The Fuzzy Revolution: Goodbye to the Aristotelian - IIIA - CSIC

25 Pages·2000·0.18 MB·English

ArtificialIntelligenceinMedicine21(2001)1–25
The Fuzzy Revolution:
Goodbye to the Aristotelian Weltanschauung
Kazem Sadegh-Zadeh*
TheoryofMedicineDepartment,UniversityofMu¨nsterMedicalInstitutions,
WaldeyerSt.27,48149Mu¨nster,Germany
Accepted1August2000
Because of its unorthodoxy, it has been and will continue to be controversial for some
time.Eventually,though,thetheoryoffuzzysetsislikelytoberecognizedasanaturaldevelopment
in the evolution of scientific thinking. In retrospect, the skepticism about its usefulness will be
viewedasamanifestationofthehumanattachmenttotraditionandresistancetoinnovation([91],
p.421).
1. Dear Professor Zadeh
Themedicalartificialintelligencecommunitycongratulatesyouonyour80thbirthday
on4February2001.Onbehalfofallpatientsandnon-patientswhoare,orwillinthefuture
be,enjoyingthemedicalfruitsofyourwork,wewouldliketothankyouforinitiatingand
advancing The Fuzzy Revolution in science, technology, and society. Medicine has been
among the privileged areas to early recognize this revolution and to embrace the fuzzy
theory in parallel withthe seminal Mamdani and Assilian application in the engineering
sciences and technology [35,36] (see, e.g. [2,30,73,74]).1 The tempo, the scope, and the
quantity of medical fuzzy research and technology have exponentially increased in the
meantime.Thereisnomedicalsubdomainleftoutsidethisacceleratingfuzzystreamtoday.
This advancement of medical thinking and practice we owe to your foresight already
*Tel.:(cid:135)49-251-83-55287;fax:(cid:135)49-251-83-55339.
E-mailaddress:[email protected](K.Sadegh-Zadeh).
1Inthepresentcontext,theterm‘fuzzytheory’isusedasaunifyinglabelforthetheoryoffuzzysets(cid:135)theory
of linguistic variables and hedges(cid:135)fuzzylogic(cid:135)concomitant conceptual frameworks and theories based
on these foundations. Although the term ‘fuzziness theory’ would be preferable, it may not be generally
acceptable.
0933-3657/01/$–seefrontmatter#2001ElsevierScienceB.V.Allrightsreserved.
PII:S0933-3657(00)00071-3
2 K.Sadegh-Zadeh/ArtificialIntelligenceinMedicine21(2001)1–25
communicatedin1962,3yearsbeforetheexplicitinceptionofyourfuzzytheoryin1965
[77,78]:
Infact,thereisafairlywidegapbetweenwhatmightberegardedas‘animate’system
theoristsand‘inanimate’systemtheoristsatthepresenttime,anditisnotatallcertain
thatthisgapwillbenarrowed,muchlessclosed, inthenearfuture.Therearesome
who feel this gap reflects the fundamental inadequacy of the conventional mathe-
matics — the mathematics of precisely-defined points, functions, sets, probability
measures,etc.—forcopingwiththeanalysisofbiologicalsystems,andthattodeal
effectively with such systems, which are generally orders of magnitudes more
complexthanman-madesystems,weneedaradicallydifferentkindofmathematics,
the mathematics of fuzzy or cloudy quantities which are not described in terms of
probability distributions... ([76], p. 857).
Sincethepatientasthesubjectofmedicineisananimatesystem,medicalprofessionals
inresearchandpracticeareindeedanimatesystemtheoristsand‘‘needaradicallydifferent
kind of mathematics...’’ you have created. The prospect abovewas concretized later by
explicit demonstrations such as ‘‘Biological application of the theory of fuzzy sets and
systems’’ [79], and by numerous encouragements and hints we have received from you
regularly in your ground-breaking publications. Unfortunately, initial misunderstandings
on the nature of fuzzy theory disseminated by some scholars such as Rudolf Kalman,
WilliamKahanandMyronTribushavebeenamajorobstacletotheawarenessinmedicine
that this theory is no mere addition to the traditional methods and methodologies, but a
novelWeltanschauungsteadilyandquietlybringingaboutaradicalrevolutionofscientific
thinking,reasoningandconceptandtheoryformationontheonehand,andoftechnology
andsociety,ontheother.Incontrasttoineffectiveattemptsinthepastbydifferentscholars
to criticize or refute one or the other of the Aristotelian principles in isolation, it
successfully terminates the whole Aristotelian paradigm that has been reigning over
scientificreasoningandhumancultureforthelast2300years.Inthissense,itrepresentsa
unique, unprecedented example of Thomas Kuhn’s account of scientific change by
paradigm shift [33].
