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.

?

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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