03. induction and confirmation 1. induction 2. confirmation
TRANSCRIPT
03.InductionandConfirmationTopic:Relationbetweentheoryandevidence.
1.Induction
Whatdoes"induction"referto?• Twotypesofargument:
Allmenaremortal.Socratesisaman.∴Socratesismortal.
?
- InitialResponse:Inthepast,thefuturehasresembledthepast.Soshouldn'tweexpectittocontinuetodothis?
- But:Thisiscircular!Itinfersfutureconsequencesfrompastconsequences;andthisiswhatisbeingquestionedinthefirstplace!
ProblemofInduction:Whatreasondowehaveforthinkingthefuturewillresemblethepast?
DavidHume(1711-1776)
Deductiveargument:Ifpremisesaretrue,conclusionmustbetrue.
1.Induction2.Confirmation
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Inductiveargument:Truthofpremisesdoesnotguaranteetruthofconclusion.
(a) EnumerativeInduction.Inferencefromafinitenumberofobservationstoageneralization.
(b) Projection.Inferencefromfinitenumberofobservationstothenextcase.
Swan#1observedattimet1 iswhite.Swan#2observedattimet2 iswhite.⋮
∴Thenextswanobservedwillbewhite.
90%ofsmokersgotlungcancer.∴Smokingcauseslungcancer.
!!
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(c) Explanatoryinference.Inferencefromobservationstoahypothesisthatbestexplainsthem.
• Isoneformofinductionmorefundamentalthantheothers?• Canalogicofinductionbeconstructed?
Dinosaurextinctionevent65millionyearsago.HighlevelsofiridiuminlayersofEarth'scrustfrom~65millionyears.Iridiumiscommonlyfoundinmeterorites.ImpactcrateroffYucatanpenninsuladatesto~65millionyears∴ AgiantmeteorimpactedtheEarth65millionyearsagocausingtheextinctionofthedinosaurs.
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2.Confirmation
(a)Hypothetico–Deductivism(HD)
GivenahypothesisH,Step1. DeriveapredictionE fromH.(deductiveinference)Step2. Testtheprediction.Step3. IfE istrue,thenH is"confirmed".
IfE isfalse,thenH is"disconfirmed".inductiveinference
Basicidea:"E confirmsH"means"H entailsE,andEistrue".HDmodelsconfirmationonentailment.
• Isthisthe"ScientificMethod"?
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TwoProblemswithHD
• But:ToderiveapredictionE fromH,weneedadditionalassumptions.
• Which ofH,A1,A2,...doesE confirm?
GeneralformofHDreasoning:
IfH istrue,thenE istrue.E istrue.∴H isconfirmed.
If(H &A1 &A2 &...)aretrue,thenE istrue.E istrue.∴(H &A1 &A2 &...)isconfirmed.
ImprovedformofHDreasoning:
Problem1.Duhem-QuineProblem
PierreDuhem(1861-1916)
WillardQuine(1908-2000)
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Problem2.ConfirmationasEntailmentistooweak!
Claim1:AnytrueobservationS HD-confirmsanyhypothesisT.
Claim2:IfE HD-confirmsT,thenE HD-confirmstheconjunctionofT withanyotherhypothesis.
- LetT beanyhypothesis,andS beanysentence.
- IfT entailsE,thenT &S entailsE foranyS.
- Then:T entailsT-or-S (wheneverT istrue,soisT-or-S ).- And:IfS istrue,thensoisT-or-S.- So:IfS istrue,thenT entailsT-or-S,andT-or-S istrue.
- So:IfE HD-confirmsT,thenE HD-confirmsT &S.- But:Thismakesconfirmationtooeasy!LetT =Newtonianmechanics,S =Creationism,E ="Theorbitsoftheplanetsareellipses".
"EHD-confirmsH"means"HentailsE,andEistrue".- Thus:IfS istrue,thenT-or-S HD-confirmsT.
- But:Thismakesconfirmationtooeasy!LetT =specialrelativityandS ="Therearemiceinmycupboard".
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(b)InstanceConfirmation
Basicidea:"E confirmsH"means"E isaninstance ofH".
Notionofan"instance":- Assumethatallhypothesesinscienceareoftheform"AllFsareGs".- Aninstance ofahypothesisisthenanF thatisalsoaG.
Problem:"RavensParadox"
• "AllFsareGs"islogicallyequivalent to"Allnon-Gsarenon-Fs".- Whenever"AllFsareGs"istrue,sois"Allnon-Gsarenon-Fs",andvice-versa.
Ex: H =Allravensareblack.E =Ablackraven.
• Whichmeans:Awhiteshoeinstance-confirms"Allravensareblack"!
• So:Anon-G thatisanon-F instance-confirms"Allnon-Gsarenon-Fs",andthusitinstance-confirms"AllFsareGs"!
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InitialResponse:Bitethebullet(Hempel'sresponse)• "Allravensareblack"means"Ifit'saraven,thenit'sblack",whichisaclaimabouteverything intheuniverse.
