(************** Content-type: application/mathematica ************** CreatedBy='Publicon 1.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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.\ \>", "Abstract"] }, Open ]], Cell[CellGroupData[{ Cell["Semiotics and polylogics", "Section"], Cell[CellGroupData[{ Cell["Motivation", "Subsection", FontFamily->"Verdana"], Cell[TextData[{ "Transjunction, 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.", Cell[BoxData[ FormBox[ SuperscriptBox["", ButtonBox[ CounterBox["Endnote", "computational semiotics"], ButtonData:>Cell[ "Computational semiotics is interested in modeling interactions \ in computational scenarios.\[LineSeparator]As much as there is no proper \ logic of interaction there is even much less developement in computational \ semiotics. There is not even an awareness about the conceptual lack of \ interactivity constructs in theoretical semiotics. Despite the many \ applicative approaches to semiotic interactions, e.g. in human-computer \ interface research, it seems, that theoretical and foundational resarch for a \ semiotic theory of interaction and reflection is not supported.\ \[LineSeparator]", "Endnote", CellTags -> "computational semiotics"], ButtonStyle->"NoteKey", ButtonNote->"computational semiotics"]], TraditionalForm]], "Note"], "\[LineSeparator]", "\[LineSeparator]The semiotic matrix is introduced as the \ \[OpenCurlyDoubleQuote]Cartesian product\[CloseCurlyDoubleQuote] of sub-signs \ (Bense, Toth). 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In order to build a a ", FontSize->12], StyleBox["sign class", FontSize->12, FontWeight->"Bold"], StyleBox[ ", one sub-sign has to be taken out of each of the \ three rows, the rows thus being different.\"", FontSize->12], "\[LineSeparator]", StyleBox["\"", FontSlant->"Italic"], "Therefore, ", StyleBox["sign sets", FontWeight->"Bold"], " like *(3.1 3.2 1.3), *(2.1 2.2 1.2), *(1.1 1.3 3.1) \ are not considered sign classes.\[CloseCurlyDoubleQuote] ", " (Toth, Ghost, p.9)" }], "Text", Editable->True, CellMargins->{{Inherited, 70.0625}, {Inherited, Inherited}}, TextJustification->0]} }], "TableSubGrid", GridBoxOptions->{ColumnWidths->{0.32, 0.68}}]} }], "TableMasterGrid"], TabularForm]], "Table", PageWidth->520, GridBoxOptions->{ColumnWidths->{0.32, 0.68}, GridFrame->False}], Cell[TextData[{ "Cartesian ", StyleBox["products", FontSlant->"Italic"], " as a conceptual point of contact.", Cell[BoxData[ FormBox[ SuperscriptBox["", ButtonBox[ CounterBox["Endnote", "cartesian"], ButtonData:>Cell[ TextData[ {"Independent of later steps of ", StyleBox[ "abolishing", FontSlant -> "Italic"], " restrictions in the traditional definition of sign \ classes by Toth\[CloseCurlyQuote]s studies, the concept of a Cartesian \ product remains a fundamental construction to build up a semiotic system.\ \[LineSeparator]\[LineSeparator]This fact allows to study the semiotic matrix \ under a different angle: the ", StyleBox[ "polycontextural", FontSlant -> "Italic"], " approach of dissemination, i.e. distribution and \ mediation, of sub-systems as a mechanism to construct and to deconstruct the \ semiotic matrix. In this sense, an extension of the semiotic matrix for \ complex sign systems, called polysemiotics, is introduced.\[LineSeparator]\ \[LineSeparator]To use Cartesian products doesn\[CloseCurlyQuote]t mean that \ they will remain stable in the development of a general theory of polylogics \ and polysemiotics. As shown at other places, what was a good starting point, \ became the main obstacle for further developments.\[LineSeparator]\nThis \ disseminative approach to the semiotics matrix allows to introduce a \ comparison of semiotic and logical constructions. As main operators of \ logical interaction, the polylogical ", StyleBox[ "transjunctions", FontSlant -> "Italic"], " had been studied ", StyleBox[ "in extenso.(Kaehr, 1978, 2005)", FontSlant -> "Italic"], "\n\[LineSeparator]In analogy and translation or \ transposition from the polycontextural to the semiotic topics, semiotic \ interactions between semiotic sub-systems shall be introduced. Semiotic \ sub-systems are a result of a decomposition of the semiotic matrix into its \ sub-systems. Such a decomposition is dynamic, depending on the complexity of \ the semiotic matrix. In this paer, only two cases are introduced. The \ decomposition into (2, 2)-subsystems, with S", Cell[ BoxData[ FormBox[ RowBox[ { RowBox[ { AdjustmentBox[ SubscriptBox[ "\[Null]", "1"], BoxMargins -> {{ 0, -0.10000000000000001}, {0, 0}}], "=", RowBox[ {"{", RowBox[ {"1", ",", " ", "2"}], "}"}]}], ",", " ", RowBox[ {"S", FormBox[ RowBox[ { AdjustmentBox[ SubscriptBox[ "\[Null]", "2"], BoxMargins -> {{0, -0.10000000000000001}, { 0, 0}}], "=", RowBox[ {"{", RowBox[ {"2", ",", " ", "3"}], "}"}]}], "TraditionalForm"]}], ",", " ", RowBox[ {"S", RowBox[ { FormBox[ RowBox[ { AdjustmentBox[ SubscriptBox[ "\[Null]", "3"], BoxMargins -> {{ 0, -0.10000000000000001}, {0, 0}}], "=", RowBox[ {"{", RowBox[ {"1", ",", " ", "3"}], "}"}]}], "TraditionalForm"], "."}]}]}], TraditionalForm]], "TextSubscript", SingleLetterItalics -> False], " And the decomposition into (3, 3)-subsystems of a \ polysemiotic system Sem", Cell[ BoxData[ FormBox[ RowBox[ { AdjustmentBox[ SuperscriptBox[ "\[Null]", RowBox[ {"(", RowBox[ {"4", ",", "2"}], ")"}]], BoxMargins -> {{0, -0.10000000000000001}, {0, 0}}], "."}], TraditionalForm]], "TextSuperscript", SingleLetterItalics -> False]}], "Endnote", CellTags -> "cartesian"], ButtonStyle->"NoteKey", ButtonNote->"cartesian"]], TraditionalForm]], "Note"], "\[LineSeparator]The aim of polycontextural semiotics is to design a \ dynamic sign theory without any fixation on a special or privileged n-ary and \ m-adic system. Another attempt to augment the structural and architectonic \ flexibility of semiotics is proposed by Toth\[CloseCurlyQuote]s approach to a \ 3- and 4-dimensional semiotics resulting in complex topological structures. \ (Cf. Transit-Korridor, 2009)" }], "Text", TextJustification->0, FontFamily->"Verdana", CellTags->":note:cartesian"] }, Open ]], Cell[CellGroupData[{ Cell["Is there a privileged number for dissemination?", "Subsection", FontFamily->"Verdana"], Cell[TextData[{ "An introduction of the topics of polycontextural formal systems, like \ polylogics, poly-arithmetic or polysemiotics, has to deal with the question \ of a ", StyleBox["privileged number", FontSlant->"Italic"], " of a possible extension of 2-valued logics, semiotics and arithmetic. \ This has been thematized at different places and can\[CloseCurlyQuote]t be \ exposed ", StyleBox["in extenso", FontSlant->"Italic"], " in this ", StyleBox["Short Study", FontSlant->"Italic"], " to Polysemiotics.", Cell[BoxData[ FormBox[ SuperscriptBox["", ButtonBox[ CounterBox["Endnote", "privilege"], ButtonData:>Cell[ "Nevertheless, a specific redundancy has to be repeated because \ of its established and deep-rooted stupidity. The more or less only answer or \ feed-back to my work, when I was emphasizing the importance of a number, e.g. \ 4, I got was, \[OpenCurlyDoubleQuote]Why an extension to 4 and not to 7 or 13 \ or 5112?