Summer-Edition 2017

— vordenker-archive —

Rudolf Kaehr

(1942-2016)

Title

Goedel Games: “Cloning Gödels Proofs”

Archive-Number / Categories

2_18 / K08

Publication Date

2005

Keywords

Gödel; Hehner; Günther; Reflectionality; Interactional; Polylogics

Disciplines

Artificial Intelligence and Robotics, Cybernetics, Logic and Foundations of Mathematics,

Epistemology, Systems Architecture and Theory and Algorithms, Theory of Science

Abstract

Gödel's Proofs in the context of beautifying (Hehner) and re-beautifying in polycontextural

logic. Deconstruction of the relevance. — See also Gödel's Games within Palindromes (3_24)

Citation Information / How to cite

Rudolf Kaehr: "Goedel Games: “Cloning Gödels Proofs", www.vordenker.de (Sommer Edition, 2017) J. Paul (Ed.), URL: http://www.vordenker.de/rk/rk_Goedel-Games_Cloning-Goedels-Proofs_2005.pdf

Categories of the RK-Archive K01 Gotthard Günther Studies

K02 Scientific Essays

K03 Polycontexturality – Second-Order-Cybernetics

K04 Diamond Theory

K05 Interactivity

K06 Diamond Strategies

K07 Contextural Programming Paradigm

K08 Formal Systems in Polycontextural Constellations

K09 Morphogrammatics

K10 The Chinese Challenge or A Challenge for China

K11 Memristics Memristors Computation

K12 Cellular Automata

K13 RK and friends

Rudolf Kaehr Januar 4, 2006 11/7/05

 

DRAFT 

DERRIDA‘S MACHINES 0

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BB YY TT EE SS && PP II EE CC EE SS

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PPoollyyLLooggii ccss ,, mm--LLaammbbddaa CCaall ccuu ll ii ,, CCoonnTTeeXXttuurreess

Gödel Games: "Cloning Gödel’s Proofs"

–short version–

by Rudolf Kaehr

ThinkArt Lab Glasgow Hallowe’en 2005

"Interactivity is all there is to write about:

it is the paradox and

the horizon of realization."

Rudolf Kaehr Januar 4, 2006 11/7/05

 

DRAFT 

DERRIDA‘S MACHINES 1

Gödel Games: "Cloning Gödel’s Proofs"

1 Goodbye Kurt!

2 Types of ITER in Iterability/Modi of disremption

3 Beautifying Gödel’s Proof

3.1 Hehner’s quotation mechanism of Gödel’s proof3.2 Hierarchical Stratagems: Definition, Substitution, Comparison

4 Dis-Beautifying Gödel’s Proof

4.1 General polycontextural interpretation 54.2 Modeling strategy of Gödel's games 84.3 Interactional strategy 104.4 Reflectional strategy 114.5 Combined strategies 12

5 Where has the contradiction disappeared?

5.1 Truth theoretic considerations 135.2 Bisimulation of strategies 145.3 Ludic inter-tanglings 145.4 Morphogrammatics of Gödel games 155.5 Truth, decidability, liveliness and locatedness 15

6 Is there no relevancy left for Gödel?

PDF-VERSION: http://www.thinkartlab.com/pkl/lola/Godel_Games-short.pdf

Gödel Games: "Cloning Gödel’s Proofs"

To Gotthard Günther’s 105th anniversary

It comes, maybe, like a call from hell, that the only book in the Google-print uni-verse in which my name is mentioned, is

Kurt Gödel, Collected Work, Volume IV,Correspondence A-G

edited by

Charles

Parsons et al. The occurrence of my nameis in straight order: Gödel, Günther, Kaehr. Gödel-Günther, the Correspondence.My name in the bibliography of Gotthard Günther, mentioning an appendix to

Idee und Grundriss,

second edition 1978, my PhD dissertation.There was probably also only one review of Grundriss+Appendix, mentioning

this complex a dinosaur. Günther’s work the body, the appendix its tail; what else?Both boring, not even contributing anything to the linguistic turn, but, at least, youcould jump from one boredom to another catastrophe.

Now, after all this movements have disappeared: linguistic turn, analytic philos-ophy, modal logic, deconstructivism, second-order cybernetics, critical rationalism,critical theory, radical constructivism, holism, system theory, etc., time may beready to read Günther, and especially the Gödel-Günther Correspondence, again,or for a very first time.

1 Goodbye Kurt!

Wouldn’t it be beautiful if we could celebrate Gödel’s proofs in disseminatingthem over all universes and contextures we know until now without getting blockedinto the monolithic character of negative fundamentalism we are confronted in es-tablished understanding of Gödel’s limitation theorems? Couldn’t such a dissemi-nation celebrate its ultimate universality even to worlds which had been profoundlydisturbing Gödel? Like Günther’s multi-verses and other kinds of discontexturality?

Cloning the naturality of natural numbers is still the program of DERRIDA’S MA-CHINES.

Cloned objects are more beautiful than natural objects because they originatein a mundane context, not being obscured by extra-mundane sources.

2 Types of ITER in Iterability/

Modi of disremption

Iter as

repetition

in superposing the identical functor.Iter as

accretion

in distributing the same functor over poly-contexturally differentcontextures.

Iter as

reflection

is introspecting the same functor over intra-contextural ranks ofa contexture.

Iter as

interaction

is disseminating the same functor over trans-contexturally dif-ferent contextures.

