posterior analytics-第15章

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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。







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　　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　ad　impossibile　is



not　a　proper　conclusion；　nor　are　its　antecedents　proper　premisses。



On　the　contrary：　the　constituents　of　syllogism　are　premisses　related



to　one　another　as　whole　to　part　or　part　to　whole；　whereas　the



premisses　A…C　and　A…B　are　not　thus　related　to　one　another。　Now　the



superior　demonstration　is　that　which　proceeds　from　better　known　and



prior　premisses；　and　while　both　these　forms　depend　for　credence　on　the



not…being　of　something；　yet　the　source　of　the　one　is　prior　to　that



of　the　other。　Therefore　negative　demonstration　will　have　an



unqualified　superiority　to　reductio　ad　impossibile；　and　affirmative



demonstration；　being　superior　to　negative；　will　consequently　be



superior　also　to　reductio　ad　impossibile。







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　　The　science　which　is　knowledge　at　once　of　the　fact　and　of　the



reasoned　fact；　not　of　the　fact　by　itself　without　the　reasoned　fact；　is



the　more　exact　and　the　prior　science。



　　A　science　such　as　arithmetic；　which　is　not　a　science　of　properties



qua　inhering　in　a　substratum；　is　more　exact　than　and　prior　to　a



science　like　harmonics；　which　is　a　science　of　pr；operties　inhering



in　a　substratum；　and　similarly　a　science　like　arithmetic；　which　is



constituted　of　fewer　basic　elements；　is　more　exact　than　and　prior　to



geometry；　which　requires　additional　elements。　What　I　mean　by



'additional　elements'　is　this：　a　unit　is　substance　without　position；



while　a　point　is　substance　with　position；　the　latter　contains　an



additional　element。







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　　A　single　science　is　one　whose　domain　is　a　single　genus；　viz。　all　the



subjects　constituted　out　of　the　primary　entities　of　the　genus…i。e。　the



parts　of　this　total　subject…and　their　essential　properties。



　　One　science　differs　from　another　when　their　basic　truths　have



neither　a　common　source　nor　are　derived　those　of　the　one　science



from　those　the　other。　This　is　verified　when　we　reach　the



indemonstrable　premisses　of　a　science；　for　they　must　be　within　one



genus　with　its　conclusions：　and　this　again　is　verified　if　the



conclusions　proved　by　means　of　them　fall　within　one　genus…i。e。　are



homogeneous。







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　　One　can　have　several　demonstrations　of　the　same　connexion　not　only



by　taking　from　the　same　series　of　predication　middles　which　are



other　than　the　immediately　cohering　term　e。g。　by　taking　C；　D；　and　F



severally　to　prove　A…Bbut　also　by　taking　a　middle　from　another



series。　Thus　let　A　be　change；　D　alteration　of　a　property；　B　feeling



pleasure；　and　G　relaxation。　We　can　then　without　falsehood　predicate



D　of　B　and　A　of　D；　for　he　who　is　pleased　suffers　alteration　of　a



property；　and　that　which　alters　a　property　changes。　Again；　we　can