Authors/Aristotle/priora/Liber 2/C5

Chapter 5

Greek	Latin	English
	(PL 64 0692A) CAPUT V. De circulari ostensione in prima figura	5
57b18 Τὸ δὲ κύκλωι καὶ ἐξ ἀλλήλων δείκνυσθαί ἐστι τὸ διὰ τοῦ συμπεράσματος καὶ τοῦ ἀνάπαλιν τῆι κατηγορίαι τὴν ἑτέραν λαβόντα πρότασιν συμπεράνασθαι τὴν λοιπήν, ἣν ἐλάμβανεν ἐν θατέρωι συλλογισμῶι. οἷον εἰ ἔδει δεῖξαι ὅτι τὸ Α τῶι Γ παντὶ ὑπάρχει, ἔδειξε δὲ διὰ τοῦ Β, πάλιν εἰ δεικνύοι ὅτι τὸ Α τῶι Β ὑπάρχει, λαβὼν τὸ μὲν Α τῶι Γ ὑπάρχειν τὸ δὲ Γ τῶι Β [καὶ τὸ Α τῶι Β]· πρότερον δ᾽ ἀνάπαλιν ἔλαβε τὸ Β τῶι Γ ὑπάρχον. ἢ εἰ [ὅτι] τὸ Β τῶι Γ δεῖ δεῖξαι ὑπάρχον, εἰ λάβοι τὸ Α κατὰ τοῦ Γ, ὁ ἦν συμπέ- ρασμα, τὸ δὲ Β κατὰ τοῦ Α ὑπάρχειν· πρότερον δ᾽ ἐλήφθη ἀνάπαλιν τὸ Α κατὰ τοῦ Β.	(0692B) Circulo autem, et ex se invicem ostendere est per conclusionem, et e converso praedicationem alteram sumentem propositionem concludere reliquam, quam sumpserat in altero syllogismo, ut si oportuit ostendere quoniam A inest omni C, ostendat autem per C, rursus si monstret quoniam A inest B, sumens A quidem inesse C, C autem B, et A inerit B, prius autem e converso sumpsit B inesse C, aut si quoniam B inest C, oporteat ostendere si sumat A de C, quae fuit conclusio, B autem de A esse, prius autem sumptum est e converso A de B.	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.
ἄλλως δ᾽ οὐκ ἔστιν ἐξ ἀλλήλων δεῖξαι. εἴτε γὰρ ἄλλο μέσον λήψεται, οὐ κύκλωι· οὐδὲν γὰρ λαμβάνεται τῶν αὐτῶν· εἴτε τούτων τι, ἀνάγκη θάτερον μόνον· εἰ γὰρ ἄμφω, ταὐτὸν ἔσται συμπέρασμα, δεῖ δ᾽ ἕτερον.	(0692C) Aliter vero non est ex se invicem ostendere, sive enim aliud medium sumetur, non circulo, nil enim sumitur eorumdem, sive horum quiddam, necesse est alterum solum, nam si ambo, eadem erit conclusio, at oportet diversam esse.	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.
Ἐν μὲν οὖν τοῖς μὴ ἀντιστρέφουσιν ἐξ ἀναποδείκτου τῆς ἑτέρας προτάσεως γίνεται ὁ συλλογισμός· οὐ γὰρ ἔστιν ἀποδεῖξαι διὰ τούτων τῶν ὅρων ὅτι τῶι μέσωι τὸ τρίτον ὑπάρχει ἢ τῶι πρώτωι τὸ μέσον. ἐν δὲ τοῖς ἀντιστρέφουσιν ἔστι πάντα δεικνύναι δι᾽ ἀλλήλων, οἷον εἰ τὸ Α καὶ τὸ Β καὶ τὸ Γ ἀντιστρέφουσιν ἀλλήλοις. δεδείχθω γὰρ τὸ Α Γ διὰ μέσου τοῦ Β, καὶ πάλιν τὸ Α Β διά τε τοῦ συμπεράσματος καὶ διὰ τῆς Β Γ προτάσεως ἀντιστραφείσης, ὡσαύτως δὲ καὶ τὸ Β Γ διά τε τοῦ συμπεράσματος καὶ τῆς Α Β ↵ προτάσεως ἀντεστραμμένης. δεῖ δὲ τήν τε Γ Β καὶ τὴν Β Α πρότασιν ἀποδεῖξαι· ταύταις γὰρ ἀναποδείκτοις κεχρήμεθα μόναις.	In iis igitur quae non convertuntur ex indemonstrata altera propositione fit syllogismus, non enim est demonstrare per hos terminos, quoniam medio inest tertium, aut primo medium. In iis autem quae convertuntur, erit omnia monstrare per se invicem, ut si A, et B, et C convertuntur sibi invicem: ostendatur enim A C per medium B, et rursum A B per conclusionem, et per B C propositionem conversam; similiter autem et B C, et per conclusionem, et per A B propositionem conversam; oportet autem et C B, et B A propositionem demonstrare, nam his demonstratis usi sumus solis.	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.