\[CloseCurlyDoubleQuote] Nobody ever questioned the fact that their \ question was based on the number 2 (TWO). And I surely never privileged a \ single natural number of the established number system.\[LineSeparator]\ \[LineSeparator]A criticism of such an idea of a privilege of a single \ natural number was perfectly done long before by Aristotle with his \ refutation of Pythagorean number theory. \[LineSeparator]It seems to be \ better to live and die with the number TWO than to question it.\ \[LineSeparator]As far, it was an important scientific step by Peirce to \ introduce his triadic-trichotomic semiotics.", "Endnote", CellTags -> "privilege"], ButtonStyle->"NoteKey", ButtonNote->"privilege"]], TraditionalForm]], "Note"], " ", StyleBox[ButtonBox["http://www.thinkartlab.com/pkl/media/Short%20Studies/\ Short%20Studies.pdf", ButtonData:>{ URL[ "http://www.thinkartlab.com/pkl/media/Short%20Studies/Short%20Studies.\ pdf"], None}, ButtonStyle->"Hyperlink"], FontSize->11], " " }], "Text", TextJustification->0, FontFamily->"Verdana", CellTags->{":note:INV-Toth", ":note:privilege"}], Cell[CellGroupData[{ Cell["\<\ Gunther\[CloseCurlyQuote]s approach to many-valued logics\ \>", \ "Subsubsection", FontFamily->"Verdana"], Cell[TextData[{ "In the advent of many-valued logics there was a big run to find a \ privileged number of truth-values, logical functions and their semantic \ interpretation.\[LineSeparator]", StyleBox["Gunther\[CloseCurlyQuote]s Program. ", FontSlant->"Italic"], "Each single value and each single logical function is entitled to have a \ logical meaning.\[LineSeparator]It is absurd to chase for the meaning of \ logical values and functions for arbitrary many-valued systems. Special value \ classes of some interest had been studied by logicians for 2, 3, 4, and \ infinite. \[LineSeparator]Hence, a method, like the arithmetic position \ system which is able to determine arbitrary numbers on a finite base system, \ has to be invented. This was Gunther\[CloseCurlyQuote]s approach to \ many-valued place-value systems (Stellenwertlogik).\[LineSeparator]Semiotics, \ today, is still in a pre-decompositional, i.e. conceptionally static state of \ research, not necessarily in the spirit of Peirce\[CloseCurlyQuote]s \ \[OpenCurlyQuote]speculations\[CloseCurlyQuote]." }], "Text", TextJustification->0, FontFamily->"Verdana", CellTags->":note:INV-Toth"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Gunther\[CloseCurlyQuote]s criticism of Peirce/Bense\ \[CloseCurlyQuote]s trinitarism\ \>", "Subsubsection", FontFamily->"Verdana"], Cell[TextData[{ "Gunther has taken the opportunity to write down and publish, what was \ clear at least since the advent of his place-valued logics in the 50s. That \ the restriction of Peirce and his decade long friend Max Bense is a heritage \ of Western and Christian thinking, which was conceived by Gunther as dead, at \ least since Nietzsche and American Cybernetics. ", StyleBox[" ", FontSize->10], StyleBox[ButtonBox["http://www.vordenker.de/ggphilosophy/theor-hegel-logik.\ pdf", ButtonData:>{ URL[ "http://www.vordenker.de/ggphilosophy/theor-hegel-logik.pdf"], None}, ButtonStyle->"Hyperlink"], FontSize->10], ButtonBox["\[LineSeparator]", ButtonData:>{ URL[ "http://www.vordenker.de/ggphilosophy/theor-hegel-logik.pdf"], None}, ButtonStyle->"Hyperlink"], " " }], "Text", TextJustification->0, FontFamily->"Verdana", CellTags->":note:INV-Toth"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Beyond Gunther\[CloseCurlyQuote]s stance on numbers\ \>", \ "Subsubsection", FontFamily->"Verdana"], Cell[TextData[{ "Gunther repeated the argumentation of Aristotle against a privileged \ number, say for his m-valued polycontextural logic, but was nevertheless the \ only one who himself introduced a (Neo)Pythagorean concept and some formalism \ of transclassic numbers, called \[OpenCurlyDoubleQuote]Philosophical numbers\ \[CloseCurlyDoubleQuote] (Gattungszahlen). \[LineSeparator]", StyleBox["In short:", FontWeight->"Bold"], " In polycontextural logic, no special number is privileged because each \ number has its own specific characteristics, hence its own privilege. With \ this paradoxical characterization of \[OpenCurlyQuote]privileged\ \[CloseCurlyQuote]/'unprivileged numbers, the whole idea of a privileged \ number in the traditional sense is obsolete. But this polycontextural \ magnitude of de-privileged privileges is based on a strategy of a finite \ structure, the number 4 of \[OpenCurlyQuote]", StyleBox["tetraktomai", FontSlant->"Italic"], "\[CloseCurlyQuote], i.e. of ", StyleBox["doing", FontSlant->"Italic"], " the tetraktys, also called proemial relationship or diamond strategies. \ Again, this number of the ", StyleBox["praxis", FontSlant->"Italic"], " of tetraktomai, i.e. diamodization, isn\[CloseCurlyQuote]t a member of \ any arithmetical number system.", Cell[BoxData[ FormBox[ SuperscriptBox["", ButtonBox[ CounterBox["Endnote", "tetraktys"], ButtonData:>Cell[ TextData[ { "Die systematische Auszeichnung der 4 mag \ willk\[UDoubleDot]rlich erscheinen; warum nicht die 3 oder die 11 und warum \ eine und nicht mehrere oder gar alle Zahlen? \nDie Kritik Aristoteles' an der \ pythagor\[ADoubleDot]ischen Auszeichnung der 4 bzw. der 10 setzt die Linearit\ \[ADoubleDot]t der nat\[UDoubleDot]rlichen Zahlen und das Prinzip der \ potentiellen Realisierbarkeit voraus. Erst dann entstehtein Konflikt zwischen \ der Reihe der nat\[UDoubleDot]rlichen Zahlen, d.h. einer beliebigen Zahl und \ der Auszeichnung einer Zahl dieser Reihe als Gattungszahl der Reihe selbst. \n\ Wird jedoch unter der 4 die 'Gattungszahl' der 4 Schrifttypen der Graphematik \ verstanden, also das Geviert der geschlossenen Proemialit\[ADoubleDot]t, dann \ entsteht kein Widerspruch zwischen Auszeichnung einer Zahl und der \ Zahlenreihe selbst. Die 4 er\[ODoubleDot]ffnet die Vielfalt der Zahlensysteme \ der Polykontexturalit\[ADoubleDot]t, liegt jedoch als solche nicht in der \ Reihe der nat\[UDoubleDot]rlichen Zahlen einer beliebigen Kontextur. \ Aristoteles lehnt die Auszeichnung der 4 (und mit ihr die der 10) ab, ist \ aber selbst gezwungen, die 1 auszuzeichnen. Denn die Uni\[Dash]Linearit\ \[ADoubleDot]t der Reihe der nat\[UDoubleDot]rlichen Zahlen setzt die 1 als \ Ma\[SZ] der Zahlen und als