posterior analytics-第13部分

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as　we　have　exemplified；　it　follows　that　in　the　case　of　the　other



causes　also　full　knowledge　is　attained　when　an　attribute　no　longer



inheres　because　of　something　else。　Thus；　when　we　learn　that　exterior



angles　are　equal　to　four　right　angles　because　they　are　the　exterior



angles　of　an　isosceles；　there　still　remains　the　question　'Why　has



isosceles　this　attribute？'　and　its　answer　'Because　it　is　a　triangle；



and　a　triangle　has　it　because　a　triangle　is　a　rectilinear　figure。'



If　rectilinear　figure　possesses　the　property　for　no　further　reason；　at



this　point　we　have　full　knowledge…but　at　this　point　our　knowledge



has　become　commensurately　universal；　and　so　we　conclude　that



commensurately　universal　demonstration　is　superior。



　　（6）　The　more　demonstration　becomes　particular　the　more　it　sinks　into



an　indeterminate　manifold；　while　universal　demonstration　tends　to



the　simple　and　determinate。　But　objects　so　far　as　they　are　an



indeterminate　manifold　are　unintelligible；　so　far　as　they　are



determinate；　intelligible：　they　are　therefore　intelligible　rather　in



so　far　as　they　are　universal　than　in　so　far　as　they　are　particular。



From　this　it　follows　that　universals　are　more　demonstrable：　but



since　relative　and　correlative　increase　concomitantly；　of　the　more



demonstrable　there　will　be　fuller　demonstration。　Hence　the



commensurate　and　universal　form；　being　more　truly　demonstration；　is



the　superior。



　　（7）　Demonstration　which　teaches　two　things　is　preferable　to



demonstration　which　teaches　only　one。　He　who　possesses



commensurately　universal　demonstration　knows　the　particular　as　well；



but　he　who　possesses　particular　demonstration　does　not　know　the



universal。　So　that　this　is　an　additional　reason　for　preferring



commensurately　universal　demonstration。　And　there　is　yet　this



further　argument：



　　（8）　Proof　becomes　more　and　more　proof　of　the　commensurate



universal　as　its　middle　term　approaches　nearer　to　the　basic　truth；　and



nothing　is　so　near　as　the　immediate　premiss　which　is　itself　the



basic　truth。　If；　then；　proof　from　the　basic　truth　is　more　accurate



than　proof　not　so　derived；　demonstration　which　depends　more　closely　on



it　is　more　accurate　than　demonstration　which　is　less　closely



dependent。　But　commensurately　universal　demonstration　is　characterized



by　this　closer　dependence；　and　is　therefore　superior。　Thus；　if　A　had



to　be　proved　to　inhere　in　D；　and　the　middles　were　B　and　C；　B　being　the



higher　term　would　render　the　demonstration　which　it　mediated　the



more　universal。



　　Some　of　these　arguments；　however；　are　dialectical。　The　clearest



indication　of　the　precedence　of　commensurately　universal　demonstration



is　as　follows：　if　of　two　propositions；　a　prior　and　a　posterior；　we



have　a　grasp　of　the　prior；　we　have　a　kind　of　knowledge…a　potential



grasp…of　the　posterior　as　well。　For　example；　if　one　knows　that　the



angles　of　all　triangles　are　equal　to　two　right　angles；　one　knows　in



a　sense…potentially…that　the　isosceles'　angles　also　are　equal　to　two



right　angles；　even　if　one　does　not　know　that　the　isosceles　is　a



triangle；　but　to　grasp　this　posterior　proposition　is　by　no　means　to



know　the　commensurate　universal　either　potentially　or　actually。



Moreover；　commensurately　universal　demonstration　is　through　and



through　intelligible；　particular　demonstration　issues　in



sense…perception。







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　　The　preceding　arguments　constitute　our　defence　of　the　superiority　of



commensurately　universal　to　particular　demonstration。　That　affirmative



demonstration　excels　negative　may　be　shown　as　follows。



　　（1）　We　may　assume　the　superiority　ceteris　paribus　of　the



demonstration　which　derives　from　fewer　postulates　or　hypotheses…in



short　from　fewer　premisses；　for；　given　that　all　these　are　equally　well



known；　where　they　are　fewer　knowledge　will　be　more　speedily



acquired；　and　that　is　a　desideratum。　The　argument　implied　in　our



contention　that　demonstration　from　fewer　assumptions　is　superior　may



be　set　out　in　universal　form　as　follows。　Assuming　that　in　both　cases



alike　the　middle　terms　are　known；　and　that　middles　which　are　prior　are



better　known　than　such　as　are　posterior；　we　may　suppose　two



demonstrations　of　the　inherence　of　A　in　E；　the　one　proving　it



through　the　middles　B；　C　and　D；　the　other　through　F　and　G。　Then　A…D　is



known　to　the　same　degree　as　A…E　（in　the　second　proof）；　but　A…D　is



better　known　than　and　prior　to　A…E　（in　the　first　proof）；　since　A…E



is　proved　through　A…D；　and　the　ground　is　more　certain　than　the



conclusion。



　　Hence　demonstration　by　fewer　premisses　is　ceteris　paribus



superior。　Now　both　affirmative　and　negative　demonstration　operate



through　three　terms　and　two　premisses；　but　whereas　the　former



assumes　only　that　something　is；　the　latter　assumes　both　that　something



is　and　that　something　else　is　not；　and　thus　operating　through　more



kinds　of　premiss　is　inferior。



　　（2）　It　has　been　proved　that　no　conclusion　follows　if　both



premisses　are　negative；　but　that　one　must　be　negative；　the　other



affirmative。　So　we　are　compelled　to　lay　down　the　following



additional　rule：　as　the　demonstration　expands；　the　affirmative



premisses　must　increase　in　number；　but　there　cannot　be　more　than　one



negative　premiss　in　each　complete　proof。　Thus；　suppose　no　B　is　A；



and　all　C　is　B。　Then　if　both　the　premisses　are　to　be　again　expanded；　a



middle　must　be　interposed。　Let　us　interpose　D　between　A　and　B；　and　E



between　B　and　C。　Then　clearly　E　is　affirmatively　related　to　B　and　C；



while　D　is　affirmatively　related　to　B　but　negatively　to　A；　for　all　B



is　D；　but　there　must　be　no　D　which　is　A。　Thus　there　proves　to　be　a



single　negative　premiss；　A…D。　In　the　further　prosyllogisms　too　it　is



the　same；　because　in　the　terms　of　an　affirmative　syllogism　the



middle　is　always　related　affirmatively　to　both　extremes；　in　a　negative



syllogism　it　must　be　negatively　related　only　to　one　of　them；　and　so



this　negation　comes　to　be　a　single　negative　premiss；　the　other



premisses　being　affirmative。　If；　then；　that　through　which　a　truth　is



proved　is　a　better　known　and　more　certain　truth；　and　if　the　negative



proposition　is　proved　through　the　affirmative　and　not　vice　versa；



affirmative　demonstration；　being　prior　and　better　known　and　more



certain；　will　be　superior。



　　（3）　The　basic　truth　of　demonstrative　syllogism　is　the　universal



immediate　premiss；　and　the　universal　premiss　asserts　in　affirmative



demonstration　and　in　negative　denies：　and　the　affirmative



proposition　is　prior　to　and　better　known　than　the　negative　（since



affirmation　explains　denial　and　is　prior　to　denial；　just　as　being　is



prior　to　not…being）。　It　follows　that　the　basic　premiss　of



affirmative　demonstration　is　superior　to　that　of　negative



demonstration；　and　the　demonstration　which　uses　superior　basic



premisses　is　superior。



　　（4）　Affirmative　demonstration　is　more　of　the　nature　of　a　basic



form　of　proof；　because　it　is　a　sine　qua　non　of　negative　demonstration。







　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　26







　　Since　affirmative　demonstration　is　superior　to　negative；　it　is



clearly　superior　also　to　reductio　ad　impossibile。　We　must　first　make



certain　what　is　the　difference　between　negative　demonstration　and



reductio　ad　impossibile。　Let　us　suppose　that　no　B　is　A；　and　that　all　C



is　B：　the　conclusion　necessarily　follows　that　no　C　is　A。　If　these



premisses　are　assumed；　therefore；　the　negative　demonstration　that　no　C



is　A　is　direct。　Reductio　ad　impossibile；　on　the　other　hand；　proceeds



as　follows。　Supposing　we　are　to　prove　that　does　not　inhere　in　B；　we



have　to　assume　that　it　does　inhere；　and　further　that　B　inheres　in　C；



with　the　resulting　inference　that　A　inheres　in　C。　This　we　have　to



suppose　a　known　and　admitted　impossibility；　and　we　then　infer　that　A



cannot　inhere　in　B。　Thus　if　the　inherence　of　B　in　C　is　not　questioned；



A's　inherence　in　B　is　impossible。



　　The　order　of　the　terms　is　the　same　in　both　proofs：　they　differ



according　to　which　of　the　negative　propositions　is　the　better　known；



the　one　denying　A　of　B　or　the　one　denying　A　of　C。　When　the　falsity



of　the　conclusion　is　the　better　known；　we　use　reductio　ad



impossible；　when　the　major　premiss　of　the　syllogism　is　the　more



obvious；　we　use　direct　demonstration。　All　the　same　the　proposition



denying　A　of　B　is；　in　the　order　of　being；　prior　to　that　denying　A　of



C；　for　premisses　are　prior　to　the　conclusion　which　follows　from



them；　and　'no　C　is　A'　is　the　conclusion；　'no　B　is　A'　one　of　its



premisses。　For　the　destructive　result　of　reductio