ἐὰν οὖν ληφθῆι τὸ Β παντὶ τῶι Γ ὑπάρχειν καὶ τὸ Γ παντὶ τῶι Α, συλλογισμὸς ἔσται τοῦ Β πρὸς τὸ Α. πάλιν ἐὰν ληφθῆι τὸ μὲν Γ παντὶ τῶι Α, τὸ δὲ Α παντὶ τῶι Β, παντὶ τῶι Β τὸ Γ ἀνάγκη ὑπάρχειν. ἐν ἀμφοτέροις δὴ τούτοις τοῖς συλλογισμοῖς ἡ Γ Α πρότασις εἴληπται ἀναπόδεικτος· αἱ γὰρ ἕτεραι δεδειγμέναι ἦσαν. ὥστ᾽ ἂν ταύτην ἀποδείξωμεν, ἅπασαι ἔσονται δεδειγμέναι δι᾽ ἀλλήλων. ἐὰν οὖν ληφθῆι τὸ Γ παντὶ τῶι Β καὶ τὸ Β παντὶ τῶι Α ὑπάρχειν, ἀμφότεραί τε αἱ προτάσεις ἀποδεδειγμέναι λαμβάνονται, καὶ τὸ Γ τῶι Α ἀνάγκη ὑπάρχειν.	(0692D) Si ergo sumatur B omni C inesse, et C omni A, syllogismus erit eius quod est B ad A. Rursus si sumatur C omni A inesse, et A omni B, necesse est C inesse omni B. In utrisque ergo syllogismis C A propositio sumpta est indemonstrata, nam aliae probatae erant: quare si hanc ostenderimus, omnes erunt approbatae per se invicem; si ergo sumatur C omni B, et B omni A inesse, utraeque propositiones demonstratae sumuntur, et C necesse est inesse A.	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.
φανερὸν οὖν ὅτι ἐν μόνοις τοῖς ἀντιστρέφουσι κύκλωι καὶ δι᾽ ἀλλήλων ἐνδέχεται γίνεσθαι τὰς ἀποδείξεις, ἐν δὲ τοῖς ἄλλοις ὡς πρότερον εἴπομεν. συμβαίνει δὲ καὶ ἐν τούτοις αὐτῶι τῶι δεικνυμένωι χρῆσθαι πρὸς τὴν ἀπόδειξιν· τὸ μὲν γὰρ Γ κατὰ τοῦ Β καὶ τὸ Β κατὰ τοῦ Α δείκνυται ληφθέντος τοῦ Γ κατὰ τοῦ Α λέγεσθαι, τὸ δὲ Γ κατὰ τοῦ Α διὰ τούτων δείκνυται τῶν προτάσεων, ὥστε τῶι συμπεράσματι χρώμεθα πρὸς τὴν ἀπόδειξιν.	Manifestum est ergo quoniam in solis iis quae convertuntur, circulo et per se invicem contingit fieri demonstrationes, in aliis vero quemadmodum prius diximus. (0693A) Accidit autem et in iis eodem quod monstratur uti ad demonstrationem, nam C de B, et B de A monstratur sumpto C de A dici, C autem de A per has ostenditur propositiones: quare conclusione utimur ad demonstrationem.	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.