-----------------BACK GROUNDS:http://www.thinkartlab.com/pkl/lola/Godel_Games-short.pdfhttp://www.thinkartlab.com/pkl/lola/poly-Lambda_Calculus.pdfhttp://www.thinkartlab.com/pkl/lola/PolyLogics.pdfhttp://www.thinkartlab.com/pkl/lola/ConTeXtures.pdfhttp://www.thinkartlab.com/pkl/media/SUSHIS_LOGICS.pdf

y g

Rudolf Kaehr Januar 4, 2006 7/27/05

 

DRAFT 

DERRIDA‘S MACHINES 2

3 Beautifying Gödel’s Proof

"Our beautification is made in stages. It is tempting to skip the stages and present just thefinal form: a three line proof." Hehner

3.1 Hehner’s quotation mechanism of Gödel’s proof

"Q equals a completely known 3-character string. Our presentation of Gödel's argumentwas parallel to his, but using strings instead of numbers.The heart of the argument is a trans-formation from one level of quotes to two levels, and back down to one, with a “¬” appear-ing in the process. We can simplify the argument without loss of content by going from zeroto one and back. So are the properties necessary to make the argument: a theory must allowus to replace something with its equal, and it must include or allow us to define its own in-terpreter. To save such a theory from inconsistency, we could suspend the ability to replacesomething with its equal under certain circumstances, but that is a distasteful option. Insteadwe leave the interpreter incomplete. In particular, if “¬ I Q ” = ¬ I Q is a theorem then wehave inconsistency, and if it is an antitheorem then I is not an interpreter, so we leave itunclassified." Hehner http://www.cs.toronto.edu/~hehner/BAUA.pdf

3.2 First comments

Q = "non I Q". For reasons of notation, I replace Q by G and "_" by Q. Q=quoting, G=sentence, I=interpreter.Q is the string Q which says (quotes: "_") it is (interpret: I) not (non) a string (Q),becomes:G is the string G

1

which says (quotes: Q

1

) it is (interpreted: I

1

) not (non

1

) a string (G

2

).

Bracket notation of the sentence.G = "non I G" = Q

1

(non

1

(I

1

(G

2

)), thus, and omitting theindices:

I (G) = I (Q(non (I (G)) = non( I (G)).

Comparison shows logical contradiction [

I(G)

; non( I (G))].

"In this situation

"I can not unquote “¬ I G ”"

because it would produce a

contradic-tion

, which would be fatal for the interpreter. In this sense, strict

self-referentiality

is excluded. To save such a theory from inconsistency, we could suspend the ability toreplace something with its equal under certain circumstances, but that is a distastefuloption. Instead we leave the interpreter

incomplete

. In particular, if “¬ I G ” = ¬ I Gis a theorem then we have

inconsistency

, and if it is an antitheorem then I is not aninterpreter, so we leave it

unclassified

." Hehner

contradiction

But this situation makes sense only for a mono-contextural framework. In a poly-contextural setting with its distinction of identity/sameness, the formula can be un-quoted by change of contexture without producing a conflictive contradiction inthe complex system. And the "unclassified" situation can be given a positive mean-ing and an own logical status in a different logical system. Thus, self-referentialitycan be handled without contradiction in a poly-contextural language. And thesame for contradiction. Contradiction in one contexture can be modeled, quotedas a contradiction, in another contexture without producing a collapse of the log-ical system as a such.

incompletness

Incompleteness seems not to be as distasteful as to suspend axioms of equalityto save the system. But, for systematic reasons, both are structurally equal. Thus,equal distasteful. Again, incompleteness and not being classified, enables the illu-sion of some freedom of choice. But it is only the other side of the coin, thereforea question of taste not necessarily of thinking.

inconsistency

Inconsistency is depending on negation or its equivalents. Polycontextural logicsare multi-negational systems, thus, the notion of inconsistency is changing. It seemsthat inconsistency becomes a special case of incomparability.

self-referentiality

In this small game of quotation/interpretation interesting terms are involved:

con-tradiction

,

self-referentiality

,

incompleteness

and

unclassified

. Quite substantialstuff for a theory of computation. The emphasis is optimism, interpreting negativeresults positive; negative teleology.

I G

I

Q

I G

I G

( )

¬ ( )( )

¬ ( )

�

�

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Rudolf Kaehr Januar 4, 2006 7/27/05

 

DRAFT 

DERRIDA‘S MACHINES 4

3.3 The Hierarchic Stratagems: Definition, Substitution, Comparison

Define a formal sentence and the laws of its quotation.Define the formal laws of substitution (transitivity).Compare the result of the substitution with the start postulation of the procedure. There are only two possible outcomes: a contradiction or equality.

Definition

Quotations are logical sentences like all other logical sentences too. They can beevaluated to the value true or the value false. E.g. Eval(Quote(true)) = true.

Substitution: Equal vs. same

"So are the properties necessary to make the argument: a theory must allow us to replacesomething with its equal, and it must include or allow us to define its own interpreter."

"Taking a liberty with substitution inside quotes..."

The defined theory

"allow us to replace something with its equal"

but it is not ableto disallow a distributed substitution because equality of terms is defined in the theoryonly "up to isomorphism". Such theories are identifying the terms "equal" with "same".The polycontextural approach offers a different option to the difference of equality andsameness. Equality in this sense is an intra-contextural term but sameness is a trans-con-textural term. Because substitution is generally defined in a theory only up to isomor-phism we always have the possibility to interpret the action also in a trans-contexturalway. As long as the definitions of the theory are not disallowing this way to use substi-tution there is always some degree of freedom to interpret the terms in another similartheory. In other words, in a polycontextural situation a substitutional action has to de-cide if it is of intra- or a trans-contextural kind. In a highly complex situation it is evenpossible that both kinds of substitutions are reusable at once and that the trans-contex-tural substitution can even be distributed over several neighbor theories instead of onlyone.

Thus, the formal conditions for this new game are: non-characterizablity ("up to iso-morphism") of the action or relation of substitution and polycontextural distribution offormal systems.

Comparison

Obviously, the terms or formulas which have to be compared have to belong to theexact same, i.e., identical formal system. Between the steps of the development of theequation there is now jump to another system. This is guaranteed by the strict identityof the terms and the equation relation. In other words, transitivity of the equation rela-tion holds strictly.

