Quantum physics, fuzzy sets and logic; steps towards a many valued interpretation of quantum mechanics

SpringerBriefsinPhysics
SeriesEditors
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JarosławPykacz

QuantumPhysics,FuzzySetsandLogic
StepsTowardsaMany-ValuedInterpretationofQuantum
Mechanics

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JarosławPykacz
InstituteofMathematics,UniversityofGdańsk,Gdańsk,Poland
ISSN2191-5423

e-ISSN2191-5431

ISBN978-3-319-19383-0

e-ISBN978-3-319-19384-7

DOI10.1007/978-3-319-19384-7
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2015940987
©TheAuthor(s)2015
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Contents
1Introduction
References
2ABriefSurveyofMainInterpretationsofQuantumMechanics
2.​1EnsembleInterpretation
2.​2CopenhagenInterpretation
2.​3Pilot-WaveInterpretation
2.​4Many-WorldsInterpretation
2.​5ConsistentHistoriesInterpretation
2.​6ModalInterpretations
2.​7RelationalQuantumMechanics
2.​8Other,LessPopularInterpretations
2.​9Summary
References
3ABriefSurveyofMany-ValuedLogics
References
4FuzzySetsandMany-ValuedLogics
4.​1RudimentsoftheFuzzySetTheory
4.​2FuzzySetsandInfinite-ValuedŁukasiewiczLogic
References
5Many-ValuedLogicsinQuantumMechanics
References
6Birkhoff-vonNeumannQuantumLogic
6.​1TheTraditionalAlgebraicModel
6.2Ma̧czyński’sFunctionalModel
6.​3TheGeneralFuzzySetModel

6.​4TwoPairsofBinaryOperations
References
7B-vNQuantumLogicas∞-ValuedŁukasiewiczLogic
7.​1TheNecessityofUsingMany-ValuedLogicforDescriptionofFutureNon-certain
Events
7.​2IsB-vNQuantumLogicTwo-Valued?​

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7.​3TheMany-ValuedModelofB-vNQuantumLogic
7.​4Application:​AnalysisofaTwo-SlitExperiment
References
8Perspectives
8.​1FuzzySetModelsofQuantumProbability
8.​2FuzzyPhaseSpaceRepresentationofQuantumMechanics
References
9TheMany-ValuedInterpretationofQuantumMechanics
References
Index

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©TheAuthor(s)2015
JarosławPykacz,QuantumPhysics,FuzzySetsandLogic,SpringerBriefsinPhysics,DOI10.1007/978-3-319-193847_1

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1.Introduction
JarosławPykacz1
(1) InstituteofMathematics,UniversityofGdańsk,Gdańsk,Poland


JarosławPykacz
Email:
AftervonWeizsäcker’sworknoseriousattemptseemsevertohavebeenmadeto
elaboratefurtheramany-valuedlogicalapproachtoquantummechanics
(MaxJammer,ThePhilosophyofQuantumMechanics,(1974),p.379)
Quantummechanicsisanextremelyefficienttheoryandup-to-nownoexperiment
designedtocheckityieldedresultsindicatingthatitcouldbewrong.Evenmore:present
technologicalprogressapproachesthepointatwhichitwillbepossibletogainfullcontrol
onsinglequanta,whichwouldmakemarvellouspredictionsoftwonewly-bornbranches
ofquantummechanics:quantuminformationandquantumcomputation[1]fully
realizable.However,thistheoreticalandtechnologicalprogressisnotaccompaniedbythe
progressinour“understanding”(whateveritcouldmean)ofquantumphenomena.
AccordingtoWebster’sThirdNewInternationalDictionary“interpretation”means
“explanationofwhatisnotimmediatelyplainorexplicit”.Indeed,quantummechanicsis
fullofconcepts,symbolsandobjectsthatarenotimmediatelyplainorexplicitsincethey
havenocounterpartsinoureverydaylife.Actually,howcouldweimagine,forexample,a
materialobjectbeingsimultaneouslyintwodistinctplacesorbeingatthesametimea
particleandawave,whenallour“macroscopic”experiencesaysthatthisisimpossible?
An“interpretation”ofquantummechanicsshouldexplainatleastsomeofsuch
conundrumsinawaythatcouldbeacceptedbyus:macroscopicbeingswhoseintuition
growsupexclusivelyonmacroscopicphenomena.
Theinterpretationofquantummechanicsproposedinthisworkisbasedonmanyvaluedlogicandobservationthatpeople,usuallyunconsciously,usethislogicwhile
consideringfutureevents,theoccurenceornon-occurenceofwhichisnotsureatpresent.
JanŁukasiewiczformalizedthisideainhisnumerouspapers[2],andarguedthattruthvaluesofnon-certainstatementsconcerningfutureevents(futurecontingents)equalthe
probability(possibility?likelihood?)thatthesestatementswill,atduetime,occurtobe