unum der Unizit\[ADoubleDot]t der Reihe voraus. \ Die Auszeichnung der 4 unter der Voraussetzung der Uni\[Dash]Linearit\ \[ADoubleDot]t hei\[SZ]t, da\[SZ] die vertikale Sprachachse der Graphematik \ auf die horizontale Linie der nat\[UDoubleDot]rlichen Zahlen projiziert wird. \ \nDer Widerspruch zwischen 'Gattungszahl' und 'Reihenzahl' ist somit das \ Produkt einer Verdeckung, einer Koinzidenz der beiden 'Zahlenachsen'. Dabei \ wird auch stillschweigend vorausgesetzt, da\[SZ] die Zahlziffern selbst \ eindeutig und nicht einer \[CapitalUDoubleDot]berdetermination ausgesetzt \ sind. Aristoteles' Kritik verf\[ADoubleDot]ngt auch dann nicht, wenn sich die \ 4 vertikalen Sprachschichten nicht legitimieren lassen und ihre Anzahl vergr\ \[ODoubleDot]\[SZ]ert oder verkleinert werden mu\[SZ]. \nDie Kritik an der \ Auszeichnung einer bestimmten Zahl vor der anderen durch die transklassische \ Arithmetik, kann sich jedoch nicht auf Aristoteles berufen, denn seine Kritik \ umfa\[SZ]t generell die Mehrlinigkeit der platonischen Zahlen und diese \ wiederum ist ein wesentlicher Charakter der transklassischen Zahlentheorie. \n\ So argumentiert G\[UDoubleDot]nther: \:201eAristoteles ist im Recht. Es ist \ notwendig, konsequent zu sein. Entweder sehen wir uns gezwungen, nicht nur \ der Monas, der Dyas, der Triade usw., kurz jeder pythgagor\[ADoubleDot]ischen \ n\[Dash]Zahl den Rang einer ontologischen Idealit\[ADoubleDot]t zuzubilligen \ oder aber die ganze Problemsicht ist verfehlt und keine Zahl hat die W\ \[UDoubleDot]rde einer Idee\[Dash]au\[SZ]er vielleicht die Einheit und die \ aoristos duas, die man aber beide nicht als Zahlen zu betrachten braucht. Da\ \[SZ] die zweite Auffassung nicht haltbar ist, lehrt die Geistesgeschichte \ vergangener Epochen.\[OpenCurlyDoubleQuote] (70) \nG\[UDoubleDot]nther \ insistiert also auf der Auszeichnung jeder Zahl und nicht nur der pythagor\ \[ADoubleDot]ischen Tetraktys. D.h. jede Zahl hat die W\[UDoubleDot]rde einer \ Idee und erh\[ADoubleDot]lt somit eine logisch\[Dash]strukturelle Relevanz in \ der Polykontexturalit\[ADoubleDot]tstheorie. Dort entspricht jeder nat\ \[UDoubleDot]rlichen Zahl m eine bestimmte irreduzible m\[Dash]kontexturale \ Qualit\[ADoubleDot]t. \nDamit geht aber die Idee der Auszeichnung, des \ Abschlusses und die Dialektik von offenem und geschlossenem System, wie sie \ sonst in der Kenogrammatik von Relevanz ist, verloren. L\[ADoubleDot]\[SZ]t \ sich keine Zahl auszeichnen, sondern m\[UDoubleDot]ssen umgekehrt alle Zahlen \ einer Auszeichnung w\[UDoubleDot]rdig sein, so f\[UDoubleDot]hrt sich die \ Idee der Auszeichnung ad absurdum. Da\[SZ] alle nat\[UDoubleDot]rlichen \ Zahlen logisch\[Dash]strukturell ausgezeichnet werden k\[ODoubleDot]nnen, ist \ aber das Resultat einer vollst\[ADoubleDot]ndigen Dekonstruktion der \ Konzeption der uni\[Dash]linearen aristo-telischen Arithmetik wie sie in der \ Kenogrammatik und der Polykontexturalit\[ADoubleDot]tstheorie vollzogen \ wurde. Mit der isolierten Thematisierung der Iterierbarkeit der \ m\[Dash]kontexturalen Zahlensysteme wird das wenig dialektische Moment der \ schlechten Unendlichkeit zugelassen. (Kaehr)\n", ButtonBox[ "ww.thinkartlab.com/pkl/media/DISSEM-final.pdf\ \[LineSeparator]", ButtonData :> { FrontEnd`FileName[ {"ww.thinkartlab.com", "pkl", "media"}, "DISSEM-final.pdf", CharacterEncoding -> "MacintoshRoman"], None}, ButtonStyle -> "Hyperlink"]}], "Endnote", CellTags -> "tetraktys"], ButtonStyle->"NoteKey", ButtonNote->"tetraktys"]], TraditionalForm]], "Note"] }], "Text", TextJustification->0, FontFamily->"Verdana", CellTags->{":note:INV-Toth", ":note:tetraktys"}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Toth\[CloseCurlyQuote]s criticism of Bense\[CloseCurlyQuote]s \ triadic-trichotomic semiotics\ \>", "Subsubsection", FontFamily->"Verdana"], Cell[TextData[{ "\"Um es kurz zu sagen: Bense hatte - es ist fast nicht zu glauben - ", StyleBox["n-\[ADoubleDot]re und n-adische", FontSlant->"Italic"], " Logiken verwechselt: Obwohl die Peirce-Bense-Semiotik triadisch ist, \ bleibt sie dennoch bin\[ADoubleDot]r, und das, obwohl sie einen zehnfach \ ausdifferenzierten Realit\[ADoubleDot]tsbegriff besitzt.\ \[CloseCurlyDoubleQuote] (Toth, Semiotische Strukturen und Prozesse, 2008). \ This, and other ebooks by Alfred Toth at: ", StyleBox[ButtonBox["http://www.uni-klu.ac.at/iff-tewi/inhalt/280.htm", ButtonData:>{ URL[ "http://www.uni-klu.ac.at/iff-tewi/inhalt/280.htm"], None}, ButtonStyle->"Hyperlink"], FontSize->10], StyleBox[" ", FontSize->10] }], "Text", TextJustification->0, FontFamily->"Verdana", CellTags->":note:INV-Toth"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Dissemination of semiotics", "Section"], Cell[TextData[{ "Interaction between different logical or semiotic systems is depending on \ the ", StyleBox["architectonics", FontSlant->"Italic"], " of the framework. In the proposed case, only two cases are presented. \ \[LineSeparator]\[LineSeparator]", StyleBox["First", FontSlant->"Italic"], " an architectonics based on a decomposition of the system into (2, \ 2)-subsystem. \[LineSeparator]And ", StyleBox["second", FontSlant->"Italic"], ", an architectonics based on the decomposition of the system into (3, 2)- \ and subsystems. \[LineSeparator]\[LineSeparator]The decomposition into (2, \ 2)-subsystems of 3-contextural systems corresponds to the usual \ polycontextural approach as introduced by Gotthard Gunther for his ", StyleBox["place-valued ", FontSlant->"Italic"], "logic. It can be understood as a dissemination of ", StyleBox["contextures", FontSlant->"Italic"], " towards polycontexturality as the base for polycontextural logics in \ general. \[LineSeparator]\[LineSeparator]This strategy of decomposing \ Peirce/Bense/Toth-semiotics into its dyadic-dichotomic parts opens up the \ possibility for a ", StyleBox["polycontextural", FontSlant->"Italic"], " approach to a logic, arithmetic and categorification of semiotics as a \ mediation of semiotically, logically and categorically independent elementary \ contextures of a mediated compound. This approach is in strict contrast to a \ modeling of triadic-trichotomic semiotics with methods of classical relation, \ set and category theory.