Deductive reasoning - Wikipedia

Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its ... Deductivereasoning FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Formofreasoning Deductivereasoningisthementalprocessofdrawingdeductiveinferences.Aninferenceisdeductivelyvalidifitsconclusionfollowslogicallyfromitspremises,i.e.ifitisimpossibleforthepremisestobetrueandtheconclusiontobefalse.Forexample,theinferencefromthepremises"allmenaremortal"and"Socratesisaman"totheconclusion"Socratesismortal"isdeductivelyvalid.Anargumentissoundifitisvalidandallitspremisesaretrue.Sometheoristsdefinedeductionintermsoftheintentionsoftheauthor:theyhavetointendforthepremisestoofferdeductivesupporttotheconclusion.Withthehelpofthismodification,itispossibletodistinguishvalidfrominvaliddeductivereasoning:itisinvalidiftheauthor'sbeliefaboutthedeductivesupportisfalse,buteveninvaliddeductivereasoningisaformofdeductivereasoning. Psychologyisinterestedindeductivereasoningasapsychologicalprocess,i.e.howpeopleactuallydrawinferences.Logic,ontheotherhand,focusesonthedeductiverelationoflogicalconsequencebetweenthepremisesandtheconclusionorhowpeopleshoulddrawinferences.Therearedifferentwaysofconceptualizingthisrelation.Accordingtothesemanticapproach,anargumentisdeductivelyvalidifandonlyifthereisnopossibleinterpretationofthisargumentwhereitspremisesaretrueanditsconclusionisfalse.Thesyntacticapproach,ontheotherhand,holdsthatanargumentisdeductivelyvalidifandonlyifitsconclusioncanbededucedfromitspremisesusingavalidruleofinference.Aruleofinferenceisaschemaofdrawingaconclusionfromasetofpremisesbasedonlyontheirlogicalform.Therearevariousrulesofinference,likethemodusponensandthemodustollens.Invaliddeductivearguments,whichdonotfollowaruleofinference,arecalledformalfallacies.Rulesofinferencearedefinitoryrulesandcontrasttostrategicrules,whichspecifywhatinferencesoneneedstodrawinordertoarriveatanintendedconclusion.Deductivereasoningcontrastswithnon-deductiveorampliativereasoning.Forampliativearguments,likeinductiveorabductivearguments,thepremisesofferweakersupporttotheirconclusion:theymakeitmorelikelybuttheydonotguaranteeitstruth.Theymakeupforthisdrawbackbybeingabletoprovidegenuinelynewinformationnotalreadyfoundinthepremises,unlikedeductivearguments. Cognitivepsychologyinvestigatesthementalprocessesresponsiblefordeductivereasoning.Oneofitstopicsconcernsthefactorsdeterminingwhetherpeopledrawvalidorinvaliddeductiveinferences.Onefactoristheformoftheargument:forexample,peoplearemoresuccessfulforargumentsoftheformmodusponensthanformodustollens.Anotheristhecontentofthearguments:peoplearemorelikelytobelievethatanargumentisvalidiftheclaimmadeinitsconclusionisplausible.Ageneralfindingisthatpeopletendtoperformbetterforrealisticandconcretecasesthanforabstractcases.Psychologicaltheoriesofdeductivereasoningaimtoexplainthesefindingsbyprovidinganaccountoftheunderlyingpsychologicalprocesses.Mentallogictheoriesholdthatdeductivereasoningisalanguage-likeprocessthathappensthroughthemanipulationofrepresentationsusingrulesofinference.Mentalmodeltheories,ontheotherhand,claimthatdeductivereasoninginvolvesmodelsofpossiblestatesoftheworldwithoutthemediumoflanguageorrulesofinference.Accordingtodual-processtheoriesofreasoning,therearetwoqualitativelydifferentcognitivesystemsresponsibleforreasoning. Theproblemofdeductivereasoningisrelevanttovariousfieldsandissues.Epistemologytriestounderstandhowjustificationistransferredfromthebeliefinthepremisestothebeliefintheconclusionintheprocessofdeductivereasoning.Probabilitylogicstudieshowtheprobabilityofthepremisesofaninferenceaffectstheprobabilityofitsconclusion.Thecontroversialthesisofdeductivismdeniesthatthereareothercorrectformsofinferencebesidesdeduction.Naturaldeductionisatypeofproofsystembasedonsimpleandself-evidentrulesofinference.Inphilosophy,thegeometricalmethodisawayofphilosophizingthatstartsfromasmallsetofself-evidentaxiomsandtriestobuildacomprehensivelogicalsystemusingdeductivereasoning. Contents 1Definition 2Conceptionsofdeduction 3Rulesofinference 3.1Prominentrulesofinference 3.1.1Modusponens 3.1.2Modustollens 3.1.3Hypotheticalsyllogism 3.2Fallacies 3.3Definitoryandstrategicrules 4Validityandsoundness 5Differencefromampliativereasoning 6Invariousfields 6.1Cognitivepsychology 6.1.1Psychologicaltheoriesofdeductivereasoning 6.1.2Intelligence 6.2Epistemology 6.3Probabilitylogic 7History 8Relatedconceptsandtheories 8.1Deductivism 8.2Naturaldeduction 8.3Geometricalmethod 9Seealso 10References 11Furtherreading 12Externallinks Definition[edit] Deductivereasoningisthepsychologicalprocessofdrawingdeductiveinferences.Aninferenceisasetofpremisestogetherwithaconclusion.Thispsychologicalprocessstartsfromthepremisesandreasonstoaconclusionbasedonandsupportedbythesepremises.Ifthereasoningwasdonecorrectly,itresultsinavaliddeduction:thetruthofthepremisesensuresthetruthoftheconclusion.[1][2][3][4]Forexample,inthesyllogisticargument"allfrogsarereptiles;nocatsarereptiles;therefore,nocatsarefrogs"theconclusionistruebecauseitstwopremisesaretrue.Butevenargumentswithwrongpremisescanbedeductivelyvalidiftheyobeythisprinciple,asin"allfrogsaremammals;nocatsaremammals;therefore,nocatsarefrogs".Ifthepremisesofavalidargumentaretrue,thenitiscalledasoundargument.[5] Therelationbetweenthepremisesandtheconclusionofadeductiveargumentisusuallyreferredtoas"logicalconsequence".AccordingtoAlfredTarski,logicalconsequencehas3essentialfeatures:itisnecessary,formal,andknowableapriori.