The all-embracing paradigm shift caused by fuzzy theory that we are excitedly
witnessing is too far-reaching to allow any of the Aristotelian foundations to survive.
It,thus,exercisesanunfuzzybreakwithalong-standinganddeeplyentrenchedtradition.
Toassesstheimpactofthisdepartureonhumancultureandcivilizationingeneral,andon
medicine in particular, we will have to carefully consider what in this process is being
replaced with what.
2. Goodbye to the Aristotelian Weltanschauung
The Polish physician Ludwik Fleck published in 1935 a remarkable book in German
entitled Entstehung und Entwicklung einer wissenschaftlichen Tatsache, i.e. genesis and
developmentofascientificfact,whichwasneglectedbyhiscontemporaries[19,20].Like
Karl Popper’s influential monograph on ‘The Logic ofScientific Discovery’ launched in
thesameyear[40],theViennaCircle’slogical-empiricistviewonthenatureofscientific
K.Sadegh-Zadeh/ArtificialIntelligenceinMedicine21(2001)1–25 3
knowledge was its target.2 Unlike Popper and by a detailed historical-philosophical
analysis of research on syphilis, however, Fleck developed in his book a fascinating,
original,relativisticepistemologyaccordingtowhichscientificknowledgeisconstructed
bya‘thoughtcollective’,whichwewouldcall‘scientificcommunity’today,actinginthe
realm of aparticular‘thought style’. Viewed from within different thought styles, things
look different [19–22].
As Thomas Kuhn briefly acknowledges in the foreword to his acclaimed book on the
structureofscientificrevolutions([33],pp.vi–vii),Fleck’swidelyunrecognizedpublica-
tionhasservedasthemainsourceofinspirationofhisownparadigm-shifttheory.Kuhn’s
paradigmsareinfactFleck’sthoughtstyles.SinceKuhnhadusedhisnotionofaparadigm
extremelylooselyandreceivedvarious,incisivecriticisms[37,64],helatersubstitutedfor
itthenotionofa‘disciplinarymatrix’[34].Thedisciplinarymatrixinaparticularscientific
areaisinessencethesetofsharedbasicrules,methods,andbeliefswhichunderlietheory
formation and knowledge acquisition in this area.
Althoughalloftheseconceptsarestilltoovagueandinadequatetobeveryuseful,3we
may,nevertheless,learnfromthesestudiesthatincontrasttoouraccustomedviewsonthe
development of science and scientific knowledge, this very development is not a cumu-
lativeprocess.Sciencedoesnotprogresscontinuouslyandbyaccumulatingknowledge.It
does notadd toanantecedentknowledge ortheory T asubsequent knowledge ortheory
i
T ofthesametypesuchthatonecouldreasonablyconsiderscienceastheopen,ordered
i(cid:135)1
seriesofrelatedtheoriesT ;T ;...;T .Scientificideas,theories,andworldviewsevolve
1 2 i(cid:135)1
discontinuouslyinthatabodyofknowledgeortheoryT,whichisheldoveraparticular
i
period of time, is dislodged by another body of knowledge or theory T because the
j
disciplinary matrix within which the former theory T had grown, changes to another
i
disciplinary matrix which gives rise to the new theory, T, that is incompatible and
j
incommensurable with its predecessor T. For example, the Hippocratic and Galenic
i
humoral pathology rooted in the pre-anatomical era of antiquity considered illness as an
imbalanceoffourhumorsinthebody,i.e.bile,phlegm,bloodandurine,andlasteduntilthe
eighteenth century. After Andreas Vesalius’ anatomy, De Humanis Corporis Fabrica
(1543),andthethen-emergingearlyempiricismconceptualizedbyFrancisBaconandJohn
Lockehadmadeanovel,empirical-anatomicaldisciplinarymatrixavailablewithinwhich
illnessappearedtohavesomethingtodowithsolidpartsofthebody,humoralpathology
wasreplacedwiththelocalizedpathologyofDeSedibusetCausisMorborum(1761)by
GiovanniBattistaMorgagni.Bytheendoftheeighteenthcentury,itwascomplementedby
2TheViennaCircle(1923–1936)wasasmallgroupofphilosophers,naturalscientists,socialscientists,and
mathematicianssuchasMoritzSchlick,OttoNeurath,RudolfCarnap,HansHahn,FriedrichWaismann,Herbert
Feigl, Karl Menger, Kurt Go¨del and a few additional ones. It constituted the germ cell of a philosophical
movementdubbed‘logicalempiricism’propagatingtheviewthatknowledgeoftheworldisperceptionplus
logic(cf.[9,32]).LudwikFleck(1896–1961)was a Polishphysicianandmicrobiologistandlivedin Lwow,
Poland. In June 1941, hewas deported to the Jewish ghetto in his city and later, in December 1943, to the
concentration camp Buchenwald in Germany to do research on typhus serum. He survived and returned to
Polandin 1945andemigratedto Israel in 1957.He is widelyrecognizedtoday as the founderof thesocial
constructionisttheoryofscience.Foracomprehensiveaccountofhislifeandwork,see[11].
