Authors/Aristotle/priora/Liber 2/C14

Chapter 14

Greek	Latin	English
	(PL 64 0699D) CAPUT XIV. Quo rusta, et quae ad impossibile ducit demonstratio, differant.	14
62b29 Διαφέρει δ᾽ ἡ εἰς τὸ ἀδύνατον ἀπόδειξις τῆς δεικτικῆς τῶι τιθέναι ὁ βούλεται ἀναιρεῖν ἀπάγουσα εἰς ὁμολογούμε- νον ψεῦδος· ἡ δὲ δεικτικὴ ἄρχεται ἐξ ὁμολογουμένων θέσεων. λαμβάνουσι μὲν οὖν ἀμφότεραι δύο προτάσεις ὁμολογουμένας· ἀλλ᾽ ἡ μὲν ἐξ ὧν ὁ συλλογισμός, ἡ δὲ μίαν μὲν τούτων, μίαν δὲ τὴν ἀντίφασιν τοῦ συμπεράσματος. καὶ ἔνθα μὲν οὐκ ἀνάγκη γνώριμον εἶναι τὸ συμπέρασμα, οὐδὲ προϋπολαμβάνειν ὡς ἔστιν ἢ οὔ· ἔνθα δὲ ἀνάγκη ὡς οὐκ ἔστιν. διαφέρει δ᾽ οὐδὲν φάσιν ἢ ἀπόφασιν εἶναι τὸ συμπέρασμα, ἀλλ᾽ ὁμοίως ἔχει περὶ ἀμφοῖν.	(0700A) Differt autem quae ad impossibile demonstratio ab ea quae est ostensiva, eo quod ponat quod vult interimere, deducens ad confessum falsum, ostensiva autem incipit A confessis positionibus veris. Sumunt ergo utraeque duas propositiones confessas, sed haec quidem ex quibus est syllogismus, illa vero unam quidem harum, alteram vero contradictionem conclusionis. Et hinc quidem non necesse est notam esse conclusionem, neque prius opinari quoniam est, aut non est; illinc vero necesse est, quoniam non est. Differt autem nihil affirmativam, vel negativam esse conclusionem, sed similiter se habet in utrisque.	Demonstration per impossibile differs from ostensive proof in that it posits what it wishes to refute by reduction to a statement admitted to be false; whereas ostensive proof starts from admitted positions. Both, indeed, take two premisses that are admitted, but the latter takes the premisses from which the syllogism starts, the former takes one of these, along with the contradictory of the original conclusion. Also in the ostensive proof it is not necessary that the conclusion should be known, nor that one should suppose beforehand that it is true or not: in the other it is necessary to suppose beforehand that it is not true. It makes no difference whether the conclusion is affirmative or negative; the method is the same in both cases.
Ἅπαν δὲ τὸ δεικτικῶς περαινόμενον καὶ διὰ τοῦ ἀδυνάτου δειχθήσεται, καὶ τὸ διὰ τοῦ ἀδυνάτου δεικτικῶς διὰ τῶν αὐτῶν ὅρων [οὐκ ἐν τοῖς αὐτοῖς δὲ σχήμασιν]. ὅταν μὲν γὰρ ὁ συλλο↵γισμὸς ἐν τῶι πρώτωι σχήματι γένηται, τὸ ἀληθὲς ἔσται ἐν τῶι μέσωι ἢ τῶι ἐσχάτωι, τὸ μὲν στερητικὸν ἐν τῶι μέσωι, τὸ δὲ κατηγορικὸν ἐν τῶι ἐσχάτωι. ὅταν δ᾽ ἐν τῶι μέσωι ὁ συλλογισμός, τὸ ἀληθὲς ἐν τῶι πρώτωι ἐπὶ πάντων τῶν προβλημάτων. ὅταν δ᾽ ἐν τῶι ἐσχάτωι ὁ συλλογισμός, τὸ ἀληθὲς ἐν τῶι πρώτωι καὶ τῶι μέσωι, τὰ μὲν καταφατικὰ ἐν τῶι πρώτωι, τὰ δὲ στερητικὰ ἐν τῶι μέσωι.	(0700B) Omnis enim quae ostensive concluditur, et per impossibile monstrabitur, et quae per impossibile ostensive, et per eosdem terminos, non autem in eisdem figuris. Nam quando per impossibile syllogismus fit in prima figura, quod verum est in media erit, aut in postrema, privativum quidem in media, praedicativum autem in postrema. Quando autem syllogismus in media fit, quod verum est erit in prima figura in omnibus propositionibus, quando autem in postrema syllogismus, quod verum est erit in prima et in media, affirmativa quidem in prima, privativa autem in media.	Everything which is concluded ostensively can be proved per impossibile, and that which is proved per impossibile can be proved ostensively, through the same terms. Whenever the syllogism is formed in the first figure, the truth will be found in the middle or the last figure, if negative in the middle, if affirmative in the last. Whenever the syllogism is formed in the middle figure, the truth will be found in the first, whatever the problem may be. Whenever the syllogism is formed in the last figure, the truth will be found in the first and middle figures, if affirmative in first, if negative in the middle.