Ἐπὶ δὲ τῶν στερητικῶν συλλογισμῶν ὧδε δείκνυται ἐξ ἀλλήλων. ἔστω τὸ μὲν Β παντὶ τῶι Γ ὑπάρχειν, τὸ δὲ Α οὐ- δενὶ τῶι Β· συμπέρασμα ὅτι τὸ Α οὐδενὶ τῶι Γ. εἰ δὴ πάλιν δεῖ συμπεράνασθαι ὅτι τὸ Α οὐδενὶ τῶι Β, ὁ πάλαι ἔλαβεν, ἔστω τὸ μὲν Α μηδενὶ τῶι Γ, τὸ δὲ Γ παντὶ τῶι Β· οὕτω γὰρ ἀνάπαλιν ἡ πρότασις. εἰ δ᾽ ὅτι τὸ Β τῶι Γ δεῖ συμπεράνασθαι, οὐκέθ᾽ ὁμοίως ἀντιστρεπτέον τὸ Α Β (ἡ γὰρ αὐτὴ πρότασις, τὸ Β μηδενὶ τῶι Α καὶ τὸ Α μηδενὶ τῶι Β ὑπάρχειν), ἀλλὰ ληπτέον, ὧι τὸ Α μηδενὶ ὑπάρχει, τὸ Β παντὶ ὑπάρχειν. ἔστω τὸ Α μηδενὶ τῶι Γ ὑπάρχειν, ὅπερ ἦν τὸ συμπέρασμα· ὧι δὲ τὸ Α μηδενί, τὸ Β εἰλήφθω παντὶ ὑπάρχειν· ἀνάγκη οὖν τὸ Β παντὶ τῶι Γ ὑπάρχειν. ὥστε τριῶν ὄντων ἕκαστον συμπέρασμα γέγονε, καὶ τὸ κύκλωι ἀποδεικνύναι τοῦτ᾽ ἔστι, τὸ τὸ συμπέρασμα λαμβάνοντα καὶ ἀνάπαλιν τὴν ἑτέραν πρότασιν τὴν λοιπὴν συλλογίζεσθαι.	In privativis autem syllogismis hoc modo monstratur ex se invicem: sit B quidem omni C inesse, A autem nulli B, conclusio autem quoniam A nulli C. Si ergo rursum oporteat concludere quoniam A nulli B, quod prius sumptum erat, erit A quidem nulli C, C autem omni B, sic enim e converso propositio. Si autem quoniam B inest C, oporteat concludere, non iam similiter convertendum A B, nam eadem propositio est B nulli A, et A nulli B inesse, sed sumendum, cui A nulli inest, huic B omni inesse. (0693B) Sit enim A nulli C inesse, quod quidem fuit conclusio, cui autem A nulli B, si sumatur omni inesse, necesse est ergo B omni C inesse: quare cum sint tria, unumquodque conclusio est facta, et circulo demonstrare, hoc est conclusionem sumentem et e converso alteram propositionem, reliquam syllogizare.	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.
Ἐπὶ δὲ τῶν ἐν μέρει συλλογισμῶν τὴν μὲν καθόλου πρότασιν οὐκ ἔστιν ἀποδεῖξαι διὰ τῶν ἑτέρων, τὴν δὲ κατὰ μέρος ἔστιν. ὅτι μὲν οὖν οὐκ ἔστιν ἀποδεῖξαι τὴν καθόλου, φανερόν· τὸ μὲν γὰρ καθόλου δείκνυται διὰ τῶν καθόλου, τὸ δὲ συμπέρασμα οὐκ ἔστι καθόλου, δεῖ δ᾽ ἐκ τοῦ συμπεράσματος δεῖξαι καὶ τῆς ἑτέρας προτάσεως. ἔτι ὅλως οὐδὲ γίνεται ↵ συλλογισμὸς ἀντιστραφείσης τῆς προτάσεως· ἐν μέρει γὰρ ἀμφότεραι γίνονται αἱ προτάσεις.	In particularibus autem syllogismis universalem quidem propositionem non est demonstrare per alias, particularem autem est; quoniam autem non est demonstrare universalem, manifestum, nam universale monstratur per universalia, conclusio autem non est universalis, oportet autem ostendere ex conclusione et altera propositione. Amplius, omnino non fit syllogismus conversa propositione, nam particulares fiunt utraeque propositiones.	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.
τὴν δ᾽ ἐπὶ μέρους ἔστιν. δεδείχθω γὰρ τὸ Α κατὰ τινὸς τοῦ Γ διὰ τοῦ Β. ἐὰν οὖν ληφθῆι τὸ Β παντὶ τῶι Α καὶ τὸ συμπέρασμα μένηι, τὸ Β τινὶ τῶι Γ ὑπάρξει· γίνεται γὰρ τὸ πρῶτον σχῆμα, καὶ τὸ Α μέσον. εἰ δὲ στερητικὸς ὁ συλλογισμός, τὴν μὲν καθόλου πρότασιν οὐκ ἔστι δεῖξαι, δι᾽ ὁ καὶ πρότερον ἐλέχθη· τὴν δ᾽ ἐν μέρει ἔστιν, ἐὰν ὁμοίως ἀντιστραφῆι τὸ Α Β ὥσπερ κἀπὶ τῶν καθόλου, [οὐκ ἔστι, διὰ προσλήψεως δ᾽ ἔστιν,] οἷον ὧι τὸ Α τινὶ μὴ ὑπάρχει, τὸ Β τινὶ ὑπάρχειν· ἄλλως γὰρ οὐ γίνεται συλλογισμὸς διὰ τὸ ἀποφατικὴν εἶναι τὴν ἐν μέρει πρότασιν.	Particulare autem est, ostendatur enim A de aliquo C per B, si ergo sumatur B omni A, et conclusio maneat, B alicui C inerit, fit enim prima figura, et est A medium. (0693C) Si autem fit privativus syllogismus, universalem quidem propositionem non est ostendere, propter hoc quod prius dictum est, particularem (si simpliciter convertatur A B quemadmodum et in universalibus) non est, per assumptionem autem est, ut cui A alicui non insit, B alicui inesse; nam aliter se habentibus non fit syllogismus, eo quod negativa est particularis propositio.	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.