Authors/Thomas Aquinas/posteriorum/L1/Lect40

Lecture 40 Negative ostensive demonstration is stronger than demonstration leading to the impossible

Latin	English
	Lecture 40 (87a1-30) NEGATIVE OSTENSIVE DEMONSTRATION IS STRONGER THAN DEMONSTRATION LEADING TO THE IMPOSSIBLE
lib. 1 l. 40 n. 1 Postquam ostendit philosophus quod demonstratio universalis dignior est particulari, et affirmativa negativa, hic tertio ostendit quod ostensiva potior est ea quae ducit ad impossibile. Et circa hoc tria facit: primo, proponit quod intendit; secundo, praemittit quaedam necessaria ad propositum ostendendum; ibi: oportet autem scire etc.; tertio, probat propositum; ibi: natura autem prior et cetera.	After showing that universal demonstration ranks higher than particular, and affirmative higher than negative, the Philosopher here shows, thirdly, that ostensive demonstration is more powerful than that which leads to the impossible. Concerning this he does three things. First, he proposes what he intends. Secondly, he prefaces certain matters needed for showing the proposition (87a2). Thirdly, he proves his proposition (87a17).
Dicit ergo primo quod, quia ostensum est quod affirmativa demonstratio est potior quam negativa, ex hoc ulterius sequitur quod affirmativa demonstratio ostensiva sit potior ea quae ducit ad impossibile.	He says therefore first (87a1) that since we have shown that affirmative, demonstration is more powerful than negative, from this it further follows that an affirmative ostensive demonstration is more powerful than one which leads to the impossible.
lib. 1 l. 40 n. 2 Deinde cum dicit: oportet autem scire etc., praemittit quaedam quae sunt necessaria ad propositum ostendendum. Et circa hoc tria facit: primo, ostendit quae sit demonstratio negativa; secundo, ostendit quae sit demonstratio ducens ad impossibile; ibi: quae vero ad impossibile etc.; tertio, concludit comparationem unius ad alteram; ibi: termini igitur et cetera.	Then (87a2) he lays down certain things that are necessary for showing his proposition. In regard to this he does three things. First, he shows what a negative demonstration is. Secondly, he shows what a demonstration to the impossible is (87a7). Thirdly, he concludes the comparison of the one with the other (87a13).
Dicit ergo primo, quod ad propositum ostendendum oportet considerare differentiam ipsarum, scilicet demonstrationis negativae et ducentis ad impossibile. Si ergo accipiatur quod a in nullo b sit, et b sit in omni c, et concludatur a esse in nullo c, erit demonstratio negativa.	He says therefore first (87a2) that in order to manifest the proposition it is necessary to point out the difference between them, i.e., between the negative demonstration and the one leading to the impossible. If, therefore, one assumes that A is in No B, and B is in every C, and concludes that A is in no C, it will be a negative demonstration.
lib. 1 l. 40 n. 3 Deinde cum dicit: quae vero est ad impossibile etc., manifestat quae sit demonstratio ducens ad impossibile. Et dicit quod demonstratio ducens ad impossibile hoc modo se habet. Sit ita quod oporteat demonstrare quod a non sit in b, et accipiamus oppositum eius quod probare volumus, scilicet omne b est a, et accipiatur quod b sit in c per hanc propositionem, omne c est b; ex quibus sequitur conclusio, omne c est a: et sit ita quod notum sit et concessum apud omnes quod hoc est impossibile. Et ex hoc concludimus primam propositionem esse falsam, scilicet omne b est a. Et ita oportebit vel quod nullum b sit a, vel saltem quod quoddam b non sit a.	Then (87a7) he elucidates what a demonstration leading to the impossible is. And he says that a demonstration leading to the impossible is as follows: Suppose we are to prove that A is not in B. Let us assume the opposite of what we wish to prove, namely, assume that every B is A; then assume that B is in C, using the proposition, “Every C is B”; from these the conclusion follows that “Every C is A,” which is such that everyone knows and admits that it is impossible. From this we conclude that the first proposition, namely, “Every B is A,” is false. Consequently, it will be necessary either that no B is A or at least that some B is not A.