Hehner’s conclusion

"We believe that our presentation of Gödel's Incompleteness Theorem(s) nicely illustratesE.W.Dijkstra's contention that computing science can now repay with interest its debt tomathematics. Specifically, the distinction between program and data, the use of the char-acter string data type, the use of an interpreter, and counting from zero, reduced the proofto three lines. " Hehner, 1990, p. 6

4 Dis-Beautifying Gödel’s Proof

"In a slogan: "Quotes don’t know their mates."" George Boolos, Logic, Logic, and Logic

4.1 General polycontextural interpretation

In a polycontextural setting we are free to choose a more flexible interplay be-tween quotation and interpretation. To quote means to put the quoted sentence ona higher level of a reflectional order or to another heterachical actional level. Weare not forced to limit ourselves to any kind of the well known intra-contexturalmeta-language hierarchies. Those local procedures are nevertheless not excludedat all but localized to their internal place.

To start the argumentation I simply map an index i to the sentences. A quotationis augmenting and an interpretation is reducing its value, say by 1.

This setting is neutral to reflectional and interac-tional definitions of interpretation and quota-tion. It reflects the main idea that quotation/interpretation is an epistemic action and is

therefore, from a polycontextural point of view, involved in locating its propositionsinto the contextural grid of cognition.

Quotational Calculus

A first step to a polycontextural formalization of the quotation/interpretationmechanism is introduced with the

Quotational Calculus

. This calculus is modelingthe quotation/interpretation mechanism along the dimensions of reflectionalityand interactionality of polycontextural logics. On this level of formalization fea-tures of transjunction are not involved. Transjunctional quotation/interpretationwould introduce a bifurcation into the operators I and Q allowing to operate onthe sentence at once at different locations.

From singularity to ubiquitous multitudes

On this level of reflection, to make a start, we will study Gödel sentences in theirsingularity, involved in reflectional and interactional interpretations, but not as si-multaneous multitudes, occurring at all knots of our contextural grid. But this surely,will be the real thing: Ubiquity of Gödel’s games.

� �� ,� ��Q X X I X Xi i i i( ) = ( ) =+ −1 1

y g

Rudolf Kaehr Januar 4, 2006 7/27/05

 

DRAFT 

DERRIDA‘S MACHINES 6

General

pattern of quotation and interpretation. The terms

s

and

t

are changing theirindex depending on the index of the operators. To emphasize on the essentials of theconstruction only the relevant changes of the formula are notated.

Quotational Calculus

general

i j N

�

:

����� �, ,�∀ ∈ ∀ii jrefl

i j i jrefls I s I s. . .:� �� ���� � ��

��

¬( ) = ¬ ( )����� �, ,� :� �� ����. . .∀ ∈ ∀ ¬( ) = ¬i j N s I si j

acti j i j�� ��

����� �, ,� :� ��. .

I s

i j N s Q

actl

i jrefl

i

( )∀ ∈ ∀ ¬ jj i j

refl

i

s Q s

i j N

���� � ��

������ �, ,�

.

.

( ) = ¬ ( )∀ ∈ ∀ jj

acti j i j

actls Q s Q s

negat

:� �� ���� � ��. .¬( ) = ¬ ( )iions

i j N s I Q si jrefl refl

i j

:

������ �, ,� : �. .∀ ∈ ∀ ¬(( )( ) = ¬ ( )∀ ∈ ∀

−� �

������ �, ,� :

.

.

i jrefl

i j

I s

i j N s

1

II Q s I srefl acti j i j

refl¬( )( ) = ¬ ( )+. .� � �

������

1

∀∀ ∈ ∀ ¬( )( ) = ¬ −�, ,� : � �. . .i j N s I Q si jact refl

i j i j1 II s

i j N s I Q

act

i jact act

i j

�

������ �, ,� : �. .

( )∀ ∈ ∀ ¬ ss I s

junctions transjuncti

i jact( )( ) = ¬ ( )

+

+� �. 1

oons

i j N s t I Q si jrefl refl

:

������� �, ,� ,� : �.∀ ∈ ∀ ⊗ii jrefl

i jreflt I s I t. .� �� � ��

������� �

( )( ) = ⊗

∀

−1

ii j N s t I Q s t Ii jrefl act

i jref, ,� ,� : � �. .∈ ∀ ⊗( )( ) = ll

i jrefl

i j

s I t

i j N s t

�� �

������� �, ,� ,� :

.

.

⊗

∀ ∈ ∀

+1

II Q s t I s I tact refli j

acti j

act� � �� � �. .⊗( )( ) = ⊗ −1

�������� �, ,� ,� : � �. .∀ ∈ ∀ ⊗(i j N s t I Q s ti jact act

i j ))( ) = ⊗ +I s I tacti j

act�� ����

����������������

. 1

������������������������������������������������ , , , ,� ,� ������

:�

⊗ = ∧ ∨ → <>{ }reflectional

������������ �, �:� � �

�

. .∀ ∈ ( ) = −i j N I X Xrefl i j i j 1

����������� �, �:� � �. .∀ ∈ ( ) = +i j N Q X X

in

refl i j i j 1

tteractional

i j N I Xact i j

:

������������ �, �:� .∀ ∈ (( ) =

∀ ∈ ( )−��

����������� �, �:�

.

.

X

i j N Q X

i j

act i j

1

�� �

:

����������� �, : :

.=

∀ ∈ = =

+X

classic

i j N i j

i j1

0 �� ��,�

�����������������������������

I I Q Q0 0≡ ≡

��� ��

�����������������������

I Q true true( )( ) =

������� �� .I Q false false( )( ) =

Composed

terms consists of junctions and transjunctions like in PolyLogics.

Reflectional

operations are changing the index j and

interactional

operations theindex i of the level of the tabular terms X

i.j

. This has to be applied to the generalrules for negations and binary operations.