3Foranextensiveandinspiringexplicationandelucidation,see[65].
4 K.Sadegh-Zadeh/ArtificialIntelligenceinMedicine21(2001)1–25
Francois Xavier Bichat’s tissue pathology. After the development of the microscope had
enabled Theodor Schwann to discover the animal cell around 1838, localized pathology
wasreplacedwithRudolfVirchow’scellularpathology(1858),whichconsidereddiseases
ascellularchangesanddisorders.Withsomealterationsandadditions,thisviewhasbeen
dominating medicine since. We are currently witnessing the emergence of a competing
molecular pathology, e.g. genomics and pathobiochemistry, which explains and treats
diseases as molecular processes in the body. Maybe our descendants will encounter
quantum pathology or something like that in the near future [59].
Inthecourseofsuchadiscontinuousevolution,aprecedingtheoryT isabandonedanda
i
subsequenttheoryT isusedinstead.ThisissonotbecauseT wasdemonstratedtobefalse
j i
or T was shown to be true, but because concepts and methods have changed which are
j
employed in constructing and acquiring knowledge, i.e. theory T in the present case.
j
Our examples above and also comparable cases dealt with in the literature, such as
Newtonian mechanics or Einstein’s theory of relativity and the like, exemplify only
particulartheoriesandsingularpiecesofknowledge,eachofwhichemergesfromwithina
specificdisciplinarymatrix.Interestingly,however,therearealsodisciplinarymatrixesof
highergenerality,suchastheconceptoftwo-valuedness,whichareresponsiblenotonlyfor
the emergence of individual theories, but for the very mode of scientific thinking and
inquiry in all fields. At the highest level of generality we presently encounter, to our
surprise, a particular disciplinary matrix which has been nourishing all sciences and
theoriesforthelast2300years,i.e.theAristoteliandisciplinarymatrix,becauseitcontains
thetwo-valued,classicallogicwithwhichresearchersreasonanddefendtheirwork.What
is being eradicated by fuzzy theory is just this universal disciplinary matrix. One may,
therefore, easily imagine what may happen when this deeply entrenched Aristotelian
thought style will be replaced with the thought style of fuzzy theory. No conceptual
structure, method, knowledge, theory, and research program will survivewhich does not
accord with the new disciplinary matrix.
The foundations of the Aristotelian disciplinary matrix had been laid by Aristotle
himselfinhisMetaphysics,Organon,andDeInterpretatione.Theywouldconstituteover
more than two millennia the basic principles of classical reasoning in science, mathe-
matics, philosophy, religion, politics, law, ethics, and all other areas. The following six
passages A–F stated in the Metaphysics, together with a seventh concept, G, a modern
derivative from his logic, sufficiently represent these principles:
(A)Thiswillbeplainifwefirstdefinetruthandfalsehood.Tosaythatwhatisisnot,or
that what is not is, is false; butto say that what is is, and what is not is not, is true (The
Metaphysics, Book IV, 1011 b 26–27) [3].
(B) By demonstrativeI mean..., e.g. ‘‘everythingmustbe either affirmed or denied’’,
and ‘‘it is impossible at once to be and not to be’’ (ibid., B III, 996 b 27–30).
(C)...thatisthemostcertainofallprinciples.Letusnextstatewhatthisprincipleis.‘‘It
isimpossibleforthesameattributeatoncetobelongandnottobelongtothesamething
and in the same relation’’ (ibid, B IV, 1005 b 19–23).
