Dissemination_ Introducing the Proemial Relationship

7/7/2015 Dissemination:Introducingtheproemialrelationship

http://www.thinkartlab.com/pkl/media/DERRIDA/Proemial%20Relationship.html 1/23

ThinkArtLabHogmanay2004

ThinkArtLabAnimation:A.T.Kelemen

November12,1998Dr.RudolfKaehr

Dissemination:Introducingtheproemialrelationship

Therearemanywaysofcombiningabstractobjectsorinstitutions.Forexample,giventwoinstitutionsINS1andINS2which,intuitively,areindependentwecanformtheirproduct.ThisproductinstitutionhasallpairsofsignaturesfromINS1andINS2,respectively,asmodels,andsentenceswhichareeithersentencesfromINS1orfromINS2withtheobvioussatisfactionrelation."Cat.,p.357

Itisshown,thatthecategoryofinstitutionsiscomplete.

Theideaofdisseminationtriestoexplicateandformalizeaquitedifferentintuitionofcombininginstitutionswhichisnotproducingdiversityandmultiplicitybycombiningabasicsystemasaproductorsumorwhateverconstructionbutintroducesmultipledifferencesintheveryconceptofthebasicsystemitself.Afterthisconstructionapolylogicalorpolycontexturalsystemcanbecombinedinmanyways.Thisideaofmultitudesofbasicdifferencesinthenotionofformality,takenseriously,isinfundamentalcontrasttotheexistingconceptsofformalityinmathematics.Obviosly,thesemultitudesaremorefundamentalthanalltypesofmanysortedtheories,ortypedlogics,orpluralitiesofregionalontologies,domainsandcontexts.

1TheideaofproemialityAveryfirststepinthisdirectionwasmadebythephilosopher



GotthardGuntherwithhisideaofaproemialrelationship"introducedinhispaperCognitionandVolition"(1970)aboutaCyberneticTheoryofSubjectivity.

Inordertoobtainageneralformulafortheconnectionbetweencognitionandvolitionwewillhavetoaskafinalquestion.Itis:Howcouldthedistinctionbetweenformandcontentbereflectedinanysortoflogicalalgorithmiftheclassictraditionoflogicinsiststhatinalllogicalrelationsthatareusedinabstractcalculithedivisionbetweenformandcontentisabsolute?Theansweris:wehavetointroduceanoperator(notadmissibleinclassiclogic)whichexchangesformandcontent.Inordertodosowehavetodistinguishclearlybetweenthreebasicconcepts.Wemustnotconfuse

arelation

arelationship(therelator)

therelatum.

Therelataaretheentitieswhichareconnectedbyarelationship,therelator,andthetotalofarelationshipandtherelataformsarelation.Thelatterconsequentlyincludesboth,arelatorandtherelata.

However,ifwelettherelatorassumetheplaceofarelatumtheexchangeisnotmutual.Therelatormaybecomearelatum,notintherelationforwhichitformerlyestablishedtherelationship,butonlyrelativetoarelationshipofhigherorder.Andviceversatherelatummaybecomearelator,notwithintherelationinwhichithasfiguredasarelationalmemberorrelatumbutonlyrelativetorelataoflowerorder.

If:

Ri+1(xi,yi)isgivenandtherelaturn(xory)becomesarelator,weobtain

Ri(xi1,yi1)whereRi=xioryi.Butiftherelatorbecomesarelatum,weobtain

Ri+2(xi+1,yi+1)whereRi+1=xi+1oryi+1.Thesubscriptisignifieshigheror

lowerlogicalorders.

Weshallcallthisconnectionbetweenrelatorandrelatumthe'proemial'relationship,forit'prefaces'thesymmetricalexchangerelationandtheorderedrelationandforms,asweshallsee,their



commonbasis."

Neitherexchangenororderedrelationwouldbeconceivabletousunlessoursubjectivitycouldestablisharelationshipbetweenarelatoringeneralandanindividualrelatum.Thustheproemialrelationshipprovidesadeeperfoundationoflogicasanabstractpotentialfromwhichtheclassicrelationsofsymmetricalexchangeandproportionedorderemerge.

Itdoesso,becausetheproemialrelationshipconstitutesrelationassuchitdefinesthedifferencebetweenrelationandunityor,whichisthesamebetweenadistinctionandwhatisdistinguished,whichisagainthesameasthedifferencebetweensubjectandobject.

Itshouldbeclearfromwhathasbeensaidthattheproemialrelationshipcrossesthedistinctionbetweenformandmatter,itrelativizestheirdifferencewhatismatter(content)maybecomeform,andwhatisformmaybereducedtothestatusofmeremateriality"."

Westatedthattheproemialrelationshippresentsitselfasaninterlockingmechanismofexchangeandorder.Thisgaveustheopportunitytolookatitinadoubleway.Wecaneithersaythatproemialityisanexchangefoundedonorderbutsincetheorderisonlyconstitutedbythefactthattheexchangeeithertransportsarelator(asrelatum)toacontextofhigherlogicalcomplexitiesordemotesarelatumtoalowerlevel,wecanalsodefineproemialityasanorderedrelationonthebaseofanexchange.Ifweapplythattotherelationwhichasystemofsubjectivityhaswithitsenvironmentwemaysaythatcognitionandvolitionareforasubjectexchangeableattitudestoestablishcontactbutalsokeepdistancefromtheworldintowhichitisborn.Buttheexchangeisnotadirectone.

Ifweswitchinthesummerfromoursnowskistowaterskisandinthenextwinterbacktosnowskis,thisisadirectexchange.Buttheswitchintheproemialrelationshipalwaysinvolvesnottworelatabutfour!"Gunther

1.1Someexplanationsoftheideaofproemiality

Theproemialrelationshipisthereforeatfirstaninterlockingmechanismofthetwoconceptsofexchangeandorderorsymmetryandasymmetry.

Diagramm1



cascadicrepresentation

Afurtherexplicationoftheintuitionofproemialityisachievedifweconsiderthefactthattheobjects,therelatorandtherelataoftherelations,havetofittogetherinacategoricalsense.Thereisasimilarityoftherelatorsofdifferentlevelsaswellasfortherelataofdifferentlevelsinthesensethatthedifferentrelatorsarerelatorsandnotsomethingelse.Andtherelataoneachlevelarerelataandnotrelators.ForthatIintroducethecoincidencerelation,whichdesignatescategoricalsameness(likeness,similtude).

TofinishthepictureIintroducetheexchangerelationbetweenthefirst"andthelast"elementoftheinterlockingmechanismoforderandexchangerelations.AsalaststepImentiontheposition,thelogicallocus,oftheorderrelationsaccordingtothehigherorlowerlogicalorders".

PrObj=(ObjOrd,Exch,Coin,Pos)

Diagramm2

Butthisexplanationstillexcludesthethirdtermofthedefinitionofarelation,therelationitself.Remember:Wemustnotconfusearelation,arelationship(therelator),therelatum.

AndfinallyIconsiderthefactthatthereisoneandonlyoneconceptofrelationandrelationalityunderconsideration.thereforetheconceptofrelationisbasedonunicity(uniqueness),representedby1.Thisissurelynotaharmlessstatement,itsupposesomethinglikeacommonintuitionofrelationalityoroperativitywhichfindsitselfexplainedandformalizedinsomemathematicalconstructionswhichareacceptedbythescientificcommunity.Therefore,Guntherschain"arelation,arelationship(therelator),therelatum"hastobecompletedbytheveryconceptofrelation,thatis,relationalitybasedin



unicity.

Thefullfledgedexplanation,withoutthearrow"relation>relationality",oftheproemialrelationovertwolociisgivenbyitsconceptualgraph.Thescenarioisthesameforthedistributionandmediationofotherconcepts,likeoperations,functions,categories,institutionsetc.

Thusthedefinitionhastobeexpandedto:

PrObj=(ObjOrd,Exch,Coin,Pos)

withObj={relator,relatum,relation,relationality,unicity}

Inthiscontextitisnotmytasktodefendthisconstructionagainstthemanyattemptstoreduceittosomethingelse.TogofurtherinthegameImaketheoptionthatitwillbeusefulfordevelopingsomenewmechanismsofcombiningabstractobjectslikeinstitutions,logics,arithmetics,categorytheoriesandmore.Inexercisingthisgamethenewintuitionwillshapeitselfintoamoreacademicform.

Afterhavingintroducedtheideaofproemialityitwouldbepossibletoformalizeitfurtherandtodevelopapreliminarytheoryofproemiality,alsosometimescalledchiasticsortheoryofmediation.Themainthesis,therefore,isthatproemialityoffersamechanismofcombininginstitutionswhichdoesn'tbelongtotheuniverseofcombiningcategories.Thismechanismofcombininginstitutions,e.g.distributionandmediation,isfundamentallydifferentfromtheclassicalones.Despiteofthisdifferencethisstrategyisinnocontradictionoroppositiontotheknownprinciplesofcombiningsystemsoflogics.

Itissimplysomethingdifferentandtheclouwouldbetoexplainthisdifferenceinfull.Dontconfusetheexchangeofrelatorandrelatumofarelationinthemechanismoftheproemialrelationshipwiththesuperpositionofrelatorandrelationinrelationallogics.Thereisnoproblemtoapplyarelator,oraoperatororafunctortotheresultofarelationoroperationorfunctionase.g.inrecursiontheoryorinmetalevelhierarchies.



Metaphor

Ifweproemializethelinguisticsubjectobjectrelationofasentenceweshouldn'thesitatetobestrictlystructural.TheexampleisborrowedfromHeinzvonFoerster.

"Thehorseisgallopping"(DasPferdgallopiert),theinterchangedsentencecanonlybe"Thegallopishorsing"(DerGalloppferdet).

Nobodysupposedthatwearedoinganalyticphilosophy.

1.2ProemialityandArchitectonics

1.2.1Abouttheascategoryinproemiality

WhatIhavedevelopedsofarisonlythehalfofthestory.Alsoitmightbeobviousthatthewordingofe.g."theoperator(ofonesystem)becomesanoperator(ofanothersystem)"isinstrongconflictwiththeidentityofitstermsthereforethissituationneedsamorepreciseexplication.Itshouldbeclearthatatermwhichisinonesystemanoperatorandsimultaneouslyanoperandinanothersystemsissplitinitsownidentity.Itisatonceitselfandsomethingelse.Thistermhasatoncetwofunctions,tobeanoperatorandtobeanoperand.Therefore,fromthepointofviewofidentityanditslogic,thistermisinitselfneitheranoperatornoranoperand.

Whatthenisit?Howcanwedefineitmoreaccurate?Itispartofanchiasticinterplayandwehavetobemoreexplicitwithourwording.Insteadofspeakingofan"operator"orofan"operand",weshouldusetheascategoryandusethewording"anobjectXasanobjectYisanobjectZ".Thus,anoperatorasanoperatorissimultaneouslyanoperand.

Anoperatorasanoperandisanoperand(ofanotheroperator)

1.2.2Aboutthearchitectonicsoftheascategory

TomakethiswordingmorepreciseIintroduceadiagramwhichiswellknownfromthetableauxmethodofformalizedpolycontexturallogic.



Thistypeofdiagramswasfirstintroducedtodealinaproperwaywiththetableauxmethodinpolycontexturallogic.Especiallytounderstandthefunctioning,andthisgivesprobablyalsosomelightonitsmeaning,ofthesocalledtransjunctions,Iintroducedthistabulationofthestepwisedecompositionofsignedformulasintableauxproofs.Transjunctionshavereachedindifferentscientificandartisticareassomedegreeofacceptanceandarewidelyusedasimportantmechanismofsubversivethinkingandmodeling.Alsothenumberofoccurrenceofthisterminliteratureisquiteimpressivethereisnotmuchscientificunderstandingtofind.

Transjunctionsarelogicalfunctionsoroperatorswhichareinvolvedinsomesortsofbifurcationsandaresplitintodifferentpartsbelongingatoncetodifferentlogicalsystems.Theyarethereforecomposedofpartialfunctionsincontrasttothetotalfunctionsofclassicallogicaljunctionslikeconjunction,disjunction,implicationandsoon.

Thischangeoflogicalsystembybifurcationwhichpresupposethedifferenceofaninsideandanoutsideofalogicalsystemisruledbytheproemialrelationbetweenthepartsofthetransjunctionandthedifferentlogicalsystemsinvolved.Tothestepwisedecompositionofatransjunctionalformulacorrespondsanorderrelation,tothebifucationtoothersystemstheexchangerelationbecauseofitsinside/outsidedifference,andtothecomponentsandthestepsofdecompositionofthetransjunctionalformulaasawholetherelationofcoincidence.Therefore,theoperationoftransjunctioncanbeunderstoodasaproemialobject.

Thisdiagramwhichgivessomefirststepsinthedesignofpolycontexturalarchitectonicscannowbeusedforfurtherexplicationsofthemechanismofproemiality.

Theexchangebetweenoperatorandoperandhastobedescribedsimultaneouslyfrombothpositions.Thatiswhywehavetorealizeadoubledescription,adoublegestureofinscribingtheproemialityoftheconstellation.Tovisualizethisprozedurewehavetorealizeadoubledescriptionofthediagram



Thefirstdiagramsarecorrectinsofarastheydescribethestructureofproemiality.Butatthesametimetheyareabbreviationsinsofarastheprocessofreadingthem,thatistoreadthematoncefrombothsides,isnotinscribed.Thisprocessofreadinghastobedonebyareader.Butwehavetomakeitexplicitandtovisualizeit.Therefore,evenifitseemstobeobvious,ithastoberealizedandnotonlybementioned.Thenewdiagramisfocussingmoretheprocessofproemialitythanonitsgeneralstructure.TonottooverloadtheschemeIreducedittothedistributionoftheIF/THENrelation.Maybewithallthatinmindwearenowreachingslowlythefamousproemialcube.

Diagramm3

Theproemialcube

Again,thegreendoublearrowrepresentstheexchangerelation,theredlinethecoincidencerelation,theblackarrowtheorderrelation,and,new,thebluelinerepresentsthedistributionofthetwoproemialrelationsinacommonarchitecture.

Idontcommentthefullcombinatoricsbetweenallknotsofthediagram.Also,Iwouldliketoleavethestudyoffurtherdimensionsofvisualizationsandtheirexplanationsasaninterestingjobtotheprogrammers.InthistextDERRIDASMACHINESIwillreducemypresentationtothegraphicallymoresimplecaseofthevisualizationofthestructureoftheconceptofproemialityanditsapplications,thatis,tothetwodimensionaldiamonddiagraminsteadofthecube.

1.3ProemialityandHeterarchyinaUMLFramework

TogiveamoretransparentmodelingoftheproemialrelationshipitmaybehelpfultosetthewholeconstructionandwordingintoanUMLdiagramandtousethemodelingofheterarchyworkedoutbyEdwardLeeasahelpfultooltoexplicateproemialityintermsofUMLmodeling.

AlsotheproemialrelationshipisnotrestrictedtoontologyandthedistributionofhierarchicalontologiesinaheterarchicframeworkanddespitethefactthatUMLhasnomechanismsofcategorychange,metamorphosisandmediationitseemstobeahelpfulexercisetofindacorrespondencebetweentheUMLheterarchydiagramandthe



constructionofproemialitywhichismorebasedonelementarytermsofrelationality.Theheterarchydiagramisaclassdiagramwhichmodelsthestaticstructureofthesystem.Proemialityhas,alsoitisfundamentallydynamic,itsstaticaspects.ItisthisstaticaspectwecanmodelwiththehelpoftheUMLheterarchydiagram.

AfurtherstepofUMLmodelingofproemialitywillhavetoinvolvemoredynamicmodelslikeinteractionandactivitydiagrams.

Diagramm4

UMLdiagramofheterarchyin:EdwardA.Lee,OrthogonalizingtheIssues,UCBerkeley

WhatisthedifferenceinmodellingbetweenconceptualgraphsandUMLdiagrams?

Aconceptualgraphof

theUMLheterarchy

diagram.

1.4Complementarityofdisseminationandtogetherness

Complementarytothenotionandprocedureofdissemination,whichismotivatedbythenecessityofconstructingcomplexandpolycontexturalsystemsoutofsimpleones,thatis,monocontextural



systems,wehavetoconsiderthepolycontexturalityofthecomplexsystemassuch.Onefirstcategoryweobserveisthecategoryoftogethernessofthelocalsystemsinthecomplexandinteractingwholeness.

Anothercategorythatemergesnaturallyoutofthedisseminatedsystemsisthecategoryofwholenessormoreprecisethecategoryofsuperadditivityofdisseminatedsystems.

Inthissense,disseminationisaprocessofdisseminatingsinglesystemsandatthesametimeitisthewholeness,thetogethernessofthedisseminatedsystems.Thisisalsoincludedinthenotionofdisseminationasaprocessofdistributionandmediationofsystems.Disseminationisalwaysboth:multitudeandwholeness.

2Combinatoricsofchiasticchangesofcategories

2.1ConservativemappingsorCategorytheoreticcombinations

Ifthecontexturaldifferencesbetweentwoobjectsaredeniedwecanmodeltherelationshipbetweenthemintermsofmorphismsinthecategorytheoreticsense.Thesemorphismsarethestructurepreservingmappingsofnamestonames,sortstosorts,operationstooperations,equalitiestoequalities,andunitytounity.,etc.oftheabstractobjects.Butagain,inthiscaseweareneglectingthefact,thattheybelongtodifferentlogicalcontextures.

Ontheotherhand,ifwetaketheircontexturaldifferencesintoconsideration,thesemappingsarepreservingthetectonicalstructureofthesystems,despitetheirlogicalincompatibility.Intermsofproemialitythesemappingsarenotofthesortoforderrelations,likemorphisms,butofthesortofcoincindencerelation.Inacategorytheoreticalmodeltheywouldbesomeidentitymorphismsorisomorphisms.

2.2MetamorphosisorProemialcombinationsinabstractobjects

1Chiasmofsortsandnames:CHI(sorts,names)

Thisissimilartothechiasmofsortsandtheuniverse(ofsorts)inamanysortedlogic.

Itseemsnottobeunnaturalthatasortcanchangeintoanameofanewobjectandontheothersideanameasbeinghierarchicallysuperiortothesortscanchangeintoalowerlevelobjectasasortinanothercontexture.



Butthisseemstobeanordinaryprocedureforinteractingsystems.Theconceptualizingprocessofdifferentagentscandifferexactlyinthesensethatforoneagentthesetofsortsorofoneofthesortsoftheotheragentcorrespondstothename,thatis,thewholeorcontextureofhisownsystem.Incontrast,whatisthewholescopeofoneagentcanbeasortwithmanyothersortsforanotheragent.Thereisnothingmagicwiththat.Andthereisalsonoreasonforunsolvableconflictsifbothareawareaboutthissituationandunderstandthemechanismofchangebetweeneachother.Thiscommonunderstandingcanbemodelledorrealizedinafurthersystem,withoutbeingforcedtonegatethedifferencesbetweenthetwoagents.

Sortsandnamesoccursondifferentlevelsoftheconceptualhierarchy.Themechanismisgeneralizationandreductionorspecializationofconcepts.

2Chiasmofsortsandoperations:CHI(sorts,opns)

3Chiasmofoperationsandequations:CHI(opns,eqns)

4Chiasmofnamesandoperations:CHI(names,opns)

5Chiasmofnamesandequations:CHI(names,eqns)

6Chiasmofunicityandnames:CHI(unicity,names)

Unicitycanbeunderstoodasthecontextureofthelocalabstractalgebra.Classicaltheorieshavenottobeconcernedwiththeircontextureanduniquenessbecausetheyareuniqueperse,thatistheyaremonocontextural.Becauseoftheiruniquenessthereisnoreasontonotifyitbyaspecialtermlike1.

Becausetheunicity(unity)isabsolute,everypossiblechangeofithasfundamentalconsequencesforthewholeframeworkofreasoning.Thechiasmbetweentheabsoluteunicity(uniqueness)andtherelativityofthenamesdeniesasimplemappingofthelociofthedifferentsystemsontothelinearityofnaturalnumbers.Thechiasmbetweenunicityandtheotherhasnobeginningandnoend.

Thechiasmisthemechanismofchange.Toconnectthedifferentunitizeswithnumberswehavetoabandontheideaofaninitialobject,astartingpointofthenumberseries.Naturalnumbers,asweunderstandthem,areconstructedbyalgebras,inductionandinitiality.Asafirststep,wecantrytomodelthechiasticsituationinthecontextofcoalgebras,coinductivityandfinality.Thischiasticwayofthinkingisclosertothemetaphorsofstreamsandflows,andthelackofultimatebeginningsandendingsasoriginsandtelos.

Moreprecisely,weshouldthinkofthechiasticparadigmasaninterlockingplayofalgebraicandcoalgebraicstrategiesandmethods.

Withthisinmind,allattemptstoformalizepolycontexturalsystems,logicsandarithmetics,withthemethodsofcategorytheoryalonehave



toberelativized.Itisneverthelessofgreatimportancetostarttheprocessofformalizationofpolycontexturalitywiththemethodswhichareaccessible.Oneverystrongmethod,whichiswellaccessible,isthemethodoffiberingorindexing(Pfalzgraf,Gabbay).Inotherterms,themethodofmappinglocalsystemstoanindexsetasavehicleofdistributionofformalsystems.Butthisprocedureinvolvesthewholeapparatusofthealgebraicparadigm:equality,identity,linearity,initiality,inductivity,etc.Which,asItriedtomakeclear,isinstrongconflicttotheveryideaofproemialityanditschiasticmechanisms.

Thechiasmbetweennamesandcontextures(unicity)isofgreatimportanceforaseriousmodelingofreflectionalcomputationbecauseitopensupthepossibilityofadistributedselfreferentialitybetweensystemsaswholes.Furthermore,namesinacontexturecanbeinterpretedasthereflectionalmappingofothercontexturesintothereflectingcontexture.

2.3Chiasms,metamorphosisandsuperoperators

ThesuperoperationCHIcanbeinterpretedastheoperatorofchangesofcategoricalperspectives,contextsorcontexturesandpointsofview.

Thesepossibilitiesofchangingthecategoricaltermsisexactlywhatmakesthedifferencebetweenchiasmsandcategorytheoreticmorphismswhicharepreservingtheconceptualstructuresofthesystemintheprocessofmappingitintoanothersystem.

Proemialityincorporatesboth,categorytheoreticalandchiasticmorphisms.

ChiasticmorphismsarenotconservativeinthesensethattheyarepreservingthetectonicalorconceptualstructureofasystembutmoresubversiveinthesensequiteanalogtothecatastrophesinThomsCatastropheTheorythattheyarechangingandnotpreservingtheconceptualorder.Thesemorphismsareinastrictsensenotonlyforgetfulmappingsbutrulesofmetamorphosis.

###ChaoticLogicsChaoticlogicsarenotthelogicsofchaosbutthelogicsofchange.Changeinchaoticsystemsisnotacontinuosprocessbutthe



switchfromonemodetoanothermodeofasystembysomechangesofthestatesofthesystem.Chaoticlogicsarethelogicsofinteractinglogicalsystems.Changesinchaoticlogicsaremodeledbytranscontexturaljumpsfromonesystemtoanothersystemandaredefinedinsharpcontrasttotheintracontexturalstepsoftheexpansionruleinasingularsystem.Transjunctionaljumpsdon'texcludethepossibilitytostayintheprimarysystematthesametimeofthejump.CyberneticOntologyOrderfromNoise.####

Asaconsequenceofthesefirstinsights,inthischiasticpartoftheproemialrelationship,thecategorytheoreticlawsofidentityandassociativityarelost,oratleastfundamentallytransformed.

Thepossibilityofmetamorphosisisgivenbytheinterlockingmechanismofthechiasm.Alsothesuperoperatorshadbeenintroducedprimarilytodealwithcontexturesassuchthereisnoreasontonottoapplytheseoperatorstotheinternalstructureofthecontexture,thatishere,totheinternalstructureoftheabstractobjects.Thereforethegeneraloperatorofmetamorphosisiscomposed,atfirst,bythesuperoperators{ID,PERM,RED,BIF).

Thisallows,thattheremaybeanidentityrelationIDbetweentocontexturesandchangesintheirinternalstructurewithe.g.sort1incontexture1becomessort2incontexture2producedbythesuperoperatorPERM.Or,thecontexturesandthesortsarestable,buttheinternaloperationsofthecontexturesmaychange.

Itisnotexcludedinthischiasticconceptofarchitecturesofdifferentsystems,thatforonesystemallthedifferencesoftheothersystemboilsdowntoonenotion.Thiswouldbeafurtherstepinmappingthearchitectureofonesystemintoanothersystem.MaybethattheinterlockingmechanismbetweenthesystemswouldbereducedtoastrongreductionproducedbytheextensiveapplicationofthesuperoperatorREDtoallcategoriesofthesysteminconsideration.

Fromthepointofviewofproemiality,metamorphosisisnotasimpleconfusionofthecategoricalframeworkbutawellruledoratleastruleguidedchangeofcategoriesintheprocessofchange,emanationandevolutionorothertypesoftransformations.Thistypeofmetamorphosisisnotwildinthesenseoftheabsolutenovum,becauseitsscenarioisfoundedontheknowncategories(names,sorts,operations,etc.)ofthesystemsintransformation.Ifwewouldchooseanothersettinginsteadofalgebras,wewouldhaveasimilarscenarioofchangewithintheframeworkofthedefiningconcepts.Anothertypeofchangecouldbethoughtforthecasewherethetransformationchangestocategoriesunknownbefore.Forthiscasewewouldbeforcedtoadtoourframeworkofproemialchangebetweencategoriessomethinglikeanemptyboxfortheunknown.



Whynot?

Again,theprocessoftransformationruledbytheproemialrelationshiphasnottohappenonlybetweenobjectsofthesamearchitecture,likealgebrastoalgebras.Italsocanhappenbetweenobjectsofdifferentarchitectures.Aninterestingcasecouldbethechangebetweenalgebrasandcoalgebras.Thesamesituationistoobservebetweendistributedcategorysystems.Morphisminonesystemcanchangetoobjectsinanothersystemofcategories.Oreventheveryconceptofcategoryofonesystemcanbetransformedinamereobjectofanothersystem.Andsoon.

Usualmathematicalpractice?

Computerscientistshavefarmoreflexibleviewofformalismandsematicsthantraditionallogicians.Whatisregardedasasemanticdomainatonemomentmaylaterberegardedasaformalisminneedofsemantics."

M.P.Fourman,TheoriesasCategories,in:CategoryTheoryandComputerpogramming,SpringerLNCS240,p.435,1986

Idon'tsaythatthisisnotthewaymathematiciansareanywayworking.Butitseemstobeobviousthattheyarenotreflectingorevenformalizingthisprocess,thisuseoftermsandmethods,thatistheiractualpracticeofdoingcreativelymathematics.Withoutevermentioningwhatthismeansandhowitisformalized,theas".

Maybecomputerscientisthaveamoreflexibleuseofformalismsthanlogicians.Butlogicianshavenotonlyproducedmostoftheseformalismslongbeforebutalsoknowverywellthattheyaredealingwithhighlyidealizedsituationsgovernedbytheprincipleofidentity.

Ontheotherside,philosophersandphilosophicallogicianshavedevelopedmuchworkinexplainingtheascategoryofthinkingandbeing(analogy).Butwhatiscalled,especiallyinEuropeanphilosophy,hermeneutics,deniesanypossibilityofformalizationoftheascategory.Wealsoshouldn'tconfusetheascategorywiththemorepopularasifcategoryoffictionalism(HansVaihinger)andconstructivism.

Itwouldbeveryinterestingtostartsomecasestudiesofthispracticeofcomputerscientistsandmathematicians.Averyinterestingcasewouldbethewayorworkingwithswingingtypes,thatistheswitchfromalgebrastocoalgebrasandback,inthesenseofPeterPadawitz.

Ormoretraditional:InthesummertermyougetLogicsasalgebras,inthewintertermtheyofferyouAlgebrasaslogics.Andinbetweenyouenjoythesummerholidaystoforgetanypossibleconflicts.

Translations,GoguensSemioticAlgebras

Itturnsoutthatcorrecttranslationsareconservativemetamorphosis.



Maybethemainproblemofmachinetranslationisjustthisdecision,tostartwithconservativetranslationsandtotrytomodelcommonsensetexts,whicharefullofgamesofviolatingthisconservativity,withthisrestrictedapproach.Inotherword,conservativetranslationsarebasedondisambiguatedandcontextfreesemantics.Acasewhichisveryartificialanddoesn'tmatchnaturallanguageatall.

Aconservativeexample:conflictsinthetreeofdataobjects

Allprogramminglanguagesarebasedonverystrictandstableconceptualstructures.Ifthedataobjectsareintroducedasanorderedsystemlikethetreeofdataobjects",thisstructurewillneverbechangedintheprocessorexecutionofaprogram(Programmablauf).Ifsomethingwouldbechangedinthisorderitwouldautomaticallyproduceseriousconflicts.

Becauseofthefact,thatclassicalprogramsareessentiallymonologic,thereisnospaceforconflictsinapositivesense.Butrealsystems,thatisinteractingsystemsastodaycomputing,arepermanentlyconfrontedwithconflicts.Whynotintroducingdialogsintheverystructureofprogramminglanguagesandsystems?I'mnotwritinghereaboutspecialinteractiveprograms,e.g.,butonthearchitectureandfundamentalconceptualityordefinitionofprogramminglanguagesassuchandnotofspecialapplicationsoftheselanguages.Likeinteractiveproofsystemsorinteractivegames.

Thereisaneasywayofproducingconflictsinadialogicalsystem,ife.g.L1declaresAasasimpleobjectandL2declaressimultaneouslyAasacomplexobject,thatisastructure.Obviouslyitispossible,inthepolycontexturalapproach,tomodelthisconflictandtoresolveitinanotherlogicalsystem,sayL3,thiswithoutproducingametasystemsubordinatingL1andL2.

Diagramm5

Treeofdataobjects

Furthermore,theconflicthasaclearstructure,itisametamorphosisofthetermssimpleobject"inL1andstructure"inL2.Thismetamorphosisisasimplepermutationbetweensortsovertwodifferentcontexturesbasedonthechiasticstructureofthemediationofthesystems.Butitrespectsthesimultaneouscorrectnessofbothpointsofviewinrespectofbeingasimpleobject"andbeinga



structure".Inthissenseitcanbecalledasymmetricalmetamorphosis.

Todaycomputingisoftencharacterizedbyitsinteractivity.Buttheprogramminglanguageshavenotchangedtorespondtothissituation.Theyarestill,inprinciple,monologic.

Afurtherexampleofaninterchangebetweenprogramminglanguageswouldbethechiasmbetweendataobjectsandcontrolstructures.

Averyshyimplementationofthisinterlockingmechanism,withfarreachingconsequences,isatthebasisofallartificialintelligenceattempts,theinternaldifferenceandpossibleambiguityinLISPbetweendataandprogramsruledbytheQUOTE/EVALfunction.

Theseexamplesshouldnotbeconfusedwithcontradictionsarisingbyaconflictinattributesbetweendifferentinformations.Thisimpliesalogicalandlinguisticlevelofcommunicationanddoesn'ttouchthecategoricalframeworkofinteraction.

AfterWegner,interactionsareparaconsistent,oratleastbelongtoaparaconsitenttypeoflogic.Thismaybetrueonalinguisticlogicallevel,butitisnotincorrespondencewithamoreachitectonicandchiasticviewofinteractivity.

blindspots

Strategiesofdetectingtheontological,logical,computational,epistemological,reflectional,andwhatever,blindspotofaninteractingagent.

2.4Asimpletypologyofchiasms

Tostudysomeaspectsofchiasmswecanrestrictourselftothestudyoftheinterplaybetweenrelatorsandrelata,neglectingthefullfledgedexpositionofthechiasmwithitsconceptrelationandunicity(uniqueness).

Inpracticeitiseasytodiscoverthatmanyvariantsofrealizationsofchiasmareintheepistemologicalplay.Mostly,chiasmarenotfullydesigned,reductionsareusedandsometimestheuseisoverdeterminated.

Wecanclassifythesinglechiasmsasbalanced,underandoverbalanced.Asdistributedandembeddedchiasmswecandistinguishtwomodiofdistribution,iterationandaccretionanditscombinations.

2.4.1Iterationsofchiasms

2.4.2Accretionsofchiasm

2.4.3Mediationofiterationandaccretionofchiasms



2.4.4Overdeterminationofchiasms

2.4.5Examplesofunderbalancedchiasms

Diagramm6

Examplesofchiasms



3Proemialitybetweenstructuralandprocessualunderstanding

3.1FormalLogic,TotalityandTheSuperadditivePrinciple

in:GotthardGntherBeitrgezurGrundlegungeineroperationsfhigenDialektik,Band1,MeinerVerlag,Hamburg,1976,p.329351,firstpubl.:BCLReport,1966

Wehavegiventhemainreasonabove:iftherelationbetweenthoughtanditsobjectisbasicallyunderstoodasasymmetricexchangerelationthephenomenonofsubjectivitydisappears.Buta"totality"inwhicheverythingisreducedtoobjectivitycanneverbetotalbecausesomethingismissing.Atotalityis,inHegelsterminology:1)aniteratedselfreflectionof2)anoniteratedselfreflection,and3)aheteroreflection.

Ifwepermit,forthedescriptionofthisstructure,onlylogicaloperationswhichleadtoreflectionsymmetrythen1)iseliminated,and2)and3)turnouttobeindistinguishableandlogicallyidentical...because1)isnothingelsebutthecapacityofkeeping2)and3)apart.

(...)

However,iftheconceptoftheuniversalsubject,i.e.ofBewussteinberhaupt(Kant),iseliminatedthelogicalconstrainttoreduceeverythingtoultimateparityrelationsdisappears.WewillstillhavereflectionsymmetrybetweenSSandSObutnotlongerbetweenSandOingeneral.Inotherwords:itwillturnoutthatthefoundingrelationbetweensubjectandobjectorbetweenThoughtandBeingisnotasymmetricalexchangerelationbutsomethingelse.ThisisthepointwherethetransitionismadefromformalclassiclogicofAristoteliantypetoatheoryoftransclassic,nonAristotelianRationality.WebeginbyredrawingFigure1omittingSUandhavingthephalanxoftheSOreplacedbyasingleSwiththeindexO.WeindicatetherelationsbetweenSS,SOandObyarrowsoffourdifferentshapes.Accordingtothelogicalcharacteroftherelationanarrowwilleitherbedoublepointedoritwillhaveoneshaftorbedoubleshaftedhavingeithercontinuousordottedlines.Figure5willthenshowthefollowingconfiguration:



IfSSdesignatesathinkingsubjectandOitsobjectingeneral(i.e.theUniverse)therelationbetweenSSandOisundoubtedlyanorderedonebecauseOmustbeconsideredthecontentofthereflectiveprocessofSS.Ontheotherhand,seenfromtheviewpointofSSanyothersubject(theThou)isanobservedsubjectanditisobservedashavingitsplaceintheUniverse.ButifSSis(partof)thecontentoftheUniverseweobtainagainanorderedrelation,thistimebetweenOandSO.ThereremainsthedirectrelationbetweenSSandSO.Thisisobviouslyofadifferenttype.SOisnotonlythepassiveobjectofthereflectiveprocessofSS.Itisinitsturnitselfanactivesubjectwhichmayviewthefirstsubject(andeverythingelse)fromitsvantagepoint.InotherwordsSOmayassumetheroleofSSthusrelegatingtheoriginalsubject,theSelf,tothepositionoftheThou.AndthereisneitheronearthnorinheaventheslightestindicationthatweshouldpreferonesubjectivevantagepointforviewingtheUniversetoanother.Inshort,therelationbetweenSSandSOisnotanorderedrelation.Itisacompletelysymmetricalexchangerelation,like"left"and"right".Anorderedrelationbetweendifferentcentersofsubjectivereflectioncomesintoplayonlyifwereintroducetheconceptofauniversalsubjectwhichcontainsallhuman"souls"ascomputingsubcenters.Ofthetworelationswehavesofarconsidered,theexchangerelationissymmetricalandtheorderedrelationrepresentsnonsymmetry.

ThisinvestigationintendsonlytoshowthattheconceptofTotalityorGanzheitiscloselylinkedtotheproblemofsubjectivityandtransclassiclogicandthatitisbasedonthreebasicstructuralrelations:

anexchangerelationbetweenlogicalpositionsanorderedrelationbetweenlogicalpositionsafoundingrelationwhichholdsbetweenthememberofarelationandarelationitself.

Itmaybesaidthatthehierarchyoflogicalthemesasindicatedintable(II)representsanhierarchyofimplicationalpower.Allthemeshaveincommonthattheyareselfimplicationstheyimplythemselves.Howeverthefirsttheme(objectiveexistence)impliesonlyitselfandnothingelse.Inthisrespectit



differsfromanysucceedingthemewhichimpliesitselfaswellasallsubordinatedthemes.Forthisreasonitispropertocalltheinitialtheme"irreflexive"andallthefollowing"reflexive".Irreflexivitymeansthatsomethingwethinkofisonlyanimplicatebutnotanimplicandforsomethingelse.Ontheotherhandifwereferlogicallytoreflexivitywemeanthatour(pseudo)objectofthoughtisanimplicandrelativetoalowerorderandaswellanimplicaterelativetoathemethatfollowsitinthehierarchyoftable(II).

Wearenowabletoestablishthefundamentallawthatgovernstheconnectionsbetweenexchange,orderedandfoundingrelation.Wediscoverfirstinclassictwovaluedlogicthataffirmationandnegationformanorderedrelation.Thepositivevalueimpliesitselfandonlyitself.Thenegativevalueimpliesitselfandthepositive.Inotherwords:affirmationisneveranythingbutimplicateandnegationisalwaysimplication.Thisiswhywespeakhereofanorderedrelationbetweentheimplicateandtheimplicand.Thenameofthisrelationinclassictwovaluedlogicisinference.

Itisnownecessarytorememberthatthepossibilityofcoexistenceoftwoindependentsubjects(IandThou)intheUniverseisbasedonanexchangerelationbetweenequipollentcentersofreflection.Moreover,thesesubjectsareallcapableofbeingimplicands.Moreobjectsdonotoperateinferentially.Thatmeanstheydonotimplyanythingelse.

IfwenowconsiderthefoundingrelationinwhichasubjectconstitutesitselfasdiametricallyposedrelativetoallobjectsandthetotalobjectiveconceptoftheUniversewewilldiscoverthatthisrelationrepresentsaninterestingsynthesisofanexchangerelationbetweenlogicalpositionsanorderedrelationbetweenlogicalpositionsafoundingrelationwhichholdsbetweenthememberofarelationandarelationitself.10exchangeandorder.Thefoundingrelationisinitselfanexchangerelationinsofarasthelinkingsubject(SS)mayassumethelogicalpositionoftheothersubjectwhichisthoughtof(SO).SOmayinitsturnassumetherankofSS.Anytwocentersofsubjectivereflectionofthesameordermutuallyimplyeachother.ButsuchanexchangedoesnotoperatebetweenSandO.Aswepointedoutbefore:thebonafideobjectcannotinferthesubjectandbydoingsousurptheroleofasubject.Ifitcoulditwouldimplythatsubjectsareirreflexiveentitieswhichforasubjectisacontradictioinadjecto.Itfollowsthattherelationbetweenimplicateandimplicandhastwodifferentaspects:betweentwosubjectsthisrelationassumestheroleofasymmetricalexchange.Betweensubjectandobjectitappearshoweverasanorderedrelation.Thefoundingrelationisthereforealsoanorderedrelation.Ortoputitdifferently:thefoundingrelationisacombinationof



exchangeandorder.Whatistheimplicand(SS)maybecometheimplicatenotrelativetoObuttoourimpartialobserverSSS.Wemightsaythatthefoundingrelationisaconcatenationofsequencesofexchangeandsequencesoforderedrelations.

ThediagramofFig._6willillustratewhatwemean:

Fig._6indicatesasequenceofsinglepointedandasecondsequenceofdoublepointedarrowssuchthatasinglepointedarrowalwaysalternateswithadoublepointedone.Aconcreteexampleofwhatthefigureillustratesisthefathersonrelation.Thisisfirstanorderedrelation.Butthesoncanalsobecomeafather.Inthissensefathersonisalsoanexchangerelation.Butthesondoesnotacquirethestatusoffatherrelativetohisownfatherbutrelativetothegrandsonofhisfather.Inabstractterms:whatismember(orargument)oftheorderedrelationOSS,namelySS,maybecomeanargumentofanexchangerelationnotrelativetoObutrelativetoSSSwhichimpliesthisexchangeSSSO.

Thuswemaysay:thefoundingrelationisanexchangerelationbasedonanorderedrelation.Butsincetheexchangerelationscanestablishthemselvesonlybetweenorderedrelationswemightalsosay:thefoundingrelationisanorderedrelationbasedonthesuccessionofexchangerelations.Whenwestatedthatthefoundingrelationestablishessubjectivitywereferredtothefactthataselfreflectingsystemmustalwaysbe:selfreflectionof(selfandheteroreflection).AsHegelpointedoutinhisdialecticlogiconeandahalfcenturiesago,theoppositionofheteroandselfreflectionisnotaparityrelationbecauseitrequiresaniterationofselfreflectionincontrasttothenoniterativecharacterofheteroreflection.Itfollowsaswaspointedoutabove,thatonevalueissufficienttodesignateinheteroreflectionbuttwovaluesarerequiredapartfromthevalueSSSOOSSSSOSSSFig_6

forobjectdesignationtoseparateselfreflectionfromthe



object.Thisisconfirmedbythecharacterofthefoundingrelation.Table(VI)clearlyshowsthatitrequiresaminimumofthreevaluesforitsownestablishment.Buttheintroductionofathirdvaluegeneratesanewprincipleofsuperadditivity.

3.2Irreflexivityastheultimatebeginning

Incontrasttomyworkinghypothesis"Thereisnoorigin,onlyamultitudeofbeginnings"irreflexivityinGunthersapproachtothefoundingrelationhasthevalueofanultimatebeginning,whichistheorigininitsunicity.Thisoriginischaracterizedasaselfimplication.

Itmaybesaidthatthehierarchyoflogicalthemesasindicatedintable(II)representsanhierarchyofimplicationalpower.Allthemeshaveincommonthattheyareselfimplicationstheyimplythemselves.Howeverthefirsttheme(objectiveexistence)impliesonlyitselfandnothingelse.Inthisrespectitdiffersfromanysucceedingthemewhichimpliesitselfaswellasallsubordinatedthemes.Forthisreasonitispropertocalltheinitialtheme"irreflexive"andallthefollowing"reflexive".Irreflexivitymeansthatsomethingwethinkofisonlyanimplicatebutnotanimplicandforsomethingelse.

Tostartproemiality(foundingrelation)withabeginninginthesenseofanoriginisnotincludedinthegeneraldefinitionofthefoundingrelation.Itisanadditionaldecision,basedonspecialontologicalinterests.

Neithertheabstractformulationnortheexamplegiven,fathersonrelationship,involvesanultimativebeginning.Otherwisethefathersonrelationshipconnotedwithanoriginwouldforceustoaccepta"Urfather".MaybeGod.Butthisisnotphilosophicalthinking.

Tointerpretproemialityashavingabeginningisguidedbytheprincipleofwellfoundedness.Thisprincipleisnecessaryforanalgebraicorconstructivistapproach.Incontrasttothisinterpretationofthefoundingrelationitisequallypossibletounderstandthismechanisminanonfoundedwayofcoalgebraiccoinduction.

AsanexamplewemaythinkofachainofalternatingXsandYswithaoutanoriginnoranend:

...XYXYXYXYX...

IsitreasonabletotakeXoralternativelyYasthestartelementofthechain?Obviouslynot.Itmaybe,insomespecialsituations,areasonabledecisiontotakeYasthestart.

Wemightsaythatthefoundingrelationisaconcatenationofsequencesofexchangeandsequencesoforderedrelations.

Thesameistruefortheconcatenationchainoforderandexchange



ThinkArtLabhttp://www.ThinkArtLab.com

[email protected]

relations.Butthisdecisionisarbitrarilyandnotpartofthemechanismofthefoundingrelation.

Tomakethesetwointerpretationsmoreclear,IintroducedinmyMaterialien197375thedistinctionbetweenopenandclosedproemialrelationship.

Evenifweacceptthattheenvironmentofalivingsystemhasincontrasttoitsmodelingofitanirreflexivecharacterforthemodelingsystem,itisimportanttoseethatthisirreflexivityisofrelativenature.Otherwiseitwillbeverydifficultforacognitivesystemtohavedifferentinterpretationsofitsenvironmentandtochangeitsontology.

Manyconstructivistshaveintroducedthedistinctionbetweenrealityandobjectivity(Maturana)todealwiththisdifficulty.Intheirapproachirreflexivityispurereality,whichassuchescapesanyknowledge.Ontheothersideobjectivityisaresultoftheprocessofinterpretation.ButsinceKantweshouldknowthatthistrickisnotproperlyworking.

Documents

Dissemination_ Introducing the Proemial Relationship