posterior analytics-第13章

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and　these　cannot　be　infinite；　the　ascending　series　will　terminate；　and



consequently　the　descending　series　too。



　　If　this　is　so；　it　follows　that　the　intermediates　between　any　two



terms　are　also　always　limited　in　number。　An　immediately　obvious



consequence　of　this　is　that　demonstrations　necessarily　involve　basic



truths；　and　that　the　contention　of　some…referred　to　at　the　outset…that



all　truths　are　demonstrable　is　mistaken。　For　if　there　are　basic



truths；　（a）　not　all　truths　are　demonstrable；　and　（b）　an　infinite



regress　is　impossible；　since　if　either　（a）　or　（b）　were　not　a　fact；



it　would　mean　that　no　interval　was　immediate　and　indivisible；　but　that



all　intervals　were　divisible。　This　is　true　because　a　conclusion　is



demonstrated　by　the　interposition；　not　the　apposition；　of　a　fresh



term。　If　such　interposition　could　continue　to　infinity　there　might



be　an　infinite　number　of　terms　between　any　two　terms；　but　this　is



impossible　if　both　the　ascending　and　descending　series　of



predication　terminate；　and　of　this　fact；　which　before　was　shown



dialectically；　analytic　proof　has　now　been　given。







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　　It　is　an　evident　corollary　of　these　conclusions　that　if　the　same



