Category Archives: fourfolds

The Ware Tetralogy

Here’s another science-fiction tetralogy for you to read that is both entertaining and thought-provoking. The  four books in Rudy Rucker’s “Ware” Tetralogy are:

  • Software
  • Wetware
  • Freeware
  • Realware

From Wikipedia:

The closest to the cyberpunk genre of all his works, the tetralogy explores themes such as rapid technological change, generational differences, consciousness, mortality and recreational drug use.

Pretty wild stuff!

Further Reading:



A Crystal Structure of the Main Protease of COVID-19

One can gain knowledge from words, but wisdom only from things.

— George Wald, Harvard biochemist and 1967 Nobel Laureate

Many images of the coronavirus SARS-CoV-2 (a.k.a. 2019/nCoV a.k.a. COVID-19) are in the media, with the colorful spiky ball motif being the most frequent. Above is a representation of one of its proteins, its main protease, which allows the virus to process the proteins created after it splices its own RNA genetic material into your cells.

The larger three lobed spiky proteins from which these types of viruses gets the name “corona” are responsible for grabbing onto and opening up the surface of (in this case) lung cells, so that this smaller protein can perform its function within the cell. In more detailed images, you may have seen it as small pairs of spheres on the viral surface.

Several different crystal structures of  various proteins of COVID-19 have been solved and released recently, including this main protease and the spiky protein peplomers. Studies of these structures will hopefully lead scientists to discover inhibitors to their functions and thus treatments and preventive measures. Go science!

Unfortunately, these results will come much too late for many of us. Science could have also helped us with the initial defense against this deadly virus, but the powers that be deigned to consider the gravity of our plight. And even today many such leaders and spokespersons are ignoring important information and spreading misinformation.

Further Reading:




The Frauchiger-Renner Paradox

“The new experiment shows that, in a quantum world, two people can end up disagreeing about a seemingly irrefutable result, such as the outcome of a coin toss, suggesting something is amiss with the assumptions we make about quantum reality.”

— From the Quanta article by Anil Ananthaswamy

As above, so below.

As we face the deadly onslaught of electron-microscopy-sized agents, remember to wash your hands for twenty seconds, follow physical distancing rules of six feet or more, and please be safe.

Further Reading:



The Book of the New Sun

Gene Wolfe’s “Book of the New Sun” tetralogy is said to be rife with alchemical symbolism. I have not read it yet, although it has been on my to-read stack for years. Nonetheless, here is a diagram for you, partly because it’s an easy post, but also for me, so that I can add interesting links as I might find them.

  • The Shadow of the Torturer
  • The Claw of the Conciliator
  • The Sword of the Lictor
  • The Citadel of the Autarch

Of course, there are a few other tetralogies that exist, although they are not as common as trilogies. One already mentioned on this blog is the Aeygpt Tetralogy by John Crowley.

Further Reading:

An Analysis Of The Alchemical Tradition Behind BOOK OF THE NEW SUN, Part I: Urth, Ushas, And Unus Mundus

An Analysis Of The Alchemical Tradition Behind BOOK OF THE NEW SUN, Part II: The Solve et Coagula of Terminus Est

An Analysis Of The Alchemical Tradition Behind BOOK OF THE NEW SUN, Part III: The Piteous Gate and A Dark Albedo

An Analysis of the Alchemical Tradition Behind BOOK OF THE NEW SUN, Part IV: Hethor, Typhon and the Temptation of Severian

An Analysis of the Alchemical Tradition Behind BOOK OF THE NEW SUN, Part V: The Aestivation Hypothesis and the Hierodules

An Analysis of the Alchemical Tradition Behind BOOK OF THE NEW SUN, Part VI: Tzadkiel and the Mystery of Yesod


Four Valued Logic

Logic is not as absolute as we would like it to be. For example, linear logic breaks down normal logic into a realm of substructurality. There seems to be several ways to consider expanding classical two-valued logic to four values.

Let this post be a placeholder for considering expansion of classical two-valued logic to four values. For example, one might have:

  • True
  • False
  • Both
  • None

Further Reading:

J. Michael Dunn / Two, Three, Four, Infinity: The Path to the Four-Valued Logic and Beyond

Katalin Bimbo & J. Michael Dunn / Four-valued Logic

J. Ulisses Ferreira / A Four-Valued Logic



Schopenhauer’s Four Laws of Thought

The first three of Arthur Schopenhauer’s Four Laws of Thought are pretty much the same as the classical three laws of thought. Schopenhauer added a fourth law that was basically for his Principle of Sufficient Reason.

  • Identity
  • Non-contradiction
  • Excluded middle
  • Sufficient reason

These Four Laws are often given in two flavors: the first, in fairly concrete terms of subjects and predicates, and the second, more glib in terms of existence and being and such (isness).

  • A subject is equal to the sum of its predicates. Everything that is, exists. (Identity)
  • No predicate can be simultaneously attributed and denied to a subject. Nothing can simultaneously be and not be. (Non-contradiction)
  • Of every two contradictorily opposite predicates one must belong to every subject. Each and every thing either is or is not. (Excluded middle)
  • Truth is the reference of a judgment to something outside it as its sufficient reason or ground. Of everything that is, it can be found why it is. (Sufficient reason)

The phrase ‘it can be found’ sounds like a constructive method rather than a mere existence proof, but the common theological technique that combines both by saying “everything happens for a reason” avers the reason to an ineffable deity. (I bet Schopenhauer would have disliked this view because from what I understand he was an atheist.)

Moving on, I would like to represent these four laws in even more concrete terms of logical expressions. In the following attempt, let a, b be subjects (or objects), and P, Q be predicates (or qualities):

  • ∀a (a ≡ ∀P P(a))
  • ∀a ¬∃P (P(a) ∧ ¬P(a))
  • ∀a ∀P (P(a) ∨ ¬P(a))
  • ∀a ∃b (b → a)

When detailed in this way, these four laws don’t seem very complete, or don’t quite form a unity, as implication and equivalence are each in only one of them. Even though it doesn’t help that criticism, perhaps one can succinctly say:

  • Things can be reduced to (all) their qualities.
  • Qualities are disjoint from their opposites.
  • Qualities and their opposites are sufficient.
  • Things are entailed by some thing (possibly same).

In addition, I quite liked this Goodread review which aligns Aristotle’s Four Causes with Schopenhauer’s Fourfold Root. So then:

  • Material Cause : Becoming : Identity
  • Final Cause : Knowing : Non-contradiction
  • Formal Cause : Being : Excluded-middle
  • Efficient Cause : Acting : Sufficient reason

Further Reading:

[*11.196, *11.197]


At some point, I need to understand the difference between the law of the excluded middle and the principle of bivalence.


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]