\[LineSeparator]\[LineSeparator]The (3, 3)-subsystem \ decomposition of 4-contextural systems, albeit it goes back to my early \ studies of polycontexturality, has been introduced recently for a new \ formalization of semiotics towards polysemiotics. \[LineSeparator]\ \[LineSeparator]Polysemiotics are disseminating, in a first step, classical \ triadic-trichotomic semiotics, Sem", Cell[BoxData[ FormBox[ RowBox[{ AdjustmentBox[\(\[Null]\^\((3, \ 2)\)\), BoxMargins->{{0, -0.1}, {0, 0}}], ","}], TraditionalForm]], "TextSuperscript", SingleLetterItalics->False], " over different kenomic places to build more complex configurations." }], "Text", TextJustification->0, FontFamily->"Verdana"], Cell[CellGroupData[{ Cell["Contextural decomposition triadic systems", "Subsection"], Cell[CellGroupData[{ Cell[TextData[{ "Unary matrix", Cell[BoxData[ 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There is no \ reason to stop conceptualizing Peircean semiotics by set and category \ theoretical concepts (Walter, Bense, Toth) only because there, at this time, \ was no better method at hand. Conceptually it seems to be obvious that each \ sign relation deserve its own contextural and logical appreciation. \ \[LineSeparator]Toth is parallelizing, i.e. constructing a mapping between \ Peircean relations build (M, I, O) and Guntherian concepts of Subjectivity. \ But Gunther\[CloseCurlyQuote]s concepts, in all steps of their developments, \ are conceived as non-Aristotelian, transclassic and polycontextural. Albeit \ the fact, that Gunther himself didn\[CloseCurlyQuote]t emphasize the use of \ proemial relations in all those constructions, they are nevertheless the \ implicit force of his general theory of mediation (for logic, arithmetic, \ systems).\[LineSeparator]This observation seems to be in some contrast to \ Toth\[CloseCurlyQuote]s statement of the incompatibility of semiotics and \ polycontextural logic. \n", StyleBox[ "\"In a series of publications (cf., e.g., Toth 2003 and \ Toth 2008a), I have shown that the basic problem that is responsible for the \ incompatibility of semiotics and polycontextural theory, the lack of proemial \ relations in classical semiotics, can be avoided by introducing semiotic \ transpositions.\[CloseCurlyDoubleQuote] (Toth, 2009)", FontSlant -> "Italic"], "\nHence, Toth\[CloseCurlyQuote]s observation, albeit \ correct without doubt, is depending on the traditional understanding of \ Peircean semiotics. Taking the fact of Peirce connection to Hegel and the \ lack of any formal theory of mediation, a new thematization and formalization \ of Peircean semiotics isn\[CloseCurlyQuote]t to absurd at \ all.\[LineSeparator]"}], "Endnote", CellTags -> "incomparability"], ButtonStyle->"NoteKey", ButtonNote->"incomparability"]], TraditionalForm]], "Note"], "\[LineSeparator]\[LineSeparator]With the strategy of ", StyleBox["decomposing", FontSlant->"Italic"], " classical semiotics into its sub-systems, classical semiotics is \ understood and thematized as a mediation of its sub-systems by the \ involvement of chiastic procedures and proemial relationship. There is no \ reason to stop conceptualizing Peircean semiotics by set and category \ theoretical concepts (Walter, Bense, Toth) only because there, at this time, \ was no better method at hand. Conceptually it seems to be obvious that each \ sign relation deserve its own contextural and logical appreciation. \ \[LineSeparator]Toth is parallelizing, i.e. constructing a mapping between \ Peircean relations build (M, I, O) and Guntherian concepts of Subjectivity. \ But Gunther\[CloseCurlyQuote]s concepts, in all steps of their developments, \ are conceived as non-Aristotelian, transclassic and polycontextural. Albeit \ the fact, that Gunther himself didn\[CloseCurlyQuote]t emphasize the use of \ proemial relations in all those constructions, they are nevertheless the \ implicit force of his general theory of mediation (for logic, arithmetic, \ systems).\[LineSeparator]This observation seems to be in some contrast to \ Toth\[CloseCurlyQuote]s statement of the incompatibility of semiotics and \ polycontextural logic. \n", StyleBox["\"In a series of publications (cf., e.g., Toth 2003 and Toth \ 2008a), I have shown that the basic problem that is responsible for the \ incompatibility of semiotics and polycontextural theory, the lack of proemial \ relations in classical semiotics, can be avoided by introducing semiotic \ transpositions.\[CloseCurlyDoubleQuote] (Toth, 2009)", FontSlant->"Italic"], "\nHence, Toth\[CloseCurlyQuote]s observation, albeit correct without \ doubt, is depending on the traditional understanding of Peircean semiotics. \ Taking the fact of Peirce connection to Hegel and the lack of any formal \ theory of mediation, a new thematization and formalization of Peircean \ semiotics isn\[CloseCurlyQuote]t to absurd at all.\[LineSeparator]" }], "Text", TextJustification->0, FontFamily->"Verdana", CellTags->":note:incomparability"], Cell[BoxData[ \(TraditionalForm\`\(\(\[LineSeparator]\)\(\[LineSeparator]\)\(\ \[LineSeparator]\)\(For\ systems, \ m = 3, \ n = 2, \ the\ matrix\[Null]\^\((3, \ 2)\)\ and\ scheme\[Null]\^\((3, \ 2)\)\ \ representation\ \(\(coincide\)\(.\)\(\[LineSeparator]\)\)\)\)\)], \ "EquationGroup", TextJustification->0, FontFamily->"Verdana"], Cell["", "Text"], Cell[CellGroupData[{ Cell["\<\ Sign classes for classical Semiotics\ \>", "Subsubsubsection"], Cell[BoxData[{ \(TraditionalForm\`Sign\ classes\ are\ traditionally\ defined\ \(by\ : \ \[LineSeparator]ZR\)\ = \ \((a, \ \((a\ \[DoubleRightArrow] \ b)\), \ \((a\ \[DoubleRightArrow] \ b\ \[DoubleRightArrow] \ c)\))\)\), "\[LineSeparator]", \(TraditionalForm\`for\), "\[LineSeparator]", \(TraditionalForm\`a\ = \ {1.1, \ 1.2, \ 1.3}\), "\n", \(TraditionalForm\`b\ = \ {2.1, \ 2.2, \ 2.3}\), "\n", \(TraditionalForm\`c\ = \ {3.1, \ 3.2, \ 3.3}\), "\[LineSeparator]", \(TraditionalForm\`General\ sign\ \(relation : \[LineSeparator]ZR\)\ = \ \ \(\(<\)\(3. x\)\), \ 2. y, \ 1. z > \ mit\ x, \ y, \ z\ \[Element] {1, \ 2, \ 3}\), "\[LineSeparator]", \(TraditionalForm\`with\ x \[LessEqual] y \[LessEqual] z . \[LineSeparator]\[LineSeparator]Resulting\ in\ the\ 10\ \ sign\ \ \(classes : \[LineSeparator]3.1\ 2.1\ 1.1\ \ \ \ \ \ 3.1\ 2.3\ 1.3\)\), "\n", \ \(TraditionalForm\`3.1\ 2.1\ 1.2\ \ \ \ \ \ 3.2\ 2.2\ 1.2\), "\n", \(TraditionalForm\`3.1\ 2.1\ 1.3\ \ \ \ \ \ 3.2\ 2.2\ 1.3\), "\n", \(TraditionalForm\`3.1\ 2.2\ 1.2\ \ \ \ \ \ 3.2\ 2.3\ 1.3\), "\n", \(TraditionalForm\`3.1\ 2.2\ 1.3\ \ \ \ \ \ 3.3\ 2.3\ 1.3\)}], \ "EquationGroupAligned", FontFamily->"Verdana"], Cell[TextData[{ "Classical semiotics is not ", StyleBox["mediating", FontSlant->"Italic"], " its sub-systems, hence, no matching conditions are required. 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The logical and arithmetical status \ of semiotics, mono- and polycontextural, remains undetermined if its \ corresponding logics are not determined.\[LineSeparator]There are many ways \ open to formalize, logically and arithmetically, semiotics and polysemiotics. \ Good candidates are the logics from the modal logic pool. Nevertheless, they \ have all to be classified as \ mono-contextural.