Transitions from a triadic-trichotomic semiotic and epistemological paradigm to a
quadralectic diamond approach.
In several papers, like SKIZZE-0.9.5, I developed a philosophical theory of the four-foldness of beginnings. A first mathematical realization was supported by the idea of chiasms and proemial relationships, finally formalzed as diamond categories.
An application to kenogrammatics of those thoughts was presented as a diamondization of the act of beginning of kenomic successor systems. A first sketch “New beginnings for kenogrammatics?” was published in “Morphogrammatics of Change”.
“At such a start of kenogrammatics there is no need for a dualism of iterativity and accretion but a chiasm is involved of the terms internal/external and accretion/iteration, producing the double determination of situations by the wording of iterative iterativity, accretive iterativity and iterative accretion, accretive accretion. Hence, basic terms in kenogrammatics are reflectional and second-order figures. In other terms, proemiality is opening up the beginnings of kenogrammatics.
"To choose a beginning with a mark is putting a difference into the possibility of a choice for another mark of beginning. Such a difference in the notion of representation of a beginning by a mark shall be inscribed as a double beginning. The question is: As which representation is a kenogram inscribed? If it is inscribed as “a” then it is differentiated from another possible inscription, say “b”. If it is inscribed as “b” then it is differentiated from another possibility, say “a”. A further differentiation, say into “c” , would be redundant and irrelevant for the characterisation of a kenomic beginning.
On the other hand, philosophically, with double beginnings, the necessity of a unique and ultimate “coincidentia oppositorum” (Cusanus, Hegel, Gunther) is differentiated and dissolved."
A further application of those insights on the project of formalizing kenogrammatics might be realized simply by accepting the distinctions of a kenogram and its “environment”, “A | a”, as parts of a second-order diamond [A | a; a | A] and its inter-relationality.
Hence, the simple introduction of a ‘beginning’ kenogram to an inscription of the act of introducing it has to be closed by its complementary step towards a full inscription of the diamond of beginnings. The diamond approach is well known but there was not yet a direct formal application of it to the beginnings of kenogrammatics systems. Diamond applications had been elaborated for category theory and for mathematical semiotics.
Hence, the act of introducing a single kenogram into a trito-structure as the beginning of the trito-structure by a start rule, “ => O “, has to be deconstructed.
It seems that the whole ambiguity, paradoxy and circularity with its blind spot of introducing a distinction and its mark in the Laws of Form is repeated with the singular act of introducing a kenomic start by a kenogram.
Gunther’s postulation: => O .
First-order diamondization: O ≡ O --> O, hence [O | O].
Second-order diamondization: O ≡ O --> O o O --> O | O <-- O, hence [O | O]|[O | O ].
Gunther’s postulation of a beginning is simply putting a kenogram into the game, and declaring it as a kenogram and as the start kenogram of the trito-structure of kenogrammatics. Its further definition happens outside this kenogrammatic start formalism. The beginning of the kenogrammatic system on the level of the trito-structure is not telling any properties of the system as such. All the features of trito-kenogrammatics are following secundarily.
The first-order diamondization is taking the kenogram as an automorphism, “O --> O” , and its matching condition are thematized as the environment of the morphism by "O”. Therefore, the automorphism is defined by its own place with the matching conditions as its environment. This was called “in-sourcing” of the matching conditions into the calculus itself. Hence, if the matching conditions are changing in the process of applications the calculus and the character of its compositions is changing too. But this second-order behavior is formalized properly only in the next step of second-order diamondization, i.e. the diamondization of the diamond.
Reflection on diamondization
A second-order diamondization is additionally to the first-order diamondization taking the diamond structure of the environment into account. This procedure becomes more plausible with the full notation of the automorphism as "O --> O o O --> O”, and therefore the environment as "O <-- O”. Hence there are two settings in the game: one is " [O | O]" as the kenogram with its environment and the other "[O | O ]" as the environment with its kenogram.
