posterior analytics-第7章

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not…man　of　itstill　grant　simply　that　man　is　animal　and　not



not…animal；　and　the　conclusion　follows：　for　it　will　still　be　true　to



say　that　Calliaseven　if　it　be　also　true　to　say　that



not…Calliasis　animal　and　not　not…animal。　The　reason　is　that　the



major　term　is　predicable　not　only　of　the　middle；　but　of　something



other　than　the　middle　as　well；　being　of　wider　application；　so　that　the



conclusion　is　not　affected　even　if　the　middle　is　extended　to　cover　the



original　middle　term　and　also　what　is　not　the　original　middle　term。



　　The　law　that　every　predicate　can　be　either　truly　affirmed　or　truly



denied　of　every　subject　is　posited　by　such　demonstration　as　uses



reductio　ad　impossibile；　and　then　not　always　universally；　but　so　far



as　it　is　requisite；　within　the　limits；　that　is；　of　the　genus…the



genus；　I　mean　（as　I　have　already　explained）；　to　which　the　man　of



science　applies　his　demonstrations。　In　virtue　of　the　common　elements



of　demonstration…I　mean　the　common　axioms　which　are　used　as



premisses　of　demonstration；　not　the　subjects　nor　the　attributes



demonstrated　as　belonging　to　them…all　the　sciences　have　communion　with



one　another；　and　in　communion　with　them　all　is　dialectic　and　any



science　which　might　attempt　a　universal　proof　of　axioms　such　as　the



law　of　excluded　middle；　the　law　that　the　subtraction　of　equals　from



equals　leaves　equal　remainders；　or　other　axioms　of　the　same　kind。



Dialectic　has　no　definite　sphere　of　this　kind；　not　being　confined　to　a



single　genus。　Otherwise　its　method　would　not　be　interrogative；　for　the



interrogative　method　is　barred　to　the　demonstrator；　who　cannot　use　the



opposite　facts　to　prove　the　same　nexus。　This　was　shown　in　my　work　on



the　syllogism。







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　　If　a　syllogistic　question　is　equivalent　to　a　proposition　embodying



