Edward Ockham: Created page with "==Chapter 11== {| border=1 cellpadding=10 !valign = top width=33.33%|Greek !valign = top width=33.33%|Latin !valign = top width=33.33%|English |- valign = top | |(PL 64 0650A..."

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Created page with "==Chapter 11== {| border=1 cellpadding=10 !valign = top width=33.33%|Greek !valign = top width=33.33%|Latin !valign = top width=33.33%|English |- valign = top | |(PL 64 0650A..."

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==Chapter 11==
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|(PL 64 0650A) CAPUT XI. De syllogismis mixtis ex altera necessaria et altera absoluta in tertia figura.

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|{{a|bk31a18|31a18}} Ἐν δὲ τῶι τελευταίωι σχήματι καθόλου μὲν ὄντων τῶν ὅρων πρὸς τὸ μέσον καὶ κατηγορικῶν ἀμφοτέρων τῶν προ τάσεων, ἐὰν ὁποτερονοῦν ἦι ἀναγκαῖον, καὶ τὸ συμπέρασμα ἔσται ἀναγκαῖον. ἐὰν δὲ τὸ μὲν ἦι στερητικὸν τὸ δὲ κατηγορικόν, ὅταν μὲν τὸ στερητικὸν ἀναγκαῖον ἦι, καὶ τὸ συμπέρασμα ἔσται ἀναγκαῖον, ὅταν δὲ τὸ κατηγορικόν, οὐκ ἔσται ἀναγκαῖον. ἔστωσαν γὰρ ἀμφότεραι κατηγορικαὶ πρῶτον αἱ προ τάσεις, καὶ τὸ Α καὶ τὸ Β παντὶ τῶι Γ ὑπαρχέτω, ἀναγ- καῖον δ᾽ ἔστω τὸ Α Γ. ἐπεὶ οὖν τὸ Β παντὶ τῶι Γ ὑπάρχει, καὶ τὸ Γ τινὶ τῶι Β ὑπάρξει διὰ τὸ ἀντιστρέφειν τὸ καθόλου τῶι κατὰ μέρος, ὥστ᾽ εἰ παντὶ τῶι Γ τὸ Α ἐξ ἀνάγκης ὑπάρχει καὶ τὸ Γ τῶι Β τινί, καὶ τῶι Β τινὶ ἀναγκαῖον ὑπάρχειν τὸ Α· τὸ γὰρ Β ὑπὸ τὸ Γ ἐστίν. γίγνεται οὖν τὸ πρῶτον σχῆμα. ὁμοίως δὲ δειχθήσεται καὶ εἰ τὸ Β Γ ἐστὶν ἀναγκαῖον· ἀντιστρέφει γὰρ τὸ Γ τῶι Α τινί, ὥστ᾽ εἰ παντὶ τῶι Γ τὸ Β ἐξ ἀνάγκης ὑπάρχει, καὶ τῶι Α τινὶ ὑπάρξει ἐξ ἀνάγκης.

|(0650B) In postrema autem figura terminis quidem universalibus ad medium, et praedicativis utrisque propositionibus, si utralibet sit necessaria, et conclusio erit necessaria. Si autem haec quidem sit privativa, illa vero praedicativa, quando privativa quidem fuerit necessaria, et conclusio erit necessaria, quando autem praedicativa, non erit necessaria. Sint enim primum utraeque praedicativae propositiones, et A et B omni C insint, necessaria autem sit A C, quoniam ergo B omni C inest, et C alicui B inerit, eo quod convertitur universalis particulariter. Quare si A inest omni C ex necessitate, et C alicui B, et A alicui B necessarium inesse, nam B sub C est. Fit igitur prima figura. Similiter autem ostendetur, et si B C est necessaria, convertitur enim C alicui A, quare si omni C inest B ex necessitate, et A alicui B inerit ex necessitate.

|In the last figure when the terms are related universally to the middle, and both premisses are affirmative, if one of the two is necessary, then the conclusion will be necessary. But if one is negative, the other affirmative, whenever the negative is necessary the conclusion also will be necessary, but whenever the affirmative is necessary the conclusion will not be necessary. First let both the premisses be affirmative, and let A and B belong to all C, and let AC be necessary. Since then B belongs to all C, C also will belong to some B, because the universal is convertible into the particular: consequently if A belongs necessarily to all C, and C belongs to some B, it is necessary that A should belong to some B also. For B is under C. The first figure then is formed. A similar proof will be given also if BC is necessary. For C is convertible with some A: consequently if B belongs necessarily to all C, it will belong necessarily also to some A.

