prior analytics-第26章

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in　negative。　For　it　makes　no　difference　to　the　setting　out　of　the



terms；　whether　one　assumes　that　what　belongs　to　none　belongs　to　all　or



that　what　belongs　to　some　belongs　to　all。　The　same　applies　to　negative



statements。



　　It　is　clear　then　that　if　the　conclusion　is　false；　the　premisses　of



the　argument　must　be　false；　either　all　or　some　of　them；　but　when　the



conclusion　is　true；　it　is　not　necessary　that　the　premisses　should　be



true；　either　one　or　all；　yet　it　is　possible；　though　no　part　of　the



syllogism　is　true；　that　the　conclusion　may　none　the　less　be　true；



but　it　is　not　necessitated。　The　reason　is　that　when　two　things　are



so　related　to　one　another；　that　if　the　one　is；　the　other　necessarily



is；　then　if　the　latter　is　not；　the　former　will　not　be　either；　but　if



the　latter　is；　it　is　not　necessary　that　the　former　should　be。　But　it



is　impossible　that　the　same　thing　should　be　necessitated　by　the



being　and　by　the　not…being　of　the　same　thing。　I　mean；　for　example；



that　it　is　impossible　that　B　should　necessarily　be　great　since　A　is



white　and　that　B　should　necessarily　be　great　since　A　is　not　white。　For



whenever　since　this；　A；　is　white　it　is　necessary　that　that；　B；



should　be　great；　and　since　B　is　great　that　C　should　not　be　white；　then



it　is　necessary　if　is　white　that　C　should　not　be　white。　And　whenever



it　is　necessary；　since　one　of　two　things　is；　that　the　other　should　be；



it　is　necessary；　if　the　latter　is　not；　that　the　former　（viz。　A）　should



not　be。　If　then　B　is　not　great　A　cannot　be　white。　But　if；　when　A　is



not　white；　it　is　necessary　that　B　should　be　great；　it　necessarily



results　that　if　B　is　not　great；　B　itself　is　great。　（But　this　is



impossible。）　For　if　B　is　not　great；　A　will　necessarily　not　be　white。



If　then　when　this　is　not　white　B　must　be　great；　it　results　that　if　B



is　not　great；　it　is　great；　just　as　if　it　were　proved　through　three



terms。







　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　5







　　Circular　and　reciprocal　proof　means　proof　by　means　of　the



conclusion；　i。e。　by　converting　one　of　the　premisses　simply　and



inferring　the　premiss　which　was　assumed　in　the　original　syllogism：



e。g。　suppose　it　has　been　necessary　to　prove　that　A　belongs　to　all　C；



and　it　has　been　proved　through　B；　suppose　that　A　should　now　be



proved　to　belong　to　B　by　assuming　that　A　belongs　to　C；　and　C　to　B…so　A



belongs　to　B：　but　in　the　first　syllogism　the　converse　was　assumed；



viz。　that　B　belongs　to　C。　Or　suppose　it　is　necessary　to　prove　that　B



belongs　to　C；　and　A　is　assumed　to　belong　to　C；　which　was　the



conclusion　of　the　first　syllogism；　and　B　to　belong　to　A　but　the



converse　was　assumed　in　the　earlier　syllogism；　viz。　that　A　belongs



to　B。　In　no　other　way　is　reciprocal　proof　possible。　If　another　term　is



taken　as　middle；　the　proof　is　not　circular：　for　neither　of　the



propositions　assumed　is　the　same　as　before：　if　one　of　the　accepted



terms　is　taken　as　middle；　only　one　of　the　premisses　of　the　first



syllogism　can　be　assumed　in　the　second：　for　if　both　of　them　are



taken　the　same　conclusion　as　before　will　result：　but　it　must　be



different。　If　the　terms　are　not　convertible；　one　of　the　premisses　from



which　the　syllogism　results　must　be　undemonstrated：　for　it　is　not



possible　to　demonstrate　through　these　terms　that　the　third　belongs



to　the　middle　or　the　middle　to　the　first。　If　the　terms　are



convertible；　it　is　possible　to　demonstrate　everything　reciprocally；



e。g。　if　A　and　B　and　C　are　convertible　with　one　another。　Suppose　the





proposition　AC　has　been　demonstrated　through　B　as　middle　term；　and



again　the　proposition　AB　through　the　conclusion　and　the　premiss　BC



converted；　and　similarly　the　proposition　BC　through　the　conclusion　and



the　premiss　AB　converted。　But　it　is　necessary　to　prove　both　the