Diamond category theory has established a complementarity between categories and saltatories in diamonds. The whole configuration has to be thematized at once in both directions: from categories to saltatories and from saltatories to categories. This reflectional feature applied on the beginning of kenogrammatic systems is installing the double design of the beginning as a full chiasm of the inside and the outside of a kenogram.
Such second-order construction of diamond category theory had been sketched in “Diamondization of Diamonds” of “Diamond Theory".
Metaphorically, what is achieved is a formalization for the wording:” [Inside of the inside | Outside of inside] | [outside of the outside | inside of the outside]" as the metaphorical meaning of "[O | O] | [O | O ]".
Because the simultaneity of “Inside of the inside” and "Outside of inside” marked by "|" and the complementarity of the whole formula: "[O | O] | [O | O ]", a further formal explication is succeed by the mechanism of functorial interchangeability.
This might hint to the concept of a complementarity of “inverse duals" (Kent Palmer).
In the process of deconstructing the classical subject/object-model for subjectivity in Western philosophy Gotthard Gunther interoduced the fundamental distinctions of “subjective subjectivity" (sS), objective subjectivity”, (oS), and “objective objectivity”, (oO), for the common linguistic terms: I, Thou and It. Epsitemologically this corresponds to subjectivity, “knowledge” and reality.
Later, the combinatorics got some completenes with “objective subject” (Mitterauer, Toth).
In his metaphysical work of “pre-semiotic tetradic” semiotics, i.e. pre-semiotics, Alfred Toth studied all the combinatorial possibilities for semiotics, ontology, epistemeology and logic as “real polycontextural pre-semiotics as a Theory of the Night”.
The formal terminology of the quadralectics (Kent Palmer) of a Diamond Calculus might be involved for a first step towards an operational implementation of quadralectic semiotics
The terms (oO), (sO), (oS), (sS) shall be considered as the constituents of a quadralctic diamond and building a generative system (Erzeugenden System) for the interaction and reflection of such systems.
The hierarchical quadralectic diamond of epistemological distinctions insist on a (linearly) ordered universe of its epistemological constituents but tries to keep a kind of a operational balance of the constituents.
In contrast, the heterarchical quadralectic diamond of epistemological distinctions insists on the ‘metaphysical’ sameness, i.e. equi-valence of its constituents. This is called by Heidegger “Gleichursprünglichkeit” ("equiprimordiality (Dreyfus)). Quadralectic diamonds of distinctions are playing with the equi-primordiality of the distinction of quadralectic polycontexturality.
Each primordial distinction of the tetradic constellation, marked as a matrix, is opening up the framework of a calculus of the domain of such a distinction. Therefore, at each place of the matrix, a distinctional calculus has to be implemented. In other words, the marks which are building the matrix of the framework, are not themselves involved into the calculi they enable. Otherwise it would be possible to eliminate the matrix by the application of its distinctions. Thus, the primordial distinction, building the matrix, are the conditions of the possibility of systems of distinctions.
This difference in the double function of distinctions as enabeling systems or modi of distinction and as being part of a distinctional calculus is not reflected in the Calculus of Indication (G. Spencer Brown).
Toth gives a complete combinatorial description of all terms involved in the quadralectics of the primordial terms (sO), (oS), (oS), (sS).
Also Toth’s new semiotic approach is fundamentally designed as an action system, a “handlungstheoretische Semiotik” we are still missing an operational definition and a formal apparatus which would be able to generate the different quadralectic semiotic constellations of the actional systems.
In a further crucial step Toth is interpreting the ‘epistemological’ configurations with the help of Conway's “surreal numbers".