The indices are marking the location of the formula as a whole. It would be tomuch of notation to mark all parts of a formula with indices.

Reflectional and interactional operators, like I and Q, are defining a simple cel-lular automaton with the movements M={left, right, up, down} and a correspond-ing set of rules. OK, a tessellation of kenomic loci is surely not what Wolfram’sworld or Fredkin’s universe are constructing, simply because they are entirelymono-contextural, preserving and guarding their Gödel sentence jealously.

Depending on the strength of reflectionality and interactionality involved in thegame, the operators I and Q can realize wider jumps inside the matrix. Thus, sin-gle operators can be seen as iterations and have not to be restricted to a pus-1/minus-1 jump. This is a further step to dynamize the game of interaction and reflec-tion. Further, but not considered here, transjunctional constellations for quotation/interpretation would have to enter the game to make it more dis-contextural.

Short version

negations

i j s m i

�

:

������ �, ,�∀ ∈ ( ) ∀ .. . .� : � � �

������ �

jrefl

i j i js I Q s I s¬( )( ) = ¬ ( )

∀

+1

ii j s m s I Q s Ii jact

i j i j, ,� � : � � �. . .∈ ( ) ∀ ¬( )( ) = ¬ +1 ss

junctions transjunctions

i j s m

( )+

∀ ∈

:

������ �, (( ) ∀ ⊗( )( ) = ⊗ +,� ,� : � � ��. . .i jrefl

i j i js t I Q s t I s 1����

������ �, ,� ,� : �. .

I t

i j s m s t I Q si jact

i∀ ∈ ( ) ∀ ⊗ jj i jt I s I t

interpreter

� �� �

:���������

.( )( ) = ⊗ +1

������������������

������ �, �:�∀ ∈ ( )i j s m I Xrefl ii j i j

act i j

X

i j s m I X

. .

.

� �

������ �, �:�

( ) =

∀ ∈ ( ) (−1

)) =

∈ = { } ⊗ = ∧ ∨→ <>

−��

,� ,� ,� , , ,

.X

m N I I I

i j

refl act

1

�� ,�{ }

I G G

I G G

Q

acti j i j

refli j i j

:� �

�:� �

. .

. .

→

→

−

−

1

1

aacti j i j

refli j i j

G G

Q G G

:� �

:� �

. .

. .

→

→

+

+

1

1

G

Q

G I G Q G

I

G

i j

refl

i jact

i jact

i j

refl

i

.

. . .

.

+

− +

1

1 1

jj−1

y g

Rudolf Kaehr Januar 4, 2006 7/27/05

 

DRAFT 

DERRIDA‘S MACHINES 8

4.2 Modeling the strategy of Gödel's Game

Despite the final

meaning

of Gödel's proof I focus on the operational steps, not onthe

what

of the construction but on

how

it is realized. What are the strategies ofGödel’s game? The meaning of the construction remains in the mind, the operationsare inscribed on the wall.

G = "non I G"

Let’s read this formula as the sentence:

G is the string G

1

which says (quotes: Q

1

) it is (interpreted: I

1

) not (non

1

) a string (G

2

).

Q1(G1) –––––> G2

| | a. level 1 to level 2 and back to level 1,G1 <–––––– I2(G2) andG2 <––––– I3(G3)| | b. level 2 to level 3 and back to level 2,Q2(G2) ––––> G3

and finally:G1 <–––––––> G2 c. comparision of level 1 and level 2.

"The heart of the argument is a transformation from one level of quotes to two levels,and back down to one, with a “¬” appearing in the process." Hehner

Again, there is no statement by Hehner about what kind of levels are involved. There-fore, to re-beautify or dis-beautify the Gödel construction we are allowed to involve thegame into at least two additional kinds of levels: the reflectional and the interactionallevels of polycontexturality. Thus, we are free to interpret the term "level". It can be, aswell known, intra-contextural, representing the classic logical or computational situa-tion. Or additional, trans-contextural, a reflectional but also an interactional strategy.And other interpretations of the modeling strategy not yet introduced, too.

With polycontextural modeling, instead of a contradiction we get a separation. Thecomparison happens in a third system which compares the result of system S1 with theresult of system S2. From the point of view of the third system S3 it is even possible tocancel the contextural differences and to put the whole derivations with their systemswitches back into one and only one contexture, so we would have returned back tothe mono-contextural case and produced as usual the famous contradiction. But thistime we would know a little bit more about the genesis, the mechanism and the hiddenconditions of this contradiction, not by supposition and intuition but by construction.

Paleonymy of wording "Our presentation of Gödel's argument was parallel to his, but using strings [...]." Hehner

In all cases, the wording of the modeling has to be preserved. There will be nochange of the pre-given formulations. Neither in the definition of the problem nor theprocedure of its solution. What differs only are the possibilities of interpreting the no-tions involved. In the classic case there is no space for different but similar (analogues)thematizations. But this new possibilities, again, are in no violation to the pre-given no-tions, because these notions are simply not defined in strict non-ambiguous terms, thusgiving space for legitimate deviations in the frame of preserving analogies.

Symmetric and asymmetric constellations

With the classic setting there is a asymmetry between quotation and interpreta-tion. Formulas have to be interpreted by I. A quoted sentence is "passive", like astored program, an interpretation I is activating the passive sentence and changingit into an "active" one. "That is what an interpreter does: it turns passive data intoactive program." (Hehner) Therefore there is an operational asymmetry betweenquotations and interpretations. In the polycontextural setting this asymmetry is rel-ative to its local realization. Between different contextures, quotation and interpre-tation, are in a symmetric exchange relation. Asymmetric relations betweenquotation and interpretations are keeping at all loci they occur the Gödel results.They are repeating at each locus the Gödel configuration. Thus, they are guaran-teeing the ubiquity of the Gödel sentence over all contextures.

