Authors/Aristotle/priora/Liber 1/C45

Chapter 45

Greek	Latin	English
	(PL 64 0681C) CAPUT XLI. De reciproca reductione syllogismorum unius figurae in aliam.	45
50b5 Ὅσα δ᾽ ἐν πλείοσι σχήμασι δείκνυται τῶν προβλημάτων, ἢν ἐν θατέρωι συλλογισθῆι, ἔστιν ἀναγαγεῖν τὸν συλλογισμὸν εἰς θάτερον, οἷον τὸν ἐν τῶι πρώτωι στερητικὸν εἰς τὸ δεύτερον, καὶ τὸν ἐν τῶι μέσωι εἰς τὸ πρῶτον, οὐχ ἅπαντας δὲ ἀλλ᾽ ἐνίους. ἔσται δὲ φανερὸν ἐν τοῖς ἑπομένοις. εἰ γὰρ τὸ Α μηδενὶ τῶι Β, τὸ δὲ Β παντὶ τῶι Γ, τὸ Α οὐδενὶ τῶι Γ. οὕτω μὲν οὖν τὸ πρῶτον σχῆμα, ἐὰν δ᾽ ἀντιστραφῆι τὸ στερητικόν, τὸ μέσον ἔσται· τὸ γὰρ Β τῶι μὲν Α οὐδενί, τῶι δὲ Γ παντὶ ὑπάρχει. ὁμοίως δὲ καὶ εἰ μὴ καθόλου ἀλλ᾽ ἐν μέρει ὁ συλλογισμός, οἷον εἰ τὸ μὲν Α μηδενὶ τῶι Β, τὸ δὲ Β τινὶ τῶι Γ· ἀντιστραφέντος γὰρ τοῦ στερητικοῦ τὸ μέσον ἔσται σχῆμα.	Quaecunque autem in pluribus figuris monstrantur proposita, si in altera syllogizetur, est reducere syllogismum in alteram, ut eum qui in prima est privativum in secundam figuram, et eum qui in media est in primam. (0681D) Non omnes autem, sed quosdam, erit autem in sequentibus manifestum. Si enim A nulli B, B autem omni C, A nulli C, sic ergo prima figura; si autem convertatur privativa, media erit. Nam B A quidem nulli, C autem omni inerit. Similiter autem et si non universalis, sed particularis fit syllogismus, ut si A quidem nulli B, B autem alicui C, conversa enim privativa media erit figura.	Whatever problems are proved in more than one figure, if they have been established in one figure by syllogism, can be reduced to another figure, e.g. a negative syllogism in the first figure can be reduced to the second, and a syllogism in the middle figure to the first, not all however but some only. The point will be clear in the sequel. If A belongs to no B, and B to all C, then A belongs to no C. Thus the first figure; but if the negative statement is converted, we shall have the middle figure. For B belongs to no A, and to all C. Similarly if the syllogism is not universal but particular, e.g. if A belongs to no B, and B to some C. Convert the negative statement and you will have the middle figure.
Τῶν δ᾽ ἐν τῶι δευτέρωι συλλογισμῶν οἱ μὲν καθόλου ἀναχθήσονται εἰς τὸ πρῶτον, τῶν δ᾽ ἐν μέρει ἅτερος μόνος. ἔστω γὰρ τὸ Α τῶι μὲν Β μηδενὶ τῶι δὲ Γ παντὶ ὑπάρχον. ἀντιστραφέντος οὖν τοῦ στερητικοῦ τὸ πρῶτον ἔσται σχῆμα· τὸ μὲν γὰρ Β οὐδενὶ τῶι Α, τὸ δὲ Α παντὶ τῶι Γ ὑπάρξει. ἐὰν δὲ τὸ κατηγορικὸν ἦι πρὸς τῶι Β, τὸ δὲ στερητικὸν πρὸς τῶι Γ, πρῶτον ὅρον θετέον τὸ Γ· τοῦτο γὰρ οὐδενὶ τῶι Α, τὸ δὲ Α παντὶ τῶι Β· ὥστ᾽ οὐδενὶ τῶι Β τὸ Γ. οὐδ᾽ ἄρα τὸ Β τῶι Γ οὐδενί· ἀντιστρέφει γὰρ τὸ στερητικόν.	Eorum autem syllogismorum, qui sunt in secunda figura, universales quidem reducentur in primam figuram, particularium autem alter solum. Insit enim A B quidem nulli, C vero omni, conversa privativa prima erit figura, nam B quidem nulli A, A autem omni C inerit. (0682A) Si autem praedicativum quidem sit ad B, privativum autem ad C, primus terminus ponendus est C, hoc enim nulli A, A autem omni B, quare nulli B inerit C, ergo et B nulli C, convertitur enim privativa.	The universal syllogisms in the second figure can be reduced to the first, but only one of the two particular syllogisms. Let A belong to no B and to all C. Convert the negative statement, and you will have the first figure. For B will belong to no A and A to all C. But if the affirmative statement concerns B, and the negative C, C must be made first term. For C belongs to no A, and A to all B: therefore C belongs to no B. B then belongs to no C: for the negative statement is convertible.
