Rudolf Kaehr Dr. phil.^{@}

Copyright © ThinkArt Lab ISSN 2041-4358

Abstract

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.

The semiotician Alfred Toth is asking: “*Are there polycontextural signs?*” (Toth, p. 1, 26.04.2009)

Is the lack of identity a sign for polycontexturality (of signs)?*"However, representatives of polycontextural theory have often pointed out that semiotics is clearly a monocontextural system in which the logical Law of Identity (and the other 2-3 fundamental laws of classical thinking) are valid without restrictions.” *(ibd., p.1)

Distributed over some polycontextural notions, Toth shows *“that the classical semiotics has no identity and thus is polycontextural”*. Even if we could agree with this demonstration, the question remains: Is it enough for a semiotic system to be non-identical to be polycontextural?

Obviously, the first obstacle in the process to be declared as polycontextural on the base of a lack of identity is the concept of negation involved in the “non” of non-identical. Negation is too much in complicity with the system it is negating.

Toth gives his analysis an interesting turn in inverting the starting question *"Are there polycontextural signs?”**"So, from here, the question should not be if there are polycontextural signs, but if there are monocontextural signs.In classical semiotics, polycontexturality is hidden in the triadic-trichotomic structure of a seeming monocontexturality.”* (ibd., p.2)

http://www.mathematical-semiotics.com/pdf/Polyc.%20signs%20question.pdf

Instead of denying identity, a complementary gesture might manage to introduce a multitude of identities as a refutation of the dominance and hegemony of the logical and semiotic principle of identity.

Independent of the complexity of the semiotic matrix, the sign classes and their sign relations are always separated by their identity.

Traditionally there are 10 sign classes recognized. All of them are properly distinguished from each other.

A dicent is a dicent and not a rhema or a symbol or a legi-sign, etc.

In an *iterative* sense, some complications might be produced for the 10 sign classes. Following Bense, we get, e.g., an argumentative-symbolic legi-sign as the semiotic modeling of formal languages (Kalkülsprachen), or a dicentic-symbolic sin-sign for “epic languages” and a dicentic-indexical legi-sign for ”programming languages”.

To all those isolated sign classes and sign relations, semioticians are delivering more or less convincing linguistic, media-related or physical examples.

Such identity constructs are reasonable for traffic systems and other unambiguous sign-related situations.

The intention to focus on identity of definitions and examples is not necessarily a semiotic action but an action, i.e. modeling, guided and ruled by the interests of identity, i.e. of identical identification.

With identity and identification, a specific form of rationality is supposed.

Similar restrictions are introduced by Chomsky’s grammar. There is a strict distinction between meaningful and meaningless sentences. Nevertheless, all meaningless sentences are easily domesticated in a game which is opening up meaning for all.

My thesis therefore is: To all examples and to all distinctions there are always overlapping other distinctions involved that are suppressed, denied and rejected by such an act of identification.

As an example, the concept of “natural number” might be mentioned. Even for such an elementary concept like the “natural numbers” there is no identitive definition available. Most definitions (introductions, postulations) are circular or lost in the abyss of non-foundedness.

Hence, identification in the mode of identity is an ontological and epistemological procedure and follows not semiotic or sign theoretical necessity. Again, semiotics in a general sense, thematized as an identity system, is ruled by non-semiotic decisions.

In other words, semiotics as an identity system of whatever complexity is dominated by logocentric preconditions, in fact by linguistic and logical notions.

Therefore, semiotic distinctions in a polycontextural paradigm are not governed by the ontological is-abstraction but are involved into the free interplay of actional as-abstractions.

From a polycontextural point of view, signs are results of actions and actions are not necessarily reducible to single agents but might be realized as interactions between a multitude of mediated actor-systems. Each semiotic action is simultaneously involved and coupled with its environment, which contains itself a multitude of agents.

It turns out that classical semiotic systems are not actional but structural or relational and are based on a singular epistemic instance, i.e. interpretant.

