prior analytics-第13章

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third。　But　when　the　minor　premiss　is　negative　and　universal；　if　it



is　problematic　a　syllogism　can　be　formed　by　means　of　conversion；　but



if　it　is　necessary　a　syllogism　is　not　possible。　The　proof　will



follow　the　same　course　as　where　the　premisses　are　universal；　and　the



same　terms　may　be　used。



　　It　is　clear　then　in　this　figure　also　when　and　how　a　syllogism　can　be



formed；　and　when　the　conclusion　is　problematic；　and　when　it　is　pure。



It　is　evident　also　that　all　syllogisms　in　this　figure　are　imperfect；



and　that　they　are　made　perfect　by　means　of　the　first　figure。







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　　It　is　clear　from　what　has　been　said　that　the　syllogisms　in　these



figures　are　made　perfect　by　means　of　universal　syllogisms　in　the　first



figure　and　are　reduced　to　them。　That　every　syllogism　without



qualification　can　be　so　treated；　will　be　clear　presently；　when　it



has　been　proved　that　every　syllogism　is　formed　through　one　or　other　of



these　figures。



　　It　is　necessary　that　every　demonstration　and　every　syllogism



should　prove　either　that　something　belongs　or　that　it　does　not；　and



this　either　universally　or　in　part；　and　further　either　ostensively



or　hypothetically。　One　sort　of　hypothetical　proof　is　the　reductio　ad



impossibile。　Let　us　speak　first　of　ostensive　syllogisms：　for　after



these　have　been　pointed　out　the　truth　of　our　contention　will　be



clear　with　regard　to　those　which　are　proved　per　impossibile；　and　in



general　hypothetically。



　　If　then　one　wants　to　prove　syllogistically　A　of　B；　either　as　an



attribute　of　it　or　as　not　an　attribute　of　it；　one　must　assert



something　of　something　else。　If　now　A　should　be　asserted　of　B；　the



proposition　originally　in　question　will　have　been　assumed。　But　if　A



should　be　asserted　of　C；　but　C　should　not　be　asserted　of　anything；　nor



anything　of　it；　nor　anything　else　of　A；　no　syllogism　will　be　possible。



For　nothing　necessarily　follows　from　the　assertion　of　some　one　thing



concerning　some　other　single　thing。　Thus　we　must　take　another



premiss　as　well。　If　then　A　be　asserted　of　something　else；　or　something



else　of　A；　or　something　different　of　C；　nothing　prevents　a　syllogism



being　formed；　but　it　will　not　be　in　relation　to　B　through　the



premisses　taken。　Nor　when　C　belongs　to　something　else；　and　that　to



something　else　and　so　on；　no　connexion　however　being　made　with　B；　will



a　syllogism　be　possible　concerning　A　in　its　relation　to　B。　For　in



general　we　stated　that　no　syllogism　can　establish　the　attribution　of



one　thing　to　another；　unless　some　middle　term　is　taken；　which　is



somehow　related　to　each　by　way　of　predication。　For　the　syllogism　in



general　is　made　out　of　premisses；　and　a　syllogism　referring　to　this



out　of　premisses　with　the　same　reference；　and　a　syllogism　relating



this　to　that　proceeds　through　premisses　which　relate　this　to　that。　But



it　is　impossible　to　take　a　premiss　in　reference　to　B；　if　we　neither



affirm　nor　deny　anything　of　it；　or　again　to　take　a　premiss　relating



A　to　B；　if　we　take　nothing　common；　but　affirm　or　deny　peculiar



attributes　of　each。　So　we　must　take　something　midway　between　the



two；　which　will　connect　the　predications；　if　we　are　to　have　a



syllogism　relating　this　to　that。　If　then　we　must　take　something　common



in　relation　to　both；　and　this　is　possible　in　three　ways　（either　by



predicating　A　of　C；　and　C　of　B；　or　C　of　both；　or　both　of　C）；　and　these



are　the　figures　of　which　we　have　spoken；　it　is　clear　that　every



syllogism　must　be　made　in　one　or　other　of　these　figures。　The



argument　is　the　same　if　several　middle　terms　should　be　necessary　to



establish　the　relation　to　B；　for　the　figure　will　be　the　same　whether



there　is　one　middle　term　or　many。



　　It　is　clear　then　that　the　ostensive　syllogisms　are　effected　by　means