[6][7]Itisnecessaryinthesensethatthepremisesofvaliddeductiveargumentsnecessitatetheconclusion:itisimpossibleforthepremisestobetrueandtheconclusiontobefalse,independentofanyothercircumstances.[6][7]Logicalconsequenceisformalinthesensethatitdependsonlyontheformorthesyntaxofthepremisesandtheconclusion.Thismeansthatthevalidityofaparticularargumentdoesnotdependonthespecificcontentsofthisargument.Ifitisvalid,thenanyargumentwiththesamelogicalformisalsovalid,nomatterhowdifferentitisonthelevelofitscontents.[6][7]Logicalconsequenceisknowableaprioriinthesensethatnoempiricalknowledgeoftheworldisnecessarytodeterminewhetheradeductionisvalid.Soitisnotnecessarytoengageinanyformofempiricalinvestigation.[6][7]Somelogiciansdefinedeductionintermsofpossibleworlds:Adeductiveinferenceisvalidifandonlyif,thereisnopossibleworldinwhichitsconclusionisfalsewhileitspremisesaretrue.Thismeansthattherearenocounterexamples:theconclusionistrueinallsuchcases,notjustinmostcases.[1] Ithasbeenarguedagainstthisandsimilardefinitionsthattheyfailtodistinguishbetweenvalidandinvaliddeductivereasoning,i.e.theyleaveitopenwhetherthereareinvaliddeductiveinferencesandhowtodefinethem.[8][9]Someauthorsdefinedeductivereasoninginpsychologicaltermsinordertoavoidthisproblem.AccordingtoMarkVorobey,whetheranargumentisdeductivedependsonthepsychologicalstateofthepersonmakingtheargument:"Anargumentisdeductiveif,andonlyif,theauthoroftheargumentbelievesthatthetruthofthepremisesnecessitates(guarantees)thetruthoftheconclusion".[8]Asimilarformulationholdsthatthespeakerclaimsorintendsthatthepremisesofferdeductivesupportfortheirconclusion.[10][11]Thisissometimescategorizedasaspeaker-determineddefinitionofdeductionsinceitdependsalsoonthespeakerwhethertheargumentinquestionisdeductiveornot.Forspeakerlessdefinitions,ontheotherhand,onlytheargumentitselfmattersindependentofthespeaker.[9]Oneadvantageofthistypeofformulationisthatitmakesitpossibletodistinguishbetweengoodorvalidandbadorinvaliddeductivearguments:theargumentisgoodiftheauthor'sbeliefconcerningtherelationbetweenthepremisesandtheconclusionistrue,otherwiseitisbad.[8]Oneconsequenceofthisapproachisthatdeductiveargumentscannotbeidentifiedbythelawofinferencetheyuse.Forexample,anargumentoftheformmodusponensmaybenon-deductiveiftheauthor'sbeliefsaresufficientlyconfused.Thatbringswithitanimportantdrawbackofthisdefinition:itisdifficulttoapplytoconcretecasessincetheintentionsoftheauthorareusuallynotexplicitlystated.[8] Deductivereasoningisstudiedinlogic,psychology,andthecognitivesciences.[3][1]Sometheoristsemphasizeintheirdefinitionthedifferencebetweenthesefields.Onthisview,psychologystudiesdeductivereasoningasanempiricalmentalprocess,i.e.whathappenswhenhumansengageinreasoning.[3][1]Butthedescriptivequestionofhowactualreasoninghappensisdifferentfromthenormativequestionofhowitshouldhappenorwhatconstitutescorrectdeductivereasoning,whichisstudiedbylogic.[3][12][6]Thisissometimesexpressedbystatingthat,strictlyspeaking,logicdoesnotstudydeductivereasoningbutthedeductiverelationbetweenpremisesandaconclusionknownaslogicalconsequence.Butthisdistinctionisnotalwayspreciselyobservedintheacademicliterature.[3]Oneimportantaspectofthisdifferenceisthatlogicisnotinterestedinwhethertheconclusionofanargumentissensible.[1]Sofromthepremise"theprinterhasink"onemaydrawtheunhelpfulconclusion"theprinterhasinkandtheprinterhasinkandtheprinterhasink",whichhaslittlerelevancefromapsychologicalpointofview.Instead,actualreasonersusuallytrytoremoveredundantorirrelevantinformationandmaketherelevantinformationmoreexplicit.[1]Thepsychologicalstudyofdeductivereasoningisalsoconcernedwithhowgoodpeopleareatdrawingdeductiveinferencesandwiththefactorsdeterminingtheirperformance.[3][5]Deductiveinferencesarefoundbothinnaturallanguageandinformallogicalsystems,suchaspropositionallogic.[1][13] Conceptionsofdeduction[edit] Deductiveargumentsdifferfromnon-deductiveargumentsinthatthetruthoftheirpremisesensuresthetruthoftheirconclusion.[14][15][6]Therearetwoimportantconceptionsofwhatthisexactlymeans.Theyarereferredtoasthesyntacticandthesemanticapproach.[13][6][5]Accordingtothesyntacticapproach,whetheranargumentisdeductivelyvaliddependsonlyonitsform,syntax,orstructure.Twoargumentshavethesameformiftheyusethesamelogicalvocabularyinthesamearrangement,eveniftheircontentsdiffer.[13][6][5]Forexample,thearguments"ifitrainsthenthestreetwillbewet;itrains;therefore,thestreetwillbewet"and"ifthemeatisnotcooledthenitwillspoil;themeatisnotcooled;therefore,itwillspoil"havethesamelogicalform:theyfollowthemodusponens.Theirformcanbeexpressedmoreabstractlyas"ifAthenB;A;thereforeB"inordertomakethecommonsyntaxexplicit.[5]Therearevariousothervalidlogicalformsorrulesofinference,likemodustollensorthedisjunctionelimination.Thesyntacticapproachthenholdsthatanargumentisdeductivelyvalidifandonlyifitsconclusioncanbededucedfromitspremisesusingavalidruleofinference.[13][6][5]Onedifficultyforthesyntacticapproachisthatitisusuallynecessarytoexpresstheargumentinaformallanguageinordertoassesswhetheritisvalid.Butsincetheproblemofdeductionisalsorelevantfornaturallanguages,thisoftenbringswithitthedifficultyoftranslatingthenaturallanguageargumentintoaformallanguage,aprocessthatcomeswithvariousproblemsofitsown.