A closer analysis of sign processes makes it obvious that signs are always intrinsically interwoven and overlapped with other signs. Signs as representamens are representing entities of a given world; polycontexturality is opening up worlds. Thus, signs in polycontextural situations are not simply representational but evocational. Such evocativeness of signs is not yet well studied. It is also not grasped by Bense’s creativity function of signs.

But this connectedness of signs in polycontextural situations is not the classical statement of the system-dependency of signs, i.e. the statement that signs are not occurring in isolation but necessarily in or as a system.

Semiotics in the sense of Bense and Toth is build on the base of the so called semiotic matrix, i.e. the Cartesian product of the sign components.

*"Ein Zeichen ist danach eine triadische Relation, genannt, Interpretantenbezug, welche eine dyadische Relation, genannt Objektbezug, und eine monadische Relation, genannt Mittelbezug enthält. Da eine Relation eine Teilmenge eines kartesischen Produktes ist, kann man auch sagen, das Zeichen sei eineTeilmenge von Teilmengen von kartesischen Produkten.” *(Toth, Bühler, p.2, 2009)

Then, sign classes are interpreted as parts of the matrix. These parts are disjunct and well separable from each other. There is no overlapping or penetration from and by other sign classes, i.e. classical semiotics is based on disjunctively separable sign classes.

As a reasonable result, such a kind of semiotics is not dealing with the whole matrix but only with its parts, i.e. the sign classes.

Transclassical semiotics, in the sense of polycontextural semiotics, is not in such a comfortable situation. Polycontexturality is not understood simply as a multitude of semiotic contextures but by its interactivity, reflectionality and interventionality between a plurality of contextures.

It is no surprise that, e.g. in polycontextural logic, the overwhelming majority of logical functions are not uniformly separable into cleanly defined sub-systems but are highly interwoven. That is, junctions, like disjunctions, conjunctions and implications, are a minority compared to the bulk of transjunctional logical functions.

It has to be stipulated therefore, that the same situation holds for a polycontexturally conceived semiotics.

From a strict terminological point of view it might be obsolete and confusing to still call this construction semiotics. A more appropriate title would be a mediation-system for interacting semiotics.

Hence a reading of a polycontextural semiotic matrix with the aim to collect sign classes doesn’t work anymore as a separation of mono-contextural sign constellations, like (3.1 2.1 1.1).

Such a reading of a polycontextural constellation obviously is producing “wild” beasts, “verwilderte Matrizen” (Toth), of, probably, not much use.

Polycontextural semiotics is forced to accept the semiotic tissue as a whole, i.e. as a game of interplaying contextures and their semiotic operations.

Hence, polycontextural semiotics is not reducible to separable sign classes. It always has to deal, at least for triadic semiotics, with the whole matrix. For more complex semiotics, a new kind of separability has to be studied, i.e. the separability of the general matrix into its overlapping sub-matrices.

Toth’s question *“Are there polycontextural signs?”*, thus has, at first, to be denied. By definition and tradition, signs are not polycontextural. (And there is obviously no such a thing like a “keno-sign”, too.)

In contrast to the just mentioned ‘holistic’, i.e. poly-contextural interpretation of arbitrary semiotic matrices as interplaying sub-matrices, Thot is giving an interpretation of polycontextural matrices by the means of classical strategies.

Even a *pentadic* matrix gets its *separated* sign classes, i.e. 5-Zkl = (a.b c.d e.f g.h i.j) with a, ..., j ∈{1, ..., 5}, this time not triadic but pentadic. But, as Toth is observing correctly, highly wild situations are disturbing such isolative interpretations.

