prior analytics-第21章

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is　formed　about　A　and　C。　Nor　if　A　does　not　belong　to　some　B；　but



belongs　to　all　C；　will　a　syllogism　be　possible　about　B　and　C。　A



similar　proof　can　be　given　if　the　premisses　are　not　universal。　For



either　both　premisses　arrived　at　by　the　conversion　must　be　particular；



or　the　universal　premiss　must　refer　to　the　minor　extreme。　But　we　found



that　no　syllogism　is　possible　thus　either　in　the　first　or　in　the



middle　figure。　But　if　the　conclusion　is　converted　into　its



contradictory；　both　the　premisses　can　be　refuted。　For　if　A　belongs



to　no　B；　and　B　to　all　C；　then　A　belongs　to　no　C：　again　if　A　belongs　to



no　B；　and　to　all　C；　B　belongs　to　no　C。　And　similarly　if　one　of　the



premisses　is　not　universal。　For　if　A　belongs　to　no　B；　and　B　to　some　C；



A　will　not　belong　to　some　C：　if　A　belongs　to　no　B；　and　to　C；　B　will



belong　to　no　C。



　　Similarly　if　the　original　syllogism　is　negative。　Suppose　it　has　been



proved　that　A　does　not　belong　to　some　B；　BC　being　affirmative；　AC



being　negative：　for　it　was　thus　that；　as　we　saw；　a　syllogism　could



be　made。　Whenever　then　the　contrary　of　the　conclusion　is　assumed　a



syllogism　will　not　be　possible。　For　if　A　belongs　to　some　B；　and　B　to



all　C；　no　syllogism　is　possible　（as　we　saw）　about　A　and　C。　Nor；　if　A



belongs　to　some　B；　and　to　no　C；　was　a　syllogism　possible　concerning



B　and　C。　Therefore　the　premisses　are　not　refuted。　But　when　the



contradictory　of　the　conclusion　is　assumed；　they　are　refuted。　For　if　A



belongs　to　all　B；　and　B　to　C；　A　belongs　to　all　C：　but　A　was　supposed



originally　to　belong　to　no　C。　Again　if　A　belongs　to　all　B；　and　to　no



C；　then　B　belongs　to　no　C：　but　it　was　supposed　to　belong　to　all　C。　A



similar　proof　is　possible　if　the　premisses　are　not　universal。　For　AC



becomes　universal　and　negative；　the　other　premiss　particular　and



affirmative。　If　then　A　belongs　to　all　B；　and　B　to　some　C；　it　results



that　A　belongs　to　some　C：　but　it　was　supposed　to　belong　to　no　C。　Again



if　A　belongs　to　all　B；　and　to　no　C；　then　B　belongs　to　no　C：　but　it　was



assumed　to　belong　to　some　C。　If　A　belongs　to　some　B　and　B　to　some　C；



no　syllogism　results：　nor　yet　if　A　belongs　to　some　B；　and　to　no　C。



Thus　in　one　way　the　premisses　are　refuted；　in　the　other　way　they　are



not。



　　From　what　has　been　said　it　is　clear　how　a　syllogism　results　in



each　figure　when　the　conclusion　is　converted；　when　a　result　contrary



to　the　premiss；　and　when　a　result　contradictory　to　the　premiss；　is



obtained。　It　is　clear　that　in　the　first　figure　the　syllogisms　are



formed　through　the　middle　and　the　last　figures；　and　the　premiss



which　concerns　the　minor　extreme　is　alway　refuted　through　the　middle



figure；　the　premiss　which　concerns　the　major　through　the　last



figure。　In　the　second　figure　syllogisms　proceed　through　the　first



and　the　last　figures；　and　the　premiss　which　concerns　the　minor　extreme



is　always　refuted　through　the　first　figure；　the　premiss　which　concerns



the　major　extreme　through　the　last。　In　the　third　figure　the　refutation



proceeds　through　the　first　and　the　middle　figures；　the　premiss　which



concerns　the　major　is　always　refuted　through　the　first　figure；　the



premiss　which　concerns　the　minor　through　the　middle　figure。



　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　11







　　It　is　clear　then　what　conversion　is；　how　it　is　effected　in　each



figure；　and　what　syllogism　results。　The　syllogism　per　impossibile　is



proved　when　the　contradictory　of　the　conclusion　stated　and　another



premiss　is　assumed；　it　can　be　made　in　all　the　figures。　For　it



resembles　conversion；　differing　only　in　this：　conversion　takes　place



after　a　syllogism　has　been　formed　and　both　the　premisses　have　been



taken；　but　a　reduction　to　the　impossible　takes　place　not　because　the



contradictory　has　been　agreed　to　already；　but　because　it　is　clear　that



it　is　true。　The　terms　are　alike　in　both；　and　the　premisses　of　both　are



taken　in　the　same　way。　For　example　if　A　belongs　to　all　B；　C　being



middle；　then　if　it　is　supposed　that　A　does　not　belong　to　all　B　or



belongs　to　no　B；　but　to　all　C　（which　was　admitted　to　be　true）；　it



follows　that　C　belongs　to　no　B　or　not　to　all　B。　But　this　is



impossible：　consequently　the　supposition　is　false：　its　contradictory



then　is　true。　Similarly　in　the　other　figures：　for　whatever　moods　admit



of　conversion　admit　also　of　the　reduction　per　impossibile。



　　All　the　problems　can　be　proved　per　impossibile　in　all　the　figures；



excepting　the　universal　affirmative；　which　is　proved　in　the　middle　and



third　figures；　but　not　in　the　first。　Suppose　that　A　belongs　not　to　all



B；　or　to　no　B；　and　take　besides　another　premiss　concerning　either　of



the　terms；　viz。　that　C　belongs　to　all　A；　or　that　B　belongs　to　all　D；



thus　we　get　the　first　figure。　If　then　it　is　supposed　that　A　does　not



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　impos