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|Πάλιν ἔστω τὸ μὲν Α Γ στερητικόν, τὸ δὲ Β Γ καταφατικόν, ἀναγκαῖον δὲ τὸ στερητικόν. ἐπεὶ οὖν ἀντιστρέφει τινὶ τῶι Β τὸ Γ, τὸ δὲ Α οὐδενὶ τῶι Γ ἐξ ἀνάγκης, οὐδὲ τῶι Β τινὶ ὑπάρξει ἐξ ἀνάγκης τὸ Α· τὸ γὰρ Β ὑπὸ τὸ Γ ἐστίν. εἰ δὲ τὸ κατηγορικὸν ἀναγκαῖον, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον. ἔστω γὰρ τὸ Β Γ κατηγορικὸν καὶ ἀναγκαῖον, τὸ δὲ Α Γ στερητικὸν καὶ μὴ ἀναγ καῖον. ἐπεὶ οὖν ἀντιστρέφει τὸ καταφατικόν, ὑπάρξει καὶ τὸ Γ τινὶ τῶι Β ἐξ ἀνάγκης, ὥστ᾽ εἰ τὸ μὲν Α μηδενὶ τῶι Γ τὸ {{ln|bk31b1}} δὲ Γ τινὶ τῶι Β, τὸ Α τινὶ τῶι Β οὐχ ὑπάρξει· ἀλλ᾽ οὐκ ἐξ ἀνάγκης· δέδεικται γὰρ ἐν τῶι πρώτωι σχήματι ὅτι τῆς στερητικῆς προτάσεως μὴ ἀναγκαίας οὔσης οὐδὲ τὸ συμπέρασμα ἔσται ἀναγκαῖον. ἔτι κἂν διὰ τῶν ὅρων εἴη φανερόν. ἔστω γὰρ τὸ μὲν Α ἀγαθόν, τὸ δ᾽ ἐφ᾽ ὧι Β ζῶιον, τὸ δὲ Γ ἵππος. τὸ μὲν οὖν ἀγαθὸν ἐνδέχεται μηδενὶ ἵππωι ὑπάρχειν, τὸ δὲ ζῶιον ἀνάγκη παντὶ ὑπάρχειν· ἀλλ᾽ οὐκ ἀνάγκη ζῶιόν τι μὴ εἶναι ἀγαθόν, εἴπερ ἐνδέχεται πᾶν εἶναι ἀγαθόν. ἢ εἰ μὴ τοῦτο δυνατόν, ἀλλὰ τὸ ἐγρηγορέναι ἢ τὸ καθεύδειν ὅρον θετέον· ἅπαν γὰρ ζῶιον δεκτικὸν τούτων.

|(0650C) Rursum sit A C quidem privativa, B C vero affirmativa, necessaria autem privativa, quoniam ergo convertitur affirmativa, erit C alicui B, A autem nulli C ex necessitate, neque A alicui B inerit ex necessitate, nam B sub C est. Si autem praedicativa sit necessaria, non erit conclusio necessaria. Sit enim B C praedicativa et necessaria, A C autem privativa et non necessaria, quoniam ergo convertitur affirmativa, inerit et C alicui B ex necessitate. Quare si A quidem nulli eorum quae sunt C inest, C autem alicui eorum quae sunt B et A alicui eorum quae sunt B non inerit, sed non ex necessitate. Ostensum est enim in prima figura quoniam privativa propositione necessaria, nec conclusio erit necessaria. (0650D) Amplius autem et per terminos sit manifestum, sit enim A quidem bonum in quo B animal, C autem equus, ergo bonum quidem contingit nulli equo inesse, animal vero necesse est omni equo inesse, sed non necesse est aliquod animal non esse bonum, siquidem contingit omne esse bonum. Aut si non hoc possibile, sed vigilare, vel dormire terminum ponendum. Omne enim animal susceptibile est horum.

|Again let AC be negative, BC affirmative, and let the negative premiss be necessary. Since then C is convertible with some B, but A necessarily belongs to no C, A will necessarily not belong to some B either: for B is under C. But if the affirmative is necessary, the conclusion will not be necessary. For suppose BC is affirmative and necessary, while AC is negative and not necessary. Since then the affirmative is convertible, C also will belong to some B necessarily: consequently if A belongs to none of the Cs, while C belongs to some of the Bs, A will not belong to some of the Bs-but not of necessity; for it has been proved, in the case of the first figure, that if the negative premiss is not necessary, neither will the conclusion be necessary. Further, the point may be made clear by considering the terms. Let the term A be ‘good’, let that which B signifies be ‘animal’, let the term C be ‘horse’. It is possible then that the term good should belong to no horse, and it is necessary that the term animal should belong to every horse: but it is not necessary that some animal should not be good, since it is possible for every animal to be good. Or if that is not possible, take as the term ‘awake’ or ‘asleep’: for every animal can accept these.