true.In“macroscopic”casesconsideredbyŁukasiewicztheseprobabilitiesweresupposed
tobeevaluatedinasubjectiveandimpreciseway.Fortunately,inquantummechanics
theseprobabilitiesareprovidedbythetheoryandarepreciselyknown.
Quantumphysicsandmoderntheoryofmany-valuedlogicswerebornnearly
simultaneouslyinthethirddecadeofthetwentiethcentury.However,theattemptsat
applyingmany-valuedlogicstothedescriptionofquantumsystemsexpiredsoonafter
WorldWarII.Thiswasthesituationthatpersistedatleastuntiltheearlyseventieswhen
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MaxJammerinhisbook[3]publishedin1974wrotethewordswhichhavebeenchosen
asakindofanti-mottotothiswork.
Therecentlyobservedrevivalofinterestinapplyingmany-valuedlogicstothe
descriptionofquantumphenomenaiscloselyconnectedwithanewandrapidly
developingbranchofmathematics:fuzzysettheory.Fuzzysetsremaininthesame
relationtoinfinite-valuedlogicastraditionalsetstoclassicaltwo-valuedlogic.Therefore,
truth-valuesofmany-valuedstatementsaboutresultsoffutureexperimentsonquantum
objectsmaybe,equivalently,treatedasdegreestowhichtheseobjectspossessrespective
propertiesbeforetheyaremeasured.
Thebookisorganizedasfollows:Abriefsurveyofmaininterpretationsofquantum
mechanicsisgiveninChap.2.Chapter3containsintroductiontomany-valuedlogics
whileChap.4givestherudimentsofthefuzzysettheoryandshowsitslinkswiththe
infinite-valuedŁukasiewiczlogic.Chapter5containsashorthistoricalsurveyofattempts
atapplyingnon-classicallogicstothedescriptionofquantumphenomena,from
Zawirski’sattemptsintheearlythirtiestovonWeizsäcker’spaperspublishedinthelate
fiftiesofthetwentiethcentury.OutoftheseattemptsonlytheBirkhoffandvonNeumann
proposaltouseatwo-valuedbutnon-distributivelogicgainedwidepopularityandisstill
inusenowadays.Chapter6,ofrathertechnicalcharacter,isdevotedtothiskindof
“quantumlogic”andpresentsitthroughthreemodels:thetraditionalalgebraicmodel,
Ma̧czyński’sfunctionalmodel,andfinallythefuzzysetone,elaboratedinaseriesof

papersbythepresentauthor.ThefuzzysetmodeloftheBirkhoff–vonNeumanquantum
logicenablesittobeexpressedimmediatelyinthelanguageoftheinfinite-valued
Łukasiewiczlogic.Thisprocedure,developedinChap.7,allowstheBirkhoff–von
Neumanquantumlogictobetreatedasaspecialkindofinfinite-valuedŁukasiewicz’s
logicwithpartiallydefinedconjunctionanddisjunction.Thisunifiestwocompeting
approaches:themany-valued,andthetwo-valuedbutnon-distributiveone,whichhavecoexistedinthequantumlogictheorysinceitsverybeginning.Thisalsoclarifiesthelongstandingproblemofpropermodelsforthedisjunctionandconjunctionofexperimentally
verifiablepropositionsaboutquantumsystemsandallowsalogicalanalysistobe
performedofthetwo-slitexperiment.
Chapter8containssomespeculationsaboutthenewperspectivesopenedbythe
proposedapproach.Finally,Chap.9isdevotedtotheconciseexpositionoftheproposed
many-valuedinterpretationofquantummechanics,performedinawaysimilartotheway
inwhichotherinterpretationsofquantummechanicswerepresentedinChap.2,which
makestheircomparisonmoreeasy.
Thisbookwasfinancedbythegrant2011/03/B/HS1/04573ofthePolishNational
FoundationforScience(NCN).

