Rudy's Diamond Strategies

This complementary Blog to the Chinese Challenge Blog is presenting studies to a mathematical theory of Diamonds. Diamond theory is studying for the first time, tabular categories as an interaction of categories and saltatories.

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Name: Dr. Rudolf Kaehr
Location: Glasgow, Scotland, United Kingdom

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Tuesday, June 23, 2009

Polycontexturality of Signs?

Are there signs anyway?


How to read polycontextural sign matrices? Are there such constructs like polycontextural signs? It is argued that there are in fact no entities or processes in the “real-world” like signs in the sense of semiotics at all.

Semiotic signs are logocentric constructs realized by semioticians and defined by identity principles. This might be appropriate for a mono-contextural world-view but it is not sufficient for the experiences in a polycontextural world.

An example is given, how to construct and read a polycontextural configuration as a texteme. Also composition/decomposition of sign classes are presented.

FULL TEXT

http://www.thinkartlab.com/pkl/lola/PolySigns/PolySigns.html
http://www.thinkartlab.com/pkl/lola/PolySigns/PolySigns.pdf

Tuesday, May 19, 2009

Interpretations of the kenomic matrix

Exercises to the topics of Poly-Change

FULL TEXT
http://www.thinkartlab.com/pkl/lola/Matrix/Matrix.html
http://www.thinkartlab.com/pkl/lola/Matrix/Matrix.pdf


Abstract
Examples for the exercises, § 5.2, of the recent article “Poly-Change” are given, concerning the logical, computational and semiotic interpretation of the kenomic matrix.

http://www.thinkartlab.com/pkl/lola/Polychange/Polychange.html


4.1.4 What is the practical use of that fuss?

If there is any practical use for triadic-trichotomic semiotics, as Toth and others demonstrated in extenso, any extension of triadicity might open up some more complexity to deal with real-world matters in an operative and not reducing manner.



In sociology, cultural theory, international law, legitimations for torture and killing innocent people for good and accepted reasons, we encounter, in short, only two structural models of reasoning and acting. One is reducing complexity of what ever domain to a binary and dichotomic pattern. The other extreme is dissolving complexity into a multitude of autonomous isolated and and not-mediated dichotomous systems.



The first has the advantage of maximal operativity in technological and juridical systems, supporting nearly fully-automated surveillance systems and killing procedures.
The second is hopelessly non-operative and still based on humanistic propaganda for a better world – and even for Change.


"The genius of Michelangelo is like the genius of the Talmud, with several layers of meaning, one on top of another. So you can interpret it in terms of Christianity and Judaism, sociologically, historically and artistically. We are just adding one level that has either been ignored or covered up over the centuries.” Cathryn Drake, Did Michelangelo Have a Hidden Agenda?
http://online.wsj.com/article/SB122661765227326251.html 



"For the third millennium, the struggle against semantic disorder and perversions of the intellect should supersede, precede and be sustained in all cultures, religions, systems of thought and political systems whenever there is a historical necessity to initiate a war of liberation from oppression, domination and exclusion.”

Mohammed Arkoun, ISLAM: To Reform or to Subvert?, The rule of law and civil society in Muslim context, Beyond Dualist Thinking, 2006, p. 381

Hence, the academic question still remains: 


Wouldn’t it be worth to support a development of a cultural paradigm in which pluriversity and operativity could co-operate together?

FULL TEXT
http://www.thinkartlab.com/pkl/lola/Matrix/Matrix.html
http://www.thinkartlab.com/pkl/lola/Matrix/Matrix.pdf

Tuesday, May 5, 2009

Polycontextural and diamond dynamics

Sketches and exercises for dynamics and metamorphosis for formal systems

FULL TEXT
http://www.thinkartlab.com/pkl/lola/Polychange/Polychange.html
http://www.thinkartlab.com/pkl/lola/Polychange/Polychange.pdf

Abstract

The Ancient Chinese idea of a permanently changing world in which stable formulations, i.e. axioms in logic, are obsolete is thematized by the polycontextural strategy of permanently changing complexity. As a framework to realize complexity change for formal systems the kenomic matrix is involved. Examples for such formal notations are given and exercises to learn more about polycontextural diamond systems are proposed.

2. A remainder from Chinese Ontology

"Traveler, there are no path. Path are made by walking.” Antonio Machado

"A good mathematician is one who is good at expanding categories or kinds (tong lei)."