Sed hoc tamen intelligendum est, quod sequitur a non esse in b, quando manifestum est b esse in c: quia si manifestum esset hanc esse falsam, omne c est a, et non esset manifestum hanc esse veram, omne c est b, non esset per consequens manifestum hanc esse falsam, omne b est a; quia falsitas conclusionis poterat procedere ex alterutra praemissarum, ut ex supra dictis patet.	But it must be understood that “A is not in B” follows, when it is clear that B is in C, because if it is obvious that “Every C is A” is false, but not obvious that “Every C is B” is true, it would not, just in virtue of the inference, be plain that “Every B is A” is false; for the falsity of the conclusion might have proceeded from either premise, as was stated above.
lib. 1 l. 40 n. 4 Deinde cum dicit: termini quidem similiter etc., concludit comparationem utriusque demonstrationis praemanifestatae. Et primo ostendit in quo conveniunt, quia in simili ordinatione terminorum. Nam sicut in demonstratione negativa accipitur b medium inter a et c, ita in ea quae ducit ad impossibile.	Then (87a13) he concludes the comparison of each of the aforesaid demonstrations. First, he shows wherein they agree, namely, in having a like ordering of their terms. For just as in the negative demonstration, B is taken as the middle between A and C, so too in that which leads to the impossible.
Secundo autem ostendit differentiam, quia differt in utraque demonstratione quae negativa propositio sit notior, utrum scilicet ista propositio, nullum b est a, vel ista, nullum c est a: quia in demonstratione ducente ad impossibile accipitur ista propositio, c non est a, ut notior; quia ex hoc quod est a non esse in c, ostenditur a non esse in b, unde haec, c non est a, sumitur ut notior. Sed quando illa quae ponitur ut praemissa in syllogismo, accipitur ut notior, tunc est demonstrativa, idest ostensiva demonstratio negativa.	Secondly, he shows wherein they differ, because it makes a difference which negative proposition in each demonstration is better known, namely, whether it be the proposition, “No B is A,” or the proposition, “No C is A”: for in a demonstration leading to the impossible, the proposition “C is not A” is taken to be better known, since from the fact that A is not in C one shows that A is not in B. Consequently, the proposition, “C is not A,” is taken as better known. But when that which is set down as a premise of the syllogism is taken as better known, there is a demonstrative, i.e., an ostensive, negative demonstration.
lib. 1 l. 40 n. 5 Deinde cum dicit: natura autem prior est etc., ostendit propositum hoc modo. Ista propositio, b non est a, est naturaliter prior quam ista propositio, c non est a. Et hoc probat per hoc, quod praemissa, ex quibus infertur conclusio, sunt naturaliter priora conclusione. Sed in ordine syllogismi, c non est a ponitur ut conclusio, sed b non est a ponitur ut id ex quo conclusio infertur; ergo b non est a est naturaliter prior.	Then (87a17) he shows his proposition in the following way: The proposition, “B is not A,” is naturally prior to the proposition, “C is not A.” And this is proved by the fact that the premises, from which the conclusion is inferred, are naturally prior to the conclusion. But in the order of the syllogism, “C is not A” is set down as the conclusion, whereas “B is not A” is set down as that from which the conclusion is inferred. Therefore, “B is not A” is naturally prior.
lib. 1 l. 40 n. 6 Consequenter cum dicit: non enim si contingit etc., removet quamdam obviationem. Posset enim aliquis dicere quod etiam ista negativa, c non est a, est id ex quo concluditur in demonstratione ad impossibile quod b non est a. Sed hoc excludit, dicens quod per hoc quod conclusio interimitur et ex eius interemptione interimitur aliquod praemissorum, non efficitur quod id quod prius erat conclusio sit principium et e converso, simpliciter et secundum naturam, sed solum quoad aliquem. Nam ista est habitudo conclusionis et principiorum, quod interempta conclusione, interimitur principium. Sed illud quidem est sicut principium, ex quo syllogismus procedit, quod se habet ad conclusionem ut totum ad partem; et conclusio se habet ad principium ut pars ad totum. Nam subiectum conclusionis negativae sumitur sub subiecto primae propositionis. Non autem ita se habent ac et ab propositiones ad invicem, quod ac comparetur ad ab ut totum ad partem. Non enim ba accipitur sub ca, sed potius e converso. Unde relinquitur quod licet interempto ca, concludatur interemptio eius quod est ba, naturaliter tamen ca est conclusio et ba est principium; et per consequens b non est a est naturaliter notior quam c non est a.	Then (87a19) he