References
1. Nielsen,M.A.andI.L.Chuang,QuantumComputationandQuantumInformation(CambridgeUniversityPress,
Cambridge,2000).
2. Łukasiewicz,J.SelectedWorks,ed.byL.Borkowski(North-Holland,Amsterdam,andPWN—PolishScientific
Publishers,Warszawa,1970).
3. Jammer,M.ThePhilosophyofQuantumMechanics(Wiley-Interscience,NewYork,1974).

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JarosławPykacz,QuantumPhysics,FuzzySetsandLogic,SpringerBriefsinPhysics,DOI10.1007/978-3-319-193847_2

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2.ABriefSurveyofMainInterpretationsof
QuantumMechanics
JarosławPykacz1
(1) InstituteofMathematics,UniversityofGdańsk,Gdańsk,Poland




JarosławPykacz
Email:

Sincedescriptionsandcommentsontheplethoraofvariousinterpretationsofquantum
mechanicsarewidelyaccessible(see,e.g.,[1,2]),wegiveinthischapteronlyaverybrief
surveyofthemostpopularofthem.Westressthatourpresentationandevaluationof
variousinterpretationsishighlysubjective.Inparticular,inouropiniononticdeterminism
precludestheexistenceoffreewill,whichwetreasure,thereforeindeterminismisinour
opinionavirtue,notadrawbackofaninterpretation.
TheotherdifficultyinpresentingsuchabriefsurveyofvariousinterpretationsofQM
iscausedbythefactthatmostofthemarenotuniquelydefined.Wetriedineachcaseto
extractabunchofideasthatcouldbetreatedasa“commondenominator”byvarious
adherentsofaninterpretation,butinmanycasesthisoccurredtobeadifficulttask.
Itshouldbealsonoticedthatnotall“interpretationsofQM”presentedintheliterature
areinterpretationsinthestrictsenseofthisword,i.e.,interpretationsofthe“bare”
mathematical(Hilbertspace)formalismoftheorthodoxquantumtheory.Inmanycasesan
“interpretation”introducesoratleastforeseesvariousmodificationsoftheusual
mathematicalformalismofQM,soitshouldberathercalleda“theory”.Sinceinthisbrief
surveywedecidedtoconfineto“interpretationsofQM”inthestrictsenseofthisword,
wedonotmentionheresuchimportantproposalsasGhirardietal.[3],orother“Objective

CollapseTheories”,or“HiddenVariablesInterpretations”.
Thesimplesttestthatallowstodistinguishbetweenaninterpretationandatheoryis
theexistenceornonexistenceofexperimentalproposalsthatcould,atleasttheoretically,
distinguishitfromtheotherones,sincenotwointerpretationsofQM,bythevery
definitionofthisnotion,couldbedistinguishedinthisway.Therefore,ifasetofideas
pertainingtoQMallowsforitsexperimentaldiscriminationfromtheotherones,itshould
berathercalledatheory,notaninterpretation.However,inmanycasesthisissueisnot
settledevenamongvariousproponentsofaspecificinterpretation,whichcausestheissue
offilteringoutinterpretationsfromtheoriesanextremelydifficulttask.