ἔστω γὰρ δεδειγμένον τὸ Α μηδενὶ ἢ μὴ παντὶ τῶι Β διὰ τοῦ πρώτου σχήματος. οὐκοῦν ἡ μὲν ὑπόθεσις ἦν τινὶ τῶι Β ὑπάρχειν τὸ Α, τὸ δὲ Γ ἐλαμβάνετο τῶι μὲν Α παντὶ ὑπάρχειν, τῶι δὲ Β οὐδενί· οὕτω γὰρ ἐγίνετο ὁ συλλογισμὸς καὶ τὸ ἀδύνατον. τοῦτο δὲ τὸ μέσον σχῆμα, εἰ τὸ Γ τῶι μὲν Α παντὶ τῶι δὲ Β μηδενὶ ὑπάρχει. καὶ φανερὸν ἐκ τούτων ὅτι οὐδενὶ τῶι Β ὑπάρχει τὸ Α. ὁμοίως δὲ καὶ εἰ μὴ παντὶ δέδεικται ὑπάρχον. ἡ μὲν γὰρ ὑπόθεσίς ἐστι παντὶ ὑπάρχειν, τὸ δὲ Γ ἐλαμβάνετο τῶι μὲν Α παντί, τῶι δὲ Β οὐ παντί. καὶ εἰ στερητικὸν λαμβάνοιτο τὸ Γ Α, ὡσαύτως· καὶ γὰρ οὕτω γίνεται τὸ μέσον σχῆμα.	(0700C) Sit enim ostensum A nulli aut non omni B per primam figuram, ergo hypothesis quidem erat alicui B inesse A, C autem sumebatur A quidem omni inesse, B autem nulli, sic enim fiebat syllogismus ad impossibile. Hoc autem media figura, si C A quidem omni, B autem nulli inest, et manifestum ex his quoniam B nulli inest A. Similiter autem et si non omni ostensum sit inesse, nam hypothesis quidem est omni B A inesse, C autem sumebatur A quidem omni, B autem non omni, et si privativa sit sumpta C A, similiter etenim sic fit in media figura.	Suppose that A has been proved to belong to no B, or not to all B, through the first figure. Then the hypothesis must have been that A belongs to some B, and the original premisses that C belongs to all A and to no B. For thus the syllogism was made and the impossible conclusion reached. But this is the middle figure, if C belongs to all A and to no B. And it is clear from these premisses that A belongs to no B. Similarly if has been proved not to belong to all B. For the hypothesis is that A belongs to all B; and the original premisses are that C belongs to all A but not to all B. Similarly too, if the premiss CA should be negative: for thus also we have the middle figure.
πάλιν δεδείχθω τινὶ ὑπάρχον τῶι Β τὸ Α. ἡ μὲν οὖν ὑπόθεσις μηδενὶ ὑπάρχειν, τὸ δὲ Β ἐλαμβάνετο παντὶ τῶι Γ ὑπάρχειν καὶ τὸ Α ἢ παντὶ ἢ τινὶ τῶι Γ· οὕτω γὰρ ἔσται τὸ ἀδύνατον. τοῦτο δὲ τὸ ἔσχατον σχῆμα, εἰ τὸ Α καὶ τὸ Β παντὶ τῶι Γ. καὶ φανερὸν ἐκτούτων ὅτι ἀνάγκη τὸ Α τινὶ τῶι Β ὑπάρχειν. ὁμοίως δὲκαὶ εἰ τινὶ τῶι Γ ληφθείη ὑπάρχον τὸ Β ἢ τὸ Α.	Rursum sit ostensum alicui B inesse A, ergo hypothesis quidem est nulli inesse, B autem sumebatur omni C inesse, et A vel omni vel alicui C, sic enim erit impossibile. Hoc autem postrema figura, si A et B omni C, et manifestum ex his quia necesse est A alicui B inesse, similiter autem et si alicui C sumatur inesse B vel A.	Again suppose it has been proved that A belongs to some B. The hypothesis here is that is that A belongs to no B; and the original premisses that B belongs to all C, and A either to all or to some C: for in this way we shall get what is impossible. But if A and B belong to all C, we have the last figure. And it is clear from these premisses that A must belong to some B. Similarly if B or A should be assumed to belong to some C.