ἐὰν δ᾽ ἐν μέρει ἦι ὁ συλλογισμός, ὅταν μὲν ἦι τὸ στερητικὸν πρὸς τῶι μείζονι ἄκρωι, ἀναχθήσεται εἰς τὸ πρῶτον, οἷον εἰ τὸ Α μηδενὶ τῶι Β, τῶι δὲ Γ τινί· ἀντιστραφέντος γὰρ τοῦ στερητικοῦ τὸ πρῶτον ἔσται σχῆμα· τὸ μὲν γὰρ Β οὐδενὶ τῶι Α, τὸ δὲ Α τινὶ τῶι Γ. ὅταν δὲ τὸ κατηγορικόν, οὐκ ἀναλυθήσεται, οἷον εἰ τὸ Α τῶι μὲν Β παντί, τῶι δὲ Γ οὐ παντί· οὔτε γὰρ δέχεται ἀντιστροφὴν τὸ Α Β, οὔτε γενομένης ἔσται συλλογισμός.	Si autem particularis sit syllogismus, quando privativum quidem erit ad maiorem extremitatem, resolvetur in primam figuram, ut si A nulli B, B autem alicui C, conversa enim privativa prima erit figura, nam B quidem nulli A, A autem alicui C. Quando vero praedicativum, non resolvetur, ut si A quidem omni B, C vero non omni, non enim suscipit conversionem A B, neque cum fit, erit syllogismus.	But if the syllogism is particular, whenever the negative statement concerns the major extreme, reduction to the first figure will be possible, e.g. if A belongs to no B and to some C: convert the negative statement and you will have the first figure. For B will belong to no A and A to some C. But when the affirmative statement concerns the major extreme, no resolution will be possible, e.g. if A belongs to all B, but not to all C: for the statement AB does not admit of conversion, nor would there be a syllogism if it did.
Πάλιν οἱ μὲν ἐν τῶι τρίτωι σχήματι οὐκ ἀναλυθήσονται πάντες εἰς τὸ πρῶτον, οἱ δ᾽ ἐν τῶι πρώτωι πάντες εἰς τὸ τρίτον. ὑπαρχέτω γὰρ τὸ Α παντὶ τῶι Β, τὸ δὲ Β τινὶ τῶι Γ. οὐκοῦν ἐπειδὴ ἀντιστρέφει τὸ ἐν μέρει κατηγορικόν, ὑπάρ- ξει τὸ Γ τινὶ τῶι Β· τὸ δὲ Α παντὶ ὑπῆρχεν, ὥστε γίνεται τὸ τρίτον σχῆμα. καὶ εἰ στερητικὸς ὁ συλλογισμός, ὡσαύτως· ἀντιστρέφει γὰρ τὸ ἐν μέρει κατηγορικόν, ὥστε τὸ μὲν Α οὐδενὶ τῶι Β, τὸ δὲ Γ τινὶ ὑπάρξει.	Rursus qui in tertia quidem sunt figura, non resolvuntur omnes in primam, qui autem sunt in prima, omnes in tertiam. Insit enim A quidem omni B, B autem alicui C, ergo quia convertitur particularis praedicativa, inerit et C alicui B, A vero omni B inerat, quare fit tertia figura. (0682B) Et si privativus sit syllogismus, similiter: convertitur enim particularis affirmativa, quare A quidem nulli B, C autem alicui inerit.	Again syllogisms in the third figure cannot all be resolved into the first, though all syllogisms in the first figure can be resolved into the third. Let A belong to all B and B to some C. Since the particular affirmative is convertible, C will belong to some B: but A belonged to all B: so that the third figure is formed. Similarly if the syllogism is negative: for the particular affirmative is convertible: therefore A will belong to no B, and to some C.