[13]Anotherdifficultyisduetothefactthatthesyntacticapproachdependsonthedistinctionbetweenformalandnon-formalfeatures.Whilethereisawideagreementconcerningtheparadigmaticcases,therearealsovariouscontroversialcaseswhereitisnotclearhowthisdistinctionistobedrawn.[16][12] Thesemanticapproachsuggestsanalternativedefinitionofdeductivevalidity.Itisbasedontheideathatthesentencesconstitutingthepremisesandconclusionshavetobeinterpretedinordertodeterminewhethertheargumentisvalid.[13][6][5]Thismeansthatoneascribessemanticvaluestotheexpressionsusedinthesentences,suchasthereferencetoanobjectforsingulartermsortoatruth-valueforatomicsentences.Thesemanticapproachisalsoreferredtoasthemodel-theoreticapproachsincethebranchofmathematicsknownasmodeltheoryisoftenusedtointerpretthesesentences.[13][6]Usually,manydifferentinterpretationsarepossible,suchaswhetherasingulartermreferstooneobjectortoanother.Accordingtothesemanticapproach,anargumentisdeductivelyvalidifandonlyifthereisnopossibleinterpretationwhereitspremisesaretrueanditsconclusionisfalse.[13][6][5]Someobjectionstothesemanticapproacharebasedontheclaimthatthesemanticsofalanguagecannotbeexpressedinthesamelanguage,i.e.thatarichermetalanguageisnecessary.Thiswouldimplythatthesemanticapproachcannotprovideauniversalaccountofdeductionforlanguageasanall-encompassingmedium.[13][12] Rulesofinference[edit] Deductivereasoningusuallyhappensbyapplyingrulesofinference.Aruleofinferenceisawayorschemaofdrawingaconclusionfromasetofpremises.[17]Thishappensusuallybasedonlyonthelogicalformofthepremises.Aruleofinferenceisvalidif,whenappliedtotruepremises,theconclusioncannotbefalse.Aparticularargumentisvalidifitfollowsavalidruleofinference.Deductiveargumentsthatdonotfollowavalidruleofinferencearecalledformalfallacies:thetruthoftheirpremisesdoesnotensurethetruthoftheirconclusion.[18][14] Insomecases,whetheraruleofinferenceisvaliddependsonthelogicalsystemoneisusing.Thedominantlogicalsystemisclassicallogicandtherulesofinferencelistedhereareallvalidinclassicallogic.Butso-calleddeviantlogicsprovideadifferentaccountofwhichinferencesarevalid.Forexample,theruleofinferenceknownasdoublenegationelimination,i.e.thatifapropositionisnotnottruethenitisalsotrue,isacceptedinclassicallogicbutrejectedinintuitionisticlogic.[19][20] Prominentrulesofinference[edit] Modusponens[edit] Mainarticle:Modusponens Modusponens(alsoknownas"affirmingtheantecedent"or"thelawofdetachment")istheprimarydeductiveruleofinference.Itappliestoargumentsthathaveasfirstpremiseaconditionalstatement( P → Q {\displaystyleP\rightarrowQ} )andassecondpremisetheantecedent( P {\displaystyleP} )oftheconditionalstatement.Itobtainstheconsequent( Q {\displaystyleQ} )oftheconditionalstatementasitsconclusion.Theargumentformislistedbelow: P → Q {\displaystyleP\rightarrowQ}  (Firstpremiseisaconditionalstatement) P {\displaystyleP}  (Secondpremiseistheantecedent) Q {\displaystyleQ}  (Conclusiondeducedistheconsequent) Inthisformofdeductivereasoning,theconsequent( Q {\displaystyleQ} )obtainsastheconclusionfromthepremisesofaconditionalstatement( P → Q {\displaystyleP\rightarrowQ} )anditsantecedent( P {\displaystyleP} ).However,theantecedent( P {\displaystyleP} )cannotbesimilarlyobtainedastheconclusionfromthepremisesoftheconditionalstatement( P → Q {\displaystyleP\rightarrowQ} )andtheconsequent( Q {\displaystyleQ} ).Suchanargumentcommitsthelogicalfallacyofaffirmingtheconsequent. Thefollowingisanexampleofanargumentusingmodusponens: Ifitisraining,thentherearecloudsinthesky. Itisraining. Thus,therearecloudsinthesky. Modustollens[edit] Mainarticle:Modustollens Modustollens(alsoknownas"thelawofcontrapositive")isadeductiveruleofinference.Itvalidatesanargumentthathasaspremisesaconditionalstatement(formula)andthenegationoftheconsequent( ¬ Q {\displaystyle\lnotQ} )andasconclusionthenegationoftheantecedent( ¬ P {\displaystyle\lnotP} ).Incontrasttomodusponens,reasoningwithmodustollensgoesintheoppositedirectiontothatoftheconditional.Thegeneralexpressionformodustollensisthefollowing: P → Q {\displaystyleP\rightarrowQ} .(Firstpremiseisaconditionalstatement) ¬ Q {\displaystyle\lnotQ} .(Secondpremiseisthenegationoftheconsequent) ¬ P {\displaystyle\lnotP} .(Conclusiondeducedisthenegationoftheantecedent) Thefollowingisanexampleofanargumentusingmodustollens: Ifitisraining,thentherearecloudsinthesky. Therearenocloudsinthesky. Thus,itisnotraining. Hypotheticalsyllogism[edit] Mainarticle:hypotheticalsyllogism Ahypotheticalsyllogismisaninferencethattakestwoconditionalstatementsandformsaconclusionbycombiningthehypothesisofonestatementwiththeconclusionofanother.Hereisthegeneralform: P → Q {\displaystyleP\rightarrowQ} Q → R {\displaystyleQ\rightarrowR} Therefore, P → R {\displaystyleP\rightarrowR} . Intherebeingasubformulaincommonbetweenthetwopremisesthatdoesnotoccurintheconsequence,thisresemblessyllogismsintermlogic,althoughitdiffersinthatthissubformulaisapropositionwhereasinAristotelianlogic,thiscommonelementisatermandnotaproposition. Thefollowingisanexampleofanargumentusingahypotheticalsyllogism: Iftherehadbeenathunderstorm,itwouldhaverained. Ifithadrained,thingswouldhavegottenwet. Thus,iftherehadbeenathunderstorm,thingswouldhavegottenwet.[21] Fallacies[edit] Variousformalfallacieshavebeendescribed.Theyareinvalidformsofdeductivereasoning.[18][14]Anadditionalaspectofthemisthattheyappeartobevalidonsomeoccasionsoronthefirstimpression.Theymaytherebyseducepeopleintoacceptingandcommittingthem.