premiss　CB；　and　the　premiss　BA：　for　we　have　used　these　alone　without



demonstrating　them。　If　then　it　is　assumed　that　B　belongs　to　all　C；　and



C　to　all　A；　we　shall　have　a　syllogism　relating　B　to　A。　Again　if　it



is　assumed　that　C　belongs　to　all　A；　and　A　to　all　B；　C　must　belong　to



all　B。　In　both　these　syllogisms　the　premiss　CA　has　been　assumed



without　being　demonstrated：　the　other　premisses　had　ex　hypothesi



been　proved。　Consequently　if　we　succeed　in　demonstrating　this　premiss；



all　the　premisses　will　have　been　proved　reciprocally。　If　then　it　is



assumed　that　C　belongs　to　all　B；　and　B　to　all　A；　both　the　premisses



assumed　have　been　proved；　and　C　must　belong　to　A。　It　is　clear　then



that　only　if　the　terms　are　convertible　is　circular　and　reciprocal



demonstration　possible　（if　the　terms　are　not　convertible；　the　matter



stands　as　we　said　above）。　But　it　turns　out　in　these　also　that　we　use



for　the　demonstration　the　very　thing　that　is　being　proved：　for　C　is



proved　of　B；　and　B　of　by　assuming　that　C　is　said　of　and　C　is　proved　of



A　through　these　premisses；　so　that　we　use　the　conclusion　for　the



demonstration。



　　In　negative　syllogisms　reciprocal　proof　is　as　follows。　Let　B



belong　to　all　C；　and　A　to　none　of　the　Bs：　we　conclude　that　A　belongs



to　none　of　the　Cs。　If　again　it　is　necessary　to　prove　that　A　belongs　to



none　of　the　Bs　（which　was　previously　assumed）　A　must　belong　to　no　C；



and　C　to　all　B：　thus　the　previous　premiss　is　reversed。　If　it　is



necessary　to　prove　that　B　belongs　to　C；　the　proposition　AB　must　no



longer　be　converted　as　before：　for　the　premiss　'B　belongs　to　no　A'



is　identical　with　the　premiss　'A　belongs　to　no　B'。　But　we　must



assume　that　B　belongs　to　all　of　that　to　none　of　which　longs。　Let　A



belong　to　none　of　the　Cs　（which　was　the　previous　conclusion）　and



assume　that　B　belongs　to　all　of　that　to　none　of　which　A　belongs。　It　is



necessary　then　that　B　should　belong　to　all　C。　Consequently　each　of　the



three　propositions　has　been　made　a　conclusion；　and　this　is　circular



demonstration；　to　assume　the　conclusion　and　the　converse　of　one　of　the



premisses；　and　deduce　the　remaining　premiss。



　　In　particular　syllogisms　it　is　not　possible　to　demonstrate　the



universal　premiss　through　the　other　propositions；　but　the　particular



premiss　can　be　demonstrated。　Clearly　it　is　impossible　to　demonstrate



the　universal　premiss：　for　what　is　universal　is　proved　through



propositions　which　are　universal；　but　the　conclusion　is　not　universal；



and　the　proof　must　start　from　the　conclusion　and　the　other　premiss。



Further　a　syllogism　cannot　be　made　at　all　if　the　other　premiss　is



converted：　for　the　result　is　that　both　premisses　are　particular。　But



the　particular　premiss　may　be　proved。　Suppose　that　A　has　been　proved



of　some　C　through　B。　If　then　it　is　assumed　that　B　belongs　to　all　A　and



the　conclusion　is　retained；　B　will　belong　to　some　C：　for　we　obtain　the



first　figure　and　A　is　middle。　But　if　the　syllogism　is　negative；　it



is　not　possible　to　prove　the　universal　premiss；　for　the　reason　given



above。　But　it　is　possible　to　prove　the　particular　premiss；　if　the



proposition　AB　is　converted　as　in　the　universal　syllogism；　i。e　'B



belongs　to　some　of　that　to　some　of　which　A　does　not　belong'：　otherwise



no　syllogism　results　because　the　particular　premiss　is　negative。







　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　6







　　In　the　second　figure　it　is　not　possible　to　prove　an　affirmative



proposition　in　this　way；　but　a　negative　proposition　may　be　proved。



An　affirmative　proposition　is　not　proved　because　both　premisses　of　the



new　syllogism　are　not　affirmative　（for　the　conclusion　is　negative）　but



an　affirmative　proposition　is　（as　we　saw）　proved　from　premisses



which　are　both　affirmative。　The　negative　is　proved　as　follows。　Let　A



belong　to　all　B；　and　to　no　C：　we　conclude　that　B　belongs　to　no　C。　If



then　it　is　assume