↵ Τῶν δ᾽ ἐν τῶι τελευταίωι σχήματι συλλογισμῶν εἷς μόνος οὐκ ἀναλύεται εἰς τὸ πρῶτον, ὅταν μὴ καθόλου τεθῆι τὸ στερητικόν, οἱ δ᾽ ἄλλοι πάντες ἀναλύονται. κατηγορείσθω γὰρ παντὸς τοῦ Γ τὸ Α καὶ τὸ Β· οὐκοῦν ἀντιστρέψει τὸ Γ πρὸς ἑκάτερον ἐπὶ μέρους· ὑπάρχει ἄρα τινὶ τῶι Β. ὥστ᾽ ἔσται τὸ πρῶτον σχῆμα, εἰ τὸ μὲν Α παντὶ τῶι Γ, τὸ δὲ Γ τινὶ τῶι Β. καὶ εἰ τὸ μὲν Α παντὶ τῶι Γ, τὸ δὲ Β τινί, ὁ αὐτὸς λόγος· ἀντιστρέφει γὰρ πρὸς τὸ Β τὸ Γ. ἐὰν δὲ τὸ μὲν Β παντὶ τῶι Γ, τὸ δὲ Α τινὶ τῶι Γ, πρῶτος ὅρος θετέος τὸ Β· τὸ γὰρ Β παντὶ τῶι Γ, τὸ δὲ Γ τινὶ τῶι Α, ὥστε τὸ Β τινὶ τῶι Α. ἐπεὶ δ᾽ ἀντιστρέφει τὸ ἐν μέρει, καὶ τὸ Α τινὶ τῶι Β ὑπάρξει. καὶ εἰ στερητικὸς ὁ συλλογισμός, καθάλου τῶν ὅρων ὄντων, ὁμοίως ληπτέον.	Eorum autem sylogismorum qui sunt in postrema figura unus tantum non resolvitur in primam, quando non universalis ponitur privativa, alii autem omnes resolvuntur. Praedicentur enim de omni C, et A et B, ergo convertetur C ad utrumque particulariter; inerit ergo A alicui B, quare erit prima figura, siquidem A omni C, C vero alicui B; et si A quidem omni C, B autem alicui C, cadem ratio, convertitur enim ad B C. Si autem B quidem omni C, A autem alicui C, primus ponendus B, nam B omni C, C autem alicui A, quare B alicui A, quoniam autem convertitur particularis, et A alicui B inerit. Et si privativus sit syllogismus universalibus terminis, similiter sumendum.	Of the syllogisms in the last figure one only cannot be resolved into the first, viz. when the negative statement is not universal: all the rest can be resolved. Let A and B be affirmed of all C: then C can be converted partially with either A or B: C then belongs to some B. Consequently we shall get the first figure, if A belongs to all C, and C to some of the Bs. If A belongs to all C and B to some C, the argument is the same: for B is convertible in reference to C. But if B belongs to all C and A to some C, the first term must be B: for B belongs to all C, and C to some A, therefore B belongs to some A. But since the particular statement is convertible, A will belong to some B. If the syllogism is negative, when the terms are universal we must take them in a similar way.
ὑπαρχέτω γὰρ τὸ Β παντὶ τῶι Γ, τὸ δὲ Α μηδενί· οὐκοῦν τινὶ τῶι Β ὑπάρξει τὸ Γ, τὸ δὲ Α οὐδενὶ τῶι Γ, ὥστ᾽ ἔσται μέσον τὸ Γ. ὁμοίως δὲ καὶ εἰ τὸ μὲν στερητικὸν καθόλου, τὸ δὲ κατηγορικὸν ἐν μέρει· τὸ μὲν γὰρ Α οὐδενὶ τῶι Γ, τὸ δὲ Γ τινὶ τῶν Β ὑπάρξει. ἐὰν δ᾽ ἐν μέρει ληφθῆι τὸ στερητικόν, οὐκ ἔσται ἀνάλυσις, οἷον εἰ τὸ μὲν Β παντὶ τῶι Γ, τὸ δὲ Α τινὶ μὴ ὑπάρ χει· ἀντιστραφέντος γὰρ τοῦ Β Γ ἀμφότεραι αἱ προτάσεις ἔσονται κατὰ μέρος.	(0682C) Insit enim B omni C, A autem nulli C, ergo alicui B inerit C, A autem nulli C, quare erit medium C. Similiter autem et si privativa quidem si universalis, praedicativa autem particularis, nam A quidem nulli C, C autem alicui B inerit. Si autem particularis sumatur privativa, non erit resolutio, ut si B quidem omni C, A autem alicui C non inest, conversa enim B C, utraeque propositiones erunt particulares.	Let B belong to all C, and A to no C: then C will belong to some B, and A to no C; and so C will be middle term. Similarly if the negative statement is universal, the affirmative particular: for A will belong to no C, and C to some of the Bs. But if the negative statement is particular, no resolution will be possible, e.g. if B belongs to all C, and A not belong to some C: convert the statement BC and both premisses will be particular.
Φανερὸν δὲ καὶ ὅτι πρὸς τὸ ἀναλύειν εἰς ἄλληλα τὰ σχήματα ἡ πρὸς τῶι ἐλάττονι ἄκρωι πρότασις ἀντιστρεπτέα ἐν ἀμφοτέροις τοῖς σχήμασι· ταύτης γὰρ μετατιθεμένης ἡ μετάβασις ἐγίνετο.	Manifestum autem quoniam ad resolvendum ad se invicem figuras, quae ad minorem extremitatem est propositio, convertenda in utrisque figuris, hac conversa, transitio fit; eorum autem qui in media sunt figura, alter quidem resolvitur, alter vero non resolvitur in tertiam, nam cum sit universalis privativa, resolvitur.	It is clear that in order to resolve the figures into one another the premiss which concerns the minor extreme must be converted in both the figures: for when this premiss is altered, the transition to the other figure is made.