"Since the action schemata of the 4 monadic semiotic partial relations
(sO), (oS), (oO), (sS)
as well as of the 15 dyadic semiotic partial relations
((sO), (oS)); ((sO), (oO)); ((sO), (sS)); ((oS), (sO)); ((oO), (sO));
((sS), (sO)); ((oS), (oS)); ((oS), (oO)); ((oS), (sS)); ((oO), (oS));
((oO), (oO)); ((oO), (sS)); ((sS), (oS)); ((sS), (oO)), ((sS), (sS)).
are trivial, we restrict ourselves here to show up the 24 triadic and the 24 tetradic semiotic partial relations for all 15 pre-semiotic sign classes and their reality thematics together with the semiotic contextures from a 4-contextural 4-adic semiotic matrix. “
"Tetradic semiotic-logical partial relations :
((sS), (oO), (oS), (sO)); ((oO), (sS), (oS), (sO)); ((oO), (oS), (sS), (sO));
((oS), (oO), (sS), (sO)); ((sS), (oS), (oO), (sO)); ((oS), (sS), (oO), (sO));
((oO), (sS), (sO), (oS)); ((sS), (oO), (sO), (oS)); ((oO), (oS), (sO), (sS));
((oS), (oO), (sO), (sS)); ((sS), (oS), (sO), (oO)); ((oS), (sS), (sO), (oO));
((oO), (sO), (sS), (oS)); ((sS), (sO), (oO), (oS)); ((oO), (sO), (oS, (sS));
((oS), (sO), (oO), (sS)); ((sS), (sO), (oS), (oO)); ((oS), (sO), (sS), (oO));
((sO), (oO), (sS), (oS)); ((sO), (sS), (oO), (oS)); ((sO), (oS), (oO), (sS));
((sO), (oO), (oS), (sS)); ((sO), (sS), (oS), (oO)); ((sO), (oS), (sS), (oO)).
(Toth, Surreale Nacht, p. 7)
Diamond category theory may offer the appropriate structure to define and analyse Toth’s epistemological approach. On the other hand, the quadralectic diamond might be easier to apply and to generate the 4-foldness of action-oriented semiotics. Because the diamond calculus is offering only the abstract mechanism of transformations and not the specifications for concrete operations, say on the semiotics of ‘surreal numbers’, the specifics have to be defined additionally to the abstract diamond.
Subject/object differences are distinctions that have to be realized by cognitive and volitive actions. In the context of Toth’s actional semiotics all 4 distinctions are holding together, and building therefore a 4-fold or quadralectic structure. The internal relationality of this structure might be conceived as a diamond structure. This interpretation had been elaborated at “Triadic Diamonds” as a concretization of Gunther’s founding relations between actional distinctions albeit considered in a triadic and not yet in a tetradic setting. It has to be strongly emphasized that triadic and tetradic distinctions are not related to n-ary relations of classical relational logic of First-Order Logic (FOL).
At first there are specific diamond distictional transformation of the quadralectics.
An example might demonstrate the mechanism.
A quadralectic distinction of the distinctional semiotics might be defined by:
Recursivity of the just introduced quadralectic constructions is still extensional and not yet connected with retrogradeness.
An action-based semiotics with its involvement with a philosophical theory of cognition and volition is not yet well conceived if its recursivity is not connected with the retrogradness of kenogrammatic recursivity. Therefore the epistemological quadruple SEM = [sS, oO, sO, oS] shall be mapped onto kenomic recursivity and its further operations.
A contexture in the sense of polycontexturality might contain an action-based semiotic system SEM = [sS, oO, sO, oS].
Contextural interactions and reflections on action-based semiotics are getting accessible by kenogrammatic retrograde recursivity.
= a :
Quadralectics deals with the 4-fold structure of distinction systems. Insofar, all distinctions are performed inside the contexture of 4-fold distinctionality.
Quadralectics therefore might be thematized as a mono-contextural system or structure of distinctions.
It might also be thematized as a polycontextural system consiting of four equiprimordial distinctions each configuring a contexture.
Becaus of the proemiality of contextural systems such interplays are natural and are not giving reasons for conflics ot contradictions.
Hence, a theory of meta-distinctions is at place to thematize distinctional systems. In other words, olycontextural distinctions or meta-distinctions are distinctions between contextures and the results of thematization.
Thematization is the process of creative understanding.
A first topic obviously is the dissemination of distinction systems and the study of their behavior.
