Authors/Aristotle/priora/Liber 2/C3

Chapter 3

Greek	Latin	English
	(PL 64 0688D) CAPUT III. Quod colligatur verum ex falsis in secunda figura.	3
55b3 Ἐν δὲ τῶι μέσωι σχήματι πάντως ἐγχωρεῖ διὰ ψευδῶν ἀληθὲς συλλογίσασθαι, καὶ ἀμφοτέρων τῶν προτάσεων ὅλων ψευδῶν λαμβανομένων καὶ ἐπί τι ἑκατέρας, καὶ τῆς μὲν ἀληθοῦς τῆς δὲ ψευδοῦς οὔσης [ὅλησ] ὁποτερασοῦν ψευδοῦς τιθεμένης, [καὶ εἰ ἀμφότεραι ἐπί τι ψευδεῖς, καὶ εἰ ἡ μὲν ἁπλῶς ἀληθὴς ἡ δ᾽ ἐπί τι ψευδής, καὶ εἰ ἡ μὲν ὅλη ψευδὴς ἡ δ᾽ ἐπί τι ἀληθής,] καὶ ἐν τοῖς καθόλου καὶ ἐπὶ τῶν ἐν μέρει συλλογισμῶν. εἰ γὰρ τὸ Α τῶι μὲν Β μηδενὶ ὑπάρχει τῶι δὲ Γ παντί, οἷον ζῶιον λίθωι μὲν οὐδενὶ ἵππωι δὲ παντί, ἐὰν ἐναντίως τεθῶσιν αἱ προτάσεις καὶ ληφθῆι τὸ Α τῶι μὲν Β παντὶ τῶι δὲ Γ μηδενί, ἐκ ψευδῶν ὅλων τῶν προτάσεων ἀληθὲς ἔσται τὸ συμπέρασμα. ὁμοίως δὲ καὶ εἰ τῶι μὲν Β παντὶ τῶι δὲ Γ μηδενὶ ὑπάρχει τὸ Α· ὁ γὰρ αὐτὸς ἔσται συλλογισμός.	(0689A) In media autem figura omnino contingit per falsa verum syllogizare, et utrisque propositionibus totis falsis sumptis, et hac quidem vera, illa tota falsa, utralibet falsa posita, et si utraeque in aliquo falsae, et si haec quidem simpliciter vera, illa autem in aliquo falsa, et in universalibus, et in particularibus syllogismis. Si enim A B quidem nulli inest, C autem omni, ut lapidi animal quidem nulli, homini autem omni, si contrariae ponantur propositiones, et si sumatur A B quidem omni, C vero nulli, ex falsis totis propositionibus erit vera conclusio. Similiter autem et si A inest B quidem omni, C vero nulli, nam idem erit syllogismus.	In the middle figure it is possible in every way to reach a true conclusion through false premisses, whether the syllogisms are universal or particular, viz. when both premisses are wholly false; when each is partially false; when one is true, the other wholly false (it does not matter which of the two premisses is false); if both premisses are partially false; if one is quite true, the other partially false; if one is wholly false, the other partially true. For (1) if A belongs to no B and to all C, e.g. animal to no stone and to every horse, then if the premisses are stated contrariwise and it is assumed that A belongs to all B and to no C, though the premisses are wholly false they will yield a true conclusion. Similarly if A belongs to all B and to no C: for we shall have the same syllogism.
Πάλιν εἰ ἡ μὲν ἑτέρα ὅλη ψευδὴς ἡ δ᾽ ἑτέρα ὅλη ἀληθής· οὐδὲν γὰρ κωλύει τὸ Α καὶ τῶι Β καὶ τῶι Γ παντὶ ὑπάρχειν, τὸ μέντοι Β μηδενὶ τῶι Γ, οἷον τὸ γένος τοῖς μὴ ὑπ᾽ ἄλληλα εἴδεσιν. τὸ γὰρ ζῶιον καὶ ἵππωι παντὶ καὶ ἀνθρώπωι, καὶ οὐδεὶς ἄνθρωπος ἵππος. ἐὰν οὖν ληφθῆι τῶι μὲν παντὶ τῶι δὲ μηδενὶ ὑπάρχειν, ἡ μὲν ὅλη ψευδὴς ἔσται ἡ δ᾽ ὅλη ἀληθής, καὶ τὸ συμπέρασμα ἀληθὲς πρὸς ὁποτερωιοῦν τεθέντος τοῦ στερητικοῦ.	(0689B) Rursum si altera quidem tota falsa, altera autem tota vera. Nihil enim prohibet A et B et C omni inesse, B autem nulli C, ut genus non subalternis speciebus. Nam animal equo omni, et homini inest, et nullus homo equus; si ergo sumatur animal huic quidem omni, illi vero nulli inesse, haec quidem erit falsa, illa vero tota vera, et conclusio vera, ad quodlibet posito privativo.	(2) Again if one premiss is wholly false, the other wholly true: for nothing prevents A belonging to all B and to all C, though B belongs to no C, e.g. a genus to its co-ordinate species. For animal belongs to every horse and man, and no man is a horse. If then it is assumed that animal belongs to all of the one, and none of the other, the one premiss will be wholly false, the other wholly true, and the conclusion will be true whichever term the negative statement concerns.
καὶ εἰ ἡ ἑτέρα ἐπί τι ψευδής, ἡ δ᾽ ἑτέρα ὅλη ἀληθής. ἐγχωρεῖ γὰρ τὸ Α τῶι μὲν Β τινὶ ὑπάρχειν τῶι δὲ Γ παντί, τὸ μέντοι Β μηδενὶ τῶι Γ, οἷον ζῶιον λευκῶι μὲν τινὶ κόρακι δὲ παντί, καὶ τὸ λευκὸν οὐδενὶ κόρακι. ἐὰν οὖν ληφθῆι τὸ Α τῶι μὲν Β μηδενὶ τῶι δὲ Γ ὅλωι ὑπάρχειν, ἡ μὲν Α Β πρότασις ἐπί τι ψευδής, ἡ δ᾽ Α Γ ὅλη ἀληθής, καὶ τὸ συμπέρασμα ἀληθές. καὶ μετατιθεμένου δὲ τοῦ στερητικοῦ ὡσαύτως· διὰ γὰρ τῶν αὐτῶν ὅρων ἡ ἀπόδειξις. καὶ εἰ ἡ καταφατικὴ πρότασις ἐπί τι ψευδής, ἡ δὲ στερητικὴ ὅλη ἀληθής. οὐδὲν γὰρ κωλύει τὸ Α τῶι μὲν Β τινὶ ὑπάρχειν τῶι δὲ Γ ὅλωι μὴ ὑπάρχειν, καὶ τὸ Β μηδενὶ τῶι Γ, οἷον τὸ ζῶιον λευκῶι μὲν τινὶ πίττηι δ᾽ οὐδεμιᾶι, καὶ τὸ λευκὸν οὐδεμιᾶι πίττηι. ὥστ᾽ ἐὰν ληφθῆι τὸ Α ὅλωι τῶι Β ὑπάρχειν τῶι δὲ Γ μηδενί, ἡ μὲν Α Β ἐπί τι ψευδής, ἡ δ᾽ Α Γ ὅλη ἀληθής, καὶ τὸ συμπέρασμα ἀληθές.	Et si altera in aliquo falsa, altera autem tota vera, possibile est enim A B quidem alicui inesse, C autem omni, et B nulli C, ut animal albo quidem alicui, corvo autem omni, album vero nulli corvo. Si ergo sumatur A B quidem nulli, C autem toti inesse, A B quidem propositio in aliquo falsa est, A C autem tota vera, et conclusio vera, et transposita quidem privativa, similiter. (0689C) Nam per eosdem terminos demonstratio. Et si affirmativa quidem propositio in aliquo falsa, privativa autem tota vera, nihil enim prohibet A B quidem alicui inesse, C autem toti non inesse, et B nulli C, ut animal albo quidem alicui, pici autem nulli, album vero nulli pici: quare si sumatur A to i B inesse, C autem nulli, A B quidem in aliquo falsa, A C autem tota vera, et conclusio vera.	(3) Also if one premiss is partially false, the other wholly true. For it is possible that A should belong to some B and to all C, though B belongs to no C, e.g. animal to some white things and to every raven, though white belongs to no raven. If then it is assumed that A belongs to no B, but to the whole of C, the premiss AB is partially false, the premiss AC wholly true, and the conclusion true. Similarly if the negative statement is transposed: the proof can be made by means of the same terms. Also if the affirmative premiss is partially false, the negative wholly true, a true conclusion is possible. For nothing prevents A belonging to some B, but not to C as a whole, while B belongs to no C, e.g. animal belongs to some white things, but to no pitch, and white belongs to no pitch. Consequently if it is assumed that A belongs to the whole of B, but to no C, the premiss AB is partially false, the premiss AC is wholly true, and the conclusion is true.
καὶ εἰ ἀμφότεραι αἱ προτάσεις ἐπί τι ψευδεῖς, ἔσται τὸ συμπέρασμα ἀληθές. ἐγχωρεῖ γὰρ τὸ Α καὶ τῶι Β καὶ τῶι Γ τινὶ ὑπάρχειν, τὸ δὲ Β μηδενὶ τῶι Γ, οἷον ζῶιον καὶ ↵ λευκῶι τινὶ καὶ μέλανί τινι, τὸ δὲ λευκὸν οὐδενὶ μέλανι. ἐὰν οὖν ληφθῆι τὸ Α τῶι μὲν Β παντὶ τῶι δὲ Γ μηδενί, ἄμφω μὲν αἱ προτάσεις ἐπί τι ψευδεῖς, τὸ δὲ συμπέρασμα ἀληθές. ὁμοίως δὲ καὶ μετατεθείσης τῆς στερητικῆς διὰ τῶν αὐτῶν ὅρων.	Et si utraeque propositiones in aliquo falsae, erit conclusio vera, possibile est enim A, et B, et C alicui inesse, B autem nulli C, ut animal, et albo alicui, et nigro alicui, album vero nulli nigro. Si ergo sumator A B quidem omni, C autem nulli, ambae quidem propositiones in aliquo falsae, conclusio autem vera; similiter autem transposita privativa per terminos.	(4) And if both the premisses are partially false, the conclusion may be true. For it is possible that A should belong to some B and to some C, and B to no C, e.g. animal to some white things and to some black things, though white belongs to nothing black. If then it is assumed that A belongs to all B and to no C, both premisses are partially false, but the conclusion is true. Similarly, if the negative premiss is transposed, the proof can be made by means of the same terms.
Φανερὸν δὲ καὶ ἐπὶ τῶν ἐν μέρει συλλογισμῶν· οὐδὲν γὰρ κωλύει τὸ Α τῶι μὲν Β παντὶ τῶι δὲ Γ τινὶ ὑπάρχειν, καὶ τὸ Β τῶι Γ τινὶ μὴ ὑπάρχειν, οἷον ζῶιον παντὶ ἀνθρώπωι λευκῶι δὲ τινί, ἄνθρωπος δὲ τινὶ λευκῶι οὐχ ὑπάρξει. ἐὰν οὖν τεθῆι τὸ Α τῶι μὲν Β μηδενὶ ὑπάρχειν τῶι δὲ Γ τινὶ ὑπάρχειν, ἡ μὲν καθόλου πρότασις ὅλη ψευδής, ἡ δ᾽ ἐν μέρει ἀληθής, καὶ τὸ συμπέρασμα ἀληθές. ὡσαύτως δὲ καὶ καταφατικῆς λαμβανομένης τῆς Α Β· ἐγχωρεῖ γὰρ τὸ Α τῶι μὲν Β μηδενὶ τῶι δὲ Γ τινὶ μὴ ὑπάρχειν, καὶ τὸ Β τῶι Γ τινὶ μὴ ὑπάρχειν, οἷον τὸ ζῶιον οὐδενὶ ἀψύχωι, λευκῶι δὲ τινί οὐχ ὑπάρχει, καὶ τὸ ἄψυχον οὐχ ὑπάρξει τινὶ λευκῶι.	(0689D) Manifestum autem et in particularibus syllogismis, nihil enim prohibet A B quidem omni, C autem alicui inesse, et B alicui C non inesse, ut animal omni homini, albo autem alicui, homo vero alicui albo non inerit. Si ergo ponatur A B quidem nulli inesse, C autem alicui inesse, universalis quidem propositio tota falsa, particularis autem vera, et conclusio vera. Similiter autem et affirmativa sumpta A B, possibile est enim A B quidem nulli, C autem alicui non inesse, et B alicui C non inesse, ut animal nulli inanimato, albo autem alicui, et inanimatum non inerit alicui albo.	It is clear also that our thesis holds in particular syllogisms. For (5) nothing prevents A belonging to all B and to some C, though B does not belong to some C, e.g. animal to every man and to some white things, though man will not belong to some white things. If then it is stated that A belongs to no B and to some C, the universal premiss is wholly false, the particular premiss is true, and the conclusion is true. Similarly if the premiss AB is affirmative: for it is possible that A should belong to no B, and not to some C, though B does not belong to some C, e.g. animal belongs to nothing lifeless, and does not belong to some white things, and lifeless will not belong to some white things.
ἐὰν οὖν τεθῆι τὸ Α τῶι μὲν Β παντὶ τῶι δὲ Γ τινὶ μὴ ὑπάρχειν, ἡ μὲν Α Β πρότασις, ἡ καθόλου, ὅλη ψευδής, ἡ δὲ Α Γ ἀληθής, καὶ τὸ συμπέρασμα ἀληθές. καὶ τῆς μὲν καθόλου ἀληθοῦς τεθείσης, τῆς δ᾽ ἐν μέρει ψευδοῦς. οὐδὲν γὰρ κωλύει τὸ Α μήτε τῶι Β μήτε τῶι Γ μηδενὶ ἕπεσθαι, τὸ μέντοι Β τινὶ τῶι Γ μὴ ὑπάρχειν, οἷον ζῶιον οὐδενὶ ἀριθμῶι οὐδ᾽ ἀψύχωι, καὶ ὁ ἀριθμὸς τινὶ ἀψύχωι οὐχ ἕπεται. ἐὰν οὖν τεθῆι τὸ Α τῶι μὲν Β μηδενὶ τῶι δὲ Γ τινί, τὸ μὲν συμπέρασμα ἔσται ἀληθὲς καὶ ἡ καθόλου πρότασις, ἡ δ᾽ ἐν μέρει ψευδής.	Si ergo ponatur A B quidem omni, C vero alicui non inesse, A B quidem propositio universalis tota falsa, A C autem vera, et conclusio vera. (0690A) Et universali quidem vera posita, minori autem particulari falsa, nihil enim prohibet A nec B nec C nullum sequi, et B alicui C non inesse, ut animal nulli numero nec inanimato, et numerus aliquod inanimatum non sequitur. Si ergo ponatur A B quidem nulli, C autem alicui, et conclusio vera, et universalis propositio vera, particularis autem falsa.	If then it is stated that A belongs to all B and not to some C, the premiss AB which is universal is wholly false, the premiss AC is true, and the conclusion is true. Also a true conclusion is possible when the universal premiss is true, and the particular is false. For nothing prevents A following neither B nor C at all, while B does not belong to some C, e.g. animal belongs to no number nor to anything lifeless, and number does not follow some lifeless things. If then it is stated that A belongs to no B and to some C, the conclusion will be true, and the universal premiss true, but the particular false.
καὶ καταφατικῆς δὲ τῆς καθόλου τιθεμένης ὡσαύτως. ἐγχωρεῖ γὰρ τὸ Α καὶ τῶι Β καὶ τῶι Γ ὅλωι ὑπάρχειν, τὸ μέντοι Β τινὶ τῶι Γ μὴ ἕπεσθαι, οἷον τὸ γένος τῶι εἴδει καὶ τῆι διαφορᾶι· τὸ γὰρ ζῶιον παντὶ ἀνθρώπωι καὶ ὅλωι πεζῶι ἕπεται, ἄνθρωπος δ᾽ οὐ παντὶ πεζῶι. ὥστ᾽ ἂν ληφθῆι τὸ Α τῶι μὲν Β ὅλωι ὑπάρχειν, τῶι δὲ Γ τινὶ μὴ ὑπάρχειν, ἡ μὲν καθόλου πρότασις ἀληθής, ἡ δ᾽ ἐν μέρει ψευδής, τὸ δὲ συμπέρασμα ἀληθές.	Affirmativa autem universali similiter posita, possibile est enim A et B et C toti inesse, B autem aliquod C non sequi, ut genus speciem et differentiam. Nam animal omnem hominem et totum gressibile sequitur, homo vero non omne gressibile: quare si sumatur A B quidem toti inesse, C autem alicui non inesse, universalis quidem propositio vera, particularis falsa, conclusio autem vera.	Similarly if the premiss which is stated universally is affirmative. For it is possible that should A belong both to B and to C as wholes, though B does not follow some C, e.g. a genus in relation to its species and difference: for animal follows every man and footed things as a whole, but man does not follow every footed thing. Consequently if it is assumed that A belongs to the whole of B, but does not belong to some C, the universal premiss is true, the particular false, and the conclusion true.
Φανερὸν δὲ καὶ ὅτι ἐξ ἀμφοτέρων ψευδῶν ἔσται τὸ συμπέρασμα ἀληθές, εἴπερ ἐνδέχεται τὸ Α καὶ τῶι Β καὶ τῶι Γ ὅλωι ὑπάρχειν, τὸ μέντοι Β τινὶ τῶι Γ μὴ ἕπεσθαι. ληφθέντος γὰρ τοῦ Α τῶι μὲν Β μηδενὶ τῶι δὲ Γ τινὶ ὑπάρχειν, αἱ μὲν προτάσεις ἀμφότεραι ψευδεῖς, τὸ δὲ συμπέρασμα ἀληθές. ὁμοίως δὲ καὶ κατηγορικῆς οὔσης τῆς καθόλου προτάσεως, τῆς δ᾽ ἐν μέρει στερητικῆς. ἐγχωρεῖ γὰρ τὸ Α τῶι μὲν Β μηδενὶ τῶι δὲ Γ παντὶ ἕπεσθαι, καὶ τὸ Β τινὶ τῶι Γ μὴ ὑπάρχειν, οἷον ζῶιον ἐπιστήμηι μὲν οὐδεμιᾶι ἀνθρώπωι δὲ παντὶ ἕπεται, ἡ δ᾽ ἐπιστήμη οὐ παντὶ ἀνθρώπωι. ↵ ἐὰν οὖν ληφθῆι τὸ Α τῶι μὲν Β ὅλωι ὑπάρχειν, τῶι δὲ Γ τινὶ μὴ ἕπεσθαι, αἱ μὲν προτάσεις ψευδεῖς, τὸ δὲ συμπέρασμα ἀληθές.	(0690B) Manifestum autem quoniam et utrisque falsis erit conclusio vera, siquidem contingit A et B et C huic quidem omni, illi vero nulli inesse, B vero aliquod C non sequi, nam sumpto A B quidem nulli, C autem alicui inesse, propositiones quidem ambae falsae, conclusio autem vera. Similiter autem et cum praedicativa fuerit universalis propositio, particularis autem privativa, possibile est enim A B quidem nullum, C autem omne sequi, et B alicui C non inesse, ut animal disciplinam quidem nullam, hominem autem omnem sequitur, disciplina vero non omnem hominem. Si ergo sumatur A B quidem toti inesse, C autem aliquod non sequi, propositiones quidem falsae, conclusio autem vera.	(6) It is clear too that though both premisses are false they may yield a true conclusion, since it is possible that A should belong both to B and to C as wholes, though B does not follow some C. For if it is assumed that A belongs to no B and to some C, the premisses are both false, but the conclusion is true. Similarly if the universal premiss is affirmative and the particular negative. For it is possible that A should follow no B and all C, though B does not belong to some C, e.g. animal follows no science but every man, though science does not follow every man. If then A is assumed to belong to the whole of B, and not to follow some C, the premisses are false but the conclusion is true.