[22]Onetypeofformalfallacyisaffirmingtheconsequent,asin"ifJohnisabachelor,thenheismale;Johnismale;therefore,Johnisabachelor".[23]Thisissimilartothevalidruleofinferencenamedmodusponens,butthesecondpremiseandtheconclusionareswitchedaround,whichiswhyitisinvalid.Asimilarformalfallacyisdenyingtheantecedent,asin"ifOthelloisabachelor,thenheismale;Othelloisnotabachelor;therefore,Othelloisnotmale".[24][25]Thisissimilartothevalidruleofinferencecalledmodustollens,thedifferencebeingthatthesecondpremiseandtheconclusionareswitchedaround.Otherformalfallaciesincludeaffirmingadisjunct,denyingaconjunct,andthefallacyoftheundistributedmiddle.Allofthemhaveincommonthatthetruthoftheirpremisesdoesnotensurethetruthoftheirconclusion.Butitmaystillhappenbycoincidencethatboththepremisesandtheconclusionofformalfallaciesaretrue.[18][14] Definitoryandstrategicrules[edit] Rulesofinferencesaredefinitoryrules:theydeterminewhetheranargumentisdeductivelyvalidornot.Butreasonersareusuallynotjustinterestedinmakinganykindofvalidargument.Instead,theyoftenhaveaspecificpointorconclusionthattheywishtoproveorrefute.Sogivenasetofpremises,theyarefacedwiththeproblemofchoosingtherelevantrulesofinferencefortheirdeductiontoarriveattheirintendedconclusion.[13][26][27]Thisissuebelongstothefieldofstrategicrules:thequestionofwhichinferencesneedtobedrawntosupportone'sconclusion.Thedistinctionbetweendefinitoryandstrategicrulesisnotexclusivetologic:itisalsofoundinvariousgames.[13][26][27]Inchess,forexample,thedefinitoryrulesstatethatbishopsmayonlymovediagonallywhilethestrategicrulesrecommendthatoneshouldcontrolthecenterandprotectone'skingifoneintendstowin.Inthissense,definitoryrulesdeterminewhetheroneplayschessorsomethingelsewhereasstrategicrulesdeterminewhetheroneisagoodorabadchessplayer.[13][26]Thesameappliestodeductivereasoning:tobeaneffectivereasonerinvolvesmasteringbothdefinitoryandstrategicrules.[13] Validityandsoundness[edit] Argumentterminology Deductiveargumentsareevaluatedintermsoftheirvalidityandsoundness. Anargumentis“valid”ifitisimpossibleforitspremisestobetruewhileitsconclusionisfalse.Inotherwords,theconclusionmustbetrueifthepremisesaretrue.Anargumentcanbe“valid”evenifoneormoreofitspremisesarefalse. Anargumentis“sound”ifitisvalidandthepremisesaretrue. Itispossibletohaveadeductiveargumentthatislogicallyvalidbutisnotsound.Fallaciousargumentsoftentakethatform. Thefollowingisanexampleofanargumentthatis“valid”,butnot“sound”: Everyonewhoeatscarrotsisaquarterback. Johneatscarrots. Therefore,Johnisaquarterback. Theexample'sfirstpremiseisfalse–therearepeoplewhoeatcarrotswhoarenotquarterbacks–buttheconclusionwouldnecessarilybetrue,ifthepremisesweretrue.Inotherwords,itisimpossibleforthepremisestobetrueandtheconclusionfalse.Therefore,theargumentis“valid”,butnot“sound”.Falsegeneralizations–suchas"Everyonewhoeatscarrotsisaquarterback"–areoftenusedtomakeunsoundarguments.Thefactthattherearesomepeoplewhoeatcarrotsbutarenotquarterbacksprovestheflawoftheargument. Inthisexample,thefirststatementusescategoricalreasoning,sayingthatallcarrot-eatersaredefinitelyquarterbacks.Thistheoryofdeductivereasoning–alsoknownastermlogic–wasdevelopedbyAristotle,butwassupersededbypropositional(sentential)logicandpredicatelogic.[citationneeded] Deductivereasoningcanbecontrastedwithinductivereasoning,inregardstovalidityandsoundness.Incasesofinductivereasoning,eventhoughthepremisesaretrueandtheargumentis“valid”,itispossiblefortheconclusiontobefalse(determinedtobefalsewithacounterexampleorothermeans). Differencefromampliativereasoning[edit] Deductivereasoningisusuallycontrastedwithnon-deductiveorampliativereasoning.[13][28][29]Thehallmarkofvaliddeductiveinferencesisthatitisimpossiblefortheirpremisestobetrueandtheirconclusiontobefalse.Inthisway,thepremisesprovidethestrongestpossiblesupporttotheirconclusion.[13][28][29]Thepremisesofampliativeinferencesalsosupporttheirconclusion.Butthissupportisweaker:theyarenotnecessarilytruth-preserving.Soevenforcorrectampliativearguments,itispossiblethattheirpremisesaretrueandtheirconclusionisfalse.[11]Twoimportantformsofampliativereasoningareinductiveandabductivereasoning.[30]Sometimestheterm"inductivereasoning"isusedinaverywidesensetocoverallformsofampliativereasoning.[11]However,inamorestrictusage,inductivereasoningisjustoneformofampliativereasoning.[30]Inthenarrowsense,inductiveinferencesareformsofstatisticalgeneralization.Theyareusuallybasedonmanyindividualobservationsthatallshowacertainpattern.Theseobservationsarethenusedtoformaconclusioneitheraboutayetunobservedentityoraboutagenerallaw.[31][32][33]Forabductiveinferences,thepremisessupporttheconclusionbecausetheconclusionisthebestexplanationofwhythepremisesaretrue.[30][34] Thesupportampliativeargumentsprovidefortheirconclusioncomesindegrees:someampliativeargumentsarestrongerthanothers.[11][35][30]Thisisoftenexplainedintermsofprobability:thepremisesmakeitmorelikelythattheconclusionistrue.[13][28][29]Strongampliativeargumentsmaketheirconclusionverylikely,butnotabsolutelycertain.Anexampleofampliativereasoningistheinferencefromthepremise"everyraveninarandomsampleof3200ravensisblack"totheconclusion"allravensareblack":theextensiverandomsamplemakestheconclusionverylikely,butitdoesnotexcludethattherearerareexceptions.[36]Inthissense,ampliativereasoningisdefeasible:itmaybecomenecessarytoretractanearlierconclusionuponreceivingnewrelatedinformation.