Home technology Inductive reasoning

# Inductive reasoning

## Definition

Forexample:inaplane,thesumoftheinternalanglesofarighttriangleis180degrees;thesumoftheinternalanglesofanacutetriangleis180degrees;thesumoftheinternalanglesofanobtusetriangleis180degrees;arighttriangle,anacutetriangleAndobtusetrianglesarealltriangles;therefore,thesumoftheinternalanglesofalltrianglesintheplaneis180degrees.

Inthisexample,fromtheindividualknowledgethattheinternalanglesofrighttriangles,acutetriangles,andobtusetrianglesare180degrees,respectively,thegeneralconclusionthat"thesumofalltriangleinternalanglesis180degrees"belongstoInductivereasoning.

Moderninductivelogicmainlystudiesprobabilisticreasoningandstatisticalreasoning.Thepremiseofinductivereasoningisthenecessaryconditionofitsconclusion.

Secondly,thepremiseofinductivereasoningistrue,buttheconclusionmaynotbetrue,butmaybefalse.Ifarabbithitsatreeanddiedonacertainday,itcanbeconcludedthateverydayrabbitshitatreeanddie.Thisconclusionislikelytobefalse,unlesssomeveryspecialcircumstancesoccur,suchasanabnormalityinthegeographicalenvironment.Makestherabbitmusthitthetreeasfastaspossible.

Wecanusethestrengthofinductiontoillustratethedegreeofsupportofthepremisetotheconclusionininductivereasoning.Ifthesupportislessthan50%,thereasoningissaidtobeweakininduction;ifthesupportislessthan100%butgreaterthan50%,thereasoningissaidtobestrongininduction;ininductivereasoning,onlythepremiseofcompleteinductivereasoningsupportstheconclusionto100%,100%supportisinevitablesupport.

Thegeneralcalculationformofmathematicallogicofinductivereasoningis:s1⊆p+s2⊆p+s3⊆p+〈n〉(s⊆p)=∀×(s⊆p).

## Comparisonofdeductivereasoning

Inductivereasoninganddeductivereasoningarebothdifferentandconnected.

### Difference

1,thethinkingprocessisdifferent.

Thethinkingprocessofinductivereasoningisfromtheindividualtothegeneral,whilethethinkingprocessofdeductivereasoningisnotfromtheindividualtothegeneral,itisaninevitablethinkingprocess.

Deductivereasoningisnotfromtheindividualtothegeneralreasoning,butitisnotonlyfromthegeneraltotheindividualreasoning:deductivereasoningcanbefromthegeneraltothegeneral,suchas"Allunjustwarsareunpopular"Introduce"Allunjustwarsarenotpopular";

Itcanbefromindividualtoindividual,forexample,from"RogierBaconisnottheBaconwhoestablishedanewinductivelogictheory"tointroduce"TheestablishmentofanewinductionBaconinlogicaltheoryisnotRogerBacon";itcanbefromindividualandgeneraltoindividual,suchasfrom"thisobjectisnotconductive"and"allmetalsareconductive"todeducedthat"thisobjectisnotmetal";itcanalsobefromindividualandgeneraltoIngeneral,forexample,from"youarecompetentforthisjob"and"wherethereisawill,youcandoitorifyouarenotcapableofthisjob",theintroductionof"wherethereisawill,youcandoit."Here,weshouldpayspecialattentiontothefactthatthethoughtprocessofcompleteinductivereasoningininductivereasoningisnotonlyfromindividualtogeneral,butalsoinevitable.

2,therequirementsfortheauthenticityofthepremisearedifferent.

Inductivereasoningrequiresmajorpremises,andminorpremisesmustbetrue.Deductivereasoningdoesnothavethisrequirement.

3,thescopeofknowledgedeterminedbytheconclusionisdifferent.Theconclusionofdeductivereasoningdoesnotexceedthescopeofknowledgedeterminedbythepremise.Inductivereasoningexceptforcompleteinductivereasoning,theconclusionsarebeyondthescopeofknowledgedeterminedbythepremise.

4,thedegreeofconnectionbetweenthepremiseandtheconclusionisdifferent.Theconnectionbetweenthepremiseandtheconclusionofdeductivereasoningisinevitable,thatistosay,ifthepremiseistrueandthereasoningformiscorrect,theconclusionmustbetrue.Inductivereasoning,exceptthattheconnectionbetweenthepremiseandtheconclusionofthecompleteinductivereasoningisinevitable,theconnectionbetweenthepremiseandtheconclusionisprobable.

### Contact

1,ifdeductivereasoningisbasedongeneralknowledge,(deductivereasoningdoesnotnecessarilyhavetobebasedongeneralknowledge),itusuallydependsoninductivereasoningToprovidegeneralknowledge.

