prior analytics-第26节

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belong　to　all　B；　no　syllogism　results　whichever　term　the　assumed



premiss　concerns；　but　if　it　is　supposed　that　A　belongs　to　no　B；　when



the　premiss　BD　is　assumed　as　well　we　shall　prove　syllogistically



what　is　false；　but　not　the　problem　proposed。　For　if　A　belongs　to　no　B；



and　B　belongs　to　all　D；　A　belongs　to　no　D。　Let　this　be　impossible：



it　is　false　then　A　belongs　to　no　B。　But　the　universal　affirmative　is



not　necessarily　true　if　the　universal　negative　is　false。　But　if　the



premiss　CA　is　assumed　as　well；　no　syllogism　results；　nor　does　it　do　so



when　it　is　supposed　that　A　does　not　belong　to　all　B。　Consequently　it



is　clear　that　the　universal　affirmative　cannot　be　proved　in　the



first　figure　per　impossibile。



　　But　the　particular　affirmative　and　the　universal　and　particular



negatives　can　all　be　proved。　Suppose　that　A　belongs　to　no　B；　and　let



it　have　been　assumed　that　B　belongs　to　all　or　to　some　C。　Then　it　is



necessary　that　A　should　belong　to　no　C　or　not　to　all　C。　But　this　is



impossible　（for　let　it　be　true　and　clear　that　A　belongs　to　all　C）：



consequently　if　this　is　false；　it　is　necessary　that　A　should　belong　to



some　B。　But　if　the　other　premiss　assumed　relates　to　A；　no　syllogism



will　be　possible。　Nor　can　a　conclusion　be　drawn　when　the　contrary　of



the　conclusion　is　supposed；　e。g。　that　A　does　not　belong　to　some　B。



Clearly　then　we　must　suppose　the　contradictory。



　　Again　suppose　that　A　belongs　to　some　B；　and　let　it　have　been　assumed



that　C　belongs　to　all　A。　It　is　necessary　then　that　C　should　belong



to　some　B。　But　let　this　be　impossible；　so　that　the　supposition　is



false：　in　that　case　it　is　true　that　A　belongs　to　no　B。　We　may



proceed　in　the　same　way　if　the　proposition　CA　has　been　taken　as



negative。　But　if　the　premiss　assumed　concerns　B；　no　syllogism　will



be　possible。　If　the　contrary　is　supposed；　we　shall　have　a　syllogism



and　an　impossible　conclusion；　but　the　problem　in　hand　is　not　proved。



Suppose　that　A　belongs　to　all　B；　and　let　it　have　been　assumed　that　C



belongs　to　all　A。　It　is　necessary　then　that　C　should　belong　to　all



B。　But　this　is　impossible；　so　that　it　is　false　that　A　belongs　to　all



B。　But　we　have　not　yet　shown　it　to　be　necessary　that　A　belongs　to　no



B；　if　it　does　not　belong　to　all　B。　Similarly　if　the　other　premiss



taken　concerns　B；　we　shall　have　a　syllogism　and　a　conclusion　which



is　impossible；　but　the　hypothesis　is　not　refuted。　Therefore　it　is



the　contradictory　that　we　must　suppose。



　　To　prove　that　A　does　not　belong　to　all　B；　we　must　suppose　that　it



belongs　to　all　B：　for　if　A　belongs　to　all　B；　and　C　to　all　A；　then　C



belongs　to　all　B；　so　that　if　this　is　impossible；　the　hypothesis　is



false。　Similarly　if　the　other　premiss　assumed　concerns　B。　The　same



results　if　the　original　proposition　CA　was　negative：　for　thus　also



we　get　a　syllogism。　But　if　the　negative　proposition　concerns　B；



nothing　is　proved。　If　the　hypothesis　is　that　A　belongs　not　to　all



but　to　some　B；　it　is　not　proved　that　A　belongs　not　to　all　B；　but



that　it　belongs　to　no　B。　For　if　A　belongs　to　some　B；　and　C　to　all　A；



then　C　will　belong　to　some　B。　If　then　this　is　impossible；　it　is



false　that　A　belongs　to　some　B；　consequently　it　is　true　that　A　belongs



to　no　B。　But　if　this　is　proved；　the　truth　is　refuted　as　well；　for



the　original　conclusion　was　that　A　belongs　to　some　B；　and　does　not



belong　to　some　B。　Further　the　impossible　does　not　result　from　the



hypothesis：　for　then　the　hypothesis　would　be　false；　since　it　is



impossible　to　draw　a　false　conclusion　from　true　premisses：　but　in　fact



it　is　true：　for　A　belongs　to　some　B。　Consequently　we　must　not



suppose　that　A　belongs　to　some　B；　but　that　it　belongs　to　all　B。



Similarly　if　we　should　be　proving　that　A　does　not　belong　to　some　B：



for　if　'not　to　belong　to　some'　and　'to　belong　not　to　all'　have　the



same　meaning；　the　demonstration　of　both　will　be　identical。



　　It　is　clear　then　that　not　the　contrary　but　the　contradictory　ought



to　be　supposed　in　all　the　syllogisms。　For　thus　we　shall　have　necessity



of　inference；　and　the　claim　we　make　is　one　that　will　be　generally



accepted。　For　if　of　everything　one　or　other　of　two　contradictory



statements　holds　good；　then　if　it　is　proved　that　the　negation　does　not



hold；　the　affirmation　must　be　true。　Again　if　it　is　not　admitted　that