(D) Nor indeed can there be any intermediate between contrary statements, but of
onethingwe musteither assert ordenyonething, whateverit maybe(ibid., B IV, 1011
23–24).
K.Sadegh-Zadeh/ArtificialIntelligenceinMedicine21(2001)1–25 5
(E) Further, an intermediate between contraries will be intermediate either as grey is
betweenblackandwhite,oras‘‘neithermannorhorse’’isbetweenmanandhorse(ibid.,B
IV,1011b29–32)....Again,therewillalsobeanintermediateinallclassesinwhichthe
negation of a term implies the contrary assertion; e.g. among numbers there will be a
numberwhichisneitheroddnornot-odd.Butthisisimpossible...(ibid.,BIV,1012a8–11).
(F) Again, unless it is maintained merely for argument’s sake, the intermediate must
existbesideallcontraryterms;sothatonewillsaywhatisneithertruenorfalse.Anditwill
existbesidewhatisandwhatisnot;sothattherewillbeaformofchangebesidegeneration
and destruction (ibid., B IV, 1012 a 5–8).
(G)Asetofpremiseslogicallyimpliesaconclusionif,andonlyif,wheneverthepremisesare
truetheconclusionistrue.PassageAisthecorrespondenceconceptoftruthprovidingthebasis
fortherest,andbeingtherootofthecorrespondencetheoryoftruthandofTarskisemanticsof
classical,two-valuedlogic.PassageBistheprincipleoftwo-valuedness.Thesecondhalfof
thispassageandpassageCarealternativeformulationsofthelawofnon-contradiction,which
saysthatastatementoftheforma^:aiscontradictory,nevertrueandshouldthereforebe
rejected,:(a^:a).PassagesBandDgivedifferentformsofthelawofexcludedmiddle,
a_:a,whichsaythatsuchastatementisatautologyandalwaystrue.
InpassageE,two-valuednessisdefendedonceagainbecauseotherwiseonewouldhave
tosupposethatthereare—sic!—classeswithoutsharpboundariesbetweenmembersand
non-members. But this is, Aristotle says, impossible. So he may be viewed as the
progenitor of Georg Cantor’s two-valued set theory.
PassageFinthesameveinrejectstheviewthattheremaybeanintermediatebetweenbeing
andnonbeing.ThiscommonlyheldAristotelianontologywillbereferredtobelowasthe
doctrineofcrispexistence.Allofussharethisontologicaldoctrine,sinceallofusbelieveand
assertthat‘everythingisorisnot,thereisnointermediatebetweenbeingandnonbeing’.
The closing definition, G, of the concept of logical implication, consequence, or
inference, has emergedfrom the modern reconstruction, completion, and axiomatization
of Aristotelian logic in the 19 and 20th centuries by Bernhard Bolzano, Gottlob Frege,
BertrandRussellandothers.Itrepresentsthebasicconceptofthetheoryofdeductionand
proof underlying all classical mathematics, science, and technology.
Therehavebeenvariousattemptsinthepasttoinvalidateoneoranotheroftheprinciples
or concepts A–G above in isolation. None of them has attained general acceptance,
however.Forexample,many-valuedlogicsruncountertotheprincipleoftwo-valuedness.
Intuitionisticlogicdoesnotacceptthelawofexcludedmiddle.Paraconsistentlogicsreject
the law of non-contradiction.4
4Paraconsistentlogicsareinconsistencytolerantsystemsoflogicwhichdonotcontainaprincipleofnon-
contradiction.Theyoriginatedaround1910withtheRussianphysicianNikolajA.Vasiliev(1880–1940),whoat
thebeginningofthe 20thcenturytaughtphilosophy attheUniversityofKazan,Russia.Inspired byNikolaj
Lobachevski’snon-EuclideangeometriesinwhichtheEuclideanparallelpostulateisnotvalid,heattemptedto
constructnew,‘ImaginaryLogics’bydiscardingsomeofthebasiclawsofclassicallogic[4,70,71].Theselogics
would enable us to study a large class of ‘imaginary worlds’ that are impossible to classical logic, but
nevertheless,quitewell imaginable. After Stanislaw Jaskowski’sinterlude[29],specificresearchin thisnew
field of non-classical logics was initiated by the significant work of the Brazilian logician and philosopher
NewtonC.A.daCosta[12,13].Theterm‘paraconsistentlogic’wascoinedbythePeruvianphilosopherF.Miro
Quesadain1976.Foracomprehensiveaccountofthesubject,see[25,41,50].
