Authors/Aristotle/priora/Liber 2/C11

Chapter 11

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61a17 Τί μὲν οὖν ἐστὶ τὸ ἀντιστρέφειν καὶ πῶς ἐν ἑκάστωι σχήματι καὶ τίς γίνεται συλλογισμός, φανερόν. ὁ δὲ διὰ τοῦ ἀδυνάτου συλλογισμὸς δείκνυται μὲν ὅταν ἡ ἀντίφασις τεθῆι τοῦ συμπεράσματος καὶ προσληφθῆι ἄλλη πρότασις, γίνεται δ᾽ ἐν ἅπασι τοῖς σχήμασιν· ὅμοιον γάρ ἐστι τῆι ἀντιστροφῆι, πλὴν διαφέρει τοσοῦτον ὅτι ἀντιστρέφεται μὲν γεγενημένου συλλογισμοῦ καὶ εἰλημμένων ἀμφοῖν τῶν προτάσεων, ἀπάγεται δ᾽ εἰς ἀδύνατον οὐ προομολογηθέντος τοῦ ἀντικειμένου πρότερον, ἀλλὰ φανεροῦ ὄντος ὅτι ἀληθές. οἱ δ᾽ ὅροι ὁμοίως ἔχουσιν ἐν ἀμφοῖν, καὶ ἡ αὐτὴ λῆψις ἀμφοτέρων. οἷον εἰ τὸ Α τῶι Β παντὶ ὑπάρχει, μέσον δὲ τὸ Γ, ἐὰν ὑποτεθῆι τὸ Α ἢ μὴ παντὶ ἢ μηδενὶ τῶι Β ὑπάρχειν, τῶι δὲ Γ παντί, ὅπερ ἦν ἀληθές, ἀνάγκη τὸ Γ τῶι Β ἢ μηδενὶ ἢ μὴ παντὶ ὑπάρχειν. τοῦτο δ᾽ ἀδύνατον, ὥστε ψεῦδος τὸ ὑποτεθέν· ἀληθὲς ἄρα τὸ ἀντικείμενον. ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων σχημάτων· ὅσα γὰρ ἀντιστροφὴν δέχεται, καὶ τὸν διὰ τοῦ ἀδυνάτου συλλογισμόν.	Quid ergo est convertere, et quomodo in unaquaque figura, et quis fit syllogismus, manifestum. (PL 64 0697C) [CAPUT XI. De syllogismo per impossibile]. Per impossibile autem syllogismus ostenditur quidem, quando contradictio ponitur conclusionis, et assumitur altera propositio. Fit autem in omnibus figuris, simile enim est conversioni. Verumtamen differt in tantum quoniam convertitur quidem facto syllogismo, et sumptis utrisque propositionibus. Deducitur autem ad impossibile non confesso opposito prius, sed manifesto quoniam est verum. (0697D) Termini vero similiter se habent in utrisque, et eadem sumptio utrorumque, ut si A inest omni B, medium autem C, si supponitur A non omni vel nulli B inesse, C vero omni, quod fuit verum, necesse est C B aut nulli aut non omni inesse, hoc autem impossibile, quare falsum est quod suppositum est. Verum ergo oppositum; similiter autem in aliis figuris, quaecunque enim conversionem suscipiunt, et per impossibile syllogismum.	It is clear then what conversion is, how it is effected in each figure, and what syllogism results. The syllogism per impossibile is proved when the contradictory of the conclusion stated and another premiss is assumed; it can be made in all the figures. For it resembles conversion, differing only in this: conversion takes place after a syllogism has been formed and both the premisses have been taken, but a reduction to the impossible takes place not because the contradictory has been agreed to already, but because it is clear that it is true. The terms are alike in both, and the premisses of both are taken in the same way. For example if A belongs to all B, C being middle, then if it is supposed that A does not belong to all B or belongs to no B, but to all C (which was admitted to be true), it follows that C belongs to no B or not to all B. But this is impossible: consequently the supposition is false: its contradictory then is true. Similarly in the other figures: for whatever moods admit of conversion admit also of the reduction per impossibile.
Τὰ μὲν οὖν ἄλλα προβλήματα πάντα δείκνυται διὰ τοῦ ἀδυνάτου ἐν ἅπασι τοῖς σχήμασι, τὸ δὲ καθόλου κατηγορικὸν ἐν μὲν τῶι μέσωι καὶ τῶι τρίτωι δείκνυται, ἐν δὲ τῶι πρώτωι οὐ δείκνυται. ὑποκείσθω γὰρ τὸ Α τῶι Β μὴ παντὶ ἢ μηδενὶ ὑπάρχειν, καὶ προσειλήφθω ἄλλη πρότασις ὁποτερωθενοῦν, εἴτε τῶι Α παντὶ ὑπάρχειν τὸ Γ εἴτε τὸ Β παντὶ τῶι Δ· οὕτω γὰρ ἂν εἴη τὸ πρῶτον σχῆμα. εἰ μὲν οὖν ὑπόκειται μὴ παντὶ ὑπάρχειν τὸ Α τῶι Β, οὐ γίνεται συλλο↵γισμὸς ὁποτερωθενοῦν τῆς προτάσεως λαμβανομένης,	(0698A) Ergo alia quidem proposita omnia ostenduntur per impossibile in omnibus figuris, universale autem praedicativum in media et in tertia monstratur, in prima autem non monstratur: supponatur enim A non omni B aut nulli inesse, et assumatur alia propositio, utrolibet modo, sive A omni inest C, sive B omni D (sic enim erat prima figura); si ergo supponatur A non omni B inesse, non fiet syllogismus quomodolibet sumpta propositione.	All the problems can be proved per impossibile in all the figures, excepting the universal affirmative, which is proved in the middle and third figures, but not in the first. Suppose that A belongs not to all B, or to no B, and take besides another premiss concerning either of the terms, viz. that C belongs to all A, or that B belongs to all D; thus we get the first figure. If then it is supposed that A does not belong to all B, no syllogism results whichever term the assumed premiss concerns; but if it is supposed that A belongs to no B, when the premiss BD is assumed as well we shall prove syllogistically what is false, but not the problem proposed.
εἰ δὲ μηδενί, ὅταν μὲν ἡ Β Δ προσληφθῆι, συλλογισμὸς μὲν ἔσται τοῦ ψεύδους, οὐ δείκνυται δὲ τὸ προκείμενον. εἰ γὰρ τὸ Α μηδενὶ τῶι Β, τὸ δὲ Β παντὶ τῶι Δ, τὸ Α οὐδενὶ τῶι Δ. τοῦτο δ᾽ ἔστω ἀδύνατον· ψεῦδος ἄρα τὸ μηδενὶ τῶι Β τὸ Α ὑπάρχειν. ἀλλ᾽ οὐκ εἰ τὸ μηδενὶ ψεῦδος, τὸ παντὶ ἀληθές. ἐὰν δ᾽ ἡ Γ Α προσληφθῆι, οὐ γίνεται συλλογισμός, οὐδ᾽ ὅταν ὑποτεθῆι μὴ παντὶ τῶι Β τὸ Α ὑπάρχειν. ὥστε φανερὸν ὅτι τὸ παντὶ ὑπάρχειν οὐ δείκνυται ἐν τῶι πρώτωι σχήματι διὰ τοῦ ἀδυνάτου.	Si autem nulli B, D quidem assumatur, syllogismus quidem erit falsi, non ostenditur autem propositum; nam si A nulli B, B autem omni D, A nulli D, hoc autem sit impossibile, falsum igitur est nulli B inesse A, sed non si nulli falsum, omni verum. Si autem C A assumatur, non fit syllogismus, nec quando supponitur non omni B inesse A; quare manifestum quoniam omni inesse non ostenditur in prima figura per impossibile.	For if A belongs to no B, and B belongs to all D, A belongs to no D. Let this be impossible: it is false then A belongs to no B. But the universal affirmative is not necessarily true if the universal negative is false. But if the premiss CA is assumed as well, no syllogism results, nor does it do so when it is supposed that A does not belong to all B. Consequently it is clear that the universal affirmative cannot be proved in the first figure per impossibile.
Τὸ δέ γε τινὶ καὶ τὸ μηδενὶ καὶ μὴ παντὶ δείκνυται. ὑποκείσθω γὰρ τὸ Α μηδενὶ τῶι Β ὑπάρχειν, τὸ δὲ Β εἰλήφθω παντὶ ἢ τινὶ τῶι Γ. οὐκοῦν ἀνάγκη τὸ Α μηδενὶ ἢ μὴ παντὶ τῶι Γ ὑπάρχειν. τοῦτο δ᾽ ἀδύνατον – ἔστω γὰρ ἀληθὲς καὶ φανερὸν ὅτι παντὶ ὑπάρχει τῶι Γ τὸ Α – ὥστ᾽ εἰ τοῦτο ψεῦδος, ἀνάγκη τὸ Α τινὶ τῶι Β ὑπάρχειν. ἐὰν δὲ πρὸς τῶι Α ληφθῆι ἡ ἑτέρα πρότασις, οὐκ ἔσται συλλογισμός. οὐδ᾽ ὅταν τὸ ἐναντίον τῶι συμπεράσματι ὑποτεθῆι, οἷον τὸ τινὶ μὴ ὑπάρχειν. φανερὸν οὖν ὅτι τὸ ἀντικείμενον ὑποθετέον.	Alicui autem, et nulli, et non omni ostenditur. Supponatur enim A nulli B inesse, B autem sumptum sit omni aut alicui C, ergo necesse est A nulli aut non omni C inesse, hoc autem impossibile. Sit enim verum et manifestum quoniam omni C inest A, quare si hoc falsum, necesse est A alicui B inesse. (0698B) Si autem ad A sumatur altera propositio, non erit syllogismus, neque quando subcontrarium conclusioni supponitur ut alicui non inesse; manifestum ergo quoniam oppositum sumendum est.	But the particular affirmative and the universal and particular negatives can all be proved. Suppose that A belongs to no B, and let it have been assumed that B belongs to all or to some C. Then it is necessary that A should belong to no C or not to all C. But this is impossible (for let it be true and clear that A belongs to all C): consequently if this is false, it is necessary that A should belong to some B. But if the other premiss assumed relates to A, no syllogism will be possible. Nor can a conclusion be drawn when the contrary of the conclusion is supposed, e.g. that A does not belong to some B. Clearly then we must suppose the contradictory.
Πάλιν ὑποκείσθω τὸ Α τινὶ τῶι Β ὑπάρχειν, εἰλήφθω δὲ τὸ Γ παντὶ τῶι Α. ἀνάγκη οὖν τὸ Γ τινὶ τῶι Β ὑπάρχειν. τοῦτο δ᾽ ἔστω ἀδύνατον, ὥστε ψεῦδος τὸ ὑποτεθέν. εἰ δ᾽ οὕτως, ἀληθὲς τὸ μηδενὶ ὑπάρχειν. ὁμοίως δὲ καὶ εἰ στερητικὸν ἐλήφθη τὸ Γ Α. εἰ δ᾽ ἡ πρὸς τῶι Β εἴληπται πρότασις, οὐκ ἔσται συλλογισμός. ἐὰν δὲ τὸ ἐναντίον ὑποτεθῆι, συλλογισμὸς μὲν ἔσται καὶ τὸ ἀδύνατον, οὐ δείκνυται δὲ τὸ προτεθέν. ὑποκείσθω γὰρ παντὶ τῶι Β τὸ Α ὑπάρχειν, καὶ τὸ Γ τῶι Α εἰλήφθω παντί. οὐκοῦν ἀνάγκη τὸ Γ παντὶ τῶι Β ὑπάρχειν. τοῦτο δ᾽ ἀδύνατον, ὥστε ψεῦδος τὸ παντὶ τῶι Β τὸ Α ὑπάρχειν. ἀλλ᾽ οὔπω γε ἀναγκαῖον, εἰ μὴ παντί, μηδενὶ ὑπάρχειν. ὁμοίως δὲ καὶ εἰ πρὸς τῶι Β ληφθείη ἡ ἑτέρα πρότασις· συλλογισμὸς μὲν γὰρ ἔσται καὶ τὸ ἀδύνατον, οὐκ ἀναιρεῖται δ᾽ ἡ ὑπόθεσις· ὥστε τὸ ἀντικείμενον ὑποθετέον.	Rursum supponatur A alicui B inesse, sumptum autem sit C omni A, necesse est igitur C alicui B inesse, hoc autem sit impossibile, quare falsum quidem suppositum est; si autem sic, verum est nulli inesse. Similiter autem et si privativa sumpta sit C A. Si autem ad B sumpta sit propositio, non erit syllogismus. Si autem contrarium supponatur, syllogismus erit et impossibile, non tamen ostenditur quod est propositum: supponatur enim A omni B, et C sumptum sit omni A, ergo necesse est C omni B inesse: hoc autem impossibile, quare falsum est omni B inesse A, sed nondum erit necessarium, si non omni, nulli inesse. (0698C) Similiter autem et si A D B sumatur altera propositio: nam syllogismus quidem erit et impossibile, non interimitur autem hypothesis, quare oppositum supponendum.	Again suppose that A belongs to some B, and let it have been assumed that C belongs to all A. It is necessary then that C should belong to some B. But let this be impossible, so that the supposition is false: in that case it is true that A belongs to no B. We may proceed in the same way if the proposition CA has been taken as negative. But if the premiss assumed concerns B, no syllogism will be possible. If the contrary is supposed, we shall have a syllogism and an impossible conclusion, but the problem in hand is not proved. Suppose that A belongs to all B, and let it have been assumed that C belongs to all A. It is necessary then that C should belong to all B. But this is impossible, so that it is false that A belongs to all B. But we have not yet shown it to be necessary that A belongs to no B, if it does not belong to all B. Similarly if the other premiss taken concerns B; we shall have a syllogism and a conclusion which is impossible, but the hypothesis is not refuted. Therefore it is the contradictory that we must suppose.
Πρὸς δὲ τὸ μὴ παντὶ δεῖξαι ὑπάρχον τῶι Β τὸ Α, ὑποθετέον παντὶ ὑπάρχειν· εἰ γὰρ τὸ Α παντὶ τῶι Β καὶ τὸ Γ παντὶ τῶι Α, τὸ Γ παντὶ τῶι Β, ὥστ᾽ εἰ τοῦτο ἀδύνατον, ψεῦδος τὸ ὑποτεθέν. ὁμοίως δὲ καὶ εἰ πρὸς τῶι Β ἐλήφθη ἡ ἑτέρα πρότασις. καὶ εἰ στερητικὸν ἦν τὸ Γ Α, ὡσαύτως· καὶ γὰρ οὕτω γίνεται συλλογισμός. ἐὰν δὲ πρὸς τῶι Β ἦι τὸ στερητικόν, οὐδὲν δείκνυται. ἐὰν δὲ μὴ παντὶ ἀλλὰ τινὶ ὑπάρχειν ὑποτεθῆι, οὐ δείκνυται ὅτι οὐ παντὶ ἀλλ᾽ ὅτι οὐδενί. εἰ γὰρ τὸ Α τινὶ τῶι Β, τὸ δὲ Γ παντὶ τῶι Α, τινὶ ↵ τῶι Β τὸ Γ ὑπάρξει. εἰ οὖν τοῦτ᾽ ἀδύνατον, ψεῦδος τὸ τινὶ ὑπάρχειν τῶι Β τὸ Α, ὥστ᾽ ἀληθὲς τὸ μηδενί. τούτου δὲ δειχθέντος προσαναιρεῖται τὸ ἀληθές· τὸ γὰρ Α τῶι Β τινὶ μὲν ὑπῆρχε, τινὶ δ᾽ οὐχ ὑπῆρχεν.	Ad ostendendum autem non omni B inesse A, supponendum omni inesse, nam si A omni B, et C omni A, omni B inerit C; si ergo hoc impossibile, falsum quod suppositum est; similiter autem et si ad B sumpta sit altera propositio. Et si privativa sit C A, similiter, nam et sic fit syllogismus. Si autem ad B sumpta sit privativa, nihil ostenditur. (0698D) Si autem non omni, sed alicui inesse supponatur, non ostenditur quoniam non omni, sed quoniam nulli: si enim A alicui B, C autem omni A, alicui B inerit C; si ergo hoc impossibile, falsum est alicui B inesse A, quare verum nulli; hoc autem ostenso, interimitur verum, nam A alicui quidem B inerat, alicui vero non inerat.	To prove that A does not belong to all B, we must suppose that it belongs to all B: for if A belongs to all B, and C to all A, then C belongs to all B; so that if this is impossible, the hypothesis is false. Similarly if the other premiss assumed concerns B. The same results if the original proposition CA was negative: for thus also we get a syllogism. But if the negative proposition concerns B, nothing is proved. If the hypothesis is that A belongs not to all but to some B, it is not proved that A belongs not to all B, but that it belongs to no B. For if A belongs to some B, and C to all A, then C will belong to some B. If then this is impossible, it is false that A belongs to some B; consequently it is true that A belongs to no B. But if this is proved, the truth is refuted as well; for the original conclusion was that A belongs to some B, and does not belong to some B.
ἔτι οὐδὲν παρὰ τὴν ὑπόθεσιν συμβαίνει [τὸ] ἀδύνατον· ψεῦδος γὰρ ἂν εἴη, εἴπερ ἐξ ἀληθῶν μὴ ἔστι ψεῦδος συλλογίσασθαι· νῦν δ᾽ ἐστὶν ἀληθές· ὑπάρχει γὰρ τὸ Α τινὶ τῶι Β. ὥστ᾽ οὐχ ὑποθετέον τινὶ ὑπάρχειν, ἀλλὰ παντί. ὁμοίως δὲ καὶ εἰ τινὶ μὴ ὑπάρχον τῶι Β τὸ Α δεικνύοιμεν· εἰ γὰρ ταὐτὸ τὸ τινὶ μὴ ὑπάρχειν καὶ μὴ παντὶ ὑπάρχειν, ἡ αὐτὴ ἀμφοῖν ἀπόδειξις.	Amplius autem non tam propter hypothesin accidit impossibile, falsa enim erit, siquidem ex veris non est falsum syllogizare: nunc autem est vera, inest enim A alicui B, quare non supponendum alicui inesse, sed omni. Similiter autem et si alicui B non inest A, ostenderemus; si enim idem est alicui non inesse, et non omni inesse, eadem in utrisque demonstratio.	Further the impossible does not result from the hypothesis: for then the hypothesis would be false, since it is impossible to draw a false conclusion from true premisses: but in fact it is true: for A belongs to some B. Consequently we must not suppose that A belongs to some B, but that it belongs to all B. Similarly if we should be proving that A does not belong to some B: for if ‘not to belong to some’ and ‘to belong not to all’ have the same meaning, the demonstration of both will be identical.
Φανερὸν οὖν ὅτι οὐ τὸ ἐναντίον ἀλλὰ τὸ ἀντικείμενον ὑποθετέον ἐν ἅπασι τοῖς συλλογισμοῖς. οὕτω γὰρ τό τε ἀναγκαῖον ἔσται καὶ τὸ ἀξίωμα ἔνδοξον. εἰ γὰρ κατὰ παντὸς ἡ φάσις ἢ ἡ ἀπόφασις, δειχθέντος ὅτι οὐχ ἡ ἀπόφασις, ἀνάγκη τὴν κατάφασιν ἀληθεύεσθαι. πάλιν εἰ μὴ τίθησιν ἀληθεύεσθαι τὴν κατάφασιν, ἔνδοξον τὸ ἀξιῶσαι τὴν ἀπόφασιν. τὸ δ᾽ ἐναντίον οὐδετέρως ἁρμόττει ἀξιοῦν· οὔτε γὰρ ἀναγκαῖον, εἰ τὸ μηδενὶ ψεῦδος, τὸ παντὶ ἀληθές, οὔτ᾽ ἔνδοξον ὡς εἰ θάτερον ψεῦδος, ὅτι θάτερον ἀληθές.	(0699A) Manifestum ergo quoniam non contrarium, sed oppositum supponendum in omnibus syllogismis, sic enim necessarium erit et axioma probabile; nam si de omni vel affirmatio vel negatio, ostenso quoniam non negatio, necesse est affirmationem veram esse; rursum si non ponant veram esse affirmationem, constat veram esse negationem; contrariam vero neutro modo contingit ratum facere. enim necessarium, si nulli falsum, omni verum, neque probabile ut sit alterum falsum, quoniam alterum verum. Manifestum ergo quoniam in prima figura alia quidem proposita omnia ostenduntur per impossibile, universale autem affirmativum non ostenditur.	It is clear then that not the contrary but the contradictory ought to be supposed in all the syllogisms. For thus we shall have necessity of inference, and the claim we make is one that will be generally accepted. For if of everything one or other of two contradictory statements holds good, then if it is proved that the negation does not hold, the affirmation must be true. Again if it is not admitted that the affirmation is true, the claim that the negation is true will be generally accepted. But in neither way does it suit to maintain the contrary: for it is not necessary that if the universal negative is false, the universal affirmative should be true, nor is it generally accepted that if the one is false the other is true.