posterior analytics-第3章

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figure　selected　at　haphazard；　nor　in　demonstrating　does　one　take　a



figure　at　random…a　square　is　a　figure　but　its　angles　are　not　equal



to　two　right　angles。　On　the　other　hand；　any　isosceles　triangle　has　its



angles　equal　to　two　right　angles；　yet　isosceles　triangle　is　not　the



primary　subject　of　this　attribute　but　triangle　is　prior。　So　whatever



can　be　shown　to　have　its　angles　equal　to　two　right　angles；　or　to



possess　any　other　attribute；　in　any　random　instance　of　itself　and



primarily…that　is　the　first　subject　to　which　the　predicate　in　question



belongs　commensurately　and　universally；　and　the　demonstration；　in



the　essential　sense；　of　any　predicate　is　the　proof　of　it　as



belonging　to　this　first　subject　commensurately　and　universally：



while　the　proof　of　it　as　belonging　to　the　other　subjects　to　which　it



attaches　is　demonstration　only　in　a　secondary　and　unessential　sense。



Nor　again　（2）　is　equality　to　two　right　angles　a　commensurately



universal　attribute　of　isosceles；　it　is　of　wider　application。







　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　5







　　We　must　not　fail　to　observe　that　we　often　fall　into　error　because



our　conclusion　is　not　in　fact　primary　and　commensurately　universal



in　the　sense　in　which　we　think　we　prove　it　so。　We　make　this　mistake



（1）　when　the　subject　is　an　individual　or　individuals　above　which　there



is　no　universal　to　be　found：　（2）　when　the　subjects　belong　to　different



species　and　there　is　a　higher　universal；　but　it　has　no　name：　（3）



when　the　subject　which　the　demonstrator　takes　as　a　whole　is　really



only　a　part　of　a　larger　whole；　for　then　the　demonstration　will　be　true



of　the　individual　instances　within　the　part　and　will　hold　in　every



instance　of　it；　yet　the　demonstration　will　not　be　true　of　this　subject



primarily　and　commensurately　and　universally。　When　a　demonstration



is　true　of　a　subject　primarily　and　commensurately　and　universally；



that　is　to　be　taken　to　mean　that　it　is　true　of　a　given　subject



primarily　and　as　such。　Case　（3）　may　be　thus　exemplified。　If　a　proof



were　given　that　perpendiculars　to　the　same　line　are　parallel；　it　might



be　supposed　that　lines　thus　perpendicular　were　the　proper　subject　of



the　demonstration　because　being　parallel　is　true　of　every　instance



of　them。　But　it　is　not　so；　for　the　parallelism　depends　not　on　these



angles　being　equal　to　one　another　because　each　is　a　right　angle；　but



simply　on　their　being　equal　to　one　another。　An　example　of　（1）　would　be



as　follows：　if　isosceles　were　the　only　triangle；　it　would　be　thought



to　have　its　angles　equal　to　two　right　angles　qua　isosceles。　An



instance　of　（2）　would　be　the　law　that　proportionals　alternate。



Alternation　used　to　be　demonstrated　separately　of　numbers；　lines；



solids；　and　durations；　though　it　could　have　been　proved　of　them　all　by



a　single　demonstration。　Because　there　was　no　single　name　to　denote



that　in　which　numbers；　lengths；　durations；　and　solids　are　identical；



and　because　they　differed　specifically　from　one　another；　this　property



was　proved　of　each　of　them　separately。　To…day；　however；　the　proof　is



commensurately　universal；　for　they　do　not　possess　this　attribute　qua



lines　or　qua　numbers；　but　qua　manifesting　this　generic　character　which



they　are　postulated　as　possessing　universally。　Hence；　even　if　one



prove　of　each　kind　of　triangle　that　its　angles　are　equal　to　two



right　angles；　whether　by　means　of　the　same　or　different　proofs；　still；



as　long　as　one　treats　separately　equilateral；　scalene；　and



isosceles；　one　does　not　yet　know；　except　sophistically；　that



triangle　has　its　angles　equal　to　two　right　angles；　nor　does　one　yet



know　that　triangle　has　this　property　commensurately　and　universally；



even　if　there　is　no　other　species　of　triangle　but　these。　For　one



does　not　know　that　triangle　as　such　has　this　property；　nor　even　that



'all'　triangles　have　it…unless　'all'　means　'each　taken　singly'：　if



'all'　means　'as　a　whole　class'；　then；　though　there　be　none　in　which



one　does　not　recognize　this　property；　one　does　not　know　it　of　'all



triangles'。



　　When；　then；　does　our　knowledge　fail　of　commensurate　universality；



and　when　it　is　unqualified　knowledge？　If　triangle　be　identical　in



essence　with　equilateral；　i。e。　with　each　or　all　equilaterals；　then



clearly　we　have　unqualified　knowledge：　if　on　the　other　hand　it　be　not；



and　the　attribute　belongs　to　equilateral　qua　triangle；　then　our



knowledge　fails　of　commensurate　universality。　'But'；　it　will　be　asked；



'does　this　attribute　belong　to　the　subject　of　which　it　has　been



demonstrated　qua　triangle　or　qua　isosceles？　What　is　the　point　at　which



the　subject。　to　which　it　belongs　is　primary？　（i。e。　to　what　subject　can



it　be　demonstrated　as　belonging　commensurately　and　universally？）'