[12][30]Ampliativereasoningisverycommonineverydaydiscourseandthesciences.[13][37] Animportantdrawbackofdeductivereasoningisthatitdoesnotleadtogenuinelynewinformation.[5]Thismeansthattheconclusiononlyrepeatsinformationalreadyfoundinthepremises.Ampliativereasoning,ontheotherhand,goesbeyondthepremisesbyarrivingatgenuinelynewinformation.[13][28][29]Onedifficultyforthischaracterizationisthatitmakesdeductivereasoningappearuseless:ifdeductionisuninformative,itisnotclearwhypeoplewouldengageinitandstudyit.[13][38]Ithasbeensuggestedthatthisproblemcanbesolvedbydistinguishingbetweensurfaceanddepthinformation.Onthisview,deductivereasoningisuninformativeonthedepthlevel,incontrasttoampliativereasoning.Butitmaystillbevaluableonthesurfacelevelbypresentingtheinformationinthepremisesinanewandsometimessurprisingway.[13][5] Apopularmisconceptionoftherelationbetweendeductionandinductionidentifiestheirdifferenceonthelevelofparticularandgeneralclaims.[2][9][39]Onthisview,deductiveinferencesstartfromgeneralpremisesanddrawparticularconclusions,whileinductiveinferencesstartfromparticularpremisesanddrawgeneralconclusions.Thisideaisoftenmotivatedbyseeingdeductionandinductionastwoinverseprocessesthatcomplementeachother:deductionistop-downwhileinductionisbottom-up.Butthisisamisconceptionthatdoesnotreflecthowvaliddeductionisdefinedinthefieldoflogic:adeductionisvalidifitisimpossibleforitspremisestobetruewhileitsconclusionisfalse,independentofwhetherthepremisesortheconclusionareparticularorgeneral.[2][9][1][5][3]Becauseofthis,somedeductiveinferenceshaveageneralconclusionandsomealsohaveparticularpremises.[2] Invariousfields[edit] Cognitivepsychology[edit] Cognitivepsychologystudiesthepsychologicalprocessesresponsiblefordeductivereasoning.[3][5]Itisconcerned,amongotherthings,withhowgoodpeopleareatdrawingvaliddeductiveinferences.Thisincludesthestudyofthefactorsaffectingtheirperformance,theirtendencytocommitfallacies,andtheunderlyingbiasesinvolved.[3][5]Anotablefindinginthisfieldisthatthetypeofdeductiveinferencehasasignificantimpactonwhetherthecorrectconclusionisdrawn.[3][5][40][41]Inameta-analysisof65studies,forexample,97%ofthesubjectsevaluatedmodusponensinferencescorrectly,whilethesuccessrateformodustollenswasonly72%.Ontheotherhand,evensomefallacieslikeaffirmingtheconsequentordenyingtheantecedentwereregardedasvalidargumentsbythemajorityofthesubjects.[3]Animportantfactorforthesemistakesiswhethertheconclusionseemsinitiallyplausible:themorebelievabletheconclusionis,thehigherthechancethatasubjectwillmistakeafallacyforavalidargument.[3][5] Animportantbiasisthematchingbias,whichisoftenillustratedusingtheWasonselectiontask.[5][3][42][43]Inanoften-citedexperimentbyPeterWason,4cardsarepresentedtotheparticipant.Inonecase,thevisiblesidesshowthesymbolsD,K,3,and7onthedifferentcards.Theparticipantistoldthat"[e]verycardwhichhasaDononesidehasa3ontheotherside".Theirtaskistoidentifywhichcardsneedtobeturnedaroundinordertoconfirmorrefutethisconditionalclaim.Thecorrectanswer,onlygivenbyabout10%,isthecardsDand7.Manyselectcard3instead,eventhoughtheconditionalclaimdoesnotinvolveanyrequirementsonwhatsymbolscanbefoundontheoppositesideofcard3.[3][5]Butthisresultcanbedrasticallychangedifdifferentsymbolsareused:thevisiblesidesshow"drinkingabeer","drinkingacoke","16yearsofage",and"22yearsofage"andtheparticipantsareaskedtoevaluatetheclaim"[i]fapersonisdrinkingbeer,thenthepersonmustbeover19yearsofage".Inthiscase,74%oftheparticipantsidentifiedcorrectlythatthecards"drinkingabeer"and"16yearsofage"havetobeturnedaround.[3][5]Thesefindingssuggestthatthedeductivereasoningabilityisheavilyinfluencedbythecontentoftheinvolvedclaimsandnotjustbytheabstractlogicalformofthetask:themorerealisticandconcretethecasesare,thebetterthesubjectstendtoperform.[3][5] Anotherbiasiscalledthe"negativeconclusionbias",whichhappenswhenoneofthepremiseshastheformofanegativematerialconditional,[5][44][45]asin"IfthecarddoesnothaveanAontheleft,thenithasa3ontheright.Thecarddoesnothavea3ontheright.Therefore,thecardhasanAontheleft".Theincreasedtendencytomisjudgethevalidityofthistypeofargumentisnotpresentforpositivematerialconditionals,asin"IfthecardhasanAontheleft,thenithasa3ontheright.Thecarddoesnothavea3ontheright.Therefore,thecarddoesnothaveanAontheleft".[5] Psychologicaltheoriesofdeductivereasoning[edit] Variouspsychologicaltheoriesofdeductivereasoninghavebeenproposed.Thesetheoriesaimtoexplainhowdeductivereasoningworksinrelationtotheunderlyingpsychologicalprocessesresponsible.Theyareoftenusedtoexplaintheempiricalfindings,suchaswhyhumanreasonersaremoresusceptibletosometypesoffallaciesthantoothers.[3][1][46] Animportantdistinctionisbetweenmentallogictheories,sometimesalsoreferredtoasruletheories,andmentalmodeltheories.Mentallogictheoriesseedeductivereasoningasalanguage-likeprocessthathappensthroughthemanipulationofrepresentations.[3][1][47][46]Thisisdonebyapplyingsyntacticrulesofinferenceinawayverysimilartohowsystemsofnaturaldeductiontransformtheirpremisestoarriveataconclusion.[46]Onthisview,somedeductionsaresimplerthanotherssincetheyinvolvefewerinferentialsteps.[3]Thisideacanbeused,forexample,toexplainwhyhumanshavemoredifficultieswithsomedeductions,likethemodustollens,thanwithothers,likethemodusponens:becausethemoreerror-proneformsdonothaveanativeruleofinferencebutneedtobecalculatedbycombiningseveralinferentialstepswithotherrulesofinference.Insuchcases,theadditionalcognitivelabormakestheinferencesmoreopentoerror.