A second topic is the study of enaction and its memristive properties in polycontextural constellations of quadralectic distinction systems.
What are actions on ‘equiprimordial’ elements of the quadralectic matrix? Obviously, they are not distinctions in the sense of the Calculus of Indication (CI) of the Laws of Form. Otherwise the double cross action would annihilate the matrix and eliminate the distinctions like in the case of the GSP double cross ⌀.
The quadralectic matrix is the framework of epistemological reflection of distinction systems.
Hence, the operational system of description with:
is not to understand in an indicational way. Albeit Howe is not explaining his matrix epistemologically, an observation of an observation, , or a relation of a relation, , is not indicating a reduction to nil, but an elevation to a higher order of epistemological reflection.
Just as a rhetorical remark: Albeit that those typographic exercises might be recognized by the experts as an utter brain-fuck, it wouldn’t be a superfluous recommendation to check the miserability of the distinctions used in corresponding scientific endeavours.
A simple interpretation might hint to the possibility to understand isolated societal agents as self-reflexive systems with quadralectic properties of cognitive/volitive behavior towards their environment. With the presupposition that for a social system, or society, a singular self-reflexive agent is not yet enabling neither the existence of itself nor the existence of a societal system, a multitude of interacting, reflecting and intervening quadralectic agents have to be involved to run the game.
Hence, the epistemological and semiotic quadralectic of [(sS), (oS), (sO, (oO)] is not defining a societal system but a singular reflexive agent of a society. A society has at least to disseminate such quadralectic agents to build a society.
With the design of the quadralectic observer model there was probably some hope to realize it for a multitude of agents by a recursive repetition of the quadralectic structure of the distinction scheme, say in the sense that a structure of structure! might contain a full system with its own command structure!. But this happens in a singular framework of quadralectic distinction and the option to distribute the whole scheme as such over different loci is not yet recognized. In this sense, there is no fundamental difference between recursive repetitions of dyadic or triadic configurations.
At a first glace the scheme is a quadralectic structure but by a secaond glace it uncovers itself as a pentalectic system with d for the whole as description . Hence, [d; o, R, r, s]. Howe’s last paragraph hints to an ultimate state of (solipsitic) recursivity: “and finally the o elements are brought to the n-n=0 level, and drop out, leaving the expression: (Howe, 1970).
An operation like enaction which is shifting its object from one contexture to another obviously is possible only in a polycontextural framework.
Successor operations in kenogrammatics are retrograde recursive. This property for non-contextualized morphograms is fully reflected by diamond successor systems of morphograms including the kernel and the ’environment’ of the diamond morphogram.
m=2: [(S), (O)]
m=3: [(sS), (oS), (oO)]
m=4: [(sS), (oS), (sO)), (oO)]
[(ssS), (oS), (sO)), (oO)]
[(sS), (ooS), (sO)), (oO)]
[(sS), (oS), (ssO)), (oO)]
[(sS), (oS), (sO)), (ooO)]
[(sS, sS), (oS), (sO)), (oO)]
[(sS), (oS, oS), (sO)), (oO)]
[(sS), (oS), (sO, sO)), (oO)]
[(sS), (oS), (sO)), (oO, oO)]
Because of reasons of symmetry most environments are as results morphogrammatically equal to their counter-part. Non-symmetric morphograms are mirrored by inverse environments produced by a reflector. Because of the inverse construction of the environments, this simple structure is supported also by more complex morphograms. It holds trivially for symmetric morphograms.
Second-order diamond strategies shall be applied on the diamondization of the initials J1 and J2 of calculus of distincions.
Diamond coalition is defined as monomorphic morphogrammatic coalition with double environments. Hence, the kernel of the procedure is equal to the common monomorphic addition of morphograms without environments. The Coalition building is defining the environments too. Therefore, some morphogrammatic environments have to be put back into normal form. Thus, a form like becomes:
The operations on diamodized morphograms, like coalition and cooperation, are not just adding or multiplying morphograms but are reflecting their environments, too.