The Chinese philosopher Jinmei Yuan has given some crucial hints to the understanding of ancient Chinese mathematical thinking:

Chinese mathematical art aims to clarify practical problems by examining their relations; it puts problems and answers in a system of mutual relation--a yin-yang structure for all the things in a changing world. The mutual relations are determined by the lei (kind), which represents a group of associations, and the lei (kind) is determined by certain kinds of mutual relations.

"Chinese logicians in ancient times presupposed no fixed order in the world. Things are changing all the time. If this is true, then universal rules that aim to represent fixed order in the world for all time are not possible." (Jinmei Yuan)

An Aperçu
Chinese ontology (cosmology) can be put into two main statements:
A. Everything in the world is changing.
B. The world, in which everything is changing, doesn't change.
This two main statements are designing a paradoxical constellation.

Polycontexturality is complementing this ancient Chinese world model of harmony by dynamizing the concept of world-models:
C. A multitude of worlds are interplaying together.

The paradox to formulate mathematical rules in an ever changing world is very puzzling.
Many attempts to shed some light into it or even to solve the problem had been proposed.

It is not my intention to solve this ‘unsolvable’ problem.

Polycontextural logic attempts to formulate formal laws for an ever changing world. Nevertheless, we first have to abandon a Western interpretation of ‘change’. The Book of Change has nothing to do with Heraklit’s or Leibniz’s flux of things.

Many aspects about a philosophy of logic and time had been studied profoundly by the philosopher Gotthard Gunther. The connection of time and logic in polycontextural systems is not to confuse with any attempts of time or tense logics or physical time systems of any kind.

My own attempt to deal with the formal structure of changing first-order ontologies can be reduced, at this place, to two propositions:

Strategies of change
1. Diamond strategies: Each move is involved with its simultaneous counter-move.
2. Complexity strategies: Each move has to decide (elect/select) its intra-/trans-contextural continuation depending on the actual complexity encountered or created.

Because the strategies of change happens on the most fundamental levels of formal systems (logic, arithmetic, mathematics, ontology, semiotics, computability) a real combination of the antagonistic features of permanent change and formal operativity is opened up and accessible to realization.

One mechanism to realize change is given by the proemiality or chiasm between intra-contextural ‘parts’ and trans-contextural ‘whole’. A predicate defined inside a contexture can become the criteria for a new contexture which is augmenting the complexity of the contextural constellation.

For the sake of simplicity, 3 constellations of change are considered:
a) balanced constellation between formalism and application, with equal complexity for the formalism and the system to be formalized: compl(Form) = compl(System),
b) under-balanced constellation, with compl(Form) <= compl(System) and c) over-balanced constellation, with compl(Form) >= compl(System).

For classical Western thinking, based, shortly, on ontology and logic, only the balanced constellation with minimal complexity is available. Change is accessible in formal systems as change of complexion only. This strategy might be extremely sophisticated but it remains stable in respect to the logico-structural complexity of its paradigm.

Hence, not only every move (composition, concatenation, combination) in polycontextural diamond systems is accompanied by its hetero-morphic counter-movement but each movement is additionally determined by its polycontextural complexity-decision by election and selection.

In other words, in such a dynamic formalism, it easily can happen, that in the middle of a formal transformation (derivation, deduction, description, modeling) the complexity of the framework within those transformations happens might be changed, enlarged or reduced to legitimate a more reasonable and viable continuation of the transformations.

2.1. Exercises

2.1.1. Collect arguments - pros and cons, and beyond- and articles given in my Blog and elsewhere, which might support or reject the ‘Apercu’ of a Chinese Ontology and a Diamond World Model.

2.1.2. How are those thoughts connected to the project of Derrida’s Grammatology and the deconstruction of phono-logo-centrism in formal systems? Read and comment original texts only (if necessary translations)!
2.1.3. What can you learn from the sketches to a new rationality based on polycontexturality and the concept of Chinese scriptural paradigm for the understanding of the decline of the Western Hegemony?
2.1.4. What are the immanent limits of Western thinking and how might they influence the economic and financial crash? Connect your insights with the proposals given in my “The Logic of Bailout Strategies".
2.1.5. Create more questions and answer of this kind.
2.1.6. A good exercise to experience the patterns and strategies of polycontextural and diamond thinking for more familiar topics, like ethics, human rights, identity, pluricentrism, Web 2.0 etc. might be the reading of the ‘exercises’ I have written in the collection “Short Studies 2008".
All answers to the exercises can be written in English, German or French and posted to my Blogs. Chinese and Japanese proposals are welcomed.
3.4. Exercises
3.4.1. Write an overview of typical notational constellations for balanced formulas. Use the sketches given in ConTeXtures and From Ruby to Rudy.
3.4.2. Program features of balanced (m,n)-contextural notational systems for junctional, transjunctional connectors and quantifiers.
3.4.3. Try to define and program more efficient and ‘ergonomic’ notational approaches to general tabular syntactics.