\[LineSeparator]\[LineSeparator]For the purpose of this \ introductory sketch of a ", StyleBox["descriptive", FontSlant->"Italic"], " characterization of the idea of poly-semiotics, it might be sufficient to \ hint to the decision to use 3-contextural subsystems of 4-contextural logics \ and arithmetics. Instead of the usual decomposition into elementary \ contextures. \[LineSeparator]\[LineSeparator]As a consequence, it turns out \ that the apparatus of classical category theory is not adequate to formalize \ semiotics and polysemiotics.\[LineSeparator]\[LineSeparator]Hence, from a \ 4-contextural logic, Log", Cell[BoxData[ FormBox[ RowBox[{ AdjustmentBox[\(\[Null]\^\((4)\)\), BoxMargins->{{0, -0.1}, {0, 0}}], ",", " "}], TraditionalForm]], "TextSuperscript", SingleLetterItalics->False], "with its six 2-contextures, Log", Cell[BoxData[ FormBox[ RowBox[{ AdjustmentBox[\(\[Null]\^\((4, \ 2)\)\), BoxMargins->{{0, -0.1}, {0, 0}}], ","}], TraditionalForm]], "TextSuperscript", SingleLetterItalics->False], " its four 3-contextures, Log", Cell[BoxData[ FormBox[ RowBox[{ AdjustmentBox[\(\[Null]\^\((4, \ 3)\)\), BoxMargins->{{0, -0.1}, {0, 0}}], ","}], TraditionalForm]], "TextSuperscript", SingleLetterItalics->False], " only the four 3-contxtural subsystems are in direct correspondence to \ the 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Cell[CellGroupData[{ Cell["Interactions between diamonds", "Subsection"], Cell[TextData[{ "As introduced in ", StyleBox["Diamond Text Theory", FontSlant->"Italic"], ", special interactions between diamonds are building networks of textemes. \ In this case, interaction between semiotic systems happens mediated by their \ neighboring environments.\[LineSeparator]", StyleBox[ButtonBox["http://www.thinkartlab.com/pkl/media/Textems/Textems.\ pdf ", ButtonData:>{ URL[ "http://www.thinkartlab.com/pkl/media/Textems/Textems.pdf "], None}, ButtonStyle->"Hyperlink"], FontSize->10], StyleBox[" ", FontSize->10] }], "Text", CellMargins->{{54.4375, Inherited}, {Inherited, Inherited}}, TextJustification->0, FontFamily->"Verdana"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Notes", "EndnoteSection", FontSize->14], Cell[TextData[{ Cell[BoxData[ FormBox[ AdjustmentBox[ SuperscriptBox["", ButtonBox[ StyleBox["1", FontWeight->"Bold"], ButtonData:>":note:computational semiotics", 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As much as there is no proper logic of interaction there is even \ much less development in computational semiotics. There is not even an \ awareness about the conceptual lack of interactivity constructs in \ theoretical semiotics. Despite the many ", StyleBox["applicative", FontSlant->"Italic"], " approaches to semiotic interactions, e.g. in human-computer interface \ research, it seems, that theoretical and foundational research for a semiotic \ theory of interaction and reflection is not supported.\[LineSeparator]\ \[LineSeparator]", StyleBox["Christopher R. Longyear, Further Towards a Triadic Calculus (Part \ 1, 2, 3)\[LineSeparator]", FontSize->11], ButtonBox["http://www.vordenker.de/ggphilosophy/longyear-part_1.pdf", ButtonData:>{ URL[ "http://www.vordenker.de/ggphilosophy/longyear-part_1.pdf"], None}, ButtonStyle->"Hyperlink"], " \[LineSeparator]" }], "Endnote", FontFamily->"Verdana", FontSize->12, CellTags->"computational semiotics"], Cell[TextData[{ Cell[BoxData[ FormBox[ StyleBox[ AdjustmentBox[ SuperscriptBox["", ButtonBox[ CounterBox["Endnote", "cartesian"], ButtonData:>":note:cartesian", ButtonStyle->"EndnoteReturnLink", ButtonFrame->"None"]], BoxMargins->{{0, 0.4}, {0, 0}}], FontWeight->"Bold"], TraditionalForm]], "EndnoteLabel"], "Independent of later steps of ", StyleBox["abolishing", FontSlant->"Italic"], " restrictions in the traditional definition of sign classes by Toth\ \[CloseCurlyQuote]s studies, the concept of a Cartesian product remains a \ fundamental construction to build up a semiotic system.\[LineSeparator]\ \[LineSeparator]This fact allows to study the semiotic matrix under a \ different angle: the ", StyleBox["polycontextural", FontSlant->"Italic"], " approach of dissemination, i.e. distribution and mediation, of \ sub-systems as a mechanism to construct and to deconstruct the semiotic \ matrix. In this sense, an extension of the semiotic matrix for complex sign \ systems, called polysemiotics, is \ introduced.\[LineSeparator]\[LineSeparator]To use Cartesian products doesn\ \[CloseCurlyQuote]t mean that they will remain stable in the development of a \ general theory of polylogics and polysemiotics. As shown at other places, \ what was a good starting point, became the main obstacle for further \ developments. Here again, the abstract mathematical frame (set and category \ theory) is not always adequate for the project of formalizing transclassical \ approaches.\[LineSeparator]\n This disseminative approach to the \ semiotics matrix allows to introduce a comparison of semiotic and logical \ constructions. As main operators of logical interaction, the polylogical ", StyleBox["transjunctions", FontSlant->"Italic"], " had been studied ", StyleBox["in extenso.(Kaehr, 1978, 2005)", FontSlant->"Italic"], "\n\[LineSeparator]In analogy and translation or transposition from the \ polycontextural to the semiotic topics, semiotic interactions between \ semiotic sub-systems shall be introduced. Semiotic sub-systems are a result \ of a decomposition of the semiotic matrix into its sub-systems. Such a \ decomposition is dynamic, depending on the complexity of the semiotic matrix. \ In this paper, only two cases are introduced. The decomposition into (2, \ 2)-subsystems, with S", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ AdjustmentBox[\(\[Null]\_1\), BoxMargins->{{0, -0.1}, {0, 0}}], "=", \({1, \ 2}\)}], ",", " ", RowBox[{"S", FormBox[ RowBox[{ AdjustmentBox[\(\[Null]\_2\), BoxMargins->{{0, -0.1}, {0, 0}}], "=", \({2, \ 3}\)}], "TraditionalForm"]}], ",", " ", RowBox[{"S", RowBox[{ FormBox[ RowBox[{ AdjustmentBox[\(\[Null]\_3\), BoxMargins->{{0, -0.1}, {0, 0}}], "=", \({1, \ 3}\)}], "TraditionalForm"], "."}]}]}], TraditionalForm]], "TextSubscript", SingleLetterItalics->False], " And the decomposition into (3, 3)-subsystems of a polysemiotic system \ Sem", Cell[BoxData[ FormBox[ RowBox[{ AdjustmentBox[\(\[Null]\^\((4, 2)\)\), BoxMargins->{{0, -0.1}, {0, 0}}], "."