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2.1 EnsembleInterpretation
EnsembleInterpretation(EI),calledalsoStatisticalInterpretation,takesliterallyBorn’s
probabilisticinterpretationofsquaredmodulusofthewavefunction.Therefore,itassumes
thatthewavefunctiondoesnotrefertoanindividualquantumobject,buttoastatistical
ensembleofsuch“identicallyprepared”objects.Thisensemblecanbeeithermeant
literally,asitisinthecaseofmyriadsofidenticallypreparedphotonsemittedbyasource,
oritcanbemeant“abstractly”asan“imaginarycollection”ofmultiplecopiesofan
individualobject.ItseemsthatthisinterpretationofQMwassupportedbyEinsteinwho,
however,wentfurtherandinferredfromitthatthe“orthodox”QMshouldbe
supplementedbyhiddenvariables,whileingeneralthereisnosuchassumptionin
contemporaryexpositionsoftheEI.MorerecentlyEIwaspromotedvigorouslyby
Ballentine[4,5](seealsoextensivebibliographyatUlfKlein’swebsite[6]).
Mainidea:
Wavefunctionisanabstractconceptthatreferstoanensembleofquantumsystems.
Inparticular,theredoesnotexistanythinglike“wavefunctionofanindividual
quantumsystem”.
Virtues:

EIis“minimal”inthesensethatitdoesnotmakeuseofanymetaphysical
assumptions.
Noproblemswithmeasurements,collapse,Schrödinger’scats,etc.
Drawbacks:
EIdoesnotsatisfyourdeepdesirefor“finalanswers”.
Impossibilitytoexplain“quantumZenoeffect”.

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2.2 CopenhagenInterpretation
Outofallinterpretationsofquantummechanicsproposeduptonow,theCopenhagen
Interpretation(CI),inspiteofbeingstillthemostpopular(seetheresultsofapoll
executedbySchlosshaueretal.[7]),istheworst-definedone.AccordingtoPeres[8]:
“ThereseemstobeatleastasmanydifferentCopenhagenInterpretationsaspeoplewho
usethatterm,probablytherearemore”.
CIhasitsrootsinBohr’sandHeisenberg’sideaselaboratedinthetownof
CopenhageninthelatetwentiesoftheXXcentury.Nevertheless,theveryname
“CopenhagenInterpretation”wasattachedtothisbunchofideasnotbeforethaninthe
fifties.ItshouldbealsonoticedthatideasusuallypresentedintextbooksasCIarenot
entirelyidenticalwithoriginalideasofBohrandHeisenbergwhich,moreover,werealso
differentfromeachotherinsomedetails.
Mainideas:
Quantumobjectsdisplayeitherwave-likeorparticle-likeproperties.Itisan
experimentalarrangementthatdefineswhichpropertiescanbeobserved.
Quantummechanicsisfundamentallyaboutobservationsorresultsofmeasurements.
Itismeaninglesstotalkaboutpropertiesofquantumobjectsbeforetheyare
measured.
Wavefunctionisamathematicalconcept.Physicalmeaninghasitssquaredmodulus
which,accordingtoBorn’srule,definesprobabilitiesofobtainingvarious

experimentalresults.
Wavefunctionsevolveintwoways:
1. Deterministically,accordingtoSchrödingerequation,whennomeasurementis
made.


2. Indeterministically(“collapse”or“reduction”)whenmeasurementismade.


Hilbertspacedescriptionofquantumphenomenaistheultimateone.Inparticular,
therearenohiddenvariablesthatcouldexplainrandombehaviourofquantum
objects.Therefore,quantumprobabilitiesareontic,notepistemic.
Virtues:
Fundamentalindeterminismofthequantumworld.
Drawbacks:
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Artificialdivisionofthephysicalworldintothequantumworldandtheclassical
world.
The“objectificationproblem”,i.e.,aproblemhow“potential”propertiesbecome
“actual”inthecourseofameasurement.

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2.3 Pilot-WaveInterpretation
ThePilot-WaveInterpretation(PWI),knownalsoasCausalorOntologicalInterpretation,
deBroglie–Bohmtheory,orBohmianmechanics,isbasedontheideaspresentedbyde
Brogliein1927inapaper[9]publishedinLeJournaldePhysiqueetleRadiumandalso