Symmetric relations are spacing, giving space, to constructions which are notproducing contradictions but analogous or queer Gödel constellations being dis-tributed over different contextures. Thus, quotations become active. The relationbetween interpreter/quotation, i.e., between active and passive, is now dynam-mic and not a static pre-given hierarchic relation.

Obviously, we observe again an order relation between the asymmetry of inter-preter and quotation, and an exchange relation between interpreter/quotation forthe symmetric trans-contextural situations. And because we are dealing in all cas-es of the distribution with quotation and interpretation and not with something dif-ferent, the coincidence relations are realized. Thus, the whole dissemination of theGödel construction is well managed, i.e., mediated by the proemial relation.

The asymmetric constructions are directly conform with the wording of the con-struction. They simply have some additional indices marking their place in the con-textural grid. The asymmetric constructions are accepting the wording, too. But areinvolved in an analogous understanding of the main notions based on the coinci-dence relations of the dissemination.

Based on the passive/active chiasm,it seems, that we have now two possi-bilities to deal with the Gödel sen-tence. One needs an interpretation Iof the formula, the other is dealing di-rectly with the formula. The first usesthe passive understanding of the quo-tation and needs an active operator,the interpreter I. The second is basedon an active understanding of the op-

eration of quotation Q and is able to deal directly with the formula. The activity ofthe quotation operator is to shift the formula to another level, while the activity ofthe interpreter is to reverse this shift. As long as we don’t use indices to mark thedifferent contextures the mechanisms coincide.

In other words, what is an interpretation in one constellation is a quotation inanother, and vice versa. Quotation in a polycontextural constellation is not freez-ing a statement in contrast to the activation of the interpreter. Both operators arein a dynamic interplay.

As a result of the distribution of the Gödel sentence and the new dynamics be-tween interpreter and quotation we have to distinguish at least two new features:

1. MULTI-VERSAL UBIQUITY of Gödel, i.e., multi-versal validity and availability,2. NON-RELEVANCY OF GÖDEL, e.g., for self-referential compiler construc-

tions.

I G

I

Q

I G

I G

( )

¬ ( )( )

¬ ( )( )

¬ ( )( )

¬

���

G

Q

I G

II G( )( )

�

y g

Rudolf Kaehr Januar 4, 2006 7/27/05 DRAFT  DERRIDA‘S MACHINES 10

4.3 Interactional strategy

The interactional inter-pretation of iteration isdistributing the formu-la over different con-t ex tu res , s t i l l i naccordance with theprimary wording ofquotation and interpre-tation.

The first part of thebracket formula shows an interaction-al realization of the Gödel sentence,repeating it at the locus i, producingnot a contradiction but incomparabili-ty. The second part realizes a shift fur-ther to another contexture, producingmore distance to the original position.

This horizontal distribution of theGödel sentence shows a save interac-tional distance to its original sentencewithout any detours.

M1 M2 M3 M1 M2 M3 M1 M2 M3

O1 O2 O3

S100 S123 S023

G

Q

I G

I

i j

i j

i j i j i j

i j i

.

.

. . .

.�

¬ ( )( )( )¬

− − −

− −

1 1 1

1 1.. .

. . .�

j i j

i j i j i j

G

I G

−

− − −

( )( )¬ ( )( )

1

1 1 1

¬ ( )

− −i j i jG2 2. .

¬ (− −G Gi j i j i j. . .;� 2 2 ))( )

interactional j constant

system systemi i

,�

����

=−− −

− − −¬

1 2

1 1

����������

������ �

�

system

G

I G

i

i

i i i 11

1 1 1

1 1 1

( )( )( )¬ ( )( )

¬ ( )( )

− − −

− − −

�

� �

i i i

i i i

I G

I G ��

�

�

� ;�¬ ( )

¬ ( )( )− −

− −

i i

i i i

G

comparison

G G2 2

2 2

I G

I

Q

I G

i i

i

i

i i i

( )

¬ ( )( )

¬ ( )( )

− − −i i iI G1 1 1

¬ ( )( )¬ ( )( )

¬

− − −

− − −

−

i i i

i i i

i

I G

I G

1 1 1

1 1 1

22 2Gi−( )

��

G

G G

G G GG G

i j

i j i j

i j i ji j i j

.

. .

. .. .

+

− + +

− +−

2

1 1 1

1 12 ii j

i j i j

i j

G G

G

+

− + −

−

2

1 1 1

2

.

. .

.

4.4 Reflectional strategy

Like the interaction-al, the reflectionalinterpretation of it-eration is distribut-ing the formu laover different con-textures, still in ac-cordance with theprimary wording ofquotation and inter-pretation.

The first part of the bracketformula shows an reflection-al realization of the Gödelsentence, repeating it at thelocus i, producing not a con-tradiction but incomparabili-ty. The second part realizesa shift further to another con-texture, producing more re-flectional distance to theoriginal position.

This vertical distribution ofthe Gödel sentence shows,dual to the horizontal, asave reflectional distance toits original sentence withoutany detours.

M1 M2 M3 M1 M2 M3 M1 M2 M3

O1 O2 O3

S111 S020 S003

G

Q

I G

I G

i

i

i i i

i i i

.

.

. . .

. . .�

1

1

2 2 2

2 2 2

¬ ( )( )( )¬ ( )( )�� . . .I Gi i i2 2 2¬ ( )( )

¬ ( )

i iG. .3 3

¬ ( )( )

G Gi i i. . .;�1 3 3

reflectional i constant

G

I G

j

j j j

,�

��

=

¬ ( )(+ + +1 1 1 ))( )¬ ( )( )

¬ ( )( ) ¬

+ + +

+ + +−

�

� ��

j j j

j j jj

I G

I G

1 1 1

1 1 122 2

2 2G

comparison

G G

j

i j i j i j

−

− −( )

¬ ( )

�

�

�

�

� ;�. . .(( )

I G

I

Q

I G

i i

i

i

i i i

. .