The morphogrammatic concept of substitution with its context rules shall be applied for diamondized morphogrammatic substitution. In a first step the reformulation of the substitution rules follows automatically from both, the definition for morphogrammatic substitution and the rules for diamondization.
The new concept of enaction might be extended to a second-order concept of enaction in distinctional or distinctive diamond calculi. Enaction rules are interesting new features of memristive systems. The diamondization of the enaction rules is supporting memristivity of the contexts of the enacting distinctions. Contexts (worlds, environments) are themselves involved into enactions. Thus, a memristive behavior is not just placed at a place but involved into the enaction of the places too.
The quadralectic (tetralemmatic, diamond) notation is enabling operations on the parts of the diamond complexions consisting of Inside, Outside and inside, outside, i.e. , short:.
Those operations applied to the quadralectic complexion have to preserve the rules of retrograde recursivity.
[Inside | Outside] | [outside| inside]:
[Inside of inside | Outside of inside] | [outside of Outside | inside of Outside].
Considering this observation, kenogrammatics has to start at the very beginning with the inscription of the difference of iteration/accretion. This observation is corresponding to the concept of “bi-objects” in diamond category theory.
At such a start of kenogrammatics there is no need for a dualism of iterativity and accretion but a chiasm is involved of the terms internal/external and accretion/iteration, producing the double determination of situations by the wording of iterative iterativity, accretive iterativity and iterative accretion, accretive accretion. Hence, basic terms in kenogrammatics are reflectional and second-order figures. In other terms, proemiality is opening up the beginnings of kenogrammatics.
To choose a beginning with a mark is putting a difference into the possibility of a choice for another mark of beginning. Such a difference in the notion of representation of a beginning by a mark shall be inscribed as a double beginning. The question is: As which representation is a kenogram inscribed? If it is inscribed as “a” then it is differentiated from another possible inscription, say “b”. If it is inscribed as “b” then it is differentiated from another possibility, say “a”. A further differentiation, say into “c” , would be redundant and irrelevant for the characterisation of a kenomic beginning.
On the other hand, philosophically, with double beginnings, the necessity of a unique and ultimate “coincidentia oppositorum” (Cusanus, Hegel, Gunther) is differentiated and dissolved.
A beginning for kenogrammatics is not in the mode of an is-abstraction, i.e. a = a, but in the mode of an as-abstraction, thus, producing beginning as a complementarity of the monads “a” and “b”.
This, obviously, is not the same as an isomorphism between different sign systems which differ in their alphabet.
With that, the notational decision for a representation is represented as the choice for an iterative or an accretive notation of a kenogram. Both interpretations are, in isolation and without their chiastic interaction, isomorphic. The present "Outline of morphogrammatics” is not yet reflecting this situations of disremptive - iterative/accretive - identifications of kenoms by the as-abstraction.
Hence, is it necessary to accept the unit element as a sole interpretation of a monadic multiplication?
As an example:  x [1, 2] = [ 1 x , 1 x [ 2]] = [1, 2],
Multiplication is a repetition. The multiplication “1 x " is a single repetition of  .
But repetitions in kenomic systems might happen as iterative and as accretive repetitions .
Hence, 1 x  =  for iteration, the same for 1 x  = , and different for accretion: 1 x [ 1] = , 1 x  = .
Iterative and accretive multiplication
Because  and [ 2] are kenomically equal as isolated monads,  =the kenomic difference disappears. And with it the semiotic difference too.
But in a contextual environment of a morphogram the monadic difference is playing its part of differentiation. Hence, there is a possibility to distinguish between iterative and accretive multiplication as it is possible to distinguish iterative and accretive succession and coalition . This is not multiplicative accretion inherited by addition but a genuine multiplicative accretion. That is, a repetition of a kenogram with itself is not necessarily equal the repeated kenogram. The same monad is equal as an iteration and different as an accretion . Thus, kmul
Similar ideas are presented at: “Lambda Calculi in Situations”.XXX