}], TraditionalForm]], "TextSuperscript", SingleLetterItalics->False], "\[LineSeparator]" }], "Endnote", FontFamily->"Verdana", FontSize->12, CellTags->"cartesian"], Cell[TextData[{ Cell[BoxData[ FormBox[ AdjustmentBox[ SuperscriptBox["", StyleBox[ ButtonBox[ CounterBox["Endnote", "privilege"], ButtonData:>":note:privilege", ButtonStyle->"EndnoteReturnLink", ButtonFrame->"None"], FontWeight->"Bold"]], BoxMargins->{{0, 0.1}, {0, 0}}], TraditionalForm]], "EndnoteLabel"], "Nevertheless, a specific redundancy has to be repeated because of its \ established and deep-rooted sheepishness and stultifying ignorance. The more \ or less only answer or \[OpenCurlyQuote]feed-back\[CloseCurlyQuote] I got, \ when I was emphasizing the importance of a number, e.g. 4, was, ", StyleBox["\[OpenCurlyDoubleQuote]Why an extension to 4 and not to 7 or 13 \ or 5112?\[CloseCurlyDoubleQuote]", FontSlant->"Italic"], " Nobody ever questioned the fact that their response is based on the \ number 2 (TWO). And surely I never privileged a single natural number of the \ established number system.\[LineSeparator]\[LineSeparator]A criticism of such \ an idea of a privilege of a single natural number was perfectly done long \ before by Aristotle with his refutation of Pythagorean number theory. \ \[LineSeparator]It seems to be better to live and to die with the number TWO \ than to question it.\[LineSeparator]As far, it was an important scientific \ step by Peirce to introduce his triadic-trichotomic semiotics and first \ sketches to a trichotomic mathematics.\[LineSeparator]" }], "Endnote", FontFamily->"Verdana", FontSize->12, CellTags->"privilege"], Cell[TextData[{ Cell[BoxData[ FormBox[ StyleBox[ AdjustmentBox[ SuperscriptBox["", ButtonBox[ CounterBox["Endnote", "tetraktys"], ButtonData:>":note:tetraktys", ButtonStyle->"EndnoteReturnLink", ButtonFrame->"None"]], BoxMargins->{{0, 0.4}, {0, 0}}], FontWeight->"Bold"], TraditionalForm]], "EndnoteLabel"], "\"Die systematische Auszeichnung der 4 mag willk\[UDoubleDot]rlich \ erscheinen; warum nicht die 3 oder die 11 und warum eine und nicht mehrere \ oder gar alle Zahlen? \nDie Kritik Aristoteles' an der \ pythagor\[ADoubleDot]ischen Auszeichnung der 4 bzw. der 10 setzt die Linearit\ \[ADoubleDot]t der nat\[UDoubleDot]rlichen Zahlen und das Prinzip der \ potentiellen Realisierbarkeit voraus. Erst dann entstehtein Konflikt zwischen \ der Reihe der nat\[UDoubleDot]rlichen Zahlen, d.h. einer beliebigen Zahl und \ der Auszeichnung einer Zahl dieser Reihe als Gattungszahl der Reihe selbst. \n\ Wird jedoch unter der 4 die '", StyleBox["Gattungszahl", FontSlant->"Italic"], "' der 4 Schrifttypen der Graphematik verstanden, also das Geviert der \ geschlossenen Proemialit\[ADoubleDot]t, dann entsteht kein Widerspruch \ zwischen Auszeichnung einer Zahl und der Zahlenreihe selbst. Die 4 er\ \[ODoubleDot]ffnet die Vielfalt der Zahlensysteme der Polykontexturalit\ \[ADoubleDot]t, liegt jedoch als solche nicht in der Reihe der nat\ \[UDoubleDot]rlichen Zahlen einer beliebigen Kontextur. Aristoteles lehnt die \ Auszeichnung der 4 (und mit ihr die der 10) ab, ist aber selbst gezwungen, \ die 1 auszuzeichnen. Denn die Uni\[Dash]Linearit\[ADoubleDot]t der Reihe der \ nat\[UDoubleDot]rlichen Zahlen setzt die 1 als Ma\[SZ] der Zahlen und als \ unum der Unizit\[ADoubleDot]t der Reihe voraus. Die Auszeichnung der 4 unter \ der Voraussetzung der Uni\[Dash]Linearit\[ADoubleDot]t hei\[SZ]t, da\[SZ] die \ vertikale Sprachachse der Graphematik auf die horizontale Linie der nat\ \[UDoubleDot]rlichen Zahlen projiziert wird. \nDer Widerspruch zwischen \ 'Gattungszahl' und 'Reihenzahl' ist somit das Produkt einer Verdeckung, einer \ Koinzidenz der beiden 'Zahlenachsen'. Dabei wird auch stillschweigend \ vorausgesetzt, da\[SZ] die Zahlziffern selbst eindeutig und nicht einer \ \[CapitalUDoubleDot]berdetermination ausgesetzt sind. Aristoteles' Kritik \ verf\[ADoubleDot]ngt auch dann nicht, wenn sich die 4 vertikalen \ Sprachschichten nicht legitimieren lassen und ihre Anzahl vergr\[ODoubleDot]\ \[SZ]ert oder verkleinert werden mu\[SZ]. \nDie Kritik an der Auszeichnung \ einer bestimmten Zahl vor der anderen durch die transklassische Arithmetik, \ kann sich jedoch nicht auf Aristoteles berufen, denn seine Kritik umfa\[SZ]t \ generell die Mehrlinigkeit der platonischen Zahlen und diese wiederum ist ein \ wesentlicher Charakter der transklassischen Zahlentheorie. \nSo argumentiert \ G\[UDoubleDot]nther: ", StyleBox["\:201eAristoteles ist im Recht. Es ist notwendig, konsequent zu \ sein. Entweder sehen wir uns gezwungen, nicht nur der Monas, der Dyas, der \ Triade usw., kurz jeder pythgagor\[ADoubleDot]ischen n\[Dash]Zahl den Rang \ einer ontologischen Idealit\[ADoubleDot]t zuzubilligen oder aber die ganze \ Problemsicht ist verfehlt und keine Zahl hat die W\[UDoubleDot]rde einer Idee\ \[Dash]au\[SZ]er vielleicht die Einheit und die aoristos duas, die man aber \ beide nicht als Zahlen zu betrachten braucht. Da\[SZ] die zweite Auffassung \ nicht haltbar ist, lehrt die Geistesgeschichte vergangener Epochen.\"", FontSlant->"Italic"], "\nG\[UDoubleDot]nther insistiert also auf der Auszeichnung jeder Zahl und \ nicht nur der pythagor\[ADoubleDot]ischen Tetraktys. D.h. jede Zahl hat die W\ \[UDoubleDot]rde einer Idee und erh\[ADoubleDot]lt somit eine \ logisch\[Dash]strukturelle Relevanz in der \ Polykontexturalit\[ADoubleDot]tstheorie. Dort entspricht jeder nat\ \[UDoubleDot]rlichen Zahl m eine bestimmte irreduzible m\[Dash]kontexturale \ Qualit\[ADoubleDot]t. \nDamit geht aber die Idee der Auszeichnung, des \ Abschlusses und die Dialektik von offenem und geschlossenem System, wie sie \ sonst in der Kenogrammatik von Relevanz ist, verloren. L\[ADoubleDot]\[SZ]t \ sich keine Zahl auszeichnen, sondern m\[UDoubleDot]ssen umgekehrt alle Zahlen \ einer Auszeichnung w\[UDoubleDot]rdig sein, so f\[UDoubleDot]hrt sich die \ Idee der Auszeichnung ad absurdum. Da\[SZ] alle nat\[UDoubleDot]rlichen \ Zahlen logisch\[Dash]strukturell ausgezeichnet werden k\[ODoubleDot]nnen, ist \ aber das Resultat einer vollst\[ADoubleDot]ndigen Dekonstruktion der \ Konzeption der uni\[Dash]linearen aristotelischen Arithmetik wie sie in der \ Kenogrammatik und der Polykontexturalit\[ADoubleDot]tstheorie vollzogen \ wurde. Mit der isolierten Thematisierung der Iterierbarkeit der \ m\[Dash]kontexturalen Zahlensysteme wird das wenig dialektische Moment der \ schlechten Unendlichkeit zugelassen.\[CloseCurlyDoubleQuote] (Kaehr, \ Einschreiben in Zukunft, \[Section] 6,1981)\[LineSeparator]", ButtonBox["http://www.thinkartlab.com/pkl/media/DISSEM-final.pdf", ButtonData:>{ URL[ "http://www.thinkartlab.com/pkl/media/DISSEM-final.pdf"], None}, ButtonStyle->"Hyperlink"], " " }], "Endnote", FontFamily->"Verdana", FontSize->12, CellTags->"tetraktys"] }, Open ]], Cell[CellGroupData[{ Cell["Notes", "EndnoteSection"], Cell[TextData[{ Cell[BoxData[ FormBox[ AdjustmentBox[ SuperscriptBox["", ButtonBox[ CounterBox["Endnote", "computational semiotics"], ButtonData:>":note:computational semiotics", ButtonStyle->"EndnoteReturnLink", ButtonFrame->"None"]], BoxMargins->{{0, 0.1}, {0, 0}}], TraditionalForm]], "EndnoteLabel"], "Computational semiotics is interested in modeling interactions in \ computational scenarios.