COMPLEMENTARITYAND
PARACONSISTENCY
NewtonC.A.daCosta D´ecioKrause
DepartmentofPhilosophy
FederalUniversityofSantaCatarina
ncacosta@terra.com.br;dkrause@cfh.ufsc.br
Abstract
Bohr’sPrincipleofComplementarityiscontroversialandtherehasbeen
muchdisputeoveritsprecisemeaning.Here,withouttryingtoprovidea
detailedexegesisofBohr’sideas,wetakeaveryplausibleinterpretation
ofwhatmaybeunderstoodbyatheorywhichencompassescomplemen-
tarityinadefinitesense,whichweterm
C
-theories.Theunderlyinglogic
ofsuchtheoriesisakindoflogicwhichhasbeentermed‘paraclassical’,
obtainedfromclassicallogicbyasuitablemodificationofthenotionof
deduction.Roughlyspeaking,
C
-theoriesarenon-trivialtheorieswhich
mayhave‘physically’incompatibletheorems(and,inparticular,contra-
dictorytheorems).So,theirunderlyinglogicisakindofparaconsistent
logic.
Keywords:Complementarity,Paraconsistency,ParaclassicalLogic.
1Introduction
“Cecimeten´evidencel’apparenceirrationnelle
delacompl´ementarit´equineserationaliseque
pardessch`emeslogiquesnouveaux.”
P.F`evrier(1951)
Theconceptof‘complementarity’wasintroducedinquantummechanicsby
NielsBohrinhisfamous‘ComoLecture’,in1927(Bohr1927).Theconse-
quencesofhisideaswerefundamentalforthedevelopmentoftheCopenhagen
interpretationofquantummechanicsandconstitutes,asislargelyrecognizedin
theliterature,asoneofthemostfundamentalcontributionstothedevelopment
ofquantumtheory(seeBeller1992;Jammer1966,1974).
1
Notwithstandingtheirimportance,Bohr’sideasoncomplementarityarecon-
troversial.Inreality,itseemsthatthereisnogeneralagreementontheprecise
meaningofhis
PrincipleofComplementarity
(seeforinstanceBeller1992,p.
148);Bohr’sownwords,byposingthat“Ithinkthatitwouldbereasonableto
saythatnomanwhoiscalledaphilosopherreallyunderstandswhatismeantby
complementarydescriptions”(quotedfromCushing1994,p.32),mightsuggest
thedi±cultiesinvolvedinanyattempttosearchfora‘rationale’forhisPrin-
ciple.Anyhow,thisremarkinvitesustolookalsoatthelogico-mathematical
grounds,mainlyinconnectionwiththeparaconsistentprogram(seedaCosta
andMarconi1987;daCostaandBueno2001).
So,althoughithasalsobeenclaimedthatBohrapparentlyunderstoodthe
PrincipleofComplementarityfromanepistemologicalpointofviewonly(cf.
Jammer1974,pp.70and89),wethinkthatitispertinenttoaskforthelogical
structureofatheorywhichencompassessuchaprincipleinitsbases.Then,
takingintoaccountthattheintuitiveideaofcomplementarityresemblesthat
ofcontradiction(seebelow),theunderlyinglogicalstructureofsuchatheory
shouldbemadeexplicit.
Asahistoricalremark,werecallthatsomeauthorslikeC.vonWeizs¨acker,
M.StraussandP.F`evrieralreadytriedtoelucidateBohr’sprinciplefroma
logicalpointofview(cf.F`evrier1951;Jammer1974,pp.376®;Strauss1973,
1975);JammermentionsBohr’snegativeanswertovonWeizs¨acker’sattempt
ofinterpretingBohr’sprincipleandobservesthatthisshouldbetakenasa
warningforanalyzingthesubject(ibid.p.90).HealsomentionsthatStrauss’
intentionwastodevelopalogicinwhichtwopropositions,say
®
and
¯
(which
shouldstandforcomplementarypropositions)maybebothacceptedastrue,
butnottheirconjunction
®^¯
(ibid.,p.335);R.CarnapsuggeststhatStrauss’
logicwere‘inadvisable’(Carnap1995,p.289).
Theintroductionofsomenon-classicallogicalsystemsdevelopedmorere-
centlymayenrichthediscussion,andthisiswhatwearedoingnow.Butlet
usfirstrecallthatapparently‘complementarydescriptions’aremoreconcerned
with‘exclusivedescriptions’thanwiththeimpossibilityof‘simultaneousmea-
surement’,asimplicitlysuggestedinsomestandardbookswhenthey‘define’
complementarity(seeforinstanceOmn`es1995).
Weshallproceedasfollows.WithoutdiscussingvonWeizs¨acker’sorStrauss’
works(onlyF´evrier’sideaswillbementionedinbriefbelowinordertomotivate
thepaper),weintroducetheconceptofatheorywhichadmitsa
Complemen-
tarityInterpretation
(touseJammer’swords–seebelow).Thenwesuggestthat
underaplausibleinterpretationofwhatistobeunderstoodbycomplementar-
ity,theunderlyinglogicofsuchatheoryisa
paraclassical
logic(firstproposed
indaCostaandVernengo1999).Belowweshallsketchthemainfeaturesof
thislogicasappliedtoourpurposes.
