Thecogencyoftheargumentdependsupontheapplicabilityoftheruletothefact。Ifmenbenotmortal，or，again，ifSocratesbenotaman，theinferenceisnotvalid；andthesetwodistinctissues，theissueoflawandtheissueoffact，mayberaisedinanycase。27*Thevalueofthesyllogismisthatitraisestheseissuesdistinctly。Theargumentisthusputinsuchaformastobeabsolutelyconclusiveifthepremisesbethemselvesgranted。
Itthereforeprovidesatestofthevalidityofthelogic。
Grantingthepremises，adenialoftheinferencemustinvolveacontradiction。Thatistheonlytestinpurelogic。Thesyllogismmust，therefore，beinasensetautologous，forotherwiseitcouldnotbeconclusive。Acceptanceo？thepremisesmustbeshownfromtheformofstatementtonecessitatetheadmissionoftheinference。Thisfollows，andthelogicallinkiscompleteandirrefragable，ifthemiddletermbeidenticalinbothpremises，andnototherwise。ThisiswhatMillindicatesbysayingthat’therulesofthesyllogismarerulesforcompellingapersontobeawareofthewholeofwhathemustundertaketodefendifhepersistsinmaintaininghisconclusion。’28*Ratiocination，ashesumsuphisviewelsewhere，’doesnotconsistofsyllogisms’；
butthesyllogismisausefulformulaintowhichitcan’translateitsreasonings，’andsoguaranteetheircorrectness。29*Ifthisbegranted，wemustconsidertheessentialstepofinferencetobeembodiedin，butnotcreatedby，thesyllogism。Correctreasoningcanalwaysbethrownintothisform。Thesyllogismemergeswhenthereasoningiscomplete。
’Theuseofthesyllogismisnoother，’saysMill，’thantheuseofgeneralpropositionsinreasoning。’Itisasecurityforcorrectgeneralisation。30*Wehave，then，stilltoaskwhatisthereasoningprocessforwhichthesyllogismprovidesatest。
Generalisationimpliesclassification。Ourgeneralruleormajorpremisestatessomepropertyofaclasstowhichtheindividualbelongs。Thequestionishowthisreferencetoaclassenablesustodrawinferenceswhichwecouldnotdrawfromtheindividualcase。TothisMillgivesasimpleanswer，whichisalreadyimpliedinhistheoryofpredication。WhenIsaythatSocratesisaman，Isaythathehastheattributesconnotedbythename。Heisarational，featherlessbiped，forexample。ButIalreadyknowbyobservationthatwiththeseattributesgoestheattributeofmortality。Theessenceofthereasoningprocessisthereforethat，fromthepossessionofcertainattributes，Iinferthepossessionofanotherattributewhichhascoexistedwiththempreviously。ThatIdo，infact，reasoninthiswayincountlesscasesisundeniable。Iknowthatacertainquality，saymalleability，goesalongwithotherqualitiesofcolour，shape，andsoforth，bywhichIrecogniseasubstanceasgold。Ican，itmaybe，givenootherreasonforbelievingthefutureconjunctionofthosequalitiesthanthefactoftheirpreviousconjunction。
Thebelief，thatis，isasamatteroffactgeneratedsimplybythepreviouscoincidenceorcorrespondstoconstantassociation。
Whetherthisexhauststhewholelogicalsignificancemaystillbedisputed；but，atanyrate，uponthesetermswecanescapefromthechargeoftautology。Theruleinthemajorpremiseregistersanumberofpreviousexperiencesofcoexistence。Whenwenoticesomeoftheattributesinagivencase，wemakeanadditiontoourknowledgebyapplyingtherule，thatis，byinferringthatanotherattributemaybeaddedtotheobservedattributes。This，then，givesarationalaccountoftheadvanceinknowledgemadethroughthesyllogisminthecasewheretheclasscanbedefinedasasimplesumofattributes。
Butisthisanadequateaccountofthereasoningprocessingeneral？ThereisanotherviewwhichsuggesteddifficultiestoMill。Hissolutionofthesedifficulties，marked，aswelearnfromtheAutobiography，anessentialstageinthedevelopmentofhisdoctrine。Referencetoaclassis，uponhisinterpretation，impliedinthesyllogism；andclassificationimpliesdefinition。
Aclassmeansallthingswhichhaveacertainlistofattributesstatedinthedefinition。Maywenottheninferotherpropertiesfromthedefinition？Maynotmortality，forexample，bededuciblefromtheotherattributesofman？Theassumptionthatwecandosoisconnectedwiththefallacymostcharacteristicofthemisuseofthesyllogism。Itisplainthatwemaycreateasmanyclassesasweplease，andmakenamesforcombinationsofattributeswhichhavenoactual，orevennopossible，existence。