**Arbitrarity**

"3. Kaehrs komponierte pentadische Matrix suggeriert grösstmögliche Arbitrariät bei der Kompositionen n-adischer und m-adischer zu (n+m-1)-adischer Matrizen und umgekehrt zur Dekomposition (n+m)-adischer Matrizen in (n-1-m)-adische und/oder (n-m-1)-adische Matrizen. Die einzige Anforderung an die “Richtigkeit” der komponierten Matrix wäre dann, dass die zueinander inversen Subzeichen die gleichen kontexturellen Indizes bekommen (z.B. (2.3) und (3.2), (1.5) und (5.1), etc.). In letzter Instanz führt diese Arbitrarität also dazu, dass in Übereinstimmung mit einer obigen Festellung die abstrakte Form einer pentadischen Zeichenklasse als

5-Zkl = (a.b c.d e.f g.h i.j)

mit a, ..., j ∈{1, ..., 5}

anzusetzen ist. Da ferner die triadischen Hauptwerte a, c, e, g, i nicht mehr paarweise verschieden sein müssen, kann jede x-beliebige Folge von 6 Ziffern natürlicher Zahlen als pentadische Zeichenklasse interpretiert werden.’" (Toth, Interakt Sem1Sem2, p. 3, 2009)

This arbitrarity might lead to a wild and nonsensical use of the concept of sign matrices and sign classes. Toth gives a possible solution for a reduction of the “wilderness” of the pentadic situation for “Haupt-Zeichenklassen” and “Neben-Zkln”.

**“Haupt-Zeichenklassen” "**4. Eine Einschränkung für diese völlig

(a = 1) --> b = 1, c = 5, d = 4, e = 1

(a = 2) --> b = c = d= 2, e = 4

(a = 5) --> b = c = d = e = 5

2.2. (5-Zkl = (3.a 3.b 3.c 2.d 3.e)) --> a = e = 5, b = 4, c = d = 3

Die Reihen der Matrizen enthalten triadische Sprünge und Wiederholungen [...]." (ibd., p. 3/4)

Thus, a sign class with “strange-values” is producing some kind of *jumps* and *gaps* in the arithmetics of semiotics. Toth mentions the examples:

(1-3-3-4-5) of the colum (1),

(3-2-3-4-4) of the colum (2) and

(3-2-3-3-3) of the colum (3) of the matrix below.

The colums (4) and (5) are not disturbed by “stange-values".

The same happens to the rows of the matrix.

However whatever kind of restrictions are introduced for a reasonable handling of complex sign classes, with m>=3, the strategy of selecting a single isolated chain of *“prime signs” *out of the matrix remains the same.

Without doubt, there might be some interesting insights possible with this approach, but there will still be an overwhelming majority of situations excluded, i.e. not interpreted as reasonable semiotic constellations. With that, a new, meta-semiotic problem occurs: What are the criteria of exclusion of the non-semiotically interpreted constellations? In other words, a criterion to distinguish between acceptable and non-acceptable constellations has to be introduced.

Again, as most junctional “value-sequences” in polycontextural logic are disturbed by external values, Gunther’s “Fremdwerte”, which are getting a logical meaning only in the context of an interplay between different mediated logical systems, the “Fremdwerte” of semiotic sign classes incur the same destiny: they are members of other sign classes interacting together in the complex semiotic game.

Therefore, polycontextural semiotics has to study, at least, both directions of the interplay: the interactional (reflectional, interventional) aspect between contextures, i.e. the transcontextural interplay, and the intra-contextural aspect of the disturbance by transcontextural interpenetrations.

Instead of excluding “strange” sign classes or to stretch adventures interpretations about gaps and jumps in the chain of prime signs, their origin in the complexity of polycontextural semiotics has to be considered first.

Because such situations are fundamentally different from semiotic approaches, the idea of *textemes* had been introduced. From the position of the idea of textemes, signs in a semiotic sense, are reductions of textemes.

Therefore, a first step to a general theory of interactional semiotics on the base of the new concept of *textemes*, i.e. bi-sign systems or anchored diamonds, consisting of the semiotic intra-kernel and the semiotic internal/external environments, and its interplay, is proposed.

