Table Of Contenti i
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Fuzzy Logic and Mathematics:
A Historical Perspective
RadimBělohlávek,JosephW.Dauben,andGeorgeJ.Klir
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1
PublishedintheUnitedStatesofAmericabyOxfordUniversityPress
198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica.
©OxfordUniversityPress2017
LibraryofCongressCataloging-in-PublicationData
Names:Bělohlávek,Radim.|Dauben,JosephWarren,1944-|Klir,GeorgeJ.,1932-
Title:Fuzzylogicandmathematics:ahistoricalperspective/Radim
Bělohlávek,JosephW.Dauben,GeorgeJ.Klir.
Description:Oxford;NewYork,NY:OxfordUniversityPress,[2017]|
Includesbibliographicalreferences.
Identifiers:LCCN2016024541|ISBN9780190200015(hardcover)|
ISBN9780190200039(onlinecontent)
Subjects:LCSH:Fuzzylogic.|Logic,Symbolicandmathematical.
Classification:LCCQA9.64.B45252017|DDC511.3/13–dc23LCrecordavailable
athttps://lccn.loc.gov/2016024541
1 3 5 7 9 8 6 4 2
PrintedbySheridanBooks,Inc.,UnitedStatesofAmerica
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Contents
Preface ix
NotesfortheReader xi
1 AimsandScopeofThisBook 1
2 Prehistory,Emergence,andEvolutionofFuzzyLogic 5
2.1 Prehistoryoffuzzylogic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Emergenceoffuzzylogic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Evolvingattitudestowardfuzzylogic . . . . . . . . . . . . . . . . . . . . . 27
2.4 Documenteddebates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Evolutionofsupportinginfrastructureforfuzzylogic. . . . . . . . . . 38
3 FuzzyLogicintheBroadSense 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Basicconceptsoffuzzysets . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Operationsonfuzzysets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Fuzzyintervals,fuzzynumbers,andfuzzyarithmetic . . . . . . . . . . 55
3.5 Fuzzyrelations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6 Approximatereasoning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.7 Possibilitytheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.8 Fuzzyclustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.9 Methodsforconstructingfuzzysets. . . . . . . . . . . . . . . . . . . . . . 94
3.10 Nonstandardfuzzysets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 FuzzyLogicintheNarrowSense 105
4.1 Fromclassicaltofuzzylogic:Principalissues . . . . . . . . . . . . . . . . 106
4.2 Many-valuedlogicsuntilthe1960s . . . . . . . . . . . . . . . . . . . . . . 112
4.2.1 Łukasiewiczfinitely-andinfinitely-valuedlogics. . . . . . . . 112
4.2.2 Gödelfinitely-andinfinitely-valuedlogics. . . . . . . . . . . . 129
4.2.3 Otherpropositionallogics . . . . . . . . . . . . . . . . . . . . . . 134
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4.2.4 Predicatelogics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.2.5 Furtherdevelopments . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.2.6 Themid-1960sandZadeh’sideaoffuzzysets. . . . . . . . . . 159
4.3 Fuzzylogicswithgradedconsequence . . . . . . . . . . . . . . . . . . . . 160
4.3.1 Goguen’slogicofinexactconcepts. . . . . . . . . . . . . . . . . 161
4.3.2 Pavelka-stylefuzzylogic . . . . . . . . . . . . . . . . . . . . . . . . 165
4.3.3 Furtherdevelopments . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.4 Fuzzylogicsbasedont-normsandtheirresidua . . . . . . . . . . . . . . 177
4.4.1 Fuzzylogicsbasedont-normsuntilthemid-1990s . . . . . . 178
4.4.2 Productlogic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.4.3 Hájek’slogicBLandMetamathematicsofFuzzyLogic . . . 189
4.4.4 LogicsrelatedtoBL . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.5 Fuzzylogicanduncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
4.5.1 Degreesoftruthvs.beliefandtruthfunctionality . . . . . . 210