\[LineSeparator]As much as there is no proper logic \ of interaction there is even much less developement in computational \ semiotics. There is not even an awareness about the conceptual lack of \ interactivity constructs in theoretical semiotics. Despite the many \ applicative approaches to semiotic interactions, e.g. in human-computer \ interface research, it seems, that theoretical and foundational resarch for a \ semiotic theory of interaction and reflection is not supported.", "\[LineSeparator]", "\[LineSeparator]", StyleBox["Christopher R. Longyear, Further Towards a Triadic Calculus (Part \ 1, 2, 3)\[LineSeparator]", FontSize->11], ButtonBox["http://www.vordenker.de/ggphilosophy/longyear-part_1.pdf", ButtonData:>{ URL[ "http://www.vordenker.de/ggphilosophy/longyear-part_1.pdf"], None}, ButtonStyle->"Hyperlink"], " \[LineSeparator]" }], "Endnote", CellTags->"computational semiotics"], Cell[TextData[{ Cell[BoxData[ FormBox[ AdjustmentBox[ SuperscriptBox["", ButtonBox[ CounterBox["Endnote", "cartesian"], ButtonData:>":note:cartesian", ButtonStyle->"EndnoteReturnLink", ButtonFrame->"None"]], BoxMargins->{{0, 0.1}, {0, 0}}], TraditionalForm]], "EndnoteLabel"], "Independent of later steps of ", StyleBox["abolishing", FontSlant->"Italic"], " restrictions in the traditional definition of sign classes by Toth\ \[CloseCurlyQuote]s studies, the concept of a Cartesian product remains a \ fundamental construction to build up a semiotic system.\[LineSeparator]\ \[LineSeparator]This fact allows to study the semiotic matrix under a \ different angle: the ", StyleBox["polycontextural", FontSlant->"Italic"], " approach of dissemination, i.e. distribution and mediation, of \ sub-systems as a mechanism to construct and to deconstruct the semiotic \ matrix. In this sense, an extension of the semiotic matrix for complex sign \ systems, called polysemiotics, is \ introduced.\[LineSeparator]\[LineSeparator]To use Cartesian products doesn\ \[CloseCurlyQuote]t mean that they will remain stable in the development of a \ general theory of polylogics and polysemiotics. As shown at other places, \ what was a good starting point, became the main obstacle for further \ developments.\[LineSeparator]\nThis disseminative approach to the semiotics \ matrix allows to introduce a comparison of semiotic and logical \ constructions. As main operators of logical interaction, the polylogical ", StyleBox["transjunctions", FontSlant->"Italic"], " had been studied ", StyleBox["in extenso.(Kaehr, 1978, 2005)", FontSlant->"Italic"], "\n\[LineSeparator]In analogy and translation or transposition from the \ polycontextural to the semiotic topics, semiotic interactions between \ semiotic sub-systems shall be introduced. Semiotic sub-systems are a result \ of a decomposition of the semiotic matrix into its sub-systems. Such a \ decomposition is dynamic, depending on the complexity of the semiotic matrix. \ In this paer, only two cases are introduced. The decomposition into (2, \ 2)-subsystems, with S", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ AdjustmentBox[\(\[Null]\_1\), BoxMargins->{{0, -0.1}, {0, 0}}], "=", \({1, \ 2}\)}], ",", " ", RowBox[{"S", FormBox[ RowBox[{ AdjustmentBox[\(\[Null]\_2\), BoxMargins->{{0, -0.1}, {0, 0}}], "=", \({2, \ 3}\)}], "TraditionalForm"]}], ",", " ", RowBox[{"S", RowBox[{ FormBox[ RowBox[{ AdjustmentBox[\(\[Null]\_3\), BoxMargins->{{0, -0.1}, {0, 0}}], "=", \({1, \ 3}\)}], "TraditionalForm"], "."}]}]}], TraditionalForm]], "TextSubscript", SingleLetterItalics->False], " And the decomposition into (3, 3)-subsystems of a polysemiotic system \ Sem", Cell[BoxData[ FormBox[ RowBox[{ AdjustmentBox[\(\[Null]\^\((4, 2)\)\), BoxMargins->{{0, -0.1}, {0, 0}}], "."}], TraditionalForm]], "TextSuperscript", SingleLetterItalics->False] }], "Endnote", CellTags->"cartesian"], Cell[TextData[{ Cell[BoxData[ FormBox[ AdjustmentBox[ SuperscriptBox["", ButtonBox[ CounterBox["Endnote", "privilege"], ButtonData:>":note:privilege", ButtonStyle->"EndnoteReturnLink", ButtonFrame->"None"]], BoxMargins->{{0, 0.1}, {0, 0}}], TraditionalForm]], "EndnoteLabel"], "Nevertheless, a specific redundancy has to be repeated because of its \ established and deep-rooted stupidity. The more or less only answer or \ feed-back to my work, when I was emphasizing the importance of a number, e.g. \ 4, I got was, \[OpenCurlyDoubleQuote]Why an extension to 4 and not to 7 or 13 \ or 5112?\[CloseCurlyDoubleQuote] Nobody ever questioned the fact that their \ question was based on the number 2 (TWO). And I surely never privileged a \ single natural number of the established number system.\[LineSeparator]\ \[LineSeparator]A criticism of such an idea of a privilege of a single \ natural number was perfectly done long before by Aristotle with his \ refutation of Pythagorean number theory. \[LineSeparator]It seems to be \ better to live and die with the number TWO than to question it.\ \[LineSeparator]As far, it was an important scientific step by Peirce to \ introduce his triadic-trichotomic semiotics." }], "Endnote", CellTags->"privilege"], Cell[TextData[{ Cell[BoxData[ FormBox[ AdjustmentBox[ SuperscriptBox["", ButtonBox[ CounterBox["Endnote", "tetraktys"], ButtonData:>":note:tetraktys", ButtonStyle->"EndnoteReturnLink", ButtonFrame->"None"]], BoxMargins->{{0, 0.1}, {0, 0}}], TraditionalForm]], "EndnoteLabel"], "Die systematische Auszeichnung der 4 mag willk\[UDoubleDot]rlich \ erscheinen; warum nicht die 3 oder die 11 und warum eine und nicht mehrere \ oder gar alle Zahlen? \nDie Kritik Aristoteles' an der \ pythagor\[ADoubleDot]ischen Auszeichnung der 4 bzw. der 10 setzt die Linearit\ \[ADoubleDot]t der nat\[UDoubleDot]rlichen Zahlen und das Prinzip der \ potentiellen Realisierbarkeit voraus. Erst dann entstehtein Konflikt zwischen \ der Reihe der nat\[UDoubleDot]rlichen Zahlen, d.h. einer beliebigen Zahl und \ der Auszeichnung einer Zahl dieser Reihe als Gattungszahl der Reihe selbst. \n\ Wird jedoch unter der 4 die 'Gattungszahl' der 4 Schrifttypen der Graphematik \ verstanden, also das Geviert der geschlossenen Proemialit\[ADoubleDot]t, dann \ entsteht kein Widerspruch zwischen Auszeichnung einer Zahl und der \ Zahlenreihe selbst. Die 4 er\[ODoubleDot]ffnet die Vielfalt der Zahlensysteme \ der Polykontexturalit\[ADoubleDot]t, liegt jedoch als solche nicht in der \ Reihe der nat\[UDoubleDot]rlichen Zahlen einer beliebigen Kontextur. \ Aristoteles lehnt die Auszeichnung der 4 (und mit ihr die der 10) ab, ist \ aber selbst gezwungen, die 1 auszuzeichnen. Denn die Uni\[Dash]Linearit\ \[ADoubleDot]t der Reihe der nat\[UDoubleDot]rlichen