Πάλιν ἐν τῶι μέσωι σχήματι δεδείχθω τὸ Α παντὶ τῶιΒ ὑπάρχον. οὐκοῦν ἡ μὲν ὑπόθεσις ἦν μὴ παντὶ τῶι Β τὸΑ ὑπάρχειν, εἴληπται δὲ τὸ Α παντὶ τῶι Γ καὶ τὸ Γ παντὶ τῶι Β· οὕτω γὰρ ἔσται τὸ ἀδύνατον. τοῦτο δὲ τὸ πρῶτονσχῆμα, τὸ Α παντὶ τῶι Γ καὶ τὸ Γ παντὶ τῶι Β. ὁμοίως δὲ καὶ εἰ τινὶ δέδεικται ὑπάρχον· ἡ μὲν γὰρ ὑπόθεσις ἦν μηδενὶ τῶι Β τὸ Α ὑπάρχειν, εἴληπται δὲ τὸ Α παντὶ τῶι Γ καὶ τὸ Γ τινὶ τῶι Β.	(0700D) Rursum in media figura ostensum sit A omni B inesse, ergo hypothesis quidem fuit, non omni B inesse A, sumptum est autem A omni C, et C omni B, sic enim erit impossibile; hoc autem prima figura, si A omni C, et C omni B. Similiter autem et si ostensum sit alicui inesse, nam hypothesis quidem fuit, nulli B inesse A, sumptum est autem A omni C, et C alicui B.	Again suppose it has been proved in the middle figure that A belongs to all B. Then the hypothesis must have been that A belongs not to all B, and the original premisses that A belongs to all C, and C to all B: for thus we shall get what is impossible. But if A belongs to all C, and C to all B, we have the first figure. Similarly if it has been proved that A belongs to some B: for the hypothesis then must have been that A belongs to no B, and the original premisses that A belongs to all C, and C to some B.
εἰ δὲ στερητικὸς ὁ συλλογισμός, ἡ μὲν ὑπόθεσις τὸ Α τινὶ τῶι Β ὑπάρχειν, εἴληπται δὲ τὸ Α μηδενὶ τῶι Γ καὶ τὸ Γ παντὶ τῶι Β, ὥστε γίνεται τὸ πρῶτον σχῆμα.	Si autem privativus fit syllogismus, hypothesis quidem A alicui B inesse, sumptum est autem A nulli C, et C omni B, quare fit prima figura.	If the syllogism is negative, the hypothesis must have been that A belongs to some B, and the original premisses that A belongs to no C, and C to all B, so that the first figure results.
καὶ εἰ μὴ καθόλου ὁ συλλογισμός, ἀλλὰ τὸ Α τινὶ τῶι Β δέδεικται μὴ ὑπάρχειν, ὡσαύτως. ὑπόθεσις μὲν γὰρ παντὶ τῶι Β τὸ Α ὑπάρχειν, εἴληπται δὲ τὸ Α μηδενὶ τῶι Γ καὶ τὸ Γ τινὶ τῶι Β· οὕτω γὰρ τὸ πρῶτον σχῆμα.	Et si non universalis sit syllogismus, sed A alicui B ostensum sit non inesse, similiter: nam hypothesis quidem omni B inesse A, sumptum est autem A nulli C, et C alicui B, sic enim prima figura.	If the syllogism is not universal, but proof has been given that A does not belong to some B, we may infer in the same way. The hypothesis is that A belongs to all B, the original premisses that A belongs to no C, and C belongs to some B: for thus we get the first figure.
