i i i i Fuzzy Logic and Mathematics: A Historical Perspective RadimBělohlávek,JosephW.Dauben,andGeorgeJ.Klir 1 i i i i i i i i 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 i i i i i i i i 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 i i i i i i i i 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 i i i i i i i i 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 i i i i i i 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 i i i i i i i i 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. i i i i i i i i 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 i i i i i i i i 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 i i i i i i i i 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. i i i i