.

.

. . .

1 1

1

1

1 1 1

( )

¬ ( )( )

¬ ( )( )

i i iI G. . .2 2 2

¬ ( )( )¬ ( )( )

¬

i i i

i i i

I G

I G

. . .

. . .� �

2 2 2

2 2 2

ii iG. .3 3( )

G

G G

G G G

G

G

i j

i j i j

i j i j i ji j

.

. .

. . ..

+

− + +

− + +

2

1 1 1

1 1 2

ii j i j

i j

G

G

. .

.

− + −

−

1 1 1

2

y g

Rudolf Kaehr Januar 4, 2006 7/27/05 DRAFT  DERRIDA‘S MACHINES 12

4.5 Combined strategiesQuotations are reflectional or epistemic attributes (operators) and not semantic at-

tributes like valuations with true and false.

The construction starts in systemi.1 (in the diagram at O1M1=S1.1) by decision, thenreflects and maps it inside of systemi onto systemi.2 (O1M1 to O1M2) and interacts ina next step with systemi-1(O2M2 to O3M2). The comparison of the two results, the be-ginning and the end of development, (Gi.1, noni-1Gi-1), has itself a representation insystemi-2 (O3M3 to O1M3). This development is achieved by applying a reflectionalinterpreter Ii.1 and a interactional interpreter Ii.j+1 successively. Obviously, the resultedcomparison has not produced any contradiction. But nevertheless some kind of "deep-structure" of a contradiction is preserved: we only have to cancel the notation of loca-tions and dislocations, that is, the indices, to obtain our well known contradiction. Thusthe structure of the construction is mapping the contradiction in its behavior but not inits semantics. The comparison as an incomparability can now be studied in its owncontexture, say systemi-2. A first step can be to ask about the distance between the com-pared sub-systems and how this distance is characterizing the whole system as such.

Mixed behaviors, interactional and re-flectional, are producing, additional tothe strict horizontal and vertical moves,some diagonal behaviors. Diagonalstrategies are open up rules for strongerdistance or strangeness to their origin.Some loss of simple safety in distance isinvolved.

M1 M2 M3 M1 M2 M3 M1 M2 M3

O1 O2 O3

S113 S020 S023

G

Q

I G

I G

i

i

i i i

i i i

.

.

. . .

. . .�

1

1

2 2 2

2 2 2

¬ ( )( )( )¬ ( )( )�� . . .I Gi i i2 2 2¬ ( )( )

¬ ( )

− −i iG1 1

¬ ( )( )

− −G Gi i i. ;�1 1 1

interactional reflectional

system systi j

+. ���� eem system

G

i j i j

i j

. .

.

���������� �

������ �

�

+ −

¬

1 1

ii j i j i j

i j i j i j

I G

I G

. . .

. . .�

+ + +

+ + +

( )( )( )¬

1 1 1

1 1 1(( )( )¬ ( )( ) ¬+ + + − + −� ��. . . . .I G Gi j i j i j i j i1 1 1 1 1 1 jj

i j i j i j

comparison

G G

+

− + − +

( )

¬ ( )(1

1 1 1 1

�

� ;�. . . ))

G

G

G G G G

G

G

i j

i j

i j i j i j

i j

i j

.

.

. . .

.

.

+

+

− − +

− +

2

1

2 1 1

1 1

ii j

i j i j

i j

G G

G

+

− + −

−

2

1 1 1

2

.

. .

.

pp

Rudolf Kaehr Januar 4, 2006 7/27/05 DRAFT  DERRIDA‘S MACHINES 13

5 Where has the contradiction disappeared?

5.1 Truth theoretic considerationsSemantics/Meontics are per se static, that is, structural and algebraic systems. Well

described by algebras, governed by equality relations as their criteria of equality.These semantic formulas are hint-ing to the semantics of PolyLogics.Locally, each logic can be consid-ered as a 2-valued system, here,with strict classic negation rulesbetween truth and false. The pairsof truth-values are connected to achiastic chain ruled by the pro-emial relation over the truth-val-ues and their loci. The chain maybe structured by more complex ar-chitectonics then linearity.Because of the incomparability ofsystemi.j and systemi-2.j for exam-ple, no contradiction can hap-pen. No contradictions for theinteractional, reflectional andmixed strategies; only some sortof incomparability or incompati-bility, but a lot of additional cleanintra-contextural contradictions.Incomparability can be studiedper se and also in their relation tothe system they occur. Logical

contradiction, thus, is incomparability with zero distance, is otherness with zero toler-ance.

Contradiction is zero tolerance towards the acceptability of others.

Polycontextural semantics, understood as meontics, will have to deal not with logicaltruth and falsehood, in Tarski’s or Kripke’s or others sense, this may still be preservedlocally, but with new constellations of strangeness, closeness, distance to otherness andtogetherness and their epistemological topologies.

Logical truth vs. acceptability conditions"An adequate truth theory for a language defines the language and determines the set ofits tautologies. It does so by specifying (recursively, in a metalanguage) truth conditions forits sentences. An adequate theory of rational belief systems for a language also defines thelanguage and determines the set of its tautologies. It does so by specifying (recursively, ina metalanguage) the conditions under which a truth or falsity claim concerning sentence ofthe language may be incorporated into a rational system of beliefs of the language. Morebriefly, we may say that it does so by specifying acceptability conditions. The differencebetween truth and acceptability semantics thus reflect different analyses of understanding.On one analyses, to understand a sentence is to know what would make it true. On the oth-er, it is to know what makes it acceptable." Brian Ellis, Rational Belief Systems, 1979, p.viii

∀ ∈ ( )

( ) ∈{ }i j s m

val G t f

val G

i ji j i j

i j

, :

� �� ,�

�

.. .

.(( ) = ⇔ ¬ ( )( ) =�� � � � ��

�

.. .

.