4.3. Exercises

4.3.1. Collect the arguments and constructions given in my articles and build a systematic model of the dynamic interplay of interactionality/reflectionality and interventionality in formal systems.
Recommended articles: ConTeXtures. Programming Dynamic Complexity, Godel’s Games, Actors and Objects, From Ruby to Rudy, How to compose?
4.3.2. Compare those polycontextural and diamond models with models from modal logic, cognitive science, theory of reflection (Levebvre), reflectional programming (Smith, Maes) - and others.
4.3.3. Play around with your own ideas. Would it make fun to simulate polycontextural diamond dynamics with cellular automata models? What could we learn from such modeling, simulation and implementation? What would be lost?
4.3.4. Dynamics based on the ‘kenomic matrix’ might be studied for logical, arithmetical, categorical and semiotic systems by applying the materials proposed by now.
4.3.5. What are the structural consequences of contextural change for diamond category theory?

5. Metamorphic changes

5.1. Metamorphosis of topics

A transition from one contextural complexity to another doesn’t presuppose a pre-given existence of the new contextures. What might be presupposed is the possibility of change. And this possibility is realized by an application of the proemial mechanism between intra- and trans-contextural decisions.
An intra-contextural topic might become contextural prominence as a new contexture associated with the previous contextural constellation.

Reflection might change the meaning of an object by applying rules of chiastic metamorphosis.
Reflection is using the statement defining the object and this usage is defining the meaning of the object. Reflection and contemplation or introspection of an object can produce the insight that the meaning of the object under consideration is changing. Reflection as replication, thus, is augmenting the deepeness of the contextural complexity by a replicative, self-thematizing way. Reflection as iteration, is augmenting contextural complexity by an iterative, self-reproducing way. Alternatively, a reflection could change to an interactional augmentation of the contextural complexity. Both together, reflectional and interactional changes, are defining replicative, iterative and accretive contextural complexity of a polycontextural system.

The example below shows that the beginning reflection is interpreting an object as the number zero belonging to the topic numerals. This situation is implemented in a 1-contextural programming language. A second reflection considers the same object not as a numeral but as nil belonging to the topic of lists. Reflection has not to come to an end and can go further and with the interpretation and might realize that the object can be understood as belonging to the topic Booleans and appearing as the truth-value true.

Therefore the introduced syntactical object in its neutrality, observed and represented by an “external observer” in logtypeset structureis conceived as having simultaneously a numerical (in logtypeset structure), a symbolic (in log1.2) and a Boolean (in logtypeset structuremeaning. Hence, there is a chain of metamorphic replication from the topic Numerals, Lists to Booleans and a notation of the ‘neutral’ syntactic object “object” of Syntax. It starts with a reflection of the object “zero” of Numerals, ends with the Boolean “true” and gets a contextural abstraction as syntactic “object” in Syntax.

The example is designed for reflectional poly-topics in the experimental programming language ConTeXtures.

5.2. Exercises

5.2.1. Construct examples for reflectional, interactional and interventional constellations for poly-topics in the framework of ConTeXtures.
5.2.2. Construct further examples in the framework of ConTeXtures with topics like semiotics, logic, arithmetics.
5.2.3. Describe ‘empirical’ situations where such contextural changes of augmenting or reducing complexity seems to be unavoidable.
5.2.4. Try to develop a polycontextural measure for complexity.
FULL TEXT
http://www.thinkartlab.com/pkl/lola/Polychange/Polychange.html
http://www.thinkartlab.com/pkl/lola/Polychange/Polychange.pdf

Monday, April 6, 2009

Triadic Diamonds

Robertson’s algebra of triadic relations, Gunther’s founding relation, Toth's semiotics and diamond triads

FULL TEXT
http://www.thinkartlab.com/pkl/lola/Triadic Diamonds/Triadic Diamonds.html
http://www.thinkartlab.com/pkl/lola/Triadic Diamonds/Triadic Diamonds.pdf

Abstract
Some further thematizations and formalizations of diamond topics, especially triads, are presented. Triads, and founded triads, are presented in the context of Gunther’s epistemology, Toth’s semiotics with the help of Robertson’s “Algebra for triadic relations”. It is proposed that founding relations had been thematized externally only. An implementation of founding strategies into the system to be founded by the diamond approach is realizing the simultaneity of construction and verification of the triad.


