1) and F(i)=>A(i+1). Not all structural information or functional abilities are necessarily accessible in the higher level of parts and actions, similar to the information and method hiding in object-oriented programming, and so reducing overall complexity. And as I have argued before, parts are combined to create the structures and actions are combined to create the functions of each level, so P(i+1)=>S(i+1) and A(i+1)=>F(i+1). In this way we have a bottom-up combo-genesis leading from quarks to culture.

Are we now entering another dynamical realm, perhaps based on some technological or computational alpha-kit? But, unfortunately we have to ask, will it take us forwards or backwards?

Further Reading:

Tyler Volk / Quarks to Culture: how we came to be

Tyler Volk and Robert Wright discuss:


Edward O. Wilson / Consilience: the unity of knowledge

[*11.98, *11.99]