2,Inductivereasoningisinseparablefromdeductivereasoning.First,inordertoimprovethereliabilityofinductivereasoning,itisnecessarytouseexistingtheoreticalknowledgetoanalyzetheindividualpremisesofinductivereasoningandgraspthecausalityandinevitability.Thisrequiresdeductivereasoning.Second,inductivereasoningreliesondeductivereasoningtoverifyitsownconclusions.Forexample,theRussianchemistMendeleevdiscoveredtheperiodiclawofelementsthroughinduction,andpointedoutthatthepropertiesofelementschangeperiodicallywiththeincreaseoftheatomicweightoftheelements.Later,itwasdiscoveredbydeductivereasoningthattheatomicweightsofsomeelementsthatwereoriginallymeasuredwerewrong.Soherearrangedtheirpositionsintheperiodictableandpredictedsomeundiscoveredelements,andpointedoutthatblankspacesintheperiodictableshouldbeleftfornewelementsthathavenotbeendiscovered.

Therehavebeentwoopposingfactionsinthehistoryoflogic-thefullinductivefactionandthealldeductivefaction.Theall-inductiveschoolregardsinductionastheonlyscientificwayofthinkinganddeniestheroleofdeductionincognition.Thefulldeductionschoolsaidthatdeductionistheonlyscientificwayofthinking,denyingthemeaningofinduction.Bothoftheseviewsareone-sided.AsEngelssaid:"Inductionanddeduction,justlikeanalysisandsynthesis,areinevitablyrelatedtoeachother.Oneshouldnotbesacrificedandtheothershouldbelifteduptothesky.Eachshouldbeusedwhereitshouldbeused,butToachievethis,wecanonlypayattentiontotheirmutualconnectionandtheirmutualcomplement."

## Arrangementmethod

Experiencematerialsobtainedthroughobservation,experimentationandothermethodsneedtobeprocessedandarranged,Inordertoformascientificconclusion.Themethodsoforganizingempiricalmaterialsincludecomparison,classification,analysisandsynthesis,andabstractionandgeneralization.

### Comparison

Comparisonisamethodtodeterminethecommonanddifferentpointsofobjects.Throughcomparison,boththesimilaritiesbetweenobjectscanbeunderstood,andthedifferencesbetweenobjectscanbeunderstood,soastoprovideabasisforfurtherscientificclassification.Usingthemethodofcomparison,itisimportanttorecognize"sameness"fromobjectsthatareextremelydifferentonthesurface,ortorecognize"differentness"fromobjectsthatarethesameorsimilaronthesurface.AsHegelsaid:"Ifapersoncanseetheobviousdifferenceatpresent,forexample,candistinguishapenfromacamel,wewouldnotsaythatthispersonhasgreatintelligence.Similarly,ontheotherhand,apersonCancomparetwosimilarthings,suchasoakandlocusttree,ortempleandchurch,andknowingthattheyaresimilar,wecannotsaythathehasahighabilitytocompare.WhatwerequireistobeabletoseethesimilaritiesanddifferencesDifferencesinthesame."

Whenmakingcomparisons,youmustpayattentiontothefollowingtwopoints:

(1)Compareunderthesamerelationship.Inotherwords,theobjectsarecomparable.Ifyoutrytomakecomparisonswithincomparablethings,youwillmakethemistakeof"comparisonwith".Forexample,thelengthofwoodisthelengthofspace,andthelengthofnightisthelengthoftime.Thetwocannotbeshorterthanthelength.

(2)Selectandformulateaccurateandstablecomparisonstandards.Forexample,biologicalspecimensarewidelyusedinbiology,andmineralspecimensarewidelyusedingeology.Theyareusedtoidentifydifferentspeciesoforganismsandminerals.Thesespecimensarethestandardforcomparison.Thestudyofmeteoritesorlunarrockmaterialscollectedbylandingonthemoonalsocomparesthemwithorespecimensontheearth.

### Categorization

(1)Thedirectionofthinkingprocessisdifferent.Lexicalitemsaredividedintosmallercategoriesfromlargercategories.Theclassificationistheopposite.Itstartsfromtheindividual,risestothecategory,andthenrisestothemoregeneralcategory.

### AnalysisandSynthesis

Analysisisto"decomposethingsintosimpleelements".Synthesisis"combination,combination,andimprovisationtogether".Inotherwords,thethingsaredecomposedintocomponentsandelements,andaftertheresearchisclear,theycanbeputtogethertoshowthethingsinanewimage.Thisisthemethodofanalysisandsynthesis.