Chinese Ontology and Diamonds
A new attempt to formalize the idea of founding relations is proposed by the diamond approach which takes into account the simultaneity of the model and its foundation. It also reflects the fact, that a foundation of an operation is localized on a different level of abstraction. The activity of modeling and the activity of founding are complementary activities demanding different kinds of abstractions. Hence, any applicative iteration of the model on itself is not fulfilling the criteria of foundation. "The idea of in-sourcing the matching conditions into the definition of diamonds tries to realize the two postulates of "Chinese Ontology", the permanent change of things and the endness (finitness) or closeness of situations. That is, diamonds should be designed as structural explications of the happenstance of compositions and not as a succession of events (morphisms).

More exactly, diamond are contemplating the interplay of acceptional and rejectional thematizations. Thus, morphisms with their matching conditions and composability are in fact of secondary order for the understanding of diamonds.

The complementarity of construction and verification, which is happening at once and not in a temporal delay, is a consequence of the finiteness and dynamics postulate of polycontextural "ontology". This simultaneous interplay is based on the insight that a delayed verification (or testing in programming) would not necessarily verify the construction in question because, at least, the context will have changed in-between. Delayed verification is possible only in the very special case of frozen dynamics.

In other words, in a changing open/closed world, the activities of construction and verification (of correctness and relevance) have to happen at once. Otherwise, because the conditions might have changed, the relevancy of the construction to be verified would have to be verified itself, again, and this ad nauseam.

Obviously, the statement is not about/against the stability of the construction (program, system, agreement, contract), this might be rock solid, but about the relevancy of the rock solid construction.

(In therapy or coaching, even by constructivists, this delayed checks are called “reality check”. Nearly always, such a reality to be checked has escaped any relevance.)



In-sourcing the matching conditions

Diamond strategies are offering a fundamentally different approach.
Each step in a diamond world has simultaneously its counter-step. Hence, each operation has an environment in which a legitimation of it can be stated. The legitimation is not happening before or after the step is realized but immediately in parallel to it.

Morphisms are representing mappings between objects, seen as domains and codomains of the mapping function. Hetero-morphisms are representing the conditions of the possibility (Bedingungen der Möglichkeit) of the composition of morphisms. That is, the conditions, expressed by the matching conditions, are reflected at the place of the heteromorphisms.

Hetero-morphisms as reflections of the matching conditions of composition are therefore second-order concepts realized "inside" the diamond system. 
Morphisms and their composition are first-order concepts, which have to match the matching conditions defined by the axiomatics of the categorical composition of morphisms. But these matching conditions are not explicit in the composition of morphism but implicit, defined "outside" of the compositional system. Hence, in diamonds, the matching conditions of categories are explicit, and moved from the "outside" to the inside of the system. In this sense, the rejectional system of hetero-morphisms is a reflectional system, reflecting the interactions of the compositions of the acceptional system. Heteromorphisms are, thus, the "morphisms" of the matching conditions for morphisms.

FULL TEXT
http://www.thinkartlab.com/pkl/lola/Triadic Diamonds/Triadic Diamonds.html
http://www.thinkartlab.com/pkl/lola/Triadic Diamonds/Triadic Diamonds.pdf

Monday, March 30, 2009

Diamond Relations

Sketch of a theory of diamond relations

FULL TEXT
http://www.thinkartlab.com/pkl/lola/Diamond Relations/Diamond Relations.html
http://www.thinkartlab.com/pkl/lola/Diamond Relations/Diamond Relations.pdf

Abstract
Because of their concreteness, the complexity of relations is more structured and is not always tackled by the axioms or properties of mathematical categories. E.g. the categorical properties of commutativity and transitivity are not necessarily holding for relations.

As an application, relations and the category of PATH as proposed by Pfalzgraf is presented. Diamond relations and a diamond version of PATH, i.e. JOURN (journey), based on diamond set theory, is sketched.

Motivation
How to introduce intransitivity (non-commutativity) in category theory? Two approaches are presented: Pfalzgraf’s generalized morphisms which are re-establishing categorical commutativity on a generalized level of relations and a sketch of polycontextural diamond constructions which are introducing different types of non-commutativity on the level of a generalized (disseminated) paradigm of categoricity.

