Diamond Relations

Sketch of a theory of diamond relations

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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.

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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.

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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.

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Sketch on semiotics in diamonds

Embedding semiotics into anchored diamonds

Semiotics are embedded into diamonds in a double way. Semiotics gets a internal environment as its neighbor semiotics and an external environment by its anchors. Embedding semiotics is a process of concretization of the abstract concept of Peircean semiotics.

Peirce' trichotomics is based on his metaphysical intuition, nurtured by his studies of Kant and Hegel, and is not a product of a general generation scheme with steps from 1 to 3 as it is echoed in the semiotic literature since decades.

Such a general generation scheme wouldn't have a built-in stop function, it could go on to arbitrary magnitudes. Only for a reconstructional and didactic interest a start with 1 and an end with 3 makes sense.

“Creation thus means “that Firstness (repertory of ‘possible’ cases) must be given, so that Secondness (the ‘real’ case) in the sense of singular, concrete and innovative givenness is selectable in dependency of also given Thirdness (determining law or necessity)” (Bense and Walther 1973, p. 127)." (Toth, In Transit, p. 49)


Intuition of trichotomy
From the point of view of the primary trichotomic intuition and its realization, monadic and dyadic relations occur as reductions of the trichotomic intuition and its realization as a triadic relation.

In a diamond theoretical and polycontextural approach to an embeddement of semiotics, nothing is given. The giveness of the semiotic categories, firstness, secondness and thirdness, are a result of a speculative decision for a trichotomic paradigm of thinking and corresponding world model, initiated scientifically bei Peirce.

Semiotics in polycontextural diamond constellations are towfold embedded
1. by their neighbor-semiotics and
2. by their diamond environments.
Obviously, semiotics and diamond environments are equiprimordial (gleichursprünglich).
In a further step of concretization, the construction gets its localization as a
3. embedmend by its place-designators of the kenomic anchors.

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