Forexample,toanalyzethestructureofanEnglisharticle,firstgetsentences,words,andfinallyget26letters;conversely,synthesisiscomposedofletterstoformwords,sentences,andthensentencestoformarticles.Thesearegrammars.Thesubjectmattertobestudied.Foranotherexample,thewhitelightpassesthroughtheprismanddecomposesintosevencolorsofred,orange,yellow,green,blue,andpurple;conversely,thesevencolorsoflightaresynthesizedintowhitelight.Thisistheanalysisandsynthesisofthespectrum,whichcanexplainthecauseoftherainbow.

Analysisandsynthesisaretwodifferentmethods,andtheyareoppositeinthedirectionofunderstanding.Buttheyarecloselyintegratedandcomplementeachother.Ontheonehand,analysisisthebasisofsynthesis;ontheotherhand,analysisalsoreliesonsynthesis.Withoutacertainsynthesisasaguide,itisimpossibletomakein-depthanalysisofthings.

### AbstractionandGeneralization

Abstractionistheuseofthinkingabilityinpeople’sresearchactivitiestoexcludethesecondaryandnon-essentialfactorsoftheobject,andextractthemainandessentialfactors.Awaytoreachtheessenceoftheobject.

Summarizationisamethodofextendingtheunderstandingofthenatureandregularityoftheobjecttoallotherthingsofthesamekindinthinking.Forexample,afterdiscoveringthecommonnatureofthe"metal"that"canconductelectricity,"thiscommonnaturecanbeextendedtoallmetals,anditcanbesummarizedthatallmetalshavetheessentialpropertyof"conductingelectricity".

## Completeinductionmethod

### Concept

Completeinductionreasoningisbasedonthefactthateachobjectofacertaintypeofthinghasacertainattribute,therebydeducingthistypeofthingAllhavetheconclusionofthiskindofattribute.

### Example

Forexample:"ItisknownthattherearemineraldepositsinEurope,mineraldepositsinAsia,mineraldepositsinAfrica,mineraldepositsinNorthAmerica,mineraldepositsinSouthAmerica,andmineraldepositsinOceania.TherearemineraldepositsinAntarctica,whileEurope,Asia,Africa,NorthAmerica,SouthAmerica,Oceania,andAntarcticaareallcontinentsontheearth.Therefore,allcontinentsontheearthhavemineraldeposits."Thelogicalformisasfollows:

S1isP

S2isP

……

SnisP

S1,S2,...,SnisoftypeSAllobjects

So,allSareP

Thecharacteristicofcompleteinductivereasoningisthatallobjectsofaclassofthingsareexaminedinthepremise,andtheconclusiondoesnotexceedtheknowledgedeterminedbythepremiseScope,therefore,theconnectionbetweenitspremisesandconclusionsisinevitable.

### Function

Completelyinductivereasoninghastwofunctions:

(1)Cognitivefunction.Completelyinductivereasoningisbasedonthefactthateachobjectofacertaintypeofthingshasacertainattribute,anditisconcludedthatthistypeofthingsallhavethisattribute,whichmakespeople'sunderstandingrisefromindividualtogeneral.Forexample,theaboveconclusionbasedonthe"continentsontheearth"categoryofthingsthateachobjecthastheattribute"hasmineraldeposits",andtheconclusionthat"allcontinentsontheearthhavemineraldeposits"reflectstheunderstandingofcompleteinductivereasoning.effect.

(2)Demonstrationfunction.Becausetheconnectionbetweenthepremiseandtheconclusionofcompleteinductivereasoningisinevitable,itisoftenusedasapowerfulmethodofargumentation.Forexample,forthetopic"Thesyllogismoftwospecialpremisescannotinferaconclusion",itcanbearguedlikethis:thepremiseisthatthesyllogismofIIcannotinferaconclusion,thepremiseisthatthesyllogismofOOcannotinferaconclusion,andthepremiseisthatthesyllogismofIO(OI)isinferred.Noconclusion,thepremiseisthesyllogismofII,thepremiseisthesyllogismofOO,andthepremiseisthatthesyllogismofIO(OI)isalltheobjectsofthetwosyllogismsofspecialpremises.Therefore,thesyllogismoftwospecialpremisescannotinferaconclusion.

Completelyinductivereasoningusuallyappliestoasmallnumberofthings.Whenthenumberofthingstobeinvestigatedisextremelylarge,eveninfinite,completeinductivereasoningisnotapplicable,andanotherformofinductivereasoningneedstobeused,thatis,incompleteinductivereasoning.

