Category Archives: fourfolds

Hjelmslev’s Net

“God is a lobster, or a double pincer, a double bind.”

— From A Thousand Plateaus by Deleuze and Guattari

Linguist Louis Hjelmslev developed a semiotic model which elaborated Saussure’s two part signifier and signified into the double dual of the substance of content, the form of content, the substance of expression, and the form of expression. Contents are “formed matters”, and expressions are “functional structures”. Both are further separated into a substance and a form. The original signifier can be considered the form of expression, while the original signified can be considered the form of content. The two types of forms are like a net of warp and woof (why else a net?), dividing an undifferentiated unformed matter (Earth, purport) into two types of substances.

Deleuze and Guattari cast this net from Hjelmslev’s use in language into universal application by way of examples in geology and biology: sedimentation/folding and molecular genetics. The two planes of content and expression are the First Articulation and Second Articulation, respectively, the first of which “chooses or deducts”, and the second of which establishes “functional, compact, stable structures”. In their geology example, the First Articulation is the process of sedimentation, and the Second, folding. Generally, the two substances deal with territorialization, deterrritorialization, and reterritorialization, and the two forms are concerned with coding and decoding (and recoding?).

Additionally, there is talk of the molar versus the molecular (as continuous/discrete or unity/multiplicity?) but the molar is not form, nor is the molecular substance, nor vice versa. The First Articulation moves from molecular substances to molar forms; the Second Articulation moves from molecular forms to molar substances. How confusing! What does it all mean? One could spend a lifetime lost in these fun-house reflections!

I propose that the four basic logical operators of Linear Logic are in correspondence to the double articulation of Hjelmslev’s Net.  Content is Conjunction, Expression is Disjunction, Substance is Additive, and Form is Multiplicative. Content and Expression is Substance or Form; Conjunction and Disjunction is Additive or Multiplicative.

References:

Gilles Deleuze and Felix Guattari / A Thousand Plateaus: capitalism and schizophrenia

Manuel De Landa / The Geology of Morals: a neo-materialist interpretation http://www.t0.or.at/delanda/geology.htm

Luke Feast / The Science of Multiplicities: post-structuralism and ecological complexities in design http://researcharchive.vuw.ac.nz/handle/10063/142

[*3.170, *4.46, *4.88, *4.112, *4.146, *5.70, *5.174, *6.10]

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Kevin Kelly’s Philosophy of Technology

Kevin Kelly’s new book “What Technology Wants” is an exploration of what technology is and what it does. Technology has many of the same attributes as biological evolution, and as such, its effects cannot be fully predicted. At best, we can try to evaluate a particular technology’s advantages and dangers before it is let loose into the world; at worst, we will have no control over it at all.

Kelly describes evolution as shaped by structural, historical, and functional factors; and goes on the describe technology as dependent on structural, historical, and intentional factors. However, he also maintains that technology is an evolutionary process, and evolution in turn is a technological process. Kelly seems to say both processes have all four of the factors shown in the double dual above.

Kelly says that human language is the first big human technology (or was it fire? or stone tools?). But I also agree with him that the mechanisms of biological evolution can be considered technology. What is technique except a method that can shared and perpetuated by others? Molecular genetics grants us the ability to pass (most of) our attributes on to our progeny, including the ability to pass (most of) their attributes on to theirs. Once techniques can be shown or told to others, biology becomes the basis for the showing or telling, but not the mechanism of it.

Kelly calls the entire system of evolution/technology the technium. Because we have been continually shaped by our human technologies, they are not foreign to us. On the whole, we are better with them, than without them. One could argue that without them we wouldn’t even be human!

References:

Kevin Kelly / What Technology Wants

[*5.92, *6.60]

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Monism = Pluralism

PLURALISM = MONISM

– From A Thousand Plateaus by Deleuze and Guattari

Jeffrey Bell’s blog entry about William James’ radical empiricism reveals relations between Hjelmslev’s Net and Linear Logic. To begin with, Hume was concerned with disjunctive relations (of expression) to the exclusion of conjunctive relations (of content). In addition, James sought the solution to the problem that consciousness (here content) has between the “one and the many”, one consciousness in relation to many consciousnesses. Unable to resolve this problem, James did not realize that conjunction can come in two modes, an additive one and a multiplicative one, a substance and a form.