Non-transitivity in diamond theories, thus, is not simply a total negation or rejection of transitivity but the acceptance of a plurality of different kinds of transitivity, enabling many kind of specific non-transitive relations.

Nontransitivity appears naturally for relations. Categories are by definition transitive (commutative). Hence, intransitivity for categories can be introduced only as a secondary concept. On the other hand, intransitivity for relations might be transformed to transitivity by a kind of a generalization or an abstraction to generalized relations, i.e. “a more general type of morphism” based on the difference of direct and indirect arrows (Pfalzgraf).

It is based on a very different paradigm to ask: “How to introduce intransitivity on the epistemological level of the definition of categories as such?”

It shall be shown, say sketched, that such a basic interplay of transitivity and different forms of non-transitivity is accessible in the framework of a polycontextural diamond category theory.

Road Map Metaphor
"Let us consider, for illustration, a simple practical example of real life: Looking at general relational structures is quite natural since transitivity and even reflexivity are not always existent in applications.
As a practical example let us look at a road map where the nodes (objects) are towns and the arcs (arrows) are road connections, then not every pair of towns has a direct connection (arrow), in general. Therefore, generally, starting from a point we have to follow a path of direct road connections passing several nodes (towns) before we can reach a goal.” (Pfalzgraf)

Pfalzgraf gives an example about direct connections between towns. The same observation holds for most intensional verbs, like win, love, hate, etc., e.g. A loves B, B loves C. Does A loves C hold necessarily? Obviously not.

Pfalzgraf’s strategy to keep transitivity by generalization could be paraphrased as:
A loves B, B loves C, A hate C , then, by generalization from intransitivity to transitivity:
A is-in-emo-relation to B,
B is-in-emo-relation to C, hence,
A is-in- emo-relation to C .

On the other hand, if A is connected with B, and B is connected with C, then A is connected with C, too. At least in a stable world, where the definition of connection is not suddenly transforming itself.


JOURN’s catalogue of journeys
There are structurally different kinds of journeys on offer.

1. PATH is a very special type of journey. It is an intra-contextural journey in a single contexture without structural environment. Hence, properly formalized as a category.

2. This situation might be distributed. Journeys in different but mediated contextures are possible. Still isolated and each thus intra-contextural.

3. A new kind appears with possible switches (permutation) and transjunctional splitting (bifurcation) simultaneously into paths of different contextures. Still without complementary environment in the sense of diamond theory.

4. Now, each contexture, even an isolated mono-contexture, might be involved into itself and its environment. This happens for diamonds, which are containing antidromically oriented path in categorical and saltatorial systems. Such journeys ar group-journeys with running into opposite directions.

5. Here, a new and risky journey is offered by the travel agency by inviting to use the bridging rules between complementary acceptional and rejectional domains of categories and saltatories of a diamonds. All that happens intra-contexturally, i.e. diamonds are defined as the complementarity of an elementary contexture.

6. Obviously, diamond journeys might be organized for advanced travellers into polycontextural constellations. Hence, there are transcontextural transitions between diamonds to risk. Interestingly, such journeys might be involved into metamorphic changes between acceptional and rejectional domains of different contextures of the polycontextural scenario.

Further Metaphors
As a metaphor, the idea of colored contextures, each containing a full PATH-system, involved in interactions between neighboring contextures, might inspire the understanding of journeys in pluri-labyrinths of JOURN.

Such journeys are not safely connected in the spirit of secured transitivity but are challenging by jumps, salti and bridging and transjunctional bifurcations and transcontectural transitions.

This metaphor of colored categories, logics, arithmetic and set theories gets a scientific implementation with real world systems containing incommensurable and incompatible but interacting domains, like for bio- and social systems.

FULL TEXT
http://www.thinkartlab.com/pkl/lola/Diamond Relations/Diamond Relations.html
http://www.thinkartlab.com/pkl/lola/Diamond Relations/Diamond Relations.pdf

Wednesday, March 25, 2009

Elements of Diamond Set Theory

Some more parts of the mosaic towards semiotics, logic, arithmetic and category theory

Abstract
Further elements are sketched towards an interplay of polycontextural logic, semiotics, arithmetic and set theory. Basics for junctional and transjunctional quantification in polycontextural logic are presented. Hints to metamorphic changes between sets, classes and conglomerates in pluri-verses are given.