Πάλιν ἐν τῶι τρίτωι σχήματι δεδείχθω τὸ Α παντὶ τῶι Β ὑπάρχειν. οὐκοῦν ἡ μὲν ὑπόθεσις ἦν μὴ παντὶ τῶι Β τὸ ↵ Α ὑπάρχειν, εἴληπται δὲ τὸ Γ παντὶ τῶι Β καὶ τὸ Α παντὶ τῶι Γ· οὕτω γὰρ ἔσται τὸ ἀδύνατον. τοῦτο δὲ τὸ πρῶτον σχῆμα. ὡσαύτως δὲ καὶ εἰ ἐπὶ τινὸς ἡ ἀπόδειξις· ἡ μὲν γὰρ ὑπόθεσις μηδενὶ τῶι Β τὸ Α ὑπάρχειν, εἴληπται δὲ τὸ Γ τινὶ τῶι Β καὶ τὸ Α παντὶ τῶι Γ. εἰ δὲ στερητικὸς ὁ συλλογισμός, ὑπόθεσις μὲν τὸ Α τινὶ τῶι Β ὑπάρχειν, εἴληπται δὲ τὸ Γ τῶι μὲν Α μηδενί, τῶι δὲ Β παντί· τοῦτο δὲ τὸ μέσον σχῆμα.	(0701A) Rursum in tertia figura ostensum sit A inesse omni B, ergo hypothesis quidem fuit non omni B inesse A, sumptum est autem C omni B, et A omni C, sic enim erit impossibile, hoc autem prima figura. Similiter autem et si in aliquo sit demonstratio, non hypothesis quidem erit nulli B inesse A, sumptum est autem C alicui B, et A omni C. Si autem privativus sit syllogismus, hypothesis quidem A alicui B inesse, sumptum est autem C A quidem nulli, B autem omni, hoc autem media figura.	Again suppose it has been proved in the third figure that A belongs to all B. Then the hypothesis must have been that A belongs not to all B, and the original premisses that C belongs to all B, and A belongs to all C; for thus we shall get what is impossible. And the original premisses form the first figure. Similarly if the demonstration establishes a particular proposition: the hypothesis then must have been that A belongs to no B, and the original premisses that C belongs to some B, and A to all C. If the syllogism is negative, the hypothesis must have been that A belongs to some B, and the original premisses that C belongs to no A and to all B, and this is the middle figure.
ὁμοίως δὲ καὶ εἰ μὴ καθόλου ἡ ἀπόδειξις. ὑπόθεσις μὲν γὰρ ἔσται παντὶ τῶι Β τὸ Α ὑπάρχειν, εἴληπται δὲ τὸ Γ τῶι μὲν Α μηδενί, τῶι δὲ Β τινί· τοῦτο δὲ τὸ μέσον σχῆμα.	Similiter autem et si non universalis sit demonstratio, nam hypothesis quidem erit omni B inesse A, sumptum est autem C A quidem nulli, B autem alicui, hoc autem media figura.	Similarly if the demonstration is not universal. The hypothesis will then be that A belongs to all B, the premisses that C belongs to no A and to some B: and this is the middle figure.
Φανερὸν οὖν ὅτι διὰ τῶν αὐτῶν ὅρων καὶ δεικτικῶς ἔστι δεικνύναι τῶν προβλημάτων ἕκαστον [καὶ διὰ τοῦ ἀδυνάτου]. ὁμοίως δ᾽ ἔσται καὶ δεικτικῶν ὄντων τῶν συλλογισμῶν εἰς ἀδύνατον ἀπάγειν ἐν τοῖς εἰλημμένοις ὅροις, ὅταν ἡ ἀντικειμένη πρότασις τῶι συμπεράσματι ληφθῆι. γίνονται γὰρ οἱ αὐτοὶ συλλογισμοὶ τοῖς διὰ τῆς ἀντιστροφῆς, ὥστ᾽ εὐθὺς ἔχομεν καὶ τὰ σχήματα δι᾽ ὧν ἕκαστον ἔσται. δῆλον οὖν ὅτι πᾶν πρόβλημα δείκνυται κατ᾽ ἀμφοτέρους τοὺς τρόπους, διά τε τοῦ ἀδυνάτου καὶ δεικτικῶς, καὶ οὐκ ἐνδέχεται χωρίζεσθαι τὸν ἕτερον.	Manifestum ergo quoniam per eosdem terminos et ostensive est demonstrare unumquodque propositum, et per impossibile. (0701B) Similiter autem erit, et cum sint ostensivi syllogismi, ad impossibile deducere in terminis sumptis, quando opposita propositio conclusioni sumpta fuerit, nam fiunt iidem syllogismi iis qui sunt per conversionem, quare statim habemus et figuras per quas unumquodque erit. Palam ergo quoniam omne propositum ostenditur per utrosque modos et per impossibile et ostensive, et non contingit separari alterum ab altero.	It is clear then that it is possible through the same terms to prove each of the problems ostensively as well. Similarly it will be possible if the syllogisms are ostensive to reduce them ad impossibile in the terms which have been taken, whenever the contradictory of the conclusion of the ostensive syllogism is taken as a premiss. For the syllogisms become identical with those which are obtained by means of conversion, so that we obtain immediately the figures through which each problem will be solved. It is clear then that every thesis can be proved in both ways, i.e. per impossibile and ostensively, and it is not possible to separate one method from the other.