Authors/Aristotle/priora/Liber 2/C4

Chapter 4

Greek	Latin	English
	(PL 64 0690B) CAPUT IV. Quod ex falsis verum identidem colligatur in tertia figura.	4
56b4 Ἔσται δὲ καὶ ἐν τῶι ἐσχάτωι σχήματι διὰ ψευδῶν ἀληθές, καὶ ἀμφοτέρων ψευδῶν οὐσῶν ὅλων καὶ ἐπί τι ἑκατέρας, καὶ τῆς μὲν ἑτέρας ἀληθοῦς ὅλης τῆς δ᾽ ἑτέρας ψευδοῦς, καὶ τῆς μὲν ἐπί τι ψευδοῦς τῆς δ᾽ ὅλης ἀληθοῦς, καὶ ἀνάπαλιν, καὶ ὁσαχῶς ἄλλως ἐγχωρεῖ μεταλαβεῖν τὰς προτάσεις. οὐδὲν γὰρ κωλύει μήτε τὸ Α μήτε τὸ Β μηδενὶ τῶι Γ ὑπάρχειν, τὸ μέντοι Α τινὶ τῶι Β ὑπάρχειν, οἷον οὔτ᾽ ἄνθρωπος οὔτε πεζὸν οὐδενὶ ἀψύχωι ἕπεται, ἄνθρωπος μέντοι τινὶ πεζῶι ὑπάρχει. ἐὰν οὖν ληφθῆι τὸ Α καὶ τὸ Β παντὶ τῶι Γ ὑπάρχειν, αἱ μὲν προτάσεις ὅλαι ψευδεῖς, τὸ δὲ συμπέρασμα ἀληθές. ὡσαύτως δὲ καὶ τῆς μὲν στερητικῆς τῆς δὲ καταφατικῆς οὔσης. ἐγχωρεῖ γὰρ τὸ μὲν Β μηδενὶ τῶι Γ ὑπάρχειν, τὸ δὲ Α παντί, καὶ τὸ Α τινὶ τῶι Β μὴ ὑπάρχειν, οἷον τὸ μέλαν οὐδενὶ κύκνωι, ζῶιον δὲ παντί, καὶ τὸ ζῶιον οὐ παντὶ μέλανι. ὥστ᾽ ἂν ληφθῆι τὸ μὲν Β παντὶ τῶι Γ, τὸ δὲ Α μηδενί, τὸ Α τινὶ τῶι Β οὐχ ὑπάρξει· καὶ τὸ μὲν συμπέρασμα ἀληθές, αἱ δὲ προτάσεις ψευδεῖς.	(0690C) Erit autem et in postrema figura per falsas totas, et in aliquo utraque, et altera quidem vera, altera autem falsa, et haec quidem in aliquo falsa, illa autem tota vera, et e converso, et quotquot modis aliter possibile est transumere propositiones. Nihil enim prohibet nec A nec B nulli C inesse, A autem alicui B inesse, ut nec homo, nec gressibile, nullum inanimatum sequitur, homo autem alicui gressibili inest; si ergo sumatur A et B omni C inesse, propositiones quidem totae falsae, conclusio autem vera. Similiter autem et cum haec quidem est privativa, illa vero affirmativa. (0690D) Possibile est enim B quidem nulli C inesse, A autem omni, et A alicui B non inesse, ut nigrum nulli cygno, animal autem omni, et animal non omni nigro: quare si sumatur B quidem omni C, A vero nulli, A alicui B non inerit, et conclusio quidem vera, propositiones autem falsae.	In the last figure a true conclusion may come through what is false, alike when both premisses are wholly false, when each is partly false, when one premiss is wholly true, the other false, when one premiss is partly false, the other wholly true, and vice versa, and in every other way in which it is possible to alter the premisses. For (1) nothing prevents neither A nor B from belonging to any C, while A belongs to some B, e.g. neither man nor footed follows anything lifeless, though man belongs to some footed things. If then it is assumed that A and B belong to all C, the premisses will be wholly false, but the conclusion true. Similarly if one premiss is negative, the other affirmative. For it is possible that B should belong to no C, but A to all C, and that should not belong to some B, e.g. black belongs to no swan, animal to every swan, and animal not to everything black. Consequently if it is assumed that B belongs to all C, and A to no C, A will not belong to some B: and the conclusion is true, though the premisses are false.