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|Εἰ μὲν οὖν οἱ ὅροι καθόλου πρὸς τὸ μέσον εἰσίν, εἴρηται πότε ἔσται τὸ συμπέρασμα ἀναγκαῖον· εἰ δ᾽ ὁ μὲν καθόλου ὁ δ᾽ ἐν μέρει, κατηγορικῶν μὲν ὄντων ἀμφοτέρων, ὅταν τὸ καθόλου γένηται ἀναγκαῖον, καὶ τὸ συμπέρασμα ἔσται ἀναγ καῖον. ἀπόδειξις δ᾽ ἡ αὐτὴ ἣ καὶ πρότερον· ἀντιστρέφει γὰρ καὶ τὸ ἐν μέρει κατηγορικόν. εἰ οὖν ἀνάγκη τὸ Β παντὶ τῶι Γ ὑπάρχειν, τὸ δὲ Α ὑπὸ τὸ Γ ἐστίν, ἀνάγκη τὸ Β τινὶ τῶι Α ὑπάρχειν. εἰ δὲ τὸ Β τῶι Α τινί, καὶ τὸ Α τῶι Β τινὶ ὑπάρχειν ἀναγκαῖον· ἀντιστρέφει γάρ.

|Si igitur termini universaliter ad medium sint, dictum est quando erit conclusio necessaria. Si autem hic quidem universalis, ille vero particularis, praedicativus uterque, quando universalis fuerit necessarius, et conclusio erit necessaria. Demonstratio autem eadem quae prius, convertitur enim et particularis affirmativa. Si ergo necesse est B omni C inesse, A autem sub C est, necesse est B alicui A inesse. Si autem B alicui A, et A alicui B inesse necessarium, convertitur enim.

|If, then, the premisses are universal, we have stated when the conclusion will be necessary. But if one premiss is universal, the other particular, and if both are affirmative, whenever the universal is necessary the conclusion also must be necessary. The demonstration is the same as before; for the particular affirmative also is convertible. If then it is necessary that B should belong to all C, and A falls under C, it is necessary that B should belong to some A. But if B must belong to some A, then A must belong to some B: for conversion is possible.

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|ὁμοίως δὲ καὶ εἰ τὸ Α Γ εἴη ἀναγκαῖον καθόλου ὄν· τὸ γὰρ Β ὑπὸ τὸ Γ ἐστίν. εἰ δὲ τὸ ἐν μέρει ἐστὶν ἀναγκαῖον, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον. ἔστω γὰρ τὸ Β Γ ἐν μέρει τε καὶ ἀναγκαῖον, τὸ δὲ Α παντὶ τῶι Γ ὑπαρχέτω, μὴ μέντοι ἐξ ἀνάγκης. ἀντιστραφέντος οὖν τοῦ Β Γ τὸ πρῶτον γίγνεται σχῆμα, καὶ ἡ μὲν κα θόλου πρότασις οὐκ ἀναγκαία, ἡ δ᾽ ἐν μέρει ἀναγκαία. ὅτε δ᾽ οὕτως ἔχοιεν αἱ προτάσεις, οὐκ ἦν τὸ συμπέρασμα ἀναγκαῖον, ὥστ᾽ οὐδ᾽ ἐπὶ τούτων.

|(0651A) Similiter autem et si A C sit necessaria universalis, nam B sub C est. Si autem particularis est necessaria, non erit conclusio necessaria. Sit enim B C particularis et necessaria, A autem insit omni C, non tamen ex necessitate, conversa ergo B C prima fit figura, et universalis quidem propositio non necessaria, particularis autem necessaria, quando autem sic se habebant propositiones, non erat conclusio necessaria, quare nec in his.

|Similarly also if AC should be necessary and universal: for B falls under C. But if the particular premiss is necessary, the conclusion will not be necessary. Let the premiss BC be both particular and necessary, and let A belong to all C, not however necessarily. If the proposition BC is converted the first figure is formed, and the universal premiss is not necessary, but the particular is necessary. But when the premisses were thus, the conclusion (as we proved was not necessary: consequently it is not here either.

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|ἔτι δὲ καὶ ἐκ τῶν ὅρων φανερόν. ἔστω γὰρ τὸ μὲν Α ἐγρήγορσις, τὸ δὲ Β δίπουν, ἐφ᾽ ὧι δὲ τὸ Γ ζῶιον. τὸ μὲν οὖν Β τινὶ τῶι Γ ἀνάγκη ὑπάρχειν, τὸ δὲ Α τῶι Γ ἐνδέχεται, καὶ τὸ Α τῶι Β οὐκ ἀναγκαῖον· οὐ γὰρ ἀνάγκη δίπουν τι καθεύδειν ἢ ἐγρηγορέναι. ὁμοίως δὲ καὶ διὰ τῶν αὐτῶν ὅρων δειχθήσεται καὶ εἰ τὸ Α Γ εἴη ἐν μέρει τε καὶ ἀναγκαῖον.