4.5.2 Possibilisticlogic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
4.5.3 Gerla’sprobabilisticfuzzylogics . . . . . . . . . . . . . . . . . . 215
4.5.4 Belief,modality,andquantifiersinfuzzylogic . . . . . . . . . 216
4.6 Miscellaneousissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
4.6.1 Relationshiptoapplications . . . . . . . . . . . . . . . . . . . . . 222
4.6.2 Computabilityandcomplexity . . . . . . . . . . . . . . . . . . . 224
4.6.3 Furtherdevelopments . . . . . . . . . . . . . . . . . . . . . . . . . 227
5 MathematicsBasedonFuzzyLogic 231
5.1 Principalissuesandoutlineofdevelopment . . . . . . . . . . . . . . . . 231
5.1.1 Whatismathematicsbasedonfuzzylogic? . . . . . . . . . . . 231
5.1.2 Theproblemandroleoffoundations. . . . . . . . . . . . . . . 233
5.1.3 Theproblemandroleofapplications . . . . . . . . . . . . . . . 238
5.1.4 Outlineofdevelopment. . . . . . . . . . . . . . . . . . . . . . . . 241
5.2 Foundationsofmathematicsbasedonfuzzylogic . . . . . . . . . . . .246
5.2.1 Theroleoffuzzylogicinthenarrowsense . . . . . . . . . . .246
5.2.2 Higher-orderlogicapproaches . . . . . . . . . . . . . . . . . . . 250
5.2.3 Set-theoreticapproaches . . . . . . . . . . . . . . . . . . . . . . . 253
5.2.4 Category-theoreticapproaches . . . . . . . . . . . . . . . . . . . 261
5.3 Selectedareasofmathematicsbasedonfuzzylogic . . . . . . . . . . . . 267
5.3.1 Setsandrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
5.3.2 Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
5.3.3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
5.3.4 Quantitiesandmathematicalanalysis . . . . . . . . . . . . . . . 298
5.3.5 Probabilityandstatistics . . . . . . . . . . . . . . . . . . . . . . . 309
5.3.6 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
5.3.7 Furtherdevelopments . . . . . . . . . . . . . . . . . . . . . . . . . 320
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5.4 Miscellaneousissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
5.4.1 Interpretationoftruthdegrees . . . . . . . . . . . . . . . . . . . 321
5.4.2 Fuzzylogicandparadoxes . . . . . . . . . . . . . . . . . . . . . . 332
5.4.3 Fuzzylogicandvagueness. . . . . . . . . . . . . . . . . . . . . . . 339
6 ApplicationsofFuzzyLogic 347
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
6.2 Ahistoricaloverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
6.3 Engineering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
6.3.1 Fuzzycontrol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
6.3.2 Otherengineeringapplications . . . . . . . . . . . . . . . . . . . 365
6.4 Decisionmaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
6.5 Naturalsciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
6.5.1 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
6.5.2 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
6.5.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
6.6 Earthsciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
6.7 Psychology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
6.8 Socialsciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
6.8.1 Ahistoricaloverview . . . . . . . . . . . . . . . . . . . . . . . . . . 390
6.8.2 Economics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
6.8.3 Politicalscience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
6.9 Computerscience. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
6.10 Medicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .407
6.11 Managementandbusiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
6.12 Otherapplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
7 SignificanceofFuzzyLogic 421
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
7.2 Aretrospectiveoverviewoffuzzylogic . . . . . . . . . . . . . . . . . . . . 421
7.3 Paradigmshiftsinscience,mathematics,engineering,andotherareas425