## Incompletemethod

### Concept

Incompleteinductivereasoningisbasedonthefactthatsomeobjectsofacertainkindofthingshavecertainattributes,andthusitisconcludedthatallkindsofthingshavecertainattributes.Theconclusionofthisattribute.Incompleteinductivereasoningincludessimpleenumerationinductivereasoningandscientificinductivereasoning.

### Simpleinductivereasoning

17=3+3+11

41=11+13+17

77=7+17+53

461=5+7+449

10=5+5

14=7+7

18=7+11

462=5+457

Thepreviouspropositioncanbeprovedfromthisproposition,andthesetwopropositionswerelatercollectivelycalledthe"GoldbachConjecture".

Manyfolkproverbs,suchas"RuixueZhaofengYear","Thefoundationismoistandrain,themoonishazyandthewind","Birdsflylow,putontheclothes",etc.,arebasedonrepeatedmanytimesinlife.Examplesaresummarizedbysimpleenumerationandinductivereasoning.

Thelogicalformofsimpleenumerationinductivereasoningisasfollows:

S1isP

S2isP

……

SnisP

S1,S2,...,SnispartoftheSclass,andnoSisnotP

So,allSis(orisnot)P

Theconclusionofsimpleenumerationandinductivereasoningisprobable,becauseitsconclusionisbeyondthescopeofknowledgedeterminedbythepremise.MathematicianHuaLuogenggaveagooddescriptionoftheprobabilityofsimpleenumerationinductivereasoninginthebook"MathematicalInduction":

Toimprovethereliabilityofsimpleenumerationinductivereasoning,wemustpayattentiontothefollowingtworequirements:(1)Thenumberofenumerationsmustbelargeenough,andthescopeofinvestigationmustbewideenough.(2)Checkwhethertherearecounterexamples.Usually,simpleenumerationandinductivereasoningthatdoesnotpayattentiontotheabovetworequirementsandthereforehastoofewsamplesandtheconclusionisobviouslyfalseiscalled"tobepartial"or"togeneralizelightly".

Simpleenumerationinductivereasoningisthesimplestmethodofinductivereasoning.However,despitethis,itssignificancecannotbeignored.

### Scientificinductivereasoning

ScientificinductivereasoningisbasedonacertainTheanalysisofthecausalrelationshipbetweensomeobjectsinaclassofthingsandacertainattribute,andthereasoningthatthisclassofthingshasthatattribute.Forexample:

Thevolumeofgoldexpandsafterheating;

SilverisheatedAfterthevolumeexpansion;

Thevolumeexpandsafterthecopperisheated;

Thevolumeexpandsaftertheironisheated;

So,allmetalsexpandinvolumeafterbeingheated.

Thepremisenotonlyexaminesthatsomeobjectsofaclassofthingshavecertainattributes,butalsofurtherpointsoutthecausalconnectionbetweenobjectsandattributes,andconcludesfromthis.Thisisscientificinductivereasoning.

ScientificinductivereasoningTheformisasfollows:

S1isP

S2isP

……

SnisP

So,allSisallP

Comparedwithsimpleenumerationinductivereasoning,scientificinductivereasoninghasthesameanddifferentpoints.

Thecommonpointisthattheyallbelongtoincompleteinductivereasoning.Thepremisesareonlypartoftheobjectsofoneclassofthings,andtheconclusionsarealltheconclusionsofthewholegroupofthings,andthescopeoftheknowledgeoftheconclusionsexceedsthepremise.

(3)Thereliabilityoftheconclusionisdifferent.Althoughtheconnectionbetweenthepremisesandconclusionsofthetwoisprobable,theintensityofinductionisnotnecessarilyequalto1.However,scientificinductivereasoningexaminesthecausalconnectionbetweenobjectsandattributes.Therefore,theinductivestrengthofscientificinductivereasoningisstrongerthanthatofsimpleenumerationinductivereasoning.Thatistosay,scientificinductivereasoningiscomparedwithsimpleenumerationinductivereasoning.Theconclusionishighlyreliable.

Forexample,themediasometimessaythatonehundredstepsafteramealisgood;sometimestheysaythatonehundredstepsafteramealisnotgood.Foranotherexample,sometimesitissaidthatovernightteacannotbedrunk,whichisharmfultohealth;sometimesitissaidthatstudieshaveshownthatovernightteacanbedrunkjustlikenon-overnighttea.Andsoon,peoplearesimplyoverwhelmed.Themainfeatureofscientificinductivereasoningistoinvestigatethecausalconnectionbetweentheobjectandtheattribute,whichhelpstoguidepeopletoexploretheessenceofthings,discoverthelawsofthings,andmorereliablypromoteperceptualknowledgetorationalknowledge.