The substance of content (here consciousness, agency, …) is constituted incrementally from choices between actions, either thoughts (thoughts-as-action) or actual actions (actions-as-action). This is additive AND. The form of content (essence, existence) is assembled by the ordering of those choices, a multiple choice of choices. This is multiplicative AND. These are the powers of AND.

However, Hume’s disjunction (expression) also comes in two flavors: additive and multiplicative (substance and form). It also has a problem with the “one and the many”. The substance of expression is either identity or generation (accident, substance). This is additive OR. The form of expression doesn’t seem like much in Linear Logic, but it is the very form of the logic, invertible with the connective tissue of the calculus (the comma). This is multiplicative OR. These are the powers of OR.

Content and expression are dual to each other, as conjunction is logically dual to disjunction. Is content the “subjective” and expression the “objective”? Is substance the “one” and form the “many”? Each is dual to the other, not distinguishable except by perspective. Perhaps these double duals are like a Mobius Strip, which only has one side, weaving in and out and forming a unity out of multiplicity.

Note that the elements of the double dual shown here are taken from the Protreptikos page “Monism and Pluralism”. The fourfold is made up of different “compositions in being”, each in two parts. There are many echoes to other double duals in these compositions, such as potency/actuality (existence) and substance/form.

References:

Gilles Deleuze and Felix Guattari / A Thousand Plateaus: capitalism and schizophrenia

http://schizosoph.wordpress.com/2010/08/11/monism-pluralism/

Further Reading:

https://en.wikipedia.org/wiki/Potentiality_and_actuality

Aquinas: Metaphysics

https://plato.stanford.edu/entries/aristotle-metaphysics/

https://plato.stanford.edu/entries/form-matter/

https://ndpr.nd.edu/news/form-matter-substance/

Matter and Form, Substance and Accidents

[*6.40]

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The Duality of Time and Information

The states of a computing system bear information and change time, while its events bear time and change information.

from The Duality of Time and Information by Vaughan Pratt

The most promising transformational logic seems to us to be Girard’s linear logic.

— from Rational Mechanics and Natural Mathematics by Vaughan Pratt

References:

Vaughan Pratt / The Duality of Time and Information http://boole.stanford.edu/pub/dti.pdf

Vaughan Pratt / Time and Information in Sequential and Concurrent Computation http://boole.stanford.edu/pub/tppp.pdf

Vaughan Pratt / Rational Mechanics and Natural Mathematics http://chu.stanford.edu/guide.html#ratmech

 [*5.170]

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Aristotle’s Four Causes

Material: That from which something is made.

Efficient: That by which something is made.

Formal: That into which something is made.

Final: That for the sake of which something is made.

— from Aristotle for Everybody by Mortimer Adler

“Happy is he who can recognize the causes of things.”

Virgil

Aristotle’s Four Causes is likely the most familiar of all the double duals that I will present. The causes are closer to being “becauses” since they are usually thought of as the reasons or explanations for things. Why not call them the four prepositions?

The standard example of the four causes is what is needed for the building of a house. A house is built by the craftsmen, from the raw materials, into the form shown on blueprints, for the homeowner to live in. This and other usual examples are concerned with the making of something.

Formal and final causes have gotten the short shift since the beginning of the scientific revolution. Francis Bacon stated that the only scientific reasons for things were the efficient and material causes. For those critical of materialism this is often termed mere “matter in motion”. Matter can be thought to exist in space, and motion in time. Where does form or finality exist? I will say in space and time as well.

References:

Max Hocutt / Aristotle’s Four Becauses, in Philosophy, Vol. 49, No. 190. (Oct., 1974), pp. 385-399.

http://en.wikipedia.org/wiki/Four_causes

http://plato.stanford.edu/entries/aristotle-causality/

http://en.wikipedia.org/wiki/The_purpose_of_a_system_is_what_it_does

Notes:

John Sowa’s Thematic Roles: initiator, resource, essence, goal.

http://www.jfsowa.com/ontology/thematic.htm

[*4.112, *5.73, *5.162, *5.168, *7.47]

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The Here and the Now

Whosoever speculated on these four things, it were better for him if he had not come into the world —

  • what is above?
  • what is beneath?
  • what was beforetime?
  • and what will be hereafter?