[3] Mentalmodeltheories,ontheotherhand,holdthatdeductivereasoninginvolvesmodelsormentalrepresentationsofpossiblestatesoftheworldwithoutthemediumoflanguageorrulesofinference.[3][1][46]Inordertoassesswhetheradeductiveinferenceisvalid,thereasonermentallyconstructsmodelsthatarecompatiblewiththepremisesoftheinference.Theconclusionisthentestedbylookingatthesemodelsandtryingtofindacounterexampleinwhichtheconclusionisfalse.Theinferenceisvalidifnosuchcounterexamplecanbefound.[3][1][46]Inordertoreducecognitivelabor,onlysuchmodelsarerepresentedinwhichthepremisesaretrue.Becauseofthis,theevaluationofsomeformsofinferenceonlyrequirestheconstructionofveryfewmodelswhileforothers,manydifferentmodelsarenecessary.Inthelattercase,theadditionalcognitivelaborrequiredmakesdeductivereasoningmoreerror-prone,therebyexplainingtheincreasedrateoferrorobserved.[3][1]Thistheorycanalsoexplainwhysomeerrorsdependonthecontentratherthantheformoftheargument.Forexample,whentheconclusionofanargumentisveryplausible,thesubjectsmaylackthemotivationtosearchforcounterexamplesamongtheconstructedmodels.[3] Bothmentallogictheoriesandmentalmodeltheoriesassumethatthereisonegeneral-purposereasoningmechanismthatappliestoallformsofdeductivereasoning.[3][47][48]Buttherearealsoalternativeaccountsthatpositvariousdifferentspecial-purposereasoningmechanismsfordifferentcontentsandcontexts.Inthissense,ithasbeenclaimedthathumanspossessaspecialmechanismforpermissionsandobligations,specificallyfordetectingcheatinginsocialexchanges.Thiscanbeusedtoexplainwhyhumansareoftenmoresuccessfulindrawingvalidinferencesifthecontentsinvolvehumanbehaviorinrelationtosocialnorms.[3]Anotherexampleistheso-calleddual-processtheory.[5][3]Thistheorypositsthattherearetwodistinctcognitivesystemsresponsibleforreasoning.Theirinterrelationcanbeusedtoexplaincommonlyobservedbiasesindeductivereasoning.System1istheoldersystemintermsofevolution.Itisbasedonassociativelearningandhappensfastandautomaticallywithoutdemandingmanycognitiveresources.[5][3]System2,ontheotherhand,isofmorerecentevolutionaryorigin.Itisslowandcognitivelydemanding,butalsomoreflexibleandunderdeliberatecontrol.[5][3]Thedual-processtheorypositsthatsystem1isthedefaultsystemguidingmostofoureverydayreasoninginapragmaticway.Butforparticularlydifficultproblemsonthelogicallevel,system2isemployed.System2ismostlyresponsiblefordeductivereasoning.[5][3] Intelligence[edit] Theabilityofdeductivereasoningisanimportantaspectofintelligenceandmanytestsofintelligenceincludeproblemsthatcallfordeductiveinferences.[1]Becauseofthisrelationtointelligence,deductionishighlyrelevanttopsychologyandthecognitivesciences.[5]Butthesubjectofdeductivereasoningisalsopertinenttothecomputersciences,forexample,inthecreationofartificialintelligence.[1] Epistemology[edit] Deductivereasoningplaysanimportantroleinepistemology.Epistemologyisconcernedwiththequestionofjustification,i.e.topointoutwhichbeliefsarejustifiedandwhy.[49][50]Deductiveinferencesareabletotransferthejustificationofthepremisesontotheconclusion.[3]Sowhilelogicisinterestedinthetruth-preservingnatureofdeduction,epistemologyisinterestedinthejustification-preservingnatureofdeduction.Therearedifferenttheoriestryingtoexplainwhydeductivereasoningisjustification-preserving.[3]Accordingtoreliabilism,thisisthecasebecausedeductionsaretruth-preserving:theyarereliableprocessesthatensureatrueconclusiongiventhepremisesaretrue.[3][51][52]Sometheoristsholdthatthethinkerhastohaveexplicitawarenessofthetruth-preservingnatureoftheinferenceforthejustificationtobetransferredfromthepremisestotheconclusion.Oneconsequenceofsuchaviewisthat,foryoungchildren,thisdeductivetransferencedoesnottakeplacesincetheylackthisspecificawareness.[3] Probabilitylogic[edit] Probabilitylogicisinterestedinhowtheprobabilityofthepremisesofanargumentaffectstheprobabilityofitsconclusion.Itdiffersfromclassicallogic,whichassumesthatpropositionsareeithertrueorfalsebutdoesnottakeintoconsiderationtheprobabilityorcertaintythatapropositionistrueorfalse.[53][54]Theprobabilityoftheconclusionofadeductiveargumentcannotbecalculatedbyfiguringoutthecumulativeprobabilityoftheargument’spremises.Dr.TimothyMcGrew,aspecialistintheapplicationsofprobabilitytheory,andDr.ErnestW.Adams,aProfessorEmeritusatUCBerkeley,pointedoutthatthetheoremontheaccumulationofuncertaintydesignatesonlyalowerlimitontheprobabilityoftheconclusion.Sotheprobabilityoftheconjunctionoftheargument’spremisessetsonlyaminimumprobabilityoftheconclusion.Theprobabilityoftheargument’sconclusioncannotbeanylowerthantheprobabilityoftheconjunctionoftheargument’spremises.Forexample,iftheprobabilityofadeductiveargument’sfourpremisesis~0.43,thenitisassuredthattheprobabilityoftheargument’sconclusionisnolessthan~0.43.Itcouldbemuchhigher,butitcannotdropunderthatlowerlimit.[55][56] Therecanbeexamplesinwhicheachsinglepremiseismorelikelytruethannotandyetitwouldbeunreasonabletoaccepttheconjunctionofthepremises.ProfessorHenryKyburg,whowasknownforhisworkinprobabilityandlogic,clarifiedthattheissuehereisoneofclosure–specifically,closureunderconjunction.ThereareexampleswhereitisreasonabletoacceptPandreasonabletoacceptQwithoutitsbeingreasonabletoaccepttheconjunction(P&Q).Lotteriesserveasveryintuitiveexamplesofthis,becauseinabasicnon-discriminatoryfinitelotterywithonlyasinglewinnertobedrawn,itissoundtothinkthatticket1isaloser,soundtothinkthatticket2isaloser,...allthewayuptothefinalnumber.However,clearly,itisirrationaltoaccepttheconjunctionofthesestatements;theconjunctionwoulddenytheverytermsofthelotterybecause(takenwiththebackgroundknowledge)itwouldentailthatthereisnowinner.