presentedatthe5thSolvayConference,andlateronrediscoveredbyBohm[10].Itseems
thatthemajorityofadvocatesofthisinterpretation(althoughnotall)maintainthatall
experimentalpredictionsofthedeBroglie–Bohmtheoryareexactlythesameas
predictionsofthe“orthodox”QM,thereforeaccordingtothem,itisreallyan
interpretationofQMinthenarrowsenseofthisword.
Mainideas:
Both“wave-like”and“particle-like”aspectsofquantumobjectshavesimultaneous
reality:quantumparticlesmovealongdefinitetrajectoriesguidedbytheirpilot
waves.Inparticular,inatwo-slitexperimentaparticlegoesthroughoneslitonlybut
itspilotwavegoesthroughbothslits,interfereswithitself,andattractstheparticleto
theareasofconstructiveinterference.
PilotwavesarerepresentedmathematicallybysolutionsofSchrödingerequation.
Theynevercollapse.
Theactualpositionsofparticlesare“hiddenvariables”.
Virtues:
PWIprovidesa“classical-like”,visibleandeasytocomprehendimageofthe
microworld.
Nomeasurementproblem.
Drawbacks:
Manifestnonlocality.
Determinism.

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2.4 Many-WorldsInterpretation
ThecornerstoneoftheManyWorldsInterpretation(MWI)waslaiddownbyHugh
EverettIIIinhisPhDthesis[11](reprintedin[12],seealsopaper[13]basedonthis
thesis).Nevertheless,itshouldbenoticedthatEveretthimselfneverjumpedintofarreachingontologicalconclusionsdrawnbyhisfollowers,andonlystatedenigmatically:
“Fromthepresentviewpointallelementsofsuperpositionareequally‘real”’([12],pp.

116–117).
Actually,theverynameMWIandexplicitformulationoftheideathat“everyquantum
transitiontakingplaceoneverystar,ineverygalaxy,ineveryremotecornerofthe
universeissplittingourlocalworldonearthintomyriadsofcopiesofitself”isdueto
DeWitt[14].
AmongotherdistinguishedadvocatesoftheMWIareDeutsch[15,16]and
Vaidman[17].Itshouldbenoticedthataccordingtotheresultsofapollexecutedby
Schlosshaueretal.[7]),theMWIoccurredtobethesecondw.r.t.popularityafterthe
CopenhagenInterpretation.
Mainideas:
Thereexiststhe“basicphysicalentity”:theuniversalwavefunction,thatnever
collapses.
Atevery“momentofchoice”:aphotoneitherpassesthroughasemi-transparent
mirrororisreflected,Schrödinger’scatiseitherpoisonedornot,auniversethatwe
witness(whichisonlyonecopyofmyriadsofitscopiesthatformtheMultiverse)
splitsintoseparate,equallyrealcopiesinwhicheitherthisorthatcourseofevents
takesplace.AdherentsoftheMWIarenotunanimouswhetherthesedifferentcopies
cansomehow“influence”or“feeltheexistence”oftheothersornot.
Virtues:
Observersandmeasurementsplaynospecialrole.
Noproblemswithcollapse.
AccordingtoVaidman[17]“TheMWIresolvesmost,ifnotall,paradoxesof
quantummechanics.”
Drawbacks:
Extremelyweirdontology.
TheveryideaofreplacingtheuniqueUniversebymyriadsofitscopiesthatformthe
MultiverseseemstobeindeepcontradictiontotheideaoftheOckhamRazorthat
successfullyguidesWesternPhilosophyforcenturies.
Indeterminismobservedinthemicroworldisonlyapparentsincetheuniversalwave
functionevolvesdeterministically.

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2.5 ConsistentHistoriesInterpretation
TheConsistentHistoriesInterpretation(CHI)issometimesproclaimedbyitsadvocatesas
“Copenhagendoneright”.ItwasoriginatedbyGriffiths[18,19],followedbyOmnès[20,
21],andbyGell-MannandHartle[22]whousedtheterm“decoherenthistories”.Itis
basedonthenotionofahistorywhichisthoughtofasatime-sequenceofproperties
actuallypossessedbyaquantumobjectinconsecutiveinstantsoftime.Thissequenceis
mathematicallyrepresentedbyatensorproductofprojectionoperators.Boundlesofsuch
histories,calledframeworksareanalogsofsamplespacesinclassicalprobabilitytheory,
andallowtodefineonthemprobabilitiesthatcoincidewithprobabilitiesyieldedby
Born’srule.However,itshouldbestressedthattoaspecificframeworkbelongonly
historiesthatareconsistentinthesensethatatanyinstantoftimetheydonotcontain
propertiesrepresentedbynon-commutingprojectors.
Mainideas:
Wavefunctionisatoolforcalculatingprobabilities,notarepresentationofreality.
Timedevelopmentofaquantumsystemisastochasticprocess.
Allframeworksareequally“real”.
Thesingleframeworkrule:Anydiscussionaboutpropertiesofquantumobjectshas
tobeconfinedtoasingleframework.Usinginthediscussionpropertiesthatbelong
toincompatibleframeworksisthesourceofparadoxes.
Measurementsrevealactuallyexistingpropertiesofquantumobjects,howevera
propertythatexistsinsomeframeworksmaynotexistinothers.
Virtues:
Nomeasurementproblem,nosuperluminalinfluences,noparadoxes.
Fundamentalindeterminism.
Drawbacks:
Highlyunclearontology.