.

t val G f

val G

i ji j i j

i j

i jji j

i j i ji j

l

f val G t

val G

( ) = ⇔ ¬ ( )( ) =�� � � � ��

�

.. .

.

(( ) ∩ ¬ ( )( ) = ∅ ∅( )� � � ��(...� ...� ...)m k k l k

l

val G

t �� � ,� � � ��� ,�∧( ) ∧( ){ } ∈ ≠f f t incomparable l kk k l

intra contradictions

i j s m I Gi j i j

−

∀ ∈ ( ) ( ), :�� �. . ;;� �. . .¬ ( )( )( )i j i j i jI G

Incomparabilities

interractional G G

reflectio

i j i j i j:� ;�. . .¬ ( )( )− −2 2

nnal G G

mixed

i j i j i j:��� ;�

:�������

. . .¬ ( )( )− −2 2

������ ;�. . .G Gi j i j i j¬ ( )( )− + − +1 1 1 1

pp

Rudolf Kaehr Januar 4, 2006 7/27/05 DRAFT  DERRIDA‘S MACHINES 14

5.2 Bisimulation of strategiesTo be concerned about the truth of sentences like the Gödel ones, can be considered

as an intra-contextural business. Locally, this is a crucial point, globally, in the game ofpoly-thematism, it is one of the stops of a wild journey. More interesting than the truthpreserving/violating results in proof situations seems to be evolved by the strategicways, the pathways discovering labyrinthine worlds, of the behaviors in the tabulargame of a multitude of constellations. Depending on the complexity of the formula, dif-ferent choices for reflectional or interactional thematizations of the operators, quota-tion/interpreter, are possible. Some kind of zag-zagging through the tabularity of thecontextural grid emerges. Further more, many Gödel games may start at once at dif-ferent locations. These paths of behaviors are not to be studied by their truth-semanticsbut by their behavioral characteristics. The method to apply is the co-algebraic studyof bisimilarity. Thus, truth-semantic considerations are structural, algebraic and visible,behavioral features are co-algebraic and hidden.

A Gödel sentence can be placed at allloci of a contextural grid. Thus, we canstart our game at any of these locations,in isolation or all at once, or mixed, in-voking a quite wild parallelism of Gödelgames. A starting sentence like Gi.j cangive opportunity for multiple thematiza-tions at once, leading to distributed re-sults and distributed comparisons based

on a single beginning. More liberated multiplicity enters the game with the possibilityto distribute the very beginnings, leading to separated comparisons in the common ma-trix. This has not to be restricted to separated parallelism but can introduce interwovenbehaviors of Gödel games.

Nonpredictability of strategical behaviors

A new undecidability enter the game with the strategic behaviors of Gödel games.Truth functional strategies are visible, behavioral strategies are hidden. The method tostudy the behavioral aspects of a system are given by the apparatus of co-algebra.There is no recursive mechanism to follow the pathway of strategies in Gödel games.The switch from one contexture to another is not recursive but spontaneous.

Bisimulation of strategies

Thus, additional to the different truth semantic and acceptability conditions of thestructural approach, which is based on constructors, questions of comparability andequality of strategies have to be focussed. What is the criterion for the equality of strat-egies? An answer may be given by the notion of bisimulation.

„By identifying two states with same external behavior, we get an extensional notion ofequality, that can be captured by the following axiom:Axiom 2.4. Two states are considered equal if they cannot be distinguished by (a combina-tion of) observations.To a user, again, the state may remain hidden, it is irrelevant, as long as the automatonimplements the desired regular expression. Again, two states may be identified, if they be-have the same way on the same input, which is to say, if they cannot be distinguished byany observation.“ Peter Gumm

G

G

G G G

G

G G

i j

i j

i j i j

i j

i j i j

.

.

. .

.

. .

+

+

− +

− +

−

2

1

1 1

1 1

2 ii j

i j

i j

i jG

G

G

+

−

−

+ −

2

1

2

1 1

.

.

.

.

pp

Rudolf Kaehr Januar 4, 2006 7/27/05 DRAFT  DERRIDA‘S MACHINES 15

5.3 Ludic inter-tanglingsThe local/global game is not the end of the chances. We can thematize the whole

interplay of algebraic and co-algebraic features. Both at once. Which is: total accep-tance. This approach is covered by the Swinging Types, introduced with much dynam-ics by the German computer scientist Peter Padawitz.

Swinging Types providea specification and veri-fication formalism fordesigning software interms of many-sortedlogic. Current formal-isms, be they set- or or-der-theoretic, algebraicor coalgebraic, rule- ornet-based, handle eitherstatic system compo-nents (in terms of func-tions or relations) ordynamic ones (in termsof transition systems)and either structural or

behavioral aspects, while swinging types combine equational, Horn and modal logic forthe purpose of applying computation and proof rules from all three logics. Peter Padawitzhttp://issan.cs.uni-dormund.de/~peter/Swinging.html

Duality table

Also algebras and co-algebras are definedas dual conceptions and are delivering in-teresting results exploiting this duality andsymmetry, there are also highly interestingasymmetries to observe delivering evenmore interesting results. One of these topicsis bisimulation.

„But the theory is not just a simple minded dualto universal algebra. Structures such as e.g.bisimulations, that don't have a classical coun-terpart in universal algebra, but that are wellknown from computer science, figure prominent-

ly in the new theory." Peter Gumm In-between duality and paradigm change

There is a paradigm shift between algebra (constructors dual selectors, induction)and co-algebra (observators, co-induction) disturbing the duality between algebra andco-algebra. http://www.thinkartlab.com/pkl/media/SKIZZE-0.9.5-Prop-book.pdf

induction co-inductioninitialconstructortotal

final objectdestructorpartial functions

structure behaviorwell founded non well founded

visible hidden

Turing Machine Persistent TM

algebra coalgebra

Horn clauses liveness axioms

Algebra Co-Algebra

construction : constructors

destruction : selectors

observators

algebra: induction

co-algebra: coinduktion

duality

paradigm change

pp

Rudolf Kaehr Januar 4, 2006 7/27/05 DRAFT  DERRIDA‘S MACHINES 16

5.4 Morphogrammatics of Gödel gamesWhat is left is the dynamics as such realized by the swinging types. The switch from

static to dynamic models, from algebraic to co-algebraic and back is neither algebraicnor co-algebraic. It is neither visible nor hidden. This structure of neither-nor, total re-jection, is inscribing a form beyond known scriptures of algebraic or co-algebraic char-acter. These forms has presentations as morphograms, and is studied in the adventuresof morphogrammatics.

