prior analytics-第10章

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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；　pure；　problematic。　We　must　consider　also　the　other　forms　of



predication。



　　It　is　clear　also　when　a　syllogism　in　general　can　be　made　and　when　it



cannot；　and　when　a　valid；　when　a　perfect　syllogism　can　be　formed；



and　that　if　a　syllogism　is　formed　the　terms　must　be　arranged　in　one　of



the　ways　that　have　been　mentioned。







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　　It　is　clear　too　that　every　demonstration　will　proceed　through



three　terms　and　no　more；　unless　the　same　conclusion　is　established



by　different　pairs　of　propositions；　e。g。　the　conclusion　E　may　be



established　through　the　propositions　A　and　B；　and　through　the



propositions　C　and　D；　or　through　the　propositions　A　and　B；　or　A　and　C；



or　B　and　C。　For　nothing　prevents　there　being　several　middles　for　the



same　terms。　But　in　that　case　there　is　not　one　but　several



syllogisms。　Or　again　when　each　of　the　propositions　A　and　B　is　obtained



by　syllogistic　inference；　e。g。　by　means　of　D　and　E；　and　again　B　by



means　of　F　and　G。　Or　one　may　be　obtained　by　syllogistic；　the　other



by　inductive　inference。　But　thus　also　the　syllogisms　are　many；　for　the



conclusions　are　many；　e。g。　A　and　B　and　C。　But　if　this　can　be　called



one　syllogism；　not　many；　the　same　conclusion　may　be　reached　by　more



than　three　terms　in　this　way；　but　it　cannot　be　reached　as　C　is



established　by　means　of　A　and　B。　Suppose　that　the　proposition　E　is



inferred　from　the　premisses　A；　B；　C；　and　D。　It　is　necessary　then



that　of　these　one　should　be　related　to　another　as　whole　to　part：　for



it　has　already　been　proved　that　if　a　syllogism　is　formed　some　of　its



terms　must　be　related　in　this　way。　Suppose　then　that　A　stands　in



this　relation　to　B。　Some　conclusion　then　follows　from　them。　It　must



either　be　E　or　one　or　other　of　C　and　D；　or　something　other　than　these。



　　（1）　If　it　is　E　the　syllogism　will　have　A　and　B　for　its　sole



premisses。　But　if　C　and　D　are　so　related　that　one　is　whole；　the



other　part；　some　conclusion　will　follow　from　them　also；　and　it　must　be



either　E；　or　one　or　other　of　the　propositions　A　and　B；　or　something



other　than　these。　And　if　it　is　（i）　E；　or　（ii）　A　or　B；　either　（i）　the



syllogisms　will　be　more　than　one；　or　（ii）　the　same　thing　happens　to　be



inferred　by　means　of　several　terms　only　in　the　sense　which　we　saw　to



be　possible。　But　if　（iii）　the　conclusion　is　other　than　E　or　A　or　B；



the　syllogisms　will　be　many；　and　unconnected　with　one　another。　But



if　C　is　not　so　related　to　D　as　to　make　a　syllogism；　the　propositions



will　have　been　assumed　to　no　purpose；　unless　for　the　sake　of　induction



or　of　obscuring　the　argument　or　something　of　the　sort。



　　（2）　But　if　from　the　propositions　A　and　B　there　follows　not　E　but



some　other　conclusion；　and　if　from　C　and　D　either　A　or　B　follows　or



something　else；　then　there　are　several　syllogisms；　and　they　do　not



establish　the　conclusion　proposed：　for　we　assumed　that　the　syllogism



proved　E。　And　if　no　conclusion　follows　from　C　and　D；　it　turns　out　that



these　propositions　have　been　assumed　to　no　purpose；　and　the



syllogism　does　not　prove　the　original　proposition。



　　So　it　is　clear　that　every　demonstration　and　every　syllogism　will



proceed　through　three　terms　only。



　　This　being　evident；　it　is　clear　that　a　syllogistic　conclusion



follows　from　two　premisses　and　not　from　more　than　two。　For　the　three



terms　make　two　premisses；　unless　a　new　premiss　is　assumed；　as　was　said



at　the　beginning；　to　perfect　the　syllogisms。　It　is　clear　therefore



that　in　whatever　syllogistic　argument　the　premisses　through　which



the　main　conclusion　follows　（for　some　of　the　preceding　conclusions



must　be　premisses）　are　not　even　in　number；　this　argument　either　has



not　been　drawn　syllogistically　or　it　has　assumed　more　than　was



necessary　to　establish　its　thesis。



　　If　then　syllogisms　are　taken　with　respect　to　their　main　premisses；



every　syllogism　will　consist　of　an　even　number　of　premisses　and　an　odd



number　of　terms　（for　the　terms　exceed　the　premisses　by　one）；　and　the



conclusions　will　be　half　the　number　of　the　premisses。　But　whenever　a



conclusion　is　reached　by　means　of　prosyllogisms　or　by　means　of　several



continuous　middle　terms；　e。g。　the　proposition　AB　by　means　of　the



middle　terms　C　and　D；　the　number　of　the　terms　will　similarly　exceed



that　of　the　premisses　by　one　（for　the　extra　term　must　either　be



added　outside　or　inserted：　but　in　either　case　it　follows　that　the



relations　of　predication　are　one　fewer　than　the　terms　related）；　and



the　premisses　will　be　equal　in　number　to　the　relations　of　predication。



The　premisses　however　will　not　always　be　even；　the　terms　odd；　but　they



will　alternate…when　the　premisses　are　even；　the　terms　must　be　odd；



when　the　terms　are　even；　the　premisses　must　be　odd：　for　along　with　one



term　one　premiss　is　added；　if　a　term　is　added　from　any　quarter。



Consequently　since　the　premisses　were　（as　we　saw）　even；　and　the



terms　odd；　we　must　make　them　alternately　even　and　odd　at　each



addition。　But　the　conclusions　will　not　follow　the　same　arrangement



either　in　respect　to　the　terms　or　to　the　premisses。　For　if　one　term　is



added；　conclusions　will　be　added　less　by　one　than　the　pre…existing



terms：　for　the　conclusion　is　drawn　not　in　relation　to　the　single



term　last　added；　but　in　relation　to　all　the　rest；　e。g。　if　to　ABC　the



term　D　is　added；　two　conclusions　are　thereby　added；　one　in　relation　to



A；　the　other　in　relation　to　B。　Similarly　with　any　further　additions。



And　similarly　too　if　the　term　is　inserted　in　the　middle：　for　in



relation　to　one　term　only；　a　syllogism　will　not　be　constructed。



Consequently　the　conclusions　will　be　much　more　numerous　than