the　affirmation　is　true；　the　claim　that　the　negation　is　true　will　be



generally　accepted。　But　in　neither　way　does　it　suit　to　maintain　the



contrary：　for　it　is　not　necessary　that　if　the　universal　negative　is



false；　the　universal　affirmative　should　be　true；　nor　is　it　generally



accepted　that　if　the　one　is　false　the　other　is　true。







　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　12







　　It　is　clear　then　that　in　the　first　figure　all　problems　except　the



universal　affirmative　are　proved　per　impossibile。　But　in　the　middle



and　the　last　figures　this　also　is　proved。　Suppose　that　A　does　not



belong　to　all　B；　and　let　it　have　been　assumed　that　A　belongs　to　all　C。



If　then　A　belongs　not　to　all　B；　but　to　all　C；　C　will　not　belong　to　all



B。　But　this　is　impossible　（for　suppose　it　to　be　clear　that　C　belongs



to　all　B）：　consequently　the　hypothesis　is　false。　It　is　true　then



that　A　belongs　to　all　B。　But　if　the　contrary　is　supposed；　we　shall



have　a　syllogism　and　a　result　which　is　impossible：　but　the　problem



in　hand　is　not　proved。　For　if　A　belongs　to　no　B；　and　to　all　C；　C



will　belong　to　no　B。　This　is　impossible；　so　that　it　is　false　that　A



belongs　to　no　B。　But　though　this　is　false；　it　does　not　follow　that



it　is　true　that　A　belongs　to　all　B。



　　When　A　belongs　to　some　B；　suppose　that　A　belongs　to　no　B；　and　let



A　belong　to　all　C。　It　is　necessary　then　that　C　should　belong　to　no



B。　Consequently；　if　this　is　impossible；　A　must　belong　to　some　B。　But



if　it　is　supposed　that　A　does　not　belong　to　some　B；　we　shall　have



the　same　results　as　in　the　first　figure。



　　Again　suppose　that　A　belongs　to　some　B；　and　let　A　belong　to　no　C。　It



is　necessary　then　that　C　should　not　belong　to　some　B。　But　originally



it　belonged　to　all　B；　consequently　the　hypothesis　is　false：　A　then



will　belong　to　no　B。



　　When　A　does　not　belong　to　an　B；　suppose　it　does　belong　to　all　B；　and



to　no　C。　It　is　necessary　then　that　C　should　belong　to　no　B。　But　this



is　impossible：　so　that　it　is　true　that　A　does　not　belong　to　all　B。



It　is　clear　then　that　all　the　syllogisms　can　be　formed　in　the　middle



figure。







　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　13







　　Similarly　they　can　all　be　formed　in　the　last　figure。　Suppose　that



A　does　not　belong　to　some　B；　but　C　belongs　to　all　B：　then　A　does　not



belong　to　some　C。　If　then　this　is　impossible；　it　is　false　that　A



does　not　belong　to　some　B；　so　that　it　is　true　that　A　belongs　to　all　B。



But　if　it　is　supposed　that　A　belongs　to　no　B；　we　shall　have　a



syllogism　and　a　conclusion　which　is　impossible：　but　the　problem　in



hand　is　not　proved：　for　if　the　contrary　is　supposed；　we　shall　have　the



same　results　as　before。



　　But　to　prove　that　A　belongs　to　some　B；　this　hypothesis　must　be　made。



If　A　belongs　to　no　B；　and　C　to　some　B；　A　will　belong　not　to　all　C。



If　then　this　is　false；　it　is　true　that　A　belongs　to　some　B。



　　When　A　belongs　to　no　B；　suppose　A　belongs　to　some　B；　and　let　it　have



been　assumed　that　C　belongs　to　all　B。　Then　it　is　necessary　that　A



should　belong　to　some　C。　But　ex　hypothesi　it　belongs　to　no　C；　so



that　it　is　false　that　A　belongs　to　some　B。　But　if　it　is　supposed



that　A　belongs　to　all　B；　the　problem　is　not　proved。



　　But　this　hypothesis　must　be　made　if　we　are　prove　that　A　belongs



not　to　all　B。　For　if　A　belongs　to　all　B　and　C　to　some　B；　then　A



belongs　to　some　C。　But　this　we　assumed　not　to　be　so；　so　it　is　false



that　A　belongs　to　all　B。　But　in　that　case　it　is　true　that　A　belongs



not　to　all　B。　If　however　it　is　assumed　that　A　belongs　to　some　B；　we



shall　have　the　same　result　as　before。



　　It　is　clear　then　that　in　all　the　syllogisms　which　proceed　per



impossibile　the　contradictory　must　be　assumed。　And　it　is　plain　that　in



the　middle　figure　an　affirmative　conclusion；　and　in　the　last　figure



a　universal　conclusion；　are　proved　in　a　way。







　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　14







　　Demonstration　per　impossibile　differs　from　ostensive　proof　in　that



it　posits　what　it　wishes　to　refute　by　reduction　to　a　statement



admitted　to　be　false；　whereas　ostensive　proof　starts　from　admitted



positions。　Both；　indeed；　take　two　premisses　that　are　admitted；　but　the



latter　takes　the　premisses　from　which　the　syllogism　starts；　the　former



takes　one　of　these；　along　with　the　contradictory　of　the　o