• So:Awhiteshoedoes instance-confirmit,althoughveryminutely.
Twomoreresponses1. Whetherornotaninstanceconfirmsahypothesismaydependonotherfactors.
(i) Allravensareblackandtheyareextremelyrare.(ii) Allravensareverycommon,mostareblack,andafewarewhite.
• But:Awhiteshoeisalsoanon-blue,non-aardvark,soitalsoinstance-confirms"Allaardvarksareblue".
CarlHempel(1905-1997)
• Ablackravenobservedoutsideyourwindow(acommonsighting)willconfirm(ii)butnot(i).
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2. Whetherornotaninstanceconfirmsahypothesisdependsonthepotentialfortheinstancetorefute thehypothesis.
You:Yes!(Yourhypothesisis"Allravensareblack".)
You:No!(Yourhypothesisisnot"Allblackthingsareravens".)
Ihaveablackthing.Wanttoseeifit'saraven?
Ihavearaven.Wanttoseeifit'sblack?
• Whatifit'sablackraveninbothcases?• Whetheritconfirmsyourhypothesisdependsontheorderinwhichyoudiscoveritsproperties!
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• Whataboutthatwhiteshoe?
You:Yes!(Ifit'saraven,thenyourhypothesisisdoomed!)
You:No!(Yourhypothesisisnot"Allblackthingsareravens".)
Ihaveashoe.Wanttoseeifit'swhite?
Ihaveawhitething.Wanttoseeifit'sashoe?
Moral:Someblackravenobservationsconfirm"Allravensareblack".Somewhiteshoeobservationsconfirm"Allravensareblack".Others,inbothcases,don't!
Generalmoral:- Observationsarenot"automatically"relevanttohypotheses.- Whetherornottheyarereleventmaydependontheirorderandonotherinformation.
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TheSelectionTask
A B
C D
Hint:"Ifthere'sacircleontheleft,thenthere'sacircleontheright"islogicallyequivalentto"Ifthereisn'tacircleontheright,thenthereisn'tacircleontheleft".
• Whichmasksneedtoberemovedtotestthetruthof"Ifthere'sacircleontheleft,thenthere'sacircleontheright"?
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Goodman'sNewRiddleofInduction
Claim:Therecanbenoformal theoryofconfirmation.
• Idea:Deductivelogicisthelogicofargumentforms:
Allmenaremortal.Socratesisaman.ThereforeSocratesismortal.
?
AllFsareGs.a isanF.Thereforea isaG.
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Goodman'sclaim:Confirmationcan'tsimilarlybeanalyzedattheformallevel.
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Def.1. grue =greenifobservedbefore2022,orblueifnotobservedbefore2022.
• Question:Areall emeralds(thosethathavebeenobservedbefore2022andthoseyet tobeobservedafter2022)grue?
• Manythingsaregrue:
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• But:(A)seemslikeastronginductiveargument.• (B)seemsweak:Shouldwebelievethatemeraldswe'venotobservedpriorto2022willbeblueifobservedafter2022?
• Whatiswrongwith(B)?
(A) Allobservedemeraldspriorto2022havebeengreen.Thereforeallemeraldsaregreen.
(B) Allobservedemeraldspriorto2022havebeengrue.Thereforeallemeraldsaregrue.
• (A)and(B)havethesameform:
Allobservedemeraldspriorto2022havebeenG.ThereforeallemeraldsareG.
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(1) Agoodtheoryofinductionshouldn'tusetime-indexedwordslike"grue".
• But:Whetherornotawordistime-indexedislanguagerelative.
Def.1. grue =greenifobservedbefore2022,orblueifnotobservedbefore2022.
Def.2. bleen =blueifobservedbefore2022,orgreenifnotobservedbefore2022.
• InEnglish,"grue"and"bleen"aretime-indexed,and"green"and"blue"aren't.• Butwhycan'ttherebeanotherlanguage,Blinglish,inwhich"grue"and"bleen"areprimitive,and"green"and"blue"aretime-indexed?
Def.3. green =grueifobservedbefore2022,orbleenifnotobservedbefore2022.
Def.4. blue =bleenifobsevedbefore2022,orgrueifnotobservedbefore2022.
• HowcouldweknowtodaywhetherwespeakEnglishorBlinglish?15
(2) Maybethewordsweusearen'ttheproblem.Maybeit'sthepropertiestheyreferto.
• Greennessisanaturalproperty:itpicksouta"naturalkind"innature.• Gruenessisunnatural:itdoesn'tpickoutanaturalkind.
• But:Howdowecometohaveknowledgeofnaturalkinds?- elements?(hydrogen,helium,lithium,etc.)- elementaryparticles?(electron,neutrino,quark,etc.)- biologicalspecies?- economieswithveryhighinflation?- mentaldisordersintheDSMIV?
• Problemofidentifyingtherightcategoryforpredictionandextrapolation.
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