|Amplius autem et ex terminis manifestum. Sit enim A quidem vigilatio, B autem bipes, in quo autem C animal, ergo B alicui C necesse est inesse, A autem omni C contingit, et A non necessario B, non enim necesse est aliquem bipedem dormire vel vigilare. Similiter autem per eosdem terminos ostendetur etiam si A C sit particularis et necessaria.

|Further, the point is clear if we look at the terms. Let A be waking, B biped, and C animal. It is necessary that B should belong to some C, but it is possible for A to belong to C, and that A should belong to B is not necessary. For there is no necessity that some biped should be asleep or awake. Similarly and by means of the same terms proof can be made, should the proposition AC be both particular and necessary.

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|Εἰ δ᾽ ὁ μὲν κατηγορικὸς ὁ δὲ στερητικὸς τῶν ὅρων, ὅταν μὲν ἦι τὸ καθόλου στερητικόν τε καὶ ἀναγκαῖον, καὶ τὸ συμπέρασμα ἔσται ἀναγκαῖον· εἰ γὰρ τὸ Α τῶι Γ μηδενὶ ἐνδέχεται, τὸ δὲ Β τινὶ τῶι Γ ὑπάρχει, τὸ Α τινὶ τῶι Β ἀνάγκη μὴ ὑπάρχειν. ὅταν δὲ τὸ καταφατικὸν ἀναγκαῖον τεθῆι, ἢ καθόλου ὂν ἢ ἐν μέρει, ἢ τὸ στερητικὸν κατὰ μέρος, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον.

|(0651B) Si autem hic quidem terminorum sit praedicativus, ille privativus et necessarius, quando universalis fuerit privativus et necessarius, et conclusio erit necessaria. Si enim A nulli C ex necessitate contingit, B autem alicui C inest, necesse est A alicui B non inesse, quando autem affirmativa necessaria ponetur vel universalis, vel particularis, vel privativa particularis, non erit conclusio necessaria.

|But if one premiss is affirmative, the other negative, whenever the universal is both negative and necessary the conclusion also will be necessary. For if it is not possible that A should belong to any C, but B belongs to some C, it is necessary that A should not belong to some B. But whenever the affirmative proposition is necessary, whether universal or particular, or the negative is particular, the conclusion will not be necessary.

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|τὰ μὲν γὰρ ἄλλα ταὐτὰ ἃ καὶ ἐπὶ τῶν πρότερον ἐροῦμεν, ὅροι δ᾽ ὅταν μὲν ἦι καθόλου τὸ κατηγορικὸν ἀναγκαῖον, ἐγρήγορσις – ζῶιον – ἄνθρωπος, μέσον ἄν {{ln|bk32a1}} θρωπος, ὅταν δ᾽ ἐν μέρει τὸ κατηγορικὸν ἀναγκαῖον, ἐγρήγορσις – ζῶιον – λευκόν· ζῶιον μὲν γὰρ ἀνάγκη τινὶ λευκῶι ὑπάρχειν, ἐγρήγορσις δ᾽ ἐνδέχεται μηδενί, καὶ οὐκ ἀνάγκη τινὶ ζώιωι μὴ ὑπάρχειν ἐγρήγορσιν. ὅταν δὲ τὸ στερητικὸν ἐν μέ ρει ὂν ἀναγκαῖον ἦι, δίπουν – κινούμενον – ζῶιον, μέσον ζῶιον.

|Nam alia quidem eadem quae et in prioribus dicemus. Termini autem cum universalis quidem affirmativa est necessaria, vigilatio, animal, homo, medium homo: cum autem particularis praedicativa necessaria, vigilatio, animal, album. Animal enim necesse est alicui albo inesse, vigilatio autem contingit nulli, et non necesse est alicui animali non inesse vigilationem. (0651C) Quando autem privativa particularis est necessaria, bipes, motus, animal, medium animal.

|The proof of this by reduction will be the same as before; but if terms are wanted, when the universal affirmative is necessary, take the terms ‘waking’-’animal’-’man’, ‘man’ being middle, and when the affirmative is particular and necessary, take the terms ‘waking’-’animal’-’white’: for it is necessary that animal should belong to some white thing, but it is possible that waking should belong to none, and it is not necessary that waking should not belong to some animal. But when the negative proposition being particular is necessary, take the terms ‘biped’, ‘moving’, ‘animal’, ‘animal’ being middle.

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Authors/Aristotle/priora/Liber 1/C11 - Revision history

Edward Ockham: Created page with "==Chapter 11== {| border=1 cellpadding=10 !valign = top width=33.33%|Greek !valign = top width=33.33%|Latin !valign = top width=33.33%|English |- valign = top | |(PL 64 0650A..."