Clearly　this　point　is　the　first　term　in　which　it　is　found　to　inhere　as



the　elimination　of　inferior　differentiae　proceeds。　Thus　the　angles



of　a　brazen　isosceles　triangle　are　equal　to　two　right　angles：　but



eliminate　brazen　and　isosceles　and　the　attribute　remains。　'But'…you



may　say…'eliminate　figure　or　limit；　and　the　attribute　vanishes。'　True；



but　figure　and　limit　are　not　the　first　differentiae　whose



elimination　destroys　the　attribute。　'Then　what　is　the　first？'　If　it　is



triangle；　it　will　be　in　virtue　of　triangle　that　the　attribute



belongs　to　all　the　other　subjects　of　which　it　is　predicable；　and



triangle　is　the　subject　to　which　it　can　be　demonstrated　as　belonging



commensurately　and　universally。







　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　6







　　Demonstrative　knowledge　must　rest　on　necessary　basic　truths；　for　the



object　of　scientific　knowledge　cannot　be　other　than　it　is。　Now



attributes　attaching　essentially　to　their　subjects　attach



necessarily　to　them：　for　essential　attributes　are　either　elements　in



the　essential　nature　of　their　subjects；　or　contain　their　subjects　as



elements　in　their　own　essential　nature。　（The　pairs　of　opposites



which　the　latter　class　includes　are　necessary　because　one　member　or



the　other　necessarily　inheres。）　It　follows　from　this　that　premisses　of



the　demonstrative　syllogism　must　be　connexions　essential　in　the



sense　explained：　for　all　attributes　must　inhere　essentially　or　else　be



accidental；　and　accidental　attributes　are　not　necessary　to　their



subjects。



　　We　must　either　state　the　case　thus；　or　else　premise　that　the



conclusion　of　demonstration　is　necessary　and　that　a　demonstrated



conclusion　cannot　be　other　than　it　is；　and　then　infer　that　the



conclusion　must　be　developed　from　necessary　premisses。　For　though



you　may　reason　from　true　premisses　without　demonstrating；　yet　if



your　premisses　are　necessary　you　will　assuredly　demonstrate…in　such



necessity　you　have　at　once　a　distinctive　character　of　demonstration。



That　demonstration　proceeds　from　necessary　premisses　is　also　indicated



by　the　fact　that　the　objection　we　raise　against　a　professed



demonstration　is　that　a　premiss　of　it　is　not　a　necessary　truth…whether



we　think　it　altogether　devoid　of　necessity；　or　at　any　rate　so　far　as



our　opponent's　previous　argument　goes。　This　shows　how　naive　it　is　to



suppose　one's　basic　truths　rightly　chosen　if　one　starts　with　a



proposition　which　is　（1）　popularly　accepted　and　（2）　true；　such　as



the　sophists'　assumption　that　to　know　is　the　same　as　to　possess



knowledge。　For　（1）　popular　acceptance　or　rejection　is　no　criterion



of　a　basic　truth；　which　can　only　be　the　primary　law　of　the　genus



constituting　the　subject　matter　of　the　demonstration；　and　（2）　not



all　truth　is　'appropriate'。



　　A　further　proof　that　the　conclusion　must　be　the　development　of