Enpassant
,letusmentionthatonethingistoprovideanexegesisofBohr’s
ideas;anotheristopayattentiontotheunderlyinglogicalstructureofatheory
whichencompassescomplementarityinsomesense.Inthispaper,althoughwe
regardthefirsttopicasveryimportant,wearefundamentallyconcernedwith
thesecond,evenifwedonotprovideallthetechnicaldetails,whichwillbepost-
2
ponedtofuturetechnicalworks.So,thispapercanberegardedasanadjunct
tothespeculationsonthissecondpoint.Concerningthefirstpoint,seeBeller
(1992)foradetailedattemptto‘decipher’Bohr’sprinciple“byuncoveringand
describingtheunderlyingnetworkofimplicitdialoguesintheComolecture”.
Finally,letussaythatourpapermightbealsoviewedasanattemptto
investigatealineofresearchwhichwasenvisaged,butnotdeveloped,byP.
F´evrier;inshort,sheattributedathirdvalue(
impossible
)totheconjunction
ofcomplementarypropositions(
propositionsincomposables
)sothatherlogic
resemblesÃLukasiewicz’threevaluedlogic(Jammer’sbookprovidesageneral
viewontheselogics;seeJammer1974,pp.341®).Notwithstanding,F´evrier
recognizedthatwecouldalsoconsiderthattheconjunctionofcomplementary
propositionscannotbeperformed:“laconjonction‘et’nepeutleurˆetreap-
pliqu´ee”(F`evrier1951,p.33),butshedidnotconsidersuchapossibilitydue
to“raisonsdetechniquemath´ematique”(ibid.).Inthispaperwearticulatea
possiblewaytosupersedethese‘di±culties’,motivatedbytheparaconsistent
program,whichatthattimehadnotyetbeendeveloped.Ourapproachrunsin
thedirectionofnotavoidingthattheconjunctionofcomplementarysentences
canbeperformedbut,roughlyspeaking,thatsuchaconjunctioncannotbe
derivedasatheoremofthetheory.
Inouropinion,Bohr’sviewprovidesthegroundsfordefiningaverygeneral
classoftheories,whichmayberegardedastheorieswhichincorporateaxioms
thatmayentailpropositionslike
°
and
:°
(thenegationof
°
),butsuchthat
thetheoryisnottrivialinthesensethatthisfactdoesnotimplythatallthe
formulasofitslanguagearetheorems,asweshallseebelow.Inotherwords,
thetheoriesweshallcharacterizebelowaresuchthatfrom
°
and
:°
wecannot
deduce
°^:°
,thatis,acontradiction.
Weshouldstillremarkthatthiskindofinvestigationhasnotonlyhistor-
icalreasons,asoneshouldinferfromthefactthatnowadaystheconceptof
complementarityseemstobenomorepopularamongphysicists.Really,the
investigationofthelogicalfoundationsofsciencehasavaluebyitself,andthe
resultingsystems(whentheyarise,asinthepresentcase),builtassometimes
motivatedbynotsoclearintuitions,notonlymayprovidethemasenseac-
cordingtoacceptablepaternsofrigour,buttheyalsomaybeusefulinother
situationsaswell,whichmayprovideotherinsightsandfurtherdevelopments.
Furthermore,ourworkshowsthatbytakingtheconceptofcomplementarityas
wehaveconsideredit(seethenextsection),thereisasenseinsayingthatthe
foundersofquantumtheory,inparticularBohr,maybereferredtoas’inconsis-
tent’,asprobablyareallthosewhoaredevelopingverycreativee®orts,butfor
suretheirfeelingswerenottrivialinthesensedefinedbelow.Maybewecould
say,takingtheduecare:theyareparaconsistent.
2Awayofunderstandingcomplementarity
Inordertoexplainthesenseaccordingtowhichweshallconsidertheterm
‘complementarity’inthispaper,letuslookathowthisconceptwasanalyzed
3
bysomeauthors.
Ofcourse,afewisolatedquotationscannotprovideevidencefortheun-
derstandingofconcepts,especiallyregardingthepresentcase,butperhapswe
couldreinforceourpointbyshowingthatcomplementaritystandsmorefor‘in-
compatibility’insomesense(the‘sense’beingexplainedinthenextsections)
thanforimpossibilityof‘simultaneouslymeasuring’,anexpressionwhichcould
resembletheuseofsomekindoftemporallogic.
Anyway,itshouldberemarkedthatwemayalsofindBohrspeakingabout
complementaryconceptswhichcannotbeused
atthesametime
(aswecansee
inseveralpapersinBohr1985),butthesesituationsaccordingtohimdemand
isolatedanalyses,andperhapsitisnotpossibletoprovideageneraldescription
whichallowsustodealwithallofthesecases:accordingtoBohr,“Onemustbe
verycareful,therefore,inanalyzingwhichconceptsactuallyunderlylimitations”
(ibid.,p.369).