Morphogrammatics is thematizing the very patterns of the behavior of Gödel games,abstracting, i.e., subversing the modi of their thematizations as formal systems and in-scribing as a result of such a subversion the pre-semiotic dynamics of its placing, i.e.,its locations, its loci, its chora.

5.5 Truth, decidability, liveliness and locatednessNew topics of meta-theoretical studies emerge. To each proposed approach we have

to study the topics analogue to the classic questions of truth, decidability, consistency,etc. in the light of a new design of formal systems and computation.

Liveliness and undecidability

Jumping between trees creates new forms of undecidability, new kinds of unpredict-ability as liveliness. Liveliness is the new kind of undecability in complexions of formalsystems. Liveliness is not a loss of control and security but a guaranty of viability.

The rule that a quotation shouldn’t be changed by the quoter is reasonable only forlimited cases, like legally relevant quotations. But self-referential computing systems arenot to be restricted by this metaphor of legality but should be more based on the met-aphor of living systems. Life is not only repetition but also change, and change is notequal destruction and chaos but creative transformations.

DiamondStrategies

In the terminology of Graphematics and DiamondStrategies, attributes of sentencesare decidable or nondecidable. They belong either to the position or the oppositionlocus of a diamond. Polycontexturality is not based on sentences (propositions, state-ments) but on textuality and its reflectional deepness and interactional broadness. Thedistance of textual constructions, say in Gödel games, can be reflected as a neither-nor situation in respect of decidability and nondecidability. Two Gödel games whichdon’t coincide have a distance, are separated and in some sense incomparable. EachGödel game has its own Gödel sentence, thus its own distinction of decidable/nonde-cidable sentences. This situation takes place at the both-at-once locus of the diamond.

Thus, additional to the classic hierarchy of decdability, a heterarchy of neither-nor-and at-once-situations have to be studied.

Types of abstractions

Propositional abstraction: from truth-value semantics to systemic meontics,Behavioral abstraction: from similarity of movements to bisimulation patterns,Chiastic abstraction: from system changes to swinging types of dynamics,Morphogrammatic abstraction: from invariance to patterns of kenomic loci.

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Rudolf Kaehr Januar 4, 2006 7/27/05 DRAFT  DERRIDA‘S MACHINES 17

6 Is there no relevancy left for Gödel?

To find back home to our story, we have to restrict and focus ourselves again on anextremely narrow game.

The whole Gödel construction lives from the fact that it doesn’t leave its contexturalbox. It is boxed strictly inside of a contexture and remains mono-contextural despitesome meta-theoretical detours. The paradoxical result is that the box is open exactlywhen the box is closed. And this, obviously, is in strict conflict with the underlying logic.

Inside of a polycontextural scenario, reflectional, interactional or mixed or neither-nor, there is no strictness which forces to the Gödel results. They simply belong to an-other world of thinking and formalizing. And by the way, this world was not evenGödel’s own paradise, he always tried to surpass it. But how? Gotthard Günther hasgiven him some hope, evoked some kind of an ultimate resolution, but at the end he/it was to late...

"Evidently Gödel entertained hopes that the kind of analysis of the structure ofself-consciousness that Günther undertakes and finds precedent for in the classicalidealists would yield not only philosophical illumination but even a basis forconstructing a type-free logic that would be philosophically better motivated thanwhat was then available." Parsons

In the Gödel case the comparison of the results of the formula are producing a logicalcontradiction, the polycontextural demonstrations are showing no contradiction butonly incompatibility/incomparability and separation of the parts of the formula in ques-tion. This is by no means only a negative result! We easily can deal, that is thematize,this result in an own contexture and produce some further results about separabilityand incompatibilities of our construction, leading to new insights about the relation-ship, say of a program and its interpreter in self-referential situations.

There is for sure nothing wrong with Gödel’s results. From a polycontextural view-point there is no criticism to add, they are simply loosing their relevance for trans-com-putational argumentations and constructions.

Ubiquity of Gödel’s Ghosts

Ubiquity is not universality nor naturality, it is much more, more than Gödel was ableto dream: Gödel is everywhere, even in worlds which are not allowed to exist inGödel’s paranoid Platonism. Multi-verses of any complexity are inhabited by Gödel’sGhost. The price he paid is the loss of his relevancy. There is nothing ultimate anymore.He is a ghost as many others, too.

We have not only the joy to clone Gödel but also, and this is even more intriguing,the fun to clone Gödel’s Ghost.

Like paradoxes, antinomies and all kind of contradictions and other monsters, Gödelis domesticated and has his cards to play, as anybody too. The wine of negative spec-ulations is thin and old. To celebrate human creativity exploiting Gödel’s negative re-sults is obsolete. What we need is an understanding of creativity and computationbeyond positive and negative attitudes.

Emil Post: "The limitations of mathematizing power of home sapiens."

http://www.thinkartlab.com/pkl/archive/GUNTHER-GODEL/GUNTHER-GODEL.htmhttp://www.thinkartlab.com/pkl/archive/Gunther-Godel_german_enlish.pdf

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Rudolf Kaehr Januar 4, 2006 7/27/05 DRAFT  DERRIDA‘S MACHINES 18