1. Diamond set theory

2. Quantification in polycontextural logics


3. Interplay of semiotics, logics, set theory and arithmetic
A study of polycontextural semiotics, focused on semiotics alone, is not yet guaranteeing its polycontexturality.

The logical, arithmetical and set theoretical status of semiotics, mono- and polycontextural, remains undetermined if its corresponding logics, arithmetic and set theory (incl. category theory) are not determined and explicitly developed as polycontextural systems.



On the other hand, what value would have a semiotic system without any chances to proof statements, studying its arithmetical, set and category theoretical properties?

Until now, arithmetic, e.g., in semiotics, is not recognizing semiotical complexity but is calculating some combinatorial properties which are independent of the genuine, say triadic-trichotomous structure.

Similar mismatches happens with well known inadequate combinatorial studies of morpho- and kenogrammatics.



The same situation has to be recognized for other formal systems. A formalization of polycontextural logic is easily reduced to monocontexturality by arithmetization (Gödelization) if there is not at the same time a polycontextural arithmetic at hand to defend the strategies of polycontextural logic.

And obviously, because there is no initial origin, the carousel has to go through all stations of logic, arithmetic, semiotic, category and set theory, thematization, meta- and proto-language, etc. to deliver and interplaying foundation for each other. 


Proto- and meta-languagues of formal systems, as normed natural languages, are important to rule the relation between natural and formal languages, especially in the case of the interpretation of formal terms for philosophical or applicative aims.

If proto-language-based considerations are limiting the formal possibilities of formal constructions, the reasons for the restrictive decision should be made as explicit as possible. Also should the formal possibilities be accepted even if they haven’t yet found an interpretation.



Earlier on, there was a big philosophical topic to fight against the advent of traditional many-valued logic with the argument that the natural meta-language used to motivate and to develop many-valuedness is a priori two-valued. Hence, there is no escape from the two-valuedness of human thinking with the help of many-valued logic. Today, not even the question is recognized.

FULL TEXT
http://www.thinkartlab.com/pkl/lola/Elements/Elements.html

Saturday, March 14, 2009

Interactional operators in diamond semiotics

From polylogical transjunctions to polysemiotic interactions and reflections


Abstract

Comparing polycontextural logics and semiotics, the idea of interactionality is introduced as a further step of interaction in embedded semiotics. To achieve interactionality/reflectionality for semiotics some new concepts had been introduced.

For polylogical systems, transjunctional operators are defining interactions between logics. After a sketch of polysemiotics, poly-semiotic formulations of interaction and reflection operators are introduced.


1. Semiotics and polylogics
"Such an interpretation does not exist yet. However, if we look at Peirce´s ideas on semiosis as


"an action, or influence, which is, or involves, a co-operation of three subjects, such as a sign, its object, and its interpretant, this tri-relative influence not being in any way resolvable into actions between pairs",


then we could conclude that Peirce would have used Günther´s ideas of polycontexturality if they would have been known to him in his time.”

(E. von Goldammer)

http://www.vordenker.de/ggphilosophy/la_poly.htm











2. Dissemination of semiotics

3. Interactivity in poly-semiotics
"Transjunctional operations become unavoidable as soon as a system shifts from first-order to second-order observations or, in Günther's terminology, to polycontextural observations.

This comes very close to Derrida's attempt to transcend the limitations of a metaphysical frame which allows for only two states: being and non-being.

It comes close to a rejection of
logocentrism.

But it does not imply a rejection of logics or of formalisms.

Günther is not satisfied with the fuzziness of verbal acoustics and paradoxical formulations and tries, whether successful or not, to find logical structures of higher complexity, capable of fixing new levels for the integration of ontology (for more than one subject) and logics (with more than two values)."

(Luhmann, Deconstruction as Second-Order Observing, 1993)
4. Logification of semiotics

5. Interactions in diamonds
Transjunctions, as important operators of interaction, are well known in polycontextural logics. Semiotics offers a different approach to cognitive/volitive modeling. In this paper, some steps to sketch an interactional approach in semiotics along the experiences, models and formalizations of polycontextural logic, is undertaken.

FULL TEXT
http://www.thinkartlab.com/pkl/lola/Transjunctional Semiotics/Transjunctional Semiotics.pdf
http://www.thinkartlab.com/pkl/lola/Transjunctional Semiotics/Transjunctional Semiotics.html

Best with Publicon:
http://www.thinkartlab.com/pkl/lola/Transjunctional Semiotics/Transjunctional Semiotics.nb