6 K.Sadegh-Zadeh/ArtificialIntelligenceinMedicine21(2001)1–25
However,thelethalcollapseofthe Aristoteliandisciplinarymatrix iscausedbyfuzzy
theory in that all fundamental principles and concepts A–G above are being removed at
once in the following way.
The treatment of truth in fuzzy theory as a many-valued linguistic variable with a
colorfulandinvigoratingtermsetsuchas{true,nottrue,verytrue,completelytrue,more
or less true, fairly true, false, very false,..., etc....}, and the treatment of these terms as
labelsoffuzzysetsovertheunitinterval(see[85–87]),isaningeniousandhighlyesthetic
dethronement of all existing theories of truth and of all simplistic semantics, including
Aristotele’s, Tarski’s, Carnap’s, and Kripke’s perspectives. It goes without saying that
wheneverthesimplisticconceptoftruthislost,everythingdependentwillalsovanish.That
means that following the fall of A above, B–G will automatically collapse. Fortunately,
thereisacompletesubstituteforallofthat,thefuzzytheory,whichiscapableofreigning
immediately as the new disciplinary matrix. Its availability as a more than perfect
substitute is, thus, the reason of its success.5
As a substitute, the fuzzy theory is a many-valued conceptual system. As stated in B
above, a consequence of this removal of the Aristotelian two-valuedness is that the
principlesofexcludedmiddleandnon-contradiction,B–Dabove,arealsopasse´ because
theunionofafuzzysetXanditscomplementXcisnotnecessarilythebaseset,andtheir
intersection is not necessarily empty, i.e. X[Xc 6(cid:136)O and X\Xc 6(cid:136)?.
It took approximately 2300 years to refute the Aristotelian position against fuzziness
statedinEaboveandtodemonstratethatthereare,first,intermediatesbetweenmembers
andnon-membersofaclass,andsecond,numberswhichareneitheroddnornot-odd,e.g.
the fuzzy number ‘approximately 2300’.
Thus far, the overall effect is an implosion of the classical concept of two-valued
inferencesketchedinGaboveand,asaconsequence,thedeceaseofalltwo-valuedlogics.
WewillreturntothispointinSection4.Letusfirstsupplementthenewparadigmwitha
novel facet, invalidating F above, to complete our goodbye to A–G above.
3. Fuzzy ontology
Asalreadyindicated,thehumanreasonseemstoholdatwo-valuedontologyreflectedin
thecommonbeliefthat‘everythingisorisnot,thereisnointermediatebetweenbeingand
nonbeing’.ThisAristoteliandoctrineofcrispexistencecitedinFabovethathasgoverned
Westernphilosophy,logic,science,religion,andcommonsenseforthelasttwomillennia,
nolongerapplies.Tosketchthisoutlookwewilladdtotheclassoffuzzyquantifiersthe
fuzzy existence operator ‘‘there is to some extent’’, denoted by 90. The sentence 90x(Px)
thusreadsthereistosomeextentanxsuchthatxisP.LetPbeanypredicatethatsignifiesa
corresponding set P with the membership function m . Then we may define:
P
5Merecriticismsofadisciplinarymatrixextendedfromwithinanemptyholewillneversufficetoforcea
shiftbecausethe questionwillarise: a shifttowhere?Forexample, wehave as yet encounteredmanysuch
vacuous attempts in medicine, e.g. homeopathy and psychosomatics, which have beenvainly trying to shift
modernmedicinefromitsnatural–scientificconceptsandmethodsto:where?Nobodyknows.
K.Sadegh-Zadeh/ArtificialIntelligenceinMedicine21(2001)1–25 7
Definition 1. 90x(cid:133)Px(cid:134) if and only if 9r(cid:133)m (cid:133)x(cid:134)(cid:136)r).
P
Anobjectxwhosegradedexistenceisassertedbyastatementoftheform90x(Px),isa
fuzzyobject.AnexampleisDavid.Heisyoungtotheextent0.7.Thus,thedefinitionabove
implies that there is to some extent someone who is young. The proposition that to the
extent 0.2 the earth is a big planet provides a second example, i.e. the fuzzy object the
earth. Elementary particles of quantum physics are additional examples which Werner
Heisenberg has tried to grasp by his uncertainty principle.