A *texteme* consists of two diamondized anchored signs, i.e. bi-signs, inter-playing together by their mediated external environments. Hence, a texteme is an interplay of two bi-signs. A bi-sign is a diamondized anchored sign, i.e. a sign with intrinsic environments and its anchor.

This is a kind of bottom up introduction. Because we know signs and have not yet experienced textemes, this way of building up textemes is legitimate. But nevertheless, it works only because we know how to construct textemes out of signs which are not able to offer any of the principles of textemes, that are needed to realize such a construction, like their chiastic interplay between the environments of signs and the anchoring of signs.

As we know well enough, signs lack environments, there is no chance to construct out of signs in a sign-theoretical sense a semiotic environment of the sign concept. And obviously, there is no such mechanism as a chiasm in the sense of proemiality for signs. And again, semiotics is not offering any insight and mechanism for anchoring signs. Hence, neither environments, internal and external, nor interactions between signs based on their environments and their anchoring are conceivable.

These statements are surely in conflict with the well established interactional socio-, bio-, zoo-semiotic programs as well as with the advances in computational semiotics. From the point of view of polycontextural and diamond theoretical approaches to sign theory, those programs have to be seen as *applications* of classical, *a priori* non-interactional semiotics, onto semiotics, and not as anything else. Their merits are to be communicable in a society of traditionally trained knowledge-mongers.

Hence, a decomposition chain might clarify the concept of texteme:

A *texteme* is decomposable to its interacting *bi-signs* by excluding its chiastic interactivity.

A semiotic *diamond* is a bi-sign, de-rooted from its *anchor*,.

A single *bi-sign* is disconnected from its neighbor bi-sign, hence it is a bi-sign without interaction but realizing an anchored semiotic diamond with its isolated, and hence restricted, *environment*.

A *sign* is a semiotic diamond, depraved from its *environment* and its *anchor*.

This decomposition from the texteme to the sign has no reverse: There is no semiotic mechanism per se to construct out of semiotics the concept of textemes.

The *complexity* of a basic texteme is 12, i.e. 2x3 for its “*signs*”, 2x2 for its *anchors*, and 2x1 for its *environments*.

It might be asked if such matrices exists as semiotic matrices and not merely as mathematical matrices. A possible answer might be given with a semiotic interpretation of the texteme construction.

Sign shall correspond to the sign Such 3x3-matrices are well accepted as semiotic matrices. Therefore, the step to compose such matrices to a 5x5-matrix shouldn’t cause to many problems. Semiotic matrices are occurring as numeric matrices and as matrices over M, I, O.

The idea of non-identical but polycontextural and multi-layered semiotic constellations seems to be accessible for formal treating within such a construction of interplaying semiotics. Hence, such inconsistent situations, where a dicent appears at once as a rhema, including all other kinds of overlapping and metamorphosis, are getting a formal framework for their interplay.

With that in mind, the semiotic interpretation of the texteme below follows naturally. It is an example, how to interpret matrixes for polycontexturally conceived sign-complexions, i.e. textemes.

**Procedure**

The (OMI)-matrix SCIis translated into the corresponding numeric matrix MMFor both, a matrix-composition with environments and anchors are presented. The result of this composition, with the matching conditions (overlapping) 3.3_{1}≡ 3.3_{2}, is written as Texteme The environments of Sem_{1} and Sem_{2} are reduced to _{1} ↔ _{2} corresponding to the overlapping conditions, and omitting the micro-environments of the 2-sub-systems of the triadic matrices.

More *precisely*, the matching conditions for the composition of overlapping matrices are not the matching conditions of *concatenational* composition of morphisms but *overlapping* conditions of different complexity, in this case of length 1. Hence, concatenational matching conditions for morphisms are overlapping in zero elements, while the proposed composition of matrices is overlapping in one element, here 3.3.

Again, 3.3_{1} is not the codomain of Sem_{1} and 3.3_{2} is not the domain of Sem_{2} of the composition Sem_{1} o Sem_{2}. Thus, it is just an abbreviation to call those “matching conditions”, MC, simply matching conditions and not *overlapping* matching conditions. Such overlapping conditions had been called “mediation conditions”, MC, in the “Matrix" paper.