7.4 Assessmentofthesignificanceoffuzzylogic . . . . . . . . . . . . . . . . 430
7.5 Aprospectiveviewonthe50thanniversaryoffuzzylogic . . . . . . .442
A TheEnigmaofCox’sProof 449
B OverviewofClassicalLogic 453
GlossaryofSymbols 473
References 475
SubjectIndex 521
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Preface
Theinitialideaforwritingabookofthiskindemergedfromdiscussionsat
the Eighth World Congress of the International Fuzzy Systems Association
(IFSA)inTaipeiinAugustof1999.Withtheapproachingnewmillennium,thefo-
cusofthecongresswasonreexaminingthepastandponderingthefutureoftheo-
reticaldevelopmentsaswellasapplicationsoffuzzylogicandmathematicsbasedon
fuzzylogic.Itwasrepeatedlysuggestedinthemanydiscussionsatthecongressthat
thetimewasripeforwritingabookdescribingcomprehensivelyandinsufficientde-
tailtherelativelyshorthistoryoffuzzylogicandassessingitssignificanceandlikely
prospectsinthenewmillennium.Oneofus(Klir)participatedinthesediscussions
andtheideaofwritingsuchabookwasappealingtohim. Herealized, however,
thatthescopeofthiskindofbookwastoolargeforasinglepersontowrite,andso
theideaofwritingsuchabookremainedidleinhismindformorethanadecade.In
2011,whileworkingjointlywithaformercolleague(Bělohlávek)onanotherbook
involvingfuzzylogic(ConceptsandFuzzyLogic,MITPress,2011),hementionedthe
ideaandtheyeventuallyagreedtoworkonthislargerbooktogether.However,due
tothehistoricalnatureofthebook,theyfeltitwouldbeusefultoenlistsomeone
withexpertiseinhistory,especiallythehistoryofmathematics,andsotheyextended
aninvitationtoJosephDaubentojointheminworkingonthischallengingbook,
whichheaccepted.
In2012, thethreeofusbeganbyconvertingtheinitial, somewhathaphazard
ideaintoawell-definedwritingproject. Thatis, weformulatedourvisionofthe
aimsandscopeofthebookanddescribedhowweintendedtoimplementit.Wealso
definedageneralstructureforthebookanddeterminedroughlytheenvisionedcon-
tentsofindividualchapters. Thisoveralldescriptionofthebookultimatelyserved
as the basis for our introductory chapter 1. We then explored several prospective
publishersandsettledeventuallyonOxfordUniversityPress,whichweallconsid-
eredtheidealpublisherforabooklikethis.
Althoughweintendedtoworkonthebookasateam, itwasreasonableand
practicalthateachofuswouldassumeaparticularroleintheactualwritingprocess.
Sincetwoofus—BělohlávekandKlir—hadbeenactiveparticipantsinresearch,ed-
ucation, and other activities regarding theory and applications of fuzzy logic, we
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tooktheresponsibilityformostofthewritinganddivideditaccordingtoourinter-
ests: Bělohlávekassumedresponsibilityforchapters4and5;Klirforchapters2,3,
and6.Eventhoughseveralparts—particularlyofchapters2and6—representjoint
work,theindividualchaptersdifferslightlyintheirstylesofwriting. Becausethese
distinctionsinstylealsoreflectdistinctbackgroundsandexperiences—indeed,Klir
and Bělohlávek represent two distinct generations of fuzzy-logic researchers—we
considereditdesirabletopreservethem. AlthoughtheprimaryroleofDauben—
arelativenewcomertofuzzylogic—wastoprovidehisexpertiseinthehistoryof
mathematics, he also critically evaluated the content of chapters 2–6 and partici-
patedactivelyinwritingchapter7.
OurcooperationwithOxfordUniversityPress,primarilyviaPeterOhlin—the
editorresponsibleforthisbookintheNewYorkofficeofOUP—hasbeenefficient
andcordialfromthebeginning,forwhichweareverygrateful. Initially,weunder-
estimatedtheenormousscopeofinformationtobecoveredinabooklikethis,but
weweredeterminedtoincludealloftherelevantandvaluablehistoricalinforma-
tionthatwecouldfind,someofwhichhadnowherebeenpreviouslydocumented.
Nevertheless, wehavehadtobeveryselectiveand, insomecases, wehavehadto
shortensomepartsofouroriginaltext.However,itisourintenttopublishatleast
someofthedeletedmaterialinsomeformelsewhereinthenearfuture.