[57][56] Dr.McGrewfurtheraddsthatthesolemethodtoensurethataconclusiondeductivelydrawnfromagroupofpremisesismoreprobablethannotistousepremisestheconjunctionofwhichismoreprobablethannot.Thispointisslightlytricky,becauseitcanleadtoapossiblemisunderstanding.Whatisbeingsearchedforisageneralprinciplethatspecifiesfactorsunderwhich,foranylogicalconsequenceCofthegroupofpremises,Cismoreprobablethannot.Particularconsequenceswilldifferintheirprobability.However,thegoalistostateaconditionunderwhichthisattributeisensured,regardlessofwhichconsequenceonedraws,andfulfilmentofthatconditionisrequiredtocompletethetask. Thisprinciplecanbedemonstratedinamoderatelyclearway.Suppose,forinstance,thefollowinggroupofpremises: {P,Q,R} Supposethattheconjunction((P&Q)&R)failstobemoreprobablethannot.Thenthereisatleastonelogicalconsequenceofthegroupthatfailstobemoreprobablethannot–namely,thatveryconjunction.Soitisanessentialfactorfortheargumentto“preserveplausibility”(Dr.McGrewcoinsthisphrasetomean“guarantee,frominformationabouttheplausibilityofthepremisesalone,thatanyconclusiondrawnfromthosepremisesbydeductiveinferenceisitselfmoreplausiblethannot”)thattheconjunctionofthepremisesbemoreprobablethannot.[56] History[edit] Thissectionneedsexpansion.Youcanhelpbyaddingtoit.(January2015) Aristotle,aGreekphilosopher,starteddocumentingdeductivereasoninginthe4thcenturyBC.[58]RenéDescartes,inhisbookDiscourseonMethod,refinedtheideafortheScientificRevolution.Developingfourrulestofollowforprovinganideadeductively,Descarteslaidthefoundationforthedeductiveportionofthescientificmethod.Descartes'backgroundingeometryandmathematicsinfluencedhisideasonthetruthandreasoning,causinghimtodevelopasystemofgeneralreasoningnowusedformostmathematicalreasoning.Similartopostulates,Descartesbelievedthatideascouldbeself-evidentandthatreasoningalonemustprovethatobservationsarereliable.Theseideasalsolaythefoundationsfortheideasofrationalism.[59] Relatedconceptsandtheories[edit] Deductivism[edit] Deductivismisaphilosophicalpositionthatgivesprimacytodeductivereasoningorargumentsovertheirnon-deductivecounterparts.[60][61]Itisoftenunderstoodastheevaluativeclaimthatonlydeductiveinferencesaregoodorcorrectinferences.Thistheorywouldhavewide-reachingconsequencesforvariousfieldssinceitimpliesthattherulesofdeductionare"theonlyacceptablestandardofevidence".[60]Thisway,therationalityorcorrectnessofthedifferentformsofinductivereasoningisdenied.[61][62]Someformsofdeductivismexpressthisintermsofdegreesofreasonablenessorprobability.Inductiveinferencesareusuallyseenasprovidingacertaindegreeofsupportfortheirconclusion:theymakeitmorelikelythattheirconclusionistrue.Deductivismstatesthatsuchinferencesarenotrational:thepremiseseitherensuretheirconclusion,asindeductivereasoning,ortheydonotprovideanysupportatall.[63] OnemotivationfordeductivismistheproblemofinductionintroducedbyDavidHume.Itconsistsinthechallengeofexplaininghoworwhetherinductiveinferencesbasedonpastexperiencessupportconclusionsaboutfutureevents.[61][64][63]Forexample,achickencomestoexpect,basedonallitspastexperiences,thatthepersonenteringitscoopisgoingtofeedit,untilonedaytheperson"atlastwringsitsneckinstead".[65]AccordingtoKarlPopper'sfalsificationism,deductivereasoningaloneissufficient.Thisisduetoitstruth-preservingnature:atheorycanbefalsifiedifoneofitsdeductiveconsequencesisfalse.[66][67]Sowhileinductivereasoningdoesnotofferpositiveevidenceforatheory,thetheorystillremainsaviablecompetitoruntilfalsifiedbyempiricalobservation.Inthissense,deductionaloneissufficientfordiscriminatingbetweencompetinghypothesesaboutwhatisthecase.[61]Hypothetico-deductivismisacloselyrelatedscientificmethod,accordingtowhichscienceprogressesbyformulatinghypothesesandthenaimstofalsifythembytryingtomakeobservationsthatruncountertotheirdeductiveconsequences.[68][69] Naturaldeduction[edit] Theterm"naturaldeduction"referstoaclassofproofsystemsbasedonself-evidentrulesofinference.[70][71]ThefirstsystemsofnaturaldeductionweredevelopedbyGerhardGentzenandStanislawJaskowskiinthe1930s.Thecoremotivationwastogiveasimplepresentationofdeductivereasoningthatcloselymirrorshowreasoningactuallytakesplace.[72]Inthissense,naturaldeductionstandsincontrasttootherlessintuitiveproofsystems,suchasHilbert-styledeductivesystems,whichemployaxiomschemestoexpresslogicaltruths.[70]Naturaldeduction,ontheotherhand,avoidsaxiomsschemesbyincludingmanydifferentrulesofinferencethatcanbeusedtoformulateproofs.Theserulesofinferenceexpresshowlogicalconstantsbehave.Theyareoftendividedintointroductionrulesandeliminationrules.Introductionrulesspecifyunderwhichconditionsalogicalconstantmaybeintroducedintoanewsentenceoftheproof.