Anyinferenceswhichwemakeonthestrengthofsuchclassificationmustbenugatoryorsimplytautologous。Ishowthatacertainpropositionfollowsfrommydefinition；butthatgivesnoguaranteeforitsconformitytotherealitiesbehindthedefinition。Your’proof’thatamanismortalmeanssimplythatifheisnotmortalyoudon’tcallhimaman。Thesyllogismtreatedonthatsystembecomessimplyanelaborateseriesofdevicesforbeggingthequestion。Fromsuchmethodsariseallthefutilitiesofscholasticism，andthedoctrineofessenceswhich，thoughLockeconfutedit，31*has’neverceasedtopoisonphilosophy。’32*Itmay，Isuppose，betakenforgrantedthatthesyllogismwasconstantlyappliedtocoversuchfallacies，andsofarMillisonsafeground。Thetheory，however，leadshimtoacharacteristicpoint。AlreadyintheearlyreviewJanuary1828，hehadcriticisedWhately’saccountofdefinition。A’realdefinition，’asWhatelyhadsaid，’explainsandunfolds“thenature“ofthethingdefined，whereasa“nominaldefinition“onlyexplainsthename。’Whatelygoesontopointoutthattheonlyrealdefinitionsinthissensearethemathematicaldefinitions。
Itisimpossibletodiscoverthepropertiesofathing，aman，oraplantfromthedefinition。Ifitwerepossible，wemightproceedto’evolveacamelfromthedepthsofourconsciousness，’
andnobodynowprofessestobeequaltothatfeat。When，however，we’define’acircleoralineandsoforth，wemakeassertionsfromwhichwecandeducethewholetheoryofgeometry。A
geometricalfigurerepresentsavastcomplexoftruths，mutuallyimplyingeachother，andalldeduciblefromafewsimpledefinitions。Themiddletermisnotthenameofasimplething，orofathingwhichhasacertainsetofcoexistingattributes，butawordexpressiveofawholesystemofreciprocalrelations。
Ifonepropertyentitlesmetosaythatacertainfigureisacircle，Iamvirtuallydeclaringthatithasinnumerableotherproperties，andIamthusabletomakeinferenceswhich，althoughimplicitlygiven，arenotperceivedtillexplicitlystated。Byassigningathingtoaclass，IsayinthiscasethatImaymakeanyoneofanindefinitenumberofpropositionsaboutit，allmutuallyimplyingeachother，andrequiringthehighestfacultiesforcombiningandevolving。Puremathematicsgivetheonegreatexampleofavastbodyoftruthsreachedbypurelydeductiveprocesses。Theyappeartobeevolvedfromcertainsimpleandself-evidenttruths。Canthey，then，beexplainedassimplyempirical？Doweknowthepropertiesofacircleasweknowthepropertiesofgold，simplybycombiningrecordsofpreviousexperience？Orcanweadmitthatthisgreatsystemoftruthisallevolvedoutof’definitions’？
MillscentsinWhately’sdoctrineataintofaprioriassumption，andaccordinglymeetsitbyadirectcontradiction。A
geometricaldefinition，hesays，isnomorea’real’definitionthanthedefinitionofacamel。Nodefinitionwhatevercan’unfoldthenature’ofathing。Hestatesthisinhisreview，thoughitwasatalaterperiod，33*whenmeditatinguponapassageofDugaldStewart，thatheperceivedthefullconsequencesofhisownposition。InansweringWhately，hehadsaidthatalldefinitionswere’nominal。’A’realdefinition’
meansthattothedefinitionproperweaddthestatementthatthereisathingcorrespondingtothename。34*Thedefinitionitselfisa’mereidenticalproposition，’fromwhichwecanlearnnothingastofacts。Butitmaybeaccompaniedbyapostulatewhich’covertlyassertsafact，’andfromthefactmayfollowconsequencesofanydegreeofimportance。Thisdistinctionbetweenthedefinitionandthepostulatemaybeexhibited，asheremarks，bysubstituting’means’for’is。’Ifwesay：acentaur’means’abeinghalfmanandhalfhorse，wegiveapuredefinition。Ifwesay：aman’is’afeatherlessbiped，ourstatementincludesthedefinition——man’means’featherlessbiped；butifwesaidnomore，noinferencecouldbemadeastofacts。Ifwearereallytoincreaseourknowledgebyusingthisdefinition，wemustaddthe’covert’assertionthatsuchfeatherlessbipedsexist。Themathematicalcaseisidentical，Stewarthadarguedthatgeometricalpropositionsfollowed，notfromtheaxiomsbut，fromthedefinitions。Fromthebareaxiomthatifequalsbeaddedtoequalsthewholesareequal，youcaninfernothing。Youmustalsoperceivetheparticularfigureswhicharecompared。Ofcoursethetruthoftheaxiomsmustbeadmitted；buttheydonotspecifythefirstprinciplesfromwhichgeometryisevolved。Inotherwords，geometryimplies’intuition，’nottheapriori’intuitions’towhichMillobjected，butthedirectperceptionofthespatialrelations。Wemustseethefigureaswellasadmittheself-evidentaxiom。