Weareawarethatthepublicationofthisbookcoincideswiththe50thanniver-
saryofthegenesisoffuzzylogic. Thisisfortunate,eventhoughinitiallyitwasnot
plannedtohappenthisway. However,webelievethatthebookcannowbesuc-
cinctlydescribedascoveringinconsiderabledetailthehistoryoffuzzylogicduring
thefirstfiftyyearsofitsexistence,aswellasofferingacarefullyarguedassessment
ofthesignificanceoffuzzylogicattheendofitsfirstfiftyyears.Wehopeourwork
mayalsoserveasabenchmarkinsofarasourvisionofpossiblefuturedevelopments
offuzzylogicmaybecomparedatsomefuturetimewheneveranotherhistoricalas-
sessmentoffuzzylogicanditssignificancemaybemade.
RadimBělohlávek,JosephW.Dauben,andGeorgeJ.Klir
Acknowledgments
Wewouldliketoexpressourgratitudetothefollowingpersons:EduardBartl,acol-
league of Radim Bělohlávek at Palacký University in Olomouc, Czech Republic,
whoplayedamajorroleintypesettingtheentirebookusingaLATEXstylethathecre-
ated.Similarly,EllenTilden,editorialassistanttoGeorgeKlir,readandcopyedited
virtuallytheentiremanuscriptfromastylisticpointofview,andwearegratefulto
herforimprovingtheoverallreadabilityofthisbook.
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Notes for the Reader
Thesenotesprovideinformationabouthowthematerialcoveredinthisbook
isorganized.Theaimsandscopeofthebookaswellasthecontentofitsmain
chapters,chapters2–7,arepresentedinchapter1.Inadditiontothesechapters,the
bookcontainsthreeappendices:appendixAconcernsCox’stheorem,whichisdis-
cussedinchapter2;appendixBprovidesanoverviewofclassicallogic;appendixC
consistsofphotographswithshortbiographicalsketchesofthemajorcontributors
totheearlydevelopmentoffuzzylogic.Thebookalsocontainsaglossaryofmathe-
maticalsymbolswidelyusedinthetext,alistofreferences,anameindex,andasub-
jectindex.
The list of references at the end of the book contains all of the major publi-
cations on which wehave drawn in writing this book. Additional references not
directlyconcernedwithitsmainsubjectarepresentedinfootnotes,whicharenum-
beredconsecutivelywithinindividualchapters,andalsocontainfurtherinforma-
tionrelatedtothemaintext. ForworkspublishedinlanguagesotherthanEnglish,
theoriginaltitleisgivenwithanEnglishtranslationinbrackets;wheneveranEnglish
translationofsuchworksisalsoavailable,therespectivebibliographicitemcontains
acompletereferencetothetranslatedversionaswell.
Forstylisticmatters,the16theditionoftheChicago Manual of Styleisgener-
allyfollowed. Incross-references,“p.” standsfor“page”and“n.” for“footnote.”
Inmathematicaltheorems,weuse“iff”for“ifandonlyif”and“w.r.t.” for“with
respectto.”Thefollowingabbreviationsareusedthroughout:
ACM AssociationforComputingMachinery
AMS AmericanMathematicalSociety
Ann Annals
Bull Bulletin
IEEE InstituteofElectricalandElectronicsEngineers
Int International
J Journal
Proc Proceedings
Symp Symposium
Trans Transactions
Univ. University
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AeqMath AequationesMathematicae
AmericanJMath AmericanJournalofMathematics
AmericanJPhys AmericanJournalofPhysics
AnnMath AnnalsofMathematics
AnnPureApplLogic AnnalsofPureandAppliedLogic
ArchMathLogic ArchiveforMathematicalLogic
ArtifIntellRev ArtificialIntelligenceReview
BullAMS BulletinoftheAmericanMathematicalSociety
CivilEngSyst CivilEngineeringSystems
ComputIntell ComputationalIntelligence
CommunACM CommunicationsoftheACM
EuropJOperationalResearch EuropeanJournalofOperationalResearch
FundInform FundamentaInformaticae
FundMath FundamentaMathematicae
FuzzySetsSyst FuzzySetsandSystems
IEEETransAutomatContr IEEETransactionsonAutomaticControl
IEEETransFuzzySyst IEEETransactionsonFuzzySystems
IEEETransSystSciCyb IEEETransactionsonSystemsScienceandCybernetics
IEEETransSystManCyb IEEETransactionsonSystems,ManandCybernetics
InfControl InformationandControl
InfSci InformationSciences
IntJApproxReason InternationalJournalofApproximateReasoning
IntJGenSyst InternationalJournalofGeneralSystems
IntJIntellSyst InternationalJournalofIntelligentSystems
IntJManMachStud InternationalJournalofMan-MachineStudies
IntJUFKBS InternationalJournalofUncertainty,FuzzinessandKnowledge-BasedSystems
JAppliedNon-ClassicalLogics JournalofAppliedNon-ClassicalLogics
JAmericanChemSoc JournaloftheAmericanChemicalSociety
JACM JournaloftheACM
JComputSystSci JournalofComputerandSystemSciences
JLogicComput JournalofLogicandComputation
JMathAnalAppl JournalofMathematicalAnalysisandApplications
JPureApplAlgebra JournalofPureandAppliedAlgebra
JSymbLogic JournalofSymbolicLogic
LectNotesComputSci LectureNotesinComputerScience
LectNotesLogic LectureNotesinLogic
LectNotesMath LectureNotesinMathematics
LogicJIGPL LogicJournaloftheInterestGroupofPureandAppliedLogic
MathLogicQuart MathematicalLogicQuarterly
NoticesAMS NoticesoftheAmericanMathematicalSociety
PhilosSci PhilosophyofScience
ProcLondonMathSoc ProceedingsoftheLondonMathematicalSociety
ProcNatlAcadSciUSA ProceedingsoftheNationalAcademyofSciencesoftheUnitedStatesofAmerica
TatraMtMathPubl TatraMountainsMathematicalPublications
TheorComputSci TheoreticalComputerScience
TransAMS TransactionsoftheAmericanMathematicalSociety
ZMathLogikGrundlagenMath ZeitschriftfürmathematischeLogikundGrundlagenderMathematik
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1
Chapter
Aims and Scope of This Book
Thesubjectofthisbook—fuzzylogicanditsroleinmathematics—hasarel-
ativelyshorthistoryofsomefiftyyears.Theoverallaimofthisbookistocover
thisshorthistoryascomprehensivelyaspossible.Thismeansthatweintendtocover
notonlytheoreticalandpracticalresultsemanatingfromfuzzylogic,butalsomoti-
vationsandcreativeprocessesthatledtotheseresults.Sucharetrospectivereflection
isinouropinionessentialforproperlyassessingtheoverallsignificanceandimpact
offuzzylogic,andwefeelthatthetimeisripeforit.
Itseemsreasonabletoexpectthattheaimsandscopeofanyscholarlybookbe
expressedinanutshellbyitstitle. Ourchoiceof“FuzzyLogicandMathematics:
AHistoricalPerspective”wasindeedintendedtodoso.First,although“fuzzylogic”
hasmultipleconnotations,theircommongroundistherejectionofafundamental
principleofclassicallogic—theprincipleofbivalence. Thisisbasicallyanassump-
tion,inherentinclassicallogic,thatanydeclarativesentencehasonlytwopossible
truthvalues,trueandfalse.
Byrejectingtheprincipleofbivalence,fuzzylogicdoesnotabandontheclassi-
caltruthvalues—trueandfalse—butallowsforadditionalones.Thesetruthvalues,
whichareinterpretedasdegrees of truth,maybeconstruedinvariousways. Most
commonly,theyarerepresentedbynumbersintheunitinterval[0,1]. Inthisin-
terpretation,1and0representtheboundarydegreesoftruththatcorrespond,re-
spectively,totheclassicaltruthvaluestrueandfalse.Thenumbersbetween0and1,
withtheirnaturalordering,representintermediatedegreesoftruth. Eitherallreal
numbersfrom[0,1]ortheirvarioussubsets,eachcontaining0and1,maybeem-
ployedastruthvalues. Othersetsoftruthvaluesarepossibleaswell,providedthat
theyareatleastpartiallyorderedandboundedbytheclassicaltruthvalues.
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