Authors/Aristotle/priora/Liber 2/C2

Chapter 2

Greek	Latin	English
	(PL 64 0686A) CAPUT II. Quod ex falsis in prima figura verum colligatur.	2
53b4 Ἔστι μὲν οὖν οὕτως ἔχειν ὥστ᾽ ἀληθεῖς εἶναι τὰς προ- τάσεις δι᾽ ὧν ὁ συλλογισμός, ἔστι δ᾽ ὥστε ψευδεῖς, ἔστι δ᾽ ὥστε τὴν μὲν ἀληθῆ τὴν δὲ ψευδῆ. τὸ δὲ συμπέρασμα ἢ ἀληθὲς ἢ ψεῦδος ἐξ ἀνάγκης. ἐξ ἀληθῶν μὲν οὖν οὐκ ἔστι ψεῦδος συλλογίσασθαι, ἐκ ψευδῶν δ᾽ ἔστιν ἀληθές, πλὴν οὐ διότι ἀλλ᾽ ὅτι· τοῦ γὰρ διότι οὐκ ἔστιν ἐκ ψευδῶν συλλογισμός· δι᾽ ἣν δ᾽ αἰτίαν, ἐν τοῖς ἑπομένοις λεχθήσεται.	Est ergo sic se habere, ut verae sint propositiones per quas fit syllogismus; est autem ut falsae, est vero ut haec quidem vera, illa autem falsa, conclusio autem aut vera, aut falsa ex necessitate. Ex veris ergo non est falsum syllogizare, ex falsis autem verum, tamen non propter quid, sed quia, nam eius qui est propter quid non est ex falsis syllogismus, ob quam autem causam in sequentibus dicetur.	It is possible for the premisses of the syllogism to be true, or to be false, or to be the one true, the other false. The conclusion is either true or false necessarily. From true premisses it is not possible to draw a false conclusion, but a true conclusion may be drawn from false premisses, true however only in respect to the fact, not to the reason. The reason cannot be established from false premisses: why this is so will be explained in the sequel.
Πρῶτον μὲν οὖν ὅτι ἐξ ἀληθῶν οὐχ οἷόν τε ψεῦδος συλλογίσασθαι, ἐντεῦθεν δῆλον. εἰ γὰρ τοῦ Α ὄντος ἀνάγκη τὸ Β εἶναι, τοῦ Β μὴ ὄντος ἀνάγκη τὸ Α μὴ εἶναι. εἰ οὖν ἀληθές ἐστι τὸ Α, ἀνάγκη τὸ Β ἀληθὲς εἶναι, ἢ συμβήσεται τὸ αὐτὸ ἅμα εἶναί τε καὶ οὐκ εἶναι· τοῦτο δ᾽ ἀδύνατον. μὴ ὅτι δὲ κεῖται τὸ Α εἷς ὅρος, ὑποληφθήτω ἐνδέχεσθαι ἑνός τινος ὄντος ἐξ ἀνάγκης τι συμβαίνειν· οὐ γὰρ οἷόν τε· τὸ μὲν γὰρ συμβαῖνον ἐξ ἀνάγκης τὸ συμπέρασμά ἐστι, δι᾽ ὧν δὲ τοῦτο γίνεται ἐλαχίστων, τρεῖς ὅροι, δύο δὲ διαστήματα καὶ προτάσεις. εἰ οὖν ἀληθές, ὧι τὸ Β ὑπάρχει, τὸ Α παντί, ὧι δὲ τὸ Γ, τὸ Β, ὧι τὸ Γ, ἀνάγκη τὸ Α ὑπάρχειν καὶ οὐχ οἷόν τε τοῦτο ψεῦδος εἶναι· ἅμα γὰρ ὑπάρξει ταὐτὸ καὶ οὐχ ὑπάρξει. τὸ οὖν Α ὥσπερ ἓν κεῖται, δύο προτάσεις συλληφθεῖσαι. ὁμοίως δὲ καὶ ἐπὶ τῶν στερητικῶν ἔχει· οὐ γὰρ ἔστιν ἐξ ἀληθῶν δεῖξαι ψεῦδος.	Primum ergo quoniam ex veris non possibile falsum syllogizare, hinc manifestum. (0686C) Si enim cum est A, necesse est esse B, si non est B, necesse est A non esse; si ergo verum est A, necesse est et B verum esse, aut accidet idem simul et esse et non esse, hoc autem impossibile. Non autem quoniam ponitur A unus terminus, accipiatur, contingere uno aliquo existente, ex necessitate aliquid accidere, non enim potest. Nam quod accidit ex necessitate conclusio est, per quae autem fit ad minimum tres sunt termini, duo autem intervalla et propositiones. Si ergo verum est cui omni inest B et A, cui autem C et B, cui C, necesse est A inesse, et non potest hoc falsum esse, simul enim erit idem et non inerit; ergo A ut unum, positum est duas propositiones colligere. Similiter autem se habet et in privativis, non enim est ex veris ostendere falsum.	First then that it is not possible to draw a false conclusion from true premisses, is made clear by this consideration. If it is necessary that B should be when A is, it is necessary that A should not be when B is not. If then A is true, B must be true: otherwise it will turn out that the same thing both is and is not at the same time. But this is impossible. Let it not, because A is laid down as a single term, be supposed that it is possible, when a single fact is given, that something should necessarily result. For that is not possible. For what results necessarily is the conclusion, and the means by which this comes about are at the least three terms, and two relations of subject and predicate or premisses. If then it is true that A belongs to all that to which B belongs, and that B belongs to all that to which C belongs, it is necessary that A should belong to all that to which C belongs, and this cannot be false: for then the same thing will belong and not belong at the same time. So A is posited as one thing, being two premisses taken together. The same holds good of negative syllogisms: it is not possible to prove a false conclusion from true premisses.
Ἐκ ψευδῶν δ᾽ ἀληθὲς ἔστι συλλογίσασθαι καὶ ἀμφοτέρων τῶν προτάσεων ψευδῶν οὐσῶν καὶ τῆς μιᾶς, ταύτης δ᾽ οὐχ ὁποτέρας ἔτυχεν ἀλλὰ τῆς δευτέρας, ἐάνπερ ὅλην λαμβάνηι ψευδῆ· μὴ ὅλης δὲ λαμβανομένης ἔστιν ὁποτερασοῦν. ἔστω γὰρ τὸ Α ὅλωι τῶι Γ ὑπάρχον, τῶι δὲ Β μηδενί, μηδὲ τὸ Β τῶι Γ. ἐνδέχεται δὲ τοῦτο, οἷον λίθωι οὐδενὶ ζῶιον, οὐδὲ λίθος οὐδενὶ ἀνθρώπωι. ἐὰν οὖν ληφθῆι τὸ Α παντὶ τῶι Β καὶ τὸ Β παντὶ τῶι Γ, τὸ Α παντὶ τῶι Γ ὑπάρξει, ὥστ᾽ ἐξ ἀμφοῖν ψευδῶν ἀληθὲς τὸ συμπέρασμα· πᾶς γὰρ ἄνθρωπος ζῶιον.	(0686D) Ex falsis autem est verum syllogizare, utrisque propositionibus falsis, et una; hac autem non utralibet contingit, sed secunda, si quidem totam sumamus falsam, non tota autem sumpta est utralibet. Insit enim A omni C, ei autem quod est B nulli, nec B insit C; contingit autem hoc, ut nulli lapidi animal, et lapis nulli homini; si igitur sumatur A omni B, et B omni C, A omni C inerit, quare ex utrisque falsis vera est conclusio, omnis enim homo animal.	But from what is false a true conclusion may be drawn, whether both the premisses are false or only one, provided that this is not either of the premisses indifferently, if it is taken as wholly false: but if the premiss is not taken as wholly false, it does not matter which of the two is false. (1) Let A belong to the whole of C, but to none of the Bs, neither let B belong to C. This is possible, e.g. animal belongs to no stone, nor stone to any man. If then A is taken to belong to all B and B to all C, A will belong to all C; consequently though both the premisses are false the conclusion is true: for every man is an animal.
ὡσαύτως δὲ καὶ τὸ στερητικόν. ἔστι γὰρ τῶι Γ μήτε τὸ Α ὑπάρχειν μηδενὶ μήτε τὸ Β, τὸ μέντοι Α τῶι Β παντί, οἷον ἐὰν τῶν αὐτῶν ὅρων ληφθέντων μέσον τεθῆι ὁ ἄνθρωπος· λίθωι γὰρ οὔτε ζῶιον οὔτε ἄνθρωπος οὐδενὶ ὑπάρχει, ἀνθρώπωι δὲ παντὶ ζῶιον. ὥστ᾽ ἐὰν ὧι μὲν ὑπάρχει, λάβηι μηδενὶ ὑπάρχειν, ὧι δὲ μὴ ὑπάρχει, παντὶ ὑπάρχειν, ἐκ ψευδῶν ἀμφοῖν ἀληθὲς ἔσται τὸ συμ↵ πέρασμα. ὁμοίως δὲ δειχθήσεται καὶ ἐὰν ἐπί τι ψευδὴς ἑκατέρα ληφθῆι.	Similiter autem et privativum: insit enim C nulli, nec A, nec B, A autem B omni, ut si eisdem terminis sumptis medium ponatur homo, lapidi enim nec animal, nec homo nulli inest, homini autem omni animal; quare si cui quidem omni inest, sumamus nulli inesse, cui vero non inest, omni inesse, ex falsis utrisque vera erit conclusio. Similiter autem ostendetur et si in aliquo utraque falsa sumatur.	Similarly with the negative. For it is possible that neither A nor B should belong to any C, although A belongs to all B, e.g. if the same terms are taken and man is put as middle: for neither animal nor man belongs to any stone, but animal belongs to every man. Consequently if one term is taken to belong to none of that to which it does belong, and the other term is taken to belong to all of that to which it does not belong, though both the premisses are false the conclusion will be true. (2) A similar proof may be given if each premiss is partially false.
Ἐὰν δ᾽ ἡ ἑτέρα τεθῆι ψευδής, τῆς μὲν πρώτης ὅλης ψευδοῦς οὔσης, οἷον τῆς Α Β, οὐκ ἔσται τὸ συμπέρασμα ἀληθές, τῆς δὲ Β Γ ἔσται. λέγω δ᾽ ὅλην ψευδῆ τὴν ἐναντίαν, οἷον εἰ μηδενὶ ὑπάρχον παντὶ εἴληπται ἢ εἰ παντὶ μηδενὶ ὑπάρχειν. ἔστω γὰρ τὸ Α τῶι Β μηδενὶ ὑπάρχον, τὸ δὲ Β τῶι Γ παντί. ἂν δὴ τὴν μὲν Β Γ πρότασιν λάβω ἀληθῆ, τὴν δὲ τὸ Α Β ψευδῆ ὅλην, καὶ παντὶ ὑπάρχειν τῶι Β τὸ Α, ἀδύνατον τὸ συμπέρασμα ἀληθὲς εἶναι· οὐδενὶ γὰρ ὑπῆρχε τῶν Γ, εἴπερ ὧι τὸ Β, μηδενὶ τὸ Α, τὸ δὲ Β παντὶ τῶι Γ.	(0687A) Si autem altera ponatur falsa, prima quidem tota falsa existente, ut A B, non erit conclusio vera, B C autem erit. Dico autem totam falsam quod contrariam verae, ut si quod nulli inest, omni sumptum est; aut si quod omni, nulli inesse. Insit enim A B nulli, B autem omni C; si ergo B C quidem propositionem sumamus veram, A B autem falsam totam, et omni B inesse A, impossibile est A C conclusionem veram esse, nulli enim inerat A earum quae sunt C, siquidem cui B nulli, B autem omni C.	(3) But if one only of the premisses is false, when the first premiss is wholly false, e.g. AB, the conclusion will not be true, but if the premiss BC is wholly false, a true conclusion will be possible. I mean by ‘wholly false’ the contrary of the truth, e.g. if what belongs to none is assumed to belong to all, or if what belongs to all is assumed to belong to none. Let A belong to no B, and B to all C. If then the premiss BC which I take is true, and the premiss AB is wholly false, viz. that A belongs to all B, it is impossible that the conclusion should be true: for A belonged to none of the Cs, since A belonged to nothing to which B belonged, and B belonged to all C.
ὁμοίως δ᾽ οὐδ᾽ εἰ τὸ Α τῶι Β παντὶ ὑπάρχει καὶ τὸ Β τῶι Γ, ἐλήφθη δ᾽ ἡ μὲν τὸ Β Γ ἀληθὴς πρότασις, ἡ δὲ τὸ Α Β ψευδὴς ὅλη, καὶ μηδενὶ ὧι τὸ Β, τὸ Α – τὸ συμπέρασμα ψεῦδος ἔσται· παντὶ γὰρ ὑπάρξει τῶι Γ τὸ Α, εἴπερ ὧι τὸ Β, παντὶ τὸ Α, τὸ δὲ Β παντὶ τῶι Γ. φανερὸν οὖν ὅτι τῆς πρώτης ὅλης λαμβανομένης ψευδοῦς, ἐάν τε καταφατικῆς ἐάν τε στερητικῆς, τῆς δ᾽ ἑτέρας ἀληθοῦς, οὐ γίνεται ἀληθὲς τὸ συμπέρασμα.	Similiter autem nec si A omni B inest, et B omni C, sumpta sit autem B C quidem vera propositio, A B autem falsa tota, et nulli, cui B inest A, conclusio falsa erit, omni enim C inest A, siquidem cui B omni C et A, B autem omni C. (0687B) Manifestum ergo quoniam prima tota sumpta falsa, sive affirmativa, sive privativa, altera autem vera, non fit vera conclusio.	Similarly there cannot be a true conclusion if A belongs to all B, and B to all C, but while the true premiss BC is assumed, the wholly false premiss AB is also assumed, viz. that A belongs to nothing to which B belongs: here the conclusion must be false. For A will belong to all C, since A belongs to everything to which B belongs, and B to all C. It is clear then that when the first premiss is wholly false, whether affirmative or negative, and the other premiss is true, the conclusion cannot be true.
Μὴ ὅλης δὲ λαμβανομένης ψευδοῦς ἔσται. εἰ γὰρ τὸ Α τῶι μὲν Γ παντὶ ὑπάρχει τῶι δὲ Β τινί, τὸ δὲ Β παντὶ τῶι Γ, οἷον ζῶιον κύκνωι μὲν παντὶ λευκῶι δὲ τινί, τὸ δὲ λευκὸν παντὶ κύκνωι, ἐὰν ληφθῆι τὸ Α παντὶ τῶι Β καὶ τὸ Β παντὶ τῶι Γ, τὸ Α παντὶ τῶι Γ ὑπάρξει ἀληθῶς· πᾶς γὰρ κύκνος ζῶιον. ὁμοίως δὲ καὶ εἰ στερητικὸν εἴη τὸ Α Β· ἐγχωρεῖ γὰρ τὸ Α τῶι μὲν Β τινὶ ὑπάρχειν τῶι δὲ Γ μηδενί, τὸ δὲ Β παντὶ τῶι Γ, οἷον ζῶιον τινὶ λευκῶι χίονι δ᾽ οὐδεμιᾶι, λευκὸν δὲ πάσηι χιόνι. εἰ οὖν ληφθείη τὸ μὲν Α μηδενὶ τῶι Β, τὸ δὲ Β παντὶ τῶι Γ, τὸ Α οὐδενὶ τῶι Γ ὑπάρξει.	Non tota autem sumpta falsa, erit: nam si A C quidem omni inest, B autem alicui, B autem omni C, ut animal, cygno quidem omni, albo autem alicui, album autem omni cygno, si sumatur A omni B, et B omni C, A omni C inerit vere, omnis enim cygnus animal. Similiter autem et si privativa sit A B; possibile est enim A B quidem alicui inesse, C vero nulli, B autem omni C, ut animal alicui albo, nivi vero nulli, album vero omni nivi; si ergo sumatur A quidem nulli B, B autem omni C, A nulli C inerit.	(4) But if the premiss is not wholly false, a true conclusion is possible. For if A belongs to all C and to some B, and if B belongs to all C, e.g. animal to every swan and to some white thing, and white to every swan, then if we take as premisses that A belongs to all B, and B to all C, A will belong to all C truly: for every swan is an animal. Similarly if the statement AB is negative. For it is possible that A should belong to some B and to no C, and that B should belong to all C, e.g. animal to some white thing, but to no snow, and white to all snow. If then one should assume that A belongs to no B, and B to all C, then will belong to no C.
Ἐὰν δ᾽ ἡ μὲν Α Β πρότασις ὅλη ληφθῆι ἀληθής, ἡ δὲ Β Γ ὅλη ψευδής, ἔσται συλλογισμὸς ἀληθής· οὐδὲν γὰρ κωλύει τὸ Α τῶι Β καὶ τῶι Γ παντὶ ὑπάρ- χειν, τὸ μέντοι Β μηδενὶ τῶι Γ, οἷον ὅσα τοῦ αὐτοῦ γένους εἴδη μὴ ὑπ᾽ ἄλληλα· τὸ γὰρ ζῶιον καὶ ἵππωι καὶ ἀνθρώπωι ὑπάρχει, ἵππος δ᾽ οὐδενὶ ἀνθρώπωι. ἐὰν οὖν ληφθῆι τὸ Α παντὶ τῶι Β καὶ τὸ Β παντὶ τῶι Γ, ἀληθὲς ἔσται τὸ συμπέρασμα, ψευδοῦς ὅλης οὔσης τῆς Β Γ προτάσεως. ὁμοίως δὲ καὶ στερητικῆς οὔσης τῆς Α Β προτάσεως. ἐνδέχεται γὰρ τὸ Α μήτε τῶι Β μήτε τῶι Γ μηδενὶ ὑπάρχειν, μηδὲ τὸ Β μηδενὶ τῶι Γ, οἷον τοῖς ἐξ ἄλλου γένους εἴδεσι τὸ γένος· τὸ γὰρ ζῶιον οὔτε μουσικῆι οὔτ᾽ ἰατρικῆι ὑπάρχει, οὐδ᾽ ↵ ἡ μουσικὴ ἰατρικῆι. ληφθέντος οὖν τοῦ μὲν Α μηδενὶ τῶι Β, τοῦ δὲ Β παντὶ τῶι Γ, ἀληθὲς ἔσται τὸ συμπέρασμα.	(0687C) Si autem A B quidem propositio tota sumatur vera, B C autem tota falsa, erit syllogismus verus, nihil enim prohibet A, et B et C omni inesse, B autem nulli C, ut quaecunque eiusdem generis sunt species non subalternae, nam animal et homini et equo inest, equus autem nulli homini inest; si ergo sumatur A omni B, et B omni C, conclusio vera erit, tota falsa B C propositione. Similiter autem cum universalis privativa est A B propositio, contingit enim A neque B, neque C nulli inesse, et B nulli C, ut ex alio genere speciebus diversum genus, nam animal nec musicae, nec medicinae inest, neque musica medicinae. Sumpta ergo A quidem nulli B, B autem omni C, vera erit conclusio. Et si non tota falsa sit B C, sed in aliquo, etiam sic erit conclusio vera.	(5) But if the premiss AB, which is assumed, is wholly true, and the premiss BC is wholly false, a true syllogism will be possible: for nothing prevents A belonging to all B and to all C, though B belongs to no C, e.g. these being species of the same genus which are not subordinate one to the other: for animal belongs both to horse and to man, but horse to no man. If then it is assumed that A belongs to all B and B to all C, the conclusion will be true, although the premiss BC is wholly false. Similarly if the premiss AB is negative. For it is possible that A should belong neither to any B nor to any C, and that B should not belong to any C, e.g. a genus to species of another genus: for animal belongs neither to music nor to the art of healing, nor does music belong to the art of healing. If then it is assumed that A belongs to no B, and B to all C, the conclusion will be true.
καὶ εἰ μὴ ὅλη ψευδὴς ἡ Β Γ ἀλλ᾽ ἐπί τι, καὶ οὕτως ἔσται τὸ συμπέρασμα ἀληθές. οὐδὲν γὰρ κωλύει τὸ Α καὶ τῶι Β καὶ τῶι Γ ὅλωι ὑπάρχειν, τὸ μέντοι Β τινὶ τῶι Γ, οἷον τὸ γένος τῶι εἴδει καὶ τῆι διαφορᾶι· τὸ γὰρ ζῶιον παντὶ ἀνθρώπωι καὶ παντὶ πεζῶι, ὁ δ᾽ ἄνθρωπος τινὶ πεζῶι καὶ οὐ παντί. εἰ οὖν τὸ Α παντὶ τῶι Β καὶ τὸ Β παντὶ τῶι Γ ληφθείη, τὸ Α παντὶ τῶι Γ ὑπάρξει· ὅπερ ἦν ἀληθές. ὁμοίως δὲ καὶ στερητικῆς οὔσης τῆς Α Β προτάσεως. ἐνδέχεται γὰρ τὸ Α μήτε τῶι Β μήτε τῶι Γ μηδενὶ ὑπάρχειν, τὸ μέντοι Β τινὶ τῶι Γ, οἷον τὸ γένος τῶι ἐξ ἄλλου γένους εἴδει καὶ διαφορᾶι· τὸ γὰρ ζῶιον οὔτε φρονήσει οὐδεμιᾶι ὑπάρχει οὔτε θεωρητικῆι, ἡ δὲ φρόνησις τινὶ θεωρητικῆι. εἰ οὖν ληφθείη τὸ μὲν Α μηδενὶ τῶι Β, τὸ δὲ Β παντὶ τῶι Γ, οὐδενὶ τῶι Γ τὸ Α ὑπάρξει· τοῦτο δ᾽ ἦν ἀληθές.	(0687D) Nihil enim prohibet A, et B et C toti inesse, B autem alicui C, ut genus speciei et differentiae, nam animal homini omni et omni gressibili, homo autem alicui gressibili, et non omni; si ergo A omni B, et B omni C sumatur, A omni C inerit, quod quidem erat verum. Similiter autem cum privativa est A, B propositio, contingit enim A nec B, nec C nulli inesse, B vero alicui C, at genus ex alio genere speciei et differentiae, nam animal nec sapientiae nulli inest, nec contemplationi, sapientia vero alicui contemplationi; si ergo sumatur A nulli B, B autem omni C, nulli C inerit A, hoc autem erat verum.	(6) And if the premiss BC is not wholly false but in part only, even so the conclusion may be true. For nothing prevents A belonging to the whole of B and of C, while B belongs to some C, e.g. a genus to its species and difference: for animal belongs to every man and to every footed thing, and man to some footed things though not to all. If then it is assumed that A belongs to all B, and B to all C, A will belong to all C: and this ex hypothesi is true. Similarly if the premiss AB is negative. For it is possible that A should neither belong to any B nor to any C, though B belongs to some C, e.g. a genus to the species of another genus and its difference: for animal neither belongs to any wisdom nor to any instance of ‘speculative’, but wisdom belongs to some instance of ‘speculative’. If then it should be assumed that A belongs to no B, and B to all C, will belong to no C: and this ex hypothesi is true.
Ἐπὶ δὲ τῶν ἐν μέρει συλλογισμῶν ἐνδέχεται καὶ τῆς πρώτης προτάσεως ὅλης οὔσης ψευδοῦς τῆς δ᾽ ἑτέρας ἀληθοῦς ἀληθὲς εἶναι τὸ συμπέρασμα, καὶ ἐπί τι ψευδοῦς οὔσης τῆς πρώτης τῆς δ᾽ ἑτέρας ἀληθοῦς, καὶ τῆς μὲν ἀληθοῦς τῆς δ᾽ ἐν μέρει ψευδοῦς, καὶ ἀμφοτέρων ψευδῶν. οὐδὲν γὰρ κωλύει τὸ Α τῶι μὲν Β μηδενὶ ὑπάρχειν τῶι δὲ Γ τινί, καὶ τὸ Β τῶι Γ τινί, οἷον ζῶιον οὐδεμιᾶι χιόνι λευκῶι δὲ τινὶ ὑπάρχει, καὶ ἡ χιὼν λευκῶι τινί. εἰ οὖν μέσον τεθείη ἡ χιών, πρῶτον δὲ τὸ ζῶιον, καὶ ληφθείη τὸ μὲν Α ὅλωι τῶι Β ὑπάρχειν, τὸ δὲ Β τινὶ τῶι Γ, ἡ μὲν Α Β ὅλη ψευδής, ἡ δὲ Β Γ ἀληθής, καὶ τὸ συμπέρασμα ἀληθές.	In particularibus autem syllogismis contingit, prima propositione tota falsa existente, altera autem vera, veram esse conclusionem, et A B in aliquo falsa existente, B C autem vera, et A B quidem vera, particulari autem falsa, et utrisque existentibus falsis. (0688A) Nihil enim prohibet A B quidem nulli inesse, C autem alicui, et B alicui C inesse, ut animal nulli nivi, albo autem alicui inest, et nix albo alicui. Si ergo ponatur medium nix, primum autem animal, et sumatur A quidem toti B inesse, B autem alicui C, A B tota falsa, B C autem vera, et conclusio vera.	In particular syllogisms it is possible when the first premiss is wholly false, and the other true, that the conclusion should be true; also when the first premiss is false in part, and the other true; and when the first is true, and the particular is false; and when both are false. (7) For nothing prevents A belonging to no B, but to some C, and B to some C, e.g. animal belongs to no snow, but to some white thing, and snow to some white thing. If then snow is taken as middle, and animal as first term, and it is assumed that A belongs to the whole of B, and B to some C, then the premiss BC is wholly false, the premiss BC true, and the conclusion true.
ὁμοίως δὲ καὶ στερητικῆς οὔσης τῆς Α Β προτάσεως· ἐγχωρεῖ γὰρ τὸ Α τῶι μὲν Β ὅλωι ὑπάρχειν τῶι δὲ Γ τινὶ μὴ ὑπάρχειν, τὸ μέντοι Β τινὶ τῶι Γ ὑπάρχειν, οἷον τὸ ζῶιον ἀνθρώπωι μὲν παντὶ ὑπάρχει, λευκῶι δὲ τινὶ οὐχ ἕπεται, ὁ δ᾽ ἄνθρωπος τινὶ λευκῶι ὑπάρχει, ὥστ᾽ εἰ μέσου τεθέντος τοῦ ἀνθρώπου ληφθείη τὸ Α μηδενὶ τῶι Β ὑπάρχειν, τὸ δὲ Β τινὶ τῶι Γ ὑπάρχειν, ἀληθὲς ἔσται τὸ συμπέρασμα ψευδοῦς οὔσης ὅλης τῆς Α Β προτάσεως. καὶ εἰ ἐπί τι ψευδὴς ἡ Α Β πρότασις, ἔσται τὸ συμπέρασμα ἀληθές. οὐδὲν γὰρ κωλύει τὸ Α καὶ τῶι Β καὶ τῶι Γ τινὶ ὑπάρχειν, καὶ τὸ Β τῶι Γ τινὶ ὑπάρχειν, οἷον τὸ ζῶιον τινὶ καλῶι καὶ τινὶ μεγάλωι, καὶ τὸ καλὸν τινὶ μεγάλωι ὑπάρχειν. ἐὰν οὖν ληφθῆι τὸ Α παντὶ τῶι Β καὶ τὸ Β τινὶ τῶι Γ, ↵ ἡ μὲν Α Β πρότασις ἐπί τι ψευδὴς ἔσται, ἡ δὲ Β Γ ἀληθής, καὶ τὸ συμπέρασμα ἀληθές.	Similiter autem et cum privativa est A B propositio, possibile est enim A B quidem toti inesse, C autem alicui non inesse, B vero alicui C inesse, ut animal homini quidem omni inest, album autem aliquod non sequitur, homo vero alicui albo inest; quare si medio posito homine sumatur A nulli B inesse, et B alicui C, vera fit conclusio, cum sit tota falsa A B propositio. Et si in aliquo sit falsa A B propositio, B C vera existente, erit conclusio vera. (0688B) Nihil enim prohibet A, et B, et C, alicui inesse, B autem alicui C, ut animal alicui pulchro, et alicui magno, et pulchrum alicui magno inest; si ergo sumatur A omni B, et B alicui C, et A B, quidem propositio in aliquo falsa erit, B C autem vera, et conclusio vera. Similiter autem et cum privativa est A B propositio, nam iidem erunt termini, et similiter positi ad demonstrationem.	Similarly if the premiss AB is negative: for it is possible that A should belong to the whole of B, but not to some C, although B belongs to some C, e.g. animal belongs to every man, but does not follow some white, but man belongs to some white; consequently if man be taken as middle term and it is assumed that A belongs to no B but B belongs to some C, the conclusion will be true although the premiss AB is wholly false. (If the premiss AB is false in part, the conclusion may be true. For nothing prevents A belonging both to B and to some C, and B belonging to some C, e.g. animal to something beautiful and to something great, and beautiful belonging to something great. If then A is assumed to belong to all B, and B to some C, the a premiss AB will be partially false, the premiss BC will be true, and the conclusion true. Similarly if the premiss AB is negative. For the same terms will serve, and in the same positions, to prove the point.
ὁμοίως δὲ καὶ στερητικῆς οὔσης τῆς Α Β προτάσεως· οἱ γὰρ αὐτοὶ ὅροι ἔσονται καὶ ὡσαύτως κείμενοι πρὸς τὴν ἀπόδειξιν. Πάλιν εἰ ἡ μὲν Α Β ἀληθὴς ἡ δὲ Β Γ ψευδής, ἀληθὲς ἔσται τὸ συμπέρασμα. οὐδὲν γὰρ κωλύει τὸ Α τῶι μὲν Β ὅλωι ὑπάρχειν τῶι δὲ Γ τινί, καὶ τὸ Β τῶι Γ μηδενὶ ὑπάρχειν, οἷον ζῶιον κύκνωι μὲν παντὶ μέλανι δὲ τινί, κύκνος δὲ οὐδενὶ μέλανι. ὥστ᾽ εἰ ληφθείη παντὶ τῶι Β τὸ Α καὶ τὸ Β τινὶ τῶι Γ, ἀληθὲς ἔσται τὸ συμπέρασμα ψευδοῦς ὄντος τοῦ Β Γ.	Rursum si A B quidem vera, B C autem falsa, vera erit conclusio. Nihil enim prohibet A quidem toti inesse B, C autem alicui, et B nulli C inesse: ut animal cygno quidem omni, nigro autem alicui, cygnus vero nulli nigro; quare si sumatur A omni B, et B alicui C, vera erit conclusio, cum sit falsa B C.	(9) Again if the premiss AB is true, and the premiss BC is false, the conclusion may be true. For nothing prevents A belonging to the whole of B and to some C, while B belongs to no C, e.g. animal to every swan and to some black things, though swan belongs to no black thing. Consequently if it should be assumed that A belongs to all B, and B to some C, the conclusion will be true, although the statement BC is false.
ὁμοίως δὲ καὶ στερητικῆς λαμβανομένης τῆς Α Β προτάσεως. ἐγχωρεῖ γὰρ τὸ Α τῶι μὲν Β μηδενὶ τῶι δὲ Γ τινὶ μὴ ὑπάρχειν, τὸ μέντοι Β μηδενὶ τῶι Γ, οἷον τὸ γένος τῶι ἐξ ἄλλου γένους εἴδει καὶ τῶι συμβεβηκότι τοῖς αὑτοῦ εἴδεσι· τὸ γὰρ ζῶιον ἀριθμῶι μὲν οὐδενὶ ὑπάρχει λευκῶι δὲ τινί, ὁ δ᾽ ἀριθμὸς οὐδενὶ λευκῶι· ἐὰν οὖν μέσον τεθῆι ὁ ἀριθμός, καὶ ληφθῆι τὸ μὲν Α μηδενὶ τῶι Β, τὸ δὲ Β τινὶ τῶι Γ, τὸ Α τινὶ τῶι Γ οὐχ ὑπάρξει, ὅπερ ἦν ἀληθές· καὶ ἡ μὲν Α Β πρότασις ἀληθής, ἡ δὲ Β Γ ψευδής.	(0688C) Similiter autem et privativa sumpta A B propositione, possibile enim A B quidem nulli, C autem alicui non inesse, et B nulli C, ut genus ex alio genere speciei et accidenti eius speciebus, nam animal quidem numero nulli inest, albo vero non alicui, numerus autem nulli albo; si ergo medium ponatur numerus, et sumatur A quidem nulli B, B autem alicui C, A alicui C non inerit, quod fuit verum, cum A B quidem sit propositio vera, B C autem falsa.	Similarly if the premiss AB is negative. For it is possible that A should belong to no B, and not to some C, while B belongs to no C, e.g. a genus to the species of another genus and to the accident of its own species: for animal belongs to no number and not to some white things, and number belongs to nothing white. If then number is taken as middle, and it is assumed that A belongs to no B, and B to some C, then A will not belong to some C, which ex hypothesi is true. And the premiss AB is true, the premiss BC false.
καὶ εἰ ἐπί τι ψευδὴς ἡ Α Β, ψευδὴς δὲ καὶ ἡ Β Γ, ἔσται τὸ συμπέρασμα ἀληθές. οὐδὲν γὰρ κωλύει τὸ Α τῶι Β τινὶ καὶ τῶι Γ τινὶ ὑπάρχειν ἑκα- τέρωι, τὸ δὲ Β μηδενὶ τῶι Γ, οἷον εἰ ἐναντίον τὸ Β τῶι Γ, ἄμφω δὲ συμβεβηκότα τῶι αὐτῶι γένει· τὸ γὰρ ζῶιον τινὶ λευκῶι καὶ τινὶ μέλανι ὑπάρχει, λευκὸν δ᾽ οὐδενὶ μέλανι. ἐὰν οὖν ληφθῆι τὸ Α παντὶ τῶι Β καὶ τὸ Β τινὶ τῶι Γ, ἀληθὲς ἔσται τὸ συμπέρασμα. καὶ στερητικῆς δὲ λαμβανομένης τῆς Α Β ὡσαύτως· οἱ γὰρ αὐτοὶ ὅροι καὶ ὡσαύτως τεθήσονται πρὸς τὴν ἀπόδειξιν.	Et si in aliquo sit falsa A B, falsa autem et B C, erit conclusio vera. Nihil enim prohibet A alicui B et alicui C inesse utrique, B autem nulli C, ut si B sit contrarium ipsi C, et ambo accidentia eidem generi, nam animal alicui albo et alicui nigro inest, album autem nulli nigro inest; si ergo sumatur A omni B, et B alicui C, vera erit conclusio. Et privativa quidem sumpta A B, similiter. Nam iidem termini, et similiter ponentur ad demonstrationem.	(10) Also if the premiss AB is partially false, and the premiss BC is false too, the conclusion may be true. For nothing prevents A belonging to some B and to some C, though B belongs to no C, e.g. if B is the contrary of C, and both are accidents of the same genus: for animal belongs to some white things and to some black things, but white belongs to no black thing. If then it is assumed that A belongs to all B, and B to some C, the conclusion will be true. Similarly if the premiss AB is negative: for the same terms arranged in the same way will serve for the proof.
καὶ ἀμφοτέρων δὲ ψευδῶν οὐσῶν ἔσται τὸ συμπέρασμα ἀληθές· ἐγχωρεῖ γὰρ τὸ Α τῶι μὲν Β μηδενὶ τῶι δὲ Γ τινὶ ὑπάρχειν, τὸ μέντοι Β μηδενὶ τῶι Γ, οἷον τὸ γένος τῶι ἐξ ἄλλου γένους εἴδει καὶ τῶι συμβεβηκότι τοῖς εἴδεσι τοῖς αὑτοῦ· ζῶιον γὰρ ἀριθμῶι μὲν οὐδενὶ λευκῶι δὲ τινὶ ὑπάρχει, καὶ ὁ ἀριθμὸς οὐδενὶ λευκῶι. ἐὰν οὖν ληφθῆι τὸ Α παντὶ τῶι Β καὶ τὸ Β τινὶ τῶι Γ, τὸ μὲν συμπέρασμα ἀληθές, αἱ δὲ προτάσεις ἄμφω ψευδεῖς.	(0688D) Et ex utrisque falsis erit conclusio vera. Possibile est enim A B quidem nulli, C autem alicui inesse, B vero nulli C. Ut genus ex alio genere speciei, et accidenti speciebus eius, animal enim numero quidem nulli, albo vero alicui inest, et numerus nulli albo. Si ergo sumatur A omni B, et B alicui C, conclusio quidem vera, propositiones vero ambae falsae.	(11) Also though both premisses are false the conclusion may be true. For it is possible that A may belong to no B and to some C, while B belongs to no C, e.g. a genus in relation to the species of another genus, and to the accident of its own species: for animal belongs to no number, but to some white things, and number to nothing white. If then it is assumed that A belongs to all B and B to some C, the conclusion will be true, though both premisses are false.
ὁμοίως δὲ καὶ στερητικῆς οὔσης τῆς Α Β. οὐδὲν γὰρ κωλύει τὸ Α τῶι μὲν Β ὅλωι ὑπάρχειν τῶι δὲ Γ τινὶ μὴ ὑπάρχειν, μηδὲ τὸ Β μηδενὶ τῶι Γ, οἷον ζῶιον κύκνωι μὲν παντὶ μέλανι δὲ τινὶ οὐχ ὑπάρχει, κύκνος δ᾽ οὐδενὶ μέλανι. ὥστ᾽ εἰ ληφθείη τὸ Α μηδενὶ τῶι Β, τὸ δὲ Β τινὶ τῶι Γ, τὸ Α τινὶ ↵ τῶι Γ οὐχ ὑπάρξει. τὸ μὲν οὖν συμπέρασμα ἀληθές, αἱ δὲ προτάσεις ψευδεῖς.	Similiter autem et cum privativa est A B. Nihil enim prohibet A B quidem toti inesse, C autem alicui non inesse, et neque B nulli C, ut animal cygno quidem omni, nigro autem alicui non inest, cygnus vero nulli nigro: quare si sumatur A nulli B, B autem alicui C A non inerit; ergo conclusio quidem vera, propositiones autem falsae.	Similarly also if the premiss AB is negative. For nothing prevents A belonging to the whole of B, and not to some C, while B belongs to no C, e.g. animal belongs to every swan, and not to some black things, and swan belongs to nothing black. Consequently if it is assumed that A belongs to no B, and B to some C, then A does not belong to some C. The conclusion then is true, but the premisses are false.