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Inductive reasoning



Definition

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

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

Traditionally,inductivereasoningisdividedintocompleteinductivereasoningandincompleteinductivereasoningaccordingtothedifferentscopeofobjectsexaminedbythepremise.Completeinductivereasoningexaminesallobjectsofacertainclassofthings,whileincompleteinductivereasoningonlyexaminessomeoftheobjectsofcertainclassofthings.Andfurtheraccordingtowhetherthepremiserevealsthecausalconnectionbetweentheobjectanditsattributes,incompleteinductivereasoningisdividedintosimpleenumerationinductivereasoningandscientificinductivereasoning.

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.

(3)Compareinthesubstanceoftheobjects.Forexample,tocomparetwocollegestudentswhoarebetter,youmustcomparetheirideologicalandmoralcharacter,academicperformance,practicalabilityandothersubstantiveaspects,notgender,nativeplace,familyrichorpoor,etc.

Categorization

Categorizationisamethodofdistinguishingobjectsbycategorybasedontheircommonanddifferentpoints.Throughclassification,chaoticphenomenacanbeorganized,andalargenumberoffactualmaterialscanbesystematized.Classificationisbasedoncomparison.Throughcomparison,findoutthesimilaritiesanddifferencesbetweenthings,andthenclassifythefactualmaterialswiththesamepointsintothesamecategory,anddividethefactualmaterialswiththedifferencesintodifferentcategories.Forexample,thereareabout400,000speciesofplantsintheworld,whichcanbeclassifiedintofourcategories(phyla):thephylumAlgae,bryophyta,pteridophyteandseedplant.Fromthedoortothenext,youcangettheoutline,order,family,genus,andspeciesofunitsatalllevels.

Thereisadifferencebetweentheclassificationandthedivisionofterms.

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

(2)Differentroles.Thedivisionoflexicalitemsistoclarifythelexicalitems.Classificationisamethodofsystematizingthematerialpossessed.Moreimportantly,becausethecorrectclassificationsystemreflectstherelationshipbetweentheessentialcharacteristicsofthingsandtheinternalregularity,ithasscientificforesightandcanguidepeopletofindorrecognizenewconcretethings.Forexample,thenaturalclassificationsystemoforganismsestablishedonthebasisofDarwin'stheoryofbiologicalevolutionpredictedmanytransitionalorganismsthathadnotyetbeendiscoveredatthattime.ArchaeopteryxisthekindthatDarwinpredictedandfound.Archaeopteryxisanintermediatetypebetweenreptilesandbirds.Itfillsthegapbetweenthesetwotypesofanimals,indicatingthatbirdsevolvedfromreptiles.

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.

Toobtaincorrectconclusionsusingcompleteinductivereasoning,tworequirementsmustbemet:(1)Allobjectsofaclassofthingsareexaminedinthepremise.(2)Theassertionsmadeinthepremiseforeachobjectofthistypeofthingaretrue.

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

Inaclassofthings,someoftheobjectsthathavebeenobservedhavecertainattributes,andnocounterexampleshavebeenencountered,soitcanbeconcludedthatthisclassofthingshasTheconclusionofthiskindofattributeissimpleenumerationandinductivereasoning.Forexample,the"GoldbachConjecture",knownasthe"jewelinthecrownofmathematics,"wasputforwardusingsimpleenumerationandinductivereasoning.Morethan200yearsago,GermanmathematicianBrotherDebachdiscoveredthatsomeoddnumbersareequaltothesumofthreeprimenumbers.Forexample:

17=3+3+11

41=11+13+17

77=7+17+53

461=5+7+449

Goldbachdidnotlistalltheoddnumbers(infact,itisimpossible),onlyafewexamplesConjecture:Alloddnumbersgreaterthan5canbedecomposedintothesumofthreeprimenumbers.HetoldthisconjecturetoMathematicianEuler.Eulerconfirmedhisconjectureandaddedaconjecture:Evennumbersgreaterthan4canbedecomposedintothesumoftwoprimenumbers.Forexample:

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":

"ThefirstoneoutofabagisredGlassballs,thesecondoneisredglassballs,evenwhenthethird,fourth,andfifthareredglassballs,weimmediatelyguess:'Arealltheballsinthebagredglassballs?'However,oncewetouchedoutawhiteglassball,thisconjecturefailed.Atthistime,wewillhaveanotherconjecture:'Iseverythinginthebagallglassballs?'Whenwetouchedoutawoodenballonce,Thisconjecturefailedagain.Atthattime,wewillhaveathirdconjecture:"Isthebaginthebagallballs?"Thisconjectureiscorrect,wemustcontinuetotest,tofindoutallthecontentsinthebagInordertoseeadifference."

