Authors/Aristotle/priora/Liber 1
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Chapter 1
| Greek | Latin | English |
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| CAPUT PRIMUM. De propositione, termino et syllogismo. | 1 | |
| 24a1 Πρῶτον εἰπεῖν περὶ τί καὶ τίνος ἐστὶν ἡ σκέψις, ὅτι περὶ ἀπόδειξιν καὶ ἐπιστήμης ἀποδεικτικῆς· εἶτα διορίσαι τί ἐστι πρότασις καὶ τί ὅρος καὶ τί συλλογισμός, καὶ ποῖος τέλειος καὶ ποῖος ἀτελής, μετὰ δὲ ταῦτα τί τὸ ἐν ὅλωι εἶναι ἢ μὴ εἶναι τόδε τῶιδε, καὶ τί λέγομεν τὸ κατὰ παντὸς ἢ μηδενὸς κατηγορεῖσθαι. | Primum dicendum circa quid et de quo est intentio, quoniam circa demonstrationem et de disciplina demonstrativa est. Deinde determinandum quid propositio, et quid terminus, quid syllogismus, quis perfectus, et quis imperfectus. Postea vero quid est in toto esse, vel non esse hoc in illo, et quid dicimus de omni, aut de nullo praedicari. | WE must first state the subject of our inquiry and the faculty to which it belongs: its subject is demonstration and the faculty that carries it out demonstrative science. We must next define a premiss, a term, and a syllogism, and the nature of a perfect and of an imperfect syllogism; and after that, the inclusion or noninclusion of one term in another as in a whole, and what we mean by predicating one term of all, or none, of another. |
| Πρότασις μὲν οὖν ἐστὶ λόγος καταφατικὸς ἢ ἀποφατικός τινος κατά τινος· οὗτος δὲ ἢ καθόλου ἢ ἐν μέρει ἢ ἀδιόριστος. λέγω δὲ καθόλου μὲν τὸ παντὶ ἢ μηδενὶ ὑπάρχειν, ἐν μέρει δὲ τὸ τινὶ ἢ μὴ τινὶ ἢ μὴ παντὶ ὑπάρχειν, ἀδιόριστον δὲ τὸ ὑπάρχειν ἢ μὴ ὑπάρχειν ἄνευ τοῦ καθόλου ἢ κατὰ μέρος, οἷον τὸ τῶν ἐναντίων εἶναι τὴν αὐτὴν ἐπιστήμην ἢ τὸ τὴν ἡδονὴν μὴ εἶναι ἀγαθόν. διαφέρει δὲ ἡ ἀποδεικτικὴ πρότασις τῆς διαλεκτικῆς, ὅτι ἡ μὲν ἀποδεικτικὴ λῆψις θατέρου μορίου τῆς ἀντιφάσεώς ἐστιν (οὐ γὰρ ἐρωτᾶι ἀλλὰ λαμβάνει ὁ ἀποδεικνύων), ἡ δὲ διαλεκτικὴ ἐρώτησις ἀντιφάσεώς ἐστιν. οὐδὲν δὲ διοίσει πρὸς τὸ γενέσθαι τὸν ἑκατέρου συλλογισμόν· καὶ γὰρ ὁ ἀποδεικνύων καὶ ὁ ἐρωτῶν συλλογίζεται λαβών τι κατά τινος ὑπάρχειν ἢ μὴ ὑπάρχειν.
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Propositio ergo est oratio affirmativa, vel negativa alicuius de aliquo. Haec autem aut universalis, aut particularis, aut indefinita. Dico autem universalem quidem, cum aliquid omni, aut nulli inesse; particularem vero, cum alicui, aut non alicui, aut non omni inesse. (0639C) Indefinitam autem, cum quid inesse, vel non inesse significat, sive universali, vel particulari, ut contrariorum eamdem esse disciplinam, aut voluptatem non esse bonum. Differt autem demonstrativa propositio A dialectica, quoniam demonstrativa quidem sumptio alterius partis contradictionis est. Non enim interrogat, sed sumit, qui demonstrat. Dialectica vero interrogatio contradictionis est. Nihil autem refert ut fiat ex utraque syllogismus; nam et qui demonstrat, et qui interrogat, syllogizat, sumens aliquid de aliquo esse, vel non esse.
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A premiss then is a sentence affirming or denying one thing of another. This is either universal or particular or indefinite. By universal I mean the statement that something belongs to all or none of something else; by particular that it belongs to some or not to some or not to all; by indefinite that it does or does not belong, without any mark to show whether it is universal or particular, e.g. ‘contraries are subjects of the same science’, or ‘pleasure is not good’. The demonstrative premiss differs from the dialectical, because the demonstrative premiss is the assertion of one of two contradictory statements (the demonstrator does not ask for his premiss, but lays it down), whereas the dialectical premiss depends on the adversary’s choice between two contradictories. But this will make no difference to the production of a syllogism in either case; for both the demonstrator and the dialectician argue syllogistically after stating that something does or does not belong to something else. |
| ὥστε ἔσται συλλογιστικὴ μὲν πρότασις ἁπλῶς κατάφασις ἢ ἀπόφασίς τινος κατά τινος τὸν εἰρημένον τρό-πον, ἀποδεικτικὴ δέ, ἐὰν ἀληθὴς ἦι καὶ διὰ τῶν ἐξ ἀρχῆς ↵ ὑποθέσεων εἰλημμένη, διαλεκτικὴ δὲ πυνθανομένωι μὲν ἐρώτησις ἀντιφάσεως, συλλογιζομένωι δὲ λῆψις τοῦ φαινομένου καὶ ἐνδόξου, καθάπερ ἐν τοῖς Τοπικοῖς εἴρηται. τί μὲν οὖν ἐστὶ πρότασις, καὶ τί διαφέρει συλλογιστικὴ καὶ ἀποδεικτικὴ καὶ διαλεκτική, δι᾽ ἀκριβείας μὲν ἐν τοῖς ἑπομένοις ῥηθήσεται, πρὸς δὲ τὴν παροῦσαν χρείαν ἱκανῶς ἡμῖν διωρίσθω τὰ νῦν. | Quare erit syllogistica quidem propositio, simpliciter affirmatio vel negatio alicuius de aliquo secundum dictum modum. Demonstrativa vero si vera sit, et per primas propositiones sumpta. Dialectica autem percontanti quidem interrogatio contradictionis est, syllogizanti vero sumptio apparentis et probabilis, quemadmodum in Topicis dictum est. (0639D) Quid est ergo propositio, et quid differt syllogistica A demonstrativa et dialectica, diligentius quidem in sequentibus dicetur. Ad praesentem vero utilitatem, sufficienter nobis determinata sint, quae nunc dicta sunt. | Therefore a syllogistic premiss without qualification will be an affirmation or denial of something concerning something else in the way we have described; it will be demonstrative, if it is true and obtained through the first principles of its science; while a dialectical premiss is the giving of a choice between two contradictories, when a man is proceeding by question, but when he is syllogizing it is the assertion of that which is apparent and generally admitted, as has been said in the Topics. The nature then of a premiss and the difference between syllogistic, demonstrative, and dialectical premisses, may be taken as sufficiently defined by us in relation to our present need, but will be stated accurately in the sequel. |
| Ὅρον δὲ καλῶ εἰς ὃν διαλύεται ἡ πρότασις, οἷον τό τε κατηγορούμενον καὶ τὸ καθ᾽ οὗ κατηγορεῖται, προστιθεμένου [ἢ διαιρουμένου] τοῦ εἶναι ἢ μὴ εἶναι. | Terminum autem voco, in quem resolvitur propositio, ut praedicatum, et de quo praedicatur, vel apposito, vel separato esse, vel non esse. | I call that a term into which the premiss is resolved, i.e. both the predicate and that of which it is predicated, ‘being’ being added and ‘not being’ removed, or vice versa. |
| συλλογισμὸς δέ ἐστι λόγος ἐν ὧι τεθέντων τινῶν ἕτερόν τι τῶν κειμένων ἐξ ἀνάγκης συμβαίνει τῶι ταῦτα εἶναι. λέγω δὲ τῶι ταῦτα εἶναι τὸ διὰ ταῦτα συμβαίνειν, τὸ δὲ διὰ ταῦτα συμβαίνειν τὸ μηδενὸς ἔξωθεν ὅρου προσδεῖν πρὸς τὸ γενέσθαι τὸ ἀναγκαῖον. | Syllogismus est oratio in qua, quibusdam positis, aliud quiddam ab his quae posita sunt ex necessitate accidit, eo quod haec sunt. Dico autem eo quod haec sunt, propter haec accidere. Propter haec vero accidere, est nullius extrinsecus termini indigere, ut fiat necessarium. Perfectum vero voco syllogismum, qui nullius alius indiget, praeter ea quae sumpta sunt, ut appareat necessarium. | A syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so. I mean by the last phrase that they produce the consequence, and by this, that no further term is required from without in order to make the consequence necessary. |
| τέλειον μὲν οὖν καλῶ συλλογισμὸν τὸν μηδενὸς ἄλλου προσδεόμενον παρὰ τὰ εἰλημμένα πρὸς τὸ φανῆναι τὸ ἀναγκαῖον, ἀτελῆ δὲ τὸν προσδεόμενον ἢ ἑνὸς ἢ πλειόνων, ἃ ἔστι μὲν ἀναγκαῖα διὰ τῶν ὑποκειμένων ὅρων, οὐ μὴν εἴληπται διὰ προτάσεων. | (0640A) Imperfectum vero, qui indiget aut unius aut plurium, quae sunt quidem necessaria per subiectos terminos, non autem sumpta sunt per propositiones. | I call that a perfect syllogism which needs nothing other than what has been stated to make plain what necessarily follows; a syllogism is imperfect, if it needs either one or more propositions, which are indeed the necessary consequences of the terms set down, but have not been expressly stated as premisses. |
| τὸ δὲ ἐν ὅλωι εἶναι ἕτερον ἑτέρωι καὶ τὸ κατὰ παντὸς κατηγορεῖσθαι θατέρου θάτερον ταὐτόν ἐστιν. λέγομεν δὲ τὸ κατὰ παντὸς κατηγορεῖσθαι ὅταν μηδὲν ἦι λαβεῖν [τοῦ ὑποκειμένου] καθ᾽ οὗ θάτερον οὐ λεχθήσεται· καὶ τὸ κατὰ μηδενὸς ὡσαύτως. | (0640B) In toto autem esse alterum in altero, et de omni praedicari alterum de altero idem est. Dicimus autem de omni praedicari, quando nihil est sumere subiecti, de quo non dicatur alterum, et de nullo similiter. | That one term should be included in another as in a whole is the same as for the other to be predicated of all of the first. And we say that one term is predicated of all of another, whenever no instance of the subject can be found of which the other term cannot be asserted: ‘to be predicated of none’ must be understood in the same way. |
Chapter 2
| Greek | Latin | English |
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| (PL 64 0640B) CAPUT II. De conversione absolutarum propositionum. | 2 | |
| ↵Ἐπεὶ δὲ πᾶσα πρότασίς ἐστιν ἢ τοῦ ὑπάρχειν ἢ τοῦ ἐξ ἀνάγκης ὑπάρχειν ἢ τοῦ ἐνδέχεσθαι ὑπάρχειν, τούτων δὲ αἱ μὲν καταφατικαὶ αἱ δὲ ἀποφατικαὶ καθ᾽ ἑκάστην πρόσρησιν, πάλιν δὲ τῶν καταφατικῶν καὶ ἀποφατικῶν αἱ μὲν καθόλου αἱ δὲ ἐν μέρει αἱ δὲ ἀδιόριστοι, τὴν μὲν ἐν τῶι ὑπάρχειν καθόλου στερητικὴν ἀνάγκη τοῖς ὅροις ἀντιστρέφειν, οἷον εἰ μηδεμία ἡδονὴ ἀγαθόν, οὐδ᾽ ἀγαθὸν οὐδὲν ἔσται ἡδονή· τὴν δὲ κατηγορικὴν ἀντιστρέφειν μὲν ἀναγκαῖον, οὐ μὴν καθόλου ἀλλ᾽ ἐν μέρει, οἷον εἰ πᾶσα ἡδονὴ ἀγαθόν, καὶ ἀγαθόν τι εἶναι ἡδονήν· τῶν δὲ ἐν μέρει τὴν μὲν καταφατικὴν ἀντιστρέφειν ἀνάγκη κατὰ μέρος (εἰ γὰρ ἡδονή τις ἀγαθόν, καὶ ἀγαθόν τι ἔσται ἡδονή), τὴν δὲ στερητικὴν οὐκ ἀναγκαῖον· (οὐ γὰρ εἰ ἄνθρωπος μὴ ὑπάρχει τινὶ ζώιωι, καὶ ζῶιον οὐχ ὑπάρχει τινὶ ἀνθρώπωι). | Quoniam autem omnis propositio est, aut de inesse, aut ex necessitate inesse, aut contingere inesse; harum autem, hae quidem affirmativae, illae autem negativae secundum unamquamque appellationem; rursus autem affirmativarum et negativarum, aliae sunt universales, aliae particulares, aliae indefinitae: universalem quidem privativam de eo quod est inesse, necesse est in terminis converti. Ut si nulla voluptas est bonum, neque bonum nullum, erit voluptas. (0640C) Praedicativam autem converti quidem necessarium est, non tamen universaliter, sed in parte, ut, si omnis voluptas est bonum, et bonum aliquod voluptas. Particularem autem affirmativam quidem converti necesse est particulariter. Nam si voluptas aliqua, bonum, et bonum aliquod erit voluptas. Privativam vero non est necessarium. Non enim si homo non inest alicui animali, et animal non inest alicui homini. | Every premiss states that something either is or must be or may be the attribute of something else; of premisses of these three kinds some are affirmative, others negative, in respect of each of the three modes of attribution; again some affirmative and negative premisses are universal, others particular, others indefinite. It is necessary then that in universal attribution the terms of the negative premiss should be convertible, e.g. if no pleasure is good, then no good will be pleasure; the terms of the affirmative must be convertible, not however, universally, but in part, e.g. if every pleasure,is good, some good must be pleasure; the particular affirmative must convert in part (for if some pleasure is good, then some good will be pleasure); but the particular negative need not convert, for if some animal is not man, it does not follow that some man is not animal. |
| Πρῶτον μὲν οὖν ἔστω στερητικὴ καθόλου ἡ Α Β πρότασις. εἰ οὖν μηδενὶ τῶι Β τὸ Α ὑπάρχει, οὐδὲ τῶι Α οὐδενὶ ὑπάρξει τὸ Β· εἰ γάρ τινι, οἷον τῶι Γ, οὐκ ἀληθὲς ἔσται τὸ μηδενὶ τῶι Β τὸ Α ὑπάρχειν· τὸ γὰρ Γ τῶν Β τί ἐστιν. εἰ δὲ παντὶ τὸ Α τῶι Β, καὶ τὸ Β τινὶ τῶι Α ὑπάρξει· εἰ γὰρ μηδενί, οὐδὲ τὸ Α οὐδενὶ τῶι Β ὑπάρξει· ἀλλ᾽ ὑπέκειτο παντὶ ὑπάρχειν. ὁμοίως δὲ καὶ εἰ κατὰ μέρος ἐστὶν ἡ πρότασις. εἰ γὰρ τὸ Α τινὶ τῶι Β, καὶ τὸ Β τινὶ τῶι Α ἀνάγκη ὑπάρχειν· εἰ γὰρ μηδενί, οὐδὲ τὸ Α οὐδενὶ τῶι Β. εἰ δέ γε τὸ Α τινὶ τῶι Β μὴ ὑπάρχει, οὐκ ἀνάγκη καὶ τὸ Β τινὶ τῶι Α μὴ ὑπάρχειν, οἷον εἰ τὸ μὲν Β ἐστὶ ζῶιον, τὸ δὲ Α ἄνθρωπος· ἄνθρωπος μὲν γὰρ οὐ παντὶ ζώιωι, ζῶιον δὲ παντὶ ἀνθρώπωι ὑπάρχει. | Primum ergo sit privativa universalis A B propositio, si ergo nulli B inest A, neque A nulli inerit B. Nam si alicui inest ut C, non verum erit nullum B esse A. Nam C eorum quae sunt B aliquod est. Si vero omni B inest A, et B alicui A inest, nam si nulli, neque A nulli B inerit, sed positum erat omni inesse. (0640D) Similiter autem et si particularis est propositio, nam si inest A alicui B, et B alicui eorum quae sunt A necesse est inesse; si enim nulli, nec A nulli inerit B. Si autem A alicui eorum quae sunt B non inest, non necesse est et B alicui A non inesse, ut si B quidem sit animal, A vero homo, homo enim non omni animali, animal vero omni homini inest. | First then take a universal negative with the terms A and B. If no B is A, neither can any A be B. For if some A (say C) were B, it would not be true that no B is A; for C is a B. But if every B is A then some A is B. For if no A were B, then no B could be A. But we assumed that every B is A. Similarly too, if the premiss is particular. For if some B is A, then some of the As must be B. For if none were, then no B would be A. But if some B is not A, there is no necessity that some of the As should not be B; e.g. let B stand for animal and A for man. Not every animal is a man; but every man is an animal. |
Chapter 3
| Greek | Latin | English |
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| (PL 64 0640D) CAPUT III. De conversione propositionum de modo | 3 | |
| 25a27 Τὸν αὐτὸν δὲ τρόπον ἕξει καὶ ἐπὶ τῶν ἀναγκαίων προτάσεων. ἡ μὲν γὰρ καθόλου στερητικὴ καθόλου ἀντιστρέφει, τῶν δὲ καταφατικῶν ἑκατέρα κατὰ μέρος. εἰ μὲν γὰρ ἀνάγκη τὸ Α τῶι Β μηδενὶ ὑπάρχειν, ἀνάγκη καὶ τὸ Β τῶι Α μηδενὶ ὑπάρχειν· εἰ γὰρ τινὶ ἐνδέχεται, καὶ τὸ Α τῶι Β τινὶ ἐνδέχοιτο ἄν. εἰ δὲ ἐξ ἀνάγκης τὸ Α παντὶ ἢ τινὶ τῶι Β ὑπάρχει, καὶ τὸ Β τινὶ τῶι Α ἀνάγκη ὑπάρχειν· εἰ γὰρ μὴ ἀνάγκη, οὐδ᾽ ἂν τὸ Α τινὶ τῶι Β ἐξ ἀνάγκης ὑπάρχοι. τὸ δ᾽ ἐν μέρει στερητικὸν οὐκ ἀντιστρέφει, διὰ τὴν αὐτὴν αἰτίαν δι᾽ ἣν καὶ πρότερον ἔφαμεν. | Eodem autem modo se habebit in necessariis propositionibus, nam universalis quidem privativa universaliter convertitur. Affirmativarum autem utraque particulariter. Nam si necesse est A nulli B inesse, necesse est et B nulli A inesse; si enim alicui contingit, et A alicui B continget. (0641A) Si autem ex necessitate A omni vel alicui B inest, et B alicui A necesse est inesse, nam si non ex necessitate inest, neque A alicui B ex necessitate inerit. Particularis vero privativa non convertitur, propter eamdem causam, propter quam et supra diximus. | The same manner of conversion will hold good also in respect of necessary premisses. The universal negative converts universally; each of the affirmatives converts into a particular. If it is necessary that no B is A, it is necessary also that no A is B. For if it is possible that some A is B, it would be possible also that some B is A. If all or some B is A of necessity, it is necessary also that some A is B: for if there were no necessity, neither would some of the Bs be A necessarily. But the particular negative does not convert, for the same reason which we have already stated. |
| Ἐπὶ δὲ τῶν ἐνδεχομένων, ἐπειδὴ πολλαχῶς λέγεται τὸ ἐνδέχεσθαι (καὶ γὰρ τὸ ἀναγκαῖον καὶ τὸ μὴ ἀναγκαῖον καὶ τὸ δυνατὸν ἐνδέχεσθαι λέγομεν), ἐν μὲν τοῖς καταφατικοῖς ὁμοίως ἕξει κατὰ τὴν ἀντιστροφὴν ἐν ἅπασιν. εἰ γὰρ τὸ Α ↵ παντὶ ἢ τινὶ τῶι Β ἐνδέχεται, καὶ τὸ Β τινὶ τῶι Α ἐνδέχοιτο ἄν· εἰ γὰρ μηδενί, οὐδ᾽ ἂν τὸ Α οὐδενὶ τῶι Β· δέδεικται γὰρ τοῦτο πρότερον. ἐν δὲ τοῖς ἀποφατικοῖς οὐχ ὡσαύτως, ἀλλ᾽ ὅσα μὲν ἐνδέχεσθαι λέγεται τῶι ἐξ ἀνάγκης ὑπάρχειν ἢ τῶι μὴ ἐξ ἀνάγκης μὴ ὑπάρχειν, ὁμοίως, οἷον εἴ τις φαίη τὸν ἄνθρωπον ἐνδέχεσθαι μὴ εἶναι ἵππον ἢ τὸ λευκὸν μηδενὶ ἱματίωι ὑπάρχειν (τούτων γὰρ τὸ μὲν ἐξ ἀνάγκης οὐχ ὑπάρχει, τὸ δὲ οὐκ ἀνάγκη ὑπάρχειν, καὶ ὁμοίως ἀντιστρέφει ἡ πρότασις· εἰ γὰρ ἐνδέχεται μηδενὶ ἀνθρώπωι ἵππον, καὶ ἄνθρωπον ἐγχωρεῖ μηδενὶ ἵππωι· καὶ εἰ τὸ λευκὸν ἐγχωρεῖ μηδενὶ ἱματίωι, καὶ τὸ ἱμάτιον ἐγχωρεῖ μηδενὶ λευκῶι· εἰ γάρ τινι ἀνάγκη, καὶ τὸ λευκὸν ἱματίωι τινὶ ἔσται ἐξ ἀνάγκης· τοῦτο γὰρ δέδεικται πρότερον), ὁμοίως δὲ καὶ ἐπὶ τῆς ἐν μέρει ἀποφατικῆς· ὅσα δὲ τῶι ὡς ἐπὶ τὸ πολὺ καὶ τῶι πεφυκέναι λέγεται ἐνδέχεσθαι, καθ᾽ ὃν τρόπον διορίζομεν τὸ ἐνδεχόμενον, οὐχ ὁμοίως ἕξει ἐν ταῖς στερητικαῖς ἀντιστροφαῖς, ἀλλ᾽ ἡ μὲν καθόλου στερητικὴ πρότασις οὐκ ἀντιστρέφει, ἡ δὲ ἐν μέρει ἀντιστρέφει. τοῦτο δὲ ἔσται φανερὸν ὅταν περὶ τοῦ ἐνδεχομένου λέγωμεν. | In contingentibus vero, quoniam multipliciter dicitur contingere, nam et necessarium, et non necessarium, et possibile contingere dicimus; in affirmativis quidem, similiter se habebit secundum conversionem in omnibus. Nam si A omni aut alicui B contingit, et B alicui A contingit, si enim nulli, nec A nulli B, ostensum est enim hoc prius. In negativis vero non similiter, sed quaecunque quidem contingere dicuntur, ex eo quod ex necessitate non insunt, vel in eo quod non ex necessitate insunt similiter. Ut si quis dicat hominem contingere non esse equum, aut album nulli tunicae inesse. (0641B) Horum enim hoc quidem ex necessitate inest, illud vero non ex necessitate inest, et similiter convertitur propositio. Nam si contingit nulli homini equum inesse, et hominem contingit nulli equo inesse, et si album contingit nulli tunicae, et tunica contingit nulli albo, si enim alicui necessario, et album tunicae alicui inerit ex necessitate, hoc enim ostensum est prius. Similiter autem et in particulari negativa. Quaecunque vero ut in pluribus, et in eo quod nata sunt dicuntur contingere secundum quem modum determinamus contingens, non similiter se habebit in privativis conversionibus. Sed et universalis quidem privativa propositio non convertitur, particularis vero convertitur. Hoc autem erit manifestum quando de contingenti dicemus. | In respect of possible premisses, since possibility is used in several senses (for we say that what is necessary and what is not necessary and what is potential is possible), affirmative statements will all convert in a manner similar to those described. For if it is possible that all or some B is A, it will be possible that some A is B. For if that were not possible, then no B could possibly be A. This has been already proved. But in negative statements the case is different. Whatever is said to be possible, either because B necessarily is A, or because B is not necessarily A, admits of conversion like other negative statements, e.g. if one should say, it is possible that man is not horse, or that no garment is white. For in the former case the one term necessarily does not belong to the other; in the latter there is no necessity that it should: and the premiss converts like other negative statements. For if it is possible for no man to be a horse, it is also admissible for no horse to be a man; and if it is admissible for no garment to be white, it is also admissible for nothing white to be a garment. For if any white thing must be a garment, then some garment will necessarily be white. This has been already proved. The particular negative also must be treated like those dealt with above. But if anything is said to be possible because it is the general rule and natural (and it is in this way we define the possible), the negative premisses can no longer be converted like the simple negatives; the universal negative premiss does not convert, and the particular does. This will be plain when we speak about the possible. |
| νῦν δὲ τοσοῦτον ἡμῖν ἔστω πρὸς τοῖς εἰρημένοις δῆλον, ὅτι τὸ ἐνδέχεσθαι μηδενὶ ἢ τινὶ μὴ ὑπάρχειν καταφατικὸν ἔχει τὸ σχῆμα (τὸ γὰρ ἐνδέχεται τῶι ἔστιν ὁμοίως τάττεται, τὸ δὲ ἔστιν, οἷς ἂν προσκατηγορῆται, κατάφασιν ἀεὶ ποιεῖ καὶ πάντως, οἷον τὸ ἔστιν οὐκ ἀγαθόν ἢ ἔστιν οὐ λευκόν ἢ ἁπλῶς τὸ ἔστιν οὐ τοῦτο· δειχθήσεται δὲ καὶ τοῦτο διὰ τῶν ἑπομένων), κατὰ δὲ τὰς ἀντιστροφὰς ὁμοίως ἕξουσι ταῖς ἄλλαις. | (0641C) Nunc autem nobis tantum sit cum iis quae dicta sunt, manifestum, quoniam contingere nulli aut alicui non inesse affirmativam habet figuram, nam et contingit ipsi est similiter ordinatur. Est autem, quibuscunque adiacens praedicatur, affirmationem semper facit, et omnino, ut: est non bonum, vel est non album, vel simpliciter, est non hoc. Ostendetur autem et hoc per sequentia, secundum conversiones autem similiter se habebunt in aliis. | At present we may take this much as clear in addition to what has been said: the statement that it is possible that no B is A or some B is not A is affirmative in form: for the expression ‘is possible’ ranks along with ‘is’, and ‘is’ makes an affirmation always and in every case, whatever the terms to which it is added, in predication, e.g. ‘it is not-good’ or ‘it is not-white’ or in a word ‘it is not-this’. But this also will be proved in the sequel. In conversion these premisses will behave like the other affirmative propositions. |
Chapter 4
| Greek | Latin | English |
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| (PL 64 0641C) CAPUT IV. De modis syllogisticis et asyllogistis absolutis primae figurae. | 4 | |
| 25b26 Διωρισμένων δὲ τούτων λέγωμεν ἤδη διὰ τίνων καὶ πότε καὶ πῶς γίνεται πᾶς συλλογισμός· ὕστερον δὲ λεκτέον περὶ ἀποδείξεως. πρότερον δὲ περὶ συλλογισμοῦ λεκτέον ἢ περὶ ἀποδείξεως διὰ τὸ καθόλου μᾶλλον εἶναι τὸν συλλογισμόν· ἡ μὲν γὰρ ἀπόδειξις συλλογισμός τις, ὁ συλλογισμὸς δὲ οὐ πᾶς ἀπόδειξις. | His vero determinatis dicemus iam per quae et quando et quomodo fit omnis syllogismus, postea vero dicendum de demonstratione. (0641D) Prius enim de syllogismo dicendum quam de demonstratione, eo quod universalior est syllogismus, nam demonstratio quidem syllogismus quidam est; syllogismus vero non omnis demonstratio. | After these distinctions we now state by what means, when, and how every syllogism is produced; subsequently we must speak of demonstration. Syllogism should be discussed before demonstration because syllogism is the general: the demonstration is a sort of syllogism, but not every syllogism is a demonstration. |
| Ὅταν οὖν ὅροι τρεῖς οὕτως ἔχωσι πρὸς ἀλλήλους ὥστε τὸν ἔσχατον ἐν ὅλωι εἶναι τῶι μέσωι καὶ τὸν μέσον ἐν ὅλωι τῶι πρώτωι ἢ εἶναι ἢ μὴ εἶναι, ἀνάγκη τῶν ἄκρων εἶναι συλλογισμὸν τέλειον. καλῶ δὲ μέσον μὲν ὁ καὶ αὐτὸ ἐν ἄλλωι καὶ ἄλλο ἐν τούτωι ἐστίν, ὁ καὶ τῆι θέσει γίνεται μέσον· ἄκρα δὲ τὸ αὐτό τε ἐν ἄλλωι ὂν καὶ ἐν ὧι ἄλλο ἐστίν. εἰ γὰρ τὸ Α κατὰ παντὸς τοῦ Β καὶ τὸ Β κατὰ παντὸς τοῦ Γ, ἀνάγκη τὸ Α κατὰ παντὸς τοῦ Γ κατηγορεῖσθαι· πρότερον γὰρ εἴρηται πῶς τὸ κατὰ παντὸς λέγομεν. ὁμοίως δὲ καὶ εἰ τὸ μὲν Α κατὰ μη ↵ δενὸς τοῦ Β, τὸ δὲ Β κατὰ παντὸς τοῦ Γ, ὅτι τὸ Α οὐδενὶ τῶι Γ ὑπάρξει. | Quando igitur tres termini sic se habent ad invicem, ut et postremus sit in toto medio, et medius in toto primo vel sit, vel non sit, necesse est extremitatum perfectum esse syllogismum. Voco autem medium quod et ipsum in alio, et aliud in ipso est, quod et positione medium est; extrema vero quod et ipsum in alio, et in quo aliud est. Si enim A de omni B, et B de omni C, necesse est A de omni C praedicari. Prius enim dictum est quomodo de omni dicimus. Similiter autem et si A de nullo B, B autem de omni C, quoniam A nulli C inerit. | Whenever three terms are so related to one another that the last is contained in the middle as in a whole, and the middle is either contained in, or excluded from, the first as in or from a whole, the extremes must be related by a perfect syllogism. I call that term middle which is itself contained in another and contains another in itself: in position also this comes in the middle. By extremes I mean both that term which is itself contained in another and that in which another is contained. If A is predicated of all B, and B of all C, A must be predicated of all C: we have already explained what we mean by ‘predicated of all’. Similarly also, if A is predicated of no B, and B of all C, it is necessary that no C will be A. |
| εἰ δὲ τὸ μὲν πρῶτον παντὶ τῶι μέσωι ἀκολουθεῖ, τὸ δὲ μέσον μηδενὶ τῶι ἐσχάτωι ὑπάρχει, οὐκ ἔσται συλλογισμὸς τῶν ἄκρων· οὐδὲν γὰρ ἀναγκαῖον συμβαίνει τῶι ταῦτα εἶναι· καὶ γὰρ παντὶ καὶ μηδενὶ ἐνδέχεται τὸ πρῶτον τῶι ἐσχάτωι ὑπάρχειν, ὥστε οὔτε τὸ κατὰ μέρος οὔτε τὸ καθόλου γίνεται ἀναγκαῖον·
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(0642A) Si autem primum quidem omni medio consequens est, medium vero nulli postremo, non erit syllogismus extremitatum. Nihil enim necessarium accidit, eo quod haec sunt, nam et omni et nulli contingit primum postremo inesse, quare neque particulare, neque universale fit necessarium. | But if the first term belongs to all the middle, but the middle to none of the last term, there will be no syllogism in respect of the extremes; for nothing necessary follows from the terms being so related; for it is possible that the first should belong either to all or to none of the last, so that neither a particular nor a universal conclusion is necessary. |
| μηδενὸς δὲ ὄντος ἀναγκαίου διὰ τούτων οὐκ ἔσται συλλογισμός. ὅροι τοῦ παντὶ ὑπάρχειν ζῶιον – ἄνθρωπος – ἵππος, τοῦ μηδενὶ ζῶιον – ἄνθρωπος – λίθος. οὐδ᾽ ὅταν μήτε τὸ πρῶτον τῶι μέσωι μήτε τὸ μέσον τῶι ἐσχάτωι μηδενὶ ὑπάρχηι, οὐδ᾽ οὕτως ἔσται συλλογισμός. ὅροι τοῦ ὑπάρχειν ἐπιστήμη – γραμμή – ἰατρική, τοῦ μὴ ὑπάρχειν ἐπιστήμη – γραμμή – μονάς. | Cum autem nihil est necessarium, per haec non erit syllogismus. Termini vero eius quod est omni inesse, animal, homo, equus; eius vero quod est nulli, animal, homo, lapis. Quando vero nec primum medio, nec medium postremo ulli inest, nec sic erit syllogismus. Termini vero ut inesse, scientia, linea, medicina; ut non inesse, scientia, linea, unitas. | But if there is no necessary consequence, there cannot be a syllogism by means of these premisses. As an example of a universal affirmative relation between the extremes we may take the terms animal, man, horse; of a universal negative relation, the terms animal, man, stone. Nor again can syllogism be formed when neither the first term belongs to any of the middle, nor the middle to any of the last. As an example of a positive relation between the extremes take the terms science, line, medicine: of a negative relation science, line, unit. |
| καθόλου μὲν οὖν ὄντων τῶν ὅρων, δῆλον ἐν τούτωι τῶι σχήματι πότε ἔσται καὶ πότε οὐκ ἔσται συλλογισμός, καὶ ὅτι ὄντος τε συλλογισμοῦ τοὺς ὅρους ἀναγκαῖον ἔχειν ὡς εἴπομεν, ἄν θ᾽ οὕτως ἔχωσιν, ὅτι ἔσται συλλογισμός. | (0642B) Universalibus igitur existentibus terminis, manifestum est in hac figura quando erit, et quando non erit syllogismus, et quoniam cum est syllogismus, necessarium est terminos sic se habere, ut diximus, et sic se habens manifestum quoniam erit syllogismus. | If then the terms are universally related, it is clear in this figure when a syllogism will be possible and when not, and that if a syllogism is possible the terms must be related as described, and if they are so related there will be a syllogism. |
| Εἰ δ᾽ ὁ μὲν καθόλου τῶν ὅρων ὁ δ᾽ ἐν μέρει πρὸς τὸν ἕτερον, ὅταν μὲν τὸ καθόλου τεθῆι πρὸς τὸ μεῖζον ἄκρον ἢ κατηγορικὸν ἢ στερητικόν, τὸ δὲ ἐν μέρει πρὸς τὸ ἔλαττον κατηγορικόν, ἀνάγκη συλλογισμὸν εἶναι τέλειον, ὅταν δὲ πρὸς τὸ ἔλαττον ἢ καὶ ἄλλως πως ἔχωσιν οἱ ὅροι, ἀδύνατον. λέγω δὲ μεῖζον μὲν ἄκρον ἐν ὧι τὸ μέσον ἐστίν, ἔλαττον δὲ τὸ ὑπὸ τὸ μέσον ὄν. | Si autem hic quidem terminorum universaliter, alius vero particulariter ad alium, quando universale quidem ponitur ad maiorem extremitatem vel praedicativum, vel privativum, particulare vero ad minorem praedicativum, necesse est syllogismum esse perfectum. Quando vero ad minorem vel quolibet modo aliter se habeant termini, impossibile est. Dico autem maiorem extremitatem quidem in qua medium est, minorem vero, quae sub medio est.
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But if one term is related universally, the other in part only, to its subject, there must be a perfect syllogism whenever universality is posited with reference to the major term either affirmatively or negatively, and particularity with reference to the minor term affirmatively: but whenever the universality is posited in relation to the minor term, or the terms are related in any other way, a syllogism is impossible. I call that term the major in which the middle is contained and that term the minor which comes under the middle. |
| ὑπαρχέτω γὰρ τὸ μὲν Α παντὶ τῶι Β, τὸ δὲ Β τινὶ τῶι Γ. οὐκοῦν εἰ ἔστι παντὸς κατηγορεῖσθαι τὸ ἐν ἀρχῆι λεχθέν, ἀνάγκη τὸ Α τινὶ τῶι Γ ὑπάρχειν. καὶ εἰ τὸ μὲν Α μηδενὶ τῶι Β ὑπάρχει, τὸ δὲ Β τινὶ τῶι Γ, ἀνάγκη τὸ Α τινὶ τῶι Γ μὴ ὑπάρχειν· ὥρισται γὰρ καὶ τὸ κατὰ μηδενὸς πῶς λέγομεν· ὥστε ἔσται συλλογισμὸς τέλειος. ὁμοίως δὲ καὶ εἰ ἀδιόριστον εἴη τὸ Β Γ, κατηγορικὸν ὄν· ὁ γὰρ αὐτὸς ἔσται συλλογισμὸς ἀδιορίστου τε καὶ ἐν μέρει ληφθέντος. | Insit enim A quidem omni B, B autem alicui C, ergo si est de omni praedicari, quod in principio dictum est, necesse est A alicui C inesse. (0642C) Et si A quidem nulli B inest, B vero alicui C, necesse est A alicui C non inesse, determinatum est enim et de nullo, quomodo dicimus, quare erit syllogismus perfectus. Similiter autem et si indefinitum sit B C praedicativum, nam idem erit syllogismus indefinito et particulari sumpto. | Let all B be A and some C be B. Then if ‘predicated of all’ means what was said above, it is necessary that some C is A. And if no B is A but some C is B, it is necessary that some C is not A. The meaning of ‘predicated of none’ has also been defined. So there will be a perfect syllogism. This holds good also if the premiss BC should be indefinite, provided that it is affirmative: for we shall have the same syllogism whether the premiss is indefinite or particular. |
| Ἐὰν δὲ πρὸς τὸ ἔλαττον ἄκρον τὸ καθόλου τεθῆι ἢ κατηγορικὸν ἢ στερητικόν, οὐκ ἔσται συλλογισμός, οὔτε καταφατικοῦ οὔτε ἀποφατικοῦ τοῦ ἀδιορίστου ἢ κατὰ μέρος ὄντος, οἷον εἰ τὸ μὲν Α τινὶ τῶι Β ὑπάρχει ἢ μὴ ὑπάρχει, τὸ δὲ Β παντὶ τῶι Γ ὑπάρχει· ὅροι τοῦ ὑπάρχειν ἀγαθόν – ἕξις – φρόνησις, τοῦ μὴ ὑπάρχειν ἀγαθόν – ἕξις – ἀμαθία. πάλιν εἰ τὸ μὲν Β μηδενὶ τῶι Γ, τὸ δὲ Α τινὶ τῶι Β ἢ ὑπάρχει ἢ μὴ ὑπάρχει ἢ μὴ παντὶ ὑπάρχει, οὐδ᾽ οὕτως ἔσται συλλογισμός. ὅροι λευκόν – ἵππος – κύκνος, λευκόν – ἵππος – κόραξ. οἱ αὐτοὶ δὲ καὶ εἰ τὸ Α Β ἀδιόριστον. | Si autem ad minorem extremitatem universale ponatur vel praedicativum, vel privativum, non erit syllogismus neque cum affirmativa, neque negativa, neque indefinita, neque particularis sit, ut si A quidem alicui B inest, vel non inest, B autem omni C inest. Termini ut inesse, bonum, habitus, prudentia; ubi non inesse, bonum, habitus, indisciplina. Rursum si B quidem nulli C, A vero alicui B inest, vel non inest, vel non omni inest, nec sic erit syllogismus. Termini omni inesse, album, equus, cygnus; nulli inesse, album, equus, corvus. Idem autem et si A B indefinitum sit. | But if the universality is posited with respect to the minor term either affirmatively or negatively, a syllogism will not be possible, whether the major premiss is positive or negative, indefinite or particular: e.g. if some B is or is not A, and all C is B. As an example of a positive relation between the extremes take the terms good, state, wisdom: of a negative relation, good, state, ignorance. Again if no C is B, but some B is or is not A or not every B is A, there cannot be a syllogism. Take the terms white, horse, swan: white, horse, raven. The same terms may be taken also if the premiss BA is indefinite. |
| Οὐδ᾽ ὅταν τὸ μὲν πρὸς ↵ τῶι μείζονι ἄκρωι καθόλου γένηται ἢ κατηγορικὸν ἢ στερητικόν, τὸ δὲ πρὸς τῶι ἐλάττονι στερητικὸν κατὰ μέρος, οὐκ ἔσται συλλογισμός [ἀδιορίστου τε καὶ ἐν μέρει ληφθέντοσ], οἷον εἰ τὸ μὲν Α παντὶ τῶι Β ὑπάρχει, τὸ δὲ Β τινὶ τῶι Γ μή, ἢ εἰ μὴ παντὶ ὑπάρχει· ὧι γὰρ ἄν τινι μὴ ὑπάρχηι τὸ μέσον, τούτωι καὶ παντὶ καὶ οὐδενὶ ἀκολουθήσει τὸ πρῶτον. | (0642D) Nec quando ad maiorem extremitatem quidem universale ponatur vel praedicativum, vel privativum, ad minorem vero particulare privativum, non erit syllogismus vel indefinito, vel particulari sumpto. Velut si A quidem omni B inest, B autem alicui C non inest, vel non omni inest. Cui enim alicui non inest medium, hoc omne et nullum sequatur primum. | Nor when the major premiss is universal, whether affirmative or negative, and the minor premiss is negative and particular, can there be a syllogism, whether the minor premiss be indefinite or particular: e.g. if all B is A and some C is not B, or if not all C is B. For the major term may be predicable both of all and of none of the minor, to some of which the middle term cannot be attributed. |
| ὑποκείσθωσαν γὰρ οἱ ὅροι ζῶιον – ἄνθρωπος – λευκόν· εἶτα καὶ ὧν μὴ κατηγορεῖται λευκῶν ὁ ἄνθρωπος, εἰλήφθω κύκνος καὶ χιών· οὐκοῦν τὸ ζῶιον τοῦ μὲν παντὸς κατηγορεῖται, τοῦ δὲ οὐδενός, ὥστε οὐκ ἔσται συλλογισμός. πάλιν τὸ μὲν Α μηδενὶ τῶι Β ὑπαρχέτω, τὸ δὲ Β τινὶ τῶι Γ μὴ ὑπαρχέτω· καὶ οἱ ὅροι ἔστωσαν ἄψυχον – ἄνθρωπος – λευκόν· εἶτα εἰλήφθωσαν, ὧν μὴ κατηγορεῖται λευκῶν ὁ ἄνθρωπος, κύκνος καὶ χιών· τὸ γὰρ ἄψυχον τοῦ μὲν παντὸς κατηγορεῖται, τοῦ δὲ οὐδενός. | Ponantur enim termini, animal, homo, album, deinde et de quibus albis non praedicatur homo, sumantur cygnus et nix; ergo animal de uno quidem omni praedicatur, de altero vero nullo, quare non erit syllogismus. (0643A) Rursum A quidem nulli B insit, B autem alicui C non insit, et sint termini, inanimatum, homo, album, deinde sumantur alba, de quibus non praedicatur homo, cygnus et nix; nam inanimatum de hoc quidem omni praedicatur, de illo vero nullo. | Suppose the terms are animal, man, white: next take some of the white things of which man is not predicated-swan and snow: animal is predicated of all of the one, but of none of the other. Consequently there cannot be a syllogism. Again let no B be A, but let some C not be B. Take the terms inanimate, man, white: then take some white things of which man is not predicated-swan and snow: the term inanimate is predicated of all of the one, of none of the other. |
| ἔτι ἐπεὶ ἀδιόριστον τὸ τινὶ τῶι Γ τὸ Β μὴ ὑπάρχειν, ἀληθεύεται δέ, καὶ εἰ μηδενὶ ὑπάρχει καὶ εἰ μὴ παντί, ὅτι τινὶ οὐχ ὑπάρχει, ληφθέντων δὲ τοιούτων ὅρων ὥστε μηδενὶ ὑπάρχειν οὐ γίνεται συλλογισμός (τοῦτο γὰρ εἴρηται πρότερον), φανερὸν οὖν ὅτι τῶι οὕτως ἔχειν τοὺς ὅρους οὐκ ἔσται συλλογισμός· ἦν γὰρ ἂν καὶ ἐπὶ τούτων. ὁμοίως δὲ δειχθήσεται καὶ εἰ τὸ καθόλου τεθείη στερητικόν. | Amplius: quoniam indefinitum est alicui eorum quae sunt C non inesse B, verum est autem et nulli inest, et si non omni, quoniam alicui non inest, sumptis autem his terminis velut nulli inesse, non fit syllogismus (hoc enim dictum est prius) manifestum; ergo est quoniam in eo quod sic se habent termini non erit syllogismus, esset enim et in his. Similiter autem ostendetur, et si universale ponatur privativum. | Further since it is indefinite to say some C is not B, and it is true that some C is not B, whether no C is B, or not all C is B, and since if terms are assumed such that no C is B, no syllogism follows (this has already been stated) it is clear that this arrangement of terms will not afford a syllogism: otherwise one would have been possible with a universal negative minor premiss. A similar proof may also be given if the universal premiss is negative. |
| Οὐδὲ ἐὰν ἄμφω τὰ διαστήματα κατὰ μέρος ἢ κατηγορικῶς ἢ στερητικῶς, ἢ τὸ μὲν κατηγορικῶς τὸ δὲ στερητικῶς λέγηται, ἢ τὸ μὲν ἀδιόριστον τὸ δὲ διωρισμένον, ἢ ἄμφω ἀδιόριστα, οὐκ ἔσται συλλογισμὸς οὐδαμῶς. ὅροι δὲ κοινοὶ πάντων ζῶιον – λευκόν – ἵππος, ζῶιον – λευκόν – λίθος. | Neque enim si ambo intervalla particularia praedicative, vel privative dicantur, aut hoc quidem praedicativum, illud vero privativum, vel hoc quidem indefinitum, illud vero definitum, vel ambo indefinita, non erit syllogismus nullo modo. (0643B) Termini vero communes omnium, animal, album, equus, animal, album, lapis. | Nor can there in any way be a syllogism if both the relations of subject and predicate are particular, either positively or negatively, or the one negative and the other affirmative, or one indefinite and the other definite, or both indefinite. Terms common to all the above are animal, white, horse: animal, white, stone. |
| Φανερὸν οὖν ἐκ τῶν εἰρημένων ὡς ἐὰν ἦι συλλογισμὸς ἐν τούτωι τῶι σχήματι κατὰ μέρος, ὅτι ἀνάγκη τοὺς ὅρους οὕτως ἔχειν ὡς εἴπομεν· ἄλλως γὰρ ἐχόντων οὐδαμῶς γίνεται. δῆλον δὲ καὶ ὅτι πάντες οἱ ἐν αὐτῶι συλλογισμοὶ τέλειοί εἰσι· (πάντες γὰρ ἐπιτελοῦνται διὰ τῶν ἐξ ἀρχῆς ληφθέντων), καὶ ὅτι πάντα τὰ προβλήματα δείκνυται διὰ τούτου τοῦ σχήματος· καὶ γὰρ τὸ παντὶ καὶ τὸ μηδενὶ καὶ τὸ τινὶ καὶ τὸ μή τινι ὑπάρχειν. καλῶ δὲ τὸ τοιοῦτον σχῆμα πρῶτον. | Manifestum est igitur ex iis quae dicta sunt quoniam si sit syllogismus in hac figura particularis, quoniam necesse est terminos sic se habere, ut diximus. Aliter enim se habentibus, nullo [modo] fit. Palam autem quoniam omnes qui in hac sunt syllogismi perfecti sunt, omnes enim perficiuntur per ea quae ex principio sumuntur, et quoniam omnia problemata ostenduntur per hanc figuram: etenim omni et nulli, alicui et non alicui inesse. Voco autem huiusmodi figuram, primam. | It is clear then from what has been said that if there is a syllogism in this figure with a particular conclusion, the terms must be related as we have stated: if they are related otherwise, no syllogism is possible anyhow. It is evident also that all the syllogisms in this figure are perfect (for they are all completed by means of the premisses originally taken) and that all conclusions are proved by this figure, viz. universal and particular, affirmative and negative. Such a figure I call the first. |
Chapter 5
| Greek | Latin | English |
|---|---|---|
| (PL 64 0643B) CAPUT V. De syllogismis absolutis in secunda figura. | 5 | |
| 26b34 Ὅταν δὲ τὸ αὐτὸ τῶι μὲν παντὶ τῶι δὲ μηδενὶ ὑπάρχηι, ἢ ἑκατέρωι παντὶ ἢ μηδενί, τὸ μὲν σχῆμα τὸ τοιοῦτον καλῶ δεύτερον, μέσον δὲ ἐν αὐτῶι λέγω τὸ κατηγορούμενον ἀμφοῖν, ἄκρα δὲ καθ᾽ ὧν λέγεται τοῦτο, μεῖζον δὲ ἄκρον τὸ πρὸς τῶι μέσωι κείμενον· ἔλαττον δὲ τὸ πορρωτέρω τοῦ μέσου. τίθεται δὲ τὸ μέσον ἔξω μὲν τῶν ἄκρων, πρῶτον δὲ τῆι θέσει. ↵ τέλειος μὲν οὖν οὐκ ἔσται συλλογισμὸς οὐδαμῶς ἐν τούτωι τῶι σχήματι, δυνατὸς δ᾽ ἔσται καὶ καθόλου καὶ μὴ καθόλου τῶν ὅρων ὄντων. | Quando vero idem huic omni quidem, illi vero nulli inest, vel utique omni, vel nulli, figuram quidem huiusmodi voco secundam. (0643C) Medium autem in hac dico quod de utraque praedicatur; extremitates vero de quibus dicitur hoc, maiorem quidem extremitatem, quae iuxta medium posita est, minorem vero, quae longius sita est A medio. Ponitur autem medium foras quidem extremitatum, primum vero positione. Perfectus igitur non erit syllogismus nullo modo in hac figura, possibile vero erit et universalibus, et non universalibus existentibus terminis. | Whenever the same thing belongs to all of one subject, and to none of another, or to all of each subject or to none of either, I call such a figure the second; by middle term in it I mean that which is predicated of both subjects, by extremes the terms of which this is said, by major extreme that which lies near the middle, by minor that which is further away from the middle. The middle term stands outside the extremes, and is first in position. A syllogism cannot be perfect anyhow in this figure, but it may be valid whether the terms are related universally or not. |
| καθόλου μὲν οὖν ὄντων ἔσται συλλογισμὸς ὅταν τὸ μέσον τῶι μὲν παντὶ τῶι δὲ μηδενὶ ὑπάρχηι, ἂν πρὸς ὁποτερωιοῦν ἦι τὸ στερητικόν· ἄλλως δ᾽ οὐδαμῶς. κατηγορείσθω γὰρ τὸ Μ τοῦ μὲν Ν μηδενός, τοῦ δὲ Ξ παντός. ἐπεὶ οὖν ἀντιστρέφει τὸ στερητικόν, οὐδενὶ τῶι Μ ὑπάρξει τὸ Ν· τὸ δέ γε Μ παντὶ τῶι Ξ ὑπέκειτο· ὥστε τὸ Ν οὐδενὶ τῶι Ξ· τοῦτο γὰρ δέδεικται πρότερον. | Universalibus igitur terminis erit syllogismus, quando medium huic quidem omni, illi vero nulli inerit, etsi ad utrumvis sit privativum, aliter vero nullo modo. Praedicetur enim M de N quidem nullo, de O vero omni, quoniam igitur convertitur privativa, nulli M inerit N, at M omni O supponebatur, quare N nulli O inerit: hoc enim ostensum est prius. | If then the terms are related universally a syllogism will be possible, whenever the middle belongs to all of one subject and to none of another (it does not matter which has the negative relation), but in no other way. Let M be predicated of no N, but of all O. Since, then, the negative relation is convertible, N will belong to no M: but M was assumed to belong to all O: consequently N will belong to no O. This has already been proved. |
| πάλιν εἰ τὸ Μ τῶι μὲν Ν παντὶ τῶι δὲ Ξ μηδενί, οὐδὲ τὸ Ξ τῶι Ν οὐδενὶ ὑπάρξει (εἰ γὰρ τὸ Μ οὐδενὶ τῶι Ξ, οὐδὲ τὸ Ξ οὐδενὶ τῶι Μ· τὸ δέ γε Μ παντὶ τῶι Ν ὑπῆρχεν· τὸ ἄρα Ξ οὐδενὶ τῶι Ν ὑπάρξει· γεγένηται γὰρ πάλιν τὸ πρῶτον σχῆμα)· ἐπεὶ δὲ ἀντιστρέφει τὸ στερητικόν, οὐδὲ τὸ Ν οὐδενὶ τῶι Ξ ὑπάρξει, ὥστ᾽ ἔσται ὁ αὐτὸς συλλογισμός. ἔστι δὲ δεικνύναι ταῦτα καὶ εἰς τὸ ἀδύνατον ἄγοντας. | (0643D) Rursum si M N quidem omni inest, O vero nulli, neque N O nulli inerit. Nam si M nulli O, neque O nulli N inerit, at vero M omni N inerat, quare O nulli inerit. Facta est enim rursum prima figura. Quoniam autem convertitur privativum, neque N nulli O inerit, quare erit idem syllogismus, est autem ostendere haec et ad impossibile ducentes. | Again if M belongs to all N, but to no O, then N will belong to no O. For if M belongs to no O, O belongs to no M: but M (as was said) belongs to all N: O then will belong to no N: for the first figure has again been formed. But since the negative relation is convertible, N will belong to no O. Thus it will be the same syllogism that proves both conclusions. It is possible to prove these results also by reductio ad impossibile. |
| ὅτι μὲν οὖν γίνεται συλλογισμὸς οὕτως ἐχόντων τῶν ὅρων, φανερόν, ἀλλ᾽ οὐ τέλειος· οὐ γὰρ μόνον ἐκ τῶν ἐξ ἀρχῆς ἀλλὰ καὶ ἐξ ἄλλων ἐπιτελεῖται τὸ ἀναγκαῖον. | Quoniam ergo fit syllogismus sic se habentibus terminis manifestum, sed non perfectus, non enim solum ex iis quae ab initio sumpta sunt, sed ex aliis perficitur necessarium. | It is clear then that a syllogism is formed when the terms are so related, but not a perfect syllogism; for necessity is not perfectly established merely from the original premisses; others also are needed. |
| ἐὰν δὲ τὸ Μ παντὸς τοῦ Ν καὶ τοῦ Ξ κατηγορῆται, οὐκ ἔσται συλλογισμός. ὅροι τοῦ ὑπάρχειν οὐσία - ζῶιον - ἄνθρωπος, τοῦ μὴ ὑπάρχειν οὐσία - ζῶιον - ἀριθμός· μέσον οὐσία. | Si autem M de omni N et O praedicetur, non erit syllogismus. Termini inesse, substantia, animal, ratio; non inesse, substantia, animal, lapis, medium, substantia. | But if M is predicated of every N and O, there cannot be a syllogism. Terms to illustrate a positive relation between the extremes are substance, animal, man; a negative relation, substance, animal, number-substance being the middle term. |
| οὐδ᾽ ὅταν μήτε τοῦ Ν μήτε τοῦ Ξ μηδενὸς κατηγορῆται τὸ Μ. ὅροι τοῦ ὑπάρχειν γραμμή – ζῶιον – ἄνθρωπος, τοῦ μὴ ὑπάρχειν γραμμή – ζῶιον – λίθος. | Nec quando de N nec de O nullo praedicatur M. (0644A) Termini inesse, linea, animal, homo; non inesse, linea, animal, lapis. | Nor is a syllogism possible when M is predicated neither of any N nor of any O. Terms to illustrate a positive relation are line, animal, man: a negative relation, line, animal, stone. |
| φανερὸν οὖν ὅτι ἂν ἦι συλλογισμὸς καθόλου τῶν ὅρων ὄντων, ἀνάγκη τοὺς ὅρους ἔχειν ὡς ἐν ἀρχῆι εἴπομεν· ἄλλως γὰρ ἐχόντων οὐ γίνεται τὸ ἀναγκαῖον. | Manifestum ergo quoniam si fit syllogismus ex universalibus terminis, necesse est terminos sic se habere, ut in principio diximus. Aliter enim se habentibus terminis non fit conclusio necessaria. | It is clear then that if a syllogism is formed when the terms are universally related, the terms must be related as we stated at the outset: for if they are otherwise related no necessary consequence follows. |
| Ἐὰν δὲ πρὸς τὸν ἕτερον ἦι καθόλου τὸ μέσον, ὅταν μὲν πρὸς τὸν μείζω γένηται καθόλου ἢ κατηγορικῶς ἢ στερητικῶς, πρὸς δὲ τὸν ἐλάττω κατὰ μέρος καὶ ἀντικειμένως τῶι καθόλου (λέγω δὲ τὸ ἀντικειμένως, εἰ μὲν τὸ καθόλου στερητικόν, τὸ ἐν μέρει καταφατικόν· εἰ δὲ κατηγορικὸν τὸ καθόλου, τὸ ἐν μέρει στερητικόν), ἀνάγκη γίνεσθαι συλλογισμὸν στερητικὸν κατὰ μέρος. | Si autem ad alterum sit universaliter medium, quando ad maius quidem fuerit universaliter vel praedicative, vel privative, ad minus autem et particulariter, et oppositae universali (dico autem oppositae, si universale quidem privativum particulare praedicativum, vel si universale praedicativum, particulare privativum), necesse est syllogismum fieri privativum particulariter. | If the middle term is related universally to one of the extremes, a particular negative syllogism must result whenever the middle term is related universally to the major whether positively or negatively, and particularly to the minor and in a manner opposite to that of the universal statement: by ‘an opposite manner’ I mean, if the universal statement is negative, the particular is affirmative: if the universal is affirmative, the particular is negative. |
| εἰ γὰρ τὸ Μ τῶι μὲν Ν μηδενὶ τῶι δὲ Ξ τινὶ ὑπάρχει, ἀνάγκη τὸ Ν τινὶ τῶι Ξ μὴ ὑπάρχειν. ἐπεὶ γὰρ ἀντιστρέφει τὸ στερητικόν, οὐδενὶ τῶι Μ ὑπάρξει τὸ Ν· τὸ δέ γε Μ ὑπέκειτο τινὶ τῶι Ξ ὑπάρχειν· ὥστε τὸ Ν τινὶ τῶι Ξ οὐχ ὑπάρξει· γίνεται γὰρ συλλογισμὸς διὰ τοῦ πρώτου σχήματος. πάλιν εἰ τῶι μὲν Ν παντὶ τὸ Μ, τῶι δὲ Ξ τινὶ μὴ ὑπάρχει, ἀνάγκη τὸ Ν τινὶ τῶι Ξ μὴ ὑπάρχειν· εἰ γὰρ παντὶ ὑπάρχει, κατηγορεῖται δὲ καὶ τὸ Μ παντὸς τοῦ Ν, ἀνάγκη τὸ Μ ↵ παντὶ τῶι Ξ ὑπάρχειν· ὑπέκειτο δὲ τινὶ μὴ ὑπάρχειν. καὶ εἰ τὸ Μ τῶι μὲν Ν παντὶ ὑπάρχει τῶι δὲ Ξ μὴ παντί, ἔσται συλλογισμὸς ὅτι οὐ παντὶ τῶι Ξ τὸ Ν· ἀπόδειξις δ᾽ ἡ αὐτή. ἐὰν δὲ τοῦ μὲν Ξ παντὸς τοῦ δὲ Ν μὴ παντὸς κατηγορῆται, οὐκ ἔσται συλλογισμός. ὅροι ζῶιον – οὐσία – κόραξ, ζῶιον – λευκόν – κόραξ. οὐδ᾽ ὅταν τοῦ μὲν Ξ μηδενός, τοῦ δὲ Ν τινός. ὅροι τοῦ ὑπάρχειν ζῶιον – οὐσία – μονάς, τοῦ μὴ ὑπάρχειν ζῶιον – οὐσία – ἐπιστήμη. | Nam si M nulli quidem N, O autem alicui inest, necesse est N alicui O non inesse. (0644B) Quoniam enim convertitur privativum, nulli M inerit, N M vero supponebatur alicui O inesse, quare N alicui eorum quae sunt O non inerit. Fit enim syllogismus per primam figuram. Rursus si N quidem omni M, O vero alicui non inest, necesse est N alicui O non inesse. Nam si O omni inest N, praedicatur autem et M de omni N, necesse est M omni O inesse, supponebatur autem alicui non inesse. Et si M N omni quidem inest, O autem non omni, erit syllogismus, quoniam non omni O inest N. Demonstratio autem eadem. Si autem de O quidem omni, de N vero non omni praedicatur M, non erit syllogismus. Termini inesse, animal, substantia, corvus. Non inesse, animal, album, corvus. Nec quando de O quidem nullo, de N vero aliquo. Termini inesse, animal, substantia, lapis. Non inesse, animal, substantia, scientia. | For if M belongs to no N, but to some O, it is necessary that N does not belong to some O. For since the negative statement is convertible, N will belong to no M: but M was admitted to belong to some O: therefore N will not belong to some O: for the result is reached by means of the first figure. Again if M belongs to all N, but not to some O, it is necessary that N does not belong to some O: for if N belongs to all O, and M is predicated also of all N, M must belong to all O: but we assumed that M does not belong to some O. And if M belongs to all N but not to all O, we shall conclude that N does not belong to all O: the proof is the same as the above. But if M is predicated of all O, but not of all N, there will be no syllogism. Take the terms animal, substance, raven; animal, white, raven. Nor will there be a conclusion when M is predicated of no O, but of some N. Terms to illustrate a positive relation between the extremes are animal, substance, unit: a negative relation, animal, substance, science. |
| Ὅταν μὲν οὖν ἀντικείμενον ἦι τὸ καθόλου τῶι κατὰ μέρος, εἴρηται πότ᾽ ἔσται καὶ πότ᾽ οὐκ ἔσται συλλογισμός· ὅταν δὲ ὁμοιοσχήμονες ὦσιν αἱ προτάσεις, οἷον ἀμφότεραι στερητικαὶ ἢ καταφατικαί, οὐδαμῶς ἔσται συλλογισμός. ἔστωσαν γὰρ πρῶτον στερητικαί, καὶ τὸ καθόλου κείσθω πρὸς τὸ μεῖζον ἄκρον, οἷον τὸ Μ τῶι μὲν Ν μηδενὶ τῶι δὲ Ξ τινὶ μὴ ὑπαρχέτω· ἐνδέχεται δὴ καὶ παντὶ καὶ μηδενὶ τῶι Ξ τὸ Ν ὑπάρχειν. ὅροι τοῦ μὲν μὴ ὑπάρχειν μέλαν – χιών – ζῶιον· τοῦ δὲ παντὶ ὑπάρχειν οὐκ ἔστι λαβεῖν, εἰ τὸ Μ τῶι Ξ τινὶ μὲν ὑπάρχει τινὶ δὲ μή. εἰ γὰρ παντὶ τῶι Ξ τὸ Ν, τὸ δὲ Μ μηδενὶ τῶι Ν, τὸ Μ οὐδενὶ τῶι Ξ ὑπάρξει· ἀλλ᾽ ὑπέκειτο τινὶ ὑπάρχειν. οὕτω μὲν οὖν οὐκ ἐγχωρεῖ λαβεῖν ὅρους, ἐκ δὲ τοῦ ἀδιορίστου δεικτέον· ἐπεὶ γὰρ ἀληθεύεται τὸ τινὶ μὴ ὑπάρχειν τὸ Μ τῶι Ξ καὶ εἰ μηδενὶ ὑπάρχει, μηδενὶ δὲ ὑπάρχοντος οὐκ ἦν συλλογισμός, φανερὸν ὅτι οὐδὲ νῦν ἔσται. | (0644C) Quando igitur oppositum est universale particulari, dictum est quando erit, et quando non erit syllogismus. Quando autem similis figurae fuerint propositiones, ut ambae privativae vel affirmativae, nullo modo erit syllogismus. Sint enim primum privativae, et universale ponatur ad maiorem extremitatem, ut M N quidem nulli, O autem alicui non insit: contingit ergo et omni, et nulli O inesse N. Termini quidem nulli inesse, nigrum, nix, animal. Omni vero inesse, non est sumere, si M alicui quidem O inest, alicui autem non. Nam si omni O inest N, et M nulli, N etiam M nulli O inerit; sed positum erat alicui inesse, non igitur sic sumere contingit terminos. Ex indefinito autem ostendendum est. (0644D) Quoniam enim verum est M non inesse alicui O, et si nulli inest, nulli vero cum insit non erit syllogismus, manifestum quoniam neque nunc erit. | If then the universal statement is opposed to the particular, we have stated when a syllogism will be possible and when not: but if the premisses are similar in form, I mean both negative or both affirmative, a syllogism will not be possible anyhow. First let them be negative, and let the major premiss be universal, e.g. let M belong to no N, and not to some O. It is possible then for N to belong either to all O or to no O. Terms to illustrate the negative relation are black, snow, animal. But it is not possible to find terms of which the extremes are related positively and universally, if M belongs to some O, and does not belong to some O. For if N belonged to all O, but M to no N, then M would belong to no O: but we assumed that it belongs to some O. In this way then it is not admissible to take terms: our point must be proved from the indefinite nature of the particular statement. For since it is true that M does not belong to some O, even if it belongs to no O, and since if it belongs to no O a syllogism is (as we have seen) not possible, clearly it will not be possible now either. |
| πάλιν ἔστωσαν κατηγορικαί, καὶ τὸ καθόλου κείσθω ὁμοίως, οἷον τὸ Μ τῶι μὲν Ν παντὶ τῶι δὲ Ξ τινὶ ὑπαρχέτω. ἐνδέχεται δὴ τὸ Ν τῶι Ξ καὶ παντὶ καὶ μηδενὶ ὑπάρχειν. ὅροι τοῦ μηδενὶ ὑπάρχειν λευκόν – κύκνος – λίθος τοῦ δὲ παντὶ οὐκ ἔσται λαβεῖν διὰ τὴν αὐτὴν αἰτίαν ἥνπερ πρότερον, ἀλλ᾽ ἐκ τοῦ ἀδιορίστου δεικτέον. εἰ δὲ τὸ καθόλου πρὸς τὸ ἔλαττον ἄκρον ἐστί, καὶ τὸ Μ τῶι μὲν Ξ μηδενὶ τῶι δὲ Ν τινὶ μὴ ὑπάρχει, ἐνδέχεται τὸ Ν τῶι Ξ καὶ παντὶ καὶ μηδενὶ ὑπάρχειν. ὅροι τοῦ ὑπάρχειν λευκόν – ζῶιον – κόραξ, τοῦ μὴ ὑπάρχειν λευκόν – λίθος – κόραξ. εἰ δὲ κατηγορικαὶ αἱ προτάσεις, ὅροι τοῦ μὴ ὑπάρχειν λευκόν – ζῶιον – χιών, τοῦ ὑπάρχειν λευκόν – ζῶιον – κύκνος. | Rursum si praedicativae, et universale ponatur similiter, ut M omni quidem N, O autem alicui insit, contingit ergo et omni, et nulli O inesse. Termini nulli inesse, album, cygnus, lapis. Omni vero non erit sumere terminos, propter eamdem causam quam et prius, sed ex indefinito monstrandum est. Si autem universale ad minorem extremitatem est, et M O quidem nulli, N vero alicui non inest, contingit N, et omni et nulli O inesse. Termini inesse, album, animal, corvus; non inesse, album, lapis, corvus. Similiter autem et si praedicativae fuerint propositiones. Termini non inesse, album, animal, nix; inesse, album, animal, cygnus. (0645A) | Again let the premisses be affirmative, and let the major premiss as before be universal, e.g. let M belong to all N and to some O. It is possible then for N to belong to all O or to no O. Terms to illustrate the negative relation are white, swan, stone. But it is not possible to take terms to illustrate the universal affirmative relation, for the reason already stated: the point must be proved from the indefinite nature of the particular statement. But if the minor premiss is universal, and M belongs to no O, and not to some N, it is possible for N to belong either to all O or to no O. Terms for the positive relation are white, animal, raven: for the negative relation, white, stone, raven. If the premisses are affirmative, terms for the negative relation are white, animal, snow; for the positive relation, white, animal, swan. |
| φανερὸν οὖν, ὅταν ὁμοιοσχήμονες ὦσιν αἱ προτάσεις καὶ ἡ μὲν καθόλου ἡ δ᾽ ἐν μέρει, ὅτι οὐδαμῶς γίνεται συλλογισμός. ἀλλ᾽ οὐδ᾽ εἰ τινὶ ἑκατέρωι ὑπάρχει ἢ μὴ ὑπάρχει, ἢ τῶι μὲν τῶι δὲ μή, ἢ μηδετέρωι παντί, ἢ ἀδιορίστως. ὅροι δὲ κοινοὶ πάντων λευκόν – ζῶιον – ἄνθρωπος, λευκόν – ζῶιον – ἄψυχον. ↵ Φανερὸν οὖν ἐκ τῶν εἰρημένων ὅτι ἐάν τε οὕτως ἔχωσιν οἱ ὅροι πρὸς ἀλλήλους ὡς ἐλέχθη, γίνεται συλλογισμὸς ἐξ ἀνάγκης, ἄν τ᾽ ἦι συλλογισμός, ἀνάγκη τοὺς ὅρους οὕτως ἔχειν. δῆλον δὲ καὶ ὅτι πάντες ἀτελεῖς εἰσὶν οἱ ἐν τούτωι τῶι σχήματι συλλογισμοί (πάντες γὰρ ἐπιτελοῦνται προσλαμβανομένων τινῶν, ἃ ἢ ἐνυπάρχει τοῖς ὅροις ἐξ ἀνάγκης ἢ τίθενται ὡς ὑποθέσεις, οἷον ὅταν διὰ τοῦ ἀδυνάτου δεικνύωμεν), καὶ ὅτι οὐ γίνεται καταφατικὸς συλλογισμὸς διὰ τούτου τοῦ σχήματος, ἀλλὰ πάντες στερητικοί, καὶ οἱ καθόλου καὶ οἱ κατὰ μέρος. | Manifestum est igitur quoniam si similis figurae sint propositiones, et haec quidem universalis, illa vero particularis, quoniam nullo modo fit syllogismus. Sed nec si alicui, utrique inest, vel non inest, vel huic quidem inest, illi vero non, vel neutri omni, vel indefinitae. Termini autem communes omnium, album, animal, homo, album, animal, inanimatum. Manifestum est igitur ex praedictis quoniam si sic se habent termini ad invicem, ut dictum est, fit syllogismus ex necessitate, et si fit syllogismus, necesse est terminos sic se habere. Palam autem et quoniam omnes imperfecti sunt, qui in hac figura sunt syllogismi; omnes enim perficiuntur assumptis quibusdam, quae vel insunt terminis ex necessitate, vel ponuntur velut hypotheses, ut quando per impossibile ostendimus. (0645B) Et quoniam non fit affirmativus syllogismus per hanc figuram, sed omnes privativi, et universales, et particulares. | Evidently then, whenever the premisses are similar in form, and one is universal, the other particular, a syllogism can, not be formed anyhow. Nor is one possible if the middle term belongs to some of each of the extremes, or does not belong to some of either, or belongs to some of the one, not to some of the other, or belongs to neither universally, or is related to them indefinitely. Common terms for all the above are white, animal, man: white, animal, inanimate. It is clear then from what has been said that if the terms are related to one another in the way stated, a syllogism results of necessity; and if there is a syllogism, the terms must be so related. But it is evident also that all the syllogisms in this figure are imperfect: for all are made perfect by certain supplementary statements, which either are contained in the terms of necessity or are assumed as hypotheses, i.e. when we prove per impossibile. And it is evident that an affirmative conclusion is not attained by means of this figure, but all are negative, whether universal or particular. |
Chapter 6
| Greek | Latin | English |
|---|---|---|
| (PL 64 0645B) CAPUT VI. De syllogismis absolutis tertiae figurae. | 6 | |
| 28a9 Ἐὰν δὲ τῶι αὐτῶι τὸ μὲν παντὶ τὸ δὲ μηδενὶ ὑπάρχηι, ἢ ἄμφω παντὶ ἢ μηδενί, τὸ μὲν σχῆμα τὸ τοιοῦτον καλῶ τρίτον, μέσον δ᾽ ἐν αὐτῶι λέγω καθ᾽ οὗ ἄμφω τὰ κατηγορούμενα, ἄκρα δὲ τὰ κατηγορούμενα, μεῖζον δ᾽ ἄκρον τὸ πορρώτερον τοῦ μέσου, ἔλαττον δὲ τὸ ἐγγύτερον. τίθεται δὲ τὸ μέσον ἔξω μὲν τῶν ἄκρων, ἔσχατον δὲ τῆι θέσει. τέλειος μὲν οὖν οὐ γίνεται συλλογισμὸς οὐδ᾽ ἐν τούτωι τῶι σχήματι, δυνατὸς δ᾽ ἔσται καὶ καθόλου καὶ μὴ καθόλου τῶν ὅρων ὄντων πρὸς τὸ μέσον. | Si autem eidem hoc quidem omni, illud vero nulli inest, vel ambo omni vel nulli, figuram quidem huiusmodi voco tertiam. Medium autem in hac dico, quo ambo praedicamus; extremitates vero, quae praedicantur; maiorem autem extremitatem, quae longius est medio; minorem vero, quae propius. Ponitur autem medium foras quidem extremitatum, ultimum vero positione est. Perfectus igitur non fit syllogismus, nec in hac figura, possibilis vero erit et universaliter, et non universaliter terminis existentibus ad medium. | But if one term belongs to all, and another to none, of a third, or if both belong to all, or to none, of it, I call such a figure the third; by middle term in it I mean that of which both the predicates are predicated, by extremes I mean the predicates, by the major extreme that which is further from the middle, by the minor that which is nearer to it. The middle term stands outside the extremes, and is last in position. A syllogism cannot be perfect in this figure either, but it may be valid whether the terms are related universally or not to the middle term. |
| Καθόλου μὲν οὖν ὄντων, ὅταν καὶ τὸ Π καὶ τὸ Ρ παντὶ τῶι Σ ὑπάρχηι, ὅτι τινὶ τῶι Ρ τὸ Π ὑπάρξει ἐξ ἀνάγκης· ἐπεὶ γὰρ ἀντιστρέφει τὸ κατηγορικόν, ὑπάρξει τὸ Σ τινὶ τῶι Ρ, ὥστ᾽ ἐπεὶ τῶι μὲν Σ παντὶ τὸ Π, τῶι δὲ Ρ τινὶ τὸ Σ, ἀνάγκη τὸ Π τινὶ τῶι Ρ ὑπάρχειν· γίνεται γὰρ συλλογισμὸς διὰ τοῦ πρώτου σχήματος. ἔστι δὲ καὶ διὰ τοῦ ἀδυνάτου καὶ τῶι ἐκθέσθαι ποιεῖν τὴν ἀπόδειξιν· εἰ γὰρ ἄμφω παντὶ τῶι Σ ὑπάρχει, ἂν ληφθῆι τι τῶν Σ οἷον τὸ Ν, τούτωι καὶ τὸ Π καὶ τὸ Ρ ὑπάρξει, ὥστε τινὶ τῶι Ρ τὸ Π ὑπάρξει. | (0645C) Universaliter quidem quando et p et r inerunt omni s, quoniam alicui r inerit p ex necessitate, nam quoniam convertitur praedicativa, inerit s alicui r. Quare quoniam p inest omni s, et s alicui r, necesse est p alicui r inesse. Fit enim syllogismus per primam figuram. Est autem et per impossibile, et expositione facere demonstrationem: si enim ambo omni s insunt, si sumatur aliquod eorum quae sunt s, ut N huic et p et r inerunt ex necessitate, quare alicui r inerit p. | If they are universal, whenever both P and R belong to S, it follows that P will necessarily belong to some R. For, since the affirmative statement is convertible, S will belong to some R: consequently since P belongs to all S, and S to some R, P must belong to some R: for a syllogism in the first figure is produced. It is possible to demonstrate this also per impossibile and by exposition. For if both P and R belong to all S, should one of the Ss, e.g. N, be taken, both P and R will belong to this, and thus P will belong to some R. |
| καὶ ἂν τὸ μὲν Ρ παντὶ τῶι Σ, τὸ δὲ Π μηδενὶ ὑπάρχηι, ἔσται συλλογισμὸς ὅτι τὸ Π τινὶ τῶι Ρ οὐχ ὑπάρξει ἐξ ἀνάγκης· ὁ γὰρ αὐτὸς τρόπος τῆς ἀποδείξεως ἀντιστραφείσης τῆς Ρ Σ προτάσεως. δειχθείη δ᾽ ἂν καὶ διὰ τοῦ ἀδυνάτου, καθάπερ ἐπὶ τῶν πρότερον. ἐὰν δὲ τὸ μὲν Ρ μηδενὶ τὸ δὲ Π παντὶ ὑπάρχηι τῶι Σ, οὐκ ἔσται συλλογισμός. ὅροι τοῦ ὑπάρχειν ζῶιον – ἵππος – ἄνθρωπος, τοῦ μὴ ὑπάρχειν ζῶιον – ἄψυχον – ἄνθρωπος. | Et si r omni quidem s, p autem nulli s inest, erit syllogismus, quoniam p alicui r non inerit ex necessitate. Nam idem modus erit demonstrationis, conversa r s propositione. Ostendetur autem et per impossibile, quemadmodum in prioribus. Si autem insit r, s quidem nulli, p vero omni s, non erit syllogismus. (0645D) Termini inesse, animal, equus, homo; non inesse, animal, inanimatum, homo, | If R belongs to all S, and P to no S, there will be a syllogism to prove that P will necessarily not belong to some R. This may be demonstrated in the same way as before by converting the premiss RS. It might be proved also per impossibile, as in the former cases. But if R belongs to no S, P to all S, there will be no syllogism. Terms for the positive relation are animal, horse, man: for the negative relation animal, inanimate, man. |
| οὐδ᾽ ὅταν ἄμφω κατὰ μηδενὸς τοῦ Σ λέγηται, οὐκ ἔσται συλλογισμός. ὅροι τοῦ ὑπάρχειν ζῶιον – ἵππος – ἄψυχον, τοῦ μὴ ὑπάρχειν ἄνθρωπος – ἵππος – ἄψυχον· | neque quando ambo de nullo s dicuntur, non erit syllogismus. Termini inesse, animal, equus, inanimatum; non inesse, homo, equus inanimatum, medium, inanimatum. | Nor can there be a syllogism when both terms are asserted of no S. Terms for the positive relation are animal, horse, inanimate; for the negative relation man, horse, inanimate-inanimate being the middle term. |
| μέσον ἄψυχον. φανερὸν οὖν καὶ ἐν τούτωι τῶι σχήματι πότ᾽ ἔσται καὶ πότ᾽ οὐκ ἔσται συλλογισμὸς καθόλου τῶν ὅρων ὄντων. ὅταν μὲν γὰρ ἀμφότεροι οἱ ὅροι ὦσι κατηγορικοί, ἔσται συλλογισμὸς ὅτι τινὶ ὑπάρχει τὸ ἄκρον τῶι ἄκρωι, ὅταν δὲ στερητικοί, οὐκ ↵ ἔσται. ὅταν δ᾽ ὁ μὲν ἦι στερητικὸς ὁ δὲ καταφατικός, ἐὰν μὲν ὁ μείζων γένηται στερητικὸς ἅτερος δὲ καταφατικός, ἔσται συλλογισμὸς ὅτι τινὶ οὐχ ὑπάρχει τὸ ἄκρον τῶι ἄκρωι, ἐὰν δ᾽ ἀνάπαλιν, οὐκ ἔσται. Ἐὰν δ᾽ ὁ μὲν ἦι καθόλου πρὸς τὸ μέσον ὁ δ᾽ ἐν μέρει, κατηγορικῶν μὲν ὄντων ἀμφοῖν ἀνάγκη γίνεσθαι συλλογισμόν, ἂν ὁποτεροσοῦν ἦι καθόλου τῶν ὅρων. εἰ γὰρ τὸ μὲν Ρ παντὶ τῶι Σ τὸ δὲ Π τινί, ἀνάγκη τὸ Π τινὶ τῶι Ρ ὑπάρχειν. ἐπεὶ γὰρ ἀντιστρέφει τὸ καταφατικόν, ὑπάρξει τὸ Σ τινὶ τῶι Π, ὥστ᾽ ἐπεὶ τὸ μὲν Ρ παντὶ τῶι Σ, τὸ δὲ Σ τινὶ τῶι Π, καὶ τὸ Ρ τινὶ τῶι Π ὑπάρξει· ὥστε τὸ Π τινὶ τῶι Ρ. | Manifestum est igitur et in hac figura et quando erit, et quando non erit syllogismus ex universalibus terminis. Quando enim ambo termini sunt praedicativi, erit syllogismus, quoniam inest alicui extremitas extremitati; quando vero privativi, non erit syllogismus; quando autem hic quidem privativus, ille vero affirmativus; si maior quidem fuerit privativus, alter vero affirmativus, erit syllogismus, quoniam alicui non inest extremitas extremitati. Si autem e converso, non erit. (0646A) Si autem hic quidem sit universaliter ad medium, alter vero particulariter, si uterque sit praedicativus, necesse est fieri syllogismum, et si alteruter sit universalis terminorum; nam si r omni s insit, p vero alicui s, necesse est et p alicui r inesse, nam quoniam convertitur affirmativa, inerit s alicui p, quare quoniam r omni s inest, s autem alicui p, et r alicui p inerit, quare et p alicui r. | It is clear then in this figure also when a syllogism will be possible and when not, if the terms are related universally. For whenever both the terms are affirmative, there will be a syllogism to prove that one extreme belongs to some of the other; but when they are negative, no syllogism will be possible. But when one is negative, the other affirmative, if the major is negative, the minor affirmative, there will be a syllogism to prove that the one extreme does not belong to some of the other: but if the relation is reversed, no syllogism will be possible. If one term is related universally to the middle, the other in part only, when both are affirmative there must be a syllogism, no matter which of the premisses is universal. For if R belongs to all S, P to some S, P must belong to some R. For since the affirmative statement is convertible S will belong to some P: consequently since R belongs to all S, and S to some P, R must also belong to some P: therefore P must belong to some R. |
| πάλιν εἰ τὸ μὲν Ρ τινὶ τῶι Σ τὸ δὲ Π παντὶ ὑπάρχει, ἀνάγκη τὸ Π τινὶ τῶι Ρ ὑπάρχειν· ὁ γὰρ αὐτὸς τρόπος τῆς ἀποδείξεως. ἔστι δ᾽ ἀποδεῖξαι καὶ διὰ τοῦ ἀδυνάτου καὶ τῆι ἐκθέσει, καθάπερ ἐπὶ τῶν πρότερον. Ἐὰν δ᾽ ὁ μὲν ἦι κατηγορικὸς ὁ δὲ στερητικός, καθόλου δὲ ὁ κατηγορικός, ὅταν μὲν ὁ ἐλάττων ἦι κατηγορικός, ἔσται συλλογισμός. εἰ γὰρ τὸ Ρ παντὶ τῶι Σ, τὸ δὲ Π τινὶ μὴ ὑπάρχει, ἀνάγκη τὸ Π τινὶ τῶι Ρ μὴ ὑπάρχειν. εἰ γὰρ παντί, καὶ τὸ Ρ παντὶ τῶι Σ, καὶ τὸ Π παντὶ τῶι Σ ὑπάρξει· ἀλλ᾽ οὐχ ὑπῆρχεν. δείκνυται δὲ καὶ ἄνευ τῆς ἀπαγωγῆς, ἐὰν ληφθῆι τι τῶν Σ ὧι τὸ Π μὴ ὑπάρχει. | Rursum si r alicui s, p vero omni s insit, necesse est et p alicui r inesse, nam idem modus demonstrationis. Est autem demonstrare et per impossibile, et expositione, quemadmodum in prioribus. (0646B) Si autem unus quidem sit praedicativus, alius vero privativus, universaliter autem praedicativus, quando minor quidem fuerit praedicativus, erit syllogismus; nam si r omni s, p vero alicui s non inest, necesse est p alicui r non inesse, si enim p omni r, et r omni s, et p omni s inerit, sed non inerat. Monstratur autem et sine deductione, si sumatur aliquid eorum quae sunt s, cui p non inest. | Again if R belongs to some S, and P to all S, P must belong to some R. This may be demonstrated in the same way as the preceding. And it is possible to demonstrate it also per impossibile and by exposition, as in the former cases. But if one term is affirmative, the other negative, and if the affirmative is universal, a syllogism will be possible whenever the minor term is affirmative. For if R belongs to all S, but P does not belong to some S, it is necessary that P does not belong to some R. For if P belongs to all R, and R belongs to all S, then P will belong to all S: but we assumed that it did not. Proof is possible also without reduction ad impossibile, if one of the Ss be taken to which P does not belong. |
| ὅταν δ᾽ ὁ μείζων ἦι κατηγορικός, οὐκ ἔσται συλλογισμός, οἷον εἰ τὸ μὲν Π παντὶ τῶι Σ, τὸ δὲ Ρ τινὶ τῶι Σ μὴ ὑπάρχει. ὅροι τοῦ παντὶ ὑπάρχειν ἔμψυχον – ἄνθρωπος – ζῶιον. τοῦ δὲ μηδενὶ οὐκ ἔστι λαβεῖν ὅρους, εἰ τινὶ μὲν ὑπάρχει τῶι Σ τὸ Ρ, τινὶ δὲ μή· εἰ γὰρ παντὶ τὸ Π τῶι Σ ὑπάρχει, τὸ δὲ Ρ τινὶ τῶι Σ, καὶ τὸ Π τινὶ τῶι Ρ ὑπάρξει· ὑπέκειτο δὲ μηδενὶ ὑπάρχειν. ἀλλ᾽ ὥσπερ ἐν τοῖς πρότερον ληπτέον· ἀδιορίστου γὰρ ὄντος τοῦ τινὶ μὴ ὑπάρχειν καὶ τὸ μηδενὶ ὑπάρχον ἀληθὲς εἰπεῖν τινὶ μὴ ὑπάρχειν· μηδενὶ δὲ ὑπάρχοντος οὐκ ἦν συλλογισμός. φανερὸν οὖν ὅτι οὐκ ἔσται συλλογισμός. | Quando vero maior fuerit praedicativus, non erit syllogismus, ut si p insit omni s, r autem alicui s non insit. Termini vero omni inesse, animatum, homo, animal. Nulli vero, non est sumere terminos si r inest alicui quidem s, alicui autem non. Si enim omni s inest p, r autem alicui s, et p inerit alicui r, sed positum erat nulli r inesse. (0646C) Sed quemadmodum in prioribus dicendum est; nam cum indefinitum est alicui non inesse, et quod nulli inest, verum est dicere alicui non inesse, nulli vero cum inesset, non erat syllogismus; manifestum ergo est, quoniam non erit syllogismus. | But whenever the major is affirmative, no syllogism will be possible, e.g. if P belongs to all S and R does not belong to some S. Terms for the universal affirmative relation are animate, man, animal. For the universal negative relation it is not possible to get terms, if R belongs to some S, and does not belong to some S. For if P belongs to all S, and R to some S, then P will belong to some R: but we assumed that it belongs to no R. We must put the matter as before.’ Since the expression ‘it does not belong to some’ is indefinite, it may be used truly of that also which belongs to none. But if R belongs to no S, no syllogism is possible, as has been shown. Clearly then no syllogism will be possible here. |
| ἐὰν δ᾽ ὁ στερητικὸς ἦι καθόλου τῶν ὅρων, ὅταν μὲν ὁ μείζων ἦι στερητικὸς ὁ δὲ ἐλάττων κατηγορικός, ἔσται συλλογισμός. εἰ γὰρ τὸ Π μηδενὶ τῶι Σ, τὸ δὲ Ρ τινὶ ὑπάρχει τῶι Σ, τὸ Π τινὶ τῶι Ρ οὐχ ὑπάρξει· πάλιν γὰρ ἔσται τὸ πρῶτον σχῆμα τῆς Ρ Σ προτάσεως ἀντιστραφείσης. | Si autem privativus sit universalis terminus, quando maior quidem privativus fuerit, minor autem praedicativus, erit syllogismus. Si enim p nulli s, r autem alicui inest s, et p alicui r non inerit. Rursum enim prima erit figura, r s propositione conversa. | But if the negative term is universal, whenever the major is negative and the minor affirmative there will be a syllogism. For if P belongs to no S, and R belongs to some S, P will not belong to some R: for we shall have the first figure again, if the premiss RS is converted. |
| ὅταν δὲ ὁ ἐλάττων ἦι στερητικός, οὐκ ἔσται συλλογισμός. ὅροι τοῦ ὑπάρχειν ζῶιον – ἄνθρωπος – ἄγριον, τοῦ μὴ ὑπάρχειν ζῶιον – ἐπιστήμη – ἄγριον· μέσον ἐν ἀμφοῖν τὸ ἄγριον. | Quando autem minor fuerit privativus, non erit syllogismus. Termini inesse, animal, homo, ferum. Non inesse, animal, scientia, ferum, medium in utrisque ferum. | But when the minor is negative, there will be no syllogism. Terms for the positive relation are animal, man, wild: for the negative relation, animal, science, wild-the middle in both being the term wild. |
| οὐδ᾽ ὅταν ἀμφότεροι στερητικοὶ τεθῶσιν, ἦι δ᾽ ὁ μὲν καθόλου ὁ δ᾽ ἐν μέρει. ὅροι ↵ ὅταν ὁ ἐλάττων ἦι καθόλου πρὸς τὸ μέσον, ζῶιον – ἐπιστήμη – ἄγριον, ζῶιον – ἄνθρωπος – ἄγριον· ὅταν δ᾽ ὁ μείζων, τοῦ μὲν μὴ ὑπάρχειν κόραξ – χιών – λευκόν. τοῦ δ᾽ ὑπάρχειν οὐκ ἔστι λαβεῖν, εἰ τὸ Ρ τινὶ μὲν ὑπάρχει τῶι Σ, τινὶ δὲ μὴ ὑπάρχει. εἰ γὰρ τὸ Π παντὶ τῶι Ρ, τὸ δὲ Ρ τινὶ τῶι Σ, καὶ τὸ Π τινὶ τῶι Σ· ὑπέκειτο δὲ μηδενί. ἀλλ᾽ ἐκ τοῦ ἀδιορίστου δεικτέον. | Nec quando ambo privativi ponuntur, est autem unus quidem universalis, alter vero particularis. Termini inesse, quando minor est universalis ad medium, animal, homo, ferum, non inesse, animal, scientia, ferum. (0646D) Quando autem maior, non inesse quidem, corvus, nix, album; inesse vero non est sumere si r alicui quidem inest s, alicui autem non inest. Si enim p omni r insit, r autem alicui s, et p inerit alicui s. Positum est autem nulli, sed ex indefinito monstrandum est. | Nor is a syllogism possible when both are stated in the negative, but one is universal, the other particular. When the minor is related universally to the middle, take the terms animal, science, wild; animal, man, wild. When the major is related universally to the middle, take as terms for a negative relation raven, snow, white. For a positive relation terms cannot be found, if R belongs to some S, and does not belong to some S. For if P belongs to all R, and R to some S, then P belongs to some S: but we assumed that it belongs to no S. Our point, then, must be proved from the indefinite nature of the particular statement. |
| Οὐδ᾽ ἂν ἑκάτερος τινὶ τῶι μέσωι ὑπάρχηι ἢ μὴ ὑπάρχηι, ἢ ὁ μὲν ὑπάρχηι ὁ δὲ μὴ ὑπάρχηι, ἢ ὁ μὲν τινὶ ὁ δὲ μὴ παντί, ἢ ἀδιορίστως, οὐκ ἔσται συλλογισμὸς οὐδαμῶς. ὅροι δὲ κοινοὶ πάντων ζῶιον – ἄνθρωπος – λευκόν, ζῶιον – ἄψυχον – λευκόν. | Neque si uterque alicui medio inest, vel non inest, vel unus quidem inest, alter vero non inest, vel hic quidem alicui, ille vero non omni, vel indefinite, nullo modo erit syllogismus. Termini autem communes omnium, animal, homo, album, animal, inani matum, album. | Nor is a syllogism possible anyhow, if each of the extremes belongs to some of the middle or does not belong, or one belongs and the other does not to some of the middle, or one belongs to some of the middle, the other not to all, or if the premisses are indefinite. Common terms for all are animal, man, white: animal, inanimate, white. |
| Φανερὸν οὖν καὶ ἐν τούτωι τῶι σχήματι πότ᾽ ἔσται καὶ πότ᾽ οὐκ ἔσται συλλογισμός, καὶ ὅτι ἐχόντων τε τῶν ὅρων ὡς ἐλέχθη γίνεται συλλογισμὸς ἐξ ἀνάγκης, ἄν τ᾽ ἦι συλλογισμός, ἀνάγκη τοὺς ὅρους οὕτως ἔχειν. φανερὸν δὲ καὶ ὅτι πάντες ἀτελεῖς εἰσὶν οἱ ἐν τούτωι τῶι σχήματι συλλογισμοί (πάντες γὰρ τελειοῦνται προσλαμβανομένων τινῶν) καὶ ὅτι συλλογίσασθαι τὸ καθόλου διὰ τούτου τοῦ σχήματος οὐκ ἔσται, οὔτε στερητικὸν οὔτε καταφατικόν. | Manifestum est igitur, et in hac figura, quando erit, et quando non erit syllogismus, et quoniam habentibus se terminis, ut dictum est, fit syllogismus ex necessitate, et si sit syllogismus, necesse est terminos sic se habere. (0647A) Manifestum est etiam, quia omnes imperfecti sunt in hac figura syllogismi, omnes enim perficiuntur quibusdam assumptis. Et quoniam syllogizare universale per hanc figuram non erit, neque privativum, neque affirmativum. | It is clear then in this figure also when a syllogism will be possible, and when not; and that if the terms are as stated, a syllogism results of necessity, and if there is a syllogism, the terms must be so related. It is clear also that all the syllogisms in this figure are imperfect (for all are made perfect by certain supplementary assumptions), and that it will not be possible to reach a universal conclusion by means of this figure, whether negative or affirmative. |
Chapter 7
| Greek | Latin | English |
|---|---|---|
| (PL 64 0647A) CAPUT VII. De tribus figuris et indirectis syllogismis ad invicem. | 7 | |
| 29a19 Δῆλον δὲ καὶ ὅτι ἐν ἅπασι τοῖς σχήμασιν, ὅταν μὴ γίνηται συλλογισμός, κατηγορικῶν μὲν ἢ στερητικῶν ἀμφοτέρων ὄντων τῶν ὅρων οὐδὲν ὅλως γίνεται ἀναγκαῖον, κατηγορικοῦ δὲ καὶ στερητικοῦ, καθόλου ληφθέντος τοῦ στερητικοῦ ἀεὶ γίνεται συλλογισμὸς τοῦ ἐλάττονος ἄκρου πρὸς τὸ μεῖζον, οἷον εἰ τὸ μὲν Α παντὶ τῶι Β ἢ τινί, τὸ δὲ Β μηδενὶ τῶι Γ· ἀντιστρεφομένων γὰρ τῶν προτάσεων ἀνάγκη τὸ Γ τινὶ τῶι Α μὴ ὑπάρχειν. ὁμοίως δὲ κἀπὶ τῶν ἑτέρων σχημάτων· ἀεὶ γὰρ γίνεται διὰ τῆς ἀντιστροφῆς συλλογισμός. δῆλον δὲ καὶ ὅτι τὸ ἀδιόριστον ἀντὶ τοῦ κατηγορικοῦ τοῦ ἐν μέρει τιθέμενον τὸν αὐτὸν ποιήσει συλλογισμὸν ἐν ἅπασι τοῖς σχήμασιν. | Palam autem et quoniam in omnibus figuris, aliquando non fit syllogismus. Cum praedicativi quidem, vel privativi sunt utrique termini, et particulares, nihil omnino fit necessarium. (0647B) Cum autem praedicativus, et privativus, et universaliter sumptus privativus, semper fit syllogismus minoris extremitatis ad maiorem, ut si A quidem omni B vel alicui, B autem nulli C; conversis enim propositionibus, necesse est C alicui A non inesse. Similiter autem et in aliis figuris, semper enim fit per conversionem syllogismus. Palam etiam quoniam indefinitum pro praedicativo particulari positum, eumdem faciet syllogismum in omnibus figuris. | It is evident also that in all the figures, whenever a proper syllogism does not result, if both the terms are affirmative or negative nothing necessary follows at all, but if one is affirmative, the other negative, and if the negative is stated universally, a syllogism always results relating the minor to the major term, e.g. if A belongs to all or some B, and B belongs to no C: for if the premisses are converted it is necessary that C does not belong to some A. Similarly also in the other figures: a syllogism always results by means of conversion. It is evident also that the substitution of an indefinite for a particular affirmative will effect the same syllogism in all the figures. |
| Φανερὸν δὲ καὶ ὅτι πάντες οἱ ἀτελεῖς συλλογισμοὶ τελειοῦνται διὰ τοῦ πρώτου σχήματος. ἢ γὰρ δεικτικῶς ἢ διὰ τοῦ ἀδυνάτου περαίνονται πάντες· ἀμφοτέρως δὲ γίνεται τὸ πρῶτον σχῆμα, δεικτικῶς μὲν τελειουμένων, ὅτι διὰ τῆς ἀντιστροφῆς ἐπεραίνοντο πάντες, ἡ δ᾽ ἀντιστροφὴ τὸ πρῶτον ἐποίει σχῆμα, διὰ δὲ τοῦ ἀδυνάτου δεικνυμένων, ὅτι τεθέντος τοῦ ψεύδους ὁ συλλογισμὸς γίνεται διὰ τοῦ πρώτου σχήματος, οἷον ἐν τῶι τελευταίωι σχήματι, εἰ τὸ Α καὶ τὸ Β παντὶ τῶι Γ ὑπάρχει, ὅτι τὸ Α τινὶ τῶι Β ὑπάρχει· εἰ γὰρ μηδενί, τὸ δὲ Β παντὶ τῶι Γ, οὐδενὶ τῶι Γ τὸ Α· ἀλλ᾽ ἦν παντί. ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων. | Manifestum autem et quoniam omnes imperfecti syllogismi perficiuntur per primam figuram. Aut enim ostensive, aut per impossibile clauduntur omnes. Utrinque autem fit prima figura. Et ostensive quidem perfectis, quoniam per conversionem claudebantur omnes, conversio autem primam faciebat figuram, per impossibile vero demonstratis, quoniam posito falso syllogismus fit per primam figuram. Ut in postrema figura, si A et B omni C insunt, quoniam A alicui B inest, nam si nulli et B omni C, nulli C inerit A, sed inerat omni. (0647C) Similiter autem in aliis. | It is clear too that all the imperfect syllogisms are made perfect by means of the first figure. For all are brought to a conclusion either ostensively or per impossibile. In both ways the first figure is formed: if they are made perfect ostensively, because (as we saw) all are brought to a conclusion by means of conversion, and conversion produces the first figure: if they are proved per impossibile, because on the assumption of the false statement the syllogism comes about by means of the first figure, e.g. in the last figure, if A and B belong to all C, it follows that A belongs to some B: for if A belonged to no B, and B belongs to all C, A would belong to no C: but (as we stated) it belongs to all C. Similarly also with the rest. |
| ↵ Ἔστι δὲ καὶ ἀναγαγεῖν πάντας τοὺς συλλογισμοὺς εἰς τοὺς ἐν τῶι πρώτωι σχήματι καθόλου συλλογισμούς. οἱ μὲν γὰρ ἐν τῶι δευτέρωι φανερὸν ὅτι δι᾽ ἐκείνων τελειοῦνται, πλὴν οὐχ ὁμοίως πάντες, ἀλλ᾽ οἱ μὲν καθόλου τοῦ στερητικοῦ ἀντιστραφέντος, τῶν δ᾽ ἐν μέρει ἑκάτερος διὰ τῆς εἰς τὸ ἀδύνατον ἀπαγωγῆς. οἱ δ᾽ ἐν τῶι πρώτωι, οἱ κατὰ μέρος, ἐπιτελοῦν- ται μὲν καὶ δι᾽ αὑτῶν, ἔστι δὲ καὶ διὰ τοῦ δευτέρου σχήματος δεικνύναι εἰς ἀδύνατον ἀπάγοντας, οἷον εἰ τὸ Α παντὶ τῶι Β, τὸ δὲ Β τινὶ τῶι Γ, ὅτι τὸ Α τινὶ τῶι Γ· εἰ γὰρ μηδενί, τῶι δὲ Β παντί, οὐδενὶ τῶι Γ τὸ Β ὑπάρξει· τοῦτο γὰρ ἴσμεν διὰ τοῦ δευτέρου σχήματος. | Est etiam reducere omnes syllogismos ad universales syllogismos primae figurae. Nam qui sunt in secunda figura, manifestum quoniam per illos perficiuntur, verum non similiter omnes, sed universales quidem privativa conversa; particularium autem utraque per ad impossibile reductionem. Qui vero in prima sunt particulares, perficiuntur quidem per se. Est autem et per secundam figuram ostendere ad impossibile ducentes, ut si A omni B, et B alicui C, quoniam A alicui C inerit. Si enim nulli, B autem omni, nulli C inerit B. Hoc enim scimus per secundam figuram. | It is possible also to reduce all syllogisms to the universal syllogisms in the first figure. Those in the second figure are clearly made perfect by these, though not all in the same way; the universal syllogisms are made perfect by converting the negative premiss, each of the particular syllogisms by reductio ad impossibile. In the first figure particular syllogisms are indeed made perfect by themselves, but it is possible also to prove them by means of the second figure, reducing them ad impossibile, e.g. if A belongs to all B, and B to some C, it follows that A belongs to some C. For if it belonged to no C, and belongs to all B, then B will belong to no C: this we know by means of the second figure. |
| ὁμοίως δὲ καὶ ἐπὶ τοῦ στερητικοῦ ἔσται ἡ ἀπόδειξις. εἰ γὰρ τὸ Α μηδενὶ τῶι Β, τὸ δὲ Β τινὶ τῶι Γ ὑπάρχει, τὸ Α τινὶ τῶι Γ οὐχ ὑπάρξει· εἰ γὰρ παντί, τῶι δὲ Β μηδενὶ ὑπάρχει, οὐδενὶ τῶι Γ τὸ Β ὑπάρξει· τοῦτο δ᾽ ἦν τὸ μέσον σχῆμα. ὥστ᾽ ἐπεὶ οἱ μὲν ἐν τῶι μέσωι σχήματι συλλογισμοὶ πάντες ἀνάγονται εἰς τοὺς ἐν τῶι πρώτωι καθόλου συλλογισμούς, οἱ δὲ κατὰ μέρος ἐν τῶι πρώτωι εἰς τοὺς ἐν τῶι μέσωι, φανερὸν ὅτι καὶ οἱ κατὰ μέρος ἀναχθήσονται εἰς τοὺς ἐν τῶι πρώτωι σχήματι καθόλου συλλογισμούς. | (0647D) Similiter autem et in privativo erit demonstratio; si enim A nulli B, et B alicui C inest, A alicui C non erit, nam si A omni C, B autem nulli inest, nulli C inerit B. Haec autem fuit media figura; quare quoniam qui in media sunt syllogismi, omnes reducuntur in primae figurae universales syllogismos, qui vero particulares sunt in prima, ad eos qui sunt in media, manifestum est quoniam et particulares reducentur ad eos qui in prima figura sunt universales syllogismos;
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Similarly also demonstration will be possible in the case of the negative. For if A belongs to no B, and B belongs to some C, A will not belong to some C: for if it belonged to all C, and belongs to no B, then B will belong to no C: and this (as we saw) is the middle figure. Consequently, since all syllogisms in the middle figure can be reduced to universal syllogisms in the first figure, and since particular syllogisms in the first figure can be reduced to syllogisms in the middle figure, it is clear that particular syllogisms can be reduced to universal syllogisms in the first figure. |
| οἱ δ᾽ ἐν τῶι τρίτωι καθόλου μὲν ὄντων τῶν ὅρων εὐθὺς ἐπιτελοῦνται δι᾽ ἐκείνων τῶν συλλογισμῶν, ὅταν δ᾽ ἐν μέρει ληφθῶσι, διὰ τῶν ἐν μέρει συλλογισμῶν τῶν ἐν τῶι πρώτωι σχήματι· οὗτοι δὲ ἀνήχθησαν εἰς ἐκείνους, ὥστε καὶ οἱ ἐν τῶι τρίτωι σχήματι, οἱ κατὰ μέρος. φανερὸν οὖν ὅτι πάντες ἀναχθήσονται εἰς τοὺς ἐν τῶι πρώτωι σχήματι καθόλου συλλογισμούς. | qui vero sunt in tertia, cum universales sint quidem termini, statim perficiuntur per illos syllogismos. Si autem particulares, sumuntur per particulares syllogismos primae figurae, sed hi reducti sunt ad illos, quare et tertiae figurae particulares. Manifestum ergo quoniam omnes reducentur in primae figurae universales syllogismos. | Syllogisms in the third figure, if the terms are universal, are directly made perfect by means of those syllogisms; but, when one of the premisses is particular, by means of the particular syllogisms in the first figure: and these (we have seen) may be reduced to the universal syllogisms in the first figure: consequently also the particular syllogisms in the third figure may be so reduced. It is clear then that all syllogisms may be reduced to the universal syllogisms in the first figure. |
| Οἱ μὲν οὖν τῶν συλλογισμῶν ὑπάρχειν ἢ μὴ ὑπάρχειν δεικνύντες εἴρηται πῶς ἔχουσι, καὶ καθ᾽ ἑαυτοὺς οἱ ἐκ τοῦ αὐτοῦ σχήματος καὶ πρὸς ἀλλήλους οἱ ἐκ τῶν ἑτέρων. | (0648A) Igitur syllogismi inesse vel non inesse ostendentes, dictum est quomodo se habent, et ad eos qui ex eadem sunt figura, et ad invicem, et ad eos qui ex aliis sunt figuris. | We have stated then how syllogisms which prove that something belongs or does not belong to something else are constituted, both how syllogisms of the same figure are constituted in themselves, and how syllogisms of different figures are related to one another. |
Chapter 8
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|---|---|---|
| (PL 64 0648A) CAPUT VIII. De syllogismis ex necessario in tribus figuris. | 8 | |
| 29b29 Ἐπεὶ δ᾽ ἕτερόν ἐστιν ὑπάρχειν τε καὶ ἐξ ἀνάγκης ὑπάρχειν καὶ ἐνδέχεσθαι ὑπάρχειν (πολλὰ γὰρ ὑπάρχει μέν, οὐ μέντοι ἐξ ἀνάγκης· τὰ δ᾽ οὔτ᾽ ἐξ ἀνάγκης οὔθ᾽ ὑπάρχει ὅλως, ἐνδέχεται δ᾽ ὑπάρχειν), δῆλον ὅτι καὶ συλλογισμὸς ἑκάστου τούτων ἕτερος ἔσται, καὶ οὐχ ὁμοίως ἐχόντων τῶν ὅρων, ἀλλ᾽ ὁ μὲν ἐξ ἀναγκαίων, ὁ δ᾽ ἐξ ὑπαρχόντων, ὁ δ᾽ ἐξ ἐνδεχομένων. | Quoniam autem diversum est inesse, et ex necessitate inesse, et contingere inesse (nam multa insunt quidem, non tamen ex necessitate, alia vero neque ex necessitate, neque insunt omnino, contingit autem inesse), manifestum quoniam et syllogismus in unoquoque horum diversus est, et non similiter habentibus se terminis, sed hic quidem ex necessariis, ille vero ex iis quae simpliciter insunt, ille autem ex contingentibus. | Since there is a difference according as something belongs, necessarily belongs, or may belong to something else (for many things belong indeed, but not necessarily, others neither necessarily nor indeed at all, but it is possible for them to belong), it is clear that there will be different syllogisms to prove each of these relations, and syllogisms with differently related terms, one syllogism concluding from what is necessary, another from what is, a third from what is possible. |
| Ἐπὶ μὲν οὖν τῶν ἀναγκαίων σχεδὸν ὁμοίως ἔχει καὶ ἐπὶ τῶν ὑπαρχόντων· ὡσαύτως γὰρ τιθεμένων τῶν ὅρων ἔν τε τῶι ὑπάρχειν καὶ τῶι ἐξ ἀνάγκης ὑπάρχειν ἢ μὴ ὑπάρχειν ἔσται τε καὶ οὐκ ἔσται συλλογισμός, πλὴν διοίσει τῶι ↵ προσκεῖσθαι τοῖς ὅροις τὸ ἐξ ἀνάγκης ὑπάρχειν ἢ μὴ ὑπάρχειν. τό τε γὰρ στερητικὸν ὡσαύτως ἀντιστρέφει, καὶ τὸ ἐν ὅλωι εἶναι καὶ τὸ κατὰ παντὸς ὁμοίως ἀποδώσομεν. | Ergo in necessariis quidem fere similiter se habet, et in iis qui insunt. (0648B) Similiter enim positis terminis, et in iis quae insunt, et in iis quae ex necessitate insunt vel non insunt, et erit, et non erit syllogismus. Verum distabit in eo quod adiacet terminis ex necessitate inesse, vel non inesse, nam et privativum similiter convertitur, et in toto esse, et de omni similiter assignabimus. | There is hardly any difference between syllogisms from necessary premisses and syllogisms from premisses which merely assert. When the terms are put in the same way, then, whether something belongs or necessarily belongs (or does not belong) to something else, a syllogism will or will not result alike in both cases, the only difference being the addition of the expression ‘necessarily’ to the terms. For the negative statement is convertible alike in both cases, and we should give the same account of the expressions ‘to be contained in something as in a whole’ and ‘to be predicated of all of something’. |
| ἐν μὲν οὖν τοῖς ἄλλοις τὸν αὐτὸν τρόπον δειχθήσεται διὰ τῆς ἀντιστροφῆς τὸ συμπέρασμα ἀναγκαῖον, ὥσπερ ἐπὶ τοῦ ὑπάρχειν· ἐν δὲ τῶι μέσωι σχήματι, ὅταν ἦι τὸ καθόλου καταφατικὸν τὸ δ᾽ ἐν μέρει στερητικόν, καὶ πάλιν ἐν τῶι τρίτωι, ὅταν τὸ μὲν καθόλου κατηγορικὸν τὸ δ᾽ ἐν μέρει στερητικόν, οὐχ ὁμοίως ἔσται ἡ ἀπόδειξις, ἀλλ᾽ ἀνάγκη ἐκθεμένους ὧι τινὶ ἑκάτερον μὴ ὑπάρχει, κατὰ τούτου ποιεῖν τὸν συλλογισμόν· ἔσται γὰρ ἀναγκαῖος ἐπὶ τούτων· εἰ δὲ κατὰ τοῦ ἐκτεθέντος ἐστὶν ἀναγκαῖος, καὶ κατ᾽ ἐκείνου τινός· τὸ γὰρ ἐκτεθὲν ὅπερ ἐκεῖνό τί ἐστιν. γίνεται δὲ τῶν συλλογισμῶν ἑκάτερος ἐν τῶι οἰκείωι σχήματι. | Ergo in aliis quidem eodem modo ostendetur per conversionem, quoniam conclusio necessaria, quomodo in eo quod est inesse. In media autem figura quando fuerit universalis affirmativa, particularis vero privativa, et rursum in tertia quando universalis quidem praedicativa, particularis vero privativa, non similiter erit demonstratio, sed necesse est exponentes, cui alicui utrumque non inest, de hoc facere syllogismum. Erit enim necessarius in hoc. Si autem de exposito est necessarius, erit et de illo aliquo. (0648C) Nam hoc quod est expositum, ipsum quidem illud aliquid est. Fit autem uterque syllogismus in propria figura. | With the exceptions to be made below, the conclusion will be proved to be necessary by means of conversion, in the same manner as in the case of simple predication. But in the middle figure when the universal statement is affirmative, and the particular negative, and again in the third figure when the universal is affirmative and the particular negative, the demonstration will not take the same form, but it is necessary by the ‘exposition’ of a part of the subject of the particular negative proposition, to which the predicate does not belong, to make the syllogism in reference to this: with terms so chosen the conclusion will necessarily follow. But if the relation is necessary in respect of the part taken, it must hold of some of that term in which this part is included: for the part taken is just some of that. And each of the resulting syllogisms is in the appropriate figure. |
Chapter 9
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| (PL 64 0648C) CAPUT IX. De mixtis ex una necessaria et altera absoluta in prima figura. | 9 | |
| 30a15 Συμβαίνει δέ ποτε καὶ τῆς ἑτέρας προτάσεως ἀναγκαίας οὔσης ἀναγκαῖον γίνεσθαι τὸν συλλογισμόν, πλὴν οὐχ ὁποτέρας ἔτυχεν, ἀλλὰ τῆς πρὸς τὸ μεῖζον ἄκρον, οἷον εἰ τὸ μὲν Α τῶι Β ἐξ ἀνάγκης εἴληπται ὑπάρχον ἢ μὴ ὑπάρχον, τὸ δὲ Β τῶι Γ ὑπάρχον μόνον· οὕτως γὰρ εἰλημμένων τῶν προτάσεων ἐξ ἀνάγκης τὸ Α τῶι Γ ὑπάρξει ἢ οὐχ ὑπάρξει. | Accidit autem quandoque et altera propositione necessaria, necessarium fieri syllogismum, verum non utralibet, sed quae ad maiorem extremitatem est, ut si A quidem, B ex necessitate sumptum est inesse, vel non inesse, B autem C inesse tantum; sic enim sumptis propositionibus ex necessitate A inerit C, vel non erit. | It happens sometimes also that when one premiss is necessary the conclusion is necessary, not however when either premiss is necessary, but only when the major is, e.g. if A is taken as necessarily belonging or not belonging to B, but B is taken as simply belonging to C: for if the premisses are taken in this way, A will necessarily belong or not belong to C. |
| ἐπεὶ γὰρ παντὶ τῶι Β ἐξ ἀνάγκης ὑπάρχει ἢ οὐχ ὑπάρχει τὸ Α, τὸ δὲ Γ τι τῶν Β ἐστί, φανερὸν ὅτι καὶ τῶι Γ ἐξ ἀνάγκης ἔσται θάτερον τούτων. εἰ δὲ τὸ μὲν Α Β μὴ ἔστιν ἀναγκαῖον, τὸ δὲ Β Γ ἀναγκαῖον, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον. εἰ γὰρ ἔστι, συμβήσεται τὸ Α τινὶ τῶι Β ὑπάρχειν ἐξ ἀνάγκης διά τε τοῦ πρώτου καὶ διὰ τοῦ τρίτου σχήματος. τοῦτο δὲ ψεῦδος· ἐνδέχεται γὰρ τοιοῦτον εἶναι τὸ Β ὧι ἐγχωρεῖ τὸ Α μηδενὶ ὑπάρχειν. ἔτι καὶ ἐκ τῶν ὅρων φανερὸν ὅτι οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον, οἷον εἰ τὸ μὲν Α εἴη κίνησις, τὸ δὲ Β ζῶιον, ἐφ᾽ ὧι δὲ τὸ Γ ἄνθρωπος· ζῶιον μὲν γὰρ ὁ ἄνθρωπος ἐξ ἀνάγκης ἐστί, κινεῖται δὲ τὸ ζῶιον οὐκ ἐξ ἀνάγκης, οὐδ᾽ ὁ ἄνθρωπος. ὁμοίως δὲ καὶ εἰ στερητικὸν εἴη τὸ Α Β· ἡ γὰρ αὐτὴ ἀπόδειξις. | (0648D) Nam quoniam omni B ex necessitate inest, vel non inest A, C autem aliquid eorum quae sunt B, est manifestum quoniam et C ex necessitate erit alterum horum. Si autem A B quidem non necessaria, B C autem necessaria, non erit conclusio necessaria. Nam si est, accidit A alicui B inesse ex necessitate, per primam et tertiam figuram, hoc autem falsum, contingit enim tale esse B cui possibile est A nulli inesse. Amplius autem et ex terminis manifestum quoniam non erit conclusio necessaria; ut si A quidem sit motus, B autem sit animal, in que autem C homo, namque homo animal est ex necessitate, movetur autem animal non ex necessitate, quare nec homo. Similiter autem et si privativa sit A B; nam eadem demonstratio. | For since necessarily belongs, or does not belong, to every B, and since C is one of the Bs, it is clear that for C also the positive or the negative relation to A will hold necessarily. But if the major premiss is not necessary, but the minor is necessary, the conclusion will not be necessary. For if it were, it would result both through the first figure and through the third that A belongs necessarily to some B. But this is false; for B may be such that it is possible that A should belong to none of it. Further, an example also makes it clear that the conclusion not be necessary, e.g. if A were movement, B animal, C man: man is an animal necessarily, but an animal does not move necessarily, nor does man. Similarly also if the major premiss is negative; for the proof is the same. |
| ἐπὶ δὲ τῶν ἐν μέρει συλλογισμῶν, εἰ μὲν τὸ καθόλου ἐστὶν ἀναγκαῖον, καὶ τὸ συμπέρασμα ἔσται ἀναγκαῖον, εἰ δὲ τὸ κατὰ μέρος, οὐκ ἀναγκαῖον, οὔτε στερητικῆς οὔτε κατηγορικῆς οὔσης τῆς καθόλου προ- τάσεως. | (0649A) In particularibus autem syllogismis, si universalis quidem est necessaria, et conclusio erit necessaria; si autem particularis, non necessaria, sive privativa, sive praedicativa fuerit universalis propositio. | In particular syllogisms, if the universal premiss is necessary, then the conclusion will be necessary; but if the particular, the conclusion will not be necessary, whether the universal premiss is negative or affirmative. |
| ἔστω δὴ πρῶτον τὸ καθόλου ἀναγκαῖον, καὶ τὸ μὲν Α παντὶ τῶι Β ὑπαρχέτω ἐξ ἀνάγκης, τὸ δὲ Β τινὶ τῶι Γ ὑπαρχέτω μόνον· ἀνάγκη δὴ τὸ Α τινὶ τῶι Γ ὑπάρχειν ἐξ ἀνάγκης· τὸ γὰρ Γ ὑπὸ τὸ Β ἐστί, τῶι δὲ Β παντὶ ↵ ὑπῆρχεν ἐξ ἀνάγκης, ὁμοίως δὲ καὶ εἰ στερητικὸς εἴη ὁ συλλογισμός· ἡ γὰρ αὐτὴ ἔσται ἀπόδειξις. εἰ δὲ τὸ κατὰ μέρος ἐστὶν ἀναγκαῖον, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον (οὐδὲν γὰρ ἀδύνατον συμπίπτει), καθάπερ οὐδ᾽ ἐν τοῖς καθόλου συλλογισμοῖς. ὁμοίως δὲ κἀπὶ τῶν στερητικῶν. ὅροι κίνησις – ζῶιον – λευκόν. | Sit autem primo universalis necessaria, et A quidem omni B insit ex necessitate, B autem alicui C insit solum, necesse est ergo A alicui C inesse ex necessitate, nam C sub B est, B autem omni A inerat ex necessitate. Similiter autem et si privativus syllogismus sit, nam eadem erit demonstratio. Si autem particularis est necessaria, non erit conclusio necessaria, nihil enim impossibile evenit, quemadmodum nec in universalibus syllogismis, similiter autem et in privativis. Termini, motus, animal, album. | First let the universal be necessary, and let A belong to all B necessarily, but let B simply belong to some C: it is necessary then that A belongs to some C necessarily: for C falls under B, and A was assumed to belong necessarily to all B. Similarly also if the syllogism should be negative: for the proof will be the same. But if the particular premiss is necessary, the conclusion will not be necessary: for from the denial of such a conclusion nothing impossible results, just as it does not in the universal syllogisms. The same is true of negative syllogisms. Try the terms movement, animal, white. |
Chapter 10
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| (PL 64 0649A) CAPUT X. De mixtis ex una necessaria et altera absoluta in secunda figura. | 10 | |
| 30b7 Ἐπὶ δὲ τοῦ δευτέρου σχήματος, εἰ μὲν ἡ στερητικὴ πρότασίς ἐστιν ἀναγκαία, καὶ τὸ συμπέρασμα ἔσται ἀναγκαῖον, εἰ δ᾽ ἡ κατηγορική, οὐκ ἀναγκαῖον. ἔστω γὰρ πρῶτον ἡ στερητικὴ ἀναγκαία, καὶ τὸ Α τῶι μὲν Β μηδενὶ ἐνδεχέσθω, τῶι δὲ Γ ὑπαρχέτω μόνον. ἐπεὶ οὖν ἀντιστρέφει τὸ στερητικόν, οὐδὲ τὸ Β τῶι Α οὐδενὶ ἐνδέχεται· τὸ δὲ Α παντὶ τῶι Γ ὑπάρχει, ὥστ᾽ οὐδενὶ τῶι Γ τὸ Β ἐνδέχεται· τὸ γὰρ Γ ὑπὸ τὸ Α ἐστίν. ὡσαύτως δὲ καὶ εἰ πρὸς τῶι Γ τεθείη τὸ στερητικόν· εἰ γὰρ τὸ Α μηδενὶ τῶι Γ ἐνδέχεται, οὐδὲ τὸ Γ οὐδενὶ τῶι Α ἐγχωρεῖ· τὸ δὲ Α παντὶ τῶι Β ὑπάρχει, ὥστ᾽ οὐδενὶ τῶι Β τὸ Γ ἐνδέχεται· γίνεται γὰρ τὸ πρῶτον σχῆμα πάλιν. οὐκ ἄρα οὐδὲ τὸ Β τῶι Γ· ἀντιστρέφει γὰρ ὁμοίως. | (0649B) In secunda autem figura si privativa quidem propositio universalis sit et necessaria, conclusio erit necessaria. Si autem praedicativa, non necessaria. Sit enim primum privativa necessaria, et A B quidem nulli contingat, C autem insit tantum; quoniam ergo convertitur privativa, et B nulli A contingit, A autem omni C inest, quare nulli C contingit B, nam C sub A est. Similiter autem et si ad C ponatur privativum, nam si A C nulli contingit, et C nulli A poterit inesse, A autem omni B inest. Quare nulli eorum quae sunt B contingit C, fit enim prima figura. Rursum non ergo neque B ipsi C, convertitur enim similiter. | In the second figure, if the negative premiss is necessary, then the conclusion will be necessary, but if the affirmative, not necessary. First let the negative be necessary; let A be possible of no B, and simply belong to C. Since then the negative statement is convertible, B is possible of no A. But A belongs to all C; consequently B is possible of no C. For C falls under A. The same result would be obtained if the minor premiss were negative: for if A is possible be of no C, C is possible of no A: but A belongs to all B, consequently C is possible of none of the Bs: for again we have obtained the first figure. Neither then is B possible of C: for conversion is possible without modifying the relation. |
| Εἰ δὲ ἡ κατηγορικὴ πρότασίς ἐστιν ἀναγκαία, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον. ὑπαρχέτω γὰρ τὸ Α παντὶ τῶι Β ἐξ ἀνάγκης, τῶι δὲ Γ μηδενὶ ὑπαρχέτω μόνον. ἀντιστραφέντος οὖν τοῦ στερητικοῦ τὸ πρῶτον γίνεται σχῆμα· δέδεικται δ᾽ ἐν τῶι πρώτωι ὅτι μὴ ἀναγκαίας οὔσης τῆς πρὸς τὸ μεῖζον στερητικῆς οὐδὲ τὸ συμπέρασμα ἔσται ἀναγκαῖον, ὥστ᾽ οὐδ᾽ ἐπὶ τούτων ἔσται ἐξ ἀνάγκης. | Si autem praedicativa propositio est necessaria, non erit conclusio necessaria, insit enim A omni B ex necessitate, C autem nulli insit tantum, conversa ergo privativa, fit prima figura. (0649C) Ostensum est autem in prima quoniam cum non est necessaria quae ad maiorem est privativa, nec conclusio erit necessaria, quare nec in his erit ex necessitate. | But if the affirmative premiss is necessary, the conclusion will not be necessary. Let A belong to all B necessarily, but to no C simply. If then the negative premiss is converted, the first figure results. But it has been proved in the case of the first figure that if the negative major premiss is not necessary the conclusion will not be necessary either. Therefore the same result will obtain here. |
| ἔτι δ᾽ εἰ τὸ συμπέρασμά ἐστιν ἀναγκαῖον, συμβαίνει τὸ Γ τινὶ τῶι Α μὴ ὑπάρχειν ἐξ ἀνάγκης. εἰ γὰρ τὸ Β τῶι Γ μηδενὶ ὑπάρχει ἐξ ἀνάγκης, οὐδὲ τὸ Γ τῶι Β οὐδενὶ ὑπάρξει ἐξ ἀνάγκης. τὸ δέ γε Β τινὶ τῶι Α ἀνάγκη ὑπάρχειν, εἴπερ καὶ τὸ Α παντὶ τῶι Β ἐξ ἀνάγκης ὑπῆρχεν. ὥστε τὸ Γ ἀνάγκη τινὶ τῶι Α μὴ ὑπάρχειν. ἀλλ᾽ οὐδὲν κωλύει τὸ Α τοιοῦτον λη- φθῆναι ὧι παντὶ τὸ Γ ἐνδέχεται ὑπάρχειν. | Amplius autem si conclusio est necessaria, accidit C alicui A non inesse ex necessitate, si enim B nulli C inest ex necessitate, neque C nulli B inerit ex necessitate, B autem alicui A necesse est inesse, siquidem et A omni B ex necessitate inerat, quare C necesse est alicui A non inesse, sed nihil prohibet A huiusmodi accipere, cui omni C contingat inesse. | Further, if the conclusion is necessary, it follows that C necessarily does not belong to some A. For if B necessarily belongs to no C, C will necessarily belong to no B. But B at any rate must belong to some A, if it is true (as was assumed) that A necessarily belongs to all B. Consequently it is necessary that C does not belong to some A. But nothing prevents such an A being taken that it is possible for C to belong to all of it. |
| ἔτι κἂν ὅρους ἐκθέμενον εἴη δεῖξαι ὅτι τὸ συμπέρασμα οὐκ ἔστιν ἀναγκαῖον ἁπλῶς, ἀλλὰ τούτων ὄντων ἀναγκαῖον. οἷον ἔστω τὸ Α ζῶιον, τὸ δὲ Β ἄνθρωπος, τὸ δὲ Γ λευκόν, καὶ αἱ προτάσεις ὁμοίως εἰλήφθωσαν· ἐνδέχεται γὰρ τὸ ζῶιον μηδενὶ λευκῶι ὑπάρχειν. οὐχ ὑπάρξει δὴ οὐδ᾽ ὁ ἄνθρωπος οὐδενὶ λευκῶι, ἀλλ᾽ οὐκ ἐξ ἀνάγκης· ἐνδέχεται γὰρ ἄνθρωπον γενέσθαι λευκόν, οὐ μέντοι ἕως ἂν ζῶιον μηδενὶ λευκῶι ὑπάρχηι. ὥστε τούτων μὲν ὄντων ἀναγκαῖον ἔσται τὸ συμπέρασμα, ἁπλῶς δ᾽ οὐκ ἀναγκαῖον. | Amplius et si terminos ponentes sit ostendere, quoniam conclusio non est necessaria simpliciter. (0649D) Et his existentibus, necessarium ut sit A animal, B vero homo, C autem album, et similiter propositiones sumptae sint, contingit enim animal nulli albo inesse, non inerit ergo nec homo nulli albo, sed non ex necessitate. Contingit enim hominem fieri album, non tamen donec animal nulli albo insit, quare cum haec sint, necessaria erit conclusio, simpliciter autem non necessaria. | Further one might show by an exposition of terms that the conclusion is not necessary without qualification, though it is a necessary conclusion from the premisses. For example let A be animal, B man, C white, and let the premisses be assumed to correspond to what we had before: it is possible that animal should belong to nothing white. Man then will not belong to anything white, but not necessarily: for it is possible for man to be born white, not however so long as animal belongs to nothing white. Consequently under these conditions the conclusion will be necessary, but it is not necessary without qualification. |
| ↵ Ὁμοίως δ᾽ ἕξει καὶ ἐπὶ τῶν ἐν μέρει συλλογισμῶν. ὅταν μὲν γὰρ ἡ στερητικὴ πρότασις καθόλου τ᾽ ἦι καὶ ἀναγκαία, καὶ τὸ συμπέρασμα ἔσται ἀναγκαῖον· ὅταν δὲ ἡ κατηγορικὴ καθόλου, ἡ δὲ στερητικὴ κατὰ μέρος, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον. ἔστω δὴ πρῶτον ἡ στερητικὴ καθόλου τε καὶ ἀναγκαία, καὶ τὸ Α τῶι μὲν Β μηδενὶ ἐνδεχέσθω ὑπάρχειν, τῶι δὲ Γ τινὶ ὑπαρχέτω. ἐπεὶ οὖν ἀντιστρέφει τὸ στερητικόν, οὐδὲ τὸ Β τῶι Α οὐδενὶ ἐνδέχοιτ᾽ ἂν ὑπάρχειν· τὸ δέ γε Α τινὶ τῶι Γ ὑπάρχει, ὥστ᾽ ἐξ ἀνάγκης τινὶ τῶι Γ οὐχ ὑπάρξει τὸ Β. | Similiter autem se habebit et in particularibus syllogismis, quando privativa quidem propositio, et universalis fuerit, et necessaria, et conclusio erit necessaria. Quando autem praedicativa universalis fuerit necessaria, privativa vero particularis non necessaria, non erit conclusio necessaria. (0650A) Sit enim primum privativa, et universalis necessaria, et A B quidem nulli contingat inesse, C autem alicui insit, quoniam ergo convertitur privativa, et B nulli A continget inesse, A autem alicui C inest, quare ex necessitate alicui eorum quae sunt, C non inerit B. | Similar results will obtain also in particular syllogisms. For whenever the negative premiss is both universal and necessary, then the conclusion will be necessary: but whenever the affirmative premiss is universal, the negative particular, the conclusion will not be necessary. First then let the negative premiss be both universal and necessary: let it be possible for no B that A should belong to it, and let A simply belong to some C. Since the negative statement is convertible, it will be possible for no A that B should belong to it: but A belongs to some C; consequently B necessarily does not belong to some of the Cs. |
| πάλιν ἔστω ἡ κατηγορικὴ καθόλου τε καὶ ἀναγκαία, καὶ κείσθω πρὸς τῶι Β τὸ κατηγορικόν. εἰ δὴ τὸ Α παντὶ τῶι Β ἐξ ἀνάγκης ὑπάρχει, τῶι δὲ Γ τινὶ μὴ ὑπάρχει, ὅτι μὲν οὐχ ὑπάρξει τὸ Β τινὶ τῶι Γ, φανερόν, ἀλλ᾽ οὐκ ἐξ ἀνάγκης· οἱ γὰρ αὐτοὶ ὅροι ἔσονται πρὸς τὴν ἀπόδειξιν οἵπερ ἐπὶ τῶν καθόλου συλλογισμῶν. ἀλλ᾽ οὐδ᾽ εἰ τὸ στερητικὸν ἀναγκαῖόν ἐστιν ἐν μέρει ληφθέν, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον· διὰ γὰρ τῶν αὐτῶν ὅρων ἡ ἀπόδειξις. | Rursum sit praedicativa, et universalis, et necessaria, et ponatur ad B quidem praedicativum, si ergo A omni B ex necessitate inest, C autem alicui non inest, quoniam non inerit B alicui C manifestum, sed non ex necessitate. Nam iidem termini erunt ad demonstrationem, qui in universalibus syllogismis: sed nec si privativa necessaria est particulariter sumpta, erit conclusio necessaria. Nam per eosdem terminos demonstratio. | Again let the affirmative premiss be both universal and necessary, and let the major premiss be affirmative. If then A necessarily belongs to all B, but does not belong to some C, it is clear that B will not belong to some C, but not necessarily. For the same terms can be used to demonstrate the point, which were used in the universal syllogisms. Nor again, if the negative statement is necessary but particular, will the conclusion be necessary. The point can be demonstrated by means of the same terms. |
Chapter 11
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| (PL 64 0650A) CAPUT XI. De syllogismis mixtis ex altera necessaria et altera absoluta in tertia figura. | 11 | |
| 31a18 Ἐν δὲ τῶι τελευταίωι σχήματι καθόλου μὲν ὄντων τῶν ὅρων πρὸς τὸ μέσον καὶ κατηγορικῶν ἀμφοτέρων τῶν προ τάσεων, ἐὰν ὁποτερονοῦν ἦι ἀναγκαῖον, καὶ τὸ συμπέρασμα ἔσται ἀναγκαῖον. ἐὰν δὲ τὸ μὲν ἦι στερητικὸν τὸ δὲ κατηγορικόν, ὅταν μὲν τὸ στερητικὸν ἀναγκαῖον ἦι, καὶ τὸ συμπέρασμα ἔσται ἀναγκαῖον, ὅταν δὲ τὸ κατηγορικόν, οὐκ ἔσται ἀναγκαῖον. ἔστωσαν γὰρ ἀμφότεραι κατηγορικαὶ πρῶτον αἱ προ τάσεις, καὶ τὸ Α καὶ τὸ Β παντὶ τῶι Γ ὑπαρχέτω, ἀναγ- καῖον δ᾽ ἔστω τὸ Α Γ. ἐπεὶ οὖν τὸ Β παντὶ τῶι Γ ὑπάρχει, καὶ τὸ Γ τινὶ τῶι Β ὑπάρξει διὰ τὸ ἀντιστρέφειν τὸ καθόλου τῶι κατὰ μέρος, ὥστ᾽ εἰ παντὶ τῶι Γ τὸ Α ἐξ ἀνάγκης ὑπάρχει καὶ τὸ Γ τῶι Β τινί, καὶ τῶι Β τινὶ ἀναγκαῖον ὑπάρχειν τὸ Α· τὸ γὰρ Β ὑπὸ τὸ Γ ἐστίν. γίγνεται οὖν τὸ πρῶτον σχῆμα. ὁμοίως δὲ δειχθήσεται καὶ εἰ τὸ Β Γ ἐστὶν ἀναγκαῖον· ἀντιστρέφει γὰρ τὸ Γ τῶι Α τινί, ὥστ᾽ εἰ παντὶ τῶι Γ τὸ Β ἐξ ἀνάγκης ὑπάρχει, καὶ τῶι Α τινὶ ὑπάρξει ἐξ ἀνάγκης. | (0650B) In postrema autem figura terminis quidem universalibus ad medium, et praedicativis utrisque propositionibus, si utralibet sit necessaria, et conclusio erit necessaria. Si autem haec quidem sit privativa, illa vero praedicativa, quando privativa quidem fuerit necessaria, et conclusio erit necessaria, quando autem praedicativa, non erit necessaria. Sint enim primum utraeque praedicativae propositiones, et A et B omni C insint, necessaria autem sit A C, quoniam ergo B omni C inest, et C alicui B inerit, eo quod convertitur universalis particulariter. Quare si A inest omni C ex necessitate, et C alicui B, et A alicui B necessarium inesse, nam B sub C est. Fit igitur prima figura. Similiter autem ostendetur, et si B C est necessaria, convertitur enim C alicui A, quare si omni C inest B ex necessitate, et A alicui B inerit ex necessitate. | In the last figure when the terms are related universally to the middle, and both premisses are affirmative, if one of the two is necessary, then the conclusion will be necessary. But if one is negative, the other affirmative, whenever the negative is necessary the conclusion also will be necessary, but whenever the affirmative is necessary the conclusion will not be necessary. First let both the premisses be affirmative, and let A and B belong to all C, and let AC be necessary. Since then B belongs to all C, C also will belong to some B, because the universal is convertible into the particular: consequently if A belongs necessarily to all C, and C belongs to some B, it is necessary that A should belong to some B also. For B is under C. The first figure then is formed. A similar proof will be given also if BC is necessary. For C is convertible with some A: consequently if B belongs necessarily to all C, it will belong necessarily also to some A. |
| Πάλιν ἔστω τὸ μὲν Α Γ στερητικόν, τὸ δὲ Β Γ καταφατικόν, ἀναγκαῖον δὲ τὸ στερητικόν. ἐπεὶ οὖν ἀντιστρέφει τινὶ τῶι Β τὸ Γ, τὸ δὲ Α οὐδενὶ τῶι Γ ἐξ ἀνάγκης, οὐδὲ τῶι Β τινὶ ὑπάρξει ἐξ ἀνάγκης τὸ Α· τὸ γὰρ Β ὑπὸ τὸ Γ ἐστίν. εἰ δὲ τὸ κατηγορικὸν ἀναγκαῖον, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον. ἔστω γὰρ τὸ Β Γ κατηγορικὸν καὶ ἀναγκαῖον, τὸ δὲ Α Γ στερητικὸν καὶ μὴ ἀναγ καῖον. ἐπεὶ οὖν ἀντιστρέφει τὸ καταφατικόν, ὑπάρξει καὶ τὸ Γ τινὶ τῶι Β ἐξ ἀνάγκης, ὥστ᾽ εἰ τὸ μὲν Α μηδενὶ τῶι Γ τὸ ↵ δὲ Γ τινὶ τῶι Β, τὸ Α τινὶ τῶι Β οὐχ ὑπάρξει· ἀλλ᾽ οὐκ ἐξ ἀνάγκης· δέδεικται γὰρ ἐν τῶι πρώτωι σχήματι ὅτι τῆς στερητικῆς προτάσεως μὴ ἀναγκαίας οὔσης οὐδὲ τὸ συμπέρασμα ἔσται ἀναγκαῖον. ἔτι κἂν διὰ τῶν ὅρων εἴη φανερόν. ἔστω γὰρ τὸ μὲν Α ἀγαθόν, τὸ δ᾽ ἐφ᾽ ὧι Β ζῶιον, τὸ δὲ Γ ἵππος. τὸ μὲν οὖν ἀγαθὸν ἐνδέχεται μηδενὶ ἵππωι ὑπάρχειν, τὸ δὲ ζῶιον ἀνάγκη παντὶ ὑπάρχειν· ἀλλ᾽ οὐκ ἀνάγκη ζῶιόν τι μὴ εἶναι ἀγαθόν, εἴπερ ἐνδέχεται πᾶν εἶναι ἀγαθόν. ἢ εἰ μὴ τοῦτο δυνατόν, ἀλλὰ τὸ ἐγρηγορέναι ἢ τὸ καθεύδειν ὅρον θετέον· ἅπαν γὰρ ζῶιον δεκτικὸν τούτων. | (0650C) Rursum sit A C quidem privativa, B C vero affirmativa, necessaria autem privativa, quoniam ergo convertitur affirmativa, erit C alicui B, A autem nulli C ex necessitate, neque A alicui B inerit ex necessitate, nam B sub C est. Si autem praedicativa sit necessaria, non erit conclusio necessaria. Sit enim B C praedicativa et necessaria, A C autem privativa et non necessaria, quoniam ergo convertitur affirmativa, inerit et C alicui B ex necessitate. Quare si A quidem nulli eorum quae sunt C inest, C autem alicui eorum quae sunt B et A alicui eorum quae sunt B non inerit, sed non ex necessitate. Ostensum est enim in prima figura quoniam privativa propositione necessaria, nec conclusio erit necessaria. (0650D) Amplius autem et per terminos sit manifestum, sit enim A quidem bonum in quo B animal, C autem equus, ergo bonum quidem contingit nulli equo inesse, animal vero necesse est omni equo inesse, sed non necesse est aliquod animal non esse bonum, siquidem contingit omne esse bonum. Aut si non hoc possibile, sed vigilare, vel dormire terminum ponendum. Omne enim animal susceptibile est horum. | Again let AC be negative, BC affirmative, and let the negative premiss be necessary. Since then C is convertible with some B, but A necessarily belongs to no C, A will necessarily not belong to some B either: for B is under C. But if the affirmative is necessary, the conclusion will not be necessary. For suppose BC is affirmative and necessary, while AC is negative and not necessary. Since then the affirmative is convertible, C also will belong to some B necessarily: consequently if A belongs to none of the Cs, while C belongs to some of the Bs, A will not belong to some of the Bs-but not of necessity; for it has been proved, in the case of the first figure, that if the negative premiss is not necessary, neither will the conclusion be necessary. Further, the point may be made clear by considering the terms. Let the term A be ‘good’, let that which B signifies be ‘animal’, let the term C be ‘horse’. It is possible then that the term good should belong to no horse, and it is necessary that the term animal should belong to every horse: but it is not necessary that some animal should not be good, since it is possible for every animal to be good. Or if that is not possible, take as the term ‘awake’ or ‘asleep’: for every animal can accept these. |
| Εἰ μὲν οὖν οἱ ὅροι καθόλου πρὸς τὸ μέσον εἰσίν, εἴρηται πότε ἔσται τὸ συμπέρασμα ἀναγκαῖον· εἰ δ᾽ ὁ μὲν καθόλου ὁ δ᾽ ἐν μέρει, κατηγορικῶν μὲν ὄντων ἀμφοτέρων, ὅταν τὸ καθόλου γένηται ἀναγκαῖον, καὶ τὸ συμπέρασμα ἔσται ἀναγ καῖον. ἀπόδειξις δ᾽ ἡ αὐτὴ ἣ καὶ πρότερον· ἀντιστρέφει γὰρ καὶ τὸ ἐν μέρει κατηγορικόν. εἰ οὖν ἀνάγκη τὸ Β παντὶ τῶι Γ ὑπάρχειν, τὸ δὲ Α ὑπὸ τὸ Γ ἐστίν, ἀνάγκη τὸ Β τινὶ τῶι Α ὑπάρχειν. εἰ δὲ τὸ Β τῶι Α τινί, καὶ τὸ Α τῶι Β τινὶ ὑπάρχειν ἀναγκαῖον· ἀντιστρέφει γάρ. | Si igitur termini universaliter ad medium sint, dictum est quando erit conclusio necessaria. Si autem hic quidem universalis, ille vero particularis, praedicativus uterque, quando universalis fuerit necessarius, et conclusio erit necessaria. Demonstratio autem eadem quae prius, convertitur enim et particularis affirmativa. Si ergo necesse est B omni C inesse, A autem sub C est, necesse est B alicui A inesse. Si autem B alicui A, et A alicui B inesse necessarium, convertitur enim. | If, then, the premisses are universal, we have stated when the conclusion will be necessary. But if one premiss is universal, the other particular, and if both are affirmative, whenever the universal is necessary the conclusion also must be necessary. The demonstration is the same as before; for the particular affirmative also is convertible. If then it is necessary that B should belong to all C, and A falls under C, it is necessary that B should belong to some A. But if B must belong to some A, then A must belong to some B: for conversion is possible. |
| ὁμοίως δὲ καὶ εἰ τὸ Α Γ εἴη ἀναγκαῖον καθόλου ὄν· τὸ γὰρ Β ὑπὸ τὸ Γ ἐστίν. εἰ δὲ τὸ ἐν μέρει ἐστὶν ἀναγκαῖον, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον. ἔστω γὰρ τὸ Β Γ ἐν μέρει τε καὶ ἀναγκαῖον, τὸ δὲ Α παντὶ τῶι Γ ὑπαρχέτω, μὴ μέντοι ἐξ ἀνάγκης. ἀντιστραφέντος οὖν τοῦ Β Γ τὸ πρῶτον γίγνεται σχῆμα, καὶ ἡ μὲν κα θόλου πρότασις οὐκ ἀναγκαία, ἡ δ᾽ ἐν μέρει ἀναγκαία. ὅτε δ᾽ οὕτως ἔχοιεν αἱ προτάσεις, οὐκ ἦν τὸ συμπέρασμα ἀναγκαῖον, ὥστ᾽ οὐδ᾽ ἐπὶ τούτων. | (0651A) Similiter autem et si A C sit necessaria universalis, nam B sub C est. Si autem particularis est necessaria, non erit conclusio necessaria. Sit enim B C particularis et necessaria, A autem insit omni C, non tamen ex necessitate, conversa ergo B C prima fit figura, et universalis quidem propositio non necessaria, particularis autem necessaria, quando autem sic se habebant propositiones, non erat conclusio necessaria, quare nec in his. | Similarly also if AC should be necessary and universal: for B falls under C. But if the particular premiss is necessary, the conclusion will not be necessary. Let the premiss BC be both particular and necessary, and let A belong to all C, not however necessarily. If the proposition BC is converted the first figure is formed, and the universal premiss is not necessary, but the particular is necessary. But when the premisses were thus, the conclusion (as we proved was not necessary: consequently it is not here either. |
| ἔτι δὲ καὶ ἐκ τῶν ὅρων φανερόν. ἔστω γὰρ τὸ μὲν Α ἐγρήγορσις, τὸ δὲ Β δίπουν, ἐφ᾽ ὧι δὲ τὸ Γ ζῶιον. τὸ μὲν οὖν Β τινὶ τῶι Γ ἀνάγκη ὑπάρχειν, τὸ δὲ Α τῶι Γ ἐνδέχεται, καὶ τὸ Α τῶι Β οὐκ ἀναγκαῖον· οὐ γὰρ ἀνάγκη δίπουν τι καθεύδειν ἢ ἐγρηγορέναι. ὁμοίως δὲ καὶ διὰ τῶν αὐτῶν ὅρων δειχθήσεται καὶ εἰ τὸ Α Γ εἴη ἐν μέρει τε καὶ ἀναγκαῖον. | Amplius autem et ex terminis manifestum. Sit enim A quidem vigilatio, B autem bipes, in quo autem C animal, ergo B alicui C necesse est inesse, A autem omni C contingit, et A non necessario B, non enim necesse est aliquem bipedem dormire vel vigilare. Similiter autem per eosdem terminos ostendetur etiam si A C sit particularis et necessaria. | Further, the point is clear if we look at the terms. Let A be waking, B biped, and C animal. It is necessary that B should belong to some C, but it is possible for A to belong to C, and that A should belong to B is not necessary. For there is no necessity that some biped should be asleep or awake. Similarly and by means of the same terms proof can be made, should the proposition AC be both particular and necessary. |
| Εἰ δ᾽ ὁ μὲν κατηγορικὸς ὁ δὲ στερητικὸς τῶν ὅρων, ὅταν μὲν ἦι τὸ καθόλου στερητικόν τε καὶ ἀναγκαῖον, καὶ τὸ συμπέρασμα ἔσται ἀναγκαῖον· εἰ γὰρ τὸ Α τῶι Γ μηδενὶ ἐνδέχεται, τὸ δὲ Β τινὶ τῶι Γ ὑπάρχει, τὸ Α τινὶ τῶι Β ἀνάγκη μὴ ὑπάρχειν. ὅταν δὲ τὸ καταφατικὸν ἀναγκαῖον τεθῆι, ἢ καθόλου ὂν ἢ ἐν μέρει, ἢ τὸ στερητικὸν κατὰ μέρος, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον. | (0651B) Si autem hic quidem terminorum sit praedicativus, ille privativus et necessarius, quando universalis fuerit privativus et necessarius, et conclusio erit necessaria. Si enim A nulli C ex necessitate contingit, B autem alicui C inest, necesse est A alicui B non inesse, quando autem affirmativa necessaria ponetur vel universalis, vel particularis, vel privativa particularis, non erit conclusio necessaria. | But if one premiss is affirmative, the other negative, whenever the universal is both negative and necessary the conclusion also will be necessary. For if it is not possible that A should belong to any C, but B belongs to some C, it is necessary that A should not belong to some B. But whenever the affirmative proposition is necessary, whether universal or particular, or the negative is particular, the conclusion will not be necessary. |
| τὰ μὲν γὰρ ἄλλα ταὐτὰ ἃ καὶ ἐπὶ τῶν πρότερον ἐροῦμεν, ὅροι δ᾽ ὅταν μὲν ἦι καθόλου τὸ κατηγορικὸν ἀναγκαῖον, ἐγρήγορσις – ζῶιον – ἄνθρωπος, μέσον ἄν ↵ θρωπος, ὅταν δ᾽ ἐν μέρει τὸ κατηγορικὸν ἀναγκαῖον, ἐγρήγορσις – ζῶιον – λευκόν· ζῶιον μὲν γὰρ ἀνάγκη τινὶ λευκῶι ὑπάρχειν, ἐγρήγορσις δ᾽ ἐνδέχεται μηδενί, καὶ οὐκ ἀνάγκη τινὶ ζώιωι μὴ ὑπάρχειν ἐγρήγορσιν. ὅταν δὲ τὸ στερητικὸν ἐν μέ ρει ὂν ἀναγκαῖον ἦι, δίπουν – κινούμενον – ζῶιον, μέσον ζῶιον. | Nam alia quidem eadem quae et in prioribus dicemus. Termini autem cum universalis quidem affirmativa est necessaria, vigilatio, animal, homo, medium homo: cum autem particularis praedicativa necessaria, vigilatio, animal, album. Animal enim necesse est alicui albo inesse, vigilatio autem contingit nulli, et non necesse est alicui animali non inesse vigilationem. (0651C) Quando autem privativa particularis est necessaria, bipes, motus, animal, medium animal. | The proof of this by reduction will be the same as before; but if terms are wanted, when the universal affirmative is necessary, take the terms ‘waking’-’animal’-’man’, ‘man’ being middle, and when the affirmative is particular and necessary, take the terms ‘waking’-’animal’-’white’: for it is necessary that animal should belong to some white thing, but it is possible that waking should belong to none, and it is not necessary that waking should not belong to some animal. But when the negative proposition being particular is necessary, take the terms ‘biped’, ‘moving’, ‘animal’, ‘animal’ being middle. |
Chapter 12
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| 12 | ||
| 32a6 Φανερὸν οὖν ὅτι τοῦ μὲν ὑπάρχειν οὐκ ἔστι συλλογισμός, ἐὰν μὴ ἀμφότεραι ὦσιν αἱ προτάσεις ἐν τῶι ὑπάρχειν, τοῦ δ᾽ ἀναγκαίου ἔστι καὶ τῆς ἑτέρας μόνον ἀναγκαίας οὔσης. ἐν ἀμφοτέροις δέ, καὶ καταφατικῶν καὶ στερητικῶν ὄντων τῶν συλλογισμῶν, ἀνάγκη τὴν ἑτέραν πρότασιν ὁμοίαν εἶναι τῶι συμπεράσματι. λέγω δὲ τὸ ὁμοίαν, εἰ μὲν ὑπάρχον, ὑπάρχουσαν, εἰ δ᾽ ἀναγκαῖον, ἀναγκαίαν. ὥστε καὶ τοῦτο δῆλον, ὅτι οὐκ ἔσται τὸ συμπέρασμα οὔτ᾽ ἀναγκαῖον οὔθ᾽ ὑπάρχον εἶναι μὴ ληφθείσης ἀναγκαίας ἢ ὑπαρχούσης προτάσεως. | Manifestum igitur quoniam inesse quidem non est syllogismus, si utraeque propositiones non sunt in eo quod est inesse, necessaria vero est, et altera solum existente necessaria. In utrisque autem affirmativis et privativis existentibus syllogismis necesse est alteram propositionem similem esse conclusioni. Dico autem similem, si inesse quidem, inexistentem, si autem necessaria, necessariam. Quare et hoc palam, quoniam non erit conclusio neque necessaria, neque inesse, non sumpta vel necessaria, vel quae inesse significet propositione. | It is clear then that a simple conclusion is not reached unless both premisses are simple assertions, but a necessary conclusion is possible although one only of the premisses is necessary. But in both cases, whether the syllogisms are affirmative or negative, it is necessary that one premiss should be similar to the conclusion. I mean by ‘similar’, if the conclusion is a simple assertion, the premiss must be simple; if the conclusion is necessary, the premiss must be necessary. Consequently this also is clear, that the conclusion will be neither necessary nor simple unless a necessary or simple premiss is assumed. |
| Περὶ μὲν οὖν τοῦ ἀναγκαίου, πῶς γίγνεται καὶ τίνα διαφορὰν ἔχει πρὸς τὸ ὑπάρχον, εἴρηται σχεδὸν ἱκανῶς· | Igitur de necessario quomodo fit, et quam differentiam habeat ad inesse, sufficienter pene dictum est. | Perhaps enough has been said about the proof of necessity, how it comes about and how it differs from the proof of a simple statement. |
Chapter 13
| Greek | Latin | English |
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| (PL 64 0651C) CAPUT XII. De contingenti non necessario. | 13 | |
| 32a16 περὶ δὲ τοῦ ἐνδεχομένου μετὰ ταῦτα λέγωμεν πότε καὶ πῶς καὶ διὰ τίνων ἔσται συλλογισμός. λέγω δ᾽ ἐνδέχεσθαι καὶ τὸ ἐνδεχόμενον, οὗ μὴ ὄντος ἀναγκαίου, τεθέντος δ᾽ ὑπάρχειν, οὐδὲν ἔσται διὰ τοῦτ᾽ ἀδύνατον· τὸ γὰρ ἀναγκαῖον ὁμωνύμως ἐνδέχεσθαι λέγομεν. [ὅτι δὲ τοῦτ᾽ ἔστι τὸ ἐνδεχόμενον, φανερὸν ἔκ τε τῶν ἀποφάσεων καὶ τῶν καταφάσεων τῶν ἀντικειμένων· | (0651D) De contingente autem post haec dicemus, quando, et quomodo, et per quae erit syllogismus. Dico autem contingere, et contingens, quo non existente necessario, posito autem inesse, nihil erit propter hoc impossibile. Nam necessarium aequivoce contingere dicitur. Quoniam autem hoc est contingens, manifestum ex affirmationibus et negationibus oppositis. | We proceed to discuss that which is possible, when and how and by what means it can be proved. I use the terms ‘to be possible’ and ‘the possible’ of that which is not necessary but, being assumed, results in nothing impossible. We say indeed ambiguously of the necessary that it is possible. But that my definition of the possible is correct is clear from the phrases by which we deny or on the contrary affirm possibility. |
| τὸ γὰρ οὐκ ἐνδέχεται ὑπάρχειν καὶ ἀδύνατον ὑπάρχειν καὶ ἀνάγκη μὴ ὑπάρχειν ἤτοι ταὐτά ἐστιν ἢ ἀκολουθεῖ ἀλλήλοις, ὥστε καὶ τὰ ἀντικείμενα, τὸ ἐνδέχεται ὑπάρχειν καὶ οὐκ ἀδύνατον ὑπάρχειν καὶ οὐκ ἀνάγκη μὴ ὑπάρχειν, ἤτοι ταὐτὰ ἔσται ἢ ἀκολουθοῦντα ἀλλήλοις· κατὰ παντὸς γὰρ ἡ φάσις ἢ ἡ ἀπόφασις.
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(0652A) Nam non contingit esse, non possibile esse, et impossibile esse, et necesse est non esse, vel eadem sunt, vel sequuntur se invicem, quare et opposito his contingit esse, et non impossibile esse, et non necesse non esse, eadem erunt, vel sequentia se invicem. De omni enim affirmatio, vel negatio vera. | For the expressions ‘it is not possible to belong’, ‘it is impossible to belong’, and ‘it is necessary not to belong’ are either identical or follow from one another; consequently their opposites also, ‘it is possible to belong’, ‘it is not impossible to belong’, and ‘it is not necessary not to belong’, will either be identical or follow from one another. For of everything the affirmation or the denial holds good. |
| ἔσται ἄρα τὸ ἐνδεχόμενον οὐκ ἀναγκαῖον καὶ τὸ μὴ ἀναγκαῖον ἐνδεχόμενον.] συμβαίνει δὲ πάσας τὰς κατὰ τὸ ἐνδέχεσθαι προτάσεις ἀντιστρέφειν ἀλλήλαις. | Erit ergo contingens necessarium, et non necessarium contingens. Accidit autem omnes quae secundum contingere sunt propositiones converti sibi invicem, | That which is possible then will be not necessary and that which is not necessary will be possible. It results that all premisses in the mode of possibility are convertible into one another. |
| λέγω δὲ οὐ τὰς καταφατικὰς ταῖς ἀποφατικαῖς, ἀλλ᾽ ὅσαι καταφατικὸν ἔχουσι τὸ σχῆμα κατὰ τὴν ἀντίθεσιν, οἷον τὸ ἐνδέχεσθαι ὑπάρχειν τῶι ἐνδέχεσθαι μὴ ὑπάρχειν, καὶ τὸ παντὶ ἐνδέχεσθαι τῶι ἐνδέχεσθαι μηδενὶ καὶ μὴ παντί, καὶ τὸ τινὶ τῶι μὴ τινί. τὸν αὐτὸν δὲ τρόπον καὶ ἐπὶ τῶν ἄλλων. | dico autem non affirmativas negativis sed quaecunque affirmativam habent figuram secundum oppositionem, ut ea quae est contingit esse ei quae est contingit non esse, et ea quae est contingit omni ei quae est contingit nulli, vel non omni, et quae alicui, et quae non alicui, eodem autem modo et in aliis.
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I mean not that the affirmative are convertible into the negative, but that those which are affirmative in form admit of conversion by opposition, e.g. ‘it is possible to belong’ may be converted into ‘it is possible not to belong’, and ‘it is possible for A to belong to all B’ into ‘it is possible for A to belong to no B’ or ‘not to all B’, and ‘it is possible for A to belong to some B’ into ‘it is possible for A not to belong to some B’. And similarly the other propositions in this mode can be converted. |
| ἐπεὶ γὰρ τὸ ἐνδεχόμενον οὐκ ἔστιν ἀναγκαῖον, τὸ δὲ μὴ ἀναγκαῖον ἐγχωρεῖ μὴ ὑπάρχειν, φανερὸν ὅτι, εἰ ἐνδέχεται τὸ Α τῶι Β ὑπάρχειν, ἐνδέχεται καὶ μὴ ὑπάρχειν· καὶ εἰ παντὶ ἐνδέχεται ὑπάρχειν, καὶ παντὶ ἐνδέχεται μὴ ὑπάρχειν. | (0652B) Quoniam enim quod est contingens non est necessarium, et quod non est necessarium possibile est non esse, manifestum quoniam si contingit A inesse B, contingit et non inesse, et si omni contingit inesse, et omni contingit non inesse. | For since that which is possible is not necessary, and that which is not necessary may possibly not belong, it is clear that if it is possible that A should belong to B, it is possible also that it should not belong to B: and if it is possible that it should belong to all, it is also possible that it should not belong to all. |
| ὁμοίως δὲ κἀπὶ τῶν ἐν μέρει καταφάσεων· ἡ γὰρ αὐτὴ ↵ ἀπόδειξις. εἰσὶ δ᾽ αἱ τοιαῦται προτάσεις κατηγορικαὶ καὶ οὐ στερητικαί· τὸ γὰρ ἐνδέχεσθαι τῶι εἶναι ὁμοίως τάττεται, καθάπερ ἐλέχθη πρότερον. | Similiter autem et in particularibus affirmationibus, nam eadem demonstratio. Sunt autem huiusmodi propositiones praedicativae, nam contingere ei quod est esse similiter ponitur, quemadmodum dictum est prius. | The same holds good in the case of particular affirmations: for the proof is identical. And such premisses are affirmative and not negative; for ‘to be possible’ is in the same rank as ‘to be’, as was said above. |
| Διωρισμένων δὲ τούτων πάλιν λέγωμεν ὅτι τὸ ἐνδέχε σθαι κατὰ δύο λέγεται τρόπους, ἕνα μὲν τὸ ὡς ἐπὶ τὸ πολὺ γίνεσθαι καὶ διαλείπειν τὸ ἀναγκαῖον, οἷον τὸ πολιοῦσθαι ἄνθρωπον ἢ τὸ αὐξάνεσθαι ἢ φθίνειν, ἢ ὅλως τὸ πεφυκὸς ὑπάρχειν (τοῦτο γὰρ οὐ συνεχὲς μὲν ἔχει τὸ ἀναγκαῖον διὰ τὸ μὴ ἀεὶ εἶναι ἄνθρωπον, ὄντος μέντοι ἀνθρώπου ἢ ἐξ ἀνάγκης ἢ ὡς ἐπὶ τὸ πολύ ἐστιν), ἄλλον δὲ τὸ ἀόριστον, ὁ καὶ οὕτως καὶ μὴ οὕτως δυνατόν, οἷον τὸ βαδίζειν ζῶιον ἢ βαδίζοντος γενέσθαι σεισμόν, ἢ ὅλως τὸ ἀπὸ τύχης γινόμενον· οὐδὲν γὰρ μᾶλλον οὕτως πέφυκεν ἢ ἐναντίως. | Determinatis autem his, rursum dicimus quoniam contingere duobus modis dicitur: uno quidem, quod plerumque fit et deficit, necessarium, ut canescere hominem, vel augeri, vel minui, vel omnino quod natum est esse. Hoc enim non continuum habet necessarium, eo quod non semper est homo, cum tamen homo est, aut ex necessitate, aut ut in pluribus est. (0652C) Alio autem modo infinitum, quod et sic, et non sic possibile, ut animal ambulare, vel ambulante fieri motum terrae, vel omnino quod casu fit, nihil enim magis sic natum est, vel econtrario. | Having made these distinctions we next point out that the expression ‘to be possible’ is used in two ways. In one it means to happen generally and fall short of necessity, e.g. man’s turning grey or growing or decaying, or generally what naturally belongs to a thing (for this has not its necessity unbroken, since man’s existence is not continuous for ever, although if a man does exist, it comes about either necessarily or generally). In another sense the expression means the indefinite, which can be both thus and not thus, e.g. an animal’s walking or an earthquake’s taking place while it is walking, or generally what happens by chance: for none of these inclines by nature in the one way more than in the opposite. |
| ἀντιστρέφει μὲν οὖν καὶ κατὰ τὰς ἀντικειμένας προτάσεις ἑκάτερον τῶν ἐνδεχομένων, οὐ μὴν τὸν αὐτόν γε τρόπον, ἀλλὰ τὸ μὲν πεφυκὸς εἶναι τῶι μὴ ἐξ ἀνάγκης ὑπάρχειν (οὕτω γὰρ ἐνδέχεται μὴ πολιοῦσθαι ἄνθρωπον), τὸ δ᾽ ἀόριστον τῶι μηδὲν μᾶλλον οὕτως ἢ ἐκείνως. ἐπιστήμη δὲ καὶ συλλογισμὸς ἀποδεικτικὸς τῶν μὲν ἀορίστων οὐκ ἔστι διὰ τὸ ἄτακτον εἶναι τὸ μέσον, τῶν δὲ πεφυκότων ἔστι, καὶ σχεδὸν οἱ λόγοι καὶ αἱ σκέψεις γίνονται περὶ τῶν οὕτως ἐνδεχομένων· ἐκείνων δ᾽ ἐγχωρεῖ μὲν γενέσθαι συλλογισμόν, οὐ μὴν εἴωθέ γε ζητεῖσθαι. | Convertitur ergo et secundum oppositas propositiones utrumque contingens, non tamen eodem modo, sed quod natum quidem est esse ei quod non ex necessitate esse. Sic enim contingit non canescere hominem. Infinitum autem ei quod nihil magis sic, vel illo modo. Disciplina autem, et syllogismus demonstrativus, ex infinitis quidem non est, eo quod inordinatum est medium, ex iis vero quae nata sunt esse, pene orationes et considerationes fiunt de sic contingentibus, ex illis autem possibile quidem est fieri syllogismum, non tamen solet quaeri. | That which is possible in each of its two senses is convertible into its opposite, not however in the same way: but what is natural is convertible because it does not necessarily belong (for in this sense it is possible that a man should not grow grey) and what is indefinite is convertible because it inclines this way no more than that. Science and demonstrative syllogism are not concerned with things which are indefinite, because the middle term is uncertain; but they are concerned with things that are natural, and as a rule arguments and inquiries are made about things which are possible in this sense. Syllogisms indeed can be made about the former, but it is unusual at any rate to inquire about them. |
| Ταῦτα μὲν οὖν διορισθήσεται μᾶλλον ἐν τοῖς ἑπομένοις· νῦν δὲ λέγωμεν πότε καὶ πῶς καὶ τίς ἔσται συλλογισμὸς ἐκ τῶν ἐνδε χομένων προτάσεων. ἐπεὶ δὲ τὸ ἐνδέχεσθαι τόδε τῶιδε ὑπάρχειν διχῶς ἔστιν ἐκλαβεῖν· ἢ γὰρ ὧι ὑπάρχει τόδε ἢ ὧι ἐνδέχεται αὐτὸ ὑπάρχειν – τὸ γάρ, καθ᾽ οὗ τὸ Β, τὸ Α ἐνδέχεσθαι τούτων σημαίνει θάτερον, ἢ καθ᾽ οὗ λέγεται τὸ Β ἢ καθ᾽ οὗ ἐνδέχεται λέγεσθαι· τὸ δέ, καθ᾽ οὗ τὸ Β, τὸ Α ἐνδέχεσθαι ἢ παντὶ τῶι Β τὸ Α ἐγχωρεῖν οὐδὲν διαφέρει – | (0652D) Haec ergo definientur magis in sequentibus, nunc autem dicemus quando et quomodo, et quis erit syllogismus ex contingentibus propositionibus. Quoniam autem contingere hoc huic inesse dupliciter est accipere, aut enim cui inest hoc, aut cui contingit ipsum inesse, nam de quo B, A contingere, horum alterum significat, aut de quo dicitur B, aut de quo contingit dici, de quo autem B, A contingere, aut omni B possibile inesse A, nihil differt. | These matters will be treated more definitely in the sequel; our business at present is to state the moods and nature of the syllogism made from possible premisses. The expression ‘it is possible for this to belong to that’ may be understood in two senses: ‘that’ may mean either that to which ‘that’ belongs or that to which it may belong; for the expression ‘A is possible of the subject of B’ means that it is possible either of that of which B is stated or of that of which B may possibly be stated. It makes no difference whether we say, A is possible of the subject of B, or all B admits of A. |
| φανερὸν ὅτι διχῶς ἂν λέγοιτο τὸ Α τῶι Β παντὶ ἐνδέχεσθαι ὑπάρχειν. πρῶτον οὖν εἴπωμεν, εἰ καθ᾽ οὗ τὸ Γ τὸ Β ἐνδέχεται, καὶ καθ᾽ οὗ τὸ Β τὸ Α, τίς ἔσται καὶ ποῖος συλλογισμός· οὕτω γὰρ αἱ προτάσεις ἀμφότεραι λαμβάνονται κατὰ τὸ ἐνδέχεσθαι, ὅταν δὲ καθ᾽ οὗ τὸ Β ὑπάρχει τὸ Α ἐνδέχηται, ἡ μὲν ὑπάρχουσα ἡ δ᾽ ἐνδεχομένη. ὥστ᾽ ἀπὸ τῶν ὁμοιοσχημόνων ἀρκτέον, καθάπερ καὶ ἐν τοῖς ἄλλοις. | Manifestum igitur quoniam dupliciter dicetur A omni B inesse contingere. Primum ergo dicemus si de quo C contingit B, et de quo B contingit A, quis erit, et qualis syllogismus, sic enim utraeque propositiones sumuntur secundum contingere, quando autem de quo B est contingit A, haec quidem inesse, illa vero contingens, quare A similibus figuris incipiendum, quemadmodum et in aliis. | It is clear then that the expression ‘A may possibly belong to all B’ might be used in two senses. First then we must state the nature and characteristics of the syllogism which arises if B is possible of the subject of C, and A is possible of the subject of B. For thus both premisses are assumed in the mode of possibility; but whenever A is possible of that of which B is true, one premiss is a simple assertion, the other a problematic. Consequently we must start from premisses which are similar in form, as in the other cases. |
Chapter 14
| Greek | Latin | English |
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| CAPUT XIII. De syllogismis ex ambabus contingentibus in prima figura. | 14 | |
| 32b38 Ὅταν οὖν τὸ Α παντὶ τῶι Β ἐνδέχηται καὶ τὸ Β παντὶ τῶι Γ, συλλογισμὸς ἔσται τέλειος ὅτι τὸ Α παντὶ τῶι Γ ἐνδέχεται ὑπάρχειν. τοῦτο δὲ φανερὸν ἐκ τοῦ ὁρισμοῦ· τὸ γὰρ ↵ ἐνδέχεσθαι παντὶ ὑπάρχειν οὕτως ἐλέγομεν. ὁμοίως δὲ καὶ εἰ τὸ μὲν Α ἐνδέχεται μηδενὶ τῶι Β, τὸ δὲ Β παντὶ τῶι Γ, ὅτι τὸ Α ἐνδέχεται μηδενὶ τῶι Γ· | Quando ergo A contingit omni B, et B omni C, syllogismus erit perfectus, quoniam A contingit omni C inesse. Hoc autem manifestum est ex definitione, nam contingere omni inesse sic dicebamus. Similiter autem et si A quidem contingit nulli B, B autem omni C, quoniam A contingit nulli C. | Whenever A may possibly belong to all B, and B to all C, there will be a perfect syllogism to prove that A may possibly belong to all C. This is clear from the definition: for it was in this way that we explained ‘to be possible for one term to belong to all of another’. Similarly if it is possible for A to belong no B, and for B to belong to all C, then it is possible for A to belong to no C. |
| τὸ γὰρ καθ᾽ οὗ τὸ Β ἐνδέχεται, τὸ Α μὴ ἐνδέχεσθαι, τοῦτ᾽ ἦν τὸ μηδὲν ἀπολείπειν τῶν ὑπὸ τὸ Β ἐνδεχομένων. ὅταν δὲ τὸ Α παντὶ τῶι Β ἐνδέχηται, τὸ δὲ Β ἐνδέχηται μηδενὶ τῶι Γ, διὰ μὲν τῶν εἰλημμένων προτάσεων οὐδεὶς γίνεται συλλογισμός, ἀντιστραφείσης δὲ τῆς Β Γ κατὰ τὸ ἐνδέχεσθαι γίνεται ὁ αὐτὸς ὅσπερ πρότερον. ἐπεὶ γὰρ ἐνδέχεται τὸ Β μηδενὶ τῶι Γ ὑπάρ χειν, ἐνδέχεται καὶ παντὶ ὑπάρχειν· τοῦτο δ᾽ εἴρηται πρότερον. ὥστ᾽ εἰ τὸ μὲν Β παντὶ τῶι Γ, τὸ δ᾽ Α παντὶ τῶι Β, πάλιν ὁ αὐτὸς γίνεται συλλογισμός. | Nam de quo B contingit, A non contingere, hoc erat nullum dimittere sub B contingentium. (0653B) Quando autem A contingit omni B, B autem nulli C, per sumptas quidem propositiones nullus fit syllogismus, conversa autem B C secundum contingere, fit idem quemadmodum et prius, quoniam enim contingit B nulli C inesse, contingit et omni inesse. Hoc autem dictum prius. Quare si B quidem omni C, A autem omni B, rursum idem fit syllogismus. | For the statement that it is possible for A not to belong to that of which B may be true means (as we saw) that none of those things which can possibly fall under the term B is left out of account. But whenever A may belong to all B, and B may belong to no C, then indeed no syllogism results from the premisses assumed, but if the premiss BC is converted after the manner of problematic propositions, the same syllogism results as before. For since it is possible that B should belong to no C, it is possible also that it should belong to all C. This has been stated above. Consequently if B is possible for all C, and A is possible for all B, the same syllogism again results. |
| ὁμοίως δὲ καὶ εἰ πρὸς ἀμφοτέρας τὰς προτάσεις ἡ ἀπόφασις τεθείη μετὰ τοῦ ἐνδέχεσθαι. λέγω δ᾽ οἷον εἰ τὸ Α ἐνδέχεται μηδενὶ τῶι Β καὶ τὸ Β μηδενὶ τῶι Γ· διὰ μὲν γὰρ τῶν εἰλημμένων προτάσεων οὐδεὶς γίνεται συλλογισμός, ἀντιστρεφομένων δὲ πάλιν ὁ αὐτὸς ἔσται ὅσπερ καὶ πρότερον. φανερὸν οὖν ὅτι τῆς ἀποφάσεως τιθεμένης πρὸς τὸ ἔλαττον ἄκρον ἢ πρὸς ἀμφοτέρας τὰς προτάσεις ἢ οὐ γίνεται συλλογισμὸς ἢ γίνεται μὲν ἀλλ᾽ οὐ τέλειος· ἐκ γὰρ τῆς ἀντιστροφῆς περαίνεται τὸ ἀναγκαῖον. | Similiter autem etsi ad utrasque propositiones negatio ponatur cum contingere (dico autem ut si A contingit nulli B, et B nulli C ), igitur per sumptas quidem propositiones nullus fit syllogismus, conversis autem rursus idem erit qui et prius. Manifestum est igitur quoniam negatione posita ad minorem extremitatem, vel ad utrasque propositiones, aut non fit syllogismus, aut fit quidem, sed non perfectus, ex conversione enim fit necessarium. | Similarly if in both the premisses the negative is joined with ‘it is possible’: e.g. if A may belong to none of the Bs, and B to none of the Cs. No syllogism results from the assumed premisses, but if they are converted we shall have the same syllogism as before. It is clear then that if the minor premiss is negative, or if both premisses are negative, either no syllogism results, or if one it is not perfect. For the necessity results from the conversion. |
| Ἐὰν δ᾽ ἡ μὲν καθόλου τῶν προτάσεων ἡ δ᾽ ἐν μέρει ληφθῆι, πρὸς μὲν τὸ μεῖζον ἄκρον κειμένης τῆς καθόλου συλλογισμὸς ἔσται [τέλειοσ]. εἰ γὰρ τὸ Α παντὶ τῶι Β ἐνδέχεται, τὸ δὲ Β τινὶ τῶι Γ, τὸ Α τινὶ τῶι Γ ἐνδέχεται. τοῦτο δὲ φανερὸν ἐκ τοῦ ὁρισμοῦ τοῦ ἐνδέχεσθαι. πάλιν εἰ τὸ Α ἐνδέχεται μηδενὶ τῶι Β, τὸ δὲ Β τινὶ τῶι Γ ἐνδέχεται ὑπάρχειν, ἀνάγκη τὸ Α ἐνδέχεσθαί τινι τῶν Γ μὴ ὑπάρχειν. ἀπόδειξις δ᾽ ἡ αὐτή. ἐὰν δὲ στερητικὴ ληφθῆι ἡ ἐν μέρει πρότασις, ἡ δὲ καθόλου καταφατική, τῆι δὲ θέσει ὁμοίως ἔχωσιν (οἷον τὸ μὲν Α παντὶ τῶι Β ἐνδέ χεται, τὸ δὲ Β τινὶ τῶι Γ ἐνδέχεται μὴ ὑπάρχειν), διὰ μὲν τῶν εἰλημμένων προτάσεων οὐ γίνεται φανερὸς συλλογισμός, ἀντιστραφείσης δὲ τῆς ἐν μέρει καὶ τεθέντος τοῦ Β τινὶ τῶι Γ ἐνδέχεσθαι ὑπάρχειν τὸ αὐτὸ ἔσται συμπέρασμα ὁ καὶ πρότερον, καθάπερ ἐν τοῖς ἐξ ἀρχῆς. | Si autem haec quidem propositionum universalis, illa vero particularis sumatur, ad maiorem quidem extremitatem posita universali, syllogismus erit perfectus. (0653C) Nam si A omni B contingit, B autem alicui C, A alicui C contingit, hoc autem manifestum ex definitione contingentis. Rursum si A contingit nulli B, B autem contingit alicui C inesse, necesse est A contingere alicui C non inesse. Demonstratio autem eadem quae in his. Si autem privativa sumatur particularis propositio, universalis autem affirmativa, positione autem similiter se habeant (ut A quidem omni B contingat, B autem alicui C contingat non inesse), per sumptas quidem propositiones non fit manifestus syllogismus, conversa autem particulari, et posito B alicui C contingere inesse, eadem erit conclusio quae et prius, quemadmodum in iis quae ex principio. | But if one of the premisses is universal, the other particular, when the major premiss is universal there will be a perfect syllogism. For if A is possible for all B, and B for some C, then A is possible for some C. This is clear from the definition of being possible. Again if A may belong to no B, and B may belong to some of the Cs, it is necessary that A may possibly not belong to some of the Cs. The proof is the same as above. But if the particular premiss is negative, and the universal is affirmative, the major still being universal and the minor particular, e.g. A is possible for all B, B may possibly not belong to some C, then a clear syllogism does not result from the assumed premisses, but if the particular premiss is converted and it is laid down that B possibly may belong to some C, we shall have the same conclusion as before, as in the cases given at the beginning. |
| Ἐὰν δ᾽ ἡ πρὸς τὸ μεῖζον ἄκρον ἐν μέρει ληφθῆι, ἡ δὲ πρὸς τὸ ἔλαττον καθόλου, ἐάν τ᾽ ἀμφότεραι καταφατικαὶ τεθῶσιν ἐάν τε στερητικαὶ ἐάν τε μὴ ὁμοιοσχήμονες, ἐάν τ᾽ ἀμφότεραι ἀδιόριστοι ἢ κατὰ μέρος, οὐδαμῶς ἔσται συλλογισμός· οὐδὲν γὰρ κωλύει τὸ Β ὑπερτείνειν τοῦ Α καὶ μὴ κατηγορεῖσθαι ἐπ᾽ ἴσων· ὧι δ᾽ ὑπερτείνει τὸ Β τοῦ Α, εἰλήφθω τὸ Γ· τούτωι γὰρ οὔτε παντὶ ↵ οὔτε μηδενὶ οὔτε τινὶ οὔτε μή τινι ἐνδέχεται τὸ Α ὑπάρχειν, εἴπερ ἀντιστρέφουσιν αἱ κατὰ τὸ ἐνδέχεσθαι προτάσεις καὶ τὸ Β πλείοσιν ἐνδέχεται ἢ τὸ Α ὑπάρχειν.
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(0653D) Si autem quae ad maiorem extremitatem particularis sumatur, quae ad minorem universalis, sive utraeque sumantur affirmativae, sive privativae, sive non similis figurae, sive utraeque indefinitae, vel particulares, nullo modo erit syllogismus. Nihil enim prohibet B transcendere A, et non praedicari de aequis, in quo enim B transcendit A sumat C, huic neque omni, neque nulli, neque alicui, neque non alicui contingit A inesse, siquidem convertuntur secundum contingere propositiones, et B pluribus contingit quam A inesse. | But if the major premiss is the minor universal, whether both are affirmative, or negative, or different in quality, or if both are indefinite or particular, in no way will a syllogism be possible. For nothing prevents B from reaching beyond A, so that as predicates cover unequal areas. Let C be that by which B extends beyond A. To C it is not possible that A should belong-either to all or to none or to some or not to some, since premisses in the mode of possibility are convertible and it is possible for B to belong to more things than A can.
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| ἔτι δὲ καὶ ἐκ τῶν ὅρων φανερόν· οὕτω γὰρ ἐχουσῶν τῶν προτάσεων τὸ πρῶτον τῶι ἐσχάτωι καὶ οὐδενὶ ἐνδέχεται καὶ παντὶ ὑπάρχειν ἀναγκαῖον. ὅροι δὲ κοινοὶ πάντων τοῦ μὲν ὑπάρχειν ἐξ ἀνάγκης ζῶιον – λευκόν – ἄνθρωπος, τοῦ δὲ μὴ ἐνδέχεσθαι ζῶιον – λευκόν – ἱμάτιον. | Amplius autem ex terminis manifestum est, nam sic se habentibus propositionibus primum postremo et nulli contingit, et omni ex necessitate inesse. Termini autem communes omnium, inesse quidem ex necessitate, animal, album, homo, non contingere vero, animal, album, vestis. | Further, this is obvious if we take terms; for if the premisses are as assumed, the major term is both possible for none of the minor and must belong to all of it. Take as terms common to all the cases under consideration ‘animal’-’white’-’man’, where the major belongs necessarily to the minor; ‘animal’-’white’-’garment’, where it is not possible that the major should belong to the minor. |
| φανερὸν οὖν τοῦτον τὸν τρόπον ἐχόντων τῶν ὅρων ὅτι οὐδεὶς γίνεται συλλογισμός. ἢ γὰρ τοῦ ὑπάρχειν ἢ τοῦ ἐξ ἀνάγκης ἢ τοῦ ἐνδέχεσθαι πᾶς ἐστὶ συλλογισμός. τοῦ μὲν οὖν ὑπάρχειν καὶ τοῦ ἀναγκαίου φανερὸν ὅτι οὐκ ἔστιν· ὁ μὲν γὰρ καταφατικὸς ἀναιρεῖται τῶι στερητικῶι, ὁ δὲ στερητικὸς τῶι καταφατικῶι. λείπεται δὴ τοῦ ἐνδέχεσθαι εἶναι· τοῦτο δ᾽ ἀδύνατον· δέδεικται γὰρ ὅτι οὕτως ἐχόντων τῶν ὅρων καὶ παντὶ τῶι ἐσχάτωι τὸ πρῶτον ἀνάγκη καὶ οὐδενὶ ἐνδέχεται ὑπάρχειν. ὥστ᾽ οὐκ ἂν εἴη τοῦ ἐνδέχεσθαι συλλογισμός· τὸ γὰρ ἀναγκαῖον οὐκ ἦν ἐνδεχόμενον. | (0654A) Manifestum igitur quoniam hoc modo habentibus se terminis, nullus fit syllogismus, nam omnis syllogismus vel eius quod est inesse est, vel ex necessitate vel contingere, non est autem eius quod est inesse, neque necessarii, manifestum quoniam non est, nam affirmativus interimitur privativo, et privativus affirmativo, relinquitur ergo eius quod contingere esse, hoc autem impossibile. Ostensum est enim quoniam sic se habentibus terminis, et omni postremo primum necesse inesse, et nulli contingere inesse, quare non erit eius quod est contingere syllogismus, nam necessarium uno [sic] erat contingens. | It is clear then that if the terms are related in this manner, no syllogism results. For every syllogism proves that something belongs either simply or necessarily or possibly. It is clear that there is no proof of the first or of the second. For the affirmative is destroyed by the negative, and the negative by the affirmative. There remains the proof of possibility. But this is impossible. For it has been proved that if the terms are related in this manner it is both necessary that the major should belong to all the minor and not possible that it should belong to any. Consequently there cannot be a syllogism to prove the possibility; for the necessary (as we stated) is not possible. |
| Φανερὸν δὲ ὅτι καθόλου τῶν ὅρων ὄντων ἐν ταῖς ἐνδεχομέναις προτάσεσιν ἀεὶ γίνεται συλλογισμὸς ἐν τῶι πρώ τωι σχήματι, καὶ κατηγορικῶν καὶ στερητικῶν ὄντων, πλὴν κατηγορικῶν μὲν τέλειος, στερητικῶν δὲ ἀτελής. δεῖ δὲ τὸ ἐνδέχεσθαι λαμβάνειν μὴ ἐν τοῖς ἀναγκαίοις, ἀλλὰ κατὰ τὸν εἰρημένον διορισμόν. ἐνίοτε δὲ λανθάνει τὸ τοιοῦτον. | Manifestum autem et quoniam cum universales sunt termini in contingentibus propositionibus, semper fit syllogismus in prima figura, sive sunt praedicativi, sive privativi. (0654B) Verum ex praedicativis quidem perfectus, ex privativis autem imperfectus. Oportet autem contingere sumere non in necessariis, sed secundum dictam definitionem, aliquoties autem latet huiusmodi. | It is clear that if the terms are universal in possible premisses a syllogism always results in the first figure, whether they are affirmative or negative, only a perfect syllogism results in the first case, an imperfect in the second. But possibility must be understood according to the definition laid down, not as covering necessity. This is sometimes forgotten. |
Chapter 15
| Greek | Latin | English |
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| (PL 64 0654B) CAPUT XIV. De syllogismis ex una absoluta et altera contingente in prima figura. | 15 | |
| 33b25 Ἐὰν δ᾽ ἡ μὲν ὑπάρχειν ἡ δ᾽ ἐνδέχεσθαι λαμβάνηται τῶν προτάσεων, ὅταν μὲν ἡ πρὸς τὸ μεῖζον ἄκρον ἐνδέχεσθαι σημαίνηι, τέλειοί τ᾽ ἔσονται πάντες οἱ συλλογισμοὶ καὶ τοῦ ἐνδέχεσθαι κατὰ τὸν εἰρημένον διορισμόν, ὅταν δ᾽ ἡ πρὸς τὸ ἔλαττον, ἀτελεῖς τε πάντες, καὶ οἱ στερητικοὶ τῶν συλλογισ μῶν οὐ τοῦ κατὰ τὸν διορισμὸν ἐνδεχομένου, ἀλλὰ τοῦ μηδενὶ ἢ μὴ παντὶ ἐξ ἀνάγκης ὑπάρχειν· εἰ γὰρ μηδενὶ ἢ μὴ παντὶ ἐξ ἀνάγκης, ἐνδέχεσθαί φαμεν καὶ μηδενὶ καὶ μὴ παντὶ ὑπάρχειν. | (0654C) Si autem haec quidem inesse, illa vero contingere sumatur propositionum, quando quae ad maiorem quidem extremitatem contingere significaverit perfecti erunt omnes syllogismi, et contingentis secundum dictam determinationem, quando autem quae ad minorem, et imperfecti omnes, et privativi syllogismi, non contingentis secundum dictam determinationem, sed eius quod est nulli, aut non omni ex necessitate inesse. Si enim nulli, aut non omni ex necessitate contingere dicimus, et nulli, et non omni inesse. | If one premiss is a simple proposition, the other a problematic, whenever the major premiss indicates possibility all the syllogisms will be perfect and establish possibility in the sense defined; but whenever the minor premiss indicates possibility all the syllogisms will be imperfect, and those which are negative will establish not possibility according to the definition, but that the major does not necessarily belong to any, or to all, of the minor. For if this is so, we say it is possible that it should belong to none or not to all.
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| ἐνδεχέσθω γὰρ τὸ Α παντὶ τῶι Β, τὸ δὲ Β παντὶ τῶι Γ κείσθω ὑπάρχειν. ἐπεὶ οὖν ὑπὸ τὸ Β ἐστὶ τὸ Γ, τῶι δὲ Β παντὶ ἐνδέχεται τὸ Α, φανερὸν ὅτι καὶ τῶι Γ παντὶ ἐνδέχεται. γίνεται δὴ τέλειος συλλογισμός· ὁμοίως δὲ καὶ στερητικῆς οὔσης τῆς Α Β προτάσεως, τῆς δὲ Β Γ καταφατικῆς, καὶ τῆς μὲν ἐνδέχεσθαι τῆς δ᾽ ὑπάρχειν λαμβανομένης, τέλειος ἔσται συλλογισμὸς ὅτι τὸ Α ἐνδέχεται μηδενὶ τῶι Γ ὑπάρχειν. | Contingat enim A omni B, B autem omni C ponatur inesse, quoniam igitur sub B est C, A autem contingit omni B, manifestum quoniam et C omni contingit A, fit ergo perfectus syllogismus. Similiter autem et cum privativa est A B propositio, B C autem affirmativa, et haec quidem contingere, illa vero inesse sumetur, perfectus erit syllogismus, quoniam A contingit nulli C inesse. | Let A be possible for all B, and let B belong to all C. Since C falls under B, and A is possible for all B, clearly it is possible for all C also. So a perfect syllogism results. Likewise if the premiss AB is negative, and the premiss BC is affirmative, the former stating possible, the latter simple attribution, a perfect syllogism results proving that A possibly belongs to no C. |
| ↵ Ὅτι μὲν οὖν τοῦ ὑπάρχειν τιθεμένου πρὸς τὸ ἔλαττον ἄκρον τέλειοι γίγνονται συλλογισμοί, φανερόν· ὅτι δ᾽ ἐναντίως ἔχοντος ἔσονται συλλογισμοί, διὰ τοῦ ἀδυνάτου δεικτέον. ἅμα δ᾽ ἔσται δῆλον καὶ ὅτι ἀτελεῖς· ἡ γὰρ δεῖξις οὐκ ἐκ τῶν εἰ λημμένων προτάσεων. | Quoniam ergo inesse posito ad minorem extremitatem, perfecti syllogismi fiunt, manifestum. Quod autem contrariae se habentes erunt syllogismi, per impossibile monstrandum est, simul autem erit manifestum et quoniam imperfecti, nam ostensio non ex sumptis propositionibus. | It is clear that perfect syllogisms result if the minor premiss states simple belonging: but that syllogisms will result if the modality of the premisses is reversed, must be proved per impossibile. At the same time it will be evident that they are imperfect: for the proof proceeds not from the premisses assumed. |
| πρῶτον δὲ λεκτέον ὅτι εἰ τοῦ Α ὄντος ἀνάγκη τὸ Β εἶναι, καὶ δυνατοῦ ὄντος τοῦ Α δυνατὸν ἔσται καὶ τὸ Β ἐξ ἀνάγκης. ἔστω γὰρ οὕτως ἐχόντων τὸ μὲν ἐφ᾽ ὧι τὸ Α δυνατόν, τὸ δ᾽ ἐφ᾽ ὧι τὸ Β ἀδύνατον. εἰ οὖν τὸ μὲν δυνατόν, ὅτε δυνατὸν εἶναι, γένοιτ᾽ ἄν, τὸ δ᾽ ἀδύνατον, ὅτ᾽ ἀδύ νατον, οὐκ ἂν γένοιτο, ἅμα δ᾽ εἴη τὸ Α δυνατὸν καὶ τὸ Β ἀδύνατον, ἐνδέχοιτ᾽ ἂν τὸ Α γενέσθαι ἄνευ τοῦ Β, εἰ δὲ γενέσθαι, καὶ εἶναι· | (0654D) Primum autem dicendum quoniam si cum est A, necesse est esse B, et cum possibile est esse A, possibile erit B ex necessitate. Sit enim sic se habentibus rebus ut in quo quidem A possibile, in quo autem B impossibile, si ergo aliud possibile quidem est, cum possibile esse, ipsum fiet, hoc vero impossibile, quoniam impossibile, non utique fiet, simul autem si A possibile, et B impossibile, continget fieri praeter B, si autem fieri et esse. | First we must state that if B’s being follows necessarily from A’s being, B’s possibility will follow necessarily from A’s possibility. Suppose, the terms being so related, that A is possible, and B is impossible. If then that which is possible, when it is possible for it to be, might happen, and if that which is impossible, when it is impossible, could not happen, and if at the same time A is possible and B impossible, it would be possible for A to happen without B, and if to happen, then to be. |
| τὸ γὰρ γεγονός, ὅτε γέγονεν, ἔστιν. δεῖ δὲ λαμβάνειν μὴ μόνον ἐν τῆι γενέσει τὸ ἀδύνατον καὶ δυνατόν, ἀλλὰ καὶ ἐν τῶι ἀληθεύεσθαι καὶ ἐν τῶι ὑπάρχειν, καὶ ὁσα χῶς ἄλλως λέγεται τὸ δυνατόν· ἐν ἅπασι γὰρ ὁμοίως ἕξει. | Nam quod fit, quando factum est, est. Oportet autem accipere non solum in generatione possibile et impossibile, sed et in verum esse, et in quod actu est, et quocunque modo simpliciter aliter dicitur possibile, in omnibus enim similiter se habebit. | For that which has happened, when it has happened, is. But we must take the impossible and the possible not only in the sphere of becoming, but also in the spheres of truth and predicability, and the various other spheres in which we speak of the possible: for it will be alike in all. |
| ἔτι τὸ ὄντος τοῦ Α τὸ Β εἶναι, οὐχ ὡς ἑνός τινος ὄντος τοῦ Α τὸ Β ἔσται δεῖ ὑπολαβεῖν· οὐ γὰρ ἔστιν οὐδὲν ἐξ ἀνάγκης ἑνός τινος ὄντος, ἀλλὰ δυοῖν ἐλαχίστοιν, οἷον ὅταν αἱ προτάσεις οὕτως ἔχωσιν ὡς ἐλέχθη κατὰ τὸν συλλογισμόν. εἰ γὰρ τὸ Γ κατὰ τοῦ Δ, τὸ δὲ Δ κατὰ τοῦ Ζ, καὶ τὸ Γ κατὰ τοῦ Ζ ἐξ ἀνάγκης· καὶ εἰ δυνατὸν ἑκάτερον, καὶ τὸ συμπέρασμα δυνατόν. ὥσπερ οὖν εἴ τις θείη τὸ μὲν Α τὰς προτάσεις, τὸ δὲ Β τὸ συμπέρασμα, συμβαίνοι ἂν οὐ μόνον ἀναγκαίου τοῦ Α ὄντος ἅμα καὶ τὸ Β εἶναι ἀναγκαῖον, ἀλλὰ καὶ δυνατοῦ δυνατόν. | (0655A) Amplius cum est A, B esse, non tanquam uno aliquo existente A, erit B, oportet opinari, nihil enim est ex necessitate uno aliquo existente, sed duobus ad minus, ut quando propositiones sic se habent (ut dictum est) secundum syllogismum, nam sic dicitur de D, D autem de E, et C de E ex necessitate, et si utrumque possibile, et conclusio erit possibilis. Quemadmodum ergo si quis ponat A quidem propositiones, B autem conclusionem, accidet non solum A existente necessario, et B simul esse necessarium, sed etiam possibili possibile. | Further we must understand the statement that B’s being depends on A’s being, not as meaning that if some single thing A is, B will be: for nothing follows of necessity from the being of some one thing, but from two at least, i.e. when the premisses are related in the manner stated to be that of the syllogism. For if C is predicated of D, and D of F, then C is necessarily predicated of F. And if each is possible, the conclusion also is possible. If then, for example, one should indicate the premisses by A, and the conclusion by B, it would not only result that if A is necessary B is necessary, but also that if A is possible, B is possible. |
| Τούτου δὲ δειχθέντος, φανερὸν ὅτι ψεύδους ὑποτεθέντος καὶ μὴ ἀδυνάτου καὶ τὸ συμβαῖνον διὰ τὴν ὑπόθεσιν ψεῦδος ἔσται καὶ οὐκ ἀδύνατον. οἷον εἰ τὸ Α ψεῦδος μέν ἐστι μὴ μέντοι ἀδύνατον, ὄντος δὲ τοῦ Α τὸ Β ἔστι, καὶ τὸ Β ἔσται ψεῦδος μὲν οὐ μέντοι ἀδύνατον. ἐπεὶ γὰρ δέδεικται ὅτι εἰ τοῦ Α ὄντος τὸ Β ἔστι, καὶ δυνατοῦ ὄντος τοῦ Α ἔσται τὸ Β δυνατόν, ὑπόκειται δὲ τὸ Α δυνατὸν εἶναι, καὶ τὸ Β ἔσται δυνατόν· εἰ γὰρ ἀδύνατον, ἅμα δυνατὸν ἔσται τὸ αὐτὸ καὶ ἀδύνατον. | (0655B) Hoc autem ostenso manifestum est quoniam falso posito, et non impossibili, et quod accidit propter positionem falsum erit, et non impossibile, ut si A falsum quidem est, non tamen impossibile, cum autem sit A et B, et B erit falsum quidem, non tamen impossibile. Nam ostensum est quoniam cum est A, est B, et cum possibile est A, possibile est B. Positum autem est A possibile esse, et B erit possibile, si enim impossibile est B, simul idem erit possibile et impossibile. | Since this is proved it is evident that if a false and not impossible assumption is made, the consequence of the assumption will also be false and not impossible: e.g. if A is false, but not impossible, and if B is the consequence of A, B also will be false but not impossible. For since it has been proved that if B’s being is the consequence of A’s being, then B’s possibility will follow from A’s possibility (and A is assumed to be possible), consequently B will be possible: for if it were impossible, the same thing would at the same time be possible and impossible. |
| Διωρισμένων δὴ τούτων ὑπαρχέτω τὸ Α παντὶ τῶι Β, τὸ δὲ Β παντὶ τῶι Γ ἐνδεχέσθω· ἀνάγκη οὖν τὸ Α παντὶ τῶι Γ ἐνδέχεσθαι ὑπάρχειν. μὴ γὰρ ἐνδεχέσθω, τὸ δὲ Β παντὶ τῶι Γ κείσθω ὡς ὑπάρχον· τοῦτο δὲ ψεῦδος μέν, οὐ μέντοι ἀδύνατον. εἰ οὖν τὸ μὲν Α μὴ ἐνδέχεται παντὶ τῶι Γ, τὸ δὲ Β παντὶ ὑπάρχει τῶι Γ, τὸ Α οὐ παντὶ τῶι Β ἐνδέχεται· γί νεται γὰρ συλλογισμὸς διὰ τοῦ τρίτου σχήματος. ἀλλ᾽ ὑπέκειτο παντὶ ἐνδέχεσθαι ὑπάρχειν. ἀνάγκη ἄρα τὸ Α παντὶ ↵ τῶι Γ ἐνδέχεσθαι· ψεύδους γὰρ τεθέντος καὶ οὐκ ἀδυνάτου τὸ συμβαῖνόν ἐστιν ἀδύνατον. [ἐγχωρεῖ δὲ καὶ διὰ τοῦ πρώτου σχήματος ποιῆσαι τὸ ἀδύνατον, θέντας τῶι Γ τὸ Β ὑπάρχειν. εἰ γὰρ τὸ Β παντὶ τῶι Γ ὑπάρχει, τὸ δὲ Α παντὶ τῶι Β ἐνδέχεται, κἂν τῶι Γ παντὶ ἐνδέχοιτο τὸ Α. ἀλλ᾽ ὑπέκειτο μὴ παντὶ ἐγχωρεῖν.] | Determinatis autem iis, insit A omni B, B autem contingit omni C, necesse est A igitur contingere omni C inesse. Non enim contingat, B autem omni C ponatur inesse, hoc autem falsum quidem, non tamen impossibile, si ergo A quidem non contingit omni C, B autem omni C insit, A non omni B contingit. Fit enim syllogismus per tertiam figuram. Sed positum erat omni C contingere inesse, necesse est ergo A omni C contingere. Falso enim posito, et non impossibili, quod accidit est impossibile. (0655C) Possibile est autem et primam figuram facere impossibile ponentes B inesse C, nam si B omni C inest, A autem omni B contingit, et omni C continget A, sed positum erat non omni possibile inesse. | Since we have defined these points, let A belong to all B, and B be possible for all C: it is necessary then that should be a possible attribute for all C. Suppose that it is not possible, but assume that B belongs to all C: this is false but not impossible. If then A is not possible for C but B belongs to all C, then A is not possible for all B: for a syllogism is formed in the third degree. But it was assumed that A is a possible attribute for all B. It is necessary then that A is possible for all C. For though the assumption we made is false and not impossible, the conclusion is impossible. It is possible also in the first figure to bring about the impossibility, by assuming that B belongs to C. For if B belongs to all C, and A is possible for all B, then A would be possible for all C. But the assumption was made that A is not possible for all C. |
| Δεῖ δὲ λαμβάνειν τὸ παντὶ ὑπάρχον μὴ κατὰ χρόνον ὁρίσαντας, οἷον νῦν ἢ ἐν τῶιδε τῶι χρόνωι, ἀλλ᾽ ἁπλῶς· διὰ τοιούτων γὰρ προτάσεων καὶ τοὺς συλλογισμοὺς ποιοῦμεν, ἐπεὶ κατά γε τὸ νῦν λαμβανομένης τῆς προτάσεως οὐκ ἔσται συλλογισμός· οὐδὲν γὰρ ἴσως κωλύει ποτὲ καὶ παντὶ κινου- μένωι ἄνθρωπον ὑπάρχειν, οἷον εἰ μηδὲν ἄλλο κινοῖτο· τὸ δὲ κινούμενον ἐνδέχεται παντὶ ἵππωι· ἀλλ᾽ ἄνθρωπον οὐδενὶ ἵππωι ἐνδέχεται. ἔτι ἔστω τὸ μὲν πρῶτον ζῶιον, τὸ δὲ μέσον κινού μενον, τὸ δ᾽ ἔσχατον ἄνθρωπος. αἱ μὲν οὖν προτάσεις ὁμοίως ἕξουσι, τὸ δὲ συμπέρασμα ἀναγκαῖον, οὐκ ἐνδεχόμενον· ἐξ ἀνάγκης γὰρ ὁ ἄνθρωπος ζῶιον. φανερὸν οὖν ὅτι τὸ καθόλου ληπτέον ἁπλῶς, καὶ οὐ χρόνωι διορίζοντας. | Oportet autem accipere omni inesse non secundum tempus determinantes, ut nunc, aut in hoc tempore, sed simpliciter (per huiusmodi enim propositiones et syllogismos facimus), quoniam secundum nunc sumpta propositione, non erit syllogismus. Nihil enim fortasse prohibet quandoque et omni moventi hominem inesse, ut si nihil aliud moveatur, movens autem contingit omni equo, sed homo nulli equo contingit. Amplius: sit primum quidem animal, medium vero movens, postremum vero homo, ergo propositiones quidem similiter se habebunt, conclusio vero erit necessaria, non contingens. (0655D) Ex necessitate enim homo est animal, manifestum igitur quoniam universale sumendum simpliciter, et non tempore determinantes. | We must understand ‘that which belongs to all’ with no limitation in respect of time, e.g. to the present or to a particular period, but simply without qualification. For it is by the help of such premisses that we make syllogisms, since if the premiss is understood with reference to the present moment, there cannot be a syllogism. For nothing perhaps prevents ‘man’ belonging at a particular time to everything that is moving, i.e. if nothing else were moving: but ‘moving’ is possible for every horse; yet ‘man’ is possible for no horse. Further let the major term be ‘animal’, the middle ‘moving’, the the minor ‘man’. The premisses then will be as before, but the conclusion necessary, not possible. For man is necessarily animal. It is clear then that the universal must be understood simply, without limitation in respect of time. |
| Πάλιν ἔστω στερητικὴ πρότασις καθόλου ἡ Α Β, καὶ εἰλήφθω τὸ μὲν Α μηδενὶ τῶι Β ὑπάρχειν, τὸ δὲ Β παντὶ ἐνδεχέσθω ὑπάρχειν τῶι Γ. τούτων οὖν τεθέντων ἀνάγκη τὸ Α ἐνδέχεσθαι μηδενὶ τῶι Γ ὑπάρχειν. μὴ γὰρ ἐνδεχέσθω, τὸ δὲ Β τῶι Γ κείσθω ὑπάρχον, καθάπερ πρότερον. ἀνάγκη δὴ τὸ Α τινὶ τῶι Β ὑπάρχειν· γίνεται γὰρ συλλογισμὸς διὰ τοῦ τρίτου σχήματος· τοῦτο δὲ ἀδύνατον. ὥστ᾽ ἐνδέχοιτ᾽ ἂν τὸ Α μηδενὶ τῶι Γ· ψεύδους γὰρ τεθέντος ἀδύνατον τὸ συμβαῖνον.
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Rursum: sit privativa propositio universalis A B, et sumatur A quidem nulli B inesse, B autem contingat omni C inesse. His igitur positis necesse est A contingere nulli C inesse, non enim contingat, B autem ponatur inesse C sicut prius, necesse est igitur A alicui B inesse, fit enim syllogismus per tertiam figuram. Hoc autem impossibile, quare contingit A, nulli C. Posito enim falso, et non impossibili, impossibile est quod accidit. | Again let the premiss AB be universal and negative, and assume that A belongs to no B, but B possibly belongs to all C. These propositions being laid down, it is necessary that A possibly belongs to no C. Suppose that it cannot belong, and that B belongs to C, as above. It is necessary then that A belongs to some B: for we have a syllogism in the third figure: but this is impossible. Thus it will be possible for A to belong to no C; for if at is supposed false, the consequence is an impossible one. |
| οὗτος οὖν ὁ συλλογισμὸς οὐκ ἔστι τοῦ κατὰ τὸν διορισμὸν ἐνδεχομένου, ἀλλὰ τοῦ μηδενὶ ἐξ ἀνάγκης (αὕτη γάρ ἐστιν ἡ ἀντίφασις τῆς γενομένης ὑποθέσεως· ἐτέθη γὰρ ἐξ ἀνάγ κης τὸ Α τινὶ τῶι Γ ὑπάρχειν, ὁ δὲ διὰ τοῦ ἀδυνάτου συλλογισμὸς τῆς ἀντικειμένης ἐστὶν φάσεωσ). | Hic ergo syllogismus non est contingentis secundum definitionem, sed nulli inesse ex necessitate. Haec est contradictio factae hypothesis. (0656A) Positum est enim ex necessitate A alicui C inesse, syllogismus autem per impossibile, oppositae est contradictionis. | This syllogism then does not establish that which is possible according to the definition, but that which does not necessarily belong to any part of the subject (for this is the contradictory of the assumption which was made: for it was supposed that A necessarily belongs to some C, but the syllogism per impossibile establishes the contradictory which is opposed to this). |
| ἔτι δὲ καὶ ἐκ τῶν ὅρων φανερὸν ὅτι οὐκ ἔσται τὸ συμπέρασμα ἐνδεχόμενον. ἔστω γὰρ τὸ μὲν Α κόραξ, τὸ δ᾽ ἐφ᾽ ὧι Β διανοούμενον, ἐφ᾽ ὧι δὲ Γ ἄνθρωπος. οὐδενὶ δὴ τῶι Β τὸ Α ὑπάρχει· οὐδὲν γὰρ διανοούμενον κόραξ. τὸ δὲ Β παντὶ ἐνδέχεται τῶι Γ· παντὶ γὰρ ἀνθρώπωι τὸ διανοεῖσθαι. ἀλλὰ τὸ Α ἐξ ἀνάγκης οὐδενὶ τῶι Γ· οὐκ ἄρα τὸ συμπέρασμα ἐνδεχόμενον. ἀλλ᾽ οὐδ᾽ ἀναγκαῖον ἀεί. ἔστω γὰρ τὸ μὲν Α κινούμενον, τὸ δὲ Β ἐπιστήμη, τὸ δ᾽ ἐφ᾽ ὧι Γ ἄνθρωπος. τὸ μὲν οὖν Α οὐδενὶ τῶι Β ὑπάρξει, τὸ δὲ Β παντὶ τῶι Γ ἐνδέχεται, καὶ οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον· οὐ γὰρ ἀνάγκη μηδένα κινεῖσθαι ἄνθρωπον, ἀλλ᾽ ↵ οὐκ ἀνάγκη τινά. δῆλον οὖν ὅτι τὸ συμπέρασμά ἐστι τοῦ μηδενὶ ἐξ ἀνάγκης ὑπάρχειν. ληπτέον δὲ βέλτιον τοὺς ὅρους. | Amplius autem et ex terminis manifestum quoniam non erit conclusio contingens, sit enim A quidem corvus, in quo autem B intelligens, in quo autem C homo, nulli ergo B inest A, nam nullum intelligens, corvus, B autem contingit omni C, omni enim homini inest intelligere, sed A ex necessitate nulli C, non igitur conclusio contingens. Sed nec necessaria semper: sit enim A quidem movens, B autem scientia, in quo autem C homo, ergo A quidem nulli B inerit, B autem omni C contingit, et non erit conclusio necessaria, non enim necesse est nullum hominem moveri, sed necesse est aliquem. (0656B) Manifestum igitur quoniam est conclusio eius quod est nulli ex necessitate inesse. Sumendum autem melius terminos. | Further, it is clear also from an example that the conclusion will not establish possibility. Let A be ‘raven’, B ‘intelligent’, and C ‘man’. A then belongs to no B: for no intelligent thing is a raven. But B is possible for all C: for every man may possibly be intelligent. But A necessarily belongs to no C: so the conclusion does not establish possibility. But neither is it always necessary. Let A be ‘moving’, B ‘science’, C ‘man’. A then will belong to no B; but B is possible for all C. And the conclusion will not be necessary. For it is not necessary that no man should move; rather it is not necessary that any man should move. Clearly then the conclusion establishes that one term does not necessarily belong to any instance of another term. But we must take our terms better. |
| Ἐὰν δὲ τὸ στερητικὸν τεθῆι πρὸς τὸ ἔλαττον ἄκρον ἐνδέχεσθαι σημαῖνον, ἐξ αὐτῶν μὲν τῶν εἰλημμένων προτάσεων οὐδεὶς ἔσται συλλογισμός, ἀντιστραφείσης δὲ τῆς κατὰ τὸ ἐνδέχεσθαι προτάσεως ἔσται, καθάπερ ἐν τοῖς πρότερον. ὑπαρχέτω γὰρ τὸ Α παντὶ τῶι Β, τὸ δὲ Β ἐνδεχέσθω μηδενὶ τῶι Γ. οὕτω μὲν οὖν ἐχόντων τῶν ὅρων οὐδὲν ἔσται ἀναγκαῖον· ἐὰν δ᾽ ἀντιστραφῆι τὸ Β Γ καὶ ληφθῆι τὸ Β παντὶ τῶι Γ ἐν δέχεσθαι, γίνεται συλλογισμὸς ὥσπερ πρότερον· ὁμοίως γὰρ ἔχουσιν οἱ ὅροι τῆι θέσει. τὸν αὐτὸν δὲ τρόπον καὶ στερητικῶν ὄντων ἀμφοτέρων τῶν διαστημάτων, ἐὰν τὸ μὲν Α Β μὴ ὑπάρχειν, τὸ δὲ Β Γ μηδενὶ ἐνδέχεσθαι σημαίνηι· δι᾽ αὐτῶν μὲν γὰρ τῶν εἰλημμένων οὐδαμῶς γίνεται τὸ ἀναγκαῖον, ἀν τιστραφείσης δὲ τῆς κατὰ τὸ ἐνδέχεσθαι προτάσεως ἔσται συλλογισμός. εἰλήφθω γὰρ τὸ μὲν Α μηδενὶ τῶι Β ὑπάρχειν, τὸ δὲ Β ἐνδέχεσθαι μηδενὶ τῶι Γ. διὰ μὲν οὖν τούτων οὐδὲν ἀναγκαῖον· | Si autem privativum ponatur ad maiorem extremitatem contingere significans, ex ipsis quidem sumptis propositionibus, nullus erit syllogismus, conversa autem secundum contingens propositione, erit quemadmodum in prioribus. Insit enim A omni B, B autem contingat nulli C, sic ergo habentibus se terminis, nihil erit necessarium. Si autem convertatur B C, et sumatur B contingere omni C, fiet syllogismus quemadmodum prius, similiter enim habent se termini positione. (0656C) Eodem autem modo et cum privativa sunt utraque intervalla, si A B quidem non inesse, B C autem nulli, contingere significat, nam per ea quidem quae sumpta sunt nullo modo fit necessarium, conversa autem secundum contingens propositione erit syllogismus, sumatur enim A quidem, nulli B inesse, B autem contingere nulli C, per haec quidem nihil necessarium. | If the minor premiss is negative and indicates possibility, from the actual premisses taken there can be no syllogism, but if the problematic premiss is converted, a syllogism will be possible, as before. Let A belong to all B, and let B possibly belong to no C. If the terms are arranged thus, nothing necessarily follows: but if the proposition BC is converted and it is assumed that B is possible for all C, a syllogism results as before: for the terms are in the same relative positions. Likewise if both the relations are negative, if the major premiss states that A does not belong to B, and the minor premiss indicates that B may possibly belong to no C. Through the premisses actually taken nothing necessary results in any way; but if the problematic premiss is converted, we shall have a syllogism. Suppose that A belongs to no B, and B may possibly belong to no C. Through these comes nothing necessary. |
| ἐὰν δὲ ληφθῆι τὸ Β παντὶ τῶι Γ ἐνδέχεσθαι, ὅπερ ἐστὶν ἀληθές, ἡ δὲ Α Β πρότασις ὁμοίως ἔχηι, πάλιν ὁ αὐτὸς ἔσται συλλογισμός. ἐὰν δὲ μὴ ὑπάρχειν τεθῆι τὸ Β παντὶ τῶι Γ καὶ μὴ ἐνδέχεσθαι μὴ ὑπάρχειν, οὐκ ἔσται συλλογισμὸς οὐδαμῶς, οὔτε στερητικῆς οὔσης οὔτε καταφατικῆς τῆς Α Β προτάσεως. ὅροι δὲ κοινοὶ τοῦ μὲν ἐξ ἀνάγκης ὑπάρχειν λευκόν – ζῶιον – χιών, τοῦ δὲ μὴ ἐνδέχεσθαι λευκόν – ζῶιον – πίττα. | Si autem sumatur B omni C contingere, quod verum est, A B autem propositio similiter se habeat, rursus erit idem syllogismus. Si autem non inesse ponatur B omni C, et non contingere non inesse, non erit syllogismus nullo modo, sive privativa sit, sive affirmativa A B propositio. Termini autem communes ex necessitate quidem inesse, album, animal, nix. Non contingere autem, album, animal, pix. | But if B is assumed to be possible for all C (and this is true) and if the premiss AB remains as before, we shall again have the same syllogism. But if it be assumed that B does not belong to any C, instead of possibly not belonging, there cannot be a syllogism anyhow, whether the premiss AB is negative or affirmative. As common instances of a necessary and positive relation we may take the terms white-animal-snow: of a necessary and negative relation, white-animal-pitch. |
| Φανερὸν οὖν ὅτι καθόλου τῶν ὅρων ὄντων, καὶ τῆς μὲν ὑπάρχειν τῆς δ᾽ ἐνδέχεσθαι λαμβανομένης τῶν προτάσεων, ὅταν ἡ πρὸς τὸ ἔλαττον ἄκρον ἐνδέχεσθαι λαμβάνηται πρότασις, ἀεὶ γίνεται συλλογισμός, πλὴν ὁτὲ μὲν ἐξ αὐτῶν ὁτὲ δ᾽ ἀντιστραφείσης τῆς προτάσεως. πότε δὲ τούτων ἑκάτε ρος καὶ διὰ τίν᾽ αἰτίαν, εἰρήκαμεν. Ἐὰν δὲ τὸ μὲν καθόλου τὸ δ᾽ ἐν μέρει ληφθῆι τῶν διαστημάτων, ὅταν μὲν τὸ πρὸς τὸ μεῖζον ἄκρον καθόλου τεθῆι καὶ ἐνδεχόμενον, εἴτ᾽ ἀποφατικὸν εἴτε καταφατικόν, τὸ δ᾽ ἐν μέρει καταφατικὸν καὶ ὑπάρχον, ἔσται συλλογισμὸς τέλειος, καθάπερ καὶ καθόλου τῶν ὅρων ὄντων. ἀπόδειξις δ᾽ ἡ αὐτὴ ἣ καὶ πρότερον. | (0656D) Manifestum est igitur quoniam cum universales sunt termini, et haec quidem propositionum inesse, illa vero sumitur contingens, quando quae ad minorem est extremitatem contingere sumitur propositio, semper fit syllogismus, verumtamen quandoquidem ex ipsis, quando autem propositione conversa, quando vero utrumque horum, et ob quam causam, diximus. Si autem hoc quidem universale, illud vero particulare sumitur intervallorum, quando ad maiorem quidem extremitatem universale ponitur, et contingens sive negativum, sive affirmativum, particulare autem affirmativum et inesse, erit syllogismus perfectus, quemadmodum et cum universales sunt termini, demonstratio autem eadem quae et prius. | Clearly then if the terms are universal, and one of the premisses is assertoric, the other problematic, whenever the minor premiss is problematic a syllogism always results, only sometimes it results from the premisses that are taken, sometimes it requires the conversion of one premiss. We have stated when each of these happens and the reason why. But if one of the relations is universal, the other particular, then whenever the major premiss is universal and problematic, whether affirmative or negative, and the particular is affirmative and assertoric, there will be a perfect syllogism, just as when the terms are universal. The demonstration is the same as before. |
| ὅταν δὲ καθόλου μὲν ἦι τὸ πρὸς τὸ μεῖζον ἄκρον, ὑπάρχον δὲ καὶ μὴ ἐνδεχόμενον, θάτερον δ᾽ ἐν μέρει καὶ ἐνδεχόμενον, ἐάν τ᾽ ἀποφατικαὶ ἐάν τε καταφατικαὶ τεθῶσιν ἀμφότεραι, ἐάν τε ἡ μὲν ἀποφατικὴ ἡ δὲ καταφατική, πάντως ἔσται συλλογισμὸς ἀτελής. πλὴν οἱ μὲν διὰ τοῦ ἀδυνάτου δειχθήσονται, ↵ οἱ δὲ καὶ διὰ τῆς ἀντιστροφῆς τῆς τοῦ ἐνδέχεσθαι, καθάπερ ἐν τοῖς πρότερον. ἔσται δὲ συλλογισμὸς διὰ τῆς ἀντιστροφῆς [καὶ] ὅταν ἡ μὲν καθόλου πρὸς τὸ μεῖζον ἄκρον τεθεῖσα σημαίνηι τὸ ὑπάρχειν [ἢ μὴ ὑπάρχειν], ἡ δ᾽ ἐν μέρει στερητικὴ οὖσα τὸ ἐνδέχεσθαι λαμβάνηι, οἷον εἰ τὸ μὲν Α παντὶ τῶι Β ὑπάρχει ἢ μὴ ὑπάρχει, τὸ δὲ Β τινὶ τῶι Γ ἐνδέχεται μὴ ὑπάρχειν· ἀντιστραφέντος γὰρ τοῦ Β Γ κατὰ τὸ ἐνδέχεσθαι γίνεται συλλογισμός. ὅταν δὲ τὸ μὴ ὑπάρχειν λαμβάνηι ἡ κατὰ μέρος τεθεῖσα, οὐκ ἔσται συλλογισμός. | (0657A) Quando autem universale quidem fuerit, ad maiorem extremitatem inesse, et non contingens, alterum vero particulare, et contingens, sive affirmative, sive negative ponantur utraeque, sive haec quidem negativa, illa vero affirmativa, omnino erit syllogismus imperfectus. Verum hi quidem per impossibile ostenduntur, illi vero per conversionem contingentis, quemadmodum in prioribus. Erit autem syllogismus per conversionem, et quando universalis quidem ad maiorem extremitatem posita significaverit inesse, vel non inesse, particularis vero cum sit privativa, sumatur contingens, ut si A quidem omni B inest, vel non inest, B autem alicui contingit non inesse, conversa enim B C, secundum contingere fit syllogismus. Quando autem non inesse sumetur particulariter posita propositio, non erit syllogismus. | But whenever the major premiss is universal, but assertoric, not problematic, and the minor is particular and problematic, whether both premisses are negative or affirmative, or one is negative, the other affirmative, in all cases there will be an imperfect syllogism. Only some of them will be proved per impossibile, others by the conversion of the problematic premiss, as has been shown above. And a syllogism will be possible by means of conversion when the major premiss is universal and assertoric, whether positive or negative, and the minor particular, negative, and problematic, e.g. if A belongs to all B or to no B, and B may possibly not belong to some C. For if the premiss BC is converted in respect of possibility, a syllogism results. But whenever the particular premiss is assertoric and negative, there cannot be a syllogism. |
| ὅροι τοῦ μὲν ὑπάρχειν λευκόν – ζῶιον – χιών, τοῦ δὲ μὴ ὑπάρχειν λευκόν – ζῶιον – πίττα· διὰ γὰρ τοῦ ἀδιορίστου ληπτέον τὴν ἀπόδειξιν. | Termini inesse, album, animal, nix; non inesse autem, album, animal, pix, per indefinitum enim est sumenda demonstratio. | As instances of the positive relation we may take the terms white-animal-snow; of the negative, white-animal-pitch. For the demonstration must be made through the indefinite nature of the particular premiss. |
| ἐὰν δὲ τὸ καθόλου τεθῆι πρὸς τὸ ἔλαττον ἄκρον, τὸ δ᾽ ἐν μέρει πρὸς τὸ μεῖζον, ἐάν τε στερητικὸν ἐάν τε καταφατικόν, ἐάν τ᾽ ἐνδεχόμενον ἐάν θ᾽ ὑπάρχον ὁποτερονοῦν, οὐδαμῶς ἔσται συλλογισμός. Οὐδ᾽ ὅταν ἐν μέρει ἢ ἀδιόριστοι τεθῶσιν αἱ προτάσεις, εἴτ᾽ ἐνδέχεσθαι λαμβάνουσαι εἴθ᾽ ὑπάρχειν εἴτ᾽ ἐναλλάξ, οὐδ᾽ οὕτως ἔσται συλλογισμός. ἀπόδειξις δ᾽ ἡ αὐτὴ ἥπερ κἀπὶ τῶν πρότερον. ὅροι δὲ κοινοὶ τοῦ μὲν ὑπάρχειν ἐξ ἀνάγκης ζῶιον – λευκόν – ἄνθρωπος, τοῦ δὲ μὴ ἐνδέχεσθαι ζῶιον – λευκόν – ἱμάτιον. φανερὸν οὖν ὅτι τοῦ μὲν πρὸς τὸ μεῖζον ἄκρον καθόλου τεθέντος ἀεὶ γίνεται συλλογισμός, τοῦ δὲ πρὸς τὸ ἔλαττον οὐδέποτ᾽ οὐδενός. | (0657B) Si autem universale quidem ponatur ad minorem extremitatem, particulare autem ad maiorem sive privativum, sive affirmativum, sive contingens, sive inesse utrumvis, nullo modo erit syllogismus. Nec cum particulares, vel indefinitae ponentur propositiones, sive contingere sumptae, sive inesse, seu permutatim, nec sic erit syllogismus, demonstratio autem eadem quae in prioribus. Termini autem communes inesse quidem, ex necessitate, animal, album, homo; non contingere vero, animal, album, tunica. Manifestum est igitur quoniam universali posito ad maiorem extremitatem semper erit syllogismus, ad minorem autem nunquam. | But if the minor premiss is universal, and the major particular, whether either premiss is negative or affirmative, problematic or assertoric, nohow is a syllogism possible. Nor is a syllogism possible when the premisses are particular or indefinite, whether problematic or assertoric, or the one problematic, the other assertoric. The demonstration is the same as above. As instances of the necessary and positive relation we may take the terms animal-white-man; of the necessary and negative relation, animal-white-garment. It is evident then that if the major premiss is universal, a syllogism always results, but if the minor is universal nothing at all can ever be proved. |
Chapter 16
| Greek | Latin | English |
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| (PL 64 0657B) CAPUT XV. Mixtio necessarii et contingentis in prima figura. | 16 | |
| 35b23 Ὅταν δ᾽ ἡ μὲν ἐξ ἀνάγκης ὑπάρχειν ἡ δ᾽ ἐνδέχεσθαι σημαίνηι τῶν προτάσεων, ὁ μὲν συλλογισμὸς ἔσται τὸν αὐτὸν τρόπον ἐχόντων τῶν ὅρων, καὶ τέλειος ὅταν πρὸς τῶι ἐλάττονι ἄκρωι τεθῆι τὸ ἀναγκαῖον· τὸ δὲ συμπέρασμα κατηγορικῶν μὲν ὄντων τῶν ὅρων τοῦ ἐνδέχεσθαι καὶ οὐ τοῦ ὑπάρχειν ἔσται, καὶ καθόλου καὶ μὴ καθόλου τιθεμένων, ἐὰν δ᾽ ἦι τὸ μὲν καταφατικὸν τὸ δὲ στερητικόν, ὅταν μὲν ἦι τὸ καταφατικὸν ἀναγκαῖον, τοῦ ἐνδέχεσθαι καὶ οὐ τοῦ μὴ ὑπάρχειν, ὅταν δὲ τὸ στερητικόν, καὶ τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν καὶ τοῦ μὴ ὑπάρχειν, καὶ καθόλου καὶ μὴ καθόλου τῶν ὅρων ὄντων· τὸ δ᾽ ἐνδέχεσθαι ἐν τῶι συμπεράσματι τὸν αὐτὸν τρόπον ληπτέον ὅνπερ καὶ ἐν τοῖς πρότερον. τοῦ δ᾽ ἐξ ἀνάγκης μὴ ὑπάρχειν οὐκ ἔσται συλλογισμός· ἕτερον γὰρ τὸ μὴ ἐξ ἀνάγκης ὑπάρχειν καὶ τὸ ἐξ ἀνάγκης μὴ ὑπάρχειν. | (0657C) Quando autem haec quidem propositionum ex necessitate inesse, vel non inesse, illa vero contingere significat, syllogismus quidem erit hoc modo habentibus se terminis. Et perfectus, quando ad minorem extremitatem ponetur necessaria. Conclusio autem, si praedicativi sunt quidem termini, contingentis, et non inesse erit, sive universaliter, sive non universaliter ponantur, si autem sint hoc quidem affirmativum, illud vero privativum, quando affirmativum quidem fuerit necessarium, et contingentis erit conclusio, et non eius quod est non inesse. Quando autem privativum necessarium, et contingentis non esse, et non inesse, sive universales, sive non universales sint termini. Contingere autem in conclusione eodem modo accipiendum est quo in prioribus. (0657D) Eius autem quod est ex necessitate non inesse, non erit syllogismus, aliud enim est non ex necessitate inesse, et ex necessitate non inesse.
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Whenever one premiss is necessary, the other problematic, there will be a syllogism when the terms are related as before; and a perfect syllogism when the minor premiss is necessary. If the premisses are affirmative the conclusion will be problematic, not assertoric, whether the premisses are universal or not: but if one is affirmative, the other negative, when the affirmative is necessary the conclusion will be problematic, not negative assertoric; but when the negative is necessary the conclusion will be problematic negative, and assertoric negative, whether the premisses are universal or not. Possibility in the conclusion must be understood in the same manner as before. There cannot be an inference to the necessary negative proposition: for ‘not necessarily to belong’ is different from ‘necessarily not to belong’. |
| Ὅτι μὲν οὖν καταφατικῶν ὄντων τῶν ὅρων οὐ γίνεται τὸ συμπέρασμα ἀναγκαῖον, φανερόν. ὑπαρχέτω γὰρ τὸ μὲν Α παντὶ τῶι Β ἐξ ἀνάγκης, τὸ δὲ Β ἐνδεχέσθω παντὶ τῶι Γ. ἔσται δὴ συλλογισμὸς ἀτελὴς ὅτι ἐνδέχεται τὸ Α παντὶ τῶι Γ ↵ ὑπάρχειν. ὅτι δ᾽ ἀτελής, ἐκ τῆς ἀποδείξεως δῆλον· τὸν αὐτὸν γὰρ τρόπον δειχθήσεται ὅνπερ κἀπὶ τῶν πρότερον. πάλιν τὸ μὲν Α ἐνδεχέσθω παντὶ τῶι Β, τὸ δὲ Β παντὶ τῶι Γ ὑπαρχέτω ἐξ ἀνάγκης. ἔσται δὴ συλλογισμὸς ὅτι τὸ Α παντὶ τῶι Γ ἐνδέχεται ὑπάρχειν, ἀλλ᾽ οὐχ ὅτι ὑπάρχει, καὶ τέλειος, ἀλλ᾽ οὐκ ἀτελής· εὐθὺς γὰρ ἐπιτελεῖται διὰ τῶν ἐξ ἀρχῆς προτάσεων. | Quoniam igitur universalibus affirmativis existentibus terminis non fit conclusio necessaria, manifestum: insit enim A omni B ex necessitate, B autem contingat omni C, erit igitur syllogismus imperfectus, quoniam A contingit omni C inesse. Quoniam autem imperfectus, ex demonstratione palam, eodem enim modo ostendetur quo et in prioribus. Rursum A quidem contingat omni B inesse, B autem omni C insit ex necessitate, erit itaque syllogismus, quoniam A contingat omni C inesse, sed non quoniam inest, et perfectus quidem, sed non imperfectus, statim enim perficitur ex principio propositionis. | If the premisses are affirmative, clearly the conclusion which follows is not necessary. Suppose A necessarily belongs to all B, and let B be possible for all C. We shall have an imperfect syllogism to prove that A may belong to all C. That it is imperfect is clear from the proof: for it will be proved in the same manner as above. Again, let A be possible for all B, and let B necessarily belong to all C. We shall then have a syllogism to prove that A may belong to all C, not that A does belong to all C: and it is perfect, not imperfect: for it is completed directly through the original premisses. |
| Εἰ δὲ μὴ ὁμοιοσχήμονες αἱ προτάσεις, ἔστω πρῶτον ἡ στερητικὴ ἀναγκαία, καὶ τὸ μὲν Α μηδενὶ ἐνδεχέσθω τῶι Β ἐξ ἀνάγκης, , τὸ δὲ Β παντὶ τῶι Γ ἐνδεχέ σθω. ἀνάγκη δὴ τὸ Α μηδενὶ τῶι Γ ὑπάρχειν. κείσθω γὰρ ὑπάρχειν ἢ παντὶ ἢ τινί· τῶι δὲ Β ὑπέκειτο μηδενὶ ἐνδέχεσθαι. ἐπεὶ οὖν ἀντιστρέφει τὸ στερητικόν, οὐδὲ τὸ Β τῶι Α οὐδενὶ ἐνδέχεται· τὸ δὲ Α τῶι Γ ἢ παντὶ ἢ τινὶ κεῖται ὑπάρχειν· ὥστ᾽ οὐδενὶ ἢ οὐ παντὶ τῶι Γ τὸ Β ἐνδέχοιτ᾽ ἂν ὑπάρχειν· ὑπέκειτο δὲ παντὶ ἐξ ἀρχῆς. φανερὸν δ᾽ ὅτι καὶ τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν γίγνεται συλλογισμός, εἴπερ καὶ τοῦ μὴ ὑπάρχειν. | (0658A) Si autem non similis figurae sint propositiones, sit primum privativa necessaria, et A quidem nulli contingat B ex necessitate, B autem contingat omni C, necesse est igitur A nulli C inesse. Ponatur enim A inesse aut omni, aut alicui, positum autem est A nulli contingere B, quoniam ergo convertitur privativa, et B nulli A contingit, A autem positum est inesse C aut omni, aut alicui, quare nulli, aut non omni C continget B inesse, sed supponebatur omni ex principio. Manifestum autem quoniam et eius quod est contingere non inesse fit syllogismus, siquidem non inesse. | But if the premisses are not similar in quality, suppose first that the negative premiss is necessary, and let necessarily A not be possible for any B, but let B be possible for all C. It is necessary then that A belongs to no C. For suppose A to belong to all C or to some C. Now we assumed that A is not possible for any B. Since then the negative proposition is convertible, B is not possible for any A. But A is supposed to belong to all C or to some C. Consequently B will not be possible for any C or for all C. But it was originally laid down that B is possible for all C. And it is clear that the possibility of belonging can be inferred, since the fact of not belonging is inferred. |
| πάλιν ἔστω ἡ καταφατικὴ πρότασις ἀναγκαία, καὶ τὸ μὲν Α ἐνδεχέσθω μηδενὶ τῶι Β ὑπάρχειν, τὸ δὲ Β παντὶ τῶι Γ ὑπαρχέτω ἐξ ἀνάγκης. ὁ μὲν οὖν συλλογισμὸς ἔσται τέλειος, ἀλλ᾽ οὐ τοῦ μὴ ὑπάρχειν ἀλλὰ τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν· ἥ τε γὰρ πρότασις οὕτως ἐλήφθη ἡ ἀπὸ τοῦ μείζονος ἄκρου, καὶ εἰς τὸ ἀδύνατον οὐκ ἔστιν ἀγαγεῖν· εἰ γὰρ ὑποτεθείη τὸ Α τῶι Γ τινὶ ὑπάρχειν, κεῖται δὲ καὶ τῶι Β ἐνδέχεσθαι μηδενὶ ὑπάρχειν, οὐδὲν συμβαίνει διὰ τούτων ἀδύνατον. | Rursum sit affirmativa quidem propositio necessaria, et A quidem contingat nulli B inesse, B autem insit omni C ex necessitate. Ergo fit syllogismus quidem perfectus, sed non eius quod est non inesse, sed eius quod est contingere non inesse. (0658B) Nam et propositio sic sumpta est, quae ad maiorem est extremitatem, et ad impossibile non est ducere: nam si ponatur A inesse ulli C, positum est autem et A B contingere nulli inesse, nihil accidit per haec impossibile. | Again, let the affirmative premiss be necessary, and let A possibly not belong to any B, and let B necessarily belong to all C. The syllogism will be perfect, but it will establish a problematic negative, not an assertoric negative. For the major premiss was problematic, and further it is not possible to prove the assertoric conclusion per impossibile. For if it were supposed that A belongs to some C, and it is laid down that A possibly does not belong to any B, no impossible relation between B and C follows from these premisses. |
| ἐὰν δὲ πρὸς τῶι ἐλάττονι ἄκρωι τεθῆι τὸ στερητικόν, ὅταν μὲν ἐνδέχεσθαι σημαίνηι, συλλογισμὸς ἔσται διὰ τῆς ἀντιστροφῆς, καθάπερ ἐν τοῖς πρότερον, ὅταν δὲ μὴ ἐνδέχεσθαι, οὐκ ἔσται. οὐδ᾽ ὅταν ἄμφω μὲν τεθῆι στερητικά, μὴ ἦι δ᾽ ἐνδεχόμενον τὸ πρὸς τὸ ἔλαττον. ὅροι δ᾽ οἱ αὐτοί, τοῦ μὲν ὑπάρχειν λευκόν – ζῶιον – χιών, τοῦ δὲ μὴ ὑπάρχειν λευκόν – ζῶιον – πίττα. | Si autem ad minorem extremitatem ponatur privativum quando contingere quidem significaverit, syllogismus erit per conversionem, quemadmodum in prioribus. Quando autem non contingere, non erit ex necessitate, nec quando utraque quidem propositio privativa, non est autem contingens quod ad minorem est. Termini autem inesse quidem, album, animal, nix; non inesse quidem, album, animal, pix. | But if the minor premiss is negative, when it is problematic a syllogism is possible by conversion, as above; but when it is necessary no syllogism can be formed. Nor again when both premisses are negative, and the minor is necessary. The same terms as before serve both for the positive relation-white-animal-snow, and for the negative relation-white-animal-pitch. |
| Τὸν αὐτὸν δὲ τρόπον ἕξει κἀπὶ τῶν ἐν μέρει συλλογισμῶν. ὅταν μὲν γὰρ ἦι τὸ στερητικὸν ἀναγκαῖον, καὶ τὸ συμπέρασμα ἔσται τοῦ μὴ ὑπάρχειν. οἷον εἰ τὸ μὲν Α μηδενὶ τῶι Β ἐνδέχεται ὑπάρ χειν, τὸ δὲ Β τινὶ τῶι Γ ἐνδέχεται ὑπάρχειν, ἀνάγκη τὸ Α τινὶ τῶι Γ μὴ ὑπάρχειν. εἰ γὰρ παντὶ ὑπάρχει, τῶι δὲ Β μηδενὶ ἐνδέχεται, οὐδὲ τὸ Β οὐδενὶ τῶι Α ἐνδέχεται ὑπάρχειν. ὥστ᾽ εἰ τὸ Α παντὶ τῶι Γ ὑπάρχει, οὐδενὶ τῶι Γ τὸ Β ἐνδέχεται. ἀλλ᾽ ὑπέκειτό τινι ἐνδέχεσθαι. ὅταν δὲ τὸ ἐν μέρει καταφατικὸν ἀναγκαῖον ἦι, τὸ ἐν τῶι στερητικῶι συλλογισμῶι, οἷον τὸ Β Γ, ἢ τὸ κα ↵ θόλου τὸ ἐν τῶι κατηγορικῶι, οἷον τὸ Α Β, οὐκ ἔσται τοῦ ὑπάρχειν συλλογισμός. ἀπόδειξις δ᾽ ἡ αὐτὴ ἣ καὶ ἐπὶ τῶν πρότερον. ἐὰν δὲ τὸ μὲν καθόλου τεθῆι πρὸς τὸ ἔλαττον ἄκρον, ἢ καταφατικὸν ἢ στερητικόν, ἐνδεχόμενον, τὸ δ᾽ ἐν μέρει ἀναγ καῖον [πρὸς τῶι μείζονι ἄκρωι], οὐκ ἔσται συλλογισμός | Eodem autem modo se habebit, et in particularibus syllogismis. (0658C) Quando enim fuerit privativa necessaria, et conclusio erit eius quod est non inesse, ut si A quidem nulli B contingit inesse ex necessitate, B autem alicui C contingat inesse, necesse est A alicui eorum quae sunt C non inesse. Si enim A omni C inest, nulli autem contingit B, et B nulli A contingit inesse: quare si omni C inest A, nulli C contingit B, sed positum erat alicui contingere. Quando autem particularis affirmativa necessaria fuerit, quae in privativo est syllogismo, ut B C, aut universalis in affirmativo, ut A B, non erit inesse syllogismus. Demonstratio autem eadem quae in prioribus. Si autem universale quidem ponatur ad minorem extremitatem vel affirmativum vel privativum contingens, particulare autem necessarium, non erit syllogismus. | The same relation will obtain in particular syllogisms. Whenever the negative proposition is necessary, the conclusion will be negative assertoric: e.g. if it is not possible that A should belong to any B, but B may belong to some of the Cs, it is necessary that A should not belong to some of the Cs. For if A belongs to all C, but cannot belong to any B, neither can B belong to any A. So if A belongs to all C, to none of the Cs can B belong. But it was laid down that B may belong to some C. But when the particular affirmative in the negative syllogism, e.g. BC the minor premiss, or the universal proposition in the affirmative syllogism, e.g. AB the major premiss, is necessary, there will not be an assertoric conclusion. The demonstration is the same as before. But if the minor premiss is universal, and problematic, whether affirmative or negative, and the major premiss is particular and necessary, there cannot be a syllogism. |
| (ὅροι δὲ τοῦ μὲν ὑπάρχειν ἐξ ἀνάγκης ζῶιον – λευκόν – ἄνθρωπος, τοῦ δὲ μὴ ἐνδέχεσθαι ζῶιον – λευκόν – ἱμάτιον)· ὅταν δ᾽ ἀναγκαῖον ἦι τὸ καθόλου, τὸ δ᾽ ἐν μέρει ἐνδεχόμενον, στερητικοῦ μὲν ὄντος τοῦ καθόλου τοῦ μὲν ὑπάρχειν ὅροι ζῶιον – λευκόν – κόραξ, τοῦ δὲ μὴ ὑπάρχειν ζῶιον – λευκόν – πίττα, καταφατικοῦ δὲ τοῦ μὲν ὑπάρχειν ζῶιον – λευκόν – κύκνος, τοῦ δὲ μὴ ἐνδέχεσθαι ζῶιον – λευκόν – χιών. | Termini autem inesse quidem ex necessitate, animal, album, homo; non contingere autem, animal, album, tunica. (0658D) Quando similiter universale quidem est necessarium, particulare autem contingens, cum privativum quidem est universale, inesse quidem termini, animal, album, corvus; non inesse, animal, album, pix. Cum autem affirmativum, inesse quidem, animal, album, cygnus; non contingere autem, animal, album, nix. | Premisses of this kind are possible both where the relation is positive and necessary, e.g. animal-white-man, and where it is necessary and negative, e.g. animal-white-garment. But when the universal is necessary, the particular problematic, if the universal is negative we may take the terms animal-white-raven to illustrate the positive relation, or animal-white-pitch to illustrate the negative; and if the universal is affirmative we may take the terms animal-white-swan to illustrate the positive relation, and animal-white-snow to illustrate the negative and necessary relation. |
| οὐδ᾽ ὅταν ἀδιόριστοι ληφθῶσιν αἱ προτάσεις ἢ ἀμφότεραι κατὰ μέρος, οὐδ᾽ οὕτως ἔσται συλλογισμός. ὅροι δὲ κοινοὶ τοῦ μὲν ὑπάρχειν ζῶιον – λευκόν – ἄνθρωπος, τοῦ δὲ μὴ ὑπάρχειν ζῶιον – λευκόν – ἄψυχον. καὶ γὰρ τὸ ζῶιον τινὶ λευκῶι καὶ τὸ λευκὸν ἀψύχωι τινὶ καὶ ἀναγκαῖον ὑπάρ- χειν καὶ οὐκ ἐνδέχεται ὑπάρχειν. κἀπὶ τοῦ ἐνδέχεσθαι ὁμοίως, ὥστε πρὸς ἅπαντα χρήσιμοι οἱ ὅροι. | Nec quando indefinitae sumuntur propositiones, aut utraeque particulares, non sic erit syllogismus. Termini autem communes, inesse quidem, animal, album, homo; non inesse autem, animal, album, inanimatum. Nam et animal alicui albo, et album inanimato alicui est necessarium inesse, et non contingit inesse, et in contingenti similiter, quare ad omnia utiles sunt termini. | Nor again is a syllogism possible when the premisses are indefinite, or both particular. Terms applicable in either case to illustrate the positive relation are animal-white-man: to illustrate the negative, animal-white-inanimate. For the relation of animal to some white, and of white to some inanimate, is both necessary and positive and necessary and negative. Similarly if the relation is problematic: so the terms may be used for all cases. |
| Φανερὸν οὖν ἐκ τῶν εἰρημένων ὅτι ὁμοίως ἐχόντων τῶν ὅρων ἔν τε τῶι ὑπάρχειν καὶ ἐν τοῖς ἀναγκαίοις γίνεταί τε καὶ οὐ γίνεται συλλογισμός, πλὴν κατὰ μὲν τὸ ὑπάρχειν τιθεμένης τῆς στερητικῆς προτάσεως τοῦ ἐνδέχεσθαι ἦν ὁ συλλογισμός, κατὰ δὲ τὸ ἀναγκαῖον τῆς στερητικῆς καὶ τοῦ ἐνδέχεσθαι καὶ τοῦ μὴ ὑπάρχειν. [δῆλον δὲ καὶ ὅτι πάντες ἀτελεῖς οἱ συλ λογισμοὶ καὶ ὅτι τελειοῦνται διὰ τῶν προειρημένων σχημάτων.] | (0659A) Manifestum ergo ex iis quae dicta sunt, quoniam similiter habentibus se terminis, et in eo quod est inesse, et in necessariis, et fit, et non fit syllogismus, verumtamen secundum inesse quidem posita privativa propositione, eius quod est contingere erat syllogismus, secundum necessarium autem privativa, et contingere, et non inesse. Palam autem et quoniam omnes imperfecti syllogismi, et quomodo perficiuntur per praedictas figuras. | Clearly then from what has been said a syllogism results or not from similar relations of the terms whether we are dealing with simple existence or necessity, with this exception, that if the negative premiss is assertoric the conclusion is problematic, but if the negative premiss is necessary the conclusion is both problematic and negative assertoric. [It is clear also that all the syllogisms are imperfect and are perfected by means of the figures above mentioned.] |
Chapter 17
| Greek | Latin | English |
|---|---|---|
| (PL 64 0659A) CAPUT XVI. De syllogismis ex ambabus contingentibus in secunda figura. | 17 | |
| 36b26 Ἐν δὲ τῶι δευτέρωι σχήματι ὅταν μὲν ἐνδέχεσθαι λαμβάνωσιν ἀμφότεραι αἱ προτάσεις, οὐδεὶς ἔσται συλλογισμός, οὔτε κατηγορικῶν οὔτε στερητικῶν τιθεμένων, οὔτε καθόλου οὔτε κατὰ μέρος· ὅταν δὲ ἡ μὲν ὑπάρχειν ἡ δ᾽ ἐνδέχεσθαι σημαίνηι, τῆς μὲν καταφατικῆς ὑπάρχειν σημαινούσης οὐδέποτ᾽ ἔσται, τῆς δὲ στερητικῆς τῆς καθόλου ἀεί. τὸν αὐτὸν δὲ τρόπον καὶ ὅταν ἡ μὲν ἐξ ἀνάγκης ἡ δ᾽ ἐνδέχεσθαι λαμβάνηται τῶν προτάσεων. δεῖ δὲ καὶ ἐν τούτοις λαμβάνειν τὸ ἐν τοῖς συμπεράσμασιν ἐνδεχόμενον ὥσπερ ἐν τοῖς πρότερον. | In secunda autem figura quando contingentes quidem sumuntur utraeque propositiones, nullus erit syllogismus, sive sint affirmativae, sive privativae, sive universales, sive particulares. (0659B) Quando autem haec quidem inesse, illa vero contingere significat, affirmativa quidem inesse significante, nunquam erit syllogismus, privativa universali existente, semper. Eodem modo et quando haec quidem ex necessitate, illa vero contingere assumatur, oportet autem et in his accipere quod in conclusionibus est contingens quemadmodum in prioribus. | In the second figure whenever both premisses are problematic, no syllogism is possible, whether the premisses are affirmative or negative, universal or particular. But when one premiss is assertoric, the other problematic, if the affirmative is assertoric no syllogism is possible, but if the universal negative is assertoric a conclusion can always be drawn. Similarly when one premiss is necessary, the other problematic. Here also we must understand the term ‘possible’ in the conclusion, in the same sense as before. |
| Πρῶτον οὖν δεικτέον ὅτι οὐκ ἀντιστρέφει τὸ ἐν τῶι ἐνδέχεσθαι στερητικόν, οἷον εἰ τὸ Α ἐνδέχεται μηδενὶ τῶι Β, οὐκ ἀνάγκη καὶ τὸ Β ἐνδέχεσθαι μηδενὶ τῶι Α. κείσθω γὰρ τοῦτο, καὶ ἐνδεχέσθω τὸ Β μηδενὶ τῶι Α ὑπάρχειν. οὐκοῦν ἐπεὶ ἀντιστρέφουσιν αἱ ἐν τῶι ἐνδέχεσθαι καταφάσεις ταῖς ἀποφάσεσι, καὶ αἱ ἐναντίαι καὶ αἱ ἀντικείμεναι, τὸ δὲ Β τῶι Α ἐνδέχεται μηδενὶ ὑπάρ ↵ χειν, φανερὸν ὅτι καὶ παντὶ ἂν ἐνδέχοιτο τῶι Α ὑπάρχειν. τοῦτο δὲ ψεῦδος· οὐ γὰρ εἰ τόδε τῶιδε παντὶ ἐνδέχεται, καὶ τόδε τῶιδε ἀναγκαῖον· ὥστ᾽ οὐκ ἀντιστρέφει τὸ στερητικόν. | Primum igitur ostendendum quoniam non convertitur in contingenti, privativa, ut si A contingit nulli B, non necesse est et B contingere nulli A. Ponatur enim hoc et contingat B nulli A inesse, ergo quoniam convertuntur quae sunt in eo quod est contingere affirmationes negationibus, et contrariae, et contraiacentes, B autem contingit nulli A inesse, manifestum est quoniam et omni A contingit B inesse. Hoc autem falsum est. (0659C) Non enim si hoc huic omni contingit, et hoc huic contingat necessarium, quare non convertitur privativa. | First we must point out that the negative problematic proposition is not convertible, e.g. if A may belong to no B, it does not follow that B may belong to no A. For suppose it to follow and assume that B may belong to no A. Since then problematic affirmations are convertible with negations, whether they are contraries or contradictories, and since B may belong to no A, it is clear that B may belong to all A. But this is false: for if all this can be that, it does not follow that all that can be this: consequently the negative proposition is not convertible. |
| ἔτι δ᾽ οὐδὲν κωλύει τὸ μὲν Α τῶι Β ἐνδέχεσθαι μηδενί, τὸ δὲ Β τινὶ τῶν Α ἐξ ἀνάγκης μὴ ὑπάρχειν, οἷον τὸ μὲν λευκὸν παντὶ ἀνθρώπωι ἐνδέχεται μὴ ὑπάρχειν (καὶ γὰρ ὑπάρχειν), ἄνθρωπον δ᾽ οὐκ ἀληθὲς εἰπεῖν ὡς ἐνδέχεται μηδενὶ λευκῶι· πολλοῖς γὰρ ἐξ ἀνάγκης οὐχ ὑπάρχει, τὸ δ᾽ ἀναγκαῖον οὐκ ἦν ἐνδεχόμενον. | Amplius autem nihil prohibet A quidem contingere nulli B, B autem alicui A ex necessitate non inesse, ut album quidem contingit omni homini non inesse, nam et inesse hominem autem non verum est dicere, quoniam contingit nulli albo, pluribus enim ex necessitate non inest, necessarium autem non inerat contingens. | Further, these propositions are not incompatible, ‘A may belong to no B’, ‘B necessarily does not belong to some of the As’; e.g. it is possible that no man should be white (for it is also possible that every man should be white), but it is not true to say that it is possible that no white thing should be a man: for many white things are necessarily not men, and the necessary (as we saw) other than the possible. |
| Ἀλλὰ μὴν οὐδ᾽ ἐκ τοῦ ἀδυνάτου δειχθήσε ται ἀντιστρέφον, οἷον εἴ τις ἀξιώσειεν, ἐπεὶ ψεῦδος τὸ ἐνδέχεσθαι τὸ Β τῶι Α μηδενὶ ὑπάρχειν, ἀληθὲς τὸ μὴ ἐνδέχεσθαι μηδενί (φάσις γὰρ καὶ ἀπόφασισ), εἰ δὲ τοῦτ᾽, ἀληθὲς ἐξ ἀνάγκης τινὶ τῶι Α ὑπάρχειν· ὥστε καὶ τὸ Α τινὶ τῶι Β· τοῦτο δ᾽ ἀδύνατον. οὐ γὰρ εἰ μὴ ἐνδέχεται μηδενὶ τὸ Β τῶι Α, ἀνάγκη τινὶ ὑπάρχειν. τὸ γὰρ μὴ ἐνδέχεσθαι μηδενὶ διχῶς λέγεται, τὸ μὲν εἰ ἐξ ἀνάγκης τινὶ ὑπάρχει, τὸ δ᾽ εἰ ἐξ ἀνάγκης τινὶ μὴ ὑπάρχει· τὸ γὰρ ἐξ ἀνάγκης τινὶ τῶν Α μὴ ὑπάρχον οὐκ ἀληθὲς εἰπεῖν ὡς παντὶ ἐνδέχεται μὴ ὑπάρχειν, ὥσπερ οὐδὲ τὸ τινὶ ὑπάρχον ἐξ ἀνάγκης ὅτι παντὶ ἐνδέχεται ὑπάρχειν. | Sed nec ex impossibili ostendet convertens, ut si quis putet quoniam falsum est B contingere nulli A inesse, verum non contingere nulli A, affirmatio enim et negatio, si autem hoc verum, ex necessitate alicui A inesse B, quare et A alicui B inesse, hoc autem impossibile. Non enim si A non contingit nulli B, necesse est A alicui B inesse. (0659D) Nam non contingere nulli dicitur dupliciter, hoc quidem si ex necessitate alicui inest, illud vero si ex necessitate alicui non inest. Nam quod ex necessitate alicui eorum quae sunt A non inest, non est verum dicere quoniam omni contingit non inesse, quemadmodum nec alicui inest ex necessitate, quoniam omni contingit inesse. | Moreover it is not possible to prove the convertibility of these propositions by a reductio ad absurdum, i.e. by claiming assent to the following argument: ‘since it is false that B may belong to no A, it is true that it cannot belong to no A, for the one statement is the contradictory of the other. But if this is so, it is true that B necessarily belongs to some of the As: consequently A necessarily belongs to some of the Bs. But this is impossible.’ The argument cannot be admitted, for it does not follow that some A is necessarily B, if it is not possible that no A should be B. For the latter expression is used in two senses, one if A some is necessarily B, another if some A is necessarily not B. For it is not true to say that that which necessarily does not belong to some of the As may possibly not belong to any A, just as it is not true to say that what necessarily belongs to some A may possibly belong to all A. |
| εἰ οὖν τις ἀξιοίη, ἐπεὶ οὐκ ἐνδέχεται τὸ Γ τῶι Δ παντὶ ὑπάρχειν, ἐξ ἀνάγκης τινὶ μὴ ὑπάρχειν αὐτό, ψεῦδος ἂν λαμβάνοι· παντὶ γὰρ ὑπάρχει, ἀλλ᾽ ὅτι ἐνίοις ἐξ ἀνάγκης ὑπάρχει, διὰ τοῦτό φαμεν οὐ παντὶ ἐνδέχεσθαι. | Si ergo aliquis putet quoniam contingit C omni D inesse, ex necessitate alicui non inesse ipsum, falsum sumet, omni enim inest, si contingat, sed quoniam quibusdam ex necessitate inest, propter hoc dicimus non omni contingere. | If any one then should claim that because it is not possible for C to belong to all D, it necessarily does not belong to some D, he would make a false assumption: for it does belong to all D, but because in some cases it belongs necessarily, |
| ὥστε τῶι ἐνδέχεσθαι παντὶ ὑπάρχειν τό τ᾽ ἐξ ἀνάγ κης τινὶ ὑπάρχειν ἀντίκειται καὶ τὸ ἐξ ἀνάγκης τινὶ μὴ ὑπάρχειν. ὁμοίως δὲ καὶ τῶι ἐνδέχεσθαι μηδενί. δῆλον οὖν ὅτι πρὸς τὸ οὕτως ἐνδεχόμενον καὶ μὴ ἐνδεχόμενον ὡς ἐν ἀρχῆι διωρίσαμεν οὐ τὸ ἐξ ἀνάγκης τινὶ ὑπάρχειν ἀλλὰ τὸ ἐξ ἀνάγκης τινὶ μὴ ὑπάρχειν ληπτέον. τούτου δὲ ληφθέντος οὐδὲν συμβαίνει ἀδύνατον, ὥστ᾽ οὐ γίνεται συλλογισμός. φανερὸν οὖν ἐκ τῶν εἰρημένων ὅτι οὐκ ἀντιστρέφει τὸ στερητικόν. | Quare ei quod est contingere omni inesse, et ea quae est ex necessitate alicui inesse, opponitur, et ea quae est ex necessitate alicui non inesse, similiter autem et ei quae est contingere nulli. (0660A) Palam ergo quoniam ad sic contingens, et non contingens, ut in principio definivimus, non solum ex necessitate alicui inesse, sed et ex necessitate alicui non inesse sumendum. Hoc autem sumpto, nihil accidit impossibile, quare non fit syllogismus. Manifestum ergo ex iis quae dicta sunt quoniam non convertitur privativa. | therefore we say that it is not possible for it to belong to all. Hence both the propositions ‘A necessarily belongs to some B’ and ‘A necessarily does not belong to some B’ are opposed to the proposition ‘A belongs to all B’. Similarly also they are opposed to the proposition ‘A may belong to no B’. It is clear then that in relation to what is possible and not possible, in the sense originally defined, we must assume, not that A necessarily belongs to some B, but that A necessarily does not belong to some B. But if this is assumed, no absurdity results: consequently no syllogism. It is clear from what has been said that the negative proposition is not convertible. |
| Τούτου δὲ δειχθέντος κείσθω τὸ Α τῶι μὲν Β ἐνδέχεσθαι μηδενί, τῶι δὲ Γ παντί. διὰ μὲν οὖν τῆς ἀντιστροφῆς οὐκ ἔσται συλλογισμός· εἴρηται γὰρ ὅτι οὐκ ἀντιστρέφει ἡ τοιαύτη πρό τασις. ἀλλ᾽ οὐδὲ διὰ τοῦ ἀδυνάτου· τεθέντος γὰρ τοῦ Β ‹μὴ› παντὶ τῶι Γ ἐνδέχεσθαι ‹μὴ› ὑπάρχειν οὐδὲν συμβαίνει ψεῦδος· ἐνδέχοιτο γὰρ ἂν τὸ Α τῶι Γ καὶ παντὶ καὶ μηδενὶ ὑπάρχειν. | Hoc autem ostenso ponatur A, B quidem contingere nulli, C vero omni, per conversionem ergo non erit syllogismus. Dictum est enim quoniam non convertitur huiusmodi propositio. Sed nec per impossibile, nam posito B omni C contingere inesse, nihil accidit falsum, continget enim A et omni et nulli C inesse. | This being proved, suppose it possible that A may belong to no B and to all C. By means of conversion no syllogism will result: for the major premiss, as has been said, is not convertible. Nor can a proof be obtained by a reductio ad absurdum: for if it is assumed that B can belong to all C, no false consequence results: for A may belong both to all C and to no C. |
| ὅλως δ᾽ εἰ ἔστι συλλογισμός, δῆλον ὅτι τοῦ ἐνδέχεσθαι ἂν εἴη διὰ τὸ μηδετέραν τῶν προτάσεων εἰλῆφθαι ἐν τῶι ὑπάρ- χειν, καὶ οὗτος ἢ καταφατικὸς ἢ στερητικός· οὐδετέρως δ᾽ ἐγ ↵ χωρεῖ. καταφατικοῦ μὲν γὰρ τεθέντος δειχθήσεται διὰ τῶν ὅρων ὅτι οὐκ ἐνδέχεται ὑπάρχειν, στερητικοῦ δέ, ὅτι τὸ συμπέρασμα οὐκ ἐνδεχόμενον ἀλλ᾽ ἀναγκαῖόν ἐστιν. ἔστω γὰρ τὸ μὲν Α λευκόν, τὸ δὲ Β ἄνθρωπος, ἐφ᾽ ὧι δὲ Γ ἵππος. τὸ δὴ Α, τὸ λευκόν, ἐνδέχεται τῶι μὲν παντὶ τῶι δὲ μηδενὶ ὑπάρχειν. ἀλλὰ τὸ Β τῶι Γ οὔτε ὑπάρχειν ἐνδέχεται οὔτε μὴ ὑπάρχειν. | (0660B) Omnino autem si est syllogismus, palam quoniam contingens erit, eo quod neutra propositionum sumpta est in eo quod est inesse, et hic vel affirmativus, vel privativus: neutro autem modo possibile est, affirmativo enim posito, ostendetur per terminos quoniam non contingit inesse; privativo autem, quoniam conclusio non est contingens, sed necessaria. Sit enim A quidem album, B autem homo, in quo autem C equus, ergo album A contingit huic quidem omni, illi vero nulli inesse, sed B neque inesse contingit C, neque non inesse. | In general, if there is a syllogism, it is clear that its conclusion will be problematic because neither of the premisses is assertoric; and this must be either affirmative or negative. But neither is possible. Suppose the conclusion is affirmative: it will be proved by an example that the predicate cannot belong to the subject. Suppose the conclusion is negative: it will be proved that it is not problematic but necessary. Let A be white, B man, C horse. It is possible then for A to belong to all of the one and to none of the other. But it is not possible for B to belong nor not to belong to C. |
| ὅτι μὲν οὖν ὑπάρχειν οὐκ ἐγχωρεῖ, φανερόν· οὐδεὶς γὰρ ἵππος ἄνθρωπος. ἀλλ᾽ οὐδ᾽ ἐνδέχεσθαι μὴ ὑπάρχειν· ἀνάγκη γὰρ μηδένα ἵππον ἄνθρωπον εἶναι, τὸ δ᾽ ἀναγκαῖον οὐκ ἦν ἐνδεχόμενον. οὐκ ἄρα γίνεται συλλογισμός. | Quoniam igitur inesse non possibile, est manifestum, nullus enim equus homo, sed neque contingere non inesse, necesse est enim nullum equum hominem esse, necessarium autem non erat contingens, non igitur fit syllogismus. | That it is not possible for it to belong, is clear. For no horse is a man. Neither is it possible for it not to belong. For it is necessary that no horse should be a man, but the necessary we found to be different from the possible. No syllogism then results. |
| ὁμοίως δὲ δειχθήσεται καὶ ἂν ἀνάπαλιν τεθῆι τὸ στερητικόν, κἂν ἀμφότεραι καταφατικαὶ ληφθῶσιν ἢ στερητικαί (διὰ γὰρ τῶν αὐτῶν ὅρων ἔσται ἡ ἀπόδειξισ)· καὶ ὅταν ἡ μὲν καθόλου ἡ δ᾽ ἐν μέρει, ἢ ἀμφότεραι κατὰ μέρος ἢ ἀδιόριστοι, ἢ ὁσα χῶς ἄλλως ἐνδέχεται μεταλαβεῖν τὰς προτάσεις· ἀεὶ γὰρ ἔσται διὰ τῶν αὐτῶν ὅρων ἡ ἀπόδειξις. φανερὸν οὖν ὅτι ἀμφοτέρων τῶν προτάσεων κατὰ τὸ ἐνδέχεσθαι τιθεμένων οὐδεὶς γίνεται συλλογισμός. | Similiter autem ostendetur, et si e converso ponatur privativa, et si utraeque affirmative ponantur, vel privative, nam per eosdem terminos erit demonstratio. (0660C) Et quando haec quidem universalis, illa vero particularis, vel utraeque particulares, vel indefinitae, aut quolibet modo aliter contingit permutari propositiones, semper enim erit per eosdem terminos demonstratio. Manifestum ergo quoniam utrisque propositionibus secundum contingere positis, nullus fit syllogismus. | A similar proof can be given if the major premiss is negative, the minor affirmative, or if both are affirmative or negative. The demonstration can be made by means of the same terms. And whenever one premiss is universal, the other particular, or both are particular or indefinite, or in whatever other way the premisses can be altered, the proof will always proceed through the same terms. Clearly then, if both the premisses are problematic, no syllogism results. |
Chapter 18
| Greek | Latin | English |
|---|---|---|
| (PL 64 0660C) CAPUT XVII/ XVIII. Mixtio absoluti et contingentis in secunda figura | 18 | |
| 37b19 Εἰ δ᾽ ἡ μὲν ὑπάρχειν ἡ δ᾽ ἐνδέχεσθαι σημαίνει, τῆς μὲν κατηγορικῆς ὑπάρχειν τεθείσης τῆς δὲ στερητικῆς ἐνδέχεσθαι οὐδέποτ᾽ ἔσται συλλογισμός, οὔτε καθόλου τῶν ὅρων οὔτ᾽ ἐν μέρει λαμβανομένων (ἀπόδειξις δ᾽ ἡ αὐτὴ καὶ διὰ τῶν αὐτῶν ὅρων)· ὅταν δ᾽ ἡ μὲν καταφατικὴ ἐνδέχεσθαι ἡ δὲ στερητικὴ ὑπάρχειν, ἔσται συλλογισμός. εἰλήφθω γὰρ τὸ Α τῶι μὲν Β μηδενὶ ὑπάρχειν, τῶι δὲ Γ παντὶ ἐνδέχεσθαι. ἀντιστραφέντος οὖν τοῦ στερητικοῦ τὸ Β τῶι Α οὐδενὶ ὑπάρξει· τὸ δὲ Α παντὶ τῶι Γ ἐνεδέχετο· γίνεται δὴ συλλογισμὸς ὅτι ἐνδέχεται τὸ Β μηδενὶ τῶι Γ διὰ τοῦ πρώτου σχήματος. ὁμοίως δὲ καὶ εἰ πρὸς τῶι Γ τεθείη τὸ στερητικόν. | Si autem altera quidem inesse, altera vero contingere significat, praedicativa quidem inesse posita, privativa vero contingere, nunquam erit syllogismus, sive universaliter, sive particulariter sumantur termini, demonstratio autem eadem, et per eosdem terminos. Quando autem affirmativa quidem contingere, privativa inesse, erit syllogismus. (0660D) Sumatur enim A B quidem nulli inesse, C vero omnia contingere, conversa ergo privativa, B inest nulli A, A autem omni C contingebat, fit ergo syllogismus, quoniam B contingit nulli C, per primam figuram. Similiter autem et si ad C ponatur privativa. | But if one premiss is assertoric, the other problematic, if the affirmative is assertoric and the negative problematic no syllogism will be possible, whether the premisses are universal or particular. The proof is the same as above, and by means of the same terms. But when the affirmative premiss is problematic, and the negative assertoric, we shall have a syllogism. Suppose A belongs to no B, but can belong to all C. If the negative proposition is converted, B will belong to no A. But ex hypothesi can belong to all C: so a syllogism is made, proving by means of the first figure that B may belong to no C. Similarly also if the minor premiss is negative.
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| ἐὰν δ᾽ ἀμ φότεραι μὲν ὦσι στερητικαί, σημαίνηι δ᾽ ἡ μὲν μὴ ὑπάρχειν ἡ δ᾽ ἐνδέχεσθαι, δι᾽ αὐτῶν μὲν τῶν εἰλημμένων οὐδὲν συμβαίνει ἀναγκαῖον, ἀντιστραφείσης δὲ τῆς κατὰ τὸ ἐνδέχεσθαι προτάσεως γίγνεται συλλογισμὸς ὅτι τὸ Β τῶι Γ ἐνδέχεται μηδενὶ ὑπάρχειν, καθάπερ ἐν τοῖς πρότερον· ἔσται γὰρ πάλιν τὸ πρῶτον σχῆμα. ἐὰν δ᾽ ἀμφότεραι τεθῶσι κατηγορικαί, οὐκ ἔσται συλλογισμός. ὅροι τοῦ μὲν ὑπάρχειν ὑγίεια – ζῶιον – ἄνθρωπος, τοῦ δὲ μὴ ὑπάρχειν ὑγίεια – ἵππος – ἄνθρωπος. | Si autem utraeque sint privativae, significet autem haec quidem non inesse, illa vero contingere non inesse, per ea quidem quae sumpta sunt nihil accidit necessarium, conversa autem secundum contingere propositione fit syllogismus, quoniam B contingit nulli C inesse, quemadmodum in prioribus, erit enim rursum prima figura. Si autem utraeque ponantur praedicativae, non erit syllogismus. Termini quidem inesse sanitas, equus, homo. | But if both premisses are negative, one being assertoric, the other problematic, nothing follows necessarily from these premisses as they stand, but if the problematic premiss is converted into its complementary affirmative a syllogism is formed to prove that B may belong to no C, as before: for we shall again have the first figure. But if both premisses are affirmative, no syllogism will be possible. This arrangement of terms is possible both when the relation is positive, e.g. health, animal, man, and when it is negative, e.g. health, horse, man. |
| Τὸν αὐτὸν δὲ τρόπον ἕξει κἀπὶ τῶν ἐν μέρει συλλογισμῶν. ὅταν μὲν γὰρ ἦι τὸ καταφατικὸν ὑπάρχον, εἴτε κα ↵ θόλου εἴτ᾽ ἐν μέρει ληφθέν, οὐδεὶς ἔσται συλλογισμός (τοῦτο δ᾽ ὁμοίως καὶ διὰ τῶν αὐτῶν ὅρων δείκνυται τοῖς πρότερον), ὅταν δὲ τὸ στερητικόν, ἔσται διὰ τῆς ἀντιστροφῆς, καθάπερ ἐν τοῖς πρότερον. | Eodem autem modo se habebit et in particularibus syllogismis. (0661A) Quando autem erit affirmativa inesse, sive universaliter, sive particulariter sumpta, nullus erit syllogismus; hoc autem similiter, et per eosdem terminos demonstratur, quibus et prius. Quando autem et privativa, erit per conversionem, quemadmodum in prioribus. | The same will hold good if the syllogisms are particular. Whenever the affirmative proposition is assertoric, whether universal or particular, no syllogism is possible (this is proved similarly and by the same examples as above), but when the negative proposition is assertoric, a conclusion can be drawn by means of conversion, as before. |
| πάλιν ἐὰν ἄμφω μὲν τὰ διαστήματα στερη τικὰ ληφθῆι, καθόλου δὲ τὸ μὴ ὑπάρχειν, ἐξ αὐτῶν μὲν τῶν προτάσεων οὐκ ἔσται τὸ ἀναγκαῖον, ἀντιστραφέντος δὲ τοῦ ἐνδέχεσθαι καθάπερ ἐν τοῖς πρότερον ἔσται συλλογισμός. ἐὰν δὲ ὑπάρχον μὲν ἦι τὸ στερητικόν, ἐν μέρει δὲ ληφθῆι, οὐκ ἔσται συλλογισμός, οὔτε καταφατικῆς οὔτε στερητικῆς οὔσης τῆς ἑτέρας προτάσεως. οὐδ᾽ ὅταν ἀμφότεραι ληφθῶσιν ἀδιόριστοι – ἢ καταφατικαὶ ἢ ἀποφατικαί – ἢ κατὰ μέρος. ἀπόδειξις δ᾽ ἡ αὐτὴ καὶ διὰ τῶν αὐτῶν ὅρων. | Rursum si ambo quidem intervalla privativa sumantur, universaliter autem quod non inesse, ex ipsis quidem propositionibus non erit necessarium, conversa autem contingenti sicut in prioribus, erit syllogismus. Si autem inesse quidem sit privativa, particulariter quidem sumpta, non erit syllogismus, neque praedicativa, neque privativa existente altera propositione. Nec quando utraeque ponuntur indefinitae, vel affirmativae, vel negativae, aut particulares; demonstratio autem eadem et per eosdem terminos. | Again if both the relations are negative, and the assertoric proposition is universal, although no conclusion follows from the actual premisses, a syllogism can be obtained by converting the problematic premiss into its complementary affirmative as before. But if the negative proposition is assertoric, but particular, no syllogism is possible, whether the other premiss is affirmative or negative. Nor can a conclusion be drawn when both premisses are indefinite, whether affirmative or negative, or particular. The proof is the same and by the same terms. |
Chapter 19
| Greek | Latin | English |
|---|---|---|
| (PL 64 0661A) CAPUT XVIII/XIX. Mixtio necessarii et contingentis in secunda figura. | 19 | |
| 38a13 Ἐὰν δ᾽ ἡ μὲν ἐξ ἀνάγκης ἡ δ᾽ ἐνδέχεσθαι σημαίνηι τῶν προτάσεων, τῆς μὲν στερητικῆς ἀναγκαίας οὔσης ἔσται συλλογισμός, οὐ μόνον ὅτι ἐνδέχεται μὴ ὑπάρχειν, ἀλλὰ καὶ ὅτι οὐχ ὑπάρχει, τῆς δὲ καταφατικῆς οὐκ ἔσται. | (0661B) Si autem haec quidem propositionum ex necessitate, illa vero contingere significat, privativa quidem necessaria, erit syllogismus, non solum quoniam contingit non inesse, sed et quoniam non inest, affirmativa autem non erit. | If one of the premisses is necessary, the other problematic, then if the negative is necessary a syllogistic conclusion can be drawn, not merely a negative problematic but also a negative assertoric conclusion; but if the affirmative premiss is necessary, no conclusion is possible. |
| κείσθω γὰρ τὸ Α τῶι μὲν Β ἐξ ἀνάγκης μηδενὶ ὑπάρχειν, τῶι δὲ Γ παντὶ ἐνδέχεσθαι. ἀντιστραφείσης οὖν τῆς στερητικῆς οὐδὲ τὸ Β τῶι Α οὐδενὶ ὑπάρξει· τὸ δὲ Α παντὶ τῶι Γ ἐνεδέχετο· γίνεται δὴ πάλιν διὰ τοῦ πρώτου σχήματος ὁ συλλογισμὸς ὅτι τὸ Β τῶι Γ ἐνδέχεται μηδενὶ ὑπάρχειν. ἅμα δὲ δῆλον ὅτι οὐδ᾽ ὑπάρξει τὸ Β οὐδενὶ τῶι Γ. κείσθω γὰρ ὑπάρχειν· οὐκοῦν εἰ τὸ Α τῶι Β μηδενὶ ἐνδέχεται, τὸ δὲ Β ὑπάρχει τινὶ τῶι Γ, τὸ Α τῶι Γ τινὶ οὐκ ἐνδέχεται· ἀλλὰ παντὶ ὑπέ κειτο ἐνδέχεσθαι. τὸν αὐτὸν δὲ τρόπον δειχθήσεται καὶ εἰ πρὸς τῶι Γ τεθείη τὸ στερητικόν. | Ponatur autem A B quidem nulli inesse ex necessitate, C autem omni contingere, conversa ergo privativa, et B nulli A inerit, A autem omni E contingebat. Fit igitur rursum per primam figuram syllogismus, quoniam B contingit nulli C inesse. Simul autem manifestum quoniam neque inest B nulli C, ponatur enim inesse, ergo si A nulli B contingit, B autem inest alicui C, A alicui C non contingit, sed omni ponebatur contingere. (0661C) Eodem autem modo ostendetur, et si ad C ponatur privativum. | Suppose that A necessarily belongs to no B, but may belong to all C. If the negative premiss is converted B will belong to no A: but A ex hypothesi is capable of belonging to all C: so once more a conclusion is drawn by the first figure that B may belong to no C. But at the same time it is clear that B will not belong to any C. For assume that it does: then if A cannot belong to any B, and B belongs to some of the Cs, A cannot belong to some of the Cs: but ex hypothesi it may belong to all. A similar proof can be given if the minor premiss is negative. |
| Πάλιν ἔστω τὸ κατηγορικὸν ἀναγκαῖον, θάτερον δ᾽ ἐνδεχόμενον, καὶ τὸ Α τῶι μὲν Β ἐν- δεχέσθω μηδενί, τῶι δὲ Γ παντὶ ὑπαρχέτω ἐξ ἀνάγκης. οὕτως οὖν ἐχόντων τῶν ὅρων οὐδεὶς ἔσται συλλογισμός. συμ βαίνει γὰρ τὸ Β τῶι Γ ἐξ ἀνάγκης μὴ ὑπάρχειν. | Rursum. Sit praedicativa quidem necessaria, altera autem privativa, et contingens, et A B contingat nulli, C autem omni insit ex necessitate, sic ergo habentibus se terminis, nullus erit syllogismus, accidit enim B ex necessitate non inesse. | Again let the affirmative proposition be necessary, and the other problematic; i.e. suppose that A may belong to no B, but necessarily belongs to all C. When the terms are arranged in this way, no syllogism is possible. For (1) it sometimes turns out that B necessarily does not belong to C. |
| ἔστω γὰρ τὸ μὲν Α λευκόν, ἐφ᾽ ὧι δὲ τὸ Β ἄνθρωπος, ἐφ᾽ ὧι δὲ τὸ Γ κύκνος. τὸ δὴ λευκὸν κύκνωι μὲν ἐξ ἀνάγκης ὑπάρχει, ἀνθρώπωι δ᾽ ἐνδέχεται μηδενί· καὶ ἄνθρωπος οὐδενὶ κύκνωι ἐξ ἀνάγκης. ὅτι μὲν οὖν τοῦ ἐνδέχεσθαι οὐκ ἔστι συλλογισμός, φανερόν· τὸ γὰρ ἐξ ἀνάγκης οὐκ ἦν ἐνδεχόμενον. | Sit enim A quidem album, in quo autem B, homo, in quo vero C, cygnus, ergo album cygno quidem ex necessitate inest, homini autem contingit nulli, et homo nulli cygno ex necessitate. Quoniam igitur eius quod est contingere non est syllogismus, manifestum est, nam ex necessitate non erat contingens. | Let A be white, B man, C swan. White then necessarily belongs to swan, but may belong to no man; and man necessarily belongs to no swan; Clearly then we cannot draw a problematic conclusion; for that which is necessary is admittedly distinct from that which is possible. |
| ἀλλὰ μὴν οὐδὲ τοῦ ἀναγκαίου· τὸ γὰρ ἀναγκαῖον ἢ ἐξ ἀμφοτέρων ἀναγκαίων ἢ ἐκ τῆς στερητικῆς συνέβαινεν. ἔτι δὲ καὶ ἐγχωρεῖ τούτων κειμένων τὸ Β τῶι Γ ὑπάρχειν· οὐδὲν γὰρ κωλύει τὸ μὲν Γ ὑπὸ τὸ Β εἶναι, τὸ δὲ Α τῶι μὲν Β παντὶ ἐνδέχεσθαι, τῶι δὲ Γ ἐξ ἀνάγκης ὑπάρχειν, οἷον εἰ τὸ μὲν Γ εἴη ἐγρηγορός, τὸ δὲ Β ζῶιον, τὸ δ᾽ ἐφ᾽ ὧι τὸ Α κίνησις. τῶι μὲν γὰρ ἐγρηγορότι ἐξ ἀνάγ ↵ κης κίνησις, ζώιωι δὲ παντὶ ἐνδέχεται· καὶ πᾶν τὸ ἐγρηγορὸς ζῶιον. φανερὸν οὖν ὅτι οὐδὲ τοῦ μὴ ὑπάρχειν, εἴπερ οὕτως ἐχόντων ἀνάγκη ὑπάρχειν. οὐδὲ δὴ τῶν ἀντικειμένων καταφάσεων, ὥστ᾽ οὐδεὶς ἔσται συλλογισμός. ὁμοίως δὲ δειχθήσεται καὶ ἀνάπαλιν τεθείσης τῆς καταφατικῆς. | Sed tamen non necessarii, nam necessarium aut ex utrisque necessariis, aut ex privativa necessaria contingebat. Amplius et possibile est iis positis B inesse C. (0661D) Nihil enim prohibet C quidem sub B esse, A autem B quidem omni contingere, C vero ex necessitate inesse, ut sit quidem C vigilia, B autem animal, in quo autem A motus. Nam vigilanti quidem ex necessitate inest motus, animali autem nulli contingit, et omne vigilans animal. Manifestum ergo quoniam non eius quod est non inesse, siquidem sic se habentibus terminis, necesse est inesse, neque autem oppositarum affirmationum, quare nullus erit syllogismus. Similiter autem ostendetur, et e converso posita affirmativa. | (2) Nor again can we draw a necessary conclusion: for that presupposes that both premisses are necessary, or at any rate the negative premiss. (3) Further it is possible also, when the terms are so arranged, that B should belong to C: for nothing prevents C falling under B, A being possible for all B, and necessarily belonging to C; e.g. if C stands for ‘awake’, B for ‘animal’, A for ‘motion’. For motion necessarily belongs to what is awake, and is possible for every animal: and everything that is awake is animal. Clearly then the conclusion cannot be the negative assertion, if the relation must be positive when the terms are related as above. Nor can the opposite affirmations be established: consequently no syllogism is possible. A similar proof is possible if the major premiss is affirmative. |
| Ἐὰν δ᾽ ὁμοιοσχήμονες ὦσιν αἱ προτάσεις, στερητικῶν μὲν οὐσῶν ἀεὶ γίνεται συλλογισμὸς ἀντιστραφείσης τῆς κατὰ τὸ ἐνδέχεσθαι προτάσεως καθάπερ ἐν τοῖς πρότερον. εἰλήφθω γὰρ τὸ Α τῶι μὲν Β ἐξ ἀνάγκης μὴ ὑπάρχειν, τῶι δὲ Γ ἐνδέχεσθαι μὴ ὑπάρχειν· ἀντιστραφεισῶν οὖν τῶν προτάσεων τὸ μὲν Β τῶι Α οὐδενὶ ὑπάρχει, τὸ δὲ Α παντὶ τῶι Γ ἐνδέχεται· γίνεται δὴ τὸ πρῶτον σχῆμα. κἂν εἰ πρὸς τῶι Γ τεθείη τὸ στερητικόν, ὡσαύτως. ἐὰν δὲ κατηγορικαὶ τεθῶσιν, οὐκ ἔσται συλλογισμός. τοῦ μὲν γὰρ μὴ ὑπάρχειν ἢ τοῦ ἐξ ἀνάγκης μὴ ὑπάρχειν φανερὸν ὅτι οὐκ ἔσται διὰ τὸ μὴ εἰλῆφθαι στερητικὴν πρότασιν μήτ᾽ ἐν τῶι ὑπάρχειν μήτ᾽ ἐν τῶι ἐξ ἀνάγκης ὑπάρχειν. ἀλλὰ μὴν οὐδὲ τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν· ἐξ ἀνάγκης γὰρ οὕτως ἐχόντων τὸ Β τῶι Γ οὐχ ὑπάρξει, οἷον εἰ τὸ μὲν Α τε θείη λευκόν, ἐφ᾽ ὧι δὲ τὸ Β κύκνος, τὸ δὲ Γ ἄνθρωπος. οὐδέ γε τῶν ἀντικειμένων καταφάσεων, ἐπεὶ δέδεικται τὸ Β τῶι Γ ἐξ ἀνάγκης οὐχ ὑπάρχον. οὐκ ἄρα γίνεται συλλογισμὸς ὅλως. | Si autem similis figurae sint propositiones, cum privativae sint, semper fit syllogismus, conversa secundum contingere propositione, quemadmodum in prioribus. (0662A) Si sumatur enim A B quidem ex necessitate non inesse, C autem contingere non inesse, conversis autem propositionibus, B quidem nulli inesse A, A autem omni C contingit, fit igitur prima figura, et si ad C ponatur privativum similiter. Si autem praedicativae ponantur, non erit syllogismus, nam eius quod est non inesse, aut eius quod est ex necessitate non inesse, manifestum quoniam non erit, eo quod non sumpta sit privativa propositio, neque in eo quod est inesse, neque in eo quod est ex necessitate inesse, sed neque eius quod est contingere non inesse, ex necessitate enim sic se habentibus, B non inerit C, ut si A quidem ponatur album, in quo autem B cygnus, in quo autem C homo, neque oppositarum affirmationum, quoniam ostensum est B ex necessitate non inesse C, non ergo fit syllogismus omnino. | But if the premisses are similar in quality, when they are negative a syllogism can always be formed by converting the problematic premiss into its complementary affirmative as before. Suppose A necessarily does not belong to B, and possibly may not belong to C: if the premisses are converted B belongs to no A, and A may possibly belong to all C: thus we have the first figure. Similarly if the minor premiss is negative. But if the premisses are affirmative there cannot be a syllogism. Clearly the conclusion cannot be a negative assertoric or a negative necessary proposition because no negative premiss has been laid down either in the assertoric or in the necessary mode. Nor can the conclusion be a problematic negative proposition. For if the terms are so related, there are cases in which B necessarily will not belong to C; e.g. suppose that A is white, B swan, C man. Nor can the opposite affirmations be established, since we have shown a case in which B necessarily does not belong to C. A syllogism then is not possible at all. |
| Ὁμοίως δ᾽ ἕξει κἀπὶ τῶν ἐν μέρει συλλογισμῶν· ὅταν μὲν γὰρ ἦι τὸ στερητικὸν καθόλου τε καὶ ἀναγκαῖον, ἀεὶ συλλογισμὸς ἔσται καὶ τοῦ ἐνδέχεσθαι καὶ τοῦ μὴ ὑπάρχειν (ἀπόδειξις δὲ διὰ τῆς ἀντιστροφῆσ), ὅταν δὲ τὸ καταφατικόν, οὐδέποτε· τὸν αὐτὸν γὰρ τρόπον δειχθήσεται ὃν καὶ ἐν τοῖς καθόλου, καὶ διὰ τῶν αὐτῶν ὅρων. οὐδ᾽ ὅταν ἀμφότεραι ληφθῶσι καταφατικαί· καὶ γὰρ τούτου ἡ αὐτὴ ἀπόδειξις ἣ καὶ πρότερον. ὅταν δὲ ἀμφότεραι μὲν στερητικαί, καθόλου δὲ καὶ ἀναγκαία ἡ τὸ μὴ ὑπάρχειν σημαίνουσα, δι᾽ αὐτῶν μὲν τῶν εἰλημμένων οὐκ ἔσται τὸ ἀναγκαῖον, ἀντιστραφείσης δὲ τῆς κατὰ τὸ ἐνδέχεσθαι προτά σεως ἔσται συλλογισμός, καθάπερ ἐν τοῖς πρότερον. ἐὰν δ᾽ ἀμφότεραι ἀδιόριστοι ἢ ἐν μέρει τεθῶσιν, οὐκ ἔσται συλλογισμός. ἀπόδειξις δ᾽ ἡ αὐτὴ καὶ διὰ τῶν αὐτῶν ὅρων. | Similiter autem se habebit et in particularibus syllogismis. (0662B) Quando autem fuerit privativa, et universalis, et necessaria, semper erit syllogismus, et eius quod est contingere non inesse, et eius quod est non inesse, demonstratio autem per conversionem. Quando autem affirmativa, nunquam, eodem autem modo ostendetur quo et in universalibus, et per eosdem terminos. Nec quando utraeque sumuntur affirmative, nam et huius eadem demonstratio, quae et prius. Quando utraeque quidem privativae, universalis autem et necessaria, quae non inesse significat, per ea quidem quae sumpta sunt, non erit necessarium, conversa autem secundum contingere propositione, erit syllogismus, quemadmodum in prioribus. (0662C) Si autem utraeque indefinitae, vel particulares sumantur, non erit syllogismus, demonstratio autem eadem, et per eosdem terminos. | Similar relations will obtain in particular syllogisms. For whenever the negative proposition is universal and necessary, a syllogism will always be possible to prove both a problematic and a negative assertoric proposition (the proof proceeds by conversion); but when the affirmative proposition is universal and necessary, no syllogistic conclusion can be drawn. This can be proved in the same way as for universal propositions, and by the same terms. Nor is a syllogistic conclusion possible when both premisses are affirmative: this also may be proved as above. But when both premisses are negative, and the premiss that definitely disconnects two terms is universal and necessary, though nothing follows necessarily from the premisses as they are stated, a conclusion can be drawn as above if the problematic premiss is converted into its complementary affirmative. But if both are indefinite or particular, no syllogism can be formed. The same proof will serve, and the same terms. |
| Φανερὸν οὖν ἐκ τῶν εἰρημένων ὅτι τῆς μὲν στερητικῆς τῆς καθόλου τιθεμένης ἀναγκαίας ἀεὶ γίνεται συλλογι σμὸς οὐ μόνον τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν, ἀλλὰ καὶ τοῦ μὴ ὑπάρχειν, τῆς δὲ καταφατικῆς οὐδέποτε. καὶ ὅτι τὸν αὐτὸν τρόπον ἐχόντων ἔν τε τοῖς ἀναγκαίοις καὶ ἐν τοῖς ↵ ὑπάρχουσι γίνεταί τε καὶ οὐ γίνεται συλλογισμός. δῆλον δὲ καὶ ὅτι πάντες ἀτελεῖς οἱ συλλογισμοί, καὶ ὅτι τελειοῦνται διὰ τῶν προειρημένων σχημάτων. | Manifestum igitur ex praedictis quoniam privativa quidem universalis posita necessaria, semper fit syllogismus, non solum eius, quod est contingere non inesse, sed et non inesse, affirmativa autem nunquam. Et quoniam eodem modo se habentibus, et in necessariis, et in iis quae insunt, fit et non fit syllogismus Palam et quoniam imperfecti omnes sunt syllogismi, et quoniam omnes perficiuntur per praedictas figuras. | It is clear then from what has been said that if the universal and negative premiss is necessary, a syllogism is always possible, proving not merely a negative problematic, but also a negative assertoric proposition; but if the affirmative premiss is necessary no conclusion can be drawn. It is clear too that a syllogism is possible or not under the same conditions whether the mode of the premisses is assertoric or necessary. And it is clear that all the syllogisms are imperfect, and are completed by means of the figures mentioned. |
Chapter 20
| Greek | Latin | English |
|---|---|---|
| (PL 64 0662C) CAPUT XIX/XX. De syllogismis ex ambabus contingentibus in tertia figura. | 20 | |
| 39a4Ἐν δὲ τῶι τελευταίωι σχήματι καὶ ἀμφοτέρων ἐν δεχομένων καὶ τῆς ἑτέρας ἔσται συλλογισμός. ὅταν μὲν οὖν ἐνδέχεσθαι σημαίνωσιν αἱ προτάσεις, καὶ τὸ συμπέρασμα ἔσται ἐνδεχόμενον· καὶ ὅταν ἡ μὲν ἐνδέχεσθαι ἡ δ᾽ ὑπάρχειν. ὅταν δ᾽ ἡ ἑτέρα τεθῆι ἀναγκαία, ἐὰν μὲν ἦι καταφατική, οὐκ ἔσται τὸ συμπέρασμα οὔτε ἀναγκαῖον οὔθ᾽ ὑπάρχον, ἐὰν δ᾽ ἦι στερητική, τοῦ μὴ ὑπάρχειν ἔσται συλλογισμός, καθάπερ καὶ ἐν τοῖς πρότερον· ληπτέον δὲ καὶ ἐν τούτοις ὁμοίως τὸ ἐν τοῖς συμπεράσμασιν ἐνδεχόμενον. | In postrema autem figura, et utrisque contingentibus, et altera, erit syllogismus. (0662D) Quando ergo contingere significant propositiones, et conclusio erit contingens. Et quando haec quidem contingere, illa vero inesse, similiter erit syllogismus. Quando autem altera ponitur necessaria, si affirmativa quidem non erit conclusio, neque necessaria, neque inesse. Si autem privativa, eius quod est non inesse erit syllogismus, quemadmodum in prioribus. | In the last figure a syllogism is possible whether both or only one of the premisses is problematic. When the premisses are problematic the conclusion will be problematic; and also when one premiss is problematic, the other assertoric. But when the other premiss is necessary, if it is affirmative the conclusion will be neither necessary or assertoric; but if it is negative the syllogism will result in a negative assertoric proposition, as above. In these also we must understand the expression ‘possible’ in the conclusion in the same way as before. |
| Ἔστωσαν δὴ πρῶτον ἐνδεχόμεναι, καὶ τὸ Α καὶ τὸ Β παντὶ τῶι Γ ἐνδεχέσθω ὑπάρχειν. ἐπεὶ οὖν ἀντιστρέφει τὸ καταφατικὸν ἐπὶ μέρους, τὸ δὲ Β παντὶ τῶι Γ ἐνδέχεται, καὶ τὸ Γ τινὶ τῶι Β ἐνδέχοιτ᾽ ἄν. ὥστ᾽ εἰ τὸ μὲν Α παντὶ τῶι Γ ἐνδέχεται, τὸ δὲ Γ τινὶ τῶι Β, ἀνάγκη καὶ τὸ Α τινὶ τῶι Β ἐνδέχεσθαι· γίγνεται γὰρ τὸ πρῶτον σχῆμα. | Sumendum autem et in his similiter, quod est in conclusionibus contingens. Sint ergo primum contingentes, et A et B contingant omni C inesse, quoniam ergo convertitur affirmativa particulariter, B autem omni C contingit, et C alicui B contingit, quare si A quidem omni C contingit, C autem alicui B, et A alicui B contingit, fit enim prima figura. | First let the premisses be problematic and suppose that both A and B may possibly belong to every C. Since then the affirmative proposition is convertible into a particular, and B may possibly belong to every C, it follows that C may possibly belong to some B. So, if A is possible for every C, and C is possible for some of the Bs, then A is possible for some of the Bs. For we have got the first figure. |
| καὶ εἰ τὸ μὲν Α ἐνδέχεται μηδενὶ τῶι Γ ὑπάρχειν, τὸ δὲ Β παντὶ τῶι Γ, ἀνάγκη τὸ Α τινὶ τῶι Β ἐνδέχεσθαι μὴ ὑπάρχειν· ἔσται γὰρ πάλιν τὸ πρῶτον σχῆμα διὰ τῆς ἀντιστροφῆς. εἰ δ᾽ ἀμφότεραι στερητικαὶ τεθείησαν, ἐξ αὐτῶν μὲν τῶν εἰλημμένων οὐκ ἔσται τὸ ἀναγκαῖον, ἀντιστραφει σῶν δὲ τῶν προτάσεων ἔσται συλλογισμός, καθάπερ ἐν τοῖς πρότερον. | (0663A) Et si A quidem contingit nulli C inesse, B autem omni C contingat, necesse est A alicui cui B contingere non inesse, erit enim rursum prima figura per conversionem. Si autem utraeque privativae ponantur, ex his quidem quae sumpta sunt non erit necessarium, conversis autem propositionibus erit syllogismus, quemadmodum in prioribus. | And A if may possibly belong to no C, but B may possibly belong to all C, it follows that A may possibly not belong to some B: for we shall have the first figure again by conversion. But if both premisses should be negative no necessary consequence will follow from them as they are stated, but if the premisses are converted into their corresponding affirmatives there will be a syllogism as before. |
| εἰ γὰρ τὸ Α καὶ τὸ Β τῶι Γ ἐνδέχεται μὴ ὑπάρχειν, ἐὰν μεταληφθῆι τὸ ἐνδέχεσθαι ὑπάρχειν, πάλιν ἔσται τὸ πρῶτον σχῆμα διὰ τῆς ἀντιστροφῆς. | Si enim A et B contingunt C non inesse, si transmutatur contingere non inesse, rursum erit prima figura per conversionem. | For if A and B may possibly not belong to C, if ‘may possibly belong’ is substituted we shall again have the first figure by means of conversion. |
| εἰ δ᾽ ὁ μέν ἐστι καθόλου τῶν ὅρων ὁ δ᾽ ἐν μέρει, τὸν αὐτὸν τρόπον ἐχόντων τῶν ὅρων ὅνπερ ἐπὶ τοῦ ὑπάρχειν, ἔσται τε καὶ οὐκ ἔσται συλλογισμός. ἐνδεχέσθω γὰρ τὸ μὲν Α παντὶ τῶι Γ, τὸ δὲ Β τινὶ τῶι Γ ὑπάρχειν. ἔσται δὴ πάλιν τὸ πρῶτον σχῆμα τῆς ἐν μέρει προτάσεως ἀντιστραφείσης· εἰ γὰρ τὸ Α παντὶ τῶι Γ, τὸ δὲ Γ τινὶ τῶι Β, τὸ Α τινὶ τῶι Β ἐνδέχεται. καὶ εἰ πρὸς τῶι Β Γ τεθείη τὸ καθόλου, ὡσαύτως. | Si autem hic quidem terminorum est universalis, ille vero particularis, eodem modo se habentibus terminis quo inesse, et erit, et non erit syllogismus. Contingat enim A quidem omni C, B autem alicui C inesse, erit ergo rursum prima figura particulari propositione conversa, nam si A omni C, C autem alicui B, et A alicui B contingit. Et si ad B C ponatur universale, similiter. | But if one of the premisses is universal, the other particular, a syllogism will be possible, or not, under the arrangement of the terms as in the case of assertoric propositions. Suppose that A may possibly belong to all C, and B to some C. We shall have the first figure again if the particular premiss is converted. For if A is possible for all C, and C for some of the Bs, then A is possible for some of the Bs. Similarly if the proposition BC is universal. |
| ὁμοίως δὲ καὶ εἰ τὸ μὲν Α Γ στερητικὸν εἴη, τὸ δὲ Β Γ καταφατικόν· ἔσται γὰρ πάλιν τὸ πρῶτον σχῆμα διὰ τῆς ἀντιστροφῆς. εἰ δ᾽ ἀμφότεραι στερητικαὶ τεθείησαν, ἡ μὲν καθόλου ἡ δ᾽ ἐν μέρει, δι᾽ αὐτῶν μὲν τῶν εἰλημ ↵ μένων οὐκ ἔσται συλλογισμός, ἀντιστραφεισῶν δ᾽ ἔσται, καθάπερ ἐν τοῖς πρότερον. ὅταν δὲ ἀμφότεραι ἀδιόριστοι ἢ ἐν μέρει ληφθῶσιν, οὐκ ἔσται συλλογισμός· καὶ γὰρ παντὶ ἀνάγκη τὸ Α τῶι Β καὶ μηδενὶ ὑπάρχειν. ὅροι τοῦ ὑπάρ χειν ζῶιον – ἄνθρωπος – λευκόν, τοῦ μὴ ὑπάρχειν ἵππος – ἄνθρωπος – λευκόν, μέσον λευκόν. | (0663B) Similiter autem et si A C quidem privativa sit, B C autem affirmativa, erit unum rursum prima figura per conversionem, si autem utraeque privativae ponantur, haec quidem universaliter, illa vero particulariter, per ea quidem quae sumpta sunt non erit syllogismus, conversis autem propositionibus erit quemadmodum in prioribus. Quando autem utraeque indefinitae vel particulares sumuntur, non erit syllogismus, etenim necesse est A omni B, et nulli inesse. Termini inesse, animal, homo, album: non inesse, equus, homo, medium album. | Likewise also if the proposition AC is negative, and the proposition BC affirmative: for we shall again have the first figure by conversion. But if both premisses should be negative-the one universal and the other particular-although no syllogistic conclusion will follow from the premisses as they are put, it will follow if they are converted, as above. But when both premisses are indefinite or particular, no syllogism can be formed: for A must belong sometimes to all B and sometimes to no B. To illustrate the affirmative relation take the terms animal-man-white; to illustrate the negative, take the terms horse-man-white — white being the middle term. |
Chapter 21
| Greek | Latin | English |
|---|---|---|
| (PL 64 0663B) CAPUT XX/XXI. Mixtio contingentis et inesse in tertia figura. | 21 | |
| 39b7Ἐὰν δὲ ἡ μὲν ὑπάρχειν ἡ δ᾽ ἐνδέχεσθαι σημαίνηι τῶν προτάσεων, τὸ μὲν συμπέρασμα ἔσται ὅτι ἐνδέχεται καὶ οὐχ ὅτι ὑπάρχει, συλλογισμὸς δ᾽ ἔσται τὸν αὐτὸν τρό πον ἐχόντων τῶν ὅρων ὃν καὶ ἐν τοῖς πρότερον. ἔστωσαν γὰρ πρῶτον κατηγορικοί, καὶ τὸ μὲν Α παντὶ τῶι Γ ὑπαρχέτω, τὸ δὲ Β παντὶ ἐνδεχέσθω ὑπάρχειν. ἀντιστραφέντος οὖν τοῦ Β Γ τὸ πρῶτον ἔσται σχῆμα, καὶ τὸ συμπέρασμα ὅτι ἐνδέχεται τὸ Α τινὶ τῶι Β ὑπάρχειν· ὅτε γὰρ ἡ ἑτέρα τῶν προτάσεων ἐν τῶι πρώτωι σχήματι σημαίνοι ἐνδέχεσθαι, καὶ τὸ συμπέρασμα ἦν ἐνδεχόμενον. | (0663C) Si autem haec quidem propositionum inesse, illa autem contingere significet, conclusio quidem erit quoniam contingit, et non quoniam inest, syllogismus autem erit eodem modo se habentibus terminis, quo et in prioribus. Sint enim primum praedicativae, et A quidem omni C insit, B autem omni C contingat, conversa ergo B C erit prima figura, et conclusio quoniam contingit A alicui B inesse, cum enim altera propositionum in prima figura significabit contingere, et conclusio erit contingens. | If one premiss is pure, the other problematic, the conclusion will be problematic, not pure; and a syllogism will be possible under the same arrangement of the terms as before. First let the premisses be affirmative: suppose that A belongs to all C, and B may possibly belong to all C. If the proposition BC is converted, we shall have the first figure, and the conclusion that A may possibly belong to some of the Bs. For when one of the premisses in the first figure is problematic, the conclusion also (as we saw) is problematic. |
| ὁμοίως δὲ καὶ εἰ τὸ μὲν Β Γ ὑπάρχειν τὸ δὲ Α Γ ἐνδέχεσθαι, καὶ εἰ τὸ μὲν Α Γ στερητικὸν τὸ δὲ Β Γ κατηγορικόν, ὑπάρχοι δ᾽ ὁποτερονοῦν, ἀμφοτέρως ἐνδεχόμενον ἔσται τὸ συμπέρασμα· γίνεται γὰρ πάλιν τὸ πρῶτον σχῆμα, δέδεικται δ᾽ ὅτι τῆς ἑτέρας προτάσεως ἐνδέχεσθαι σημαινούσης ἐν αὐτῶι καὶ τὸ συμπέρασμα ἔσται ἐνδεχόμενον. εἰ δὲ τὸ στερητικὸν τεθείη πρὸς τὸ ἔλαττον ἄκρον, ἢ καὶ ἄμφω ληφθείη στερητικά, δι᾽ αὐτῶν μὲν τῶν κειμένων οὐκ ἔσται συλλογισμός, ἀντιστρα φέντων δ᾽ ἔσται, καθάπερ ἐν τοῖς πρότερον. | Similiter autem et si B C quidem inesse, A C autem contingit inesse. Et si A C quidem privativa, B C autem praedicativa, insit autem alterutra utrinque, contingens erit conclusio, fit enim rursum prima figura. Ostensum est autem quoniam si altera propositio significet contingere in prima figura, et conclusio erit contingens. (0663D) Si autem contingens privativa ponatur ad minorem extremitatem, vel si utraque ponatur privativa, per ea quidem quae posita sunt non erit syllogismus, conversis autem erit, quemadmodum et in prioribus. | Similarly if the proposition BC is pure, AC problematic; or if AC is negative, BC affirmative, no matter which of the two is pure; in both cases the conclusion will be problematic: for the first figure is obtained once more, and it has been proved that if one premiss is problematic in that figure the conclusion also will be problematic. But if the minor premiss BC is negative, or if both premisses are negative, no syllogistic conclusion can be drawn from the premisses as they stand, but if they are converted a syllogism is obtained as before. |
| Εἰ δ᾽ ἡ μὲν καθόλου τῶν προτάσεων ἡ δ᾽ ἐν μέρει, κατηγορικῶν μὲν οὐσῶν ἀμφοτέρων, ἢ τῆς μὲν καθόλου στερητικῆς τῆς δ᾽ ἐν μέρει καταφατικῆς, ὁ αὐτὸς τρόπος ἔσται τῶν συλλογισμῶν· πάντες γὰρ περαίνονται διὰ τοῦ πρώτου σχήματος. ὥστε φανερὸν ὅτι τοῦ ἐνδέχεσθαι καὶ οὐ τοῦ ὑπάρχειν ἔσται ὁ συλλογισμός. εἰ δ᾽ ἡ μὲν καταφατικὴ καθόλου ἡ δὲ στερητικὴ ἐν μέρει, διὰ τοῦ ἀδυνάτου ἔσται ἡ ἀπόδειξις. ὑπαρχέτω γὰρ τὸ μὲν Β παντὶ τῶι Γ, τὸ δὲ Α ἐνδεχέσθω τινὶ τῶι Γ μὴ ὑπάρχειν· ἀνάγκη δὴ τὸ Α ἐν δέχεσθαι τινὶ τῶι Β μὴ ὑπάρχειν. εἰ γὰρ παντὶ τῶι Β τὸ Α ὑπάρχει ἐξ ἀνάγκης, τὸ δὲ Β παντὶ τῶι Γ κεῖται ὑπάρχειν, τὸ Α παντὶ τῶι Γ ἐξ ἀνάγκης ὑπάρξει· τοῦτο γὰρ δέδεικται πρότερον. ἀλλ᾽ ὑπέκειτο τινὶ ἐνδέχεσθαι μὴ ὑπάρχειν. | Si autem haec quidem propositionum sit universalis, illa vero particularis, utrisque quidem praedicativis, aut universali quidem privativa, particulari autem affirmativa, idem modus erit syllogismorum, omnes enim clauduntur per primam figuram. Quare manifestum quoniam eius quod est contingere, et non eius quod est inesse, erit syllogismus. Si autem affirmativa quidem universalis, privativa autem particularis, per impossibile erit demonstratio. (0664A) Insit enim B quidem omni C, A autem contingat alicui C non inesse, necesse est ergo A alicui B contingere non inesse, nam si omni B inest A ex necessitate, B autem omni C positum est inesse, A omni C ex necessitate inerit. Hoc autem ostensum est prius, sed positum est alicui contingere non inesse. | If one of the premisses is universal, the other particular, then when both are affirmative, or when the universal is negative, the particular affirmative, we shall have the same sort of syllogisms: for all are completed by means of the first figure. So it is clear that we shall have not a pure but a problematic syllogistic conclusion. But if the affirmative premiss is universal, the negative particular, the proof will proceed by a reductio ad impossibile. Suppose that B belongs to all C, and A may possibly not belong to some C: it follows that may possibly not belong to some B. For if A necessarily belongs to all B, and B (as has been assumed) belongs to all C, A will necessarily belong to all C: for this has been proved before. But it was assumed at the outset that A may possibly not belong to some C. |
| ↵ Ὅταν δ᾽ ἀδιόριστοι ἢ ἐν μέρει ληφθῶσιν ἀμφότεραι, οὐκ ἔσται συλλογισμός. ἀπόδειξις δ᾽ ἡ αὐτὴ ἣ καὶ ἐν τοῖς πρότερον, καὶ διὰ τῶν αὐτῶν ὅρων. | Quando autem indefinitae, vel particulares sumuntur utraeque, non erit syllogismus, demonstratio autem eadem quae et in universis et per eosdem terminos. | Whenever both premisses are indefinite or particular, no syllogism will be possible. The demonstration is the same as was given in the case of universal premisses, and proceeds by means of the same terms. |
Chapter 22
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| (PL 64 0664A) CAPUT XXI/ XXII.Mixtio necessarii et contingentis in tertia figura. | 22 | |
| 40a4 Εἰ δ᾽ ἐστὶν ἡ μὲν ἀναγκαία τῶν προτάσεων ἡ δ᾽ ἐν δεχομένη, κατηγορικῶν μὲν ὄντων τῶν ὅρων ἀεὶ τοῦ ἐνδέχεσθαι ἔσται συλλογισμός, ὅταν δ᾽ ἦι τὸ μὲν κατηγορικὸν τὸ δὲ στερητικόν, ἐὰν μὲν ἦι τὸ καταφατικὸν ἀναγκαῖον, τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν, ἐὰν δὲ τὸ στερητικόν, καὶ τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν καὶ τοῦ μὴ ὑπάρχειν. τοῦ δ᾽ ἐξ ἀνάγ κης μὴ ὑπάρχειν οὐκ ἔσται συλλογισμός, ὥσπερ οὐδ᾽ ἐν τοῖς ἑτέροις σχήμασιν.
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Si autem est haec quidem propositionum necessaria, illa vero contingens, si praedicativi quidem sunt termini, semper eius quod est contingere erit syllogismus. (0664B) Quando autem fuerit hic quidem praedicativus, ille autem privativus, si sit affirmativus quidem necessarius, eius erit quod est contingere non inesse, si autem privativus, et eius quod est contingere non inesse, et eius quod est non inesse; eius autem quod est ex necessitate non inesse non erit syllogismus, quemadmodum et in aliis figuris. | If one of the premisses is necessary, the other problematic, when the premisses are affirmative a problematic affirmative conclusion can always be drawn; when one proposition is affirmative, the other negative, if the affirmative is necessary a problematic negative can be inferred; but if the negative proposition is necessary both a problematic and a pure negative conclusion are possible. But a necessary negative conclusion will not be possible, any more than in the other figures. |
| Ἔστωσαν δὴ κατηγορικοὶ πρῶτον οἱ ὅροι, καὶ τὸ μὲν Α παντὶ τῶι Γ ὑπαρχέτω ἐξ ἀνάγκης, τὸ δὲ Β παντὶ ἐνδεχέσθω ὑπάρχειν. ἐπεὶ οὖν τὸ μὲν Α παντὶ τῶι Γ ἀνάγκη, τὸ δὲ Γ τινὶ τῶι Β ἐνδέχεται, καὶ τὸ Α τινὶ τῶι Β ἐνδεχόμενον ἔσται καὶ οὐχ ὑπάρχον· οὕτω γὰρ συνέπιπτεν ἐπὶ τοῦ πρώτου σχήματος. ὁμοίως δὲ δειχθήσεται καὶ εἰ τὸ μὲν Β Γ τεθείη ἀναγκαῖον, τὸ δὲ Α Γ ἐνδεχόμενον.
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Sint ergo praedicativi termini primum, et A C quidem omni insit ex necessitate, B autem omni C contingat inesse, quoniam ergo A omni C necessario inest, C autem alicui B contingit, et A alicui B contingens erit, et non inerit, sic enim accidit in prima figura. Similiter autem ostendetur, et si B C quidem ponatur necessaria, A C autem contingens. | Suppose first that the premisses are affirmative, i.e. that A necessarily belongs to all C, and B may possibly belong to all C. Since then A must belong to all C, and C may belong to some B, it follows that A may (not does) belong to some B: for so it resulted in the first figure. A similar proof may be given if the proposition BC is necessary, and AC is problematic. |
| πάλιν ἔστω τὸ μὲν κατηγορικὸν τὸ δὲ στερητικόν, ἀναγκαῖον δὲ τὸ κατηγορικόν· καὶ τὸ μὲν Α ἐνδεχέσθω μη δενὶ τῶι Γ ὑπάρχειν, τὸ δὲ Β παντὶ ὑπαρχέτω ἐξ ἀνάγκης. ἔσται δὴ πάλιν τὸ πρῶτον σχῆμα· καὶ γὰρ ἡ στερητικὴ πρότασις ἐνδέχεσθαι σημαίνει· φανερὸν οὖν ὅτι τὸ συμπέρασμα ἔσται ἐνδεχόμενον· ὅτε γὰρ οὕτως ἔχοιεν αἱ προτάσεις ἐν τῶι πρώτωι σχήματι, καὶ τὸ συμπέρασμα ἦν ἐνδεχόμενον. εἰ δ᾽ ἡ στερητικὴ πρότασις ἀναγκαία, τὸ συμπέρασμα ἔσται καὶ ὅτι ἐνδέχεται τινὶ μὴ ὑπάρχειν καὶ ὅτι οὐχ ὑπάρχει. | Rursum sit hoc quidem praedicativum, illud vero privativum, necessarium autem praedicativum, et A quidem contingat nulli C inesse, B autem omni insit ex necessitate C, erit ergo rursum prima figura, et conclusio contingens, sed non inesse. Nam privativa propositio contingere significat. (0664C) Manifestum est igitur quoniam conclusio erit contingens; cum enim sic se habebant propositiones in prima figura, et conclusio erat contingens. Si autem privativa sit propositio necessaria, et conclusio erit, quoniam contingit alicui non inesse, et quoniam non inesse. | Again suppose one proposition is affirmative, the other negative, the affirmative being necessary: i.e. suppose A may possibly belong to no C, but B necessarily belongs to all C. We shall have the first figure once more: and-since the negative premiss is problematic-it is clear that the conclusion will be problematic: for when the premisses stand thus in the first figure, the conclusion (as we found) is problematic. But if the negative premiss is necessary, the conclusion will be not only that A may possibly not belong to some B but also that it does not belong to some B. |
| κείσθω γὰρ τὸ Α τῶι Γ μὴ ὑπάρχειν ἐξ ἀνάγκης, τὸ δὲ Β παντὶ ἐνδέχεσθαι. ἀντιστραφέντος οὖν τοῦ Β Γ καταφατικοῦ τὸ πρῶτον ἔσται σχῆμα, καὶ ἀναγκαία ἡ στερητικὴ πρότασις. ὅτε δ᾽ οὕτως ἔχοιεν αἱ προτάσεις, συνέβαινε τὸ Α τῶι Γ καὶ ἐνδέχεσθαι τινὶ μὴ ὑπάρχειν καὶ μὴ ὑπάρχειν, ὥστε καὶ τὸ Α τῶι Β ἀνάγκη τινὶ μὴ ὑπάρχειν. ὅταν δὲ τὸ στερητικὸν τεθῆι πρὸς τὸ ἔλαττον ἄκρον, ἐὰν μὲν ἐνδεχόμενον, ἔσται συλλογισμὸς μεταληφθείσης τῆς προτά σεως, καθάπερ ἐν τοῖς πρότερον, ἐὰν δ᾽ ἀναγκαῖον, οὐκ ἔσται· καὶ γὰρ παντὶ ἀνάγκη καὶ οὐδενὶ ἐνδέχεται ὑπάρχειν. ὅροι τοῦ παντὶ ὑπάρχειν ὕπνος – ἵππος καθεύδων – ἄνθρωπος, τοῦ μηδενὶ ὕπνος – ἵππος ἐγρηγορώς – ἄνθρωπος. | Ponatur enim A non inesse C, ex necessitate, B autem omni C contingere, conversa ergo B C affirmativa, prima erit figura, et necessaria privativa propositio. Cum autem sic se habebant propositiones, accidebat A et contingere alicui C non inesse, et non inesse, quare et A necesse est alicui B non inesse. Quando autem privativum ponitur ad minorem extremitatem, si contingens quidem, erit syllogismus transsumpta propositione, quemadmodum! et in prioribus. Si autem necessarium, non erit. (0664D) Etenim necesse est omni et nulli contingat inesse. Termini omni inesse, somnus, equus, dormiens homo. Nulli inesse, somnus, equus, vigilans homo. | For suppose that A necessarily does not belong to C, but B may belong to all C. If the affirmative proposition BC is converted, we shall have the first figure, and the negative premiss is necessary. But when the premisses stood thus, it resulted that A might possibly not belong to some C, and that it did not belong to some C; consequently here it follows that A does not belong to some B. But when the minor premiss is negative, if it is problematic we shall have a syllogism by altering the premiss into its complementary affirmative, as before; but if it is necessary no syllogism can be formed. For A sometimes necessarily belongs to all B, and sometimes cannot possibly belong to any B. To illustrate the former take the terms sleep-sleeping horse-man; to illustrate the latter take the terms sleep-waking horse-man. |
| Ὁμοίως δ᾽ ἕξει καὶ εἰ ὁ μὲν καθόλου τῶν ὅρων ὁ δ᾽ ἐν μέρει πρὸς τὸ μέσον· κατηγορικῶν μὲν γὰρ ὄντων ἀμ ↵ φοτέρων τοῦ ἐνδέχεσθαι καὶ οὐ τοῦ ὑπάρχειν ἔσται συλλογισμός, καὶ ὅταν τὸ μὲν στερητικὸν ληφθῆι τὸ δὲ καταφατικόν, ἀναγκαῖον δὲ τὸ καταφατικόν. ὅταν δὲ τὸ στερητικὸν ἀναγκαῖον, καὶ τὸ συμπέρασμα ἔσται τοῦ μὴ ὑπάρχειν· ὁ γὰρ αὐτὸς τρόπος ἔσται τῆς δείξεως καὶ καθόλου καὶ μὴ καθόλου τῶν ὅρων ὄντων. | Similiter autem se habebit, et si hic quidem terminorum sit universalis, ille autem particularis ad medium, nam si utrique sint praedicativi, eius quod est contingere, et non eius quod est inesse erit syllogismus. Et quando hoc quidem privativum sumetur, illud vero affirmativum, necessarium autem affirmativum, huius quod est contingere. Quando autem privativum necessarium, et conclusio erit quod est non inesse, nam idem modus erit demonstrationis, et cum universales et non universales sunt termini. | Similar results will obtain if one of the terms is related universally to the middle, the other in part. If both premisses are affirmative, the conclusion will be problematic, not pure; and also when one premiss is negative, the other affirmative, the latter being necessary. But when the negative premiss is necessary, the conclusion also will be a pure negative proposition; for the same kind of proof can be given whether the terms are universal or not.
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| ἀνάγκη γὰρ διὰ τοῦ πρώτου σχήματος τελειοῦσθαι τοὺς συλλογισμούς, ὥστε καθάπερ ἐν ἐκείνοις, καὶ ἐπὶ τούτων ἀναγκαῖον συμπίπτειν. ὅταν δὲ τὸ στερητικὸν καθόλου ληφθὲν τεθῆι πρὸς τὸ ἔλαττον ἄκρον, ἐὰν μὲν ἐν δεχόμενον, ἔσται συλλογισμὸς διὰ τῆς ἀντιστροφῆς, ἐὰν δ᾽ ἀναγκαῖον, οὐκ ἔσται. δειχθήσεται δὲ τὸν αὐτὸν τρόπον ὃν καὶ ἐν τοῖς καθόλου, καὶ διὰ τῶν αὐτῶν ὅρων. | (0665A) Necesse est enim per primam figuram perfici syllogismos, quare ut in illis, et in his necessarium accidere. Quando autem privativum universaliter sumptum ponitur ad minorem extremitatem, si contingens quidem, erit syllogismus per conversionem, si autem necessarium sit, non erit, ostendetur autem eodem modo quo et in universalibus, et per eosdem terminos. | For the syllogisms must be made perfect by means of the first figure, so that a result which follows in the first figure follows also in the third. But when the minor premiss is negative and universal, if it is problematic a syllogism can be formed by means of conversion; but if it is necessary a syllogism is not possible. The proof will follow the same course as where the premisses are universal; and the same terms may be used. |
| φανερὸν οὖν καὶ ἐν τούτωι τῶι σχήματι πότε καὶ πῶς ἔσται συλλογισμός, καὶ πότε τοῦ ἐνδέχεσθαι καὶ πότε τοῦ ὑπάρχειν. δῆλον δὲ καὶ ὅτι πάντες ἀτελεῖς, καὶ ὅτι τελειοῦνται διὰ τοῦ πρώτου σχήματος. | Manifestum ergo et in hac figura quando et quomodo erit syllogismus, et quando eius quod est contingere, et quando eius quod est inesse. Palam autem et quoniam omnes imperfecti, et quoniam perficiuntur per primam figuram. | It is clear then in this figure also when and how a syllogism can be formed, and when the conclusion is problematic, and when it is pure. It is evident also that all syllogisms in this figure are imperfect, and that they are made perfect by means of the first figure. |
Chapter 23
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| (PL 64 0665A) CAPUT XXII/ XXIII. De syllogismo ostensivo. | 23 | |
| 40b17 Ὅτι μὲν οὖν οἱ ἐν τούτοις τοῖς σχήμασι συλλογισμοὶ τελειοῦνταί τε διὰ τῶν ἐν τῶι πρώτωι σχήματι καθόλου συλλογισμῶν καὶ εἰς τούτους ἀνάγονται, δῆλον ἐκ τῶν εἰ ρημένων· ὅτι δ᾽ ἁπλῶς πᾶς συλλογισμὸς οὕτως ἕξει, νῦν ἔσται φανερόν, ὅταν δειχθῆι πᾶς γινόμενος διὰ τούτων τινὸς τῶν σχημάτων. | (0665B) Quoniam igitur qui in his figuris sunt syllogismi perficiuntur per eos qui in prima figura sunt universales syllogismos, et in hos reducuntur, palam ex dictis. Quoniam autem simpliciter omnis syllogismus sic se habebit, nunc erit manifestum, cum ostensus fuerit omnis qui fit, per aliquam harum figurarum fieri. | It is clear from what has been said that the syllogisms in these figures are made perfect by means of universal syllogisms in the first figure and are reduced to them. That every syllogism without qualification can be so treated, will be clear presently, when it has been proved that every syllogism is formed through one or other of these figures. |
| Ἀνάγκη δὴ πᾶσαν ἀπόδειξεν καὶ πάντα συλλογισμὸν ἢ ὑπάρχον τι ἢ μὴ ὑπάρχον δεικνύναι, καὶ τοῦτο ἢ καθόλου ἢ κατὰ μέρος, ἔτι ἢ δεικτικῶς ἢ ἐξ ὑποθέσεως. τοῦ δ᾽ ἐξ ὑποθέσεως μέρος τὸ διὰ τοῦ ἀδυνάτου. πρῶτον οὖν εἴπωμεν περὶ τῶν δεικτικῶν· τούτων γὰρ δειχθέντων φανερὸν ἔσται καὶ ἐπὶ τῶν εἰς τὸ ἀδύνατον καὶ ὅλως τῶν ἐξ ὑποθέσεως. | Necesse est ergo omnem demonstrationem et omnem syllogismum aut inesse quid, aut non inesse monstrare. Et hoc aut universaliter, aut particulariter, amplius aut ostensive, aut ex hypothesi. Eius autem quod est ex hypothesi, pars est per impossibile. Primum ergo dicemus de ostensivis, his enim ostensis, manifestum erit et de iis qui ad impossibile, et omnino de iis qui ex hypothesi. | It is necessary that every demonstration and every syllogism should prove either that something belongs or that it does not, and this either universally or in part, and further either ostensively or hypothetically. One sort of hypothetical proof is the reductio ad impossibile. Let us speak first of ostensive syllogisms: for after these have been pointed out the truth of our contention will be clear with regard to those which are proved per impossibile, and in general hypothetically. |
| Εἰ δὴ δέοι τὸ Α κατὰ τοῦ Β συλλογίσασθαι ἢ ὑπάρ- χον ἢ μὴ ὑπάρχον, ἀνάγκη λαβεῖν τι κατά τινος. εἰ μὲν οὖν τὸ Α κατὰ τοῦ Β ληφθείη, τὸ ἐξ ἀρχῆς ἔσται εἰλημμένον. εἰ δὲ κατὰ τοῦ Γ, τὸ δὲ Γ κατὰ μηδενός, μηδ᾽ ἄλλο κατ᾽ ἐκείνου, μηδὲ κατὰ τοῦ Α ἕτερον, οὐδεὶς ἔσται συλλογισμός· τῶι γὰρ ἓν καθ᾽ ἑνὸς ληφθῆναι οὐδὲν συμβαίνει ἐξ ἀνάγκης. ὥστε προσληπτέον καὶ ἑτέραν πρότασιν. ἐὰν μὲν οὖν ληφθῆι τὸ Α κατ᾽ ἄλλου ἢ ἄλλο κατὰ τοῦ Α, ἢ κατὰ τοῦ Γ ἕτερον, εἶναι μὲν συλλογισμὸν οὐδὲν κωλύει, πρὸς μέντοι τὸ Β οὐκ ἔσται διὰ τῶν εἰλημμένων. οὐδ᾽ ὅταν τὸ Γ ἑτέρωι, κἀκεῖνο ἄλλωι, καὶ τοῦτο ἑτέρωι, μὴ ↵ συνάπτηι δὲ πρὸς τὸ Β, οὐδ᾽ οὕτως ἔσται πρὸς τὸ Β συλλογισμός. | Si ergo oporteat A de B syllogizare, vel inesse, vel non inesse, necesse est sumere aliquid de aliquo. (0665C) Si ergo A sumatur de B, quod ex principio erit sumptum, si autem A de C, C autem de nullo alio, nec aliud de illo C, neque de A alterum, neque de altero A, nullus erit syllogismus, nam in eo quod unum de uno sumitur, nihil accidit ex necessitate, quare assumenda est altera propositio. Si igitur sumatur A de alio, aut aliud de A, aut de C alterum, esse quidem syllogismum nihil prohibet, ad B autem non erit per ea quae sumpta sunt, nec quando C inest alteri, et illud alii, et hoc alteri, non copuletur autem ad B, nec sic erit ad B syllogismus ipsius A. | If then one wants to prove syllogistically A of B, either as an attribute of it or as not an attribute of it, one must assert something of something else. If now A should be asserted of B, the proposition originally in question will have been assumed. But if A should be asserted of C, but C should not be asserted of anything, nor anything of it, nor anything else of A, no syllogism will be possible. For nothing necessarily follows from the assertion of some one thing concerning some other single thing. Thus we must take another premiss as well. If then A be asserted of something else, or something else of A, or something different of C, nothing prevents a syllogism being formed, but it will not be in relation to B through the premisses taken. Nor when C belongs to something else, and that to something else and so on, no connexion however being made with B, will a syllogism be possible concerning A in its relation to B. |
| ὅλως γὰρ εἴπομεν ὅτι οὐδεὶς οὐδέποτε ἔσται συλλογισμὸς ἄλλου κατ᾽ ἄλλου μὴ ληφθέντος τινὸς μέσου, ὁ πρὸς ἑκάτερον ἔχει πως ταῖς κατηγορίαις· | Omnino enim dicimus quoniam nullus nunquam erit syllogismus alius de alio, non sumpto aliquo medio, quod ad utrumque se habet quoquo modo praedicationibus. | For in general we stated that no syllogism can establish the attribution of one thing to another, unless some middle term is taken, which is somehow related to each by way of predication. |
| ὁ μὲν γὰρ συλλογισμὸς ἁπλῶς ἐκ προτάσεών ἐστιν, ὁ δὲ πρὸς τόδε συλλογισμὸς ἐκ τῶν πρὸς τόδε προτάσεων, ὁ δὲ τοῦδε πρὸς τόδε διὰ τῶν τοῦδε πρὸς τόδε προτάσεων. ἀδύνατον δὲ πρὸς τὸ Β λαβεῖν πρότασιν μηδὲν μήτε κατηγοροῦντας αὐτοῦ μήτ᾽ ἀπαρνουμένους, ἢ πάλιν τοῦ Α πρὸς τὸ Β μη δὲν κοινὸν λαμβάνοντας ἀλλ᾽ ἑκατέρου ἴδια ἄττα κατηγοροῦντας ἢ ἀπαρνουμένους. ὥστε ληπτέον τι μέσον ἀμφοῖν, ὁ συνάψει τὰς κατηγορίας, εἴπερ ἔσται τοῦδε πρὸς τόδε συλλογισμός. | (0665D) Nam syllogismus quidem simpliciter ex propositionibus est, ad hoc autem syllogismus ex propositionibus, quae ad hoc, qui autem est huius ad hoc, per propositiones huius ad hoc, impossibile est autem ad B sumere propositionem, nihil neque praedicantes de eo, neque negantes, aut rursum eius quod est A ad B, nihil commune sumentes, sed utriusque propria quaedam praedicantes, aut negantes, quare sumendum, utriusque quod copulet praedicationes, si erit huius ad hoc syllogismus. | For the syllogism in general is made out of premisses, and a syllogism referring to this out of premisses with the same reference, and a syllogism relating this to that proceeds through premisses which relate this to that. But it is impossible to take a premiss in reference to B, if we neither affirm nor deny anything of it; or again to take a premiss relating A to B, if we take nothing common, but affirm or deny peculiar attributes of each. So we must take something midway between the two, which will connect the predications, if we are to have a syllogism relating this to that. |
| εἰ οὖν ἀνάγκη μέν τι λαβεῖν πρὸς ἄμφω κοινόν, τοῦτο δ᾽ ἐνδέχεται τριχῶς (ἢ γὰρ τὸ Α τοῦ Γ καὶ τὸ Γ τοῦ Β κατηγορήσαντας, ἢ τὸ Γ κατ᾽ ἀμφοῖν, ἢ ἄμφω κατὰ τοῦ Γ), ταῦτα δ᾽ ἐστὶ τὰ εἰρημένα σχήματα, φανερὸν ὅτι πάντα συλλογισμὸν ἀνάγκη γίνεσθαι διὰ τούτων τινὸς τῶν σχημάτων. ὁ γὰρ αὐτὸς λόγος καὶ εἰ διὰ πλειόνων συνάπτοι πρὸς τὸ Β· ταὐτὸ γὰρ ἔσται σχῆμα καὶ ἐπὶ τῶν πολλῶν. Ὅτι μὲν οὖν οἱ δεικτικοὶ περαίνονται διὰ τῶν προειρημένων σχημάτων, φανερόν· | Ergo si necesse est aliquod sumere ad utrumque commune, hoc autem contingit tripliciter, aut enim A de C et de B praedicantes, aut C de utrisque, aut utraque de C, hae autem sunt tres dictae figurae. Manifestum quoniam omnem syllogismum necesse est fieri per aliquam harum figurarum. Nam eadem ratio est, etsi per plura copuletur ad B, eadem enim erit figura et in pluribus. (0666A) Quoniam igitur ostensivi terminantur per praedictas figuras, manifestum est. | If then we must take something common in relation to both, and this is possible in three ways (either by predicating A of C, and C of B, or C of both, or both of C), and these are the figures of which we have spoken, it is clear that every syllogism must be made in one or other of these figures. The argument is the same if several middle terms should be necessary to establish the relation to B; for the figure will be the same whether there is one middle term or many. It is clear then that the ostensive syllogisms are effected by means of the aforesaid figures; these considerations will show that reductiones ad also are effected in the same way. |
| (PL 64 0666A) CAPUT XXIII/ XXIV. De syllogismo ex hypothesi. | ||
| ὅτι δὲ καὶ οἱ εἰς τὸ ἀδύνατον, δῆλον ἔσται διὰ τούτων. πάντες γὰρ οἱ διὰ τοῦ ἀδυνάτου περαίνοντες τὸ μὲν ψεῦδος συλλογίζονται, τὸ δ᾽ ἐξ ἀρχῆς ἐξ ὑποθέσεως δεικνύουσιν, ὅταν ἀδύνατόν τι συμβαίνηι τῆς ἀντιφάσεως τεθείσης, οἷον ὅτι ἀσύμμετρος ἡ διάμετρος διὰ τὸ γί- νεσθαι τὰ περιττὰ ἴσα τοῖς ἀρτίοις συμμέτρου τεθείσης. τὸ μὲν οὖν ἴσα γίνεσθαι τὰ περιττὰ τοῖς ἀρτίοις συλλογίζεται, τὸ δ᾽ ἀσύμμετρον εἶναι τὴν διάμετρον ἐξ ὑποθέσεως δείκνυσιν, ἐπεὶ ψεῦδος συμβαίνει διὰ τὴν ἀντίφασιν. τοῦτο γὰρ ἦν τὸ διὰ τοῦ ἀδυνάτου συλλογίσασθαι, τὸ δεῖξαί τι ἀδύνατον διὰ τὴν ἐξ ἀρχῆς ὑπόθεσιν. | Quoniam autem et qui ad impossibile, palam erit per haec, omnes enim qui per impossibile concludunt, falsum quidem syllogizant. Quod autem ex principio erat, ex hypothesi demonstrant, quando aliquid accidit impossibile posita contradictione, ut quoniam diameter est asymeter, eo quod fiunt abundantia aequalia perfectis, posito symetro. Ergo aequalia quidem fieri abundantia perfectis syllogizant, asymetrum autem esse diametrum, ex hypothesi monstrant, quoniam falsum accidit propter contradictionem. (0666B) Hoc enim fuit per impossibile syllogizare, ostendere aliquid impossibile propter priorem hypothesin.
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For all who effect an argument per impossibile infer syllogistically what is false, and prove the original conclusion hypothetically when something impossible results from the assumption of its contradictory; e.g. that the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate. One infers syllogistically that odd numbers come out equal to evens, and one proves hypothetically the incommensurability of the diagonal, since a falsehood results through contradicting this. For this we found to be reasoning per impossibile, viz. proving something impossible by means of an hypothesis conceded at the beginning. |
| ὥστ᾽ ἐπεὶ τοῦ ψεύδους γίνεται συλλογισμὸς δεικτικὸς ἐν τοῖς εἰς τὸ ἀδύνατον ἀπαγομένοις, τὸ δ᾽ ἐξ ἀρχῆς ἐξ ὑποθέσεως δείκνυται, τοὺς δὲ δεικτικοὺς πρότερον εἴπομεν ὅτι διὰ τούτων περαίνονται τῶν σχημάτων, φανερὸν ὅτι καὶ οἱ διὰ τοῦ ἀδυνάτου συλλογισμοὶ διὰ τούτων ἔσονται τῶν σχημάτων. | Quare quoniam falsus fit syllogismus ostensivus in his quae ad impossibile deducuntur, quod autem est ex principio, ex hypothesi monstratur, ostensivos autem diximus prius, quoniam per has terminantur figuras, manifestum quoniam et per impossibile syllogismi per has erunt figuras. | Consequently, since the falsehood is established in reductions ad impossibile by an ostensive syllogism, and the original conclusion is proved hypothetically, and we have already stated that ostensive syllogisms are effected by means of these figures, it is evident that syllogisms per impossibile also will be made through these figures. |
| ὡσαύτως δὲ καὶ οἱ ἄλλοι πάντες οἱ ἐξ ὑποθέσεως· ἐν ἅπασι γὰρ ὁ μὲν συλλογισμὸς γίνεται πρὸς τὸ μεταλαμβανόμενον, τὸ δ᾽ ἐξ ἀρχῆς περαίνεται δι᾽ ὁμολογίας ἤ τινος ἄλλης ὑπο ↵ θέσεως. εἰ δὲ τοῦτ᾽ ἀληθές, πᾶσαν ἀπόδειξιν καὶ πάντα συλλογισμὸν ἀνάγκη γίνεσθαι διὰ τριῶν τῶν προειρημένων σχημάτων. τούτου δὲ δειχθέντος δῆλον ὡς ἅπας τε συλλογισμὸς ἐπιτελεῖται διὰ τοῦ πρώτου σχήματος καὶ ἀνά γεται εἰς τοὺς ἐν τούτωι καθόλου συλλογισμούς. | Similiter autem et alii omnes qui sunt ex hypothesi, in omnibus his enim syllogismus quidem fit ad transsumptum, quod autem est ex principio, terminatur per confessionem aut per aliquam aliam hypothesin. Si autem hoc verum, necesse est omnem demonstrationem et omnem syllogismum fieri per tres praedictas figuras. (0666C) Hoc autem ostenso, palam quoniam omnis syllogismus perficitur per primam figuram, et reducitur in huius universales syllogismos. | Likewise all the other hypothetical syllogisms: for in every case the syllogism leads up to the proposition that is substituted for the original thesis; but the original thesis is reached by means of a concession or some other hypothesis. But if this is true, every demonstration and every syllogism must be formed by means of the three figures mentioned above. But when this has been shown it is clear that every syllogism is perfected by means of the first figure and is reducible to the universal syllogisms in this figure. |
Chapter 24
| Greek | Latin | English |
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| (PL 64 0666C) CAPUT XXIV. De qualitate et quantitate terminorum syllogismi. | 24 | |
| 41b6 Ἔτι τε ἐν ἅπαντι δεῖ κατηγορικόν τινα τῶν ὅρων εἶναι καὶ τὸ καθόλου ὑπάρχειν· ἄνευ γὰρ τοῦ καθόλου ἢ οὐκ ἔσται συλλογισμὸς ἢ οὐ πρὸς τὸ κείμενον, ἢ τὸ ἐξ ἀρχῆς αἰτήσεται. κείσθω γὰρ τὴν μουσικὴν ἡδονὴν εἶναι σπου δαίαν. εἰ μὲν οὖν ἀξιώσειεν ἡδονὴν εἶναι σπουδαίαν μὴ προσθεὶς τὸ πᾶσαν, οὐκ ἔσται συλλογισμός· εἰ δὲ τινὰ ἡδονήν, εἰ μὲν ἄλλην, οὐδὲν πρὸς τὸ κείμενον, εἰ δ᾽ αὐτὴν ταύτην, τὸ ἐξ ἀρχῆς λαμβάνει. | Amplius autem in omnibus oportet aliquem terminorum praedicativum esse et universalem, sine universali enim non erit syllogismus, aut non ad hoc quod positum est, aut quod ex principio est petet. Ponatur enim musicam voluptatem esse studiosam, si ergo poposcerit voluptatem esse studiosam, non addens omnem, non erit syllogismus, si autem aliquam voluptatem esse studiosam, si aliam quidem, nihil ad hoc quod positum est, si autem eamdem, quod ex principio erat, sumit.
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Further in every syllogism one of the premisses must be affirmative, and universality must be present: unless one of the premisses is universal either a syllogism will not be possible, or it will not refer to the subject proposed, or the original position will be begged. Suppose we have to prove that pleasure in music is good. If one should claim as a premiss that pleasure is good without adding ‘all’, no syllogism will be possible; if one should claim that some pleasure is good, then if it is different from pleasure in music, it is not relevant to the subject proposed; if it is this very pleasure, one is assuming that which was proposed at the outset to be proved. |
| μᾶλλον δὲ γίνεται φανερὸν ἐν τοῖς διαγράμμασιν, οἷον ὅτι τοῦ ἰσοσκελοῦς ἴσαι αἱ πρὸς τῆι βάσει. ἔστωσαν εἰς τὸ κέντρον ἠγμέναι αἱ Α Β. εἰ οὖν ἴσην λαμβάνοι τὴν Α Γ γωνίαν τῆι Β Δ μὴ ὅλως ἀξιώσας ἴσας τὰς τῶν ἡμικυκλίων, καὶ πάλιν τὴν Γ τῆι Δ μὴ πᾶσαν προσλαβὼν τὴν τοῦ τμήματος, ἔτι δ᾽ ἀπ᾽ ἴσων οὐσῶν τῶν ὅλων γωνιῶν καὶ ἴσων ἀφηιρημένων ἴσας εἶναι τὰς λοιπὰς τὰς Ε Ζ, τὸ ἐξ ἀρχῆς αἰτήσε- ται, ἐὰν μὴ λάβηι ἀπὸ τῶν ἴσων ἴσων ἀφαιρουμένων ἴσα λείπεσθαι. | (0666D) Magis autem fit manifestum in figuris, ut quoniam aequicruris aequales sunt anguli, qui sunt ad basim: sint enim in centrum ductae A B, si ergo aequalem sumpserit A C angulum ei qui est B D, non omnino petens aequales eos qui sunt semicirculorum, et rursum C ei qui est D, non omnem assumens eum qui est incisionis. Amplius, ab aequalibus existentibus totis angulis, aequalibus demptis, aequales esse reliquos, scilicet E F, quod ex principio est petet, nisi sumat ab omnibus aequalibus, aequis demptis, aequalia relinqui. | This is more obvious in geometrical proofs, e.g. that the angles at the base of an isosceles triangle are equal. Suppose the lines A and B have been drawn to the centre. If then one should assume that the angle AC is equal to the angle BD, without claiming generally that angles of semicircles are equal; and again if one should assume that the angle C is equal to the angle D, without the additional assumption that every angle of a segment is equal to every other angle of the same segment; and further if one should assume that when equal angles are taken from the whole angles, which are themselves equal, the remainders E and F are equal, he will beg the thing to be proved, unless he also states that when equals are taken from equals the remainders are equal. |
| φανερὸν οὖν ὅτι ἐν ἅπαντι δεῖ τὸ καθόλου ὑπάρχειν, καὶ ὅτι τὸ μὲν καθόλου ἐξ ἁπάντων τῶν ὅρων καθόλου δείκνυται, τὸ δ᾽ ἐν μέρει καὶ οὕτως κἀκείνως, ὥστ᾽ ἐὰν μὲν ἦι τὸ συμπέρασμα καθόλου, καὶ τοὺς ὅρους ἀνάγκη καθόλου εἶναι, ἐὰν δ᾽ οἱ ὅροι καθόλου, ἐνδέχεται τὸ συμπέρασμα μὴ εἶναι καθόλου. δῆλον δὲ καὶ ὅτι ἐν ἅπαντι συλλογισμῶι ἢ ἀμφοτέρας ἢ τὴν ἑτέραν πρότασιν ὁμοίαν ἀνάγκη γίνεσθαι τῶι συμπεράσματι. λέγω δ᾽ οὐ μόνον τῶι καταφατικὴν εἶναι ἢ στερητικήν, ἀλλὰ καὶ τῶι ἀναγκαίαν ἢ ὑπάρχουσαν ἢ ἐνδεχομένην. ἐπισκέψασθαι δὲ δεῖ καὶ τὰς ἄλλας κατηγορίας. | Manifestum igitur quoniam in omni syllogismo oportet universale esse. Et quoniam universale quidem ex omnibus terminis universalibus monstratur, particulare autem et sic, et aliter. (0667A) Quare si conclusio sit universalis, et terminos necesse est universales esse, si autem universales sint termini, contingit conclusionem non universalem esse. Palam etiam quoniam in omni syllogismo aut utramque, aut alteram propositionem similem necesse est fieri conclusioni, dico autem non solum in eo quod affirmativa sit, vel negativa, sed in eo quod necessaria aut inesse, aut contingens: considerare autem oportet et alia praedicamenta. | It is clear then that in every syllogism there must be a universal premiss, and that a universal statement is proved only when all the premisses are universal, while a particular statement is proved both from two universal premisses and from one only: consequently if the conclusion is universal, the premisses also must be universal, but if the premisses are universal it is possible that the conclusion may not be universal. And it is clear also that in every syllogism either both or one of the premisses must be like the conclusion. I mean not only in being affirmative or negative, but also in being necessary, pure, problematic. We must consider also the other forms of predication. |
| Φανερὸν δὲ καὶ ἁπλῶς πότ᾽ ἔσται καὶ πότ᾽ οὐκ ἔσται συλλογισμός, καὶ πότε δυνατὸς καὶ πότε τέλειος, καὶ ὅτι συλλογισμοῦ ὄντος ἀναγκαῖον ἔχειν τοὺς ὅρους κατά τινα τῶν εἰρημένων τρόπων. | Manifestum autem et simpliciter quando erit, et quando non erit syllogismus, et quando perfectus, et quoniam si est syllogismus, necessarium est habere terminos secundum aliquem dictorum modorum. | It is clear also when a syllogism in general can be made and when it cannot; and when a valid, when a perfect syllogism can be formed; and that if a syllogism is formed the terms must be arranged in one of the ways that have been mentioned. |
Chapter 25
| Greek | Latin | English |
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| (PL 64 0667A) CAPUT XXV. De numero terminorum syllogismi. | 25 | |
| 41b36 Δῆλον δὲ καὶ ὅτι πᾶσα ἀπόδειξις ἔσται διὰ τριῶν ὅρων καὶ οὐ πλειόνων, ἐὰν μὴ δι᾽ ἄλλων καὶ ἄλλων τὸ αὐτὸ συμπέρασμα γίνηται, οἷον τὸ Ε διά τε τῶν Α Β καὶ διὰ τῶν Γ Δ, ἢ διὰ τῶν Α Β καὶ Α Γ Δ· πλείω γὰρ μέσα τῶν αὐτῶν οὐδὲν εἶναι κωλύει. τούτων δ᾽ ὄντων οὐχ εἷς ἀλλὰ ↵ πλείους εἰσὶν οἱ συλλογισμοί. ἢ πάλιν ὅταν ἑκάτερον τῶν Α Β διὰ συλλογισμοῦ ληφθῆι (οἷον τὸ Α διὰ τῶν Δ Ε καὶ πάλιν τὸ Β διὰ τῶν Ζ Θ), ἢ τὸ μὲν ἐπαγωγῆι, τὸ δὲ συλλογισμῶι. ἀλλὰ καὶ οὕτως πλείους οἱ συλλογισμοί· πλείω γὰρ τὰ συμπεράσματα ἐστιν, οἷον τό τε Α καὶ τὸ Β καὶ τὸ Γ. | (0667B) Palam autem et quoniam omnis demonstratio erit per tres terminos, et non per plures, nisi per alia et alia eadem conclusio fiat, ut E per A B, et per C D, aut per A B, et A C, et B C, plura enim media eorumdem nihil esse prohibet, haec autem cum sint, non unus, sed plures sunt syllogismi. (0667C) Aut rursum, quando utrumque A B sumitur per syllogismum, ut A per D E, et rursum B per F G, aut hoc quidem inductione, illud autem syllogismo, sed et si plures erunt syllogismi, plures enim conclusiones sunt, ut A B et C. | It is clear too that every demonstration will proceed through three terms and no more, unless the same conclusion is established by different pairs of propositions; e.g. the conclusion E may be established through the propositions A and B, and through the propositions C and D, or through the propositions A and B, or A and C, or B and C. For nothing prevents there being several middles for the same terms. But in that case there is not one but several syllogisms. Or again when each of the propositions A and B is obtained by syllogistic inference, e.g. by means of D and E, and again B by means of F and G. Or one may be obtained by syllogistic, the other by inductive inference. But thus also the syllogisms are many; for the conclusions are many, e.g. A and B and C. |
| Εἰ δ᾽ οὖν μὴ πλείους ἀλλ᾽ εἷς, οὕτω μὲν ἐνδέχεται γενέσθαι διὰ πλειόνων τὸ αὐτὸ συμπέρασμα, ὡς δὲ τὸ Γ διὰ τῶν Α Β, ἀδύνατον. ἔστω γὰρ τὸ Ε συμπεπερασμένον ἐκ τῶν Α Β Γ Δ. οὐκοῦν ἀνάγκη τι αὐτῶν ἄλλο πρὸς ἄλλο εἰλῆφθαι, τὸ μὲν ὡς ὅλον τὸ δ᾽ ὡς μέρος· τοῦτο γὰρ δέδεικται πρότερον, ὅτι ὄντος συλλογισμοῦ ἀναγκαῖον οὕτως τινὰς ἔχειν τῶν ὅρων. ἐχέτω οὖν τὸ Α οὕτως πρὸς τὸ Β. ἔστιν ἄρα τι ἐξ αὐτῶν συμπέρασμα. οὐκοῦν ἤτοι τὸ Ε ἢ τῶν Γ Δ θάτερον ἢ ἄλλο τι παρὰ ταῦτα. | Si igitur non plures, sed unus (sic autem contingit fieri per plura media eamdem conclusionem, ut E quidem per A B C D ), impossibile. Sit enim E conclusio ex A B C D, ergo necesse est aliquid eorum, aliud ad aliud sumptum esse, hoc quidem ut totum, illud vero ut pars, hoc enim ostensum est prius, quoniam si est syllogismus, necesse est sic aliquos se habere terminorum. Habeat se ergo A sic ad B, est itaque aliqua ex eis conclusio, aut ergo E, aut alterum eorum quae sunt C D, aut alterum aliud quidem praeter haec. | But if this can be called one syllogism, not many, the same conclusion may be reached by more than three terms in this way, but it cannot be reached as C is established by means of A and B. Suppose that the proposition E is inferred from the premisses A, B, C, and D. It is necessary then that of these one should be related to another as whole to part: for it has already been proved that if a syllogism is formed some of its terms must be related in this way. Suppose then that A stands in this relation to B. Some conclusion then follows from them. It must either be E or one or other of C and D, or something other than these. |
| καὶ εἰ μὲν τὸ Ε, ἐκ τῶν Α Β μό νον ἂν εἴη ὁ συλλογισμός. τὰ δὲ Γ Δ εἰ μὲν ἔχει οὕτως ὥστ᾽ εἶναι τὸ μὲν ὡς ὅλον τὸ δ᾽ ὡς μέρος, ἔσται τι καὶ ἐξ ἐκείνων, καὶ ἤτοι τὸ Ε ἢ τῶν Α Β θάτερον ἢ ἄλλο τι παρὰ ταῦτα. | Et si E quidem, ex A B tantum, erit syllogismus, C D autem quidem se habeant sic ut sit hoc quidem ut notum, illud vero ut pars, erit aliquid ex illis aut E, aut aliquid eorum quae sunt A B, aut alterum aliud quidem praeter haec. | (1) If it is E the syllogism will have A and B for its sole premisses. But if C and D are so related that one is whole, the other part, some conclusion will follow from them also; and it must be either E, or one or other of the propositions A and B, or something other than these. |
| καὶ εἰ μὲν τὸ Ε ἢ τῶν Α Β θάτερον, ἢ πλείους ἔσονται οἱ συλλογισμοί, ἢ ὡς ἐνεδέχετο ταὐτὸ διὰ πλειόνων ὅρων περαίνεσθαι συμβαίνει· εἰ δ᾽ ἄλλο τι παρὰ ταῦτα, πλείους ἔσονται καὶ ἀσύναπτοι οἱ συλλογισμοὶ πρὸς ἀλλήλους. εἰ δὲ μὴ οὕτως ἔχοι τὸ Γ πρὸς τὸ Δ ὥστε ποιεῖν συλλογισμόν, μάτην ἔσται εἰλημμένα, εἰ μὴ ἐπαγωγῆς ἢ κρύψεως ἤ τινος ἄλλου τῶν τοιούτων χάριν. | Et si E quidem, aut eorum quae sunt A B alterum, aut plures erunt syllogismi, aut (ut contingebat) idem per plures terminos concludi accidit, si autem aliud quidem praeter haec, plures erunt et inconiuncti syllogismi ad invicem, si autem non sic se habeat C ad D ut faciat syllogismum, vane erunt sumpta, nisi inductionis, aut celationis, aut alicuius alius talium gratia. | And if it is (i) E, or (ii) A or B, either (i) the syllogisms will be more than one, or (ii) the same thing happens to be inferred by means of several terms only in the sense which we saw to be possible. But if (iii) the conclusion is other than E or A or B, the syllogisms will be many, and unconnected with one another. But if C is not so related to D as to make a syllogism, the propositions will have been assumed to no purpose, unless for the sake of induction or of obscuring the argument or something of the sort. |
| Εἰ δ᾽ ἐκ τῶν Α Β μὴ τὸ Ε ἀλλ᾽ ἄλλο τι γίγνεται συμπέρασμα, ἐκ δὲ τῶν Γ Δ ἢ τούτων θάτερον ἢ ἄλλο παρὰ ταῦτα, πλείους τε οἱ συλλογισμοὶ γίνονται καὶ οὐ τοῦ ὑποκειμένου· ὑπέκειτο γὰρ εἶναι τοῦ Ε τὸν συλλογισμόν. | (0667D) Si autem ex A B non E, sed alia quaedam fiat conclusio, ex C D autem aut horum alterum, aut aliud praeter haec, et plures fiunt syllogismi, et non eius quod positum est. Ponebatur enim eius quod est E esse syllogismum. | (2) But if from the propositions A and B there follows not E but some other conclusion, and if from C and D either A or B follows or something else, then there are several syllogisms, and they do not establish the conclusion proposed: for we assumed that the syllogism proved E. |
| Εἰ δὲ μὴ γίνεται ἐκ τῶν Γ Δ μηδὲν συμπέρασμα, μάτην τε εἰλῆφθαι αὐτὰ συμβαίνει καὶ μὴ τοῦ ἐξ ἀρχῆς εἶναι τὸν συλλογισμόν. | Si autem non fiat ex C D nulla conclusio, et vane sumpta esse ea accidit, et non eius quod est ex principio esse syllogismum. | And if no conclusion follows from C and D, it turns out that these propositions have been assumed to no purpose, and the syllogism does not prove the original proposition. |
| ὥστε φανερὸν ὅτι πᾶσα ἀπόδειξις καὶ πᾶς συλλογισμὸς ἔσται διὰ τριῶν ὅρων μόνον. | Quare manifestum quoniam omnis demonstratio et omnis syllogismus erit per tres terminos solos. | So it is clear that every demonstration and every syllogism will proceed through three terms only. |
| (PL 64 0667D) CAPUT XXVI. De numero propositionum et prosyllogismis. | ||
| Τούτου δ᾽ ὄντος φανεροῦ, δῆλον ὡς καὶ ἐκ δύο προτάσεων καὶ οὐ πλειόνων (οἱ γὰρ τρεῖς ὅροι δύο προτάσεισ), εἰ μὴ προσλαμβάνοιτό τι, καθάπερ ἐν τοῖς ἐξ ἀρχῆς ἐλέχθη, πρὸς τὴν τελείωσιν τῶν συλλογισμῶν. φανερὸν οὖν ὡς ἐν ὧι λόγωι συλλογιστικῶι μὴ ἄρτιαί εἰσιν αἱ προτάσεις δι᾽ ὧν γίνεται τὸ συμπέρασμα τὸ κύριον (ἔνια γὰρ τῶν ἄνωθεν συμπερασμάτων ἀναγκαῖον εἶναι προτάσεισ), οὗτος ὁ λόγος ἢ οὐ συλλελόγισται ἢ πλείω τῶν ἀναγκαίων ἠρώτηκε πρὸς τὴν θέσιν. | (0668A) Hoc autem manifesto, palam quoniam et ex duabus propositionibus, et non pluribus, nam tres termini, duae sunt propositiones, nisi assumatur aliquid (quemadmodum in prioribus dictum est) ad perfectionem syllogismorum. Manifestum igitur quando, ut in oratione syllogistica, non pares sunt propositiones per quas fit conclusio principalis (quasdam enim superiorum conclusionum necessarium est esse propositiones), haec oratio aut non syllogistica est, aut plura necessariis interrogavit ad positionem. | This being evident, it is clear that a syllogistic conclusion follows from two premisses and not from more than two. For the three terms make two premisses, unless a new premiss is assumed, as was said at the beginning, to perfect the syllogisms. It is clear therefore that in whatever syllogistic argument the premisses through which the main conclusion follows (for some of the preceding conclusions must be premisses) are not even in number, this argument either has not been drawn syllogistically or it has assumed more than was necessary to establish its thesis. |
| ↵ Κατὰ μὲν οὖν τὰς κυρίας προτάσεις λαμβανομένων τῶν συλλογισμῶν, ἅπας ἔσται συλλογισμὸς ἐκ προτάσεων μὲν ἀρτίων ἐξ ὅρων δὲ περιττῶν· ἑνὶ γὰρ πλείους οἱ ὅροι τῶν προτάσεων. ἔσται δὲ καὶ τὰ συμπεράσματα ἡμίση τῶν προ τάσεων. | Secundum igitur principales propositiones sumptis syllogismis, omnis syllogismus erit ex propositionibus quidem perfectis, ex terminis autem abundantibus, uno enim plures termini propositionibus, erunt autem et conclusiones dimidietas propositionum. | If then syllogisms are taken with respect to their main premisses, every syllogism will consist of an even number of premisses and an odd number of terms (for the terms exceed the premisses by one), and the conclusions will be half the number of the premisses.
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| ὅταν δὲ διὰ προσυλλογισμῶν περαίνηται ἢ διὰ πλειόνων μέσων συνεχῶν, οἷον τὸ Α Β διὰ τῶν Γ Δ, τὸ μὲν πλῆθος τῶν ὅρων ὡσαύτως ἑνὶ ὑπερέξει τὰς προτάσεις (ἢ γὰρ ἔξωθεν ἢ εἰς τὸ μέσον τεθήσεται ὁ παρεμπίπτων ὅρος· ἀμφοτέρως δὲ συμβαίνει ἑνὶ ἐλάττω εἶναι τὰ διαστήματα τῶν ὅρων), αἱ δὲ προτάσεις ἴσαι τοῖς διαστήμασιν· | (0668B) Quando autem per prosyllogismos concluditur, aut per plura media non continua, ut A B per C D, multitudo quidem terminorum similiter uno superabit propositiones, aut enim extrinsecus, aut medium ponetur intercidens terminus, utrinque autem accidit uno minus esse intervalla quam terminos, propositiones autem aequales sunt intervallis.
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But whenever a conclusion is reached by means of prosyllogisms or by means of several continuous middle terms, e.g. the proposition AB by means of the middle terms C and D, the number of the terms will similarly exceed that of the premisses by one (for the extra term must either be added outside or inserted: but in either case it follows that the relations of predication are one fewer than the terms related), and the premisses will be equal in number to the relations of predication. |
| οὐ μέντοι αἰεὶ αἱ μὲν ἄρτιαι ἔσονται οἱ δὲ περιττοί, ἀλλ᾽ ἐναλλάξ, ὅταν μὲν αἱ προτάσεις ἄρτιαι, περιττοὶ οἱ ὅροι, ὅταν δ᾽ οἱ ὅροι ἄρτιοι, περιτταὶ αἱ προτάσεις· ἅμα γὰρ τῶι ὅρωι μία προστίθεται πρότασις, ἂν ὁποθενοῦν προστεθῆι ὁ ὅρος, ὥστ᾽ ἐπεὶ αἱ μὲν ἄρτιαι οἱ δὲ περιττοὶ ἦσαν, ἀνάγκη παραλλάττειν τῆς αὐτῆς προσθέσεως γινομένης. | Non tamen hae quidem semper perfectae erunt, illi vero abundantes, sed permutatim, quia cum propositiones quidem sunt perfectae, abundantes erunt termini, cum vero termini perfecti, abundantes erunt propositiones, simul enim termino addito, una additur propositio, undecunque addatur terminus. Quare quoniam hae propositiones quidem perfectae, illi vero abundantes erant, necesse est transmutare eadem, additione facta. | The premisses however will not always be even, the terms odd; but they will alternate-when the premisses are even, the terms must be odd; when the terms are even, the premisses must be odd: for along with one term one premiss is added, if a term is added from any quarter. Consequently since the premisses were (as we saw) even, and the terms odd, we must make them alternately even and odd at each addition. |
| τὰ δὲ συμπεράσματα οὐκέτι τὴν αὐτὴν ἕξει τάξιν οὔτε πρὸς τοὺς ὅρους οὔτε πρὸς τὰς προτάσεις· ἑνὸς γὰρ ὅρου προστιθεμένου συμπεράσματα προστεθήσεται ἑνὶ ἐλάττω τῶν προϋπαρχόντων ὅρων· πρὸς μόνον γὰρ τὸν ἔσχατον οὐ ποιεῖ συμπέρασμα, πρὸς δὲ τοὺς ἄλλους πάντας, οἷον εἰ τῶι Α Β Γ πρόσκειται τὸ Δ, εὐθὺς καὶ συμπεράσματα δύο πρόσκειται, τό τε πρὸς τὸ Α καὶ τὸ πρὸς τὸ Β. ὁμοίως δὲ κἀπὶ τῶν ἄλλων. κἂν εἰς τὸ μέσον δὲ παρεμπίπτηι, τὸν αὐτὸν τρόπον· πρὸς ἕνα γὰρ μόνον οὐ ποιήσει συλλογισμόν. ὥστε πολὺ πλείω τὰ συμπεράσματα καὶ τῶν ὅρων ἔσται καὶ τῶν προτάσεων. | (0668C) Conclusiones autem non etiam eum habebunt ordinem neque ad terminos, neque ad propositiones, uno enim termino addito, conclusiones adiungentur uno, pauciores praeexistentibus terminis, ad solum enim ultimum non facit conclusionem, ad alios autem omnes. Ut si eis quae sunt A B C, adiacet D, statim et conclusiones duae adiacent, quae ad A, et ad B, similiter autem et in aliis. Si autem ad medium intercidat, eodem modo, ad unum enim solum non faciet syllogismum, quare multo plures conclusiones erunt et terminis et propositionibus. | But the conclusions will not follow the same arrangement either in respect to the terms or to the premisses. For if one term is added, conclusions will be added less by one than the pre-existing terms: for the conclusion is drawn not in relation to the single term last added, but in relation to all the rest, e.g. if to ABC the term D is added, two conclusions are thereby added, one in relation to A, the other in relation to B. Similarly with any further additions. And similarly too if the term is inserted in the middle: for in relation to one term only, a syllogism will not be constructed. Consequently the conclusions will be much more numerous than the terms or the premisses. |
Chapter 26
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| (PL 64 0668C) CAPUT XXVII. De problematis, hoc est propositis in unaquaque figura facile et difficile construendis et destruendis. | 26 | |
| 42b27 Ἐπεὶ δ᾽ ἔχομεν περὶ ὧν οἱ συλλογισμοί, καὶ ποῖον ἐν ἑκάστωι σχήματι καὶ ποσαχῶς δείκνυται, φανερὸν ἡμῖν ἐστὶ καὶ ποῖον πρόβλημα χαλεπὸν καὶ ποῖον εὐεπιχείρητον· τὸ μὲν γὰρ ἐν πλείοσι σχήμασι καὶ διὰ πλειόνων πτώσεων περαινόμενον ῥᾶιον, τὸ δ᾽ ἐν ἐλάττοσι καὶ δι᾽ ἐλαττόνων δυσεπιχειρητότερον. | (0668D) Quoniam autem habemus ex quibus syllogismi, et quale in unaquaque figura, et quot modis monstratur, manifestum nobis est, et quae propositio facile, et quae difficile argumentabilis est. Nam quae in pluribus figuris et per plures casus concluditur, facilis; quae autem in paucis et per pauciores, difficilius argumentabilis. | Since we understand the subjects with which syllogisms are concerned, what sort of conclusion is established in each figure, and in how many moods this is done, it is evident to us both what sort of problem is difficult and what sort is easy to prove. For that which is concluded in many figures and through many moods is easier; that which is concluded in few figures and through few moods is more difficult to attempt. |
| τὸ μὲν οὖν καταφατικὸν τὸ καθόλου διὰ τοῦ πρώτου σχήματος δείκνυται μόνου, καὶ διὰ τούτου μοναχῶς· τὸ δὲ στερητικὸν διά τε τοῦ πρώτου καὶ διὰ τοῦ μέσου, καὶ διὰ μὲν τοῦ πρώτου μοναχῶς, διὰ δὲ τοῦ μέσου διχῶς· τὸ δ᾽ ἐν μέρει καταφατικὸν διὰ τοῦ πρώτου καὶ διὰ τοῦ ἐσχάτου, μοναχῶς μὲν διὰ τοῦ πρώτου, τριχῶς δὲ διὰ τοῦ ἐσχάτου. τὸ δὲ στερητικὸν τὸ κατὰ μέρος ἐν ἅπασι τοῖς σχήμασι δείκνυται, πλὴν ἐν μὲν τῶι πρώτωι μοναχῶς, ἐν δὲ τῶι μέσωι καὶ τῶι ἐσχάτωι ἐν τῶι μὲν διχῶς ἐν τῶι δὲ τριχῶς. | Ergo affirmativa quidem universalis per primam tantum figuram monstratur, et per hanc simpliciter. Privativa vero et per primam, et per mediam. Per primam quidem simpliciter, per mediam autem dupliciter. Particularis autem affirmativa per primam et per postremam, simpliciter quidem per primam, tripliciter autem per postremam. Privativa vero particularis in omnibus figuris monstratur, verum in prima quidem semel, in media autem et postrema, in illa quidem dupliciter, in hac vero tripliciter. | The universal affirmative is proved by means of the first figure only and by this in only one mood; the universal negative is proved both through the first figure and through the second, through the first in one mood, through the second in two. The particular affirmative is proved through the first and through the last figure, in one mood through the first, in three moods through the last. The particular negative is proved in all the figures, but once in the first, in two moods in the second, in three moods in the third. |
| φανε↵ρὸν οὖν ὅτι τὸ καθόλου κατηγορικὸν κατασκευάσαι μὲν χαλεπώτατον, ἀνασκευάσαι δὲ ῥᾶιστον. ὅλως δ᾽ ἐστὶν ἀναιροῦντι μὲν τὰ καθόλου τῶν ἐν μέρει ῥάιω· καὶ γὰρ ἢν μηδενὶ καὶ ἢν τινὶ μὴ ὑπάρχηι, ἀνήιρηται· τούτων δὲ τὸ μὲν τινὶ μὴ ἐν ἅπασι τοῖς σχήμασι δείκνυται, τὸ δὲ μηδενὶ ἐν τοῖς δυσίν. τὸν αὐτὸν δὲ τρόπον κἀπὶ τῶν στερητικῶν· καὶ γὰρ εἰ παντὶ καὶ εἰ τινί, ἀνήιρηται τὸ ἐξ ἀρχῆς· τοῦτο δ᾽ ἦν ἐν δύο σχήμασιν. ἐπὶ δὲ τῶν ἐν μέρει μοναχῶς, ἢ παντὶ ἢ μηδενὶ δείξαντα ὑπάρχειν. κατασκευάζοντι δὲ ῥάιω τὰ ἐν μέρει· καὶ γὰρ ἐν πλείοσι σχήμασι καὶ διὰ πλειόνων τρόπων. | (0669A) Manifestum ergo quoniam universalem affirmativam construere quidem difficillimum, destruere autem facillimum, omnino autem est interimenti quidem, universalia quam particularia facilius. Etenim si nulli, et si alicui non insit interemptum est, horum autem alicui quidem non in omnibus figuris monstratur, nulli autem in duabus. Eodem autem modo et in privativis, etenim si omni, et si alicui, interemptum est quod ex principio. Hoc autem fuit in duabus figuris. In particularibus autem simpliciter, aut omni, aut nulli ostendentem inesse. Construenti autem, facilius est particularia, nam in pluribus figuris, et per plures modos. | It is clear then that the universal affirmative is most difficult to establish, most easy to overthrow. In general, universals are easier game for the destroyer than particulars: for whether the predicate belongs to none or not to some, they are destroyed: and the particular negative is proved in all the figures, the universal negative in two. Similarly with universal negatives: the original statement is destroyed, whether the predicate belongs to all or to some: and this we found possible in two figures. But particular statements can be refuted in one way only-by proving that the predicate belongs either to all or to none. But particular statements are easier to establish: for proof is possible in more figures and through more moods.
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| ὅλως τε οὐ δεῖ λανθάνειν ὅτι ἀνασκευάσαι μὲν δι᾽ ἀλλήλων ἔστι καὶ τὰ καθόλου διὰ τῶν ἐν μέρει καὶ ταῦτα διὰ τῶν καθόλου, κατασκευάσαι δ᾽ οὐκ ἔστι διὰ τῶν κατὰ μέρος τὰ καθόλου, δι᾽ ἐκείνων δὲ ταῦτ᾽ ἔστιν. ἅμα δὲ δῆλον ὅτι καὶ τὸ ἀνα σκευάζειν ἐστὶ τοῦ κατασκευάζειν ῥᾶιον. | (0669B) Omnino autem non oportet latere quoniam destruere quidem per se invicem est, et universalia per particularia, et haec per universalia; construere autem non est per particularia universalia, per illa vero haec est. Nam si omni, et alicui. Simul autem manifestum quoniam destruere quam construere facilius. | And in general we must not forget that it is possible to refute statements by means of one another, I mean, universal statements by means of particular, and particular statements by means of universal: but it is not possible to establish universal statements by means of particular, though it is possible to establish particular statements by means of universal. At the same time it is evident that it is easier to refute than to establish. |
| Πῶς μὲν οὖν γίνεται πᾶς συλλογισμὸς καὶ διὰ πόσων ὅρων καὶ προτάσεων, καὶ πῶς ἐχουσῶν πρὸς ἀλλήλας, ἔτι δὲ ποῖον πρόβλημα ἐν ἑκάστωι σχήματι καὶ ποῖον ἐν πλείοσι καὶ ποῖον ἐν ἐλάττοσι δείκνυται, δῆλον ἐκ τῶν εἰρημένων. | Quomodo ergo fit omnis syllogismus, et per quot terminos et propositiones, et quomodo habentes se ad invicem, amplius autem quae propositio in unaquaque figura, et quae in pluribus, et quae in paucioribus monstratur, palam ex his quae dicta sunt. | The manner in which every syllogism is produced, the number of the terms and premisses through which it proceeds, the relation of the premisses to one another, the character of the problem proved in each figure, and the number of the figures appropriate to each problem, all these matters are clear from what has been said. |
Chapter 27
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| (PL 64 0669B) CAPUT XXVIII. De abundantia propositionum. | 27 | |
| 43a20 Πῶς δ᾽ εὐπορήσομεν αὐτοὶ πρὸς τὸ τιθέμενον ἀεὶ συλλογισμῶν, καὶ διὰ ποίας ὁδοῦ ληψόμεθα τὰς περὶ ἕκαστον ἀρχάς, νῦν ἤδη λεκτέον· οὐ γὰρ μόνον ἴσως δεῖ τὴν γένεσιν θεωρεῖν τῶν συλλογισμῶν, ἀλλὰ καὶ τὴν δύναμιν ἔχειν τοῦ ποιεῖν. | Quomodo autem idonei erimus semper syllogizare ad propositum, et per quam viam sumemus circa unumquodque principia, nunc dicendum. Non enim solum fortasse oportet generationem considerare syllogismorum, sed et potestatem habere faciendi. (0669C) | We must now state how we may ourselves always have a supply of syllogisms in reference to the problem proposed and by what road we may reach the principles relative to the problem: for perhaps we ought not only to investigate the construction of syllogisms, but also to have the power of making them. |
| Ἁπάντων δὴ τῶν ὄντων τὰ μέν ἐστι τοιαῦτα ὥστε κατὰ μηδενὸς ἄλλου κατηγορεῖσθαι ἀληθῶς καθόλου (οἷον Κλέων καὶ Καλλίας καὶ τὸ καθ᾽ ἕκαστον καὶ αἰσθητόν), κατὰ δὲ τούτων ἄλλα (καὶ γὰρ ἄνθρωπος καὶ ζῶιον ἑκάτερος τούτων ἐστί)· τὰ δ᾽ αὐτὰ μὲν κατ᾽ ἄλλων κατηγορεῖται, κατὰ δὲ τούτων ἄλλα πρότερον οὐ κατηγορεῖται· τὰ δὲ καὶ αὐτὰ ἄλλων καὶ αὐτῶν ἕτερα, οἷον ἄνθρωπος Καλλίου καὶ ἀνθρώπου ζῶιον. | Omnium igitur quae sunt, haec quidem sunt talia, ut de nullo alio praedicentur vere universaliter, ut Cleon, et Callias, et quod singulare, et sensibile, de his autem alia, nam et homo, et animal uterque horum est. Illa vero et ipsa quidem de aliis praedicantur, de illis autem alia prius non praedicantur, alia autem et ipsa de aliis, et de his alia, ut homo de Callia, et de homine animal. | Of all the things which exist some are such that they cannot be predicated of anything else truly and universally, e.g. Cleon and Callias, i.e. the individual and sensible, but other things may be predicated of them (for each of these is both man and animal); and some things are themselves predicated of others, but nothing prior is predicated of them; and some are predicated of others, and yet others of them, e.g. man of Callias and animal of man. |
| ὅτι μὲν οὖν ἔνια τῶν ὄντων κατ᾽ οὐδενὸς πέφυκε λέγεσθαι, δῆλον· τῶν γὰρ αἰσθητῶν σχεδὸν ἕκαστόν ἐστι τοιοῦτον ὥστε μὴ κατηγορεῖσθαι κατὰ μηδενός, πλὴν ὡς κατὰ συμ βεβηκός· φαμὲν γάρ ποτε τὸ λευκὸν ἐκεῖνο Σωκράτην εἶναι καὶ τὸ προσιὸν Καλλίαν. | Quoniam ergo quaedam eorum quae sunt de nullo nata sunt dici, palam: nam sensibilium pene unumquodque est huiusmodi, ut de nullo praedicetur, nisi, ut secundum accidens, dicimus enim quandoque album illud Socratem esse, et hoc veniens Calliam. | It is clear then that some things are naturally not stated of anything: for as a rule each sensible thing is such that it cannot be predicated of anything, save incidentally: for we sometimes say that that white object is Socrates, or that that which approaches is Callias. |
| ὅτι δὲ καὶ ἐπὶ τὸ ἄνω πορευομένοις ἵσταταί ποτε, πάλιν ἐροῦμεν· νῦν δ᾽ ἔστω τοῦτο κείμενον. κατὰ μὲν οὖν τούτων οὐκ ἔστιν ἀποδεῖξαι κατηγορούμενον ἕτερον, πλὴν εἰ μὴ κατὰ δόξαν, ἀλλὰ ταῦτα κατ᾽ ἄλλων· οὐδὲ τὰ καθ᾽ ἕκαστα κατ᾽ ἄλλων, ἀλλ᾽ ἕτερα κατ᾽ ἐκείνων. τὰ δὲ μεταξὺ δῆλον ὡς ἀμφοτέρως ἐνδέχεται (καὶ γὰρ αὐτὰ κατ᾽ ἄλλων καὶ ἄλλα κατὰ τούτων λεχθήσεται)· καὶ σχεδὸν οἱ λόγοι καὶ αἱ σκέψεις εἰσὶ μάλιστα περὶ τούτων. | Quoniam autem in sursum pergentibus statur quandoque, rursum dicemus. (0669D) Nunc autem sit hoc positum, de iis ergo praedicatum aliquod non est demonstrare nisi secundum opinionem, sed haec de aliis, neque singularia de aliis, sed alia de ipsis. Quae autem in medio sunt, manifestum quoniam utrumque contingit, nam et haec de aliis, et alia de his dicuntur, et pene rationes et considerationes sunt maxime de his. | We shall explain in another place that there is an upward limit also to the process of predicating: for the present we must assume this. Of these ultimate predicates it is not possible to demonstrate another predicate, save as a matter of opinion, but these may be predicated of other things. Neither can individuals be predicated of other things, though other things can be predicated of them. Whatever lies between these limits can be spoken of in both ways: they may be stated of others, and others stated of them. And as a rule arguments and inquiries are concerned with these things. |
| ↵ Δεῖ δὴ τὰς προτάσεις περὶ ἕκαστον οὕτως ἐκλαμβάνειν, ὑποθέμενον αὐτὸ πρῶτον καὶ τοὺς ὁρισμούς τε καὶ ὅσα ἴδια τοῦ πράγματός ἐστιν, εἶτα μετὰ τοῦτο ὅσα ἕπεται τῶι πράγματι, καὶ πάλιν οἷς τὸ πρᾶγμα ἀκολουθεῖ, καὶ ὅσα μὴ ἐνδέχεται αὐτῶι ὑπάρχειν. οἷς δ᾽ αὐτὸ μὴ ἐνδέχεται, οὐκ ἐκληπτέον διὰ τὸ ἀντιστρέφειν τὸ στερητικόν. | Oportet ergo propositiones circa unumquodque horum sic sumere supponentem, ipsum primum et definitiones, et quaecunque propria sunt rei, deinde post hoc quaecunque sequuntur rem. Et rursum quae res sequitur, et quaecunque non contingit ipsi inesse, quibus autem ipsa non contingit, non sumendum, eo quod convertitur privativa. | We must select the premisses suitable to each problem in this manner: first we must lay down the subject and the definitions and the properties of the thing; next we must lay down those attributes which follow the thing, and again those which the thing follows, and those which cannot belong to it. But those to which it cannot belong need not be selected, because the negative statement implied above is convertible. |
| διαιρετέον δὲ καὶ τῶν ἑπομένων ὅσα τε ἐν τῶι τί ἐστι καὶ ὅσα ἴδια καὶ ὅσα ὡς συμβεβηκότα κατηγορεῖται, καὶ τούτων ποῖα δοξαστικῶς καὶ ποῖα κατ᾽ ἀλήθειαν· ὅσωι μὲν γὰρ ἂν πλειόνων τοιούτων εὐπορῆι τις, θᾶττον ἐντεύξεται συμπεράσματι, ὅσωι δ᾽ ἂν ἀληθεστέρων, μᾶλλον ἀποδείξει. | (0670A) Dividendum autem est, et eorum quae sequuntur, quaecunque in eo quod quid est, et quaecunque ut propria, et quaecunque ut accidentia praedicantur, et horum quae secundum opinionem, et quae secundum veritatem. Quanto enim plurium talium abundaverit quis, citius inveniet conclusionem, quanto autem veriorum, magis demonstrabit. | Of the attributes which follow we must distinguish those which fall within the definition, those which are predicated as properties, and those which are predicated as accidents, and of the latter those which apparently and those which really belong. The larger the supply a man has of these, the more quickly will he reach a conclusion; and in proportion as he apprehends those which are truer, the more cogently will he demonstrate. |
| Δεῖ δ᾽ ἐκλέγειν μὴ τὰ ἑπόμενα τινί, ἀλλ᾽ ὅσα ὅλωι τῶι πράγματι ἕπεται, οἷον μὴ τί τινὶ ἀνθρώπωι ἀλλὰ τί παντὶ ἀνθρώπωι ἕπεται· διὰ γὰρ τῶν καθόλου προτάσεων ὁ συλλογισμός. ἀδιορίστου μὲν οὖν ὄν τος ἄδηλον εἰ καθόλου ἡ πρότασις, διωρισμένου δὲ φανερόν. ὁμοίως δ᾽ ἐκλεκτέον καὶ οἷς αὐτὸ ἕπεται ὅλοις, διὰ τὴν εἰρημένην αἰτίαν. | Oportet autem eligere non quae sequuntur aliquam, sed quaecunque totam rem sequuntur, ut non quod aliquem hominem, sed quod omnem hominem sequitur, per universales enim propositiones fit syllogismus. Cum autem est indefinitum, incertum si universalis est propositio, cum vero definitum, manifestum. Similiter autem eligendum et quae ipsum sequitur tota, propter dictam causam. | But he must select not those which follow some particular but those which follow the thing as a whole, e.g. not what follows a particular man but what follows every man: for the syllogism proceeds through universal premisses. If the statement is indefinite, it is uncertain whether the premiss is universal, but if the statement is definite, the matter is clear. Similarly one must select those attributes which the subject follows as wholes, for the reason given. |
| αὐτὸ δὲ τὸ ἑπόμενον οὐ ληπτέον ὅλον ἕπεσθαι, λέγω δ᾽ οἷον ἀνθρώπωι πᾶν ζῶιον ἢ μουσικῆι πᾶσαν ἐπιστήμην, ἀλλὰ μόνον ἁπλῶς ἀκολουθεῖν, καθάπερ καὶ προ τεινόμεθα· καὶ γὰρ ἄχρηστον θάτερον καὶ ἀδύνατον, οἷον πάντα ἄνθρωπον εἶναι πᾶν ζῶιον ἢ δικαιοσύνην ἅπαν ἀγαθόν. ἀλλ᾽ ὧι ἕπεται, ἐπ᾽ ἐκείνου τὸ παντὶ λέγεται. | (0670B) Ipsum autem quod sequitur, non est sumendum totum sequi, dico ut hominem omne animal, aut musicam, omnem disciplinam, sed simpliciter solum sequi quemadmodum et praetendimus, etenim inutile alterum et impossibile, ut omnem hominem esse omne animal, vel iustitiam omne bonum, sed cui consequens est, in illo omni esse dicitur. | But that which follows one must not suppose to follow as a whole, e.g. that every animal follows man or every science music, but only that it follows, without qualification, and indeed we state it in a proposition: for the other statement is useless and impossible, e.g. that every man is every animal or justice is all good. But that which something follows receives the mark ‘every’. |
| ὅταν δ᾽ ὑπό τινος περιέχηται τὸ ὑποκείμενον ὧι τὰ ἑπόμενα δεῖ λαβεῖν, τὰ μὲν τῶι καθόλου ἑπόμενα ἢ μὴ ἑπόμενα οὐκ ἐκλεκτέον ἐν τούτοις (εἴληπται γὰρ ἐν ἐκείνοις· ὅσα γὰρ ζώιωι, καὶ ἀνθρώπωι ἕπεται, καὶ ὅσα μὴ ὑπάρχει, ὡσαύτωσ), τὰ δὲ περὶ ἕκαστον ἴδια ληπτέον· ἔστι γὰρ ἄττα τῶι εἴδει ἴδια παρὰ τὸ γένος· ἀνάγκη γὰρ τοῖς ἑτέροις εἴδεσιν ἴδια ἄττα ὑπάρχειν. | Quando autem ab aliquo continetur subiectum, cuius consequentia oportet sumere, quae universale quidem sequuntur, vel non sequuntur, non eligendum in his, sumpta enim sunt in illis quaecunque animal et hominem sequuntur, et quaecunque non animali insunt, similiter. Quae autem in unoquoque sunt propria, sumendum: sunt enim quaedam speciei propria praeter genus, necesse est enim diversis speciebus propria quaedam inesse. | Whenever the subject, for which we must obtain the attributes that follow, is contained by something else, what follows or does not follow the highest term universally must not be selected in dealing with the subordinate term (for these attributes have been taken in dealing with the superior term; for what follows animal also follows man, and what does not belong to animal does not belong to man); but we must choose those attributes which are peculiar to each subject. For some things are peculiar to the species as distinct from the genus; for species being distinct there must be attributes peculiar to each. |
| οὐδὲ δὴ τῶι καθόλου ἐκλεκτέον οἷς ἕπεται τὸ περι εχόμενον, οἷον ζώιωι οἷς ἕπεται ἄνθρωπος· ἀνάγκη γάρ, εἰ ἀνθρώπωι ἀκολουθεῖ τὸ ζῶιον, καὶ τούτοις ἅπασιν ἀκολουθεῖν, οἰκειότερα δὲ ταῦτα τῆς τοῦ ἀνθρώπου ἐκλογῆς. ληπτέον δὲ καὶ τὰ ὡς ἐπὶ τὸ πολὺ ἑπόμενα καὶ οἷς ἕπεται· τῶν γὰρ ὡς ἐπὶ τὸ πολὺ προβλημάτων καὶ ὁ συλλογισμὸς ἐκ τῶν ὡς ἐπὶ τὸ πολὺ προτάσεων, ἢ πασῶν ἢ τινῶν· ὅμοιον γὰρ ἑκάστου τὸ συμπέρασμα ταῖς ἀρχαῖς. ἔτι τὰ πᾶσιν ἑπόμενα οὐκ ἐκλεκτέον· οὐ γὰρ ἔσται συλλογισμὸς ἐξ αὐτῶν. δι᾽ ἣν δ᾽ αἰτίαν, ἐν τοῖς ἑπομένοις ἔσται δῆλον. | (0670C) Neque autem universale eligendum iis quae sequitur quod continetur, ut animal iis quae sequitur homo, necesse est enim si hominem sequitur animal, et haec omnia sequi, convenientiora autem haec hominis electioni. Sumendum autem et quae plerumque sequuntur ea quae consequuntur, nam et problematibus quae plerumque, et syllogismus ex propositionibus, quae plerumque aut in omnibus, aut aliquibus, similis enim est uniuscuiusque conclusio principiis. Amplius quae omnibus sequentia sunt, non eligendum, non enim erit syllogismus ex ipsis, ob quam autem causam, in sequentibus erit manifestum. | Nor must we take as things which the superior term follows, those things which the inferior term follows, e.g. take as subjects of the predicate ‘animal’ what are really subjects of the predicate ‘man’. It is necessary indeed, if animal follows man, that it should follow all these also. But these belong more properly to the choice of what concerns man. One must apprehend also normal consequents and normal antecedents, for propositions which obtain normally are established syllogistically from premisses which obtain normally, some if not all of them having this character of normality. For the conclusion of each syllogism resembles its principles. We must not however choose attributes which are consequent upon all the terms: for no syllogism can be made out of such premisses. The reason why this is so will be clear in the sequel. |
Chapter 28
| Greek | Latin | English |
|---|---|---|
| (PL 64 0670C) CAPUT XXIX. Medii syllogismorum inveniendi regulae. | 28 | |
| 43b39 Κατασκευάζειν μὲν οὖν βουλομένοις κατά τινος ὅλου τοῦ μὲν κατασκευαζομένου βλεπτέον εἰς τὰ ὑποκείμενα καθ᾽ ὧν αὐτὸ τυγχάνει λεγόμενον, οὗ δὲ δεῖ κατηγορεῖσθαι, ὅσα τούτωι ἕπεται· ἂν γάρ τι τούτων ἦι ταὐτόν, ἀνάγκη θάτερον θατέρωι ὑπάρχειν. ἢν δὲ μὴ ὅτι παντὶ ἀλλ᾽ ὅτι τινί, οἷς ἕπε ↵ ται ἑκάτερον· εἰ γάρ τι τούτων ταὐτόν, ἀνάγκη τινὶ ὑπάρχειν. | (0670D) Construere ergo volentibus aliquid de aliquo toto, eius quidem quod construitur, inspiciendum ad subiecta de quibus ipsum dicitur, de quo autem oportet praedicari quaecunque hoc sequuntur. Si enim aliquod horum sit idem, alterum alteri necesse est inesse. Si autem non quoniam omni, sed quoniam alicui, quae sequitur utrumque, si enim aliquod horum idem fuerit, necesse est alicui inesse. | If men wish to establish something about some whole, they must look to the subjects of that which is being established (the subjects of which it happens to be asserted), and the attributes which follow that of which it is to be predicated. For if any of these subjects is the same as any of these attributes, the attribute originally in question must belong to the subject originally in question. But if the purpose is to establish not a universal but a particular proposition, they must look for the terms of which the terms in question are predicable: for if any of these are identical, the attribute in question must belong to some of the subject in question. |
| ὅταν δὲ μηδενὶ δέηι ὑπάρχειν, ὧι μὲν οὐ δεῖ ὑπάρχειν, εἰς τὰ ἑπόμενα, ὁ δὲ δεῖ μὴ ὑπάρχειν, εἰς ἃ μὴ ἐνδέχεται αὐτῶι παρεῖναι· ἢ ἀνάπαλιν, ὧι μὲν δεῖ μὴ ὑπάρχειν, εἰς ἃ μὴ ἐνδέχεται αὐτῶι παρεῖναι, ὁ δὲ μὴ ὑπάρχειν, εἰς τὰ ἑπόμενα. τούτων γὰρ ὄντων τῶν αὐτῶν ὁποτερωνοῦν, οὐδενὶ ἐνδέχεται θατέρωι θάτερον ὑπάρχειν· γίνεται γὰρ ὁτὲ μὲν ὁ ἐν τῶι πρώτωι σχήματι συλλογισμός, ὁτὲ δ᾽ ὁ ἐν τῶι μέσωι. | Quando autem nulli oporteat inesse, cui quidem oportet non inesse, ad sequentia subiecti, quod autem oportet non inesse, inspiciendum ad ea quae non contingunt illi adesse. Aut conversim cui quidem oportet non inesse, ad ea quae non contingunt eidem adesse, quod vero non inesse, inspiciendum ad sequentia. Nam si haec sint eadem utrorumque, nulli contingi alteri alterum inesse, fit enim quandoque quidem in prima figura syllogismus, quandoque autem in media. | Whenever the one term has to belong to none of the other, one must look to the consequents of the subject, and to those attributes which cannot possibly be present in the predicate in question: or conversely to the attributes which cannot possibly be present in the subject, and to the consequents of the predicate. If any members of these groups are identical, one of the terms in question cannot possibly belong to any of the other. For sometimes a syllogism in the first figure results, sometimes a syllogism in the second. |
| ἐὰν δὲ τινὶ μὴ ὑπάρχειν, ὧι μὲν δεῖ μὴ ὑπάρχειν, οἷς ἕπεται, ὁ δὲ μὴ ὑπάρχειν, ἃ μὴ δυνατὸν αὐτῶι ὑπάρχειν· εἰ γάρ τι τούτων εἴη ταὐτόν, ἀνάγκη τινὶ μὴ ὑπάρχειν. Μᾶλλον δ᾽ ἴσως ὧδ᾽ ἔσται τῶν λεγομένων ἕκαστον φανερόν. ἔστω γὰρ τὰ μὲν ἑπόμενα τῶι Α ἐφ᾽ ὧν Β, οἷς δ᾽ αὐτὸ ἕπεται, ἐφ᾽ ὧν Γ, ἃ δὲ μὴ ἐνδέχεται αὐτῶι ὑπάρχειν, ἐφ᾽ ὧν Δ· πάλιν δὲ τῶι Ε τὰ μὲν ὑπάρχοντα, ἐφ᾽ οἷς Ζ, οἷς δ᾽ αὐτὸ ἕπεται, ἐφ᾽ οἷς Η, ἃ δὲ μὴ ἐνδέχεται αὐτῶι ὑπάρχειν, ἐφ᾽ οἷς Θ.
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(0671A) Si autem alicui non inesse, cui quidem oportet non inesse, quae consequitur: quod vero non inesse, quae non possibile est illi inesse. Si enim aliquid horum sit idem, necesse est alicui non inesse. Magis autem fortasse erit sic, unumquodque eorum quae dicta sunt manifestum. Sint enim sequentia quidem A, in quibus B, quae autem ipsum sequitur, in quibus C, quae autem non contingunt ei inesse, in quibus D, rursum autem ipsi E quae quidem insunt, in quibus F, quae autem ipsum sequitur, in quibus G, quae autem non contingunt eidem inesse, in quibus H. | But if the object is to establish a particular negative proposition, we must find antecedents of the subject in question and attributes which cannot possibly belong to the predicate in question. If any members of these two groups are identical, it follows that one of the terms in question does not belong to some of the other. Perhaps each of these statements will become clearer in the following way. Suppose the consequents of A are designated by B, the antecedents of A by C, attributes which cannot possibly belong to A by D. Suppose again that the attributes of E are designated by F, the antecedents of E by G, and attributes which cannot belong to E by H. |
| εἰ μὲν οὖν ταὐτό τί ἐστι τῶν Γ τινὶ τῶν Ζ, ἀνάγκη τὸ Α παντὶ τῶι Ε ὑπάρχειν· τὸ μὲν γὰρ Ζ παντὶ τῶι Ε, τῶι δὲ Γ παντὶ τὸ Α, ὥστε παντὶ τῶι Ε τὸ Α. εἰ δὲ τὸ Γ καὶ τὸ Η ταὐτόν, ἀνάγκη τινὶ τῶι Ε τὸ Α ὑπάρχειν· τῶι μὲν γὰρ Γ τὸ Α, τῶι δὲ Η τὸ Ε παντὶ ἀκολουθεῖ. εἰ δὲ τὸ Ζ καὶ τὸ Δ ταὐτόν, οὐδενὶ τῶν Ε τὸ Α ὑπάρξει ἐκ προσυλλογισμοῦ· ἐπεὶ γὰρ ἀντιστρέφει τὸ στερητικὸν καὶ τὸ Ζ τῶι Δ ταὐτόν, οὐδενὶ τῶν Ζ ὑπάρξει τὸ Α, τὸ δὲ Ζ παντὶ τῶι Ε. | Si ergo eidem aliquid eorum quae sunt C, alicui eorum quae sunt F, necesse est A omni E inesse, nam F quidem omni E, C autem omni A, quare omni E inest. (0671B) Si autem C et G idem, necesse est alicui E inesse A, nam id quod est E A, id vero quod est G E, omne ei sequitur. Si autem F et D sint idem, nulli E inerit ex proprio syllogismo, quoniam enim convertitur privativa, et F ei quod est D idem, nulli F inerit A, F autem omni E. | If then one of the Cs should be identical with one of the Fs, A must belong to all E: for F belongs to all E, and A to all C, consequently A belongs to all E. If C and G are identical, A must belong to some of the Es: for A follows C, and E follows all G. If F and D are identical, A will belong to none of the Es by a prosyllogism: for since the negative proposition is convertible, and F is identical with D, A will belong to none of the Fs, but F belongs to all E. |
| πάλιν εἰ τὸ Β καὶ τὸ Θ ταὐτόν, οὐδενὶ τῶν Ε τὸ Α ὑπάρξει· τὸ γὰρ Β τῶι μὲν Α παντί, τῶι δ᾽ ἐφ᾽ ὧι τὸ Ε οὐδενὶ ὑπάρξει· ταὐτὸ γὰρ ἦν τῶι Θ, τὸ δὲ Θ οὐδενὶ τῶν Ε ὑπῆρχεν. εἰ δὲ τὸ Δ καὶ τὸ Η ταὐτόν, τὸ Α τινὶ τῶι Ε οὐχ ὑπάρξει· τῶι γὰρ Η οὐχ ὑπάρξει, ὅτι οὐδὲ τῶι Δ· τὸ δὲ Η ἐστὶν ὑπὸ τὸ Ε, ὥστε τινὶ τῶν Ε οὐχ ὑπάρξει. εἰ δὲ τῶι Η τὸ Β ταὐτόν, ἀντεστραμμένος ἔσται συλλογισμός· τὸ μὲν γὰρ Ε τῶι Α ὑπάρξει παντί – τὸ γὰρ Β τῶι Α, τὸ δὲ Ε τῶι Β (ταὐτὸ γὰρ ἦν τῶι Η) – τὸ δὲ Α τῶι Ε παντὶ μὲν οὐκ ἀνάγκη ὑπάρχειν, τινὶ δ᾽ ἀνάγκη διὰ τὸ ἀντιστρέφειν τὴν καθόλου κατη γορίαν τῆι κατὰ μέρος. | Rursus si B et H idem, nulli E inerit A, nam B A quidem omni, ei autem in quo E nulli inerit. Idem enim erat ei quod est H, B; H autem nulli E inerat. Si autem G et D idem, A alicui E non inerit, nam ei quod est G non inerit A, quoniam neque D, G autem sub E est, quare alicui E non inerit. (0671C) Si autem G et B idem, conversus erit syllogismus, nam G inerit omni A, nam B ei quod est A, E autem ei quod est B, idem enim erat ei quod est G, A autem ei quod est E, omni quidem non necessarium est inesse, alicui autem necessarium, eo quod convertatur universale praedicativum in particulare. | Again, if B and H are identical, A will belong to none of the Es: for B will belong to all A, but to no E: for it was assumed to be identical with H, and H belonged to none of the Es. If D and G are identical, A will not belong to some of the Es: for it will not belong to G, because it does not belong to D: but G falls under E: consequently A will not belong to some of the Es. If B is identical with G, there will be a converted syllogism: for E will belong to all A since B belongs to A and E to B (for B was found to be identical with G): but that A should belong to all E is not necessary, but it must belong to some E because it is possible to convert the universal statement into a particular. |
| Φανερὸν οὖν ὅτι εἰς τὰ προειρημένα βλεπτέον ἑκατέρου καθ᾽ ἕκαστον πρόβλημα· διὰ τούτων γὰρ ἅπαντες οἱ συλλογισμοί. δεῖ δὲ καὶ τῶν ἑπομένων, καὶ οἷς ἕπεται ἕκαστον, εἰς τὰ πρῶτα καὶ τὰ καθόλου μάλιστα βλέπειν, οἷον τοῦ μὲν Ε μᾶλλον εἰς τὸ Κ Ζ ἢ εἰς τὸ Ζ μόνον, τοῦ δὲ Α εἰς ↵ τὸ Κ Γ ἢ εἰς τὸ Γ μόνον. εἰ μὲν γὰρ τῶι Κ Ζ ὑπάρχει τὸ Α, καὶ τῶι Ζ καὶ τῶι Ε ὑπάρχει· εἰ δὲ τούτωι μὴ ἕπεται, ἐγχωρεῖ τῶι Ζ ἕπεσθαι. ὁμοίως δὲ καὶ ἐφ᾽ ὧν αὐτὸ ἀκολουθεῖ σκεπτέον· εἰ μὲν γὰρ τοῖς πρώτοις, καὶ τοῖς ὑπ᾽ ἐκεῖνα ἕπεται, εἰ δὲ μὴ τούτοις, ἀλλὰ τοῖς ὑπὸ ταῦτα ἐγχωρεῖ. | Manifestum ergo quoniam ad praedicta perspiciendum utrinque in unaquaque quaestione, per haec enim omnes syllogismi. Oportet autem et sequentium, et quibus sequitur singulum, ad prima et universalia maxime inspicere, ut E quidem magis ad k F quam ad F solum, A autem ad k C magis quam ad C solum. Si enim ei quod est k F inest A, et ei quod est F inest et ipsi E, si vero hoc non sequitur A, possibile est id quod est F sequi. Similiter autem et in quibus idem sequitur, considerandum, nam si primis, et iis quae sub ipsis sunt, sequitur; si autem non his, et iis quae sub ipsis sunt, possibile. | It is clear then that in every proposition which requires proof we must look to the aforesaid relations of the subject and predicate in question: for all syllogisms proceed through these. But if we are seeking consequents and antecedents we must look for those which are primary and most universal, e.g. in reference to E we must look to KF rather than to F alone, and in reference to A we must look to KC rather than to C alone. For if A belongs to KF, it belongs both to F and to E: but if it does not follow KF, it may yet follow F. Similarly we must consider the antecedents of A itself: for if a term follows the primary antecedents, it will follow those also which are subordinate, but if it does not follow the former, it may yet follow the latter. |
| Δῆλον δὲ καὶ ὅτι διὰ τῶν τριῶν ὅρων καὶ τῶν δύο προτάσεων ἡ σκέψις, καὶ διὰ τῶν προειρημένων σχημάτων οἱ συλλογισμοὶ πάντες. δείκνυται γὰρ ὑπάρχειν μὲν παντὶ τῶι Ε τὸ Α, ὅταν τῶν Γ καὶ Ζ ταὐτόν τι ληφθῆι. τοῦτο δ᾽ ἔσται μέσον, ἄκρα δὲ τὸ Α καὶ Ε· γίνεται οὖν τὸ πρῶτον σχῆμα. | (0671D) Palam autem quoniam per tres terminos et duas propositiones consideratio, et per praedictas figuras syllogismi omnes, monstratur enim omni quidem E inesse A, quando eorum quae sunt C F idem, quiddam sumitur, hoc autem erit medium, extremitates autem A et E, fit enim prima figura. | It is clear too that the inquiry proceeds through the three terms and the two premisses, and that all the syllogisms proceed through the aforesaid figures. For it is proved that A belongs to all E, whenever an identical term is found among the Cs and Fs. This will be the middle term; A and E will be the extremes. So the first figure is formed. |
| τινὶ δέ, ὅταν τὸ Γ καὶ τὸ Η ληφθῆι ταὐτόν. τοῦτο δὲ τὸ ἔσχατον σχῆμα· μέσον γὰρ τὸ Η γίνεται. μηδενὶ δέ, ὅταν τὸ Δ καὶ Ζ ταὐτόν. οὕτω δὲ καὶ τὸ πρῶτον σχῆμα καὶ τὸ μέσον, τὸ μὲν πρῶτον ὅτι οὐδενὶ τῶι Ζ ὑπάρχει τὸ Α (εἴπερ ἀντι- στρέφει τὸ στερητικόν), τὸ δὲ Ζ παντὶ τῶι Ε, τὸ δὲ μέσον ὅτι τὸ Δ τῶι μὲν Α οὐδενὶ τῶι δὲ Ε παντὶ ὑπάρχει. τινὶ δὲ μὴ ὑπάρχειν, ὅταν τὸ Δ καὶ Η ταὐτὸν ἦι. τοῦτο δὲ τὸ ἔσχατον σχῆμα· τὸ μὲν γὰρ Α οὐδενὶ τῶι Η ὑπάρξει, τὸ δὲ Ε παντὶ τῶι Η. | Alicui autem quando C et G sumitur idem, hoc autem postrema figura, medium enim fit G. Nulli vero quando D et F idem; sic autem et prima figura, et media: prima quidem, quoniam nulli F inest A, siquidem convertitur privativa, F autem omni E. Media autem quoniam D A quidem nulli, E autem omni inest. Alicui autem non inesse, quando D et G idem fuerit, haec autem postrema figura, nam A quidem nulli G inerit, E vero omni G; | And A will belong to some E, whenever C and G are apprehended to be the same. This is the last figure: for G becomes the middle term. And A will belong to no E, when D and F are identical. Thus we have both the first figure and the middle figure; the first, because A belongs to no F, since the negative statement is convertible, and F belongs to all E: the middle figure because D belongs to no A, and to all E. And A will not belong to some E, whenever D and G are identical. This is the last figure: for A will belong to no G, and E will belong to all G. |
| φανερὸν οὖν ὅτι διὰ τῶν προειρημένων σχη μάτων οἱ συλλογισμοὶ πάντες, καὶ ὅτι οὐκ ἐκλεκτέον ὅσα πᾶσιν ἕπεται, διὰ τὸ μηδένα γίγνεσθαι συλλογισμὸν ἐξ αὐτῶν. κατασκευάζειν μὲν γὰρ ὅλως οὐκ ἦν ἐκ τῶν ἑπομένων, ἀποστερεῖν δ᾽ οὐκ ἐνδέχεται διὰ τοῦ πᾶσιν ἑπομένου· δεῖ γὰρ τῶι μὲν ὑπάρχειν τῶι δὲ μὴ ὑπάρχειν. | manifestum igitur est quoniam per praedictas figuras omnes syllogismi. (0672A) Et quoniam non eligendum quaecunque omnibus sequuntur, eo quod nullus fiat syllogismus ex ipsis, nam construere quidem non omnino erat ex sequentibus, privare autem non contingit per ea quae omnibus sequuntur, oportet huic quidem inesse, illi vero non inesse. | Clearly then all syllogisms proceed through the aforesaid figures, and we must not select consequents of all the terms, because no syllogism is produced from them. For (as we saw) it is not possible at all to establish a proposition from consequents, and it is not possible to refute by means of a consequent of both the terms in question: for the middle term must belong to the one, and not belong to the other. |
| Φανερὸν δὲ καὶ ὅτι αἱ ἄλλαι σκέψεις τῶν κατὰ τὰς ἐκλογὰς ἄχρειοι πρὸς τὸ ποιεῖν συλλογισμόν, οἷον εἰ τὰ ἑπόμενα ἑκατέρωι ταὐτά ἐστιν, ἢ εἰ οἷς ἕπεται τὸ Α καὶ ἃ μὴ ἐνδέχεται τῶι Ε, ἢ ὅσα πάλιν μὴ ἐγχωρεῖ ἑκατέρωι ὑπάρχειν· οὐ γὰρ γίνεται συλλογισμὸς διὰ τούτων. εἰ μὲν γὰρ τὰ ἑπόμενα ταὐτά, οἷον τὸ Β καὶ τὸ Ζ, τὸ μέσον γίνεται σχῆμα κατηγορικὰς ἔχον τὰς προτάσεις· εἰ δ᾽ οἷς ἕπεται τὸ Α καὶ ἃ μὴ ἐνδέχεται τῶι Ε, οἷον τὸ Γ καὶ τὸ Θ, τὸ πρῶτον σχῆμα στερητικὴν ἔχον τὴν πρὸς τὸ ἔλαττον ἄκρον πρότασιν. εἰ δ᾽ ὅσα μὴ ἐνδέχεται ἑκατέρωι, οἷον τὸ Δ καὶ τὸ Θ, στερητικαὶ ἀμφότεραι αἱ προτάσεις, ἢ ἐν τῶι πρώτωι ἢ ἐν τῶι μέσωι σχήματι. οὕτως δ᾽ οὐδαμῶς συλλογισμός. | Manifestum autem quoniam et aliae considerationes quae secundum electiones, inutiles ad faciendum syllogismum. Ut si sequentia utrumque eadem sint, aut quae sequitur A, et quae non contingit E inesse, aut rursum quaecunque non possibile est utrique inesse, non enim fit syllogismus per haec. Nam si sequentia sunt eadem, ut B et F, media fit figura praedicativas habens utrasque propositiones. (0672B) Si autem ea quae sequitur A, et quae non contingit E, ut C, et H, prima erit figura privativam habens propositionem ad minorem extremitatem. Si autem quaecunque non contingunt utrique, ut D et H, privativae utraeque propositiones erunt vel in prima figura, vel in media, sic autem nullo modo erit syllogismus. | It is clear too that other methods of inquiry by selection of middle terms are useless to produce a syllogism, e.g. if the consequents of the terms in question are identical, or if the antecedents of A are identical with those attributes which cannot possibly belong to E, or if those attributes are identical which cannot belong to either term: for no syllogism is produced by means of these. For if the consequents are identical, e.g. B and F, we have the middle figure with both premisses affirmative: if the antecedents of A are identical with attributes which cannot belong to E, e.g. C with H, we have the first figure with its minor premiss negative. If attributes which cannot belong to either term are identical, e.g. C and H, both premisses are negative, either in the first or in the middle figure. But no syllogism is possible in this way. |
| Δῆλον δὲ καὶ ὅτι ὁποῖα ταὐτὰ ληπτέον τὰ κατὰ τὴν ἐπίσκεψιν, καὶ οὐχ ὁποῖα ἕτερα ἢ ἐναντία, πρῶτον μὲν ὅτι τοῦ μέσου χάριν ἡ ἐπίβλεψις, τὸ δὲ μέσον οὐχ ἕτερον ↵ ἀλλὰ ταὐτὸν δεῖ λαβεῖν. εἶτα ἐν ὅσοις καὶ συμβαίνει γίνεσθαι συλλογισμὸν τῶι ληφθῆναι ἐναντία ἢ μὴ ἐνδεχόμενα τῶι αὐτῶι ὑπάρχειν, εἰς τοὺς προειρημένους ἅπαντα ἀναχθήσεται τρόπους, οἷον εἰ τὸ Β καὶ τὸ Ζ ἐναντία ἢ μὴ ἐνδέχεται τῶι αὐτῶι ὑπάρχειν· ἔσται μὲν γὰρ τούτων ληφθέντων συλλογισμὸς ὅτι οὐδενὶ τῶν Ε τὸ Α ὑπάρχει, ἀλλ᾽ οὐκ ἐξ αὐτῶν ἀλλ᾽ ἐκ τοῦ προειρημένου τρόπου· τὸ γὰρ Β τῶι μὲν Α παντὶ τῶι δὲ Ε οὐδενὶ ὑπάρξει· ὥστ᾽ ἀνάγκη ταὐτὸ εἶναι τὸ Β τινὶ τῶι Θ. [πάλιν εἰ τὸ Β καὶ Η μὴ ἐγχωρεῖ τῶι αὐτῶι παρεῖναι, ὅτι τινὶ τῶι Ε οὐχ ὑπάρξει τὸ Α· καὶ γὰρ οὕτως τὸ μέσον ἔσται σχῆμα· τὸ γὰρ Β τῶι μὲν Α παντὶ τῶι δὲ Ε οὐδενὶ ὑπάρξει· ὥστ᾽ ἀνάγκη τὸ Β ταὐτόν τινι εἶναι τῶν Θ. τὸ γὰρ μὴ ἐνδέχεσθαι τὸ Β καὶ τὸ Η τῶι αὐτῶι ὑπάρχειν οὐδὲν διαφέρει ἢ τὸ Β τῶν Θ τινὶ ταὐ τὸν εἶναι· πάντα γὰρ εἴληπται τὰ μὴ ἐνδεχόμενα τῶι Ε ὑπάρχειν.] | Palam autem et quae eadem, sumendum secundum considerationem, et non quae diversa vel contraria, primum quidem quoniam medii gratia, inspectio, medium autem non diversum, sed idem oportet sumere. Deinde et in quibus accidit fieri syllogismum quod sumantur contraria, aut non contigentia eidem inesse, in praedictos omnia reducuntur modos. (0672C) Ut si B et F sint contraria, aut non contingant eidem inesse, erit enim his sumptis syllogismus, quoniam nulli E inest A, sed non ex ipsis, sed ex praedicto modo, nam B A quidem omni, E autem nulli inerit, quare necesse est B idem esse alicui eorum quae sunt H. Rursum si B et G non possint eidem adesse, erit quoniam alicui E non inerit A, nam et sic media erit figura, nam B A quidem omni, G vero nulli inerit, quare necesse est G idem esse alicui eorum quae sunt D, nam non contingere G et B eidem inesse nihil differt, aut G alicui D idem esse, omnia enim sumpta sunt in D, quae non contingunt A inesse. | It is evident too that we must find out which terms in this inquiry are identical, not which are different or contrary, first because the object of our investigation is the middle term, and the middle term must be not diverse but identical. Secondly, wherever it happens that a syllogism results from taking contraries or terms which cannot belong to the same thing, all arguments can be reduced to the aforesaid moods, e.g. if B and F are contraries or cannot belong to the same thing. For if these are taken, a syllogism will be formed to prove that A belongs to none of the Es, not however from the premisses taken but in the aforesaid mood. For B will belong to all A and to no E. Consequently B must be identical with one of the Hs. Again, if B and G cannot belong to the same thing, it follows that A will not belong to some of the Es: for then too we shall have the middle figure: for B will belong to all A and to no G. Consequently B must be identical with some of the Hs. For the fact that B and G cannot belong to the same thing differs in no way from the fact that B is identical with some of the Hs: for that includes everything which cannot belong to E. |
| Φανερὸν οὖν ὅτι ἐξ αὐτῶν μὲν τούτων τῶν ἐπιβλέψεων οὐδεὶς γίνεται συλλογισμός, ἀνάγκη δ᾽ εἰ τὸ Β καὶ τὸ Ζ ἐναντία, ταὐτόν τινι εἶναι τὸ Β τῶν Θ καὶ τὸν συλλογι σμὸν γίγνεσθαι διὰ τούτων. συμβαίνει δὴ τοῖς οὕτως ἐπισκοποῦσι προσεπιβλέπειν ἄλλην ὁδὸν τῆς ἀναγκαίας διὰ τὸ λανθάνειν τὴν ταὐτότητα τῶν Β καὶ τῶν Θ. | Manifestum ergo quoniam ex istis quidem inspectionibus nullus fit syllogismus, et si B et F sint contraria, idem esse B alicui H, et syllogismum semper fieri per haec. Accidit ergo sic inspicientibus considerare viam aliam necessariam, eo quod quandoque latet identitas horum quae sunt B et H. | It is clear then that from the inquiries taken by themselves no syllogism results; but if B and F are contraries B must be identical with one of the Hs, and the syllogism results through these terms. It turns out then that those who inquire in this manner are looking gratuitously for some other way than the necessary way because they have failed to observe the identity of the Bs with the Hs. |
Chapter 29
| Greek | Latin | English |
|---|---|---|
| (PL 64 0672C) CAPUT XXX. De syllogismis assertoriis ad impossibile, et reliquis qui ex hypothesi. | 29 | |
| 45a23 Τὸν αὐτὸν δὲ τρόπον ἔχουσι καὶ οἱ εἰς τὸ ἀδύνατον ἄγοντες συλλογισμοὶ τοῖς δεικτικοῖς· καὶ γὰρ οὗτοι γίνον ται διὰ τῶν ἑπομένων καὶ οἷς ἕπεται ἑκάτερον. καὶ ἡ αὐτὴ ἐπίσκεψις ἐν ἀμφοῖν· ὁ γὰρ δείκνυται δεικτικῶς, καὶ διὰ τοῦ ἀδυνάτου ἔστι συλλογίσασθαι διὰ τῶν αὐτῶν ὅρων, καὶ ὁ διὰ τοῦ ἀδυνάτου, καὶ δεικτικῶς, οἷον ὅτι τὸ Α οὐδενὶ τῶι Ε ὑπάρχει. κείσθω γὰρ τινὶ ὑπάρχειν· οὐκοῦν ἐπεὶ τὸ Β παντὶ τῶι Α, τὸ δὲ Α τινὶ τῶι Ε, τὸ Β τινὶ τῶν Ε ὑπάρξει· ἀλλ᾽ οὐδενὶ ὑπῆρχεν. πάλιν ὅτι τινὶ ὑπάρχει· εἰ γὰρ μηδενὶ τῶι Ε τὸ Α, τὸ δὲ Ε παντὶ τῶι Η, οὐδενὶ τῶν Η ὑπάρξει τὸ Α· ἀλλὰ παντὶ ὑπῆρχεν. | (0672D) Eodem autem modo se habent et qui ad impossibile deducunt syllogismi, ostensivis, nam et ipsi fiunt per ea quae sequuntur, et quibus sequitur utrumque. Et eadem consideratio in utrisque, nam quod monstratur ostensive, et per impossibile est syllogizare, et per eosdem terminos, et quod per impossibile et ostensive. Ut quoniam A nulli E inest, ponatur enim alicui inesse, ergo quoniam B omni A, A autem alicui E, et B alicui E inerit, sed nulli inerat. Rursum quoniam alicui E inest A, si enim nulli E inest A, E autem omni G, nulli G inerit A, sed omni inerat. | Syllogisms which lead to impossible conclusions are similar to ostensive syllogisms; they also are formed by means of the consequents and antecedents of the terms in question. In both cases the same inquiry is involved. For what is proved ostensively may also be concluded syllogistically per impossibile by means of the same terms; and what is proved per impossibile may also be proved ostensively, e.g. that A belongs to none of the Es. For suppose A to belong to some E: then since B belongs to all A and A to some of the Es, B will belong to some of the Es: but it was assumed that it belongs to none. Again we may prove that A belongs to some E: for if A belonged to none of the Es, and E belongs to all G, A will belong to none of the Gs: but it was assumed to belong to all. |
| ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων προβλημάτων· ἀεὶ γὰρ ἔσται καὶ ἐν ἅπασιν ἡ διὰ τοῦ ἀδυνάτου δεῖξις ἐκ τῶν ἑπομένων καὶ οἷς ἕπεται ἑκάτερον. καὶ καθ᾽ ἕκαστον πρόβλημα ἡ αὐτὴ σκέψις δεικτικῶς τε βουλομένωι συλλογίσασθαι καὶ εἰς ἀδύνατον ἀγαγεῖν· ἐκ γὰρ τῶν αὐτῶν ὅρων ἀμφότεραι αἱ ἀποδείξεις, οἷον εἰ δέδεικται μηδενὶ ὑπάρχειν τῶι Ε τὸ Α, ὅτι συμβαίνει καὶ τὸ Β τινὶ τῶι Ε ὑπάρχειν, ὅπερ ἀδύνατον· ἐὰν ληφθῆι τῶι μὲν Ε μηδενὶ τῶι δὲ Α παντὶ ὑπάρχειν τὸ Β, φανερὸν ↵ ὅτι οὐδενὶ τῶι Ε τὸ Α ὑπάρξει. | (0673A) Similiter autem est in aliis propositis, semper enim erit in omnibus per impossibile ostensio, ex sequentibus, et quibus sequitur utrumque. Et in uno quoque proposito, eadem consideratio et ostensive volenti syllogizare, et ad impossibile ducere, nam ex eisdem terminis utraeque demonstrationes. Ut si ostensum est nulli E inesse A, quoniam accidit et B alicui E inesse, quod est impossibile. Si sumptum sit E quidem nulli B, A autem omni B inesse, manifestum est enim quoniam nulli E inerit A. | Similarly with the other propositions requiring proof. The proof per impossibile will always and in all cases be from the consequents and antecedents of the terms in question. Whatever the problem the same inquiry is necessary whether one wishes to use an ostensive syllogism or a reduction to impossibility. For both the demonstrations start from the same terms, e.g. suppose it has been proved that A belongs to no E, because it turns out that otherwise B belongs to some of the Es and this is impossible-if now it is assumed that B belongs to no E and to all A, it is clear that A will belong to no E. |
| πάλιν εἰ δεικτικῶς συλλελόγισται τὸ Α τῶι Ε μηδενὶ ὑπάρχειν, ὑποθεμένοις ὑπάρχειν τινὶ διὰ τοῦ ἀδυνάτου δειχθήσεται οὐδενὶ ὑπάρχον. ὁμοίως δὲ κἀπὶ τῶν ἄλλων· ἐν ἅπασι γὰρ ἀνάγκη κοινόν τινα λαβεῖν ὅρον ἄλλον τῶν ὑποκειμένων, πρὸς ὃν ἔσται τοῦ ψεύδους ὁ συλλογισμός, ὥστ᾽ ἀντιστραφείσης ταύτης τῆς προτάσεως, τῆς δ᾽ ἑτέρας ὁμοίως ἐχούσης, δεικτικὸς ἔσται ὁ συλλογισμὸς διὰ τῶν αὐτῶν ὅρων. διαφέρει γὰρ ὁ δεικτικὸς τοῦ εἰς τὸ ἀδύνατον, ὅτι ἐν μὲν τῶι δεικτικῶι κατ᾽ ἀλήθειαν ἀμφότεραι τίθενται αἱ προτάσεις, ἐν δὲ τῶι εἰς τὸ ἀδύνατον ψευδῶς ἡ μία. | Rursum si ostensive syllogizatum sit A inesse nulli E, suppositis inesse per impossibile monstrabitur nulli inesse, similiter autem et in aliis. (0673B) In omnibus enim necesse est iis qui per impossibile communem aliquem sumere terminum alium A subiectis, ad quem erit mendacii syllogismus, quare conversa ea propositione, altera autem similiter se habente, ostensivus erit syllogismus per eosdem terminos. Differt autem ostensivus ab eo qui ad impossibile, quoniam in ostensivo secundum veritatem ambae propositiones ponuntur, in eo autem qui ad impossibile, falsa una. | Again if it has been proved by an ostensive syllogism that A belongs to no E, assume that A belongs to some E and it will be proved per impossibile to belong to no E. Similarly with the rest. In all cases it is necessary to find some common term other than the subjects of inquiry, to which the syllogism establishing the false conclusion may relate, so that if this premiss is converted, and the other remains as it is, the syllogism will be ostensive by means of the same terms. For the ostensive syllogism differs from the reductio ad impossibile in this: in the ostensive syllogism both premisses are laid down in accordance with the truth, in the reductio ad impossibile one of the premisses is assumed falsely. |
| Ταῦτα μὲν οὖν ἔσται μᾶλλον φανερὰ διὰ τῶν ἑπομένων, ὅταν περὶ τοῦ ἀδυνάτου λέγωμεν· νῦν δὲ τοσοῦτον ἡμῖν ἔστω δῆλον, ὅτι εἰς ταὐτὰ βλεπτέον δεικτικῶς τε βου λομένωι συλλογίζεσθαι καὶ εἰς τὸ ἀδύνατον ἄγειν. ἐν δὲ τοῖς ἄλλοις συλλογισμοῖς τοῖς ἐξ ὑποθέσεως, οἷον ὅσοι κατὰ μετάληψιν ἢ κατὰ ποιότητα, ἐν τοῖς ὑποκειμένοις, οὐκ ἐν τοῖς ἐξ ἀρχῆς ἀλλ᾽ ἐν τοῖς μεταλαμβανομένοις, ἔσται ἡ σκέψις, ὁ δὲ τρόπος ὁ αὐτὸς τῆς ἐπιβλέψεως. ἐπισκέ ψασθαι δὲ δεῖ καὶ διελεῖν ποσαχῶς οἱ ἐξ ὑποθέσεως. | Haec vero erunt magis manifesta per sequentia quando de impossibili dicemus; nunc autem tantum nobis sit manifestus, quoniam ad haec perspiciendum, et ostensive volentibus syllogizare, et ad impossibile deducere. (0673C)In aliis autem syllogismis quicunque sunt ex hypothesi, ut quicunque secundum transsumptionem, aut secundum qualitatem in subiectis, non in prioribus, sed in transsumptis erit consideratio, modus autem inspectionis idem: considerare autem oportet, et dividere quot modis sunt ex hypothesi, | These points will be made clearer by the sequel, when we discuss the reduction to impossibility: at present this much must be clear, that we must look to terms of the kinds mentioned whether we wish to use an ostensive syllogism or a reduction to impossibility. In the other hypothetical syllogisms, I mean those which proceed by substitution, or by positing a certain quality, the inquiry will be directed to the terms of the problem to be proved-not the terms of the original problem, but the new terms introduced; and the method of the inquiry will be the same as before. But we must consider and determine in how many ways hypothetical syllogisms are possible. |
| Δείκνυται μὲν οὖν ἕκαστον τῶν προβλημάτων οὕτως, ἔστι δὲ καὶ ἄλλον τρόπον ἔνια συλλογίσασθαι τούτων, οἷον τὰ καθόλου διὰ τῆς κατὰ μέρος ἐπιβλέψεως ἐξ ὑποθέσεως. εἰ γὰρ τὸ Γ καὶ τὸ Η ταὐτὰ εἴη, μόνοις δὲ ληφθείη τοῖς Η τὸ Ε ὑπάρχειν, παντὶ ἂν τῶι Ε τὸ Α ὑπάρχοι· καὶ πάλιν εἰ τὸ Δ καὶ Η ταὐτά, μόνων δὲ τῶν Η τὸ Ε κατηγοροῖτο, ὅτι οὐδενὶ τῶι Ε τὸ Α ὑπάρξει. φανερὸν οὖν ὅτι καὶ οὕτως ἐπιβλεπτέον. τὸν αὐτὸν δὲ τρόπον καὶ ἐπὶ τῶν ἀναγκαίων καὶ τῶν ἐνδεχομένων· ἡ γὰρ αὐτὴ σκέψις, καὶ διὰ τῶν αὐτῶν ὅρων ἔσται τῆι τάξει τοῦ τ᾽ ἐνδέχεσθαι καὶ τοῦ ὑπάρχειν ὁ συλλογισμός. ληπτέον δ᾽ ἐπὶ τῶν ἐνδεχομένων καὶ τὰ μὴ ὑπάρχοντα δυνατὰ δ᾽ ὑπάρχειν· δέ- δεικται γὰρ ὅτι καὶ διὰ τούτων γίνεται ὁ τοῦ ἐνδέχεσθαι συλλογισμός. ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων κατη γοριῶν. | monstratur ergo unumquodque propositorum sic. Est autem et alio modo quaedam syllogizare horum, ut universalia per particularem inspectionem ex hypothesi. Si enim C et G eadem sint, solum G autem sumatur E inesse, omni E inerit A, et rursum si G et D eadem, solum autem de G praedicetur E, quoniam nulli E inerit A, manifestum ergo quoniam sic inspiciendum. Eodem autem modo et in necessariis, et in contingentibus, nam eadem consideratio, et per eosdem terminos erit, eodemque ordine et contingentis, et inesse syllogismus. Sumendum autem et in contingentibus et quae non insunt, possibilia autem inesse. (0673D) Ostensum est enim quoniam et per haec fit contingentis syllogismus, similiter autem se habebit et in aliis praedicationibus. | Each of the problems then can be proved in the manner described; but it is possible to establish some of them syllogistically in another way, e.g. universal problems by the inquiry which leads up to a particular conclusion, with the addition of an hypothesis. For if the Cs and the Gs should be identical, but E should be assumed to belong to the Gs only, then A would belong to every E: and again if the Ds and the Gs should be identical, but E should be predicated of the Gs only, it follows that A will belong to none of the Es. Clearly then we must consider the matter in this way also. The method is the same whether the relation is necessary or possible. For the inquiry will be the same, and the syllogism will proceed through terms arranged in the same order whether a possible or a pure proposition is proved. We must find in the case of possible relations, as well as terms that belong, terms which can belong though they actually do not: for we have proved that the syllogism which establishes a possible relation proceeds through these terms as well. Similarly also with the other modes of predication. |
| (PL 64 0673D) CAPUT XXXI. Quod omnium scientiarum syllogismi superioribus praeceptis efficiantur. | ||
| Φανερὸν οὖν ἐκ τῶν εἰρημένων οὐ μόνον ὅτι ἐγχωρεῖ διὰ ταύτης τῆς ὁδοῦ γίνεσθαι πάντας τοὺς συλλογισμούς, ἀλλὰ καὶ ὅτι δι᾽ ἄλλης ἀδύνατον. ἅπας μὲν γὰρ συλλογισμὸς δέδεικται διά τινος τῶν προειρημένων σχημάτων γινόμενος, ταῦτα δ᾽ οὐκ ἐγχωρεῖ δι᾽ ἄλλων συσταθῆναι πλὴν διὰ τῶν ἑπομένων καὶ οἷς ἕπεται ἕκαστον· ἐκ τούτων γὰρ ↵ αἱ προτάσεις καὶ ἡ τοῦ μέσου λῆψις, ὥστ᾽ οὐδὲ συλλογισμὸν ἐγχωρεῖ γίνεσθαι δι᾽ ἄλλων. | Manifestum ergo ex praedictis quoniam non solum possibile est per hanc viam fieri omnes syllogismos, sed etiam quoniam per aliam impossibile. Omnis enim syllogismus ostensus est quoniam per aliquam praedictarum figurarum fit, has autem non contingit per alia constitui quam per sequentia et quae sequitur unumquodque, ex his enim propositiones, et medii sumptio, quare nec syllogismum possibile est fieri per alia. | It is clear then from what has been said not only that all syllogisms can be formed in this way, but also that they cannot be formed in any other. For every syllogism has been proved to be formed through one of the aforementioned figures, and these cannot be composed through other terms than the consequents and antecedents of the terms in question: for from these we obtain the premisses and find the middle term. Consequently a syllogism cannot be formed by means of other terms. |
Chapter 30
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| 30 | ||
| 46a3 Ἡ μὲν οὖν ὁδὸς κατὰ πάντων ἡ αὐτὴ καὶ περὶ φιλοσοφίαν καὶ περὶ τέχνην ὁποιανοῦν καὶ μάθημα· δεῖ γὰρ τὰ ὑπάρχοντα καὶ οἷς ὑπάρχει περὶ ἑκάτερον ἀθρεῖν, καὶ τούτων ὡς πλείστων εὐπορεῖν, καὶ ταῦτα διὰ τῶν τριῶν ὅρων σκοπεῖν, ἀνασκευάζοντα μὲν ὡδί, κατασκευάζοντα δὲ ὡδί, κατὰ μὲν ἀλήθειαν ἐκ τῶν κατ᾽ ἀλήθειαν διαγεγραμμένων ὑπάρχειν, εἰς δὲ τοὺς διαλεκτικοὺς συλλογισμοὺς ἐκ τῶν κατὰ δόξαν προτάσεων. | (0674A) Ergo methodus quidem de omnibus eadem est, et circa philosophiam, et circa autem quamlibet disciplinam. Oportet enim quae insunt, et quibus insunt circa unumquodque colligere, et his quamplurimis abundare, et hoc per tres terminos considerare, destruentem quidem sic, construentem vero sic, et secundum veritatem quidem, ex iis quae secundum veritatem scripta sunt inesse, ad dialecticos autem syllogismos, ex propositionibus quae sunt secundum opinionem. | The method is the same in all cases, in philosophy, in any art or study. We must look for the attributes and the subjects of both our terms, and we must supply ourselves with as many of these as possible, and consider them by means of the three terms, refuting statements in one way, confirming them in another, in the pursuit of truth starting from premisses in which the arrangement of the terms is in accordance with truth, while if we look for dialectical syllogisms we must start from probable premisses. |
| αἱ δ᾽ ἀρχαὶ τῶν συλλογισμῶν καθόλου μὲν εἴρηνται, ὃν τρόπον τ᾽ ἔχουσι καὶ ὃν τρόπον δεῖ θηρεύειν αὐτάς, ὅπως μὴ βλέπωμεν εἰς ἅπαντα τὰ λεγόμενα, μηδ᾽ εἰς ταὐτὰ κατασκευάζοντες καὶ ἀνασκευάζοντες, μηδὲ κατασκευάζοντές τε κατὰ παντὸς ἢ τινὸς καὶ ἀνασκευάζον τες ἀπὸ πάντων ἢ τινῶν, ἀλλ᾽ εἰς ἐλάττω καὶ ὡρισμένα, καθ᾽ ἕκαστον δὲ ἐκλέγειν τῶν ὄντων, οἷον περὶ ἀγαθοῦ ἢ ἐπιστήμης. ἴδιαι δὲ καθ᾽ ἑκάστην αἱ πλεῖσται. διὸ τὰς μὲν ἀρχὰς τὰς περὶ ἕκαστον ἐμπειρίας ἐστὶ παραδοῦναι, λέγω δ᾽ οἷον τὴν ἀστρολογικὴν μὲν ἐμπειρίαν τῆς ἀστρολογι κῆς ἐπιστήμης (ληφθέντων γὰρ ἱκανῶς τῶν φαινομένων οὕτως εὑρέθησαν αἱ ἀστρολογικαὶ ἀποδείξεισ), | Principia autem syllogismorum universaliter quidem dicta sunt, et quomodo se habeant, et quomodo oportet inquirere ea, quatenus non aspiciamus ad omnia quae dicuntur, neque eadem construentes et destruentes, neque construentes de omni aut de aliquo, destruentes ab omnibus aut ab aliquibus, sed ad pauciora et determinata. (0674B) Secundum singulum autem eorum quae sunt eligere, ut de bono aut disciplina. Propria autem in unaquaque sunt plurima, quare principia quidem quae sunt circa unumquodque, experimento est crescere, dico autem ut astrologicam quidem experientiam astrologicae disciplinae, sumptis enim sufficienter apparentibus, sic inventae sunt astrologicae demonstrationes. | The principles of syllogisms have been stated in general terms, both how they are characterized and how we must hunt for them, so as not to look to everything that is said about the terms of the problem or to the same points whether we are confirming or refuting, or again whether we are confirming of all or of some, and whether we are refuting of all or some. we must look to fewer points and they must be definite. We have also stated how we must select with reference to everything that is, e.g. about good or knowledge. But in each science the principles which are peculiar are the most numerous. Consequently it is the business of experience to give the principles which belong to each subject. I mean for example that astronomical experience supplies the principles of astronomical science: for once the phenomena were adequately apprehended, the demonstrations of astronomy were discovered. |
| ὁμοίως δὲ καὶ περὶ ἄλλην ὁποιανοῦν ἔχει τέχνην τε καὶ ἐπιστήμην· ὥστ᾽ ἐὰν ληφθῆι τὰ ὑπάρχοντα περὶ ἕκαστον, ἡμέτερον ἤδη τὰς ἀποδείξεις ἑτοίμως ἐμφανίζειν. εἰ γὰρ μηδὲν κατὰ τὴν ἱστορίαν παρα λειφθείη τῶν ἀληθῶς ὑπαρχόντων τοῖς πράγμασιν, ἕξομεν περὶ ἅπαντος οὗ μὲν ἔστιν ἀπόδειξις, ταύτην εὑρεῖν καὶ ἀποδεικνύναι, οὗ δὲ μὴ πέφυκεν ἀπόδειξις, τοῦτο ποιεῖν φανερόν. | Similiter autem et circa quamlibet aliam se habet et artem et disciplinam. Quare si sumantur quae insunt circa unumquodque, nostrum erit iam demonstrationes prompte declarare: si enim nihil secundum historiam omittatur eorum quae subtiliter et vere insunt rebus, habebimus de omni (cuius quidem non est demonstratio) hanc invenire et demonstrare, cuius autem non nata est demonstratio, hoc facere manifestum. | Similarly with any other art or science. Consequently, if the attributes of the thing are apprehended, our business will then be to exhibit readily the demonstrations. For if none of the true attributes of things had been omitted in the historical survey, we should be able to discover the proof and demonstrate everything which admitted of proof, and to make that clear, whose nature does not admit of proof. |
| Καθόλου μὲν οὖν, ὃν δεῖ τρόπον τὰς προτάσεις ἐκλέγειν, εἴρηται σχεδόν· δι᾽ ἀκριβείας δὲ διεληλύθαμεν ἐν τῆι πραγματείαι τῆι περὶ τὴν διαλεκτικήν. | (0674C) Universaliter ergo quo oportet modo propositiones eligere pene dictum est, per diligentiam autem pertransivimus in eo negotio quod circa dialecticam est. | In general then we have explained fairly well how we must select premisses: we have discussed the matter accurately in the treatise concerning dialectic. |
Chapter 31
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| (PL 64 0674C) CAPUT XXXII. De divisione et eius syllogismo. | 31 | |
| 46a31 Ὅτι δ᾽ ἡ διὰ τῶν γενῶν διαίρεσις μικρόν τι μόριόν ἐστι τῆς εἰρημένης μεθόδου, ῥάιδιον ἰδεῖν· ἔστι γὰρ ἡ διαίρεσις οἷον ἀσθενὴς συλλογισμός· ὁ μὲν γὰρ δεῖ δεῖξαι αἰτεῖται, συλλογίζεται δ᾽ ἀεί τι τῶν ἄνωθεν.
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Quoniam autem divisio per genera parva quaedam particula est dictae methodi facile videre: est enim divisio velut infirmus syllogismus, nam quod oporteat quidem ostendere petitur, syllogizatur vero semper aliquid superiorum. | It is easy to see that division into classes is a small part of the method we have described: for division is, so to speak, a weak syllogism; for what it ought to prove, it begs, and it always establishes something more general than the attribute in question. |
| πρῶτον δ᾽ αὐτὸ τοῦτο ἐλελήθει τοὺς χρωμένους αὐτῆι πάντας, καὶ πείθειν ἐπεχείρουν ὡς ὄντος δυνατοῦ περὶ οὐσίας ἀπόδειξιν γενέσθαι καὶ τοῦ τί ἐστιν. ὥστ᾽ οὔτε ὅ τι ἐνδέχεται συλλογίσασθαι διαιρουμένοις ξυνίεσαν, οὔτε ὅτι οὕτως ἐνεδέχετο ὥσπερ εἰρήκαμεν. ἐν μὲν οὖν ταῖς ἀποδείξεσιν, ὅταν δέηι τι συλλογίσασθαι ὑπάρχειν, δεῖ τὸ μέσον, δι᾽ οὗ γίνεται ὁ συλλο ↵ γισμός, καὶ ἧττον ἀεὶ εἶναι καὶ μὴ καθόλου τοῦ πρώτου τῶν ἄκρων· ἡ δὲ διαίρεσις τοὐναντίον βούλεται· τὸ γὰρ καθόλου λαμβάνει μέσον.
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(0674D) Primum autem idem hoc latuit omnes utentes ea, et suadere conati sunt quoniam esset possibile de substantia demonstrationem fieri, et de eo quod est quid; quare neque quoniam contingebat syllogizare eos qui dividunt, intellexerunt, neque quoniam contingebat sic quemadmodum diximus. Ergo in demonstrationibus quidem cum oporteat quid syllogizare, oportet medium per quod fit syllogismus minus semper esse, et non universaliter de prima extremitate. Divisio autem contrarium vult, nam universalius sumit medium. | First, this very point had escaped all those who used the method of division; and they attempted to persuade men that it was possible to make a demonstration of substance and essence. Consequently they did not understand what it is possible to prove syllogistically by division, nor did they understand that it was possible to prove syllogistically in the manner we have described. In demonstrations, when there is a need to prove a positive statement, the middle term through which the syllogism is formed must always be inferior to and not comprehend the first of the extremes. But division has a contrary intention: for it takes the universal as middle. |
| ἔστω γὰρ ζῶιον ἐφ᾽ οὗ Α, τὸ δὲ θνητὸν ἐφ᾽ οὗ Β, καὶ ἀθάνατον ἐφ᾽ οὗ Γ, ὁ δ᾽ ἄνθρω πος, οὗ τὸν λόγον δεῖ λαβεῖν, ἐφ᾽ οὗ τὸ Δ. ἅπαν δὴ ζῶιον λαμβάνει ἢ θνητὸν ἢ ἀθάνατον· τοῦτο δ᾽ ἐστίν, ὁ ἂν ἦι Α, ἅπαν εἶναι ἢ Β ἢ Γ. | Sit enim animal quidem in quo A, mortale autem in quo B, et immortale in quo C, homo vero cuius terminum oportet sumere in quo D, omne ergo animal accipit aut mortale, aut immortale: hoc autem est quidquid erat, omne esse aut B, aut C. | Let animal be the term signified by A, mortal by B, and immortal by C, and let man, whose definition is to be got, be signified by D. The man who divides assumes that every animal is either mortal or immortal: i.e. whatever is A is all either B or C. |
| πάλιν τὸν ἄνθρωπον ἀεὶ διαιρούμενος τίθεται ζῶιον εἶναι, ὥστε κατὰ τοῦ Δ τὸ Α λαμβάνει ὑπάρχειν. ὁ μὲν οὖν συλλογισμός ἐστιν ὅτι τὸ Δ ἢ Β ἢ Γ ἅπαν ἔσται, ὥστε τὸν ἄνθρωπον ἢ θνητὸν μὲν ἢ ἀθάνατον ἀναγκαῖον εἶναι, ζῶιον θνητὸν δὲ οὐκ ἀναγκαῖον, ἀλλ᾽ αἰτεῖται· τοῦτο δ᾽ ἦν ὁ ἔδει συλλογίσασθαι. | (0675A) Rursus hominem semper qui dividit, ponit animal esse, quare de D sumit A esse, ergo syllogismus quidem est, quoniam D, aut B, aut C omne erit, quare hominem aut mortalem, aut immortalem oportet sumere, nam mortale quidem, aut immortale esse necessarium est animal, mortale autem non necessarium est, sed petitur. Hoc autem erat quod oportebat syllogizare. | Again, always dividing, he lays it down that man is an animal, so he assumes A of D as belonging to it. Now the true conclusion is that every D is either B or C, consequently man must be either mortal or immortal, but it is not necessary that man should be a mortal animal-this is begged: and this is what ought to have been proved syllogistically. |
| καὶ πάλιν θέμενος τὸ μὲν Α ζῶιον θνητόν, ἐφ᾽ οὗ δὲ τὸ Β ὑπόπουν, ἐφ᾽ οὗ δὲ τὸ Γ ἄπουν, τὸν δ᾽ ἄνθρωπον τὸ Δ, ὡσαύτως λαμβάνει τὸ μὲν Α ἤτοι ἐν τῶι Β ἢ ἐν τῶι Γ εἶναι (ἅπαν γὰρ ζῶιον θνητὸν ἢ ὑπόπουν ἢ ἄπουν ἐστί), κατὰ δὲ τοῦ Δ τὸ Α (τὸν γὰρ ἄνθρωπον ζῶιον θνητὸν εἶναι ἔλαβεν)· ὥσθ᾽ ὑπόπουν μὲν ἢ ἄπουν εἶναι ζῶιον ἀνάγκη τὸν ἄνθρωπον, ὑπόπουν δ᾽ οὐκ ἀνάγκη, ἀλλὰ λαμβάνει· τοῦτο δ᾽ ἦν ὁ ἔδει πάλιν δεῖξαι. | Et rursus qui ponit A quidem animal mortale in quo autem B pedes habens, in quo autem C, non habens pedes, hominem vero D, similiter sumit A quidem, aut in B, aut in C esse. Omne enim animal mortale aut pedes habens, aut pedes non habens est, de D autem A, nam hominem animal mortale sumpsit esse, quare habens pedes, vel non habens pedes esse animal, necesse est hominem, pedes autem habens non necesse est, sed sumit, hoc autem erat quod oportebat rursum ostendere. | And again, taking A as mortal animal, B as footed, C as footless, and D as man, he assumes in the same way that A inheres either in B or in C (for every mortal animal is either footed or footless), and he assumes A of D (for he assumed man, as we saw, to be a mortal animal); consequently it is necessary that man should be either a footed or a footless animal; but it is not necessary that man should be footed: this he assumes: and it is just this again which he ought to have demonstrated. |
| καὶ τοῦτον δὴ τὸν τρόπον ἀεὶ διαιρουμένοις τὸ μὲν καθόλου συμβαίνει αὐτοῖς μέσον λαμβάνειν, καθ᾽ οὗ δ᾽ ἔδει δεῖξαι καὶ τὰς διαφορὰς ἄκρα. τέλος δέ, ὅτι τοῦτ᾽ ἔστιν ἄνθρωπος ἢ ὅ τι ποτ᾽ ἂν ἦι τὸ ζητούμενον, οὐδὲν λέγουσι σαφὲς ὥστ᾽ ἀναγκαῖον εἶναι· καὶ γὰρ τὴν ἄλλην ὁδὸν ποιοῦνται πᾶσαν, οὐδὲ τὰς ἐνδεχομένας εὐπορίας ὑπολαμβάνοντες ὑπάρχειν. Φανερὸν δ᾽ ὅτι οὔτ᾽ ἀνασκευάσαι ταύτηι τῆι μεθόδωι ἔστιν, οὔτε περὶ συμβεβηκότος ἢ ἰδίου συλλογίσασθαι, οὔτε περὶ γένους, οὔτ᾽ ἐν οἷς ἀγνοεῖται τὸ πότερον ὡδὶ ἢ ὡδὶ ἔχει, οἷον ἆρ᾽ ἡ διάμετρος ἀσύμμετρος ἢ σύμμετρος. ἐὰν γὰρ λάβηι ὅτι ἅπαν μῆκος ἢ σύμμετρον ἢ ἀσύμμετρον, ἡ δὲ διάμετρος μῆκος, συλλελόγισται ὅτι ἀσύμμετρος ἢ σύμμετρος ἡ διάμετρος. εἰ δὲ λήψεται ἀσύμμετρον, ὁ ἔδει συλλογίσασθαι λήψεται. οὐκ ἄρα ἔστι δεῖξαι· ἡ μὲν γὰρ ὁδὸς αὕτη, διὰ ταύτης δ᾽ οὐκ ἔστιν. τὸ ἀσύμμετρον ἢ σύμμετρον ἐφ᾽ οὗ Α, μῆκος Β, διάμετρος Γ. φανερὸν οὖν ὅτι οὔτε πρὸς πᾶσαν σκέψιν ἁρμόζει τῆς ζητήσεως ὁ τρόπος, οὔτ᾽ ἐν οἷς μάλιστα δοκεῖ πρέπειν, ἐν τούτοις ἐστὶ χρήσιμος. | (0675B) Et ad hunc modum semper dividentibus, universale quidem accidit eis medium sumere, de quo oporteat ostendere et differentias et extremitates. In fine autem quoniam hoc est homo, aut quidquid erat quod quaeritur, nihil dicunt manifestum, quare necessarium est esse, etenim aliam viam faciunt omnem, non quidem contingentes idoneitates, opinantes esse. Manifestum est autem quoniam neque destruere hac via est, neque de accidente aliquid, aut de proprio syllogizare, neque de genere, neque de quibus ignoretur utrum hoc modo aut illo se habet, ut putasne diameter est symeter, vel asymeter? si enim sumat quoniam omnis longitudo est symetros vel asymetros, diameter autem longitudo, syllogizatum est quoniam symeter vel asymeter est diameter. (0675C) Si autem sumetur incommensurabile, quod oportebat syllogizare sumetur, non ergo est ostendere, nam via quidem haec, per hanc autem non est ostendere symetrum vel asymetrum, in quo A longitudo, B autem symeter aut asymeter, diameter C. Manifestum est igitur quoniam neque ad omnem considerationem congruit inquisitionis modus, neque in quibus maxime videtur convenire, in his est utilis. | Always dividing then in this way it turns out that these logicians assume as middle the universal term, and as extremes that which ought to have been the subject of demonstration and the differentiae. In conclusion, they do not make it clear, and show it to be necessary, that this is man or whatever the subject of inquiry may be: for they pursue the other method altogether, never even suspecting the presence of the rich supply of evidence which might be used. It is clear that it is neither possible to refute a statement by this method of division, nor to draw a conclusion about an accident or property of a thing, nor about its genus, nor in cases in which it is unknown whether it is thus or thus, e.g. whether the diagonal is incommensurate. For if he assumes that every length is either commensurate or incommensurate, and the diagonal is a length, he has proved that the diagonal is either incommensurate or commensurate. But if he should assume that it is incommensurate, he will have assumed what he ought to have proved. He cannot then prove it: for this is his method, but proof is not possible by this method. Let A stand for ‘incommensurate or commensurate’, B for ‘length’, C for ‘diagonal’. It is clear then that this method of investigation is not suitable for every inquiry, nor is it useful in those cases in which it is thought to be most suitable. |
| Ex quibus ergo demonstrationes fiunt, et quomodo, et ad quae perspiciendum secundum unumquodque propositum manifestum ex dictis. | From what has been said it is clear from what elements demonstrations are formed and in what manner, and to what points we must look in each problem. |
Chapter 32
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| (PL 64 0675C) CAPUT XXXIII. De resolutione syllogismorum in propositiones. | 32 | |
| 46b38 Ἐκ τίνων μὲν οὖν αἱ ἀποδείξεις γίνονται καὶ πῶς, καὶ εἰς ὁποῖα βλεπτέον καθ᾽ ἕκαστον πρόβλημα, φανερὸν ἐκ τῶν εἰρημένων· πῶς δ᾽ ἀνάξομεν τοὺς συλλογισμοὺς εἰς ↵ τὰ προειρημένα σχήματα, λεκτέον ἂν εἴη μετὰ ταῦτα· λοιπὸν γὰρ ἔτι τοῦτο τῆς σκέψεως. εἰ γὰρ τήν τε γένεσιν τῶν συλλογισμῶν θεωροῖμεν καὶ τοῦ εὑρίσκειν ἔχοιμεν δύναμιν, ἔτι δὲ τοὺς γεγενημένους ἀναλύοιμεν εἰς τὰ προειρημένα σχήματα, τέλος ἂν ἔχοι ἡ ἐξ ἀρχῆς πρόθεσις. συμβήσεται δ᾽ ἅμα καὶ τὰ πρότερον εἰρημένα ἐπιβεβαιοῦσθαι καὶ φανερώτερα εἶναι ὅτι οὕτως ἔχει, διὰ τῶν νῦν λεχθησομένων· δεῖ γὰρ πᾶν τὸ ἀληθὲς αὐτὸ ἑαυτῶι ὁμολογούμενον εἶναι πάντηι. | (0675D) Quomodo autem reducemus syllogismos in praedictas figuras, dicendum erit post haec, restat enim consideratio haec, si enim et generationem syllogismorum inspiciamus, et inveniendi habeamus potestatem, amplius autem factos reducamus praedictas figuras, finem habebit quod ex principio propositum est, accidet etiam simul quae praedicta sunt confirmari et manifestiora esse, quoniam sic se habent per ea quae nunc dicenda sunt. Oportet enim omne quod verum est, ipsum sibi ipsi manifestum esse omnino. | Our next business is to state how we can reduce syllogisms to the aforementioned figures: for this part of the inquiry still remains. If we should investigate the production of the syllogisms and had the power of discovering them, and further if we could resolve the syllogisms produced into the aforementioned figures, our original problem would be brought to a conclusion. It will happen at the same time that what has been already said will be confirmed and its truth made clearer by what we are about to say. For everything that is true must in every respect agree with itself. |
| Πρῶτον μὲν οὖν δεῖ πειρᾶσθαι τὰς δύο προτάσεις ἐκλαμβάνειν τοῦ συλλογισμοῦ (ῥᾶιον γὰρ εἰς τὰ μείζω διελεῖν ἢ τὰ ἐλάττω, μείζω δὲ τὰ συγκείμενα ἢ ἐξ ὧν), εἶτα σκοπεῖν ποτέρα ἐν ὅλωι καὶ ποτέρα ἐν μέρει, καί, εἰ μὴ ἄμφω εἰλημμέναι εἶεν, αὐτὸν τιθέναι τὴν ἑτέραν. | Primum ergo oportet tentare duas propositiones accipere syllogismi, facilius enim in maiora dividere quam in minora: maiora autem compositiora sunt quam ea ex quibus componuntur. Deinde considerare utra in toto, et utra in parte. Et si non ambae sumptae sint, eum qui ponit alteram. | First then we must attempt to select the two premisses of the syllogism (for it is easier to divide into large parts than into small, and the composite parts are larger than the elements out of which they are made); next we must inquire which are universal and which particular, and if both premisses have not been stated, we must ourselves assume the one which is missing. |
| ἐνίοτε γὰρ τὴν καθόλου προτείναντες τὴν ἐν ταύτηι οὐ λαμβάνουσιν, οὔτε γράφοντες οὔτ᾽ ἐρωτῶντες· ἢ ταύτας μὲν προτείνουσι, δι᾽ ὧν δ᾽ αὗται περαίνονται, παραλείπουσιν, ἄλλα δὲ μάτην ἐρωτῶσιν. σκεπτέον οὖν εἴ τι περίεργον εἴληπται καὶ εἴ τι τῶν ἀναγκαίων παραλέλειπται, καὶ τὸ μὲν θετέον τὸ δ᾽ ἀφαιρετέον, ἕως ἂν ἔλθηι εἰς τὰς δύο προτάσεις· ἄνευ γὰρ τούτων οὐκ ἔστιν ἀναγαγεῖν τοὺς οὕτως ἠρωτημένους λόγους.
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(0676A) Aliquoties enim universalem protendentes, eam quae in hac est non sumunt, neque scribentes, neque interrogantes, aut has quidem protendunt, per quas autem hae concluduntur, omittunt, alia vero vane interrogant. Considerandum autem si quid superfluum sumptum sit, et si quid necessariorum omissum, et hoc quidem ponendum, illud vero auferendum, donec veniat quis ad duas propositiones, sine his enim non est reducere sic interrogatas orationes. | For sometimes men put forward the universal premiss, but do not posit the premiss which is contained in it, either in writing or in discussion: or men put forward the premisses of the principal syllogism, but omit those through which they are inferred, and invite the concession of others to no purpose. We must inquire then whether anything unnecessary has been assumed, or anything necessary has been omitted, and we must posit the one and take away the other, until we have reached the two premisses: for unless we have these, we cannot reduce arguments put forward in the way described. |
| ἐνίων μὲν οὖν ῥάιδιον ἰδεῖν τὸ ἐνδεές, ἔνιοι δὲ λανθάνουσι καὶ δοκοῦσι συλλογίζεσθαι διὰ τὸ ἀναγκαῖόν τι συμβαίνειν ἐκ τῶν κειμένων, οἷον εἰ ληφθείη μὴ οὐσίας ἀναιρουμέ νης μὴ ἀναιρεῖσθαι οὐσίαν, ἐξ ὧν δ᾽ ἐστὶν ἀναιρουμένων, καὶ τὸ ἐκ τούτων φθείρεσθαι· | In aliquibus ergo facile est videre quod minus est, aliqui vero latent, et videntur quidem syllogizare, eo quod necessarium quid accidit ex iis quae posita sunt. Ut si sumatur, non substantia interempta substantiam non interimi, ex quibus autem est, interemptis, et quod ex eis est corrumpi. | In some arguments it is easy to see what is wanting, but some escape us, and appear to be syllogisms, because something necessary results from what has been laid down, e.g. if the assumptions were made that substance is not annihilated by the annihilation of what is not substance, and that if the elements out of which a thing is made are annihilated, then that which is made out of them is destroyed: |
| τούτων γὰρ τεθέντων ἀναγκαῖον μὲν τὸ οὐσίας μέρος εἶναι οὐσίαν, οὐ μὴν συλλελόγισται διὰ τῶν εἰλημμένων, ἀλλ᾽ ἐλλείπουσι προτάσεις.
πάλιν εἰ ἀνθρώπου ὄντος ἀνάγκη ζῶιον εἶναι καὶ ζώιου οὐσίαν, ἀνθρώπου ὄντος ἀνάγκη οὐσίαν εἶναι· ἀλλ᾽ οὔπω συλλελόγισται· οὐ γὰρ ἔχουσιν αἱ προτάσεις ὡς εἴπομεν. |
(0676B) His enim positis, necessarium est substantiae partem esse substantiam, non tamen syllogizatum est quod ea quae sumpta sunt, sed desunt, propositiones. Rursum si cum est homo, necesse est esse animal, et cum est animal, substantiam, et cum est homo, necesse est esse substantiam, sed nondum syllogizatum est, non enim se habent propositiones ut diximus. | these propositions being laid down, it is necessary that any part of substance is substance; this has not however been drawn by syllogism from the propositions assumed, but premisses are wanting. Again if it is necessary that animal should exist, if man does, and that substance should exist, if animal does, it is necessary that substance should exist if man does: but as yet the conclusion has not been drawn syllogistically: for the premisses are not in the shape we required. |
| Ἀπατώμεθα δ᾽ ἐν τοῖς τοιούτοις διὰ τὸ ἀναγκαῖόν τι συμβαίνειν ἐκ τῶν κειμένων, ὅτι καὶ ὁ συλλογισμὸς ἀναγκαῖόν ἐστιν. ἐπὶ πλέον δὲ τὸ ἀναγ καῖον ἢ ὁ συλλογισμός· ὁ μὲν γὰρ συλλογισμὸς πᾶς ἀναγκαῖον, τὸ δ᾽ ἀναγκαῖον οὐ πᾶν συλλογισμός. ὥστ᾽ οὐκ εἴ τι συμβαίνει τεθέντων τινῶν, πειρατέον ἀνάγειν εὐθύς, ἀλλὰ πρῶτον ληπτέον τὰς δύο προτάσεις, | Fallimur autem in talibus eo quod necessarium quiddam accidat ex his quae posita sunt, quam et syllogismus, necessarium est, in plus autem est necessarium quam syllogismus, nam omnis syllogismus, necessarium, necessarium autem non omne syllogismus. Quare non (si quid accidat positis quibusdam) statim tentandum est reducere, sed primum secundum est duas propositiones. | We are deceived in such cases because something necessary results from what is assumed, since the syllogism also is necessary. But that which is necessary is wider than the syllogism: for every syllogism is necessary, but not everything which is necessary is a syllogism. Consequently, though something results when certain propositions are assumed, we must not try to reduce it directly, but must first state the two premisses, |
| (PL 64 0676B) CAPUT XXXIV. De resolutione in terminos. | ||
| εἶθ᾽ οὕτω διαιρετέον εἰς τοὺς ὅρους, μέσον δὲ θετέον τῶν ὅρων τὸν ἐν ἀμφοτέραις ταῖς προτάσεσι λεγόμενον· ἀνάγκη γὰρ τὸ μέσον ἐν ἀμφοτέραις ὑπάρχειν ἐν ἅπασι τοῖς σχήμασιν. | (0676C) Deinde sic dividendum in terminos. Medium autem ponendum terminorum, qui utrisque propositionibus dicitur, necesse est enim medium in utrisque esse in omnibus figuris. | then divide them into their terms. We must take that term as middle which is stated in both the remisses: for it is necessary that the middle should be found in both premisses in all the figures. |
| Ἐὰν μὲν οὖν ↵ κατηγορῆι καὶ κατηγορῆται τὸ μέσον, ἢ αὐτὸ μὲν κατηγορῆι, ἄλλο δ᾽ ἐκείνου ἀπαρνῆται, τὸ πρῶτον ἔσται σχῆμα· ἐὰν δὲ καὶ κατηγορῆι καὶ ἀπαρνῆται ἀπό τινος, τὸ μέσον· ἐὰν δ᾽ ἄλλα ἐκείνου κατηγορῆται, ἢ τὸ μὲν ἀπαρνῆται τὸ δὲ κατηγορῆται, τὸ ἔσχατον. οὕτω γὰρ εἶχεν ἐν ἑκάστωι σχήματι τὸ μέσον. ὁμοίως δὲ καὶ ἐὰν μὴ καθόλου ὦσιν αἱ προτάσεις· ὁ γὰρ αὐτὸς διορισμὸς τοῦ μέσου. φανερὸν οὖν ὡς ἐν ὧι λόγωι μὴ λέγεται ταὐτὸ πλεονάκις, ὅτι οὐ γίνεται συλλογισμός· οὐ γὰρ εἴληπται μέσον. ἐπεὶ δ᾽ ἔχομεν ποῖον ἐν ἑκάστωι σχήματι περαίνεται τῶν προβλημάτων, καὶ ἐν τίνι τὸ καθόλου καὶ ἐν ποίωι τὸ ἐν μέρει, φανερὸν ὡς οὐκ εἰς ἅπαντα τὰ σχήματα βλεπτέον, ἀλλ᾽ ἑκάστου προβλήματος εἰς τὸ οἰκεῖον. ὅσα δ᾽ ἐν πλείοσι περαίνεται, τῆι τοῦ μέσου θέσει γνωριοῦμεν τὸ σχῆμα. | Si ergo subiiciatur et praedicetur medium, aut ipsum quidem praedicetur, aliud vero illo abnegetur, prima erit figura. Si autem et praedicetur, et negetur ab aliquo, media erit figura: si vero alia de illo praedicentur, aut hoc quidem praedicetur, illud vero ab illo negetur, postrema, sic enim se habuit in postrema figura medium, similiter autem etsi non universales sint propositiones, nam est eadem determinatio medii. Manifestum igitur quoniam in qua oratione non dicitur idem frequenter, non fit syllogismus, non enim sumptum est medium. (0676D) Quoniam autem habemus quod propositorum in unaquaque figura clauditur, et in qua universale, et in qua particulare, manifestum est quoniam non ad omnes figuras perspiciendum, sed in unoquoque proposito ad propriam. Quaecunque vero in pluribus concluduntur, medii positione cognoscimus figuram. | If then the middle term is a predicate and a subject of predication, or if it is a predicate, and something else is denied of it, we shall have the first figure: if it both is a predicate and is denied of something, the middle figure: if other things are predicated of it, or one is denied, the other predicated, the last figure. For it was thus that we found the middle term placed in each figure. It is placed similarly too if the premisses are not universal: for the middle term is determined in the same way. Clearly then, if the same term is not stated more than once in the course of an argument, a syllogism cannot be made: for a middle term has not been taken. Since we know what sort of thesis is established in each figure, and in which the universal, in what sort the particular is described, clearly we must not look for all the figures, but for that which is appropriate to the thesis in hand. If the thesis is established in more figures than one, we shall recognize the figure by the position of the middle term. |
Chapter 33
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| (PL 64 0676D) CAPUT XXXV. De necessario et thesi terminorum. | 33 | |
| 47b15 Πολλάκις μὲν οὖν ἀπατᾶσθαι συμβαίνει περὶ τοὺς συλλογισμοὺς διὰ τὸ ἀναγκαῖον, ὥσπερ εἴρηται πρότερον, ἐνίοτε δὲ παρὰ τὴν ὁμοιότητα τῆς τῶν ὅρων θέσεως· ὅπερ οὐ χρὴ λανθάνειν ἡμᾶς. οἷον εἰ τὸ Α κατὰ τοῦ Β λέγεται καὶ τὸ Β κατὰ τοῦ Γ· δόξειε γὰρ ἂν οὕτως ἐχόντων τῶν ὅρων εἶναι συλλογισμός, οὐ γίνεται δ᾽ οὔτ᾽ ἀναγκαῖον οὐδὲν οὔτε συλλογισμός. | Frequenter ergo falli accidit circa syllogismos propter necessarium, quemadmodum dictum est prius: aliquoties autem propter similitudinem positionis terminorum, quod non oportet latere nos. Ut si A de B dicitur, et B de C, videbitur enim sic se habentibus terminis esse syllogismus, non fit autem neque necessarium quidquam, neque syllogismus. | Men are frequently deceived about syllogisms because the inference is necessary, as has been said above; sometimes they are deceived by the similarity in the positing of the terms; and this ought not to escape our notice. E.g. if A is stated of B, and B of C: it would seem that a syllogism is possible since the terms stand thus: but nothing necessary results, nor does a syllogism. |
| ἔστω γὰρ ἐφ᾽ ὧι Α τὸ ἀεὶ εἶναι, ἐφ᾽ ὧι δὲ Β διανοητὸς Ἀριστομένης, τὸ δ᾽ ἐφ᾽ ὧι Γ Ἀριστομένης. ἀληθὲς δὴ τὸ Α τῶι Β ὑπάρχειν· ἀεὶ γάρ ἐστι διανοητὸς Ἀριστομένης. ἀλλὰ καὶ τὸ Β τῶι Γ· ὁ γὰρ Ἀριστομένης ἐστὶ διανοητὸς Ἀριστομένης. τὸ δ᾽ Α τῶι Γ οὐχ ὑπάρχει· φθαρτὸς γάρ ἐστιν ὁ Ἀριστομένης. οὐ γὰρ ἐγίνετο συλλογισμὸς οὕτως ἐχόντων τῶν ὅρων, ἀλλ᾽ ἔδει καθόλου τὴν Α Β ληφθῆναι πρότασιν. τοῦτο δὲ ψεῦδος, τὸ ἀξιοῦν πάντα τὸν διανοητὸν Ἀριστομένην ἀεὶ εἶναι, φθαρτοῦ ὄντος Ἀριστομένους. πάλιν ἔστω τὸ μὲν ἐφ᾽ ὧι Γ Μίκκαλος, τὸ δ᾽ ἐφ᾽ ὧι Β μουσικὸς Μίκκαλος, ἐφ᾽ ὧι δὲ τὸ Α τὸ φθείρεσθαι αὔριον. ἀληθὲς δὴ τὸ Β τοῦ Γ κατηγορεῖν· ὁ γὰρ Μίκκαλός ἐστι μουσικὸς Μίκκαλος. ἀλλὰ καὶ τὸ Α τοῦ Β· φθείροιτο γὰρ ἂν αὔριον μουσικὸς Μίκκαλος. τὸ δέ γε Α τοῦ Γ ψεῦδος. τοῦτο δὴ ταὐτόν ἐστι τῶι πρότερον· οὐ γὰρ ἀληθὲς καθόλου, Μίκκαλος μουσικὸς ὅτι φθείρεται αὔριον· τούτου δὲ μὴ ληφθέντος οὐκ ἦν συλλογισμός. | (0677A) Sit enim in quo A semper esse, in quo autem B intelligibilis Aristomenes, in quo autem C Aristomenes, verum est autem A inesse B, semper enim est intelligibilis Aristomenes, sed et B de C, nam Aristomenes est intelligibilis Aristomenes, A autem non inest C, corruptibilis est enim Aristomenes; non igitur fiebat syllogismus sic se habentibus terminis, sed oportebat universaliter A B sumi propositionem: hoc vero falsum quod putabat omnem intelligibilem Aristomenem semper esse, cum Aristomenes sit corruptibilis. Rursum sit in quo quidem C Micalus, in quo autem B musicus Micalus, in quo autem A corrumpi cras. (0677B) Verum est ergo B de C praedicari, nam Micalus est musicus Micalus, sed et A de B, corrumpetur enim cras musicus Micalus, A autem de C falsum: hoc autem idem est priori, non enim verum est universaliter, Micalus musicus quoniam corrumpetur cras. Hoc autem non sumpto non erat syllogismus. | Let A represent the term ‘being eternal’, B ‘Aristomenes as an object of thought’, C ‘Aristomenes’. It is true then that A belongs to B. For Aristomenes as an object of thought is eternal. But B also belongs to C: for Aristomenes is Aristomenes as an object of thought. But A does not belong to C: for Aristomenes is perishable. For no syllogism was made although the terms stood thus: that required that the premiss AB should be stated universally. But this is false, that every Aristomenes who is an object of thought is eternal, since Aristomenes is perishable. Again let C stand for ‘Miccalus’, B for ‘musical Miccalus’, A for ‘perishing to-morrow’. It is true to predicate B of C: for Miccalus is musical Miccalus. Also A can be predicated of B: for musical Miccalus might perish to-morrow. But to state A of C is false at any rate. This argument then is identical with the former; for it is not true universally that musical Miccalus perishes to-morrow: but unless this is assumed, no syllogism (as we have shown) is possible. |
| Αὕτη μὲν οὖν ἡ ἀπάτη γίνεται ἐν τῶι παρὰ μικρόν· ὡς γὰρ οὐδὲν διαφέρον εἰπεῖν τόδε τῶιδε ὑπάρχειν ἢ τόδε τῶιδε παντὶ ὑπάρχειν, συγχωροῦμεν. | Haec ergo fallacia fit in eo quod pene, ut enim nihil differens dicere hoc huic inesse, aut hoc huic omni inesse, concedimus. | This deception then arises through ignoring a small distinction. For if we accept the conclusion as though it made no difference whether we said ‘This belong to that’ or ‘This belongs to all of that’. |
Chapter 34
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| (PL 64 0677B) CAPUT XXXVI. De ecthesi terminorum. | 34 | |
| 47b40 πολλάκις δὲ διαψεύ ↵ δεσθαι συμπεσεῖται παρὰ τὸ μὴ καλῶς ἐκτίθεσθαι τοὺς κατὰ τὴν πρότασιν ὅρους, οἷον εἰ τὸ μὲν Α εἴη ὑγίεια, τὸ δ᾽ ἐφ᾽ ὧι Β νόσος, ἐφ᾽ ὧι δὲ Γ ἄνθρωπος. ἀληθὲς γὰρ εἰπεῖν ὅτι τὸ Α οὐδενὶ τῶι Β ἐνδέχεται ὑπάρχειν (οὐδεμιᾶι γὰρ νόσωι ὑγίεια ὑπάρχει), καὶ πάλιν ὅτι τὸ Β παντὶ τῶι Γ ὑπάρχει (πᾶς γὰρ ἄνθρωπος δεκτικὸς νόσου). δόξειεν ἂν οὖν συμβαίνειν μηδενὶ ἀνθρώπωι ἐνδέχεσθαι ὑγίειαν ὑπάρχειν. τούτου δ᾽ αἴτιον τὸ μὴ καλῶς ἐκκεῖσθαι τοὺς ὅρους κατὰ τὴν λέξιν, ἐπεὶ μεταληφθέντων τῶν κατὰ τὰς ἕξεις οὐκ ἔσται συλλογισμός, οἷον ἀντὶ μὲν τῆς ὑγιείας εἰ τεθείη τὸ ὑγιαῖνον, ἀντὶ δὲ τῆς νόσου τὸ νοσοῦν. οὐ γὰρ ἀληθὲς εἰπεῖν ὡς οὐκ ἐνδέχεται τῶι νοσοῦντι τὸ ὑγιαίνειν ὑπάρξαι. τούτου δὲ μὴ ληφθέντος οὐ γίνεται συλλογισμός, εἰ μὴ τοῦ ἐνδέχεσθαι· τοῦτο δ᾽ οὐκ ἀδύνατον· ἐνδέχεται γὰρ μηδενὶ ἀνθρώπωι ὑπάρχειν ὑγίειαν. πάλιν ἐπὶ τοῦ μέσου σχήματος ὁμοίως ἔσται τὸ ψεῦδος· τὴν γὰρ ὑγίειαν νόσωι μὲν οὐδεμιᾶι ἀνθρώπωι δὲ παντὶ ἐνδέχεται ὑπάρχειν, ὥστ᾽ οὐδενὶ ἀνθρώπωι νόσον. | (0677C) Frequenter autem mentiri evenit, eo quod non bene exponuntur secundum propositionem termini, ut si A quidem sit sanitas, B autem aegritudo, C vero homo, verum est enim dicere quoniam A nulli B contingit inesse, nulli enim aegritudini sapitas inest; et rursum quoniam B inest omni C, omnis enim homo susceptibilis est aegritudinis, videbitur ergo accidere nulli homini contingere sanitatem inesse. Huius autem causa est quod non bene exponuntur termini secundum locutionem, quoniam transsumptis quae iis sunt secundum habitudines, non erit syllogismus. Ut si pro sanitate quidem ponatur sanum, pro aegritudine autem aegrum, non enim verum est dicere quoniam non contingit aegrotanti inesse sanum esse, hoc autem non sumpto, non fit syllogismus, nisi contingentis. Hoc autem non impossibile, contingit enim nulli homini inesse sanitatem. Rursum in media figura similiter erit falsum. Nam sanitatem aegritudini quidem nulli, homini vero omni contingit inesse, quare nulli homini aegritudo. | Men will frequently fall into fallacies through not setting out the terms of the premiss well, e.g. suppose A to be health, B disease, C man. It is true to say that A cannot belong to any B (for health belongs to no disease) and again that B belongs to every C (for every man is capable of disease). It would seem to follow that health cannot belong to any man. The reason for this is that the terms are not set out well in the statement, since if the things which are in the conditions are substituted, no syllogism can be made, e.g. if ‘healthy’ is substituted for ‘health’ and ‘diseased’ for ‘disease’. For it is not true to say that being healthy cannot belong to one who is diseased. But unless this is assumed no conclusion results, save in respect of possibility: but such a conclusion is not impossible: for it is possible that health should belong to no man. Again the fallacy may occur in a similar way in the middle figure: ‘it is not possible that health should belong to any disease, but it is possible that health should belong to every man, consequently it is not possible that disease should belong to any man’. |
| ἐν δὲ τῶι τρίτωι σχήματι κατὰ τὸ ἐνδέχεσθαι συμβαίνει τὸ ψεῦδος, καὶ γὰρ ὑγίειαν καὶ νόσον καὶ ἐπιστή μην καὶ ἄγνοιαν καὶ ὅλως τὰ ἐναντία τῶι αὐτῶι ἐνδέχεται ὑπάρχειν, ἀλλήλοις δ᾽ ἀδύνατον. τοῦτο δ᾽ ἀνομολογούμενον τοῖς προειρημένοις· ὅτε γὰρ τῶι αὐτῶι πλείω ἐνεδέχετο ὑπάρχειν, ἐνεδέχετο καὶ ἀλλήλοις. | (0677D) In tertia autem figura secundum contingere accidit falsum, etenim sanitatem, et aegritudinem, et disciplinam, et ignorantiam, et omnino contraria omni eidem contingit inesse, sibi vero invicem impossibile, hoc autem confessum in praedictis. Cum enim eidem plura contingere inesse, contingebant et sibi invicem. | In the third figure the fallacy results in reference to possibility. For health and diseae and knowledge and ignorance, and in general contraries, may possibly belong to the same thing, but cannot belong to one another. This is not in agreement with what was said before: for we stated that when several things could belong to the same thing, they could belong to one another. |
| Φανερὸν οὖν ὅτι ἐν ἅπασι τούτοις ἡ ἀπάτη γίνεται παρὰ τὴν τῶν ὅρων ἔκθεσιν· μεταληφθέντων γὰρ τῶν κατὰ τὰς ἕξεις οὐδὲν γίνεται ψεῦδος. δῆλον οὖν ὅτι κατὰ τὰς τοιαύτας προτάσεις ἀεὶ τὸ κατὰ τὴν ἕξιν ἀντὶ τῆς ἕξεως μεταληπτέον καὶ θετέον ὅρον. | Manifestum igitur quoniam in omnibus his fallacia fit propter terminorum expositionem, transsumptis enim his quae sunt secundum habitudines, nihil fit falsum. Palam ergo quoniam secundum huiusmodi propositiones semper quod est secundum habitum, pro habitu sumendum et ponendum terminum. | It is evident then that in all these cases the fallacy arises from the setting out of the terms: for if the things that are in the conditions are substituted, no fallacy arises. It is clear then that in such premisses what possesses the condition ought always to be substituted for the condition and taken as the term. |
Chapter 35
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| (PL 64 0677D) CAPUT XXXVII. De ecthesi terminorum compositorum et obliquorum syllogismi. | 35 | |
| 48a29 Οὐ δεῖ δὲ τοὺς ὅρους ἀεὶ ζητεῖν ὀνόματι ἐκτίθεσθαι· πολλάκις γὰρ ἔσονται λόγοι οἷς οὐ κεῖται ὄνομα· διὸ χαλεπὸν ἀνάγειν τοὺς τοιούτους συλλογισμούς. ἐνίοτε δὲ καὶ ἀπατᾶσθαι συμβήσεται διὰ τὴν τοιαύτην ζήτησιν, οἷον ὅτι τῶν ἀμέσων ἔστι συλλογισμός. ἔστω τὸ Α δύο ὀρθαί, τὸ ἐφ᾽ ὧι Β τρίγωνον, ἐφ᾽ ὧι δὲ Γ ἰσοσκελές. τῶι μὲν οὖν Γ ὑπάρχει τὸ Α διὰ τὸ Β, τῶι δὲ Β οὐκέτι δι᾽ ἄλλο (καθ᾽ αὑτὸ γὰρ τὸ τρίγωνον ἔχει δύο ὀρθάσ), ὥστ᾽ οὐκ ἔσται μέσον τοῦ Α Β, ἀποδεικτοῦ ὄντος. φανερὸν γὰρ ὅτι τὸ μέσον οὐχ οὕτως ἀεὶ ληπτέον ὡς τόδε τι, ἀλλ᾽ ἐνίοτε λόγον, ὅπερ συμβαίνει κἀπὶ τοῦ λεχθέντος. | (0678A) Non oportet autem terminos semper quaerere nomine exponi, saepe enim erunt orationes quibus non ponuntur nomina, quare et difficile erit reducere huiusmodi syllogismos, aliquot es autem et falli accidet propter huiusmodi inquisitionem, ut quoniam immediatorum erit syllogismus; sit enim A duo recti, B autem triangulus, C vero aequicrurus; ergo ei quod est C inest A propter B; ei vero quod est B, non iterum propter aliud, per se enim triangulus habet duos rectos, quare non erit medium eius quod est A B, cum sit demonstrativum. Manifestum enim quoniam medium non sic semper est sumendum ut hoc aliquid, sed aliquando orationem, quod accidit et in praedicto. | We must not always seek to set out the terms a single word: for we shall often have complexes of words to which a single name is not given. Hence it is difficult to reduce syllogisms with such terms. Sometimes too fallacies will result from such a search, e.g. the belief that syllogism can establish that which has no mean. Let A stand for two right angles, B for triangle, C for isosceles triangle. A then belongs to C because of B: but A belongs to B without the mediation of another term: for the triangle in virtue of its own nature contains two right angles, consequently there will be no middle term for the proposition AB, although it is demonstrable. For it is clear that the middle must not always be assumed to be an individual thing, but sometimes a complex of words, as happens in the case mentioned. |
Chapter 36
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| 36 | ||
| 48a40 Τὸ δὲ ὑπάρχειν τὸ πρῶτον τῶι μέσωι καὶ τοῦτο τῶι ἄκρωι οὐ δεῖ λαμβάνειν ὡς αἰεὶ κατηγορηθησομένων ἀλλή ↵ λων ἢ ὁμοίως τό τε πρῶτον τοῦ μέσου καὶ τοῦτο τοῦ ἐσχάτου. καὶ ἐπὶ τοῦ μὴ ὑπάρχειν δ᾽ ὡσαύτως. ἀλλ᾽ ὁσαχῶς τὸ εἶναι λέγεται καὶ τὸ ἀληθὲς εἰπεῖν αὐτὸ τοῦτο, τοσαυταχῶς οἴεσθαι χρὴ σημαίνειν καὶ τὸ ὑπάρχειν. οἷον ὅτι τῶν ἐναντίων ἔστι μία ἐπιστήμη. ἔστω γὰρ τὸ Α τὸ μίαν εἶναι ἐπιστήμην, τὰ ἐναντία ἀλλήλοις ἐφ᾽ οὗ Β. τὸ δὴ Α τῶι Β ὑπάρχει οὐχ ὥστε τὰ ἐναντία [τὸ] μίαν εἶναι [αὐτῶν] ἐπιστήμην, ἀλλ᾽ ὅτι ἀληθὲς εἰπεῖν κατ᾽ αὐτῶν μίαν εἶναι αὐτῶν ἐπιστήμην. | (0678B) Inesse autem primum medio, et hoc postremo non oportet sumere, ut praedicentur semper ad se invicem similiter, et primum de medio, et hoc de postremo, et in non inesse similiter, sed quoties dicitur esse et verum dicere, hoc toties arbitrari oportet significare et inesse. Ut quoniam contrariorum una est disciplina: sit enim A unam esse disciplinam, B autem contraria sibi invicem, A ergo inest B, non quoniam contraria unam esse eorum disciplinam, sed quoniam verum est dicere de ipsis unam esse eorum disciplinam. Accidit autem quandoque primum de medio dici, medium autem de tertio non dici, ut si sophia est disciplina, boni autem est sophia: conclusio, quoniam boni est disciplina, et non bonum quidem est disciplina, sophia autem est disciplina. | That the first term belongs to the middle, and the middle to the extreme, must not be understood in the sense that they can always be predicated of one another or that the first term will be predicated of the middle in the same way as the middle is predicated of the last term. The same holds if the premisses are negative. But we must suppose the verb ‘to belong’ to have as many meanings as the senses in which the verb ‘to be’ is used, and in which the assertion that a thing ‘is’ may be said to be true. Take for example the statement that there is a single science of contraries. Let A stand for ‘there being a single science’, and B for things which are contrary to one another. Then A belongs to B, not in the sense that contraries are the fact of there being a single science of them, but in the sense that it is true to say of the contraries that there is a single science of them. |
| Συμβαίνει δ᾽ ὁτὲ μὲν ἐπὶ τοῦ μέσου τὸ πρῶτον λέγεσθαι, τὸ δὲ μέσον ἐπὶ τοῦ τρίτου μὴ λέγεσθαι, οἷον εἰ ἡ σοφία ἐστὶν ἐπιστήμη, τοῦ δ᾽ ἀγαθοῦ ἐστὶν ἡ σοφία, συμπέρασμα ὅτι τοῦ ἀγαθοῦ ἔστιν ἐπιστήμη· τὸ μὲν δὴ ἀγαθὸν οὐκ ἔστιν ἐπιστήμη, ἡ δὲ σοφία ἐστὶν ἐπιστήμη. ὁτὲ δὲ τὸ μὲν μέσον ἐπὶ τοῦ τρίτου λέγεται, τὸ δὲ πρῶτον ἐπὶ τοῦ μέσου οὐ λέγεται, οἷον εἰ τοῦ ποιοῦ παντὸς ἔστιν ἐπιστήμη ἢ ἐναντίου, τὸ δ᾽ ἀγαθὸν καὶ ἐναντίον καὶ ποιόν, συμπέρασμα μὲν ὅτι τοῦ ἀγαθοῦ ἔστιν ἐπιστήμη, οὐκ ἔστι δὲ τὸ ἀγαθὸν ἐπιστήμη οὐδὲ τὸ ποιὸν οὐδὲ τὸ ἐναντίον, ἀλλὰ τὸ ἀγαθὸν ταῦτα. | Quandoque autem medium quidem de tertio dicitur, primum autem de medio non dicitur, ut si qualis omnis est disciplina, aut contrarii. (0678C) Bonum autem est, et contrarium, et quale: conclusio quidem, quoniam boni est disciplina. Non est autem bonum disciplina, neque quale, neque contrarium, sed omnium disciplina. Non est autem bonum disciplina, neque conclusio secundum rectum, neque quale, neque contrarium, sed bonum haec. | It happens sometimes that the first term is stated of the middle, but the middle is not stated of the third term, e.g. if wisdom is knowledge, and wisdom is of the good, the conclusion is that there is knowledge of the good. The good then is not knowledge, though wisdom is knowledge. Sometimes the middle term is stated of the third, but the first is not stated of the middle, e.g. if there is a science of everything that has a quality, or is a contrary, and the good both is a contrary and has a quality, the conclusion is that there is a science of the good, but the good is not science, nor is that which has a quality or is a contrary, though the good is both of these. |
| ἔστι δὲ μήτε τὸ πρῶτον κατὰ τοῦ μέσου μήτε τοῦτο κατὰ τοῦ τρίτου, τοῦ πρώτου κατὰ τοῦ τρίτου ὁτὲ μὲν λεγομένου ὁτὲ δὲ μὴ λεγομένου. οἷον εἰ οὗ ἐπιστήμη ἔστιν, ἔστι τούτου γένος, τοῦ δ᾽ ἀγαθοῦ ἔστιν ἐπιστήμη, συμπέρασμα ὅτι τοῦ ἀγαθοῦ ἔστι γένος· κατηγορεῖται δ᾽ οὐδὲν κατ᾽ οὐδενός. εἰ δ᾽ οὗ ἔστιν ἐπιστήμη, γένος ἐστὶ τοῦτο, τοῦ δ᾽ ἀγαθοῦ ἔστιν ἐπιστήμη, συμπέρασμα ὅτι τἀγαθόν ἐστι γένος· κατὰ μὲν δὴ τοῦ ἄκρου κατηγορεῖται τὸ πρῶτον, κατ᾽ ἀλλήλων δ᾽ οὐ λέγεται. | Est autem quandoque neque primum de medio, neque hoc de tertio, primo de tertio quandoque quidem dicto, quandoque autem non dicto. Ut si cuius est disciplina, huius est genus, boni autem est disciplina: conclusio, quoniam boni est genus. Praedicatur autem nullum de nullo, si autem cuius est disciplina, genus est hoc, boni autem est disciplina: conclusio, quoniam bonum est genus: ergo de extremo quidem praedicatur primum, de se autem invicem non dicuntur. | Sometimes neither the first term is stated of the middle, nor the middle of the third, while the first is sometimes stated of the third, and sometimes not: e.g. if there is a genus of that of which there is a science, and if there is a science of the good, we conclude that there is a genus of the good. But nothing is predicated of anything. And if that of which there is a science is a genus, and if there is a science of the good, we conclude that the good is a genus. The first term then is predicated of the extreme, but in the premisses one thing is not stated of another. |
| Τὸν αὐτὸν δὴ τρόπον καὶ ἐπὶ τοῦ μὴ ὑπάρχειν ληπτέον. οὐ γὰρ ἀεὶ σημαίνει τὸ μὴ ὑπάρχειν τόδε τῶιδε μὴ εἶναι τόδε τόδε, ἀλλ᾽ ἐνίοτε τὸ μὴ εἶναι τόδε τοῦδε ἢ τόδε τῶιδε, οἷον ὅτι οὐκ ἔστι κινήσεως κίνησις ἢ γενέσεως γένεσις, ἡδονῆς δ᾽ ἔστιν· οὐκ ἄρα ἡ ἡδονὴ γένεσις. ἢ πάλιν ὅτι γέλωτος μὲν ἔστι σημεῖον, σημείου δ᾽ οὐκ ἔστι σημεῖον, ὥστ᾽ οὐ σημεῖον ὁ γέλως. ὁμοίως δὲ κἀν τοῖς ἄλλοις ἐν ὅσοις ἀναιρεῖται τὸ πρόβλημα τῶι λέγεσθαί πως πρὸς αὐτὸ τὸ γένος. | (0678D) Eodem autem modo et non inesse sumendum, non enim semper significat non inesse hoc huic, non esse hoc, hoc; sed aliquando non esse hoc huius, aut hoc huic: ut quoniam non est motionis motus, aut generationis generatio, voluptatis autem est, non ergo voluptas generatio. Aut rursus quoniam risus est signum, signi autem non est signum, quare non est signum risus; similiter autem et in aliis, in quibus interimitur propositum, eo quod dicitur aliquo modo ad id genus. | The same holds good where the relation is negative. For ‘that does not belong to this’ does not always mean that ‘this is not that’, but sometimes that ‘this is not of that’ or ‘for that’, e.g. ‘there is not a motion of a motion or a becoming of a becoming, but there is a becoming of pleasure: so pleasure is not a becoming.’ Or again it may be said that there is a sign of laughter, but there is not a sign of a sign, consequently laughter is not a sign. This holds in the other cases too, in which the thesis is refuted because the genus is asserted in a particular way, in relation to the terms of the thesis. |
| πάλιν ὅτι ὁ καιρὸς οὐκ ἔστι χρόνος δέων· θεῶι γὰρ καιρὸς μὲν ἔστι, χρόνος δ᾽ οὐκ ἔστι δέων διὰ τὸ μηδὲν εἶναι θεῶι ὠφέλιμον. ὅρους μὲν γὰρ θετέον καιρὸν καὶ χρόνον δέοντα καὶ θεόν, τὴν δὲ πρότασιν ληπτέον κατὰ τὴν τοῦ ὀνόματος πτῶσιν. ἁπλῶς γὰρ τοῦτο λέγομεν κατὰ πάντων, ὅτι τοὺς μὲν ὅρους ἀεὶ θετέον κατὰ τὰς κλήσεις τῶν ὀνομάτων, οἷον ἄνθρωπος ἢ ἀγαθόν ἢ ἐναν ↵τία, οὐκ ἀνθρώπου ἢ ἀγαθοῦ ἢ ἐναντίων, τὰς δὲ προτάσεις ληπτέον κατὰ τὰς ἑκάστου πτώσεις· ἢ γὰρ ὅτι τούτωι, οἷον τὸ ἴσον, ἢ ὅτι τούτου, οἷον τὸ διπλάσιον, ἢ ὅτι τοῦτο, οἷον τὸ τύπτον ἢ ὁρῶν, ἢ ὅτι οὗτος, οἷον ὁ ἄνθρωπος ζῶιον, ἢ εἴ πως ἄλλως πίπτει τοὔνομα κατὰ τὴν πρότασιν. | Rursus quoniam occasio non est tempus opportunum, Deo enim occasio quidem est, tempus autem opportunum non est, eo quod nihil sit Deo conferens. Terminos enim ponendum est occasionem, et tempus opportunum, et Deum. (0679A) Propositio autem sumenda secundum nominis casum, simpliciter enim hoc dicimus de omnibus, quoniam terminos quidem semper ponendum secundum declinationes nominum, ut homo, aut bonum, aut contraria, aut hominis, aut boni, aut contrariorum. Propositiones autem sumendum secundum cuiusque casus, aut enim quoniam huic ut aequale, aut quoniam huius ut duplum, aut quoniam hoc ut feriens, vel videns, aut quoniam hic ut homo, animal, aut si quolibet modo aliter cadit nomen secundum propositionem, | Again take the inference ‘opportunity is not the right time: for opportunity belongs to God, but the right time does not, since nothing is useful to God’. We must take as terms opportunity-right time-God: but the premiss must be understood according to the case of the noun. For we state this universally without qualification, that the terms ought always to be stated in the nominative, e.g. man, good, contraries, not in oblique cases, e.g. of man, of a good, of contraries, but the premisses ought to be understood with reference to the cases of each term-either the dative, e.g. ‘equal to this’, or the genitive, e.g. ‘double of this’, or the accusative, e.g. ‘that which strikes or sees this’, or the nominative, e.g. ‘man is an animal’, or in whatever other way the word falls in the premiss. |
Chapter 37
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| 37 | ||
| 49a6 Τὸ δ᾽ ὑπάρχειν τόδε τῶιδε καὶ τὸ ἀληθεύεσθαι τόδε κατὰ τοῦδε τοσαυταχῶς ληπτέον ὁσαχῶς αἱ κατηγορίαι διήιρηνται, καὶ ταύτας ἢ πῆι ἢ ἁπλῶς, ἔτι ἢ ἁπλᾶς ἢ συμπεπλεγμένας· ὁμοίως δὲ καὶ τὸ μὴ ὑπάρχειν. ἐπισκεπτέον δὲ ταῦτα καὶ διοριστέον βέλτιον. | inesse autem hoc huic, et verum esse hoc de hoc, toties sumendum, quoties praedicamenta divisa sunt, et haec aut aliquo modo, aut simpliciter, amplius aut simplicia, aut complexa. Similiter autem et non inesse. Considerandum haec autem, et determinandum optimum. | The expressions ‘this belongs to that’ and ‘this holds true of that’ must be understood in as many ways as there are different categories, and these categories must be taken either with or without qualification, and further as simple or compound: the same holds good of the corresponding negative expressions. We must consider these points and define them better. |
Chapter 38
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| (PL 64 0679A) CAPUT XXXVIII. De anadiplosi et thesi syllogismorum, hoc est de geminatione et positione. | 38 | |
| 49a11 Τὸ δ᾽ ἐπαναδιπλούμενον ἐν ταῖς προτάσεσι πρὸς τῶι πρώτωι ἄκρωι θετέον, οὐ πρὸς τῶι μέσωι. λέγω δ᾽ οἷον εἰ γένοιτο συλλογισμὸς ὅτι τῆς δικαιοσύνης ἔστιν ἐπιστήμη ὅτι ἀγαθόν, τὸ ὅτι ἀγαθόν ἢ ἧι ἀγαθόν πρὸς τῶι πρώτωι θετέον. ἔστω γὰρ τὸ Α ἐπιστήμη ὅτι ἀγαθόν, ἐφ᾽ ὧι δὲ Β ἀγαθόν, ἐφ᾽ ὧι δὲ Γ δικαιοσύνη. τὸ δὴ Α ἀληθὲς τοῦ Β κατηγορῆσαι· | (0679B) Reduplicatum autem in propositionibus ad primam extremitatem ponendum, non ad medium, dico autem ut si fiat syllogismus, quoniam iustitiae est disciplina quoniam bonum, ad primam extremitatem ponendum. Sit enim A disciplina quoniam bonum, in quo autem B bonum, in quo autem C iustitia, ergo verum est A de B praedicari. | A term which is repeated in the premisses ought to be joined to the first extreme, not to the middle. I mean for example that if a syllogism should be made proving that there is knowledge of justice, that it is good, the expression ‘that it is good’ (or ‘qua good’) should be joined to the first term. Let A stand for ‘knowledge that it is good’, B for good, C for justice. It is true to predicate A of B. |
| τοῦ γὰρ ἀγαθοῦ ἔστιν ἐπιστήμη ὅτι ἀγαθόν. ἀλλὰ καὶ τὸ Β τοῦ Γ· ἡ γὰρ δικαιοσύνη ὅπερ ἀγαθόν. οὕτω μὲν οὖν γί- νεται ἀνάλυσις. εἰ δὲ πρὸς τῶι Β τεθείη τὸ ὅτι ἀγαθόν, οὐκ ἔσται· τὸ μὲν γὰρ Α κατὰ τοῦ Β ἀληθὲς ἔσται, τὸ δὲ Β κατὰ τοῦ Γ οὐκ ἀληθὲς ἔσται· τὸ γὰρ ἀγαθὸν ὅτι ἀγαθὸν κατηγορεῖν τῆς δικαιοσύνης ψεῦδος καὶ οὐ συνετόν. ὁμοίως δὲ καὶ εἰ τὸ ὑγιεινὸν δειχθείη ὅτι ἔστιν ἐπιστητὸν ἧι ἀγαθόν, ἢ τραγέλαφος ἧι μὴ ὄν, ἢ ὁ ἄνθρωπος φθαρτὸν ἧι αἰσθητόν· ἐν ἅπασι γὰρ τοῖς ἐπικατηγορουμένοις πρὸς τῶι ἄκρωι τὴν ἐπαναδίπλωσιν θετέον. | Nam boni est disciplina quoniam bonum. Sed et B de C, nam iustitia quiddam bonum est; sic ergo fit resolutio. (0679C) Si autem ad B ponatur, quoniam bonum, non erit, nam A quidem de B verum erit, B autem de C non erit verum, nam bonum quoniam bonum praedicari de iustitia falsum est, et non intelligibile. Similiter autem et si salubre ostendatur, quoniam disciplinatum est in eo quod bonum, aut hircocervus, opinabilis in eo quod existens, aut homo corruptibilis in eo quod sensibile, in omnibus enim praedicatis ad extremum reduplicationem ponendum. | For of the good there is knowledge that it is good. Also it is true to predicate B of C. For justice is identical with a good. In this way an analysis of the argument can be made. But if the expression ‘that it is good’ were added to B, the conclusion will not follow: for A will be true of B, but B will not be true of C. For to predicate of justice the term ‘good that it is good’ is false and not intelligible. Similarly if it should be proved that the healthy is an object of knowledge qua good, of goat-stag an object of knowledge qua not existing, or man perishable qua an object of sense: in every case in which an addition is made to the predicate, the addition must be joined to the extreme. |
| Οὐχ ἡ αὐτὴ δὲ θέσις τῶν ὅρων ὅταν ἁπλῶς τι συλλογισθῆι καὶ ὅταν τόδε τι ἢ πῆι ἢ πώς, λέγω δ᾽ οἷον ὅταν τἀγαθὸν ἐπιστητὸν δειχθῆι καὶ ὅταν ἐπιστητὸν ὅτι ἀγα θόν· ἀλλ᾽ εἰ μὲν ἁπλῶς ἐπιστητὸν δέδεικται, μέσον θετέον τὸ ὄν, εἰ δ᾽ ὅτι ἀγαθόν, τὸ τὶ ὄν. | Non est autem eadem positio terminorum, quando simpliciter quidem syllogizatum fuerit, et quando hoc aliquid, aut quo, aut quomodo. Dico autem ut quando bonum disciplinatum ostensum erit, et quando disciplinatum quoniam bonum. (0679D) Sed simpliciter quidem disciplinatum ostensum est medium ponendum ens, si autem quoniam bonum, quid ens. | The position of the terms is not the same when something is established without qualification and when it is qualified by some attribute or condition, e.g. when the good is proved to be an object of knowledge and when it is proved to be an object of knowledge that it is good. If it has been proved to be an object of knowledge without qualification, we must put as middle term ‘that which is’, but if we add the qualification ‘that it is good’, the middle term must be ‘that which is something’. |
| ἔστω γὰρ τὸ μὲν Α ἐπιστήμη ὅτι τὶ ὄν, ἐφ᾽ ὧι δὲ Β ὄν τι, τὸ δ᾽ ἐφ᾽ ὧι Γ ἀγαθόν. ἀληθὲς δὴ τὸ Α τοῦ Β κατηγορεῖν· ἦν γὰρ ἐπιστήμη τοῦ τινὸς ὄντος ὅτι τὶ ὄν. ἀλλὰ καὶ τὸ Β τοῦ Γ· τὸ γὰρ ἐφ᾽ ὧι Γ ὄν τι. ὥστε καὶ τὸ Α τοῦ Γ· ἔσται ἄρα ἐπιστήμη τἀγαθοῦ ὅτι ἀγαθόν· ἦν γὰρ τὸ τὶ ὂν τῆς ἰδίου σημεῖον οὐσίας. εἰ δὲ τὸ ὂν μέσον ἐτέθη καὶ πρὸς τῶι ἄκρωι τὸ ὂν ἁπλῶς καὶ μὴ τὸ τὶ ὂν ἐλέχθη, οὐκ ἂν ἦν συλλογισμὸς ὅτι ἔστιν ἐπιστήμη τἀγαθοῦ ὅτι ἀγαθόν, ἀλλ᾽ ὅτι ὄν, οἷον ἐφ᾽ ὧι τὸ Α ἐπιστήμη ↵ ὅτι ὄν, ἐφ᾽ ὧι Β ὄν, ἐφ᾽ ὧι Γ ἀγαθόν. φανερὸν οὖν ὅτι ἐν τοῖς ἐν μέρει συλλογισμοῖς οὕτως ληπτέον τοὺς ὅρους. | Sit enim A disciplina quoniam quid ens, in quo autem B ens quid, in quo autem C bonum, verum est ergo A de B praedicari, erat enim disciplina alicuius entis, quoniam quid ens, sed et B de C, nam in quo C ens quid, quare et A de C, erit ergo disciplina boni quoniam bonum, erat enim quid ens, proprie substantiae signum. Si autem ens medium positum sit, et ad extremum ens simpliciter, et non quid ens dictum sit, non erit syllogismus, quoniam est disciplina boni quoniam bonum, sed quoniam ens, ut si sit in quo A disciplina quoniam ens, in quo B ens, in quo C bonum. Manifestum igitur quoniam in particularibus syllogismis sic sumendum terminos. | Let A stand for ‘knowledge that it is something’, B stand for ‘something’, and C stand for ‘good’. It is true to predicate A of B: for ex hypothesi there is a science of that which is something, that it is something. B too is true of C: for that which C represents is something. Consequently A is true of C: there will then be knowledge of the good, that it is good: for ex hypothesi the term ‘something’ indicates the thing’s special nature. But if ‘being’ were taken as middle and ‘being’ simply were joined to the extreme, not ‘being something’, we should not have had a syllogism proving that there is knowledge of the good, that it is good, but that it is; e.g. let A stand for knowledge that it is, B for being, C for good. Clearly then in syllogisms which are thus limited we must take the terms in the way stated. |
Chapter 39
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|---|---|---|
| (PL 64 0679D) CAPUT XXXIX. Pro debita resolutione praecepta. | 39 | |
| 49b3 Δεῖ δὲ καὶ μεταλαμβάνειν ἃ τὸ αὐτὸ δύναται, ὀνόματα ἀντ᾽ ὀνομάτων καὶ λόγους ἀντὶ λόγων καὶ ὄνομα καὶ λόγον, καὶ ἀεὶ ἀντὶ τοῦ λόγου τοὔνομα λαμβάνειν· ῥάιων γὰρ ἡ τῶν ὅρων ἔκθεσις. οἷον εἰ μηδὲν διαφέρει εἰπεῖν τὸ ὑποληπτὸν τοῦ δοξαστοῦ μὴ εἶναι γένος ἢ μὴ εἶναι ὅπερ ὑποληπτόν τι τὸ δοξαστόν (ταὐτὸν γὰρ τὸ σημαινόμενον), ἀντὶ τοῦ λόγου τοῦ λεχθέντος τὸ ὑποληπτὸν καὶ τὸ δοξαστὸν ὅρους θετέον. | (0680A) Oportet autem accipere quae idem possunt nomina pro nominibus, et orationes pro orationibus, et nomen et orationem et semper pro oratione nomen suscipere, facilior est enim terminorum expositio, ut si nil differt dicere suspicabile opinabilis non esse genus, aut non esse idem quiddam suspicabile, quod opinabile, nam si idem est quod significatur, pro oratione dicta, suspicabile et opinabile terminos ponendum. | We ought also to exchange terms which have the same value, word for word, and phrase for phrase, and word and phrase, and always take a word in preference to a phrase: for thus the setting out of the terms will be easier. For example if it makes no difference whether we say that the supposable is not the genus of the opinable or that the opinable is not identical with a particular kind of supposable (for what is meant is the same in both statements), it is better to take as the terms the supposable and the opinable in preference to the phrase suggested. |
Chapter 40
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| 40 | ||
| 49b10 Ἐπεὶ δ᾽ οὐ ταὐτόν ἐστι τὸ εἶναι τὴν ἡδονὴν ἀγαθὸν καὶ τὸ εἶναι τὴν ἡδονὴν τὸ ἀγαθόν, οὐχ ὁμοίως θετέον τοὺς ὅρους, ἀλλ᾽ εἰ μέν ἐστιν ὁ συλλογισμὸς ὅτι ἡ ἡδονὴ τἀγαθόν, τἀγαθόν, εἰ δ᾽ ὅτι ἀγαθόν, ἀγαθόν. οὕτως κἀπὶ τῶν ἄλλων. | (0680B) Quoniam vero non est idem voluptatem esse bonum, et esse voluptatem quod bonum, non similiter ponendum terminos; sed si est syllogismus quoniam voluptas quod bonum, terminum ponendum quod bonum; si autem quoniam bonum, bonum, similiter autem et in aliis. | Since the expressions ‘pleasure is good’ and ‘pleasure is the good’ are not identical, we must not set out the terms in the same way; but if the syllogism is to prove that pleasure is the good, the term must be ‘the good’, but if the object is to prove that pleasure is good, the term will be ‘good’. Similarly in all other cases. |
Chapter 41
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| 41 | ||
| 49b14 Οὐκ ἔστι δὲ ταὐτὸν οὔτ᾽ εἶναι οὔτ᾽ εἰπεῖν, ὅτι ὧι τὸ Β ὑπάρχει, τούτωι παντὶ τὸ Α ὑπάρχει, καὶ τὸ εἰπεῖν τὸ ὧι παντὶ τὸ Β ὑπάρχει, καὶ τὸ Α παντὶ ὑπάρχει· οὐδὲν γὰρ κωλύει τὸ Β τῶι Γ ὑπάρχειν, μὴ παντὶ δέ. οἷον ἔστω τὸ Β καλόν, τὸ δὲ Γ λευκόν. εἰ δὴ λευκῶι τινὶ ὑπάρχει καλόν, ἀληθὲς εἰπεῖν ὅτι τῶι λευκῶι ὑπάρχει καλόν· ἀλλ᾽ οὐ παντὶ ἴσως. εἰ μὲν οὖν τὸ Α τῶι Β ὑπάρχει, μὴ παντὶ δὲ καθ᾽ οὗ τὸ Β, οὔτ᾽ εἰ παντὶ τῶι Γ τὸ Β, οὔτ᾽ εἰ μόνον ὑπάρχει, ἀνάγκη τὸ Α οὐχ ὅτι οὐ παντί, ἀλλ᾽ οὐδ᾽ ὑπάρχειν. | Non est autem idem neque esse, neque dicere quoniam cui B inest, huic quoque omni A inest, et dicere, cui omni B inest, et A inest omni, nihil enim prohibet B inesse C, non autem omni. Ut sit B pulchrum quid, C autem album, si igitur alicui albo inest pulchrum quid, verum est dicere quoniam albo inest pulchrum, sed non omni fortasse. Si ergo A inest B, non omni autem de quo B (neque si omni C, inest B, neque si solum alicui), non necesse est ei quod est C inesse A, non quia non omni, sed nec inesse ei quod est C. | It is not the same, either in fact or in speech, that A belongs to all of that to which B belongs, and that A belongs to all of that to all of which B belongs: for nothing prevents B from belonging to C, though not to all C: e.g. let B stand for beautiful, and C for white. If beauty belongs to something white, it is true to say that beauty belongs to that which is white; but not perhaps to everything that is white. If then A belongs to B, but not to everything of which B is predicated, then whether B belongs to all C or merely belongs to C, it is not necessary that A should belong, I do not say to all C, but even to C at all. |
| εἰ δὲ καθ᾽ οὗ ἂν τὸ Β λέγηται ἀληθῶς, τούτωι παντὶ ὑπάρχει, συμβήσεται τὸ Α, καθ᾽ οὗ παντὸς τὸ Β λέγεται, κατὰ τούτου παντὸς λέγεσθαι. εἰ μέντοι τὸ Α λέγεται καθ᾽ οὗ ἂν τὸ Β λέγηται κατὰ παντός, οὐδὲν κωλύει τῶι Γ ὑπάρχειν τὸ Β, μὴ παντὶ δὲ τὸ Α ἢ ὅλως μὴ ὑπάρχειν. ἐν δὴ τοῖς τρισὶν ὅροις δῆλον ὅτι τὸ καθ᾽ οὗ τὸ Β παντὸς τὸ Α λέγεσθαι τοῦτ᾽ ἔστι, καθ᾽ ὅσων τὸ Β λέγεται, κατὰ πάντων λέ γεσθαι καὶ τὸ Α. καὶ εἰ μὲν κατὰ παντὸς τὸ Β, καὶ τὸ Α οὕτως· εἰ δὲ μὴ κατὰ παντός, οὐκ ἀνάγκη τὸ Α κατὰ παντός. | Si autem de quocunque B dicatur vere, huic omni inest A, accidet A de quo omni B dicitur, de eo omni dici. (0680C) Si autem A dicitur de omni de quo B dicatur, nihil prohibet ei quod est C inesse B, non omni autem A, aut non inesse omnino. In tribus igitur terminis manifestum est quoniam de quo B quidem omni, et A dicitur, hoc est de quibuscunque B dicitur, de omnibus dicitur et A, et si B quidem de omni, et A similiter, si autem non de omni, non necesse est A inesse omni. | But if A belongs to everything of which B is truly stated, it will follow that A can be said of all of that of all of which B is said. If however A is said of that of all of which B may be said, nothing prevents B belonging to C, and yet A not belonging to all C or to any C at all. If then we take three terms it is clear that the expression ‘A is said of all of which B is said’ means this, ‘A is said of all the things of which B is said’. And if B is said of all of a third term, so also is A: but if B is not said of all of the third term, there is no necessity that A should be said of all of it. |
| Οὐ δεῖ δ᾽ οἴεσθαι παρὰ τὸ ἐκτίθεσθαί τι συμβαίνειν ἄτοπον· οὐδὲν γὰρ προσχρώμεθα τῶι τόδε τι εἶναι, ἀλλ᾽ ὥσπερ ὁ γεωμέτρης τὴν ποδιαίαν καὶ εὐθεῖαν τήνδε καὶ ἀπλατῆ εἶναι λέγει οὐκ οὔσας, ἀλλ᾽ οὐχ οὕτως χρῆται ὡς ἐκ τούτων συλλογιζόμενος. ὅλως γὰρ ὁ μὴ ἔστιν ὡς ὅλον πρὸς μέρος καὶ ἄλλο πρὸς τοῦτο ὡς μέρος πρὸς ὅλον, ἐξ οὐδενὸς τῶν τοιούτων δείκνυσιν ὁ δεικνύων, ὥστε οὐδὲ γίνεται ↵ συλλογισμός. τῶι δ᾽ ἐκτίθεσθαι οὕτω χρώμεθα ὥσπερ καὶ τῶι αἰσθάνεσθαι, τὸν μανθάνοντ᾽ ἀλέγοντες· οὐ γὰρ οὕτως ὡς ἄνευ τούτων οὐχ οἷόν τ᾽ ἀποδειχθῆναι, ὥσπερ ἐξ ὧν ὁ συλλογισμός. | Non oportet autem arbitrari propter expositionem accidere aliquod inconveniens, non enim laboramus in eo quod aliquid sit hoc, sed quemadmodum geometer pedalem, et rectam hanc esse et sine latitudine dicit quae non est, sed non sic utitur, ut eis syllogizans. (0680D) Omnino enim quod non est ut totum ad partem, et aliud ad hoc ut pars ad totum, ex nullo talium ostendit demonstrator, neque enim fit syllogismus, expositione autem sic utimur, ut et sentiat qui discit dicentes, non enim sic ut sine his non possibile sit demonstrare, quemadmodum ex quibus est syllogismus. | We must not suppose that something absurd results through setting out the terms: for we do not use the existence of this particular thing, but imitate the geometrician who says that ‘this line a foot long’ or ‘this straight line’ or ‘this line without breadth’ exists although it does not, but does not use the diagrams in the sense that he reasons from them. For in general, if two things are not related as whole to part and part to whole, the prover does not prove from them, and so no syllogism is formed. We (I mean the learner) use the process of setting out terms like perception by sense, not as though it were impossible to demonstrate without these illustrative terms, as it is to demonstrate without the premisses of the syllogism. |
Chapter 42
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| 42 | ||
| 50a5 Μὴ λανθανέτω δ᾽ ἡμᾶς ὅτι ἐν τῶι αὐτῶι συλλογισμῶι οὐχ ἅπαντα τὰ συμπεράσματα δι᾽ ἑνὸς σχήματός ἐστιν, ἀλλὰ τὸ μὲν διὰ τούτου τὸ δὲ δι᾽ ἄλλου. δῆλον οὖν ὅτι καὶ τὰς ἀναλύσεις οὕτω ποιητέον. ἐπεὶ δ᾽ οὐ πᾶν πρόβλημα ἐν ἅπαντι σχήματι ἀλλ᾽ ἐν ἑκάστωι τεταγμένα, φανερὸν ἐκ τοῦ συμπεράσματος ἐν ὧι σχήματι ζητητέον. | Non lateat autem nos, quoniam in eodem syllogismo, non omnes conclusiones per unam eamdem figuram sunt, sed haec quidem per hanc, illa vero per aliam. Palam ergo quoniam et resolutiones sic faciendum. Quoniam autem non omne propositum in omni figura, sed in unaquaque disposita sunt, manifestum est ex conclusione in qua figura sit quaerendum. | We should not forget that in the same syllogism not all conclusions are reached through one figure, but one through one figure, another through another. Clearly then we must analyse arguments in accordance with this. Since not every problem is proved in every figure, but certain problems in each figure, it is clear from the conclusion in what figure the premisses should be sought. |
Chapter 43
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| 50a11 Τούς τε πρὸς ὁρισμὸν τῶν λόγων, ὅσοι πρὸς ἕν τι τυγχάνουσι διειλεγμένοι τῶν ἐν τῶι ὅρωι, πρὸς ὁ διείλεκται θετέον ὅρον, καὶ οὐ τὸν ἅπαντα λόγον· ἧττον γὰρ συμβήσεται ταράττεσθαι διὰ τὸ μῆκος, οἷον εἰ τὸ ὕδωρ ἔδειξεν ὅτι ὑγρὸν ποτόν, τὸ ποτὸν καὶ τὸ ὕδωρ ὅρους θετέον. | (0681A) Et ad definitiones orationum quaecunque ad unum quiddam sunt argumentatae in eorum quae insunt termino, ad quod argumentatum est ponendum terminum, et non totam orationem, minus enim contingit perturbari propter longitudinem, ut si quis aquam ostendit quoniam est humidus potus, potum et aquam terminos ponendum. | In reference to those arguments aiming at a definition which have been directed to prove some part of the definition, we must take as a term the point to which the argument has been directed, not the whole definition: for so we shall be less likely to be disturbed by the length of the term: e.g. if a man proves that water is a drinkable liquid, we must take as terms drinkable and water. |
Chapter 44
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| (PL 64 0681A) CAPUT XL. De resolutione syllogismorum ad impossibile et ex hypothesi. | 44 | |
| 50a16 Ἔτι δὲ τοὺς ἐξ ὑποθέσεως συλλογισμοὺς οὐ πειρατέον ἀνάγειν· οὐ γὰρ ἔστιν ἐκ τῶν κειμένων ἀνάγειν. οὐ γὰρ διὰ συλλογισμοῦ δεδειγμένοι εἰσίν, ἀλλὰ διὰ συνθήκης ὡμολογημένοι πάντες. οἷον εἰ ὑποθέμενος, ἂν δύναμίς τις μία μὴ ἦι τῶν ἐναντίων, μηδ᾽ ἐπιστήμην μίαν εἶναι, εἶτα διαλεχθείη ὅτι οὐκ ἔστι πᾶσα δύναμις τῶν ἐναντίων, οἱονεὶ τοῦ ὑγιεινοῦ καὶ τοῦ νοσώδους· ἅμα γὰρ ἔσται τὸ αὐτὸ ὑγιεινὸν καὶ νοσῶδες. ὅτι μὲν οὖν οὐκ ἔστι μία πάντων τῶν ἐναντίων δύναμις, ἐπιδέδεικται, ὅτι δ᾽ ἐπιστήμη οὐκ ἔστιν, οὐ δέδεικται. καίτοι ὁμολογεῖν ἀναγκαῖον· ἀλλ᾽ οὐκ ἐκ συλλογισμοῦ, ἀλλ᾽ ἐξ ὑποθέσεως. τοῦτον μὲν οὖν οὐκ ἔστιν ἀναγαγεῖν, ὅτι δ᾽ οὐ μία δύναμις, ἔστιν· οὗτος γὰρ ἴσως καὶ ἦν συλλογισμός, ἐκεῖνο δ᾽ ὑπόθεσις. | Amplius autem ex hypothesi syllogismos non est tentandum reducere, nam non est ex iis quae posita sunt reducere; non enim per syllogismum ostensi sunt, sed ad placitum concessi sunt omnes. Ut si quis ponat, si una quaedam potestas non sit contrariorum, neque disciplinam esse unam; deinde disputet quoniam non est una potestas contrariorum, ut sanativi et aegrotativi, simul enim idem erit sanativum et aegrotativum. (0681B) Quoniam autem non est omnium contrariorum una potestas, ostensum est, sed quoniam disciplina non una, non est ostensum; quamvis confiteri sit necesse, at non ex syllogismo, verum ex hypothesi; hoc igitur non est reducere, quoniam non una potestas est: hic enim fortassee erat syllogismus, illud autem hypothesis. | Further we must not try to reduce hypothetical syllogisms; for with the given premisses it is not possible to reduce them. For they have not been proved by syllogism, but assented to by agreement. For instance if a man should suppose that unless there is one faculty of contraries, there cannot be one science, and should then argue that not every faculty is of contraries, e.g. of what is healthy and what is sickly: for the same thing will then be at the same time healthy and sickly. He has shown that there is not one faculty of all contraries, but he has not proved that there is not a science. And yet one must agree. But the agreement does not come from a syllogism, but from an hypothesis. This argument cannot be reduced: but the proof that there is not a single faculty can. The latter argument perhaps was a syllogism: but the former was an hypothesis. |
| Ὁμοίως δὲ καὶ ἐπὶ τῶν διὰ τοῦ ἀδυνάτου περαινομένων· οὐδὲ γὰρ τούτους οὐκ ἔστιν ἀναλύειν, ἀλλὰ τὴν μὲν εἰς τὸ ἀδύνατον ἀπαγωγὴν ἔστι (συλλογισμῶι γὰρ δείκνυται), θάτερον δ᾽ οὐκ ἔστιν· ἐξ ὑποθέσεως γὰρ περαίνεται. διαφέρουσι δὲ τῶν προειρημένων ὅτι ἐν ἐκείνοις μὲν δεῖ προδιομολογήσασθαι, εἰ μέλλει συμφήσειν, οἷον ἂν δειχθῆι μία δύναμις τῶν ἐναντίων, καὶ ἐπιστήμην εἶναι τὴν αὐτήν· ἐνταῦθα δὲ καὶ μὴ προδιομολογησάμενοι συγχωροῦσι διὰ τὸ φανερὸν εἶναι τὸ ψεῦδος, οἷον τεθείσης τῆς διαμέτρου συμμέτρου τὸ τὰ περιττὰ ἴσα εἶναι τοῖς ἀρτίοις. | Similiter autem in his qui per impossibile concluduntur, nam neque hoc est resolvere, sed ad impossibile quidem reductio est; syllogismo enim monstratur; alterum autem non est, nam ex hypothesi concluditur. Differunt autem A praedictis quoniam in illis quidem oportet prius confiteri, si debet concedere, ut si ostendatur una potestas contrariorum, et disciplinam es E eamdem; hic autem et non prius confessi concedunt, eo quod manifestum sit falsum, ut posita dian etro symetro, eo quod imparia esse aequalia paribus. (0681C) Plures autem et diversi terminantur ex conditione, quos prospicere oportet, et notare apte. | The same holds good of arguments which are brought to a conclusion per impossibile. These cannot be analysed either; but the reduction to what is impossible can be analysed since it is proved by syllogism, though the rest of the argument cannot, because the conclusion is reached from an hypothesis. But these differ from the previous arguments: for in the former a preliminary agreement must be reached if one is to accept the conclusion; e.g. an agreement that if there is proved to be one faculty of contraries, then contraries fall under the same science; whereas in the latter, even if no preliminary agreement has been made, men still accept the reasoning, because the falsity is patent, e.g. the falsity of what follows from the assumption that the diagonal is commensurate, viz. that then odd numbers are equal to evens. |
| Πολλοὶ δὲ καὶ ἕτεροι περαίνονται ἐξ ὑποθέσεως, οὓς ἐπισκέψασθαι δεῖ καὶ διασημῆναι καθαρῶς. τίνες μὲν οὖν αἱ ↵ διαφοραὶ τούτων, καὶ ποσαχῶς γίνεται τὸ ἐξ ὑποθέσεως, ὕστερον ἐροῦμεν· νῦν δὲ τοσοῦτον ἡμῖν ἔστω φανερόν, ὅτι οὐκ ἔστιν ἀναλύειν εἰς τὰ σχήματα τοὺς τοιούτους συλλογισμούς. καὶ δι᾽ ἣν αἰτίαν, εἰρήκαμεν. | Quae ergo horum differentiae, et quoties fiunt, qui sunt ex hypothesi, postea dicemus. Nunc autem tantum sit nobis manifestum quoniam non est resolvere in figuras huiusmodi syllogismos, et ob quam causam diximus. | Many other arguments are brought to a conclusion by the help of an hypothesis; these we ought to consider and mark out clearly. We shall describe in the sequel their differences, and the various ways in which hypothetical arguments are formed: but at present this much must be clear, that it is not possible to resolve such arguments into the figures. And we have explained the reason. |
Chapter 45
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| (PL 64 0681C) CAPUT XLI. De reciproca reductione syllogismorum unius figurae in aliam. | 45 | |
| 50b5 Ὅσα δ᾽ ἐν πλείοσι σχήμασι δείκνυται τῶν προβλημάτων, ἢν ἐν θατέρωι συλλογισθῆι, ἔστιν ἀναγαγεῖν τὸν συλλογισμὸν εἰς θάτερον, οἷον τὸν ἐν τῶι πρώτωι στερητικὸν εἰς τὸ δεύτερον, καὶ τὸν ἐν τῶι μέσωι εἰς τὸ πρῶτον, οὐχ ἅπαντας δὲ ἀλλ᾽ ἐνίους. ἔσται δὲ φανερὸν ἐν τοῖς ἑπομένοις. εἰ γὰρ τὸ Α μηδενὶ τῶι Β, τὸ δὲ Β παντὶ τῶι Γ, τὸ Α οὐδενὶ τῶι Γ. οὕτω μὲν οὖν τὸ πρῶτον σχῆμα, ἐὰν δ᾽ ἀντιστραφῆι τὸ στερητικόν, τὸ μέσον ἔσται· τὸ γὰρ Β τῶι μὲν Α οὐδενί, τῶι δὲ Γ παντὶ ὑπάρχει. ὁμοίως δὲ καὶ εἰ μὴ καθόλου ἀλλ᾽ ἐν μέρει ὁ συλλογισμός, οἷον εἰ τὸ μὲν Α μηδενὶ τῶι Β, τὸ δὲ Β τινὶ τῶι Γ· ἀντιστραφέντος γὰρ τοῦ στερητικοῦ τὸ μέσον ἔσται σχῆμα. | Quaecunque autem in pluribus figuris monstrantur proposita, si in altera syllogizetur, est reducere syllogismum in alteram, ut eum qui in prima est privativum in secundam figuram, et eum qui in media est in primam. (0681D) Non omnes autem, sed quosdam, erit autem in sequentibus manifestum. Si enim A nulli B, B autem omni C, A nulli C, sic ergo prima figura; si autem convertatur privativa, media erit. Nam B A quidem nulli, C autem omni inerit. Similiter autem et si non universalis, sed particularis fit syllogismus, ut si A quidem nulli B, B autem alicui C, conversa enim privativa media erit figura. | Whatever problems are proved in more than one figure, if they have been established in one figure by syllogism, can be reduced to another figure, e.g. a negative syllogism in the first figure can be reduced to the second, and a syllogism in the middle figure to the first, not all however but some only. The point will be clear in the sequel. If A belongs to no B, and B to all C, then A belongs to no C. Thus the first figure; but if the negative statement is converted, we shall have the middle figure. For B belongs to no A, and to all C. Similarly if the syllogism is not universal but particular, e.g. if A belongs to no B, and B to some C. Convert the negative statement and you will have the middle figure. |
| Τῶν δ᾽ ἐν τῶι δευτέρωι συλλογισμῶν οἱ μὲν καθόλου ἀναχθήσονται εἰς τὸ πρῶτον, τῶν δ᾽ ἐν μέρει ἅτερος μόνος. ἔστω γὰρ τὸ Α τῶι μὲν Β μηδενὶ τῶι δὲ Γ παντὶ ὑπάρχον. ἀντιστραφέντος οὖν τοῦ στερητικοῦ τὸ πρῶτον ἔσται σχῆμα· τὸ μὲν γὰρ Β οὐδενὶ τῶι Α, τὸ δὲ Α παντὶ τῶι Γ ὑπάρξει. ἐὰν δὲ τὸ κατηγορικὸν ἦι πρὸς τῶι Β, τὸ δὲ στερητικὸν πρὸς τῶι Γ, πρῶτον ὅρον θετέον τὸ Γ· τοῦτο γὰρ οὐδενὶ τῶι Α, τὸ δὲ Α παντὶ τῶι Β· ὥστ᾽ οὐδενὶ τῶι Β τὸ Γ. οὐδ᾽ ἄρα τὸ Β τῶι Γ οὐδενί· ἀντιστρέφει γὰρ τὸ στερητικόν. | Eorum autem syllogismorum, qui sunt in secunda figura, universales quidem reducentur in primam figuram, particularium autem alter solum. Insit enim A B quidem nulli, C vero omni, conversa privativa prima erit figura, nam B quidem nulli A, A autem omni C inerit. (0682A) Si autem praedicativum quidem sit ad B, privativum autem ad C, primus terminus ponendus est C, hoc enim nulli A, A autem omni B, quare nulli B inerit C, ergo et B nulli C, convertitur enim privativa. | The universal syllogisms in the second figure can be reduced to the first, but only one of the two particular syllogisms. Let A belong to no B and to all C. Convert the negative statement, and you will have the first figure. For B will belong to no A and A to all C. But if the affirmative statement concerns B, and the negative C, C must be made first term. For C belongs to no A, and A to all B: therefore C belongs to no B. B then belongs to no C: for the negative statement is convertible. |
| ἐὰν δ᾽ ἐν μέρει ἦι ὁ συλλογισμός, ὅταν μὲν ἦι τὸ στερητικὸν πρὸς τῶι μείζονι ἄκρωι, ἀναχθήσεται εἰς τὸ πρῶτον, οἷον εἰ τὸ Α μηδενὶ τῶι Β, τῶι δὲ Γ τινί· ἀντιστραφέντος γὰρ τοῦ στερητικοῦ τὸ πρῶτον ἔσται σχῆμα· τὸ μὲν γὰρ Β οὐδενὶ τῶι Α, τὸ δὲ Α τινὶ τῶι Γ. ὅταν δὲ τὸ κατηγορικόν, οὐκ ἀναλυθήσεται, οἷον εἰ τὸ Α τῶι μὲν Β παντί, τῶι δὲ Γ οὐ παντί· οὔτε γὰρ δέχεται ἀντιστροφὴν τὸ Α Β, οὔτε γενομένης ἔσται συλλογισμός. | Si autem particularis sit syllogismus, quando privativum quidem erit ad maiorem extremitatem, resolvetur in primam figuram, ut si A nulli B, B autem alicui C, conversa enim privativa prima erit figura, nam B quidem nulli A, A autem alicui C. Quando vero praedicativum, non resolvetur, ut si A quidem omni B, C vero non omni, non enim suscipit conversionem A B, neque cum fit, erit syllogismus. | But if the syllogism is particular, whenever the negative statement concerns the major extreme, reduction to the first figure will be possible, e.g. if A belongs to no B and to some C: convert the negative statement and you will have the first figure. For B will belong to no A and A to some C. But when the affirmative statement concerns the major extreme, no resolution will be possible, e.g. if A belongs to all B, but not to all C: for the statement AB does not admit of conversion, nor would there be a syllogism if it did. |
| Πάλιν οἱ μὲν ἐν τῶι τρίτωι σχήματι οὐκ ἀναλυθήσονται πάντες εἰς τὸ πρῶτον, οἱ δ᾽ ἐν τῶι πρώτωι πάντες εἰς τὸ τρίτον. ὑπαρχέτω γὰρ τὸ Α παντὶ τῶι Β, τὸ δὲ Β τινὶ τῶι Γ. οὐκοῦν ἐπειδὴ ἀντιστρέφει τὸ ἐν μέρει κατηγορικόν, ὑπάρ- ξει τὸ Γ τινὶ τῶι Β· τὸ δὲ Α παντὶ ὑπῆρχεν, ὥστε γίνεται τὸ τρίτον σχῆμα. καὶ εἰ στερητικὸς ὁ συλλογισμός, ὡσαύτως· ἀντιστρέφει γὰρ τὸ ἐν μέρει κατηγορικόν, ὥστε τὸ μὲν Α οὐδενὶ τῶι Β, τὸ δὲ Γ τινὶ ὑπάρξει. | Rursus qui in tertia quidem sunt figura, non resolvuntur omnes in primam, qui autem sunt in prima, omnes in tertiam. Insit enim A quidem omni B, B autem alicui C, ergo quia convertitur particularis praedicativa, inerit et C alicui B, A vero omni B inerat, quare fit tertia figura. (0682B) Et si privativus sit syllogismus, similiter: convertitur enim particularis affirmativa, quare A quidem nulli B, C autem alicui inerit. | Again syllogisms in the third figure cannot all be resolved into the first, though all syllogisms in the first figure can be resolved into the third. Let A belong to all B and B to some C. Since the particular affirmative is convertible, C will belong to some B: but A belonged to all B: so that the third figure is formed. Similarly if the syllogism is negative: for the particular affirmative is convertible: therefore A will belong to no B, and to some C. |
| ↵ Τῶν δ᾽ ἐν τῶι τελευταίωι σχήματι συλλογισμῶν εἷς μόνος οὐκ ἀναλύεται εἰς τὸ πρῶτον, ὅταν μὴ καθόλου τεθῆι τὸ στερητικόν, οἱ δ᾽ ἄλλοι πάντες ἀναλύονται. κατηγορείσθω γὰρ παντὸς τοῦ Γ τὸ Α καὶ τὸ Β· οὐκοῦν ἀντιστρέψει τὸ Γ πρὸς ἑκάτερον ἐπὶ μέρους· ὑπάρχει ἄρα τινὶ τῶι Β. ὥστ᾽ ἔσται τὸ πρῶτον σχῆμα, εἰ τὸ μὲν Α παντὶ τῶι Γ, τὸ δὲ Γ τινὶ τῶι Β. καὶ εἰ τὸ μὲν Α παντὶ τῶι Γ, τὸ δὲ Β τινί, ὁ αὐτὸς λόγος· ἀντιστρέφει γὰρ πρὸς τὸ Β τὸ Γ. ἐὰν δὲ τὸ μὲν Β παντὶ τῶι Γ, τὸ δὲ Α τινὶ τῶι Γ, πρῶτος ὅρος θετέος τὸ Β· τὸ γὰρ Β παντὶ τῶι Γ, τὸ δὲ Γ τινὶ τῶι Α, ὥστε τὸ Β τινὶ τῶι Α. ἐπεὶ δ᾽ ἀντιστρέφει τὸ ἐν μέρει, καὶ τὸ Α τινὶ τῶι Β ὑπάρξει. καὶ εἰ στερητικὸς ὁ συλλογισμός, καθάλου τῶν ὅρων ὄντων, ὁμοίως ληπτέον. | Eorum autem sylogismorum qui sunt in postrema figura unus tantum non resolvitur in primam, quando non universalis ponitur privativa, alii autem omnes resolvuntur. Praedicentur enim de omni C, et A et B, ergo convertetur C ad utrumque particulariter; inerit ergo A alicui B, quare erit prima figura, siquidem A omni C, C vero alicui B; et si A quidem omni C, B autem alicui C, cadem ratio, convertitur enim ad B C. Si autem B quidem omni C, A autem alicui C, primus ponendus B, nam B omni C, C autem alicui A, quare B alicui A, quoniam autem convertitur particularis, et A alicui B inerit. Et si privativus sit syllogismus universalibus terminis, similiter sumendum. | Of the syllogisms in the last figure one only cannot be resolved into the first, viz. when the negative statement is not universal: all the rest can be resolved. Let A and B be affirmed of all C: then C can be converted partially with either A or B: C then belongs to some B. Consequently we shall get the first figure, if A belongs to all C, and C to some of the Bs. If A belongs to all C and B to some C, the argument is the same: for B is convertible in reference to C. But if B belongs to all C and A to some C, the first term must be B: for B belongs to all C, and C to some A, therefore B belongs to some A. But since the particular statement is convertible, A will belong to some B. If the syllogism is negative, when the terms are universal we must take them in a similar way.
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| ὑπαρχέτω γὰρ τὸ Β παντὶ τῶι Γ, τὸ δὲ Α μηδενί· οὐκοῦν τινὶ τῶι Β ὑπάρξει τὸ Γ, τὸ δὲ Α οὐδενὶ τῶι Γ, ὥστ᾽ ἔσται μέσον τὸ Γ. ὁμοίως δὲ καὶ εἰ τὸ μὲν στερητικὸν καθόλου, τὸ δὲ κατηγορικὸν ἐν μέρει· τὸ μὲν γὰρ Α οὐδενὶ τῶι Γ, τὸ δὲ Γ τινὶ τῶν Β ὑπάρξει. ἐὰν δ᾽ ἐν μέρει ληφθῆι τὸ στερητικόν, οὐκ ἔσται ἀνάλυσις, οἷον εἰ τὸ μὲν Β παντὶ τῶι Γ, τὸ δὲ Α τινὶ μὴ ὑπάρ χει· ἀντιστραφέντος γὰρ τοῦ Β Γ ἀμφότεραι αἱ προτάσεις ἔσονται κατὰ μέρος. | (0682C) Insit enim B omni C, A autem nulli C, ergo alicui B inerit C, A autem nulli C, quare erit medium C. Similiter autem et si privativa quidem si universalis, praedicativa autem particularis, nam A quidem nulli C, C autem alicui B inerit. Si autem particularis sumatur privativa, non erit resolutio, ut si B quidem omni C, A autem alicui C non inest, conversa enim B C, utraeque propositiones erunt particulares. | Let B belong to all C, and A to no C: then C will belong to some B, and A to no C; and so C will be middle term. Similarly if the negative statement is universal, the affirmative particular: for A will belong to no C, and C to some of the Bs. But if the negative statement is particular, no resolution will be possible, e.g. if B belongs to all C, and A not belong to some C: convert the statement BC and both premisses will be particular. |
| Φανερὸν δὲ καὶ ὅτι πρὸς τὸ ἀναλύειν εἰς ἄλληλα τὰ σχήματα ἡ πρὸς τῶι ἐλάττονι ἄκρωι πρότασις ἀντιστρεπτέα ἐν ἀμφοτέροις τοῖς σχήμασι· ταύτης γὰρ μετατιθεμένης ἡ μετάβασις ἐγίνετο. | Manifestum autem quoniam ad resolvendum ad se invicem figuras, quae ad minorem extremitatem est propositio, convertenda in utrisque figuris, hac conversa, transitio fit; eorum autem qui in media sunt figura, alter quidem resolvitur, alter vero non resolvitur in tertiam, nam cum sit universalis privativa, resolvitur. | It is clear that in order to resolve the figures into one another the premiss which concerns the minor extreme must be converted in both the figures: for when this premiss is altered, the transition to the other figure is made. |
| Τῶν δ᾽ ἐν τῶι μέσωι σχήματι ἅτερος μὲν ἀναλύεται, ἅτερος δ᾽ οὐκ ἀναλύεται, εἰς τὸ τρίτον. ὅταν μὲν γὰρ ἦι τὸ καθόλου στερητικόν, ἀναλύεται. εἰ γὰρ τὸ Α μηδενὶ τῶι Β, τῶι δὲ Γ τινί, ἀμφότερα ὁμοίως ἀντιστρέφει πρὸς τὸ Α, ὥστε τὸ μὲν Β οὐδενὶ τῶι Α, τὸ δὲ Γ τινί· μέσον ἄρα τὸ Α. ὅταν δὲ τὸ Α παντὶ τῶι Β, τῶι δὲ Γ τινὶ μὴ ὑπάρχηι, οὐκ ἔσται ἀνάλυσις· οὐδετέρα γὰρ τῶν προτάσεων ἐκ τῆς ἀντιστροφῆς καθόλου. | (0682D) Si enim A nulli quidem B, alicui autem C, utraque similiter convertitur ad A, quare B quidem nulli A, C vero alicui, medium ergo A. Quando autem A omni B, C autem alicui non insit, non fit resolutio, neutra enim propositionum ex conversione universalis. | One of the syllogisms in the middle figure can, the other cannot, be resolved into the third figure. Whenever the universal statement is negative, resolution is possible. For if A belongs to no B and to some C, both B and C alike are convertible in relation to A, so that B belongs to no A and C to some A. A therefore is middle term. But when A belongs to all B, and not to some C, resolution will not be possible: for neither of the premisses is universal after conversion. |
| Καὶ οἱ ἐκ τοῦ τρίτου δὲ σχήματος ἀναλυθήσονται εἰς τὸ μέσον, ὅταν ἦι καθόλου τὸ στερητικόν, οἷον εἰ τὸ Α μηδενὶ τῶι Γ, τὸ δὲ Β τινὶ ἢ παντί. καὶ γὰρ τὸ Γ τῶι μὲν Α οὐδενί, τῶι δὲ Β τινὶ ὑπάρξει. ἐὰν δ᾽ ἐπὶ μέρους ἦι τὸ στερητικόν, οὐκ ἀναλυθήσεται· οὐ γὰρ δέχεται ἀντιστροφὴν τὸ ἐν μέρει ἀποφατικόν. | Qui autem ex tertia sunt figura, resolventur in mediam, quando fuerit universalis privativa, ut si A nulli C, B autem alicui, aut omni C, nam C, A quidem nulli, B autem alicui inerit. Si autem particularis sit privativa, non resolvetur, non enim suscipit conversionem particularis negativa. | Syllogisms in the third figure can be resolved into the middle figure, whenever the negative statement is universal, e.g. if A belongs to no C, and B to some or all C. For C then will belong to no A and to some B. But if the negative statement is particular, no resolution will be possible: for the particular negative does not admit of conversion. |
| Φανερὸν οὖν ὅτι οἱ αὐτοὶ συλλογισμοὶ οὐκ ἀναλύονται ἐν τούτοις τοῖς σχήμασιν οἵπερ οὐδ᾽ εἰς τὸ πρῶτον ἀνελύοντο, ↵ καὶ ὅτι εἰς τὸ πρῶτον σχῆμα τῶν συλλογισμῶν ἀναγομένων οὗτοι μόνοι διὰ τοῦ ἀδυνάτου περαίνονται. | Manifestum ergo quoniam iidem syllogismi non resolvuntur in his figuris, qui nec in primam resolvebantur, et quoniam in primam figuram reductis syllogismis, isti soli syllogismi per impossibile clauduntur. | It is clear then that the same syllogisms cannot be resolved in these figures which could not be resolved into the first figure, and that when syllogisms are reduced to the first figure these alone are confirmed by reduction to what is impossible. |
| Πῶς μὲν οὖν δεῖ τοὺς συλλογισμοὺς ἀνάγειν, καὶ ὅτι ἀναλύεται τὰ σχήματα εἰς ἄλληλα, φανερὸν ἐκ τῶν εἰ ρημένων. | (0683A) Quomodo ergo oportet syllogismos reducere, et quoniam resolvuntur figurae in se invicem, manifestum ex dictis. | It is clear from what we have said how we ought to reduce syllogisms, and that the figures may be resolved into one another. |
Chapter 46
| Greek | Latin | English |
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| (PL 64 0683A) CAPUT XLII. De syllogismis infinitis, et regulis consequentiarum. | 46 | |
| 50b5 διαφέρει δέ τι ἐν τῶι κατασκευάζειν ἢ ἀνασκευάζειν τὸ ὑπολαμβάνειν ἢ ταὐτὸν ἢ ἕτερον σημαίνειν τὸ μὴ εἶναι τοδὶ καὶ εἶναι μὴ τοῦτο, οἷον τὸ μὴ εἶναι λευκὸν τῶι εἶναι μὴ λευκόν. οὐ γὰρ ταὐτὸν σημαίνει, οὐδ᾽ ἔστιν ἀπόφασις τοῦ εἶναι λευκὸν τὸ εἶναι μὴ λευκόν, ἀλλὰ τὸ μὴ εἶναι λευκόν.
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Differt autem in construendo vel destruendo opinari, aut idem, aut diversum significare, non esse hoc, et esse non hoc, ut non esse album, ei quod est esse non album; non enim idem significant, nec est negatio eius quae est esse album ea quae est esse non album, sed non esse album.
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In establishing or refuting, it makes some difference whether we suppose the expressions ‘not to be this’ and ‘to be not-this’ are identical or different in meaning, e.g. ‘not to be white’ and ‘to be not-white’. For they do not mean the same thing, nor is ‘to be not-white’ the negation of ‘to be white’, but ‘not to be white’.
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| λόγος δὲ τούτου ὅδε. ὁμοίως γὰρ ἔχει τὸ δύναται βαδίζειν πρὸς τὸ δύναται οὐ βαδίζειν τῶι ἔστι λευκόν πρὸς τὸ ἔστιν οὐ λευκόν, καὶ ἐπίσταται τἀγαθόν πρὸς τὸ ἐπίσταται τὸ οὐκ ἀγαθόν. τὸ γὰρ ἐπίσταται τἀγαθόν ἢ ἔστιν ἐπιστάμενος τἀγαθόν οὐδὲν διαφέρει, οὐδὲ τὸ δύναται βαδί ζειν ἢ ἔστι δυνάμενος βαδίζειν· ὥστε καὶ τὰ ἀντικείμενα, οὐ δύναται βαδίζειν – οὐκ ἔστι δυνάμενος βαδίζειν. εἰ οὖν τὸ οὐκ ἔστι δυνάμενος βαδίζειν ταὐτὸ σημαίνει καὶ ἔστι δυνάμενος οὐ βαδίζειν ἢ μὴ βαδίζειν, ταῦτά γε ἅμα ὑπάρξει ταὐτῶι (ὁ γὰρ αὐτὸς δύναται καὶ βαδίζειν καὶ μὴ βαδί ζειν, καὶ ἐπιστήμων τἀγαθοῦ καὶ τοῦ μὴ ἀγαθοῦ ἐστί), φάσις δὲ καὶ ἀπόφασις οὐχ ὑπάρχουσιν αἱ ἀντικείμεναι ἅμα τῶι αὐτῶι. | (0683B) Ratio autem huius haec est; similiter enim se habet possibile est ambulare ad possibile non ambulare, id quae est esse album ad esse non album, et scit bonum ad scit non bonum: nam scit bonum vel sciens bonum nihil differt, neque potest ambulare vel est potens ambulare; quare et opposita, non potest ambulare et non est potens ambulare. Si igitur non est potens ambulare idem significat et est potens non ambulare, ipsa simul inerunt eidem, nam idem potest ambulare et non ambulare, et idem sciens bonum et non bonum est. Affirmatio autem et negatio non sunt oppositae simul in eodem. | The reason for this is as follows. The relation of ‘he can walk’ to ‘he can not-walk’ is similar to the relation of ‘it is white’ to ‘it is not-white’; so is that of ‘he knows what is good’ to ‘he knows what is not-good’. For there is no difference between the expressions ‘he knows what is good’ and ‘he is knowing what is good’, or ‘he can walk’ and ‘he is able to walk’: therefore there is no difference between their contraries ‘he cannot walk’-’he is not able to walk’. If then ‘he is not able to walk’ means the same as ‘he is able not to walk’, capacity to walk and incapacity to walk will belong at the same time to the same person (for the same man can both walk and not-walk, and is possessed of knowledge of what is good and of what is not-good), but an affirmation and a denial which are opposed to one another do not belong at the same time to the same thing. |
| ὥσπερ οὖν οὐ ταὐτό ἐστι τὸ μὴ ἐπίστασθαι τἀγαθὸν καὶ ἐπίστασθαι τὸ μὴ ἀγαθόν, οὐδ᾽ εἶναι μὴ ἀγαθὸν καὶ μὴ εἶναι ἀγαθὸν ταὐτόν. τῶν γὰρ ἀνάλογον ἐὰν θάτερα ἦι ἕτερα, καὶ θάτερα. οὐδὲ τὸ εἶναι μὴ ἴσον καὶ τὸ μὴ εἶναι ἴσον· τῶι μὲν γὰρ ὑπόκειταί τι, τῶι ὄντι μὴ ἴσωι, καὶ τοῦτ᾽ ἔστι τὸ ἄνισον, τῶι δ᾽ οὐδέν. διόπερ ἴσον μὲν ἢ ἄνισον οὐ πᾶν, ἴσον δ᾽ ἢ οὐκ ἴσον πᾶν. | (0683C) Quemadmodum ergo non idem est, non scire bonum et scire non bonum, nec esse non bonum et non esse bonum idem, nam proportionalium, si alterum sit, et alterum, nec esse non aequale et non esse aequale idem, huic enim quod est non aequale subiacet aliquid, et hoc est inaequale, illi vero nihil, eo quod aequale quidem vel inaequale non omne est, aequale autem vel non aequale omne; | As then ‘not to know what is good’ is not the same as ‘to know what is not good’, so ‘to be not-good’ is not the same as ‘not to be good’. For when two pairs correspond, if the one pair are different from one another, the other pair also must be different. Nor is ‘to be not-equal’ the same as ‘not to be equal’: for there is something underlying the one, viz. that which is not-equal, and this is the unequal, but there is nothing underlying the other. Wherefore not everything is either equal or unequal, but everything is equal or is not equal. |
| ἔτι τὸ ἔστιν οὐ λευκὸν ξύλον καὶ οὐκ ἔστι λευκὸν ξύλον οὐχ ἅμα ὑπάρχει. εἰ γάρ ἐστι ξύλον οὐ λευκόν, ἔσται ξύλον· τὸ δὲ μὴ ὂν λευκὸν ξύλον οὐκ ἀνάγκη ξύλον εἶναι. ὥστε φανερὸν ὅτι οὐκ ἔστι τοῦ ἔστιν ἀγαθόν τὸ ἔστιν οὐκ ἀγαθόν ἀπόφασις. εἰ οὖν κατὰ παντὸς ἑνὸς ἢ φάσις ἢ ἀπόφασις ἀληθής, εἰ μὴ ἔστιν ἀπόφασις, δῆλον ὡς κατάφασις ἄν πως εἴη. καταφάσεως δὲ πάσης ἀπόφασις ἔστιν· καὶ ταύτης ἄρα τὸ οὐκ ἔστιν οὐκ ἀγαθόν. | amplius, est non album lignum et non est album lignum non simul sunt, si enim est lignum non album, erit lignum, quod autem non est album lignum, non necesse est esse lignum: quare manifestum est quoniam non est eius quod est bonum, est non bonum, negatio; si ergo de omni uno vel affirmatio, vel negatio vera, si non est negatio, palam quoniam affirmatio aliquo modo erit; affirmationis autem omnis, negatio est, et huius ergo, ea quae est non est, non bonum. | Further the expressions ‘it is a not-white log’ and ‘it is not a white log’ do not imply one another’s truth. For if ‘it is a not-white log’, it must be a log: but that which is not a white log need not be a log at all. Therefore it is clear that ‘it is not-good’ is not the denial of ‘it is good’. If then every single statement may truly be said to be either an affirmation or a negation, if it is not a negation clearly it must in a sense be an affirmation. But every affirmation has a corresponding negation. The negation then of ‘it is not-good’ is ‘it is not not-good’. |
| Ἔχει δὲ τάξιν τήνδε πρὸς ἄλληλα. ἔστω τὸ εἶναι ἀγαθὸν ἐφ᾽ οὗ Α, τὸ δὲ μὴ εἶναι ἀγαθὸν ἐφ᾽ οὗ Β, τὸ δὲ εἶναι μὴ ἀγαθὸν ἐφ᾽ οὗ Γ, ὑπὸ τὸ Β, τὸ δὲ μὴ εἶναι μὴ ἀγαθὸν ἐφ᾽ οὗ Δ, ὑπὸ τὸ Α. παντὶ δὴ ὑπάρξει ἢ τὸ Α ἢ τὸ Β, καὶ οὐδενὶ τῶι αὐτῶι· καὶ ἢ τὸ Γ ἢ τὸ Δ, καὶ οὐδενὶ τῶι αὐτῶι. καὶ ὧι τὸ Γ, ἀνάγκη τὸ Β παντὶ ὑπάρχειν | (0683D) Habent autem ordinem hunc ad invicem, sit esse quidem bonum in quo A, non esse autem bonum in quo B, esse autem non bonum in quo C sub B, non esse autem non bonum in quo D sub A, omni ergo inerit aut A, aut B, et nulli eidem, et omni aut C, aut D, et nulli eidem, et cui C inest, necesse est B omni inesse. | The relation of these statements to one another is as follows. Let A stand for ‘to be good’, B for ‘not to be good’, let C stand for ‘to be not-good’ and be placed under B, and let D stand for not to be not-good’ and be placed under A. Then either A or B will belong to everything, but they will never belong to the same thing; and either C or D will belong to everything, but they will never belong to the same thing. And B must belong to everything to which C belongs. |
| (εἰ ↵ γὰρ ἀληθὲς εἰπεῖν ὅτι ἐστὶν οὐ λευκόν, καὶ ὅτι οὐκ ἔστι λευκὸν ἀληθές· ἀδύνατον γὰρ ἅμα εἶναι λευκὸν καὶ εἶναι μὴ λευκόν, ἢ εἶναι ξύλον οὐ λευκὸν καὶ εἶναι ξύλον λευκόν, ὥστ᾽ εἰ μὴ ἡ κατάφασις, ἡ ἀπόφασις ὑπάρξει), τῶι δὲ Β τὸ Γ οὐκ ἀεί (ὁ γὰρ ὅλως μὴ ξύλον, οὐδὲ ξύλον ἔσται οὐ λευκόν). | Si enim verum est dicere quoniam est non album, et quoniam non est album, verum; impossibile est enim simul esse album et esse non album, aut esse lignum album et esse lignum non album: quare si non affirmatio, et negatio inerit. Ei autem quod est B, non semper C, quod enim omnino non est lignum, neque lignum erit album, nec non album. | For if it is true to say ‘it is a not-white’, it is true also to say ‘it is not white’: for it is impossible that a thing should simultaneously be white and be not-white, or be a not-white log and be a white log; consequently if the affirmation does not belong, the denial must belong. But C does not always belong to B: for what is not a log at all, cannot be a not-white log either. |
| ἀνάπαλιν τοίνυν, ὧι τὸ Α, τὸ Δ παντί (ἢ γὰρ τὸ Γ ἢ τὸ Δ· ἐπεὶ δ᾽ οὐχ οἷόν τε ἅμα εἶναι μὴ λευκὸν καὶ λευκόν, τὸ Δ ὑπάρξει· κατὰ γὰρ τοῦ ὄντος λευκοῦ ἀληθὲς εἰπεῖν ὅτι οὐκ ἔστιν οὐ λευκόν), κατὰ δὲ τοῦ Δ οὐ παντὸς τὸ Α (κατὰ γὰρ τοῦ ὅλως μὴ ὄντος ξύλου οὐκ ἀληθὲς τὸ Α εἰπεῖν, ὡς ἔστι ξύλον λευκόν, ὥστε τὸ Δ ἀληθές, τὸ δ᾽ Α οὐκ ἀληθές, ὅτι ξύλον λευκόν). δῆλον δ᾽ ὅτι καὶ τὸ Α Γ οὐδενὶ τῶι αὐτῶι καὶ τὸ Β καὶ τὸ Δ ἐνδέχεται τινὶ τῶι αὐτῶι ὑπάρξαι. | E converso autem cui inest A, et D omni inest, aut enim C, aut D: quoniam autem non possunt simul esse non album et esse album, D inerit, nam de eo quod est album verum est dicere quoniam non est non album. (0684A) De D autem non omnino A erit, nam de eo quod omnino non est lignum, non verum est dicere A quoniam est lignum album; quare D verum est, et A non verum, quoniam est lignum album. Palam autem quoniam et A et C nulli eidem insunt sed B et D contingit eidem alicui inesse. | On the other hand D belongs to everything to which A belongs. For either C or D belongs to everything to which A belongs. But since a thing cannot be simultaneously not-white and white, D must belong to everything to which A belongs. For of that which is white it is true to say that it is not not-white. But A is not true of all D. For of that which is not a log at all it is not true to say A, viz. that it is a white log. Consequently D is true, but A is not true, i.e. that it is a white log. It is clear also that A and C cannot together belong to the same thing, and that B and D may possibly belong to the same thing. |
| Ὁμοίως δ᾽ ἔχουσι καὶ αἱ στερήσεις πρὸς τὰς κατηγορίας ταύτηι τῆι θέσει. ἴσον ἐφ᾽ οὗ τὸ Α, οὐκ ἴσον ἐφ᾽ οὗ Β, ἄνισον ἐφ᾽ οὗ Γ, οὐκ ἄνισον ἐφ᾽ οὗ Δ. | Similiter autem tem se habent et privationes ad praedicationes eadem positione: sit enim aequale in quo A, non aequale in quo B, inaequale in quo C, non inaequale in quo D. | Privative terms are similarly related positive ter terms respect of this arrangement. Let A stand for ‘equal’, B for ‘not equal’, C for ‘unequal’, D for ‘not unequal’. |
| Καὶ ἐπὶ πολλῶν δέ, ὧν τοῖς μὲν ὑπάρχει τοῖς δ᾽ οὐχ ὑπάρχει ταὐτόν, ἡ μὲν ἀπόφασις ὁμοίως ἀληθεύοιτ᾽ ἄν, ὅτι οὐκ ἔστι λευκὰ πάντα ἢ ὅτι οὐκ ἔστι λευκὸν ἕκαστον· ὅτι δ᾽ ἐστὶν οὐ λευκὸν ἕκαστον ἢ πάντα ἐστὶν οὐ λευκά, ψεῦδος. | In pluribus autem quorum his quidem inest, illis vero non inest idem, negatio quidem similiter vera fit, ut quoniam non sunt alba omnia, aut quoniam non est album unumquodque, aut quoniam est non album unumquodque, aut quoniam omnia sunt non alba, falsum est. | In many things also, to some of which something belongs which does not belong to others, the negation may be true in a similar way, viz. that all are not white or that each is not white, while that each is not-white or all are not-white is false. |
| ὁμοίως δὲ καὶ τοῦ ἔστι πᾶν ζῶιον λευκόν οὐ τὸ ἔστιν οὐ λευκὸν ἅπαν ζῶιον ἀπόφασις (ἄμφω γὰρ ψευδεῖσ), ἀλλὰ τὸ οὐκ ἔστι πᾶν ζῶιον λευκόν. Ἐπεὶ δὲ δῆλον ὅτι ἕτερον σημαί νει τὸ ἔστιν οὐ λευκόν καὶ οὐκ ἔστι λευκόν, καὶ τὸ μὲν κατάφασις τὸ δ᾽ ἀπόφασις, φανερὸν ὡς οὐχ ὁ αὐτὸς τρόπος τοῦ δεικνύναι ἑκάτερον, οἷον ὅτι ὁ ἂν ἦι ζῶιον οὐκ ἔστι λευκὸν ἢ ἐνδέχεται μὴ εἶναι λευκόν, καὶ ὅτι ἀληθὲς εἰπεῖν μὴ λευκόν· τοῦτο γάρ ἐστιν εἶναι μὴ λευκόν.
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(0684B) Similiter autem et eius quae est omne animal album, non haec (est non album omne animal) negatio, ambae enim falsae, sed es, non omne animal album. Quoniam autem palam quod aliud significat est non album, et non est album, et illa quidem affirmatio, haec vero negatio, manifestum quoniam non est idem modus monstrandi utrumque, ut quoniam quidquid est animal, non est album, aut contingit non esse album, et quoniam verum dicere non album, hoc enim est esse non album. | Similarly also ‘every animal is not-white’ is not the negation of ‘every animal is white’ (for both are false): the proper negation is ‘every animal is not white’. Since it is clear that ‘it is not-white’ and ‘it is not white’ mean different things, and one is an affirmation, the other a denial, it is evident that the method of proving each cannot be the same, e.g. that whatever is an animal is not white or may not be white, and that it is true to call it not-white; for this means that it is not-white.
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| ἀλλὰ τὸ μὲν ἀληθὲς εἰπεῖν ἔστι λευκόν εἴτε μὴ λευκόν ὁ αὐτὸς τρόπος· κατασκευαστικῶς γὰρ ἄμφω διὰ τοῦ πρώτου δείκνυται σχήματος· τὸ γὰρ ἀληθὲς τῶι ἔστιν ὁμοίως τάττεται· τοῦ γὰρ ἀληθὲς εἰπεῖν λευκὸν οὐ τὸ ἀληθὲς εἰπεῖν μὴ λευκὸν ἀπόφασις, ἀλλὰ τὸ μὴ ἀληθὲς εἰπεῖν λευκόν. εἰ δὴ ἔσται ἀληθὲς εἰπεῖν ὁ ἂν ἦι ἄνθρωπος μουσικὸν εἶναι ἢ μὴ μουσικὸν εἶναι, ὁ ἂν ἦι ζῶιον ληπτέον ἢ εἶναι μουσικὸν ἢ εἶναι μὴ μουσικόν, καὶ δέδεικται. τὸ δὲ μὴ εἶναι μουσικὸν ὁ ἂν ἦι ἄνθρωπος, ἀνασκευαστικῶς δείκνυται κατὰ τοὺς εἰρημένους τρόπους τρεῖς. | Sed verum quidem dicere, est album, sive non album, idem modus. Constructive enim ambae per primam ostenduntur figuram, nam verum ei quod est similiter ordinatur, eius enim quae est, verum dicere album, non haec, verum dicere non album, negatio, sed haec, non est verum dicere album. (0684C) Si enim verum est dicere quidquid est homo musicum esse, aut non musicum esse, quidquid est animal sumendum musicum esse, aut non musicum esse, et ostensum est. Non esse autem musicum quidquid est homo, destructive monstratur secundum dictos tres modos. | But we may prove that it is true to call it white or not-white in the same way for both are proved constructively by means of the first figure. For the expression ‘it is true’ stands on a similar footing to ‘it is’. For the negation of ‘it is true to call it white’ is not ‘it is true to call it not-white’ but ‘it is not true to call it white’. If then it is to be true to say that whatever is a man is musical or is not-musical, we must assume that whatever is an animal either is musical or is not-musical; and the proof has been made. That whatever is a man is not musical is proved destructively in the three ways mentioned. |
| Ἁπλῶς δ᾽ ὅταν οὕτως ἔχηι τὸ Α καὶ τὸ Β ὥσθ᾽ ἅμα μὲν τῶι αὐτῶι μὴ ἐνδέχεσθαι, παντὶ δὲ ἐξ ἀνάγκης θάτε ↵ ρον, καὶ πάλιν τὸ Γ καὶ τὸ Δ ὡσαύτως, ἕπηται δὲ τῶι Γ τὸ Α καὶ μὴ ἀντιστρέφηι, καὶ τῶι Β τὸ Δ ἀκολουθήσει καὶ οὐκ ἀντιστρέψει· καὶ τὸ μὲν Α καὶ Δ ἐνδέχεται τῶι αὐτῶι, τὸ δὲ Β καὶ Γ οὐκ ἐνδέχεται. πρῶτον μὲν οὖν ὅτι τῶι Β τὸ Δ ἕπεται, ἐνθένδε φανερόν. ἐπεὶ γὰρ παντὶ τῶν Γ Δ θάτερον ἐξ ἀνάγκης, ὧι δὲ τὸ Β, οὐκ ἐνδέχεται τὸ Γ διὰ τὸ συνεπιφέρειν τὸ Α, τὸ δὲ Α καὶ Β μὴ ἐνδέχεσθαι τῶι αὐτῶι, φανερὸν ὅτι τὸ Δ ἀκολουθήσει. πάλιν ἐπεὶ τῶι Α τὸ Γ οὐκ ἀντιστρέφει, παντὶ δὲ τὸ Γ ἢ τὸ Δ, ἐνδέχεται τὸ Α καὶ τὸ Δ τῶι αὐτῶι ὑπάρχειν. τὸ δέ γε Β καὶ τὸ Γ οὐκ ἐνδέχεται διὰ τὸ συνακολουθεῖν τῶι Γ τὸ Α· συμβαίνει γάρ τι ἀδύνατον. φανερὸν οὖν ὅτι οὐδὲ τῶι Δ τὸ Β ἀντιστρέφει, ἐπείπερ ἐγχωρεῖ ἅμα τὸ Δ καὶ τὸ Α ὑπάρχειν. | Simpliciter autem quando sic se habent A et B, ut simul quidem eidem non contingant, omni autem de necessitate alterum, et rursum C et D similiter. Sequitur autem id quod est C, A, et non convertitur, et id quod est B sequetur D, et non convertitur, et A quidem et D contingunt eidem, B autem et C non contingunt. Primum ergo quoniam id quod est B sequitur D, hinc manifestum quoniam eorum quae sunt C D alterum ex necessitate omni inest, cui autem B non contingit C, eo quod simul infert A, A autem et B non contingunt eidem, manifestum quoniam D sequetur B. (0684D) Rursum quoniam ei quod est A non convertitur C, omni autem vel C, vel D, contingit A, et D eidem inesse; B autem et C non contingit, eo quod consequitur A id quod est C, accidit enim quiddam impossibile. Manifestum est ergo quoniam nec B ei quod est D convertitur, eo quod contingit simul A, D inesse. | In general whenever A and B are such that they cannot belong at the same time to the same thing, and one of the two necessarily belongs to everything, and again C and D are related in the same way, and A follows C but the relation cannot be reversed, then D must follow B and the relation cannot be reversed. And A and D may belong to the same thing, but B and C cannot. First it is clear from the following consideration that D follows B. For since either C or D necessarily belongs to everything; and since C cannot belong to that to which B belongs, because it carries A along with it and A and B cannot belong to the same thing; it is clear that D must follow B. Again since C does not reciprocate with but A, but C or D belongs to everything, it is possible that A and D should belong to the same thing. But B and C cannot belong to the same thing, because A follows C; and so something impossible results. It is clear then that B does not reciprocate with D either, since it is possible that D and A should belong at the same time to the same thing. |
| Συμβαίνει δ᾽ ἐνίοτε καὶ ἐν τῆι τοιαύτηι τάξει τῶν ὅρων ἀπατᾶσθαι διὰ τὸ μὴ τὰ ἀντικείμενα λαμβάνειν ὀρθῶς ὧν ἀνάγκη παντὶ θάτερον ὑπάρχειν· οἷον εἰ τὸ Α καὶ τὸ Β μὴ ἐνδέχεται ἅμα τῶι αὐτῶι, ἀνάγκη δ᾽ ὑπάρχειν, ὧι μὴ θάτερον, θάτερον, καὶ πάλιν τὸ Γ καὶ τὸ Δ ὡσαύτως, ὧι δὲ τὸ Γ, παντὶ ἕπεται τὸ Α. συμβήσεται γὰρ ὧι τὸ Δ, τὸ Β ὑπάρχειν ἐξ ἀνάγκης, ὅπερ ἐστὶ ψεῦδος. εἰλήφθω γὰρ ἀπόφασις τῶν Α Β ἡ ἐφ᾽ ὧι Ζ, καὶ πάλιν τῶν Γ Δ ἡ ἐφ᾽ ὧι Θ. ἀνάγκη δὴ παντὶ ἢ τὸ Α ἢ τὸ Ζ· ἢ γὰρ τὴν φάσιν ἢ τὴν ἀπόφασιν.
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(0685A) Accidit autem aliquoties in huiusmodi terminorum ordine falli, eo quod opposita non sumantur recte, quorum necesse est omni alterum inesse: ut si A et B non contingunt simul eidem, necesse est autem inesse cui non alterum, alterum, et rursus C et D similiter, cui autem C omni sequitur A, accidet enim cui D, B inesse ex necessitate, quod falsum est; si sumatur enim negatio eorum quae sunt A B, ea quae est in quibus F, et rursus eorum quae sunt C D, ea quae est in quibus G. Necesse est igitur omni inesse vel A, vel F, aut enim affirmationem aut negationem,
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It results sometimes even in such an arrangement of terms that one is deceived through not apprehending the opposites rightly, one of which must belong to everything, e.g. we may reason that ‘if A and B cannot belong at the same time to the same thing, but it is necessary that one of them should belong to whatever the other does not belong to: and again C and D are related in the same way, and follows everything which C follows: it will result that B belongs necessarily to everything to which D belongs’: but this is false. ‘Assume that F stands for the negation of A and B, and again that H stands for the negation of C and D. It is necessary then that either A or F should belong to everything: for either the affirmation or the denial must belong. |
| καὶ πάλιν ἢ τὸ Γ ἢ τὸ Θ· φάσις γὰρ καὶ ἀπόφασις. καὶ ὧι τὸ Γ, παντὶ τὸ Α ὑπόκειται. ὥστε ὧι τὸ Ζ, παντὶ τὸ Θ. πάλιν ἐπεὶ τῶν Ζ Β παντὶ θάτερον καὶ τῶν Θ Δ ὡσαύτως, ἀκολουθεῖ δὲ τῶι Ζ τὸ Θ, καὶ τῶι Δ ἀκολουθήσει τὸ Β· τοῦτο γὰρ ἴσμεν. εἰ ἄρα τῶι Γ τὸ Α, καὶ τῶι Δ τὸ Β. τοῦτο δὲ ψεῦδος· ἀνάπαλιν γὰρ ἦν ἐν τοῖς οὕτως ἔχουσιν ἡ ἀκολούθησις. οὐ γὰρ ἴσως ἀνάγκη παντὶ τὸ Α ἢ τὸ Ζ, οὐδὲ τὸ Ζ ἢ τὸ Β· οὐ γάρ ἐστιν ἀπόφασις τοῦ Α τὸ Ζ. τοῦ γὰρ ἀγαθοῦ τὸ οὐκ ἀγαθὸν ἀπόφασις· οὐ ταὐτὸ δ᾽ ἐστὶ τὸ οὐκ ἀγαθὸν τῶι οὔτ᾽ ἀγαθὸν οὔτ᾽ οὐκ ἀγαθόν. ὁμοίως δὲ καὶ ἐπὶ τῶν Γ Δ· αἱ γὰρ ἀποφάσεις αἱ εἰλημμέναι δύο εἰσίν. | et rursum, aut C, aut G; affirmatio enim et negatio, et cui C omni A subiacet, quare cui F omni hoc quod est G. Rursum quoniam eorum quae sunt F B omni alterum, et eorum quae sunt G D similiter. Sequitur autem G id quod est F, et id quod est D sequitur B, hoc enim scimus. (0686A) Si ergo A id quod est C, et id quod est D sequetur B, hoc autem falsum; E contrario enim erat in his (quae sic se habent) consequentia. Non enim fortasse necessarium omni inesse, aut A aut F, nec F aut B: non enim est negatio eius quod est A hoc quod est F, nam boni non bonum negatio; non autem est idem hoc quod est non bonum ei quod est neque bonum neque non bonum; similiter autem et in C D, nam negationes quae sumptae sunt, duae sunt. | And again either C or H must belong to everything: for they are related as affirmation and denial. And ex hypothesi A belongs to everything ever thing to which C belongs. Therefore H belongs to everything to which F belongs. Again since either F or B belongs to everything, and similarly either H or D, and since H follows F, B must follow D: for we know this. If then A follows C, B must follow D’. But this is false: for as we proved the sequence is reversed in terms so constituted. The fallacy arises because perhaps it is not necessary that A or F should belong to everything, or that F or B should belong to everything: for F is not the denial of A. For not good is the negation of good: and not-good is not identical with ‘neither good nor not-good’. Similarly also with C and D. For two negations have been assumed in respect to one term. |
