J.-Y. Girard’s Linear Logic

Linear logic is a substructural logic invented (or discovered, if you’re a Platonist) by logician Jean-Yves Girard. Many other logics can be embedded into it, including classical and intuitionistic logic, so in a sense it is a “logic behind logic”. Linear logic can be partially derived from the rejection of the structural rules of weakening and contraction, the first of which adds arbitrary propositions and the second reduces duplicated propositions to single occurrences. Due to these changes in the logical rules, logic is transformed from being transcendental (truth transcends its use) to pragmatic or materialistic (truth is restricted by use). Therefore linear logic can be given a “resource interpretation” that makes it a logic not of truth but of things: producing and consuming, giving and taking, pushing and pulling, like the desiring machines of Deleuze and Guattari (see Hjelmslev’s Net).

The fragment of linear logic I show here is called MALL, for Multiplicative-Additive Linear Logic. The two exponentials that interconvert additive and multiplicative operations are not shown, which also allow for the weakening and contraction rules to be reintroduced.

Note that the two additive operations allow for propositions to be created and destroyed and the two multiplicative operations contain exactly the same propositions. One could say the additive operations allow for change, and the multiplicative operations allow for bearing. In the resource interpretation, note that additive disjunction () is creative and additive conjunction (&) is destructive. Both additive conjunction (&) and multiplicative disjunction () are reversible, whereas additive disjunction () and multiplicative conjunction () are irreversible.

Linear logic was a major inspiration for naming this blog “Equivalent Exchange” (see Introduction), since it is a logic of production and consumption. Linear implication, written as A –o B (and equivalent to A B), can be thought of as exchanging A for B.

Linear logic has also been adopted as the logic for “radical anti-realism”. How can it have both a physicalistic interpretation, and yet describe an anti-realism more radical than ordinary anti-realism? I will need further study to understand these claims.

References:

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

http://plato.stanford.edu/entries/logic-linear/

[*5.68-*5.70]

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The Tetralemma and Semiotic Square

The Tetralemma is a list that is supposed to exhaust all logical possibilities. Something is either X, or not X, or both X and not X, or neither X nor not X. Nagarjuna’s Fourfold Negation expresses a paradox by negating the Tetralemma, and asking what is not X, nor not X, nor both X and not X, nor neither X nor not X.

The Semiotic Square is an analytical tool to explore relationships between two semiotic signs, usually considered opposites of each other. The S1 and S2 in the figure are the signs in opposition, so that S2 is the dual of S1. S1+~S2 (S1 and not S2) would be X, ~S1+S2 would be not X, S1+S2 would be both X and not X, and ~S1+~S2 would be neither X nor not X.

References:

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

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

[*4.84, *5.182]

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Richard McKeon’s Aspects of Knowing

Richard McKeon’s system of Philosophical Semantics arises from the sixteen pairwise and ordered relations between his four aspects of knowing or cognates: knower, knowledge, the known, and the knowable. These sixteen relations can be sorted in four groups of four elements each: methods, interpretations, principles, and selections.

Between knower and knowledge, and between the knowable and the known, arise the four methods of two each: the universal and the particular.

  • From knower to knowledge, the operational method.
  • From knowledge to knower, the dialectical method.
  • From the knowable to the known, the logistic method.
  • From the known to the knowable, the problematic method.

Between knower and the known, and between the knowable and knowledge, arise the four interpretations of two each: the phenomenal and the ontic.

  • From knower to the known, the existential interpretation.
  • From the known to knower, the essential interpretation.
  • From the knowable to knowledge, the entitative interpretation.
  • From knowledge to the knowable, the ontological interpretation.

Between knower and the knowable, and between knowledge and the known, arise the four principles of two each: the meroscopic and the holoscopic.

  • From knower to the knowable, the actional principle.
  • From the knowable to knower, the simple principle.
  • From knowledge to the known, the comprehensive principle.
  • From the known to knowledge, the reflexive principle.

Between each of the aspects of knowing with itself, arise the four selections.

  • From knower to itself, the perspectival selection.
  • From knowledge to itself, the transcendental selection.
  • From the knowable to itself, the reductive selection.
  • From the known to itself, the functional selection.

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.

References:

Richard McKeon / On Knowing–The Natural Sciences

Richard McKeon / Freedom and History and Other Essays: an introduction to the thought of Richard McKeon

Sadly, the following pages are no longer available:

http://net-prophet.net/mckeon/mckeon.htm

http://forums.abrahadabra.com/showthread.php?2331-Unifying-Astrology-and-I-Ching

[*4.47, *5.184-*5.187, *6.20, *6.106]

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Arthur M. Young’s Fourfold Theory of Process

I have recently come across the philosophical work of Arthur M. Young (AMY). This is an initial impression of that work since I have only read what is available from the web links below, and even then there is a great deal to digest. In addition, there is difficulty in presenting a summary of his theory because of similarities to my ideas as well as substantial differences. I am sure I will need to return to AMY’s theory after more consideration.

I have hinted at a correspondence between several double duals presented in this blog, but I have steered away from claiming that they are all linked to each other – that they are essentially equivalently exchangeable. AMY’s theory links the four elements, the four causes, Jung’s functions of the psyche, geometrical elements and transformations, as well as several other fourfolds into a cosmic theory of reality.

Some of these same fourfolds are present in my theory, and I am considering how others may be introduced. Some not mentioned by AMY are only mentioned in earlier entries on this blog, without presentation. However, from many of these same fourfolds I have reached substantially different conclusions from AMY. I believe this is because AMY’s theory of process is essentially dualistic, whereas my theory appears to be physicalistic, although one might also say it is a process and/or relational theory.

Below is a table of some of the correspondences for AMY’s theory of process:

Aristotle’s
Four
Causes
Jung’s
Functions
of the Psyche
Four
Elements of
Empedocles
Geometric
Transform-
ations
Purpose Final Intuition Fire Rotation Spirit
Value Material Emotion Water Scale Soul
Form Formal Intellect Air Inversion Mind
Object Efficient Sensation Earth Translation Body

Below is a table of some of the correspondences for my theory:

Four Elements
of Empedocles
The Here and
the Now
Aristotle’s
Four Causes
Duality of
Time and
Information
Hjelmslev’s
Net
Fire Before Efficient Change time Substance
of content
Water After Final Bear time Form of
content
Air Above Formal Bear
information
Form of
expression
Earth Below Material Change
information
Substance
of expression

References:

Arthur M. Young / The Reflexive Universe

Arthur M. Young / The Geometry of Meaning

http://www.arthuryoung.com/index.html

http://www.arthuryoung.com/barr.html

http://www.arthuryoung.com/4levels.html

http://www.arthuryoung.com/the2exc.html

[*6.84-*6.89, *7.78, *7.79, *8.2, *8.62, *8.63]

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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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