[70][71]Forexample,theintroductionruleforthelogicalconstant" ∧ {\displaystyle\land} "(and)is" A , B ( A ∧ B ) {\displaystyle{\frac{A,B}{(A\landB)}}} ".Itexpressesthat,giventhepremises" A {\displaystyleA} "and" B {\displaystyleB} "individually,onemaydrawtheconclusion" A ∧ B {\displaystyleA\landB} "andtherebyincludeitinone'sproof.Thisway,thesymbol" ∧ {\displaystyle\land} "isintroducedintotheproof.Theremovalofthissymbolisgovernedbyotherrulesofinference,suchastheeliminationrule" ( A ∧ B ) A {\displaystyle{\frac{(A\landB)}{A}}} ",whichstatesthatonemaydeducethesentence" A {\displaystyleA} "fromthepremise" ( A ∧ B ) {\displaystyle(A\landB)} ".Similarintroductionandeliminationrulesaregivenforotherlogicalconstants,suchasthepropositionaloperator" ¬ {\displaystyle\lnot} ",thepropositionalconnectives" ∨ {\displaystyle\lor} "and" → {\displaystyle\rightarrow} ",andthequantifiers" ∃ {\displaystyle\exists} "and" ∀ {\displaystyle\forall} ".[70][71] Thefocusonrulesofinferencesinsteadofaxiomschemesisanimportantfeatureofnaturaldeduction.[70][71]Butthereisnogeneralagreementonhownaturaldeductionistobedefined.Sometheoristsholdthatallproofsystemswiththisfeatureareformsofnaturaldeduction.Thiswouldincludevariousformsofsequentcalculiortableaucalculi.Butothertheoristsusetheterminamorenarrowsense,forexample,torefertotheproofsystemsdevelopedbyGentzenandJaskowski.Becauseofitssimplicity,naturaldeductionisoftenusedforteachinglogictostudents.[70] Geometricalmethod[edit] Thegeometricalmethodisamethodofphilosophybasedondeductivereasoning.Itstartsfromasmallsetofself-evidentaxiomsandtriestobuildacomprehensivelogicalsystembasedonlyondeductiveinferencesfromthesefirstaxioms.[73]ItwasinitiallyformulatedbyBaruchSpinozaandcametoprominenceinvariousrationalistphilosophicalsystemsinthemodernera.[74]Itgetsitsnamefromtheformsofmathematicaldemonstrationfoundintraditionalgeometry,whichareusuallybasedonaxioms,definitions,andinferredtheorems.[75][76]Animportantmotivationofthegeometricalmethodistorepudiatephilosophicalskepticismbygroundingone'sphilosophicalsystemonabsolutelycertainaxioms.Deductivereasoningiscentraltothisendeavorbecauseofitsnecessarilytruth-preservingnature.Thisway,thecertaintyinitiallyinvestedonlyintheaxiomsistransferredtoallpartsofthephilosophicalsystem.[73] Onerecurrentcriticismofphilosophicalsystemsbuildusingthegeometricalmethodisthattheirinitialaxiomsarenotasself-evidentorcertainastheirdefendersproclaim.[73]Thisproblemliesbeyondthedeductivereasoningitself,whichonlyensuresthattheconclusionistrueifthepremisesaretrue,butnotthatthepremisesthemselvesaretrue.Forexample,Spinoza'sphilosophicalsystemhasbeencriticizedthiswaybasedonobjectionsraisedagainstthecausalaxiom,i.e.that"theknowledgeofaneffectdependsonandinvolvesknowledgeofitscause".[77]Adifferentcriticismtargetsnotthepremisesbutthereasoningitself,whichmayattimesimplicitlyassumepremisesthatarethemselvesnotself-evident.[73] Seealso[edit] Philosophyportal Webportal Abductivereasoning Analogicalreasoning Argument(logic) Argumentationtheory Correspondencetheoryoftruth Decisionmaking Decisiontheory Defeasiblereasoning Fallacy Faulttreeanalysis Geometry Hypothetico-deductivemethod Inference Inquiry Legalsyllogism Logicandrationality Logicalconsequence Logicalreasoning Mathematicallogic 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^Thornton,Stephen(2021)."KarlPopper:4.BasicStatements,FalsifiabilityandConvention".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved14March2022. ^Shea,Brendan."Popper,Karl:PhilosophyofScience".InternetEncyclopediaofPhilosophy.Retrieved14March2022. ^"hypothetico-deductivemethod".EncyclopediaBritannica.Retrieved14March2022. ^"hypothetico-deductivemethod".OxfordReference.Retrieved14March2022. ^abcdefIndrzejczak,Andrzej."NaturalDeduction".InternetEncyclopediaofPhilosophy.Retrieved15March2022. ^abcdPelletier,FrancisJeffry;Hazen,Allen(2021)."NaturalDeductionSystemsinLogic".TheStanfordEncyclopediaofPhilosophy.MetaphysicsResearchLab,StanfordUniversity.Retrieved15March2022. ^Gentzen,Gerhard(1934)."UntersuchungenüberdaslogischeSchließen.I".MathematischeZeitschrift.39(2):176–210.doi:10.1007/BF01201353.S2CID 121546341.IchwolltenunzunächsteinmaleinenFormalismusaufstellen,derdemwirklichenSchließenmöglichstnahekommt.Soergabsichein"KalküldesnatürlichenSchließens.(FirstIwishedtoconstructaformalismthatcomesascloseaspossibletoactualreasoning.Thusarosea"calculusofnaturaldeduction".) ^abcdDaly,Chris(2015)."IntroductionandHistoricalOverview".ThePalgraveHandbookofPhilosophicalMethods.PalgraveMacmillanUK.pp. 1–30.doi:10.1057/9781137344557_1.ISBN 978-1-137-34455-7. ^Dutton,BlakeD."Spinoza,BenedictDe".InternetEncyclopediaofPhilosophy.Retrieved16March2022. ^Goldenbaum,Ursula."GeometricalMethod".InternetEncyclopediaofPhilosophy.Retrieved17February2022. ^Nadler,Steven(2006)."Thegeometricmethod".Spinoza's'Ethics':AnIntroduction.CambridgeUniversityPress.pp. 35–51.ISBN 978-0-521-83620-3. ^Doppelt,Torin(2010)."3:TheTruthAbout1A4".Spinoza'sCausalAxiom:ADefense(PDF). Furtherreading[edit] VincentF.Hendricks,Thought2Talk:ACrashCourseinReflectionandExpression,NewYork:AutomaticPress/VIP,2005,ISBN 87-991013-7-8 PhilipJohnson-Laird,RuthM.J.Byrne,Deduction,PsychologyPress1991,ISBN 978-0-86377-149-1 Zarefsky,David,Argumentation:TheStudyofEffectiveReasoningPartsIandII,TheTeachingCompany2002 Bullemore,Thomas.ThePragmaticProblemofInduction. Externallinks[edit] LookupdeductivereasoninginWiktionary,thefreedictionary. WikiquotehasquotationsrelatedtoDeductivereasoning. LookupdeductivereasoninginWiktionary,thefreedictionary. WikiversityhaslearningresourcesaboutDeductiveLogic DeductivereasoningatPhilPapers DeductivereasoningattheIndianaPhilosophyOntologyProject "Deductivereasoning".InternetEncyclopediaofPhilosophy. 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