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

LuXunwrotein"ThePrefaceofUchiyamaFinishingWorks":"AtravelerwalkedintothestudyroomofarichofficialinXiayeandsawmanyexpensiveinkstones.ShisaidthatChinaisa'gentlecountry';whenanobservercomestoShanghai,buysseveralobscenebooksandpictures,andthenlooksforstrangethingstoobserve,hecallsChinaa'pornographiccountry'."Inthisarticle,LuXunfurtherrevealedthatbecauseoftheinsufficientnumberofenumerationsortheinsufficientscopeofinvestigation,suchpeopledidnotpayattentiontoinvestigatingwhethertherewerecounterexamples,sothattheywere"partiallycomprehensive"or"indiscreetlygeneralized"andinevitableintheend.Thedilemmatofallinto:"Ifyougotoapoorliterati’shomeorapartment,youdon’tcareaboutyourstudy,eventheinkstonecostsonlytwodimes.Onceyouseesuchathing,thepreviousconclusionwillnotpass.Soobserversarealittleembarrassed."

Simpleenumerationinductivereasoningisthesimplestmethodofinductivereasoning.However,despitethis,itssignificancecannotbeignored.

(1)Simpleenumerationandinductivereasoningcanhelpdiscovery.Whenasufficientbasisforthegeneralizationcannotbefound,butthereisconsiderablematerial,itisnecessarytousesimpleenumerationandinductivereasoningtomakeapreliminarygeneralizationandcomeupwithaprobableconclusionasastartingpointforfurtherresearch.Therefore,simpleenumerationandinductivereasoningareoftenusedwhenforminghypotheses.Forexample,inthediscoveryprocessofBoyle'sLaw,simpleenumerationandinductivereasoningplayedacertainrole.Boylesummedupthelawthat"undercertainconditions,thevolumeofagasisinverselyproportionaltothepressureitreceives"frommanyexperimentalfactshehasmastered.

(2)Simpleenumerationandinductivereasoningcanalsobeusedasamethodofargumentationandplayacertainroleintheprocessofargumentation.Forexample,HuShihadaconversationinhislateryears:"Anyonewhoisverysuccessfulissomeonewhoisextremelysmartandwillingtododumbskills.NotonlyinChina,butalsointheWest.LikeConfucius,hesaid,"Idon’teatalldayandsleepallnight,thinking,it’suseless,it’sbettertolearn.ThisistheworkofConfuciusinlearning.ChinesemathematicianandlinguistZhouHaizhongb>FormanyyearsofresearchonMersenneprimenumbers,heusedtheconnectionobservationmethodandtheincompleteinductionmethodtofirstgivethepreciseexpressionofthemersenneprimenumberdistributionin1992,thusrevealingtheimportantlawsofMersenneprimenumbers.,Whichprovidesconvenienceforpeopletoexplorethisprimenumber.Laterthisscientificresearchresultwascalled"Zhou’sguess"internationally.

Scientificinductivereasoning

ScientificinductivereasoningisbasedonacertainTheanalysisofthecausalrelationshipbetweensomeobjectsinaclassofthingsandacertainattribute,andthereasoningthatthisclassofthingshasthatattribute.Forexample:

Thevolumeofgoldexpandsafterheating;

SilverisheatedAfterthevolumeexpansion;

Thevolumeexpandsafterthecopperisheated;

Thevolumeexpandsaftertheironisheated;

Becausethemetalisheated,themolecularcohesionweakensandthemolecularmotionaccelerates,Thedistancebetweenmoleculesincreases,whichleadstoexpansion,whilegold,silver,copper,andironareallmetals;

So,allmetalsexpandinvolumeafterbeingheated.

Thepremisenotonlyexaminesthatsomeobjectsofaclassofthingshavecertainattributes,butalsofurtherpointsoutthecausalconnectionbetweenobjectsandattributes,andconcludesfromthis.Thisisscientificinductivereasoning.

ScientificinductivereasoningTheformisasfollows:

S1isP

S2isP

……

SnisP

S1,S2,...,SnarepartoftheobjectsofclassS,andthereisnoSi(1≤i≤n)whichisnotP;andscientificresearchshowsthatthereisacausallinkbetweenSandP

So,allSisallP

Comparedwithsimpleenumerationinductivereasoning,scientificinductivereasoninghasthesameanddifferentpoints.

Thecommonpointisthattheyallbelongtoincompleteinductivereasoning.Thepremisesareonlypartoftheobjectsofoneclassofthings,andtheconclusionsarealltheconclusionsofthewholegroupofthings,andthescopeoftheknowledgeoftheconclusionsexceedsthepremise.

Thedifferenceis:(1)Thereasoningisdifferent.Simpleenumerationinductivereasoningisbasedonlyonthefactthatsomeoftheobjectsthathavebeenobservedhavecertainattributesandhavenotencounteredanycounterexamples.Scientificinductivereasoningdoesnotstopatrepeatingtheexperienceofthings,butconductsin-depthscientificanalysisandgraspsConclusionsaremadeonthebasisofthecausalconnectionbetweenobjectsandattributes.