one　of　the　two　sides　of　a　contradiction；　and　if　each　science　has　its



peculiar　propositions　from　which　its　peculiar　conclusion　is　developed；



then　there　is　such　a　thing　as　a　distinctively　scientific　question；　and



it　is　the　interrogative　form　of　the　premisses　from　which　the



'appropriate'　conclusion　of　each　science　is　developed。　Hence　it　is



clear　that　not　every　question　will　be　relevant　to　geometry；　nor　to



medicine；　nor　to　any　other　science：　only　those　questions　will　be



geometrical　which　form　premisses　for　the　proof　of　the　theorems　of



geometry　or　of　any　other　science；　such　as　optics；　which　uses　the



same　basic　truths　as　geometry。　Of　the　other　sciences　the　like　is　true。



Of　these　questions　the　geometer　is　bound　to　give　his　account；　using



the　basic　truths　of　geometry　in　conjunction　with　his　previous



conclusions；　of　the　basic　truths　the　geometer；　as　such；　is　not　bound





to　give　any　account。　The　like　is　true　of　the　other　sciences。　There



is　a　limit；　then；　to　the　questions　which　we　may　put　to　each　man　of



science；　nor　is　each　man　of　science　bound　to　answer　all　inquiries　on



each　several　subject；　but　only　such　as　fall　within　the　defined　field



of　his　own　science。　If；　then；　in　controversy　with　a　geometer　qua



geometer　the　disputant　confines　himself　to　geometry　and　proves



anything　from　geometrical　premisses；　he　is　clearly　to　be　applauded；　if



he　goes　outside　these　he　will　be　at　fault；　and　obviously　cannot　even



refute　the　geometer　except　accidentally。　One　should　therefore　not



discuss　geometry　among　those　who　are　not　geometers；　for　in　such　a



company　an　unsound　argument　will　pass　unnoticed。　This　is



correspondingly　true　in　the　other　sciences。



　　Since　there　are　'geometrical'　questions；　does　it　follow　that　there



are　also　distinctively　'ungeometrical'　questions？　Further；　in　each



special　science…geometry　for　instance…what　kind　of　error　is　it　that



may　vitiate　questions；　and　yet　not　exclude　them　from　that　science？



Again；　is　the　erroneous　conclusion　one　constructed　from　premisses



opposite　to　the　true　premisses；　or　is　it　formal　fallacy　though　drawn



from　geometrical　premisses？　Or；　perhaps；　the　erroneous　conclusion　is



due　to　the　drawing　of　premisses　from　another　science；　e。g。　in　a



geometrical　controversy　a　musical　question　is　distinctively



ungeometrical；　whereas　the　notion　that　parallels　meet　is　in　one



sense　geometrical；　being　ungeometrical　in　a　different　fashion：　the



reason　being　that　'ungeometrical'；　like　'unrhythmical'；　is



equivocal；　meaning　in　the　one　case　not　geometry　at　all；　in　the　other



bad　geometry？　It　is　this　error；　i。e。　error　based　on　premisses　of



this　kind…'of'　the　science　but　false…that　is　the　contrary　of



science。　In　mathematics　the　formal　fallacy　is　not　so　common；　because



it　is　the　middle　term　in　which　the　ambiguity　lies；　since　the　major



is　predicated　of　the　whole　of　the　middle　and　the　middle　of　the　whole



of　the　minor　（the　predicate　of　course　never　has　the　prefix　'all'）；　and



in　mathematics　one　can；　so　to　speak；　see　these　middle　terms　with　an



intellectual　vision；　while　in　dialectic　the　ambiguity　may　escape



detection。　E。g。　'Is　every　circle　a　figure？'　A　diagram　shows　that



this　is　so；　but　the　minor　premiss　'Are　epics　circles？'　is　shown　by　the



diagram　to　be　false。



　　If　a　proof　has　an　inductive　minor　premiss；　one　should　not　bring　an



'objection'　against　it。　For　since　every　premiss　must　be　applicable



to　a　number　of　cases　（otherwise　it　will　not　be　true　in　every　instance；



which；　since　the　syllogism　proceeds　from　universals；　it　must　be）；　then



assuredly　the　same　is　true　of　an　'objection'；　since　premisses　and



'objections'　are　so　far　the　same　that　anything　which　can　be　validly



advanced　as　an　'objection'　must　be　such　that　it　could　take　the　form　of



a　premiss；　either　demonstrative　or　dialectical。　On　the　other　hand；



arguments　formally　illogical　do　sometimes　occur　through　taking　as



middles　mere　attributes　of　the　major　and　minor　terms。　An　instance　of



this　is　Caeneus'　proof　that　fire　increases　in　geometrical



proportion：　'Fire'；　he　argues；　'increases　rapidly；　and　so　does



geometrical　proportion'。　There　is　no　syllogism　so；　but　there　is　a



syllogism　if　the　most　rapidly　increasing　proportion　is　geometrical　and



the　most　rapidly　increasing　proportion　is　attributable　to　fire　in



its　motion。　Sometimes；　no　doubt；　it　is　impossible　to　reason　from



premisses　predicating　mere　attributes：　but　sometimes　it　is　possible；



though　the　possibility　is　overlooked。　If　false　premisses　could　never



give　true　conclusions　'resolution'　would　be　easy；　for　premisses　and



conclusion　would　in　that　case　inevitably　reciprocate。　I　might　then



argue　thus：　let　A　be　an　existing　fact；　let　the　existence　of　A　imply



such　and　such　facts　actually　known　to　me　to　exist；　which　we　may　call



B。　I　can　now；　since　they　reciprocate；　infer　A　from　B。



　　Reciprocation　of　premisses　and　conclusion　is　more　frequent　in



mathematics；　because　mathematics　takes　definitions；　but　never　an



accident；　for　its　premisses…a　second　characteristic　distinguishing



mathematical　reasoning　from　dialectical　disputations。



　　A　science　expands　not　by　the　interposition　of　fresh　middle　terms；



but　by　the　apposition　of　fresh　extreme　terms。　E。g。　A　is　predicated



of　B；　B　of　C；　C　of　D；　and　so　indefinitely。　Or　the　expansion　may　be



lateral：　e。g。　one　major　A；　may　be　proved　of　two　minors；　C　and　E。



Thus　let　A　represent　number…a　number　or　number　taken



indeterminately；　B　determinate　odd　number；　C　any　particular　odd



number。　We　can　then　predicate　A　of　C。　Next　let　D　represent　determinate



even　number；　and　E　even　number。　Then　A　is　predicable　of　E。







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　　Knowledge　of　the　fact　differs　from　knowledge　of　the　reasoned　fact。



To　begin　with；　they　differ　within　the　same　science　and　in　two　ways