The Tale of Transjunctions

Some historical steps in the explanation and implementation of transcontextural operations

Rudolf Kaehr Dr. phil@

ThinkArt Lab Glasgow ISSN 2041-4358

 

Abstract

Transjunctions had been introduced in the early 60s by the philosopher and cybernetician Gotthard Gunther at the Biological Computer Laboratory (BCL) Urbana at Illinois in his historical research report “Cybernetic Ontology and Transjunctional Operators”. (http://www.thinkartlab.com/pkl/archive/GUNTHER-BOOK/GUNTHER.htm)
A history of polycontextural logic has been sketched at: Place-valued Logics around Cybernetic Ontology, the BCL and AFOSR (http://works.bepress.com/thinkartlab/16/) or at: (http://www.scribd.com/doc/18543421/Placevalued-logics-around-CyberneticOntology-the-BCL-and-AFOSR). This paper complements the theoretical paper: "Catching Transjunctions. Steps towards an emulation of polycontextural transjunctions in memristic systems.” at: (http://memristors.memristics.com/Transjunctions/Catching Transjunctions.pdf))

1.  History of transjunctions

1.1.  Steps towards logical transjunctions

Gunther ' s truth table (Gunther) morphogrammatic interpretation (Gunther) Indexed interpretat ... ical interpretation (Pfalzgraf) functorial interpretation (Kaehr) memristic interpretatin (Kaehr)

1.1.1.  Gunther’s truth tables

 Table for transjunction, conjunction and disjunction . <br /> <br /> (⊕ ∧ ∨ ...              3                           2                           1                           3


[Graphics:HTMLFiles/index_3.gif]

1.1.2.   Gunther's “akward formula"


[Graphics:HTMLFiles/index_4.gif]
[Graphics:HTMLFiles/index_5.gif]

1.1.3.  Reformulation of the “akward formula"


[Graphics:HTMLFiles/index_6.gif]



[Graphics:HTMLFiles/index_7.gif]


[Graphics:HTMLFiles/index_8.gif]


[Graphics:HTMLFiles/index_9.gif]

1.1.4.  Gunther’s morphogrammatic interpretation of the “akward formula"

"The precise meaning of such a statement is simple that the behavioral properties of the system in question display a logical structure that includes rejection values. And the individual morphograms which come into play will indicate precisely which of the three described varieties of subjective behavior we are referring to.
The introduction of the fifteen morphograms as the basic logical units of a trans-classic system of logic has far-reaching consequences. Such units would have hardly more than decorative significance unless there exists a specific operator able to handle them and to transform one morphogram directly into another. Negation is not capable of doing this as long as we adhere to the classic concept of negation. It is traditionally a reversible exchange relation between two values. It follows that by negating values we only change the value occupancy of a morphogram, not the morphogram itself; no matter how many negations are used, the abstract pattern of value occupancy remains always the same.” (Gunther, 1962)

[Graphics:HTMLFiles/index_10.gif]

[Graphics:HTMLFiles/index_11.gif]

1.1.5.  Indexed interpretation
[Graphics:HTMLFiles/index_12.gif]
1.1.6.   Indexed negations
typeset structure
1.1.7.  Comparison of global and local
[Graphics:HTMLFiles/index_14.gif]



[Graphics:HTMLFiles/index_15.gif]

1.1.8.  Pfalzgraf’s fibre bundle approach

[Graphics:HTMLFiles/index_16.gif]

1.1.9.  Tableaux interpretation


[Graphics:HTMLFiles/index_17.gif]

1.1.10.  Term interpretation of tableaux (Bashford)


[Graphics:HTMLFiles/index_18.gif]

1.1.11.  Matrix interpretation
[Graphics:HTMLFiles/index_19.gif]

[Graphics:HTMLFiles/index_20.gif]
The Abacus of Universal Logics
http://works.bepress.com/thinkartlab/17/
1.1.12.  Contextural programming

[Graphics:HTMLFiles/index_21.gif]

1.1.13.  Functorial interpretation

   <br />      (X <> ∧ ∧ Y) - scheme : [<&g ...                                          3                              3.1                      1

   <br />     <br />     (     &nb ...                                                                      3           3.3             3

1.1.14.  Memristic speculations (matrix, functorial)

typeset structure

typeset structure

1.1.15.  Morphic abstraction

[Graphics:HTMLFiles/index_26.gif]


[Graphics:HTMLFiles/index_27.gif]

[Graphics:HTMLFiles/index_28.gif]

[Graphics:HTMLFiles/index_29.gif]

The Abacus of Universal Logics
http://works.bepress.com/thinkartlab/17/

Scheme of morphogrammatic transjunction (typeset structure)

typeset structure <==>  typeset structure  <==>   typeset structure   

1.1.16.  Interchangeability for morphic transjunctions

Transjunction as bifurcation  ⊕ = bifurcation " bif " . <br /> <br /> bif  ...  = cod ([S _ 1^2]) = dom ([S _ 2^2]) <br />    cod ([S _ 2^2]) = cod ([S _ 3^3]) <br />

 <br />         [⊕  _ 1.1 MG  _ 2.2   ...                                                3                  3.1   3                  3.2   3

2.  Catching Transjunctions

Steps towards an emulation of polycontextural transjunctions in memristic systems

http://memristors.memristics.com/Transjunctions/Catching Transjunctions.pdf