Obviously, any fuzzy set defines fuzzy objects. The fuzzy existence of such an object
maybemeasuredbyanewoperator 90 thatisread‘relativetoPthereistotheextentr’.
P r
Accordingly,theexpression 90x(cid:133)Px(cid:134)meansthatrelativetoPthereistotheextentranx
P r
suchthatxisP.InplaceofDefinition1,thismetricexistenceoperatormaybedefinedas
follows:
Definition 2. 90x(cid:133)Px(cid:134) if and only if 9r(cid:133)m (cid:133)x(cid:134)(cid:136)r(cid:134).
P r P
WithregardtoourfirstexampleabovewhichsaidthatDavidisyoungtotheextent0.7,
this new concept implies that relative to the predicate young there is to the extent 0.7
someone who is young, i.e. David.
According to this new concept, a fuzzy object exists only relative to a particular
predicateandtoaparticularextentbetween1and0.Thetransitionfrombeingtononbeing
andviceversaisthusgradualratherthanabrupt.Pantarei.Everythingexiststoanextent
r (cid:20)1.Ifthisr (cid:136)0forallpredicatesofaparticularlanguage,theobjectdoesnotexistwith
respect to this language. However, it may exist with respect to another language. That
means,first,thatalanguageinducesanontology,andsecond,thatbeingandnonbeingis
relativetolanguagesandlogics.‘‘Changeyourlanguageorlogic,andyouwillseeanother
world’’ ([48], p. 171).
Thisamountsroughlytosayingthattobeistobeafuzzyobject,i.e.amemberofafuzzy
set within a particular language [50].6
4. Fuzzy proof theory
Ithasbeensaidabovethatwithinthenewdisciplinarymatrixthetwo-valuedconceptof
classical–logicalinferenceimplodes.Asaresult,classicallogicceasestoserveasameans
of reasoning in science and technology. This decease is not only due to the above-
mentionedimplosion.Itisalsocausedbytheunparalleledpoweroffuzzylogictocopewith
vaguenessinthewidestsense,i.e.withpartiallytruestatements,fuzzyconnectives,fuzzy
predicates, predicate modifiers, and fuzzy quantifiers.
6Viewedfromthisperspective,Quine’swell-knownaccountofontologywhichsaysthat‘‘tobeistobethe
valueofavariable’’([42],p.15)appearssomewhatsimplistic.Indevelopingmyontologyovertheyears,Ihave
profitedfromdiscussionswithmyfriendProfessorNewtonC.AffonsodaCosta,UniversityofSaoPaulo,and
fromhispublicationsandmanuscripts,e.g.[14].
8 K.Sadegh-Zadeh/ArtificialIntelligenceinMedicine21(2001)1–25
Capable of dealing with these and other vague ingredients of natural languages and
commonsense reasoning, fuzzy logic started with the doctrine of being a logic of
approximate reasoning based on rules of inference whose validity is approximate rather
thanexact(cf.[83–87,90,5]).Althoughithasbecomeatrulystrongtheoryofreasoningand
controlinthemeantime,itsinitialdoctrine continues tobewidelymisunderstood.Many
traditionallogiciansandmathematiciansarestillhesitantabouttakingalogicseriouslythat
reliesonapproximaterulesofinference(cf.,e.g.[27,28]).Thisissobecauseithasnotyet
beenproperlyascertainedwhatisactuallybroughtaboutbyfuzzylogic.Oneconsidersits
surface structure and overlooks the processes in its deep structure. In the depth offuzzy
logicthefollowingdevelopmentmaybeobservedwhichrepresentsaphilosophicallymost
intriguing aspect of the revolution.
WehavealreadystatedthattheAristoteliantwo-valuednessisbeingreplacedwiththe
fuzzy-theoretical many-valuedness. This substitution is vividly reflected in the fuzzy-
theoretical treatment of truth as a many-valued linguistic variable. However, this many-
valuednessnotonlyconcernsfactualtruthsandfalsehoodswhenthetruthorfalsehoodofa
statementsuchas,forexample,‘snowiswhite’isconsidered,butalsoaffectstheconcept
oflogicaltruththatcharacterizesthemetalinguisticrelationofclassical-logicalinference
sketchedinGinSection2.Thatmeansthatinthedepthoffuzzytheorythereisahidden,
metalinguistic concept of many-valuedness, i.e. inferential many-valuedness, that will
moreandmoreemergeasafuzzyprooftheoryinthenearfuture.Withthisprooftheoryat
its disposal, fuzzy theory will conquer all mathematical sciences. To shed some light on
this imminent course, consider the following, prototypical fuzzy-logical inference:
aissmall (cid:17)statementa
1
aandbareapproximatelyequal (cid:17)statementa
2
bismoreorlesssmall (cid:17)statementb
Inferencesofthistypeareusuallybaseduponrulessuchascompositionalruleofinference
orgeneralizedmodusponensorsomethingelse,andarecommonlyviewedasimprecise.