(2)Thenumberofpremiseshasdifferentmeaningsforthetwo.Forsimpleenumerationandinductivereasoning,themoreobjectsareexaminedinpremises,Thewiderthescope,themorereliabletheconclusion.Forscientificinductivereasoning,thenumberofpremisesdoesnothaveadecisivesignificance,aslongasthecauseandeffectbetweentheobjectandtheattributeisfullyunderstoodConnection,evenifthenumberofpremisesissmall,evenifthereareonlyoneortwotypicalcases,reliableconclusionscanbedrawn.AsEngelssaid,ahundredthousandsteamengineisnobetterthanasteamenginetoexplaintheconversionofheatenergyintomechanicalenergy.

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

Scientificinductivereasoningadvocatesahabitofthinkinginthefaceofknowledgeandconclusions.Thishabitisespeciallyimportantintheageofadvancedinformation.Thinkaboutit,ourmediaoftendisseminatessomecontradictory"scientificknowledge"tous,thisisnotdifficulttounderstand.

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

ProbabilisticReasoning

M.Kleinwrotein"MathematicsinWesternCulture":"Needlesstosayaboutourfuture,evenonehourfromnow,Andthereisnothingforsure.Afteraminute,thegroundunderourfeetmaycrack.However,claimingthispossibilitycannotscareus,becauseweknowthattheprobabilityofthissituationisextremelysmall.

Inotherwords,itistheprobabilityofwhetheraneventoccursornot,whichdeterminesourattitudesandactionstowardstheevent."ThephenomenonthatmayormaynotappearundercertainconditionsiswhatwecallItisarandomeventoraccidentalevent,suchasdrawingakingofheartsfromadeckofbridgecards.Infact,whenweobservealargenumberofrandomeventsofthesamekind,wewillfindthatthereisacertainregularityamongthem.

Probabilityisthequantitativedescriptionofthelawpresentedbyalargenumberofrandomevents,usuallyexpressedbyP(A).Usingprobabilisticreasoning,wecanlearnhowlikelyaneventis,orhowlikelyitistohappen.Inthissense,itcanbesaidthatprobabilisticreasoningistheinferenceaboutchance.

Probabilityvalue

Indailylife,weareonlysatisfiedwithestimatingwhethertheprobabilityofaneventishighorlow.However,thisestimateistoobroadtomeettheneedsofmanyissuessuchasindustry,economics,insurance,medicalcare,sociology,psychology,andsoon.Becauseintheabovesituation,theexactprobabilityvaluemustbeknown.Toachievethisgoal,youneedhelpfrommathematics.Relyingontheprobabilityvaluecalculatedbymathematicscanwereliablyguideouractions.

Generally,thedefinitionofcalculatingtheprobabilityvalueis:iftherearenkindsofequalpossibilities,andthesituationthatisconducivetotheoccurrenceofaneventism,thentheprobabilityoftheoccurrenceoftheeventism/n,theprobabilityofnothappeningis(nm)/n.Underthisdefinition,iftheeventisimpossible,theprobabilityoftheeventis0/n,whichis0;iftheeventiscompletelycertain,theprobabilityisn/n,whichis1.

Therefore,theprobabilityvaluevariesfrom0to1,thatis,fromimpossibilitytocertainty.Theso-calledequalpossibilitymeansthatthelikelihoodofoccurrenceisthesame.Forexample,adicehas6sides.Ifthereisnofactorintheshapeofthediceorinthewayofthrowingthedice,thereisnofactorinfavoroftheappearanceofacertainside,thentheprobabilityoftheappearanceof6sidesofthediceisthesame,thatis,thedicehas6types,etc.possibility.

Accordingtothisdefinitionofcalculatingtheprobabilityvalue,theprobabilityofselectingacard"A"fromadeckof52ordinaryunsoapedplayingcardsis4/52,or1/13.Becausethereare52suchpossibilities,4ofthemareadvantageous.However,ifallpossibilitiesarenotequallypossible,thenthisdefinitionofcalculatedprobabilityvaluedoesnotapply.Forexample,thereareonlytwopossibilitiesforapersontocrossthestreet:eithersafelyornot.However,itcannotbeconcludedfromthisthattheprobabilityofapersoncrossingthestreetsafelyis1/2,becausethetwopossibilitiesof"passingsafely"and"passingwithoutsafety"arenotequallypossible.

Itshouldbenotedthatprobabilitytellsuswhathappensinalargeselection.Forexample,theprobabilityofchoosingan"A"from52decksofcardsis1/13.Thisdoesnotmeanthatifapersontakes13timesfromthisdeckofcards,hewilldefinitelychoosean"A".".Hemayhavefetched30or40times,butdidnotgetan"A".However,themoretimeshetakes,theratioofthenumberofAcestothetotalnumberofcardswillbecloseto1/13.Inaddition,thisdoesnotmeanthatifapersontakesan"A",forexample,ithappenstobethefirsttime,theprobabilityoftakingan"A"nexttimemustbelessthan1/13.Theprobabilitywillstillbethesame,whichis1/13,evenifthe3"A"saretakenoutinsuccession.Becauseadeckofcardshasneithermemorynorconsciousness,sowhathashappenedwillnotaffectthefuture.

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