Τῶν δ᾽ ἐν τῶι μέσωι σχήματι ἅτερος μὲν ἀναλύεται, ἅτερος δ᾽ οὐκ ἀναλύεται, εἰς τὸ τρίτον. ὅταν μὲν γὰρ ἦι τὸ καθόλου στερητικόν, ἀναλύεται. εἰ γὰρ τὸ Α μηδενὶ τῶι Β, τῶι δὲ Γ τινί, ἀμφότερα ὁμοίως ἀντιστρέφει πρὸς τὸ Α, ὥστε τὸ μὲν Β οὐδενὶ τῶι Α, τὸ δὲ Γ τινί· μέσον ἄρα τὸ Α. ὅταν δὲ τὸ Α παντὶ τῶι Β, τῶι δὲ Γ τινὶ μὴ ὑπάρχηι, οὐκ ἔσται ἀνάλυσις· οὐδετέρα γὰρ τῶν προτάσεων ἐκ τῆς ἀντιστροφῆς καθόλου.	(0682D) Si enim A nulli quidem B, alicui autem C, utraque similiter convertitur ad A, quare B quidem nulli A, C vero alicui, medium ergo A. Quando autem A omni B, C autem alicui non insit, non fit resolutio, neutra enim propositionum ex conversione universalis.	One of the syllogisms in the middle figure can, the other cannot, be resolved into the third figure. Whenever the universal statement is negative, resolution is possible. For if A belongs to no B and to some C, both B and C alike are convertible in relation to A, so that B belongs to no A and C to some A. A therefore is middle term. But when A belongs to all B, and not to some C, resolution will not be possible: for neither of the premisses is universal after conversion.
Καὶ οἱ ἐκ τοῦ τρίτου δὲ σχήματος ἀναλυθήσονται εἰς τὸ μέσον, ὅταν ἦι καθόλου τὸ στερητικόν, οἷον εἰ τὸ Α μηδενὶ τῶι Γ, τὸ δὲ Β τινὶ ἢ παντί. καὶ γὰρ τὸ Γ τῶι μὲν Α οὐδενί, τῶι δὲ Β τινὶ ὑπάρξει. ἐὰν δ᾽ ἐπὶ μέρους ἦι τὸ στερητικόν, οὐκ ἀναλυθήσεται· οὐ γὰρ δέχεται ἀντιστροφὴν τὸ ἐν μέρει ἀποφατικόν.	Qui autem ex tertia sunt figura, resolventur in mediam, quando fuerit universalis privativa, ut si A nulli C, B autem alicui, aut omni C, nam C, A quidem nulli, B autem alicui inerit. Si autem particularis sit privativa, non resolvetur, non enim suscipit conversionem particularis negativa.	Syllogisms in the third figure can be resolved into the middle figure, whenever the negative statement is universal, e.g. if A belongs to no C, and B to some or all C. For C then will belong to no A and to some B. But if the negative statement is particular, no resolution will be possible: for the particular negative does not admit of conversion.
Φανερὸν οὖν ὅτι οἱ αὐτοὶ συλλογισμοὶ οὐκ ἀναλύονται ἐν τούτοις τοῖς σχήμασιν οἵπερ οὐδ᾽ εἰς τὸ πρῶτον ἀνελύοντο, ↵ καὶ ὅτι εἰς τὸ πρῶτον σχῆμα τῶν συλλογισμῶν ἀναγομένων οὗτοι μόνοι διὰ τοῦ ἀδυνάτου περαίνονται.	Manifestum ergo quoniam iidem syllogismi non resolvuntur in his figuris, qui nec in primam resolvebantur, et quoniam in primam figuram reductis syllogismis, isti soli syllogismi per impossibile clauduntur.	It is clear then that the same syllogisms cannot be resolved in these figures which could not be resolved into the first figure, and that when syllogisms are reduced to the first figure these alone are confirmed by reduction to what is impossible.
Πῶς μὲν οὖν δεῖ τοὺς συλλογισμοὺς ἀνάγειν, καὶ ὅτι ἀναλύεται τὰ σχήματα εἰς ἄλληλα, φανερὸν ἐκ τῶν εἰ ρημένων.	(0683A) Quomodo ergo oportet syllogismos reducere, et quoniam resolvuntur figurae in se invicem, manifestum ex dictis.	It is clear from what we have said how we ought to reduce syllogisms, and that the figures may be resolved into one another.