— From the Mishnah (Hagigah 2:1)

All things have a root and a top; all events an end and a beginning. Whoever understands correctly what comes first and what follows draws nearer the Dao.

— From T’ai Hioh by Confucius

As above, so below.

— From The Emerald Tablet

I like these quotes because they show that Above, Below, Before and After are linked together. The first quote gives a warning about thinking about these concepts, but the second, encouragement. Above and below, or higher and lower, can be thought of as directions in space, but also as terms of hierarchy. Before and after can be thought of as directions in time, but also as beginnings and endings, causes and results.

Every individual is situated in space and time (see SpaceTime). Every perspective is due to expression and content (see Hjelmslev’s Net). Here is space, now is time.

References:

http://en.wikipedia.org/wiki/Emerald_Tablet

[*5.160, *6.30]

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The Four Elements of Empedocles

Into this wilde Abyss,
The Womb of nature and perhaps her Grave,
Of neither Sea, nor Shore, nor Air, nor Fire,
But all these in their pregnant causes mixt
Confus’dly, and which thus must ever fight,
Unless th’ Almighty Maker them ordain
His dark materials to create more Worlds,
Into this wilde Abyss the warie fiend
Stood on the brink of Hell and look’d a while,
Pondering his Voyage; for no narrow frith
He had to cross.

— From Paradise Lost by John Milton


The Four Elements of Empedocles is one of the earliest ontologies. While not of special interest in itself (except of course to alchemy), other double duals are comparable to it, such as the The Here and the Now and Heidegger’s Fourfold. Additionally, it is interesting to remember that it was thought that the four elements were the primitives of which everything is composed; for example bone was fire, air, water, and earth blended in a certain proportion. I am unsure of the origin of the choice of the elements: why these four and not others?

Also, note that with respect to light transmission, the four elements can be arranged in the sequence of bright, light, dim, and dark. This is not to say that fire is “good” and earth “bad”, or that fire and earth are the most different from each other. Perhaps this distinction will have some meaning later.

[*5.188, *6.32, *6.56, *7.194]

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Ken Wilber’s AQAL

References:

http://en.wikipedia.org/wiki/Ken_Wilber

http://www.kheper.net/topics/Wilber/Wilber_IV.html

http://tetrast4.blogspot.com/2008/12/wilber.html

[*5.190-*5.192, *6.58]

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Whitehead’s Criteria for Metaphysical Theories

Note that consistency and coherency are considered rational, and that applicability and adequacy are considered empirical. This has importance for Heideggar’s Fourfold since the rational is revealed,  and the empirical is concealed. For Linear Logic, additive conjunction and multiplicative disjunction are reversible, yet additive disjunction and multiplicative conjunction are irreversible.

References:

Alfred North Whitehead / Process and Reality

Frederick Ferre / Being and Value:  toward a constructive postmodern metaphysics

Mark Graves / Mind, Brain, and Elusive Soul: human systems of cognitive science and religion

Paul Reid-Bowen / Goddess as Nature: towards a philosophical theology

[*6.12, *6.70]

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Heidegger’s Fourfold (das Geviert)

We are too late for the gods and too early for Being. Being’s poem, just begun, is man.

— From Thinker as Poet by Martin Heidegger

“Every man makes a god of his own desire.”

Virgil

Note that Heidegger’s Fourfold is a perfect schema for the Tower of Babel: while mortals strive to join the earth and sky, the gods undo their efforts by turning their unity into multiplicity.

References:

Graham Harman / Tool-Being: Heidegger and the Metaphysics of Objects

http://www.cosmosandhistory.org/index.php/journal/article/view/133/276

http://en.wikipedia.org/wiki/Speculative_realism

http://culturalstudiesnow.blogspot.com/2011/05/martin-heidegger-building-dwelling.html

http://avoidingthevoid.wordpress.com/2009/10/28/heating-up-the-transmutations-of-media-beings-part-1/

Also:

Andrew J. Mitchell / The Fourfold: Reading the Late Heidegger (Studies in Phenomenology and Existential Philosophy)

Martin Heidegger’s Gods

[*6.2, *6.3, *6.56, *6.62, *9.163]

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