Actualityofpropertiesdependsonthechosenframework(“relativityofreality”).

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2.6 ModalInterpretations
Thenameofthisclassofinterpretationsreferstomodallogic,i.e.,logiccapableoftaking
intoconsiderationsentencesexpressingnecessity,possibilityandcontingency.
Originallytherewasasinglemodalinterpretation(MI)ofnon-relativisticquantum
mechanicsproposedbyvanFraassen[23].Lateronvariousresearchersinvolvedinthis
lineofinvestigationdevelopedslightlydifferentapproacheswhich,however,areusually
collectivelycalled“modalinterpretations”.
CharacteristictoallMIsisadistinctionbetweenthedynamicalstateofaquantum
system,whichdetermineswhatmaybethecaseandisjustthequantumstateofthe
orthodoxQM,andthevaluestatewhichrepresentsallpropertiesthatthesystemactually
possessesatagiveninstant.InvariousversionsofMIsvariousobservablesarechosenas
“privileged”,i.e.,alwayspossessingdefinitevalues.
Mainideas:
ThestandardformalismofQM,howeverwithouttheprojectionpostulate,is
accepted.
Quantumsystemspossessallthetimedefinitepropertiesthatdefinetheirvalue
states.
ThedynamicalstatethatalwaysevolvesaccordingtoSchrödingerequationandnever
collapsesdefineswhatthepossiblepropertiesofasystemandtheircorresponding
probabilitiesare.
Virtues:
Nomeasurementproblem.
Indeterminism.
Drawbacks:
Unclearontologywhichis,moreover,differentindifferentversionsofMIs.

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2.7 RelationalQuantumMechanics
ThemainassumptionofRelationalQuantumMechanics(RQM),originatedbyRovelli
[24],statesthatQMisnotan“absolute”descriptionofrealitybutratherdealswith
relationsbetweenvariousobjects.Consequently,thenotionof“observer-independent”
descriptionoftheworldisdeclaredasbeingunphysical.Differentobserversmaygive
differentdescriptionsofthesameevent.However,itshouldbenoticedthatthisrefersonly
to“hierarchical”setsofobservers:the“prime”observerisOthatobserveswhat’sgoing
oninanobservedsystemS,the“secondary”observerisPthatobserveswhat’sgoingon
inasystem
,andsoon…
Mainideas:
Allphysicalsystemsare,fundamentally,quantumsystems.
QMisa“complete”theory:therearenohiddenvariablesorotheritemsthatshould
beaddedtoit.
QMisnotaboutpropertiesofobjects,butaboutrelationsbetweenobjects.
Measurementisanordinaryphysicalinteraction.
“Absolute”or“observer-independent”stateofaquantumsystemhasnomeaning.
Virtues:
Ontologicalparsimony.
Itisclaimed[25]thatRQMallowsforsuchreformulationoftheoriginalEPR
conditions,thatapparentconflictbetweenQMandspecialrelativitydisappears.
Drawbacks:
Relativityofpropertiesofphysicalobjects(evenifonlyw.r.t.“hierarchical”setof
observers).
Notclearlystatedpositionw.r.t.thedeterminism/indeterminismissue.