Pauli,forinstance,hasclaimedthat,“[If]theuseofaclassicalconcept
excludesof
another
,wecallbothconcepts(
:::
)
complementary
(toeachother),
followingBohr”(Pauli1980,p.7,quotedinCushing1994,p.33).Cushinghas
alsostressedthat,“[W]hateverhistoricalroute,Bohrdidarriveatadoctrineof
mutuallyexclusive,incompatible,butnecessaryclassicalpicturesinwhichany
givenapplicationemphasizingoneclassofconcepts
must
excludetheother”
(ibid.,pp.34-5).
Thisideathatcomplementarypropositions‘exclude’eachother(incompat-
ibility)isreinforcedbyBohrhimselfinseveralpassages:
Theexistenceofdi®erentaspectsofthedescriptionofaphysical
system,seeminglyincompatiblebutbothneededforacompletede-
scriptionofthesystem. Inparticular,thewave-particleduality.
(quotedfromFrenchandKennedy1985,p.370)
Thephenomenonbywhich,intheatomicdomain,objectsexhibit
thepropertiesofbothparticleandwaves,whichinclassical,macro-
scopicphysicsaremutuallyexclusivecategories.(ibid.,pp.371-2)
Theverynatureofthequantumtheorythusforcesustoregard
thespace-timeco-ordinationandtheclaimofcausality,theunion
ofwhichcharacterizestheclassicaltheories,ascomplementarybut
exclusivefeaturesofthedescription,symbolizingtheidealizationof
observationanddefinitionrespectively.(Bohr1927,p.566)
SeveralotherpassagesfromBohrcouldbequotedfromScheibe’sbook
(1973),forinstance,thefollowing:
Theapparentlyincompatiblesortsofinformationaboutthebehavior
oftheobjectunderexaminationwhichwegetbydi®erentexperi-
mentalarrangementscanclearlynotbebroughtintoconnectionwith
eachotherintheusualway,butmay,asequallyessentialforanex-
haustiveaccountofallexperience,beregardedas‘complementary’
toeachother.(Bohr1937,p.291;Scheibe1973,p.31)
4
Scheibealsosaysthat
:::
whichisheresaidtobe‘complementary’,isalsosaidtobe‘appar-
entlyincompatible’,thereferencecanscarcelybetothoseclassical
concepts,quantitiesoraspectswhose
combination
waspreviouslyas-
sertedtobecharacteristicoftheclassicaltheories.For‘apparently
incompatible’surelymeansincompatibleonclassicalconsiderations
alone.(Scheibe1973,p.31)
Thefollowingquotationisalsorelevantforthepointwearetryingtostress
here:thecharacteristicof‘exclusion’ofcomplementarity.Bohrsays:
Informationregardingthebehaviourofanatomicobjectobtained
underdefiniteexperimentalconditionsmay,however,accordingto
aterminologyoftenusedinatomicphysics,beadequatelycharac-
terizedas
complementary
toanyinformationaboutthesameobject
obtainedbysomeotherexperimentalarrangementexcludingtheful-
fillmentofthefirstconditions.Althoughsuchkindsofinformation
cannotbecombinedintoasinglepicture
bymeansofordinarycon-
cepts,theyrepresentindeedequallyessentialaspectsofanyknowl-
edgeoftheobjectinquestionwhichcanbeobtainedinthisdomain.
(Bohr1938,p.26,quotedfromScheibe1973,p.31,seconditalic
ours).
Inotherwords,itseemsperfectlyreasonabletoregardcomplementaryas-
pectsas
incompatible
,inthesensethattheir
combination
intoasingledescrip-
tionmayleadtodi±culties.Inthissense,thequantumworldisratherdistinct
fromthe‘classical’world.
Itshouldberemarkedthatinthe‘classicalworld’,whichatfirstglancecan
bedescribedbyusingstandardlogicandmathematics,if
®
and
¯
areboth
thesesortheoremsofatheory(foundedonclassicallogic),then
®^¯
isalso
athesisofthattheory.Thisiswhatweintuitivelymeanwhenwesaythaton
thegroundsofclassicallogic,atruepropositioncannot‘exclude’anothertrue
proposition.
Inclassicallogic,iffromsomegroup¢
1
ofaxiomsofatheory
T
wededuce
°
,andiffromanothergroup¢
2
wededuce
:°
,then
°^:°
isalsodeductible
in
T
.
Normally,ourgroup¢ofaxiomsof
T
is
finite
,sothatwemaytalkofthe
conjunctionofitssentencesinsteadof¢itself.Then,if
®
and
¯
arerespectively
theconjunctionsassociatedto¢
1
and¢
2
,asabove,wearelookingforatheory
T
suchthatin
T
wemayhave
®`°
and
¯`:°
,butinwhich
°^:°
isnota
theoremof
T
.
Therefore,ourgoalistodescribeawaytoformallyavoidthat¢
1
[
¢
2
(or
®^¯
)entailsacontradiction,sincewedonotintendtoruleout‘complementary
situations’.Notwithstanding,weemphasizethatBohr’sideasarenotcompletely
clear,asthefollowingquotationshows:
5
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