Zahlen setzt die 1 als \ Ma\[SZ] der Zahlen und als unum der Unizit\[ADoubleDot]t der Reihe voraus. \ Die Auszeichnung der 4 unter der Voraussetzung der Uni\[Dash]Linearit\ \[ADoubleDot]t hei\[SZ]t, da\[SZ] die vertikale Sprachachse der Graphematik \ auf die horizontale Linie der nat\[UDoubleDot]rlichen Zahlen projiziert wird. \ \nDer Widerspruch zwischen 'Gattungszahl' und 'Reihenzahl' ist somit das \ Produkt einer Verdeckung, einer Koinzidenz der beiden 'Zahlenachsen'. Dabei \ wird auch stillschweigend vorausgesetzt, da\[SZ] die Zahlziffern selbst \ eindeutig und nicht einer \[CapitalUDoubleDot]berdetermination ausgesetzt \ sind. Aristoteles' Kritik verf\[ADoubleDot]ngt auch dann nicht, wenn sich die \ 4 vertikalen Sprachschichten nicht legitimieren lassen und ihre Anzahl vergr\ \[ODoubleDot]\[SZ]ert oder verkleinert werden mu\[SZ]. \nDie Kritik an der \ Auszeichnung einer bestimmten Zahl vor der anderen durch die transklassische \ Arithmetik, kann sich jedoch nicht auf Aristoteles berufen, denn seine Kritik \ umfa\[SZ]t generell die Mehrlinigkeit der platonischen Zahlen und diese \ wiederum ist ein wesentlicher Charakter der transklassischen Zahlentheorie. \n\ So argumentiert G\[UDoubleDot]nther: \:201eAristoteles ist im Recht. Es ist \ notwendig, konsequent zu sein. Entweder sehen wir uns gezwungen, nicht nur \ der Monas, der Dyas, der Triade usw., kurz jeder pythgagor\[ADoubleDot]ischen \ n\[Dash]Zahl den Rang einer ontologischen Idealit\[ADoubleDot]t zuzubilligen \ oder aber die ganze Problemsicht ist verfehlt und keine Zahl hat die W\ \[UDoubleDot]rde einer Idee\[Dash]au\[SZ]er vielleicht die Einheit und die \ aoristos duas, die man aber beide nicht als Zahlen zu betrachten braucht. Da\ \[SZ] die zweite Auffassung nicht haltbar ist, lehrt die Geistesgeschichte \ vergangener Epochen.\[OpenCurlyDoubleQuote] (70) \nG\[UDoubleDot]nther \ insistiert also auf der Auszeichnung jeder Zahl und nicht nur der pythagor\ \[ADoubleDot]ischen Tetraktys. D.h. jede Zahl hat die W\[UDoubleDot]rde einer \ Idee und erh\[ADoubleDot]lt somit eine logisch\[Dash]strukturelle Relevanz in \ der Polykontexturalit\[ADoubleDot]tstheorie. Dort entspricht jeder nat\ \[UDoubleDot]rlichen Zahl m eine bestimmte irreduzible m\[Dash]kontexturale \ Qualit\[ADoubleDot]t. \nDamit geht aber die Idee der Auszeichnung, des \ Abschlusses und die Dialektik von offenem und geschlossenem System, wie sie \ sonst in der Kenogrammatik von Relevanz ist, verloren. L\[ADoubleDot]\[SZ]t \ sich keine Zahl auszeichnen, sondern m\[UDoubleDot]ssen umgekehrt alle Zahlen \ einer Auszeichnung w\[UDoubleDot]rdig sein, so f\[UDoubleDot]hrt sich die \ Idee der Auszeichnung ad absurdum. Da\[SZ] alle nat\[UDoubleDot]rlichen \ Zahlen logisch\[Dash]strukturell ausgezeichnet werden k\[ODoubleDot]nnen, ist \ aber das Resultat einer vollst\[ADoubleDot]ndigen Dekonstruktion der \ Konzeption der uni\[Dash]linearen aristo-telischen Arithmetik wie sie in der \ Kenogrammatik und der Polykontexturalit\[ADoubleDot]tstheorie vollzogen \ wurde. Mit der isolierten Thematisierung der Iterierbarkeit der \ m\[Dash]kontexturalen Zahlensysteme wird das wenig dialektische Moment der \ schlechten Unendlichkeit zugelassen. (Kaehr)\n", ButtonBox["ww.thinkartlab.com/pkl/media/DISSEM-final.pdf\[LineSeparator]", ButtonData:>{ FrontEnd`FileName[ {"ww.thinkartlab.com", "pkl", "media"}, "DISSEM-final.pdf", CharacterEncoding -> "MacintoshRoman"], None}, ButtonStyle->"Hyperlink"] }], "Endnote", CellTags->"tetraktys"], Cell[TextData[{ Cell[BoxData[ FormBox[ AdjustmentBox[ SuperscriptBox["", ButtonBox[ CounterBox["Endnote", "incomparability"], ButtonData:>":note:incomparability", ButtonStyle->"EndnoteReturnLink", ButtonFrame->"None"]], BoxMargins->{{0, 0.1}, {0, 0}}], TraditionalForm]], "EndnoteLabel"], "Is the incompatibility beeen semiotics and polycontexturality compulsive?\ \[LineSeparator]With the strategy of ", StyleBox["decomposing", FontSlant->"Italic"], " classical semiotics into its sub-systems, classical semiotics is \ understood and thematized as a mediation of its sub-systems by the \ involvement of chiastic procedures and proemial relationship. There is no \ reason to stop conceptualizing Peircean semiotics by set and category \ theoretical concepts (Walter, Bense, Toth) only because there, at this time, \ was no better method at hand. Conceptually it seems to be obvious that each \ sign relation deserve its own contextural and logical appreciation. \ \[LineSeparator]Toth is parallelizing, i.e. constructing a mapping between \ Peircean relations build (M, I, O) and Guntherian concepts of Subjectivity. \ But Gunther\[CloseCurlyQuote]s concepts, in all steps of their developments, \ are conceived as non-Aristotelian, transclassic and polycontextural. Albeit \ the fact, that Gunther himself didn\[CloseCurlyQuote]t emphasize the use of \ proemial relations in all those constructions, they are nevertheless the \ implicit force of his general theory of mediation (for logic, arithmetic, \ systems).\[LineSeparator]\[LineSeparator]This observation seems to be in some \ contrast to Toth\[CloseCurlyQuote]s statement of the incompatibility of \ semiotics and polycontextural logic. ", "\[LineSeparator]", "\n", StyleBox["\"In a series of publications (cf., e.g., Toth 2003 and Toth \ 2008a), I have shown that the basic problem that is responsible for the \ incompatibility of semiotics and polycontextural theory, the lack of proemial \ relations in classical semiotics, can be avoided by introducing semiotic \ transpositions.\[CloseCurlyDoubleQuote] (Toth, 2009)", FontSlant->"Italic"], "\[LineSeparator]\nHence, Toth\[CloseCurlyQuote]s observation, albeit \ correct without doubt, is depending on a traditional understanding of \ Peircean semiotics. Taking the fact of Peirce connection to Hegel into \ account and the lack of any formal theory of mediation, a new thematization \ and formalization of Peircean semiotics isn\[CloseCurlyQuote]t to absurd at \ all.\[LineSeparator]" }], "Endnote", CellTags->"incomparability"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.2 for Macintosh", ScreenRectangle->{{0, 1276}, {0, 1002}}, Editable->True, WindowToolbars->"RulerBar", Selectable->True, WindowSize->{961, 863}, WindowMargins->{{64, Automatic}, {-1051, Automatic}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, TaggingRules:>{"Notes" -> {"tetraktys", "privilege", "computational \ semiotics", "cartesian", "incomparability"}, "NotesAsNotes" -> True}, DefaultFormatType->DefaultTextFormatType, Magnification->1.25, StyleDefinitions -> "Article2.nb" ] (******************************************************************* Cached data follows. 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