καὶ εἰ ἐπί τι ἑκατέρα ψευδής, ἔσται τὸ συμπέρασμα ἀληθές. οὐδὲν γὰρ κωλύει καὶ τὸ Α καὶ τὸ Β τινὶ τῶι Γ ὑπάρχειν, καὶ τὸ Α τινὶ τῶι Β, οἷον τὸ λευκὸν καὶ τὸ καλὸν τινὶ ζώιωι ὑπάρχει, καὶ τὸ λευκὸν τινὶ καλῶι. ἐὰν οὖν τεθῆι τὸ Α καὶ τὸ Β παντὶ τῶι Γ ὑπάρχειν, αἱ μὲν προτάσεις ἐπί τι ψευδεῖς, τὸ δὲ συμπέρασμα ἀληθές. καὶ στερητικῆς δὲ τῆς Α Γ τιθεμένης ὁμοίως. οὐδὲν γὰρ κωλύει τὸ μὲν Α τινὶ τῶι Γ μὴ ὑπάρχειν, τὸ δὲ Β τινὶ ὑπάρχειν, καὶ τὸ Α τῶι Β μὴ παντὶ ὑπάρχειν, οἷον τὸ λευκὸν τινὶ ζώιωι οὐχ ὑπάρχει, τὸ δὲ καλὸν τινὶ ὑπάρχει, καὶ τὸ λευκὸν οὐ παντὶ καλῶι. ὥστ᾽ ἂν ληφθῆι τὸ μὲν Α μηδενὶ τῶι Γ, τὸ δὲ Β παντί, ἀμφότεραι μὲν αἱ προτάσεις ἐπί τι ψευδεῖς, τὸ δὲ συμπέρασμα ἀληθές.	Et si in aliquo fuerit utraque falsa, erit conclusio vera, nihil enim prohibet et A et B alicui C inesse, et A alicui B, ut album et pulchrum alicui animali inest, et album alicui pulchro; si ergo ponatur A et B omni C inesse, propositiones quidem in aliquo falsae, conclusio autem vera. Et privativa A C posita, similiter: nihil enim prohibet A quidem alicui C non inesse, B vero alicui inesse, et A non omni B inesse, ut album alicui animali non inesse. (0691A) Pulchrum autem alicui inest, et album non omni pulchro: quare si sumatur A quidem nulli, C B autem omni, utraeque propositiones quidem in aliquo falsae, conclusio autem vera.	(2) Also if each premiss is partly false, the conclusion may be true. For nothing prevents both A and B from belonging to some C while A belongs to some B, e.g. white and beautiful belong to some animals, and white to some beautiful things. If then it is stated that A and B belong to all C, the premisses are partially false, but the conclusion is true. Similarly if the premiss AC is stated as negative. For nothing prevents A from not belonging, and B from belonging, to some C, while A does not belong to all B, e.g. white does not belong to some animals, beautiful belongs to some animals, and white does not belong to everything beautiful. Consequently if it is assumed that A belongs to no C, and B to all C, both premisses are partly false, but the conclusion is true.
Ὡσαύτως δὲ καὶ τῆς μὲν ὅλης ψευδοῦς τῆς δ᾽ ὅλης ἀληθοῦς λαμβανομένης. ἐγχωρεῖ γὰρ καὶ τὸ Α καὶ τὸ Β παντὶ τῶι Γ ἕπεσθαι, τὸ μέντοι Α τινὶ τῶι Β μὴ ὑπάρχειν, οἷον ζῶιον καὶ λευκὸν παντὶ κύκνωι ἕπεται, τὸ μέντοι ζῶιον οὐ παντὶ ὑπάρχει λευκῶι. τεθέντων οὖν ὅρων τοιούτων, ἐὰν ληφθῆι τὸ μὲν Β ὅλωι τῶι Γ ὑπάρχειν, τὸ δὲ Α ὅλωι μὴ ὑπάρχειν, ἡ μὲν Β Γ ὅλη ἔσται ἀληθής, ἡ δὲ Α Γ ὅλη ψευδής, καὶ τὸ συμπέρασμα ἀληθές. ὁμοίως δὲ καὶ εἰ τὸ μὲν Β Γ ψεῦδος, τὸ δὲ Α Γ ἀληθές· οἱ γὰρ αὐτοὶ ὅροι πρὸς τὴν ἀπό- ↵ δειξιν [μέλαν – κύκνος – ἄψυχον]. ἀλλὰ καὶ εἰ ἀμφότεραι λαμβάνοιντο καταφατικαί. οὐδὲν γὰρ κωλύει τὸ μὲν Β παντὶ τῶι Γ ἕπεσθαι, τὸ δὲ Α ὅλωι μὴ ὑπάρχειν, καὶ τὸ Α τινὶ τῶι Β ὑπάρχειν, οἷον κύκνωι παντὶ ζῶιον, μέλαν δ᾽ οὐδενὶ κύκνωι, καὶ τὸ μέλαν ὑπάρχει τινὶ ζώιωι. ὥστ᾽ ἂν ληφθῆι τὸ Α καὶ τὸ Β παντὶ τῶι Γ ὑπάρχειν, ἡ μὲν Β Γ ὅλη ἀληθής, ἡ δὲ Α Γ ὅλη ψευδής, καὶ τὸ συμπέρασμα ἀληθές. ὁμοίως δὲ καὶ τῆς Α Γ ληφθείσης ἀληθοῦς· διὰ γὰρ τῶν αὐτῶν ὅρων ἡ ἀπόδειξις.	