attribute　A　inheres　in　two　terms　C　and　D　predicable　either　not　at　all；



or　not　of　all　instances；　of　one　another；　it　does　not　always　belong



to　them　in　virtue　of　a　common　middle　term。　Isosceles　and　scalene



possess　the　attribute　of　having　their　angles　equal　to　two　right　angles



in　virtue　of　a　common　middle；　for　they　possess　it　in　so　far　as　they



are　both　a　certain　kind　of　figure；　and　not　in　so　far　as　they　differ



from　one　another。　But　this　is　not　always　the　case：　for；　were　it　so；　if



we　take　B　as　the　common　middle　in　virtue　of　which　A　inheres　in　C　and



D；　clearly　B　would　inhere　in　C　and　D　through　a　second　common　middle；



and　this　in　turn　would　inhere　in　C　and　D　through　a　third；　so　that



between　two　terms　an　infinity　of　intermediates　would　fall…an



impossibility。　Thus　it　need　not　always　be　in　virtue　of　a　common　middle



term　that　a　single　attribute　inheres　in　several　subjects；　since



there　must　be　immediate　intervals。　Yet　if　the　attribute　to　be　proved



common　to　two　subjects　is　to　be　one　of　their　essential　attributes；　the



middle　terms　involved　must　be　within　one　subject　genus　and　be



derived　from　the　same　group　of　immediate　premisses；　for　we　have　seen



that　processes　of　proof　cannot　pass　from　one　genus　to　another。



　　It　is　also　clear　that　when　A　inheres　in　B；　this　can　be



demonstrated　if　there　is　a　middle　term。　Further；　the　'elements'　of



such　a　conclusion　are　the　premisses　containing　the　middle　in　question；



and　they　are　identical　in　number　with　the　middle　terms；　seeing　that



the　immediate　propositions…or　at　least　such　immediate　propositions



as　are　universal…are　the　'elements'。　If；　on　the　other　hand；　there　is



no　middle　term；　demonstration　ceases　to　be　possible：　we　are　on　the　way



to　the　basic　truths。　Similarly　if　A　does　not　inhere　in　B；　this　can



be　demonstrated　if　there　is　a　middle　term　or　a　term　prior　to　B　in



which　A　does　not　inhere：　otherwise　there　is　no　demonstration　and　a



basic　truth　is　reached。　There　are；　moreover；　as　many　'elements'　of　the



demonstrated　conclusion　as　there　are　middle　terms；　since　it　is



propositions　containing　these　middle　terms　that　are　the　basic



premisses　on　which　the　demonstration　rests；　and　as　there　are　some



indemonstrable　basic　truths　asserting　that　'this　is　that'　or　that



'this　inheres　in　that'；　so　there　are　others　denying　that　'this　is



that'　or　that　'this　inheres　in　that'…in　fact　some　basic　truths　will



affirm　and　some　will　deny　being。



　　When　we　are　to　prove　a　conclusion；　we　must　take　a　primary



essential　predicate…suppose　it　C…of　the　subject　B；　and　then　suppose



A　similarly　predicable　of　C。　If　we　proceed　in　this　manner；　no



proposition　or　attribute　which　falls　beyond　A　is　admitted　in　the



proof：　the　interval　is　constantly　condensed　until　subject　and



predicate　become　indivisible；　i。e。　one。　We　have　our　unit　when　the



premiss　becomes　immediate；　since　the　immediate　premiss　alone　is　a



single　premiss　in　the　unqualified　sense　of　'single'。　And　as　in　other



spheres　the　basic　element　is　simple　but　not　identical　in　all…in　a



system　of　weight　it　is　the　mina；　in　music　the　quarter…tone；　and　so



onso　in　syllogism　the　unit　is　an　immediate　premiss；　and　in　the



knowledge　that　demonstration　gives　it　is　an　intuition。　In



syllogisms；　then；　which　prove　the　inherence　of　an　attribute；　nothing



falls　outside　the　major　term。　In　the　case　of　negative　syllogisms　on



the　other　hand；　（1）　in　the　first　figure　nothing　falls　outside　the



major　term　whose　inherence　is　in　question；　e。g。　to　prove　through　a



middle　C　that　A　does　not　inhere　in　B　the　premisses　required　are；　all　B



is　C；　no　C　is　A。　Then　if　it　has　to　be　proved　that　no　C　is　A；　a



middle　must　be　found　between　and　C；　and　this　procedure　will　never



vary。



　　（2）　If　we　have　to　show　that　E　is　not　D　by　means　of　the　premisses；



all　D　is　C；　no　E；　or　not　all　E；　is　C；　then　the　middle　will　never



fall　beyond　E；　and　E　is　the　subject　of　which　D　is　to　be　denied　in



the　conclusion。



　　（3）　In　the　third　figure　the　middle　will　never　fall　beyond　the　limits



of　the　subject　and　the　attribute　denied　of　it。







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　　Since　demonstrations　may　be　either　commensurately　universal　or



particular；　and　either　affirmative　or　negative；　the　question　arises；



which　form　is　the　better？　And　the　same　question　may　be　put　in　regard



to　so…called　'direct'　demonstration　and　reductio　ad　impossibile。　Let



us　first　examine　the　commensurately　universal　and　the　particular



forms；　and　when　we　have　cleared　up　this　problem　proceed　to　discuss



'direct'　demonstration　and　reductio　ad　impossibile。



　　The　following　considerations　might　lead　some　minds　to　prefer



particular　demonstration。



　　（1）　The　superior　demonstration　is　the　demonstration　which　gives　us



greater　knowledge　（for　this　is　the　ideal　of　demonstration）；　and　we



have　greater　knowledge　of　a　particular　individual　when　we　know　it　in



itself　than　when　we　know　it　through　something　else；　e。g。　we　know



Coriscus　the　musician　better　when　we　know　that　Coriscus　is　musical



than　when　we　know　only　that　man　is　musical；　and　a　like　argument



holds　in　all　other　cases。　But　commensurately　universal



demonstration；　instead　of　proving　that　the　subject　itself　actually



is　x；　proves　only　that　something　else　is　x…　e。g。　in　attempting　to



prove　that　isosceles　is　x；　it　proves　not　that　isosceles　but　only　that



triangle　is　x…　whereas　particular　demonstration　proves　that　the



subject　itself　is　x。　The　demonstration；　then；　that　a　subject；　as　such；



possesses　an　attribute　is　superior。　If　this　is　so；　and　if　the



particular　rather　than　the　commensurately　universal　forms



demonstrates；　particular　demonstration　is　superior。



　　（2）　The　universal　has　not　a　separate　being　over　against　groups　of



singulars。　Demonstration　nevertheless　creates　the　opinion　that　its



function　is　conditioned　by　something　like　this…some　separate　entity



belonging　to　the　real　world；　that；　for　instance；　of　triangle　or　of



figure　or　number；　over　against　particular　triangles；　figures；　and



numbers。　But　demonstration　which　touches　the　real　and　will　not　mislead



is　superior　to　that　which　moves　among　unrealities　and　is　delusory。　Now



commensurately　universal　demonstration　is　of　the　latter　kind：　if　we



engage　in　it　we　find　ourselves　reasoning　after　a　fashion　well



illustrated　by　the　argument　that　the　proportionate　is　what　answers



to　the　definition　of　some　entity　which　is　neither　line；　number；　solid；