necessary　premisses　is　as　follows。　Where　demonstration　is　possible；



one　who　can　give　no　account　which　includes　the　cause　has　no　scientific



knowledge。　If；　then；　we　suppose　a　syllogism　in　which；　though　A



necessarily　inheres　in　C；　yet　B；　the　middle　term　of　the　demonstration；



is　not　necessarily　connected　with　A　and　C；　then　the　man　who　argues



thus　has　no　reasoned　knowledge　of　the　conclusion；　since　this



conclusion　does　not　owe　its　necessity　to　the　middle　term；　for　though



the　conclusion　is　necessary；　the　mediating　link　is　a　contingent



fact。　Or　again；　if　a　man　is　without　knowledge　now；　though　he　still



retains　the　steps　of　the　argument；　though　there　is　no　change　in



himself　or　in　the　fact　and　no　lapse　of　memory　on　his　part；　then



neither　had　he　knowledge　previously。　But　the　mediating　link；　not　being



necessary；　may　have　perished　in　the　interval；　and　if　so；　though



there　be　no　change　in　him　nor　in　the　fact；　and　though　he　will　still



retain　the　steps　of　the　argument；　yet　he　has　not　knowledge；　and



therefore　had　not　knowledge　before。　Even　if　the　link　has　not



actually　perished　but　is　liable　to　perish；　this　situation　is



possible　and　might　occur。　But　such　a　condition　cannot　be　knowledge。



　　When　the　conclusion　is　necessary；　the　middle　through　which　it　was



proved　may　yet　quite　easily　be　non…necessary。　You　can　in　fact　infer



the　necessary　even　from　a　non…necessary　premiss；　just　as　you　can　infer



the　true　from　the　not　true。　On　the　other　hand；　when　the　middle　is



necessary　the　conclusion　must　be　necessary；　just　as　true　premisses



always　give　a　true　conclusion。　Thus；　if　A　is　necessarily　predicated　of



B　and　B　of　C；　then　A　is　necessarily　predicated　of　C。　But　when　the



conclusion　is　nonnecessary　the　middle　cannot　be　necessary　either。



Thus：　let　A　be　predicated　non…necessarily　of　C　but　necessarily　of　B；



and　let　B　be　a　necessary　predicate　of　C；　then　A　too　will　be　a



necessary　predicate　of　C；　which　by　hypothesis　it　is　not。



　　To　sum　up；　then：　demonstrative　knowledge　must　be　knowledge　of　a



necessary　nexus；　and　therefore　must　clearly　be　obtained　through　a



necessary　middle　term；　otherwise　its　possessor　will　know　neither　the



cause　nor　the　fact　that　his　conclusion　is　a　necessary　connexion。



Either　he　will　mistake　the　non…necessary　for　the　necessary　and　believe



the　necessity　of　the　conclusion　without　knowing　it；　or　else　he　will



not　even　believe　it…in　which　case　he　will　be　equally　ignorant；　whether



he　actually　infers　the　mere　fact　through　middle　terms　or　the



reasoned　fact　and　from　immediate　premisses。



　　Of　accidents　that　are　not　essential　according　to　our　definition　of



essential　there　is　no　demonstrative　knowledge；　for　since　an



accident；　in　the　sense　in　which　I　here　speak　of　it；　may　also　not



inhere；　it　is　impossible　to　prove　its　inherence　as　a　necessary



conclusion。　A　difficulty；　however；　might　be　raised　as　to　why　in



dialectic；　if　the　co