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2.8 Other,LessPopularInterpretations
Sevenmaininterpretationsoutlinedabovedefinitelydonotexhaustthelistofuptonow
proposedinterpretationsofQM.Amongtheotheroneswecanmentionthefollowing:
“ConsciousnessCausesCollapse”:aratherextremepointofviewascribedtovon
Neumann[26]andWigner[27,28].
ManyMindsInterpretation[29,30]:a“subjectiveoffspring”ofMWI,inwhichthe
multitudeof“paralleluniverses”isreplacedbythemultitudeof“minds”associated
witheachsentientbeing.
TransactionalInterpretation[31]inwhichaquantumeventisaresultofa
“transaction”betweenadvanced(backward-in-time)andretarded(forward-in-time)
waves.
InformationInterpretationwhichassumesthat“theQM-formalismdescribes
informationaboutmicrosystemsextractedbymeansofmacroscopicmeasurement
devices”[32].Thisrelativelynewinterpretationquicklygainspopularityandmost
probablywillbeconsideredasbelongingtothemainstreamsoon(see,e.g.,[33,34]).

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2.9 Summary
AllinterpretationsofQMpresentedinthisChapterarebasedon2-valuedlogic.1Thisis
notasurprise,takingintoaccountthat2-valuedlogicsuccessfullyguidedWesternScience
forcenturies.Actually,tillŁukasiewicztherewerenoalternatives,andevenlateron
many-valuedlogicswanderedonthefringesofthemainstreamofScience,andwere
regardedasamathematicalcuriositywithnorelationtothephysicalworld.
Mostprobablytothemajorityofscientiststheideaofgoingbeyondthe2-valuedlogic
inthedescriptionofthephysicalrealityisasaberrantasitwouldbetheideaof

abandoningPtolemaicsystembeforetheCopernicusorleavingthedomainofEuclidean
flatgeometrybeforeEinstein.
However,theaccumulationof“paradoxes”anddevelopmentofmoreandmoreweird
interpretationsofQMismaybeasignthatthisGordianknotshouldbecutby
transgressingtheboundariesencircledbythe2-valuedlogic.Therestofthisworkis
devotedtothepresentationandjustificationofthisproposal.

References
1. TheStanfordEncyclopediaofPhilosophy(Winter2014Edition),EdwardN.Zalta(ed.),http://​plato.​stanford.​edu/​
search/​searcher.​py?​query=​Interpretation+o​f+quantum+mechan​ics.
2. Wikipediacontributors,“Interpretationsofquantummechanics,”Wikipedia,TheFreeEncyclopedia,http://​en.​
wikipedia.​org/​w/​index.​php?​title=​Interpretations_​of_​quantum_​mechanics&​oldid=​623968383(accessedJune5,
2014).
3. Ghirardi,G.C.,A.Rimini,andT.Weber,“AModelforaUnifiedQuantumDescriptionofMacroscopicand
MicroscopicSystems”,in:L.Accardietal.(eds)QuantumProbabilityandApplications(Springer,Berlin,1985).
4. Ballentine,L.E.“Thestatisticalinterpretationofquantummechanics”,ReviewsofModernPhysics,42(1970)358–
381.
5. Ballentine,L.E.QuantumMechanics:AModernDevelopment(WorldScientific.,Singapore,1998).
6. Klein,U.TheStatisticalInterpretationofQuantumTheory,version02,11.12.2012,
statintquant.net/siq.html#siqli1.html.
7. Schlosshauer,M.,J.Kofler,andA.Zeilinger,“Asnapshotoffoundationalattitudestowardquantummechanics”,
arXiv:​1301.​1069[quant-ph].
8. Peres,A.“KarlPopperandtheCopenhageninterpretation”,StudiesinHistoryandPhilosophyofModernPhysics,
33(2002)23–34.
9. deBroglie,L.“Wavemechanicsandtheatomicstructureofmatterandofradiation”,LeJournaldePhysiqueetle
Radium,8(1927)225.
10. Bohm,D.“Asuggestedinterpretationofthequantumtheoryintermsofhiddenvariables,I”PhysicalReview,85
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Footnotes
1 Evenmodallogic,whichisabaseofmodalinterpretations,althoughnon-classicalandsometimesregardedasa
“coarsegraining”ofmany-valuedlogics,isgenerallyconsideredas2-valuedlogic.



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