Andsincewhatisimpreciseisnotgood,fuzzylogicisnotgood.Averdictofthiskindis
strange and wrong simply because it evaluates a fuzzy inference as above from the
outmoded perspective of two-valued logic where a statement b either definitely is, or
definitely is not, a consequence of a set S(cid:136)fa ;...;a g of premises:
1 n
A set S(cid:136)fa ;...;a g of premises classical–logically implies a conclusion b iff
1 n
whenever the premises are true b is true.
According tothisclassical-logicalconceptofinference, thesetofallconsequencesof
thepremisesS,denotedbycons(S),isthesetofallstatementsthatareclassical-logically
implied by S, i.e.
cons(cid:133)S(cid:134)(cid:136)fbjSclassical(cid:255)logicallyimpliesbg
The set cons(S) is crisp. This means that given a set S of n(cid:21)1 premises a ;...;a ,
1 n
traditionally a particular statement b is considered to be implied by S if it is true that
b2cons(cid:133)S(cid:134),whileitisconsiderednottobeimpliedbySifitistruethatb2=cons(cid:133)S(cid:134).And
sinceitisnotevidentifastatementsuchas‘bismoreorlesssmall’aboveisorisnotan
K.Sadegh-Zadeh/ArtificialIntelligenceinMedicine21(2001)1–25 9
element of the set cons({a is small, a and b are approximately equal}), the inference is
labeled ‘imprecise’, ‘vague’, or ‘uncertain’. A remedy for this perpetual mistake is
provided by Zadeh’s fuzzifiability principle:
Any theory, X, may be fuzzified by replacing the concept of a set in X by the more
general concept of a fuzzy set ([98], p. 816).
Thus,letusfuzzifythesetofconsequencesofastatementset,cons(S).Thistaskmaybe
accomplished by constructing a membership function m that assigns a number
cons(S)
m (cid:133)b(cid:134)2(cid:137)0;1(cid:138) to a statement b as the degree of its belonging to the set cons(S).
cons(cid:133)S(cid:134)
As a result, we obtain a concept of graded inference that we also call graded logical
implication, graded deduction, or graded reasoning (cf. [45]).
LetSbethesetofallsyntacticallyadmissiblesentencesofaparticularlanguage.Acrisp
logic L over this language is a mapping of the form
) :2S(cid:2)S!f0;1g
L
with) beingitsimplicationoperatorand2SthepowersetofSsuchthatforallS22Sand
L
b2S we have ) (cid:133)S;b(cid:134)(cid:136)1 if according to logic L the statement b follows from the
L
statement set S, and ) (cid:133)S;b(cid:134)(cid:136)0, otherwise. By contrast, a fuzzy logic FL over that
L
language is a generalized mapping of the form
) :F(cid:133)2S(cid:134)(cid:2)S!(cid:137)0;1(cid:138)
FL
withF(2S)beingthefuzzypowersetofSsuchthat) (cid:133)S;b(cid:134)(cid:136)r saysthataccordingto
FL
logicFLthestatementbfollowsfromStotheextentr 2(cid:137)0;1(cid:138).Theavailabilityofsucha
fuzzy implication operator, ) , would enable us to define our envisaged membership
FL
function m above as follows:
cons(S)
m (cid:133)b(cid:134)(cid:136)riff ) (cid:133)S;b(cid:134)(cid:136)r:
cons(cid:133)S(cid:134) FL
And that means
m (cid:133)b(cid:134)(cid:136)) (cid:133)S;b(cid:134):
cons(cid:133)S(cid:134) FL
TheextenttowhichastatementbisamemberoftheconsequencesofSis,thus,thedegree
of its deducibility from S. We would in this way obtain the fuzzy set cons(S) of
consequences of the premises S:
cons(cid:133)S(cid:134)(cid:136)f(cid:133)b;r(cid:134)jr (cid:136)m (cid:133)b(cid:134)g(cid:136)f(cid:133)b;r(cid:134)jr (cid:136)) (cid:133)S;b(cid:134)g:
cons(cid:133)S(cid:134) FL
What fuzzy logic is actually doing will give birth just to such a graded concept of
deducibility creating a quantitative logic that may be called a fuzzy proof theory. A
promisingapproachtothisgoalistheinterpretationoftherulesoffuzzyinferenceasrules
of fuzzy constraint propagation [97,100].7
7Itistemptingtointerpretfromthisperspectivethedevelopmentoflogicasahistoryofrecognizingand
handlingthephenomenonoffuzzyconstraintpropagation.Theconceptofinferenceinalltraditional,deductive
logics is a truth preserving relation between premise and conclusion. And that means that all traditional,
deductive logics are theories of the propagation of the single constraint ‘‘true’’. But there are innumerable
constraints, including the entire term set T(cid:133)truth(cid:134)(cid:136)ftrue;nottrue;verytrue;quitetrue;false;...g of the
linguistic variable truth itself, whose propagation is being studied in fuzzy logic. Also Rudolf Carnap with
hisprobability-basedinductivelogicmayhavehadsomekindofaquantitativelogicinmind[10].
10 K.Sadegh-Zadeh/ArtificialIntelligenceinMedicine21(2001)1–25
Whiletheconceptofinferenceofthequantitativelogicaboveisanumericalvariable,a
qualitativefuzzylogicwouldemergefromtreatingthenotionofinferenceasalinguistic
variable. For example, let the sentence ‘S implies b’ be written in prefix notation as
implies(S, b). Here, ‘implies’ is a binary linguistic variable. A possible term set for this
linguistic variable, T (implies), is exemplified by the following arrangement:
T(cid:133)implies(cid:134)(cid:136)fabsolutely;strongly;quitestrongly;verystrongly;notstrongly;
weakly;veryweakly;...etc:...g:
Inclinicalmedicine,forinstance,aqualitativefuzzylogicwouldenablethephysicianto
judge to what extent the patient’s data implies a particular diagnosis. For example, the
result of such a diagnostic reasoning may be:
implies(cid:133)patient data[KB; thepatienthasdiabetes(cid:134)(cid:136)strongly
implies(cid:133)patient data[KB; thepatienthashepatitis(cid:134)(cid:136)veryweakly
wherepatient_datainthepremisescompriseacollectionofstatementsaboutthepatient’s
complaints,symptoms,andsigns,andKBistheknowledgebaseused.Alternatively,the
implication relation may also be constructed as ‘b is implied by S’, expressed by the
linguistic variable is_implied(S, b) with the term set:
T(cid:133)is implied(cid:134)(cid:136)fhigh;medium;low;veryhigh;verylow;...etc:...g:
In this case, the physician may arrive, for example, at the clinical judgment:
is implied(cid:133)patient data[KB; thepatienthasdiabetes(cid:134)(cid:136)veryhigh;
is implied(cid:133)patient data[KB; thepatienthashepatitis(cid:134)(cid:136)quitelow:
Ineithercase,wewouldinclinicalreasoningbecomeabletoexploitthepowerfultheories
of linguistic variables, information granulation, and computing with words [81,82,85–
89,95,97,99,100].
5. Fuzzy-theoretical worldmaking
The development and application offuzzy theory is steadily accelerating. There is no
doubtthatinafewyearsthetheorywillconstitutetheuniversaldisciplinarymatrixinall
areas of science, humanities, technology, and human reasoning. The three main pillars
uponwhichitwasinitiallybuiltandstillrestsandgrowsarefuzzysettheory,thetheoryof
linguisticvariables,andfuzzylogic.AswasalreadyreferredtoinSection1,themotivation
behindthisinnovationhasbeentheperceptionthatforcopingwiththeanalysisofhighly
complex systems such as the animate ones ‘‘we need a radically different kind of
mathematics, the mathematics of fuzzy or cloudy quantities...’’ ([76], p. 857).
Asamathematicallyorientedsystemtheorist,Ihadbeenconditionedtobelievethat
theanalyticaltoolsbasedonsettheoryandtwo-valuedlogicwereallthatwasneeded
to build a framework for a precise, rigorous and effective body of concepts and
techniques for the analysis of almost any kind of man-made or natural system,