From a logical perspective, overlapping is producing contradictions. Therefore, a proper treatment of overlapping patterns has to consider its *morphogrammatic* foundation.

The term “polycontextural sign” is an *abbreviation* for the wording: *“Signs in polycontextural constellations”*. This means that signs and sign systems are conceived as distributed and mediated, i.e. disseminated over the kenomic matrix.

In the same sense, as there is no poly-Lambda Calculus, but lambda calculi disseminated over the kenomic matrix, and the label “poly-Lambda Calculus” is a simple, maybe misleading, abbreviation, polycontextural semiotics or poly-semiotics, are abbreviations for the dissemination of semiotics.

This is a *conservative* interpretation because it is conserving the original concept of signs, i.e. its triadic-trichotomic structure, and is not changing or deconstructing it towards another conception. But the experiences with conservative expansions are enabling new decisions which are deliberating from the acceptance of the basic system and its restrictions.

From the point of view of polycontextural dissemination, even classical semiotics appear as polycontextural, simply because it is distributed too, albeit only over a single place and not able to be aware of it. What's called its blind spot. **More exercises**For the classical matrix MM the disjunctivity of the sign classes holds.

All sign classes composed out of the matrix MMare *homogeneous*, i.e. clean of “strange values”, i.e. id(MM) = MM.

All combination of the original matrix MM are produced by *operations* on the matrix. A set of important operators had been introduced as the super-operators, sop={id, perm, iter, repl, red, bif}. The above interactions are based on the interactional operation “bif”.

Because there is no space offered by the notation of the matrix MMfor iterations and replications, the presentation is still a short for the full scheme, as it was introduced in previous papers.

The sign class (O.O_{1} O.M_{2} O.M_{1}) with its contextural type (1, 2, 1) is obviously disturbed by the “strange” value O.M, its meaning isn’t easy to determine as an isolated event. Studied as the result of an interaction between the two matrices, the meaning is well defined as a *rejectional* value from the position of the first matrix and as a *penetrational* value from the position of the second matrix.

**Composition and decomposition**Polycontexturality is mainly about

Hence, a sign class like 5-Zkl = (5.a 4.b 3.c 3.d 1.e) with a, ..., e ∈{1, ..., 5} has to be studied in a double way:

Hence, the sign class 5-Zkl is understood as a composition of the sign class 3-Zkl, which is representing the Peircean semiotics distributed over 10 different places in the sign class 5-Zkl.

The advantage of the decomposition method is clear. Each sign class m-Zkl is decomposable into its sub-systems, the distributed sign class 3-Zkl.

There is no need to invent infinite many different semiotics for arbitrary m-Zkl.

On the other hand, the distribution method is not restricted to triadic-trichotomic semiotics. For each n<m, there is a distribution of n-Zkl in m-Zkl.

**Morphograms for sign classes **If we insist that 3-Zkl is defined as <3.x 2.y 1.z>, then obviously, patterns like <3.x 4.y 5.z> or <1.x 4.y 5.z> are not defining Peircean sign classes at all.

The distribution of 3-Zkl over the places offered by m-Zkl is forcing an

With that, it is natural to introduce the remaining 4 morphograms for m=3 as

morph() = {MG}

The values of x, y, z ∈ {1, 2, 3} have not been considered.

This exercise was done with my paper “*Interactional operators in diamond semiotics*” and “*Matrix*” for the case of complexity 3 and 4. Hence, again for 4 and further results for 5.

**Generalizations**

The method of composition/decomposition holds generally for n-adic constellations of sign classes too.

m-ZR = o , m, n>=3.

**Further readings:**

http://www.thinkartlab.com/pkl/lola/Transjunctional%20Semiotics/Transjunctional%20Semiotics.html

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

http://www.thinkartlab.com/pkl/lola/Abacus.pdf