of　the　aforesaid　figures；　these　considerations　will　show　that



reductiones　ad　also　are　effected　in　the　same　way。　For　all　who　effect



an　argument　per　impossibile　infer　syllogistically　what　is　false；　and



prove　the　original　conclusion　hypothetically　when　something　impossible



results　from　the　assumption　of　its　contradictory；　e。g。　that　the



diagonal　of　the　square　is　incommensurate　with　the　side；　because　odd



numbers　are　equal　to　evens　if　it　is　supposed　to　be　commensurate。　One



infers　syllogistically　that　odd　numbers　come　out　equal　to　evens；　and



one　proves　hypothetically　the　incommensurability　of　the　diagonal；



since　a　falsehood　results　through　contradicting　this。　For　this　we



found　to　be　reasoning　per　impossibile；　viz。　proving　something



impossible　by　means　of　an　hypothesis　conceded　at　the　beginning。



Consequently；　since　the　falsehood　is　established　in　reductions　ad



impossibile　by　an　ostensive　syllogism；　and　the　original　conclusion



is　proved　hypothetically；　and　we　have　already　stated　that　ostensive



syllogisms　are　effected　by　means　of　these　figures；　it　is　evident



that　syllogisms　per　impossibile　also　will　be　made　through　these



figures。　Likewise　all　the　other　hypothetical　syllogisms：　for　in



every　case　the　syllogism　leads　up　to　the　proposition　that　is



substituted　for　the　original　thesis；　but　the　original　thesis　is



reached　by　means　of　a　concession　or　some　other　hypothesis。　But　if　this



is　true；　every　demonstration　and　every　syllogism　must　be　formed　by



means　of　the　three　figures　mentioned　above。　But　when　this　has　been



shown　it　is　clear　that　every　syllogism　is　perfected　by　means　of　the



first　figure　and　is　reducible　to　the　universal　syllogisms　in　this



figure。



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　　Further　in　every　syllogism　one　of　the　premisses　must　be　affirmative；



and　universality　must　be　present：　unless　one　of　the　premisses　is



universal　either　a　syllogism　will　not　be　possible；　or　it　will　not



refer　to　the　subject　proposed；　or　the　original　position　will　be



begged。　Suppose　we　have　to　prove　that　pleasure　in　music　is　good。　If



one　should　claim　as　a　premiss　that　pleasure　is　good　without　adding



'all'；　no　syllogism　will　be　possible；　if　one　should　claim　that　some



pleasure　is　good；　then　if　it　is　different　from　pleasure　in　music；　it



is　not　relevant　to　the　subject　proposed；　if　it　is　this　very



pleasure；　one　is　assuming　that　which　was　proposed　at　the　outset　to



be　proved。　This　is　more　obvious　in　geometrical　proofs；　e。g。　that　the



angles　at　the　base　of　an　isosceles　triangle　are　equal。　Suppose　the



lines　A　and　B　have　been　drawn　to　the　centre。　If　then　one　should　assume



that　the　angle　AC　is　equal　to　the　angle　BD；　without　claiming　generally



that　angles　of　semicircles　are　equal；　and　again　if　one　should　assume



that　the　angle　C　is　equal　to　the　angle　D；　without　the　additional



assumption　that　every　angle　of　a　segment　is　equal　to　every　other　angle



of　the　same　segment；　and　further　if　one　should　assume　that　when



equal　angles　are　taken　from　the　whole　angles；　which　are　themselves



equal；　the　remainders　E　and　F　are　equal；　he　will　beg　the　thing　to　be



proved；　unless　he　also　states　that　when　equals　are　taken　from　equals



the　remainders　are　equal。



　　It　is　clear　then　that　in　every　syllogism　there　must　be　a　universal



premiss；　and　that　a　universal　statement　is　proved　only　when　all　the



premisses　are　universal；　while　a　particular　statement　is　proved　both



from　two　universal　premisses　and　from　one　only：　consequently　if　the



conclusion　is　universal；　the　premisses　also　must　be　universal；　but



if　the　premisses　are　universal　it　is　possible　that　the　conclusion



may　not　be　universal。　And　it　is　clear　also　that　in　every　syllogism



either　both　or　one　of　the　premisses　must　be　like　the　conclusion。　I



mean　not　only　in　being　affirmative　or　negative；　but　also　in　being



necessary；