Mill，onconsideringthisargument，thoughtthatStewarthadstoppedatahalftruth。35*Heoughttohavegotridofthedefinitionsaswellastheaxioms。EverydemonstrationinEuclid，saysMill，mightbecarriedonwithoutthem。Whenwearguefromadiagraminwhichthereisacircle，wedonotreallyrefertocirclesingeneral，butonlytotheparticularcirclebeforeus。
Ifitsradiibeequalorapproximatelyequal，theconclusionsaretrue。Weafterwardsextendourreasoningtosimilarcases；butonlyoneinstanceisdemonstrated。Thedefinitionismerelya’noticetoourselvesandothers，’statingwhatassumptionswethinkourselvesentitledtomake；andinthiswayitresemblesthemajorinthesyllogism。Thedemonstrationdoesnot’dependupon’it，thoughifwedenyit，thedemonstrationfails。Bythisargument，Millconceivesthatthecaseofmathematicsisputonalevelwithothercases。Wealwaysarguefromfacts，andmoreoverfrom’particularfacts，’notfromdefinitions。Westartfromanobservationofthisparticularcircle——asensible’thing’orobject，asinarguingaboutnaturalhistorywestartfromobservationofthecamel。Hencewemaylaydownthegeneralproposition，applicabletogeometryaswellastoallordinaryobservation，that，allinferenceisfromparticularstoparticulars。’36*Thisisthe’foundation’bothofInduction，whichis’popularlysaid’toreasonfromparticularstogenerals，andofDeduction，whichissupposedtoreasonfromgeneralstoparticulars。37*ThissumsupMill’scharacteristicposition。
III。MATHEMATICALTRUTHS
ThisattempttobringallreasoningtothesametypeforcesMilltoignorewhattoothersseemstobeoftheessenceofthecase。Thereare，hesays，twostatements：’Theremayexistafigureboundedbythreestraightlines’；thatisthefruitfulstatementoffacts。’Thisfigureiscalledatriangle’；thatisthemerelynominaldefinitionorexplanationofwords。Moreover，ashesays，wemaydropthedefinitionbysubstitutingtheequivalentwordsorsimplylookingatthething。Itdoesnotfollowthatwecandispensewiththemodeofapprehensionimpliedbythedefinition。Whetherweusethewordtriangle，orthewords，’threelinesenclosingaspace，’ornowordsatall，wemustequallyhavetheconceptionsorintuitionsoflinesandspace。Alldemonstrationingeometryconsistsinmentallyrearrangingacombinationoflinesandanglessoastoshowthatonefiguremaybemadetocoincideabsolutelywithanotherfigure。Theoriginalfactremainsunaltered，butthewaysofapprehendingthefactareinnumerable。NewtonandhisdogDiamondmightbothseethesamecircularthing；buttoDiamondthecirclewasasimpleroundobject；toNewtonitwasalsoacomplexsystemofrelatedlines，capableofbeingsoregardedastoembodyavastvarietyofelaborateformulae。38*Geometry，asMillundeniablysays，dealswithfacts。NewtonandDiamondhavepreciselythesamefactbeforethem。Itremainsthesame，whetherwestopatthesimpleststageorproceedtothemostcomplexevolutionofgeometry。ThedifferencebetweentheobserversisnotthatNewtonhasseennewfacts，butthatheseesmoreinthesamefact。Thechangeisnotinthethingsbutinthemind，which，bygroupingthethingsinthewaypointedoutbythedefinitions，isabletodiscovercountlessnewrelationsinvolvedinthesameperception。Thisagainmaysuggestthateventhefactrevealedtosimpleperceptionisnotabare’fact，’something，asMillputsit，’externaltothemind，’butisinsomesenseitselfconstitutedbythefacultyofperception。Itcontainsalreadythegermofthewholeintellectualevolution。Thechangeisnotinthethingperceived，butinthemodeofperceiving。And，therefore，again，wedonotacquirenewknowledge，asweacquireitinthephysicalsciences，byobservingnewfacts，discoveringresemblancesanddifferences，andgeneralisingfromthepropertiescommontoall；butbycontemplatingthesamefact。Allgeometryisinanyparticularspace——ifonlywecanfindit。Wedonotproceedbycomparinganumberofdifferentregionsofspaces，andinquirewhetherFrenchtriangleshavethesamepropertiesasEnglishtriangles。ToMill，however，thestatementthatgeometrydealswithfactleadstoanotherconclusion。Wemustdealwiththesefactsaswithotherfacts，andfollowthemethodofothernaturalsciences。Wereallyproceedinthesamewaywhetherweareinvestigatingthepropertiesofanellipseoracamel。Ineithercasewemustdiscovertruthbyexperience。