Similiter autem et haec quidem tota falsa, illa vero tota vera sumpta. Possibile est enim A et B omne C sequi, et A alicui B non inesse, ut animal et album omne cygnum sequitur, et animal non omni inest albo; positis igitur his terminis, si sumatur B quidem toti C inesse, A vero toti non inesse, B C quidem tota erit vera, A C autem tota falsa, et conclusio vera. Similiter autem et si B C quidem falsa, A C autem vera, nam hi quidem termini ad demonstrationem, nigrum, inanimatum, cygnus. (0691B) Sed et si utraeque assumantur affirmative, nihil enim prohibet B quidem omne C sequi, A autem toti C non inesse, et A alicui B inesse, ut omni cygno animal, nigrum vero nulli cygno, et nigrum inest alicui animali: quare si sumatur A et B omni C inesse, B C quidem tota vera, A C autem tota falsa, et conclusio vera. Similiter autem et A C sumpta vera, nam per eosdem terminos demonstratio.	(3) Similarly if one of the premisses assumed is wholly false, the other wholly true. For it is possible that both A and B should follow all C, though A does not belong to some B, e.g. animal and white follow every swan, though animal does not belong to everything white. Taking these then as terms, if one assumes that B belongs to the whole of C, but A does not belong to C at all, the premiss BC will be wholly true, the premiss AC wholly false, and the conclusion true. Similarly if the statement BC is false, the statement AC true, the conclusion may be true. The same terms will serve for the proof. Also if both the premisses assumed are affirmative, the conclusion may be true. For nothing prevents B from following all C, and A from not belonging to C at all, though A belongs to some B, e.g. animal belongs to every swan, black to no swan, and black to some animals. Consequently if it is assumed that A and B belong to every C, the premiss BC is wholly true, the premiss AC is wholly false, and the conclusion is true. Similarly if the premiss AC which is assumed is true: the proof can be made through the same terms.
Πάλιν τῆς μὲν ὅλης ἀληθοῦς οὔσης, τῆς δ᾽ ἐπί τι ψευδοῦς. ἐγχωρεῖ γὰρ τὸ μὲν Β παντὶ τῶι Γ ὑπάρχειν, τὸ δὲ Α τινί, καὶ τὸ Α τινὶ τῶι Β, οἷον δίπουν μὲν παντὶ ἀνθρώπωι, καλὸν δ᾽ οὐ παντί, καὶ τὸ καλὸν τινὶ δίποδι ὑπάρχει. ἐὰν οὖν ληφθῆι καὶ τὸ Α καὶ τὸ Β ὅλωι τῶι Γ ὑπάρχειν, ἡ μὲν Β Γ ὅλη ἀληθής, ἡ δὲ Α Γ ἐπί τι ψευδής, τὸ δὲ συμπέρασμα ἀληθές. ὁμοίως δὲ καὶ τῆς μὲν Α Γ ἀληθοῦς τῆς δὲ Β Γ ἐπί τι ψευδοῦς λαμβανομένης· μετατεθέντων γὰρ τῶν αὐτῶν ὅρων ἔσται ἡ ἀπόδειξις.	Rursum hac quidem tota vera existente, illa vero in aliquo falsa, possibile est enim B quidem omni C inesse, A autem alicui C et alicui B, ut bipes quidem omni homini, pulchrum non omni, et pulchrum alicui bipedi inest. Si ergo sumatur A et B toti C inesse, B C quidem tota vera, A C autem in aliquo falsa, conclusio autem vera. Similiter autem et A C quidem vera, B C autem falsa in aliquo sumpta, transpositis enim eisdem terminis erit demonstratio.	(4) Again if one premiss is wholly true, the other partly false, the conclusion may be true. For it is possible that B should belong to all C, and A to some C, while A belongs to some B, e.g. biped belongs to every man, beautiful not to every man, and beautiful to some bipeds. If then it is assumed that both A and B belong to the whole of C, the premiss BC is wholly true, the premiss AC partly false, the conclusion true. Similarly if of the premisses assumed AC is true and BC partly false, a true conclusion is possible: this can be proved, if the same terms as before are transposed.
καὶ τῆς μὲν στερητικῆς τῆς δὲ καταφατικῆς οὔσης. ἐπεὶ γὰρ ἐγχωρεῖ τὸ μὲν Β ὅλωι τῶι Γ ὑπάρχειν, τὸ δὲ Α τινί, καὶ ὅταν οὕτως ἔχωσιν, οὐ παντὶ τῶι Β τὸ Α, ἐὰν οὖν ληφθῆι τὸ μὲν Β ὅλωι τῶι Γ ὑπάρχειν, τὸ δὲ Α μηδενί, ἡ μὲν στερητικὴ ἐπί τι ψευδής, ἡ δ᾽ ἑτέρα ὅλη ἀληθὴς καὶ τὸ συμπέρασμα. πάλιν ἐπεὶ δέδεικται ὅτι τοῦ μὲν Α μηδενὶ ὑπάρχοντος τῶι Γ, τοῦ δὲ Β τινί, ἐγχωρεῖ τὸ Α τινὶ τῶι Β μὴ ὑπάρχειν, φανερὸν ὅτι καὶ τῆς μὲν Α Γ ὅλης ἀληθοῦς οὔσης, τῆς δὲ Β Γ ἐπί τι ψευδοῦς, ἐγχωρεῖ τὸ συμπέρασμα εἶναι ἀληθές. ἐὰν γὰρ ληφθῆι τὸ μὲν Α μηδενὶ τῶι Γ, τὸ δὲ Β παντί, ἡ μὲν Α Γ ὅλη ἀληθής, ἡ δὲ Β Γ ἐπί τι ψευδής.	(0691C) Et cum haec quidem est privativa, illa vero affirmativa, quoniam possibile est B quidem toti C inesse, A autem alicui C, et quando sic se habeant, non omni B inesse A. Si ergo assumatur B quidem toti C inesse, A autem nulli, privativa quidem in aliquo falsa, altera autem tota vera, et conclusio erit vera. Rursum quoniam ostensum est quod cum A quidem nulli C inest, et B alicui, evenit A alicui B non inesse, manifestum igitur quoniam et cum A C tota est vera, B C autem in aliquo falsa, contingit conclusionem esse veram; si enim sumatur A quidem nulli C, B autem omni, A C quidem tota vera, B C autem in aliquo falsa.	Also the conclusion may be true if one premiss is negative, the other affirmative. For since it is possible that B should belong to the whole of C, and A to some C, and, when they are so, that A should not belong to all B, therefore it is assumed that B belongs to the whole of C, and A to no C, the negative premiss is partly false, the other premiss wholly true, and the conclusion is true. Again since it has been proved that if A belongs to no C and B to some C, it is possible that A should not belong to some C, it is clear that if the premiss AC is wholly true, and the premiss BC partly false, it is possible that the conclusion should be true. For if it is assumed that A belongs to no C, and B to all C, the premiss AC is wholly true, and the premiss BC is partly false.
Φανερὸν δὲ καὶ ἐπὶ τῶν ἐν μέρει συλλογισμῶν ὅτι πάντως ἔσται διὰ ψευδῶν ἀληθές. οἱ γὰρ αὐτοὶ ὅροι ληπτέοι καὶ ὅταν καθόλου ὦσιν αἱ προτάσεις, οἱ μὲν ἐν τοῖς κατηγορικοῖς κατηγορικοί, οἱ δ᾽ ἐν τοῖς στερητικοῖς στερητικοί. οὐδὲν γὰρ διαφέρει μηδενὶ ὑπάρχοντος παντὶ λαβεῖν ὑπάρ- χειν, καὶ τινὶ ὑπάρχοντος καθόλου λαβεῖν ὑπάρχειν, πρὸς τὴν τῶν ὅρων ἔκθεσιν· ὁμοίως δὲ καὶ ἐπὶ τῶν στερητικῶν.	(0691D) Manifestum autem et in particularibus syllogismis quoniam omnino per falsa erit verum, nam iidem termini sumendi, et quando universales fuerint propositiones, in praedicativis quidem praedicativi, in privativis autem privativi; nihil enim differt, cum nulli inerat, universaliter sumere inesse, et si alicui inerat, universaliter sumere ad terminorum positionem; similiter autem et in privativis.	(5) It is clear also in the case of particular syllogisms that a true conclusion may come through what is false, in every possible way. For the same terms must be taken as have been taken when the premisses are universal, positive terms in positive syllogisms, negative terms 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.
Φανερὸν οὖν ὅτι ἂν μὲν ἦι τὸ συμπέρασμα ψεῦδος, ἀνάγκη, ἐξ ὧν ὁ λόγος, ψευδῆ εἶναι ἢ πάντα ἢ ἔνια, ὅταν δ᾽ ἀληθές, οὐκ ἀνάγκη ἀληθὲς εἶναι οὔτε τὶ οὔτε πάντα, ἀλλ᾽ ἔστι μηδενὸς ὄντος ἀληθοῦς τῶν ἐν τῶι συλλογισμῶι τὸ συμπέρασμα ὁμοίως εἶναι ἀληθές· οὐ μὴν ἐξ ἀνάγκης. αἴτιον δ᾽ ↵ ὅτι ὅταν δύο ἔχηι οὕτω πρὸς ἄλληλα ὥστε θατέρου ὄντος ἐξ ἀνάγκης εἶναι θάτερον, τούτου μὴ ὄντος μὲν οὐδὲ θάτερον ἔσται, ὄντος δ᾽ οὐκ ἀνάγκη εἶναι θάτερον· τοῦ δ᾽ αὐτοῦ ὄντος καὶ μὴ ὄντος ἀδύνατον ἐξ ἀνάγκης εἶναι τὸ αὐτό·	Manifestum igitur quod quando sit conclusio falsa, necesse est ea ex quibus est oratio falsa esse, aut omnia, aut aliqua; quando autem vera, non necesse est verum esse nec aliquod quidem, nec omne. Sed est cum nullum sit verum eorum quae sunt in syllogismis, et conclusionem similiter esse veram, non tamen ex necessitate. Causa autem quoniam cum duo sic se habent ad invicem, ut cum alterum sit, ex necessitate esse alterum, hoc cum non sit quidem, nec alterum erit; cum autem sit, non necesse est esse alterum; idem autem cum sit, et non sit, impossibile ex necessitate esse idem.	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.
λέγω δ᾽ οἷον τοῦ Α ὄντος λευκοῦ τὸ Β εἶναι μέγα ἐξ ἀνάγκης, καὶ μὴ ὄντος λευκοῦ τοῦ Α τὸ Β εἶναι μέγα ἐξ ἀνάγκης. ὅταν γὰρ τουδὶ ὄντος λευκοῦ, τοῦ Α, τοδὶ ἀνάγκη μέγα εἶναι, τὸ Β, μεγάλου δὲ τοῦ Β ὄντος τὸ Γ μὴ λευκόν, ἀνάγκη, εἰ τὸ Α λευκόν, τὸ Γ μὴ εἶναι λευκόν. καὶ ὅταν δύο ὄντων θατέρου ὄντος ἀνάγκη θάτερον εἶναι, τούτου μὴ ὄντος ἀνάγκη τὸ πρῶτον μὴ εἶναι. τοῦ δὴ Β μὴ ὄντος μεγάλου τὸ Α οὐχ οἷόν τε λευκὸν εἶναι. τοῦ δὲ Α μὴ ὄντος λευκοῦ εἰ ἀνάγκη τὸ Β μέγα εἶναι, συμβαίνει ἐξ ἀνάγκης τοῦ Β μεγάλου μὴ ὄντος αὐτὸ τὸ Β εἶναι μέγα· τοῦτο δ᾽ ἀδύνατον. εἰ γὰρ τὸ Β μὴ ἔστι μέγα, τὸ Α οὐκ ἔσται λευκὸν ἐξ ἀνάγκης. εἰ οὖν μὴ ὄντος τούτου λευκοῦ τὸ Β ἔσται μέγα, συμβαίνει, εἰ τὸ Β μὴ ἔστι μέγα, εἶναι μέγα, ὡς διὰ τριῶν.	(0692A) Dico autem, cum sit A album, B esse magnum ex necessitate, et cum non sit A album, B esse magnum ex necessitate; quando enim cum hoc sit (ut A ) album, illud necesse est (ut B ) esse magnum, cum autem sit B magnum, C non esse album, necesse est, si A sit album, C non esse album. Et quando duobus existentibus, cum alterum sit, necesse est alterum esse, hoc autem cum non sit, necesse est A non esse, cum ergo B non sit magnum, A non potest album esse, cum vero A non sit album, necesse est B magnum esse, accidit ex necessitate cum B magnum non sit, idem B esse magnum: hoc autem impossibile, nam si B non est magnum, A non erit album ex necessitate; si ergo cum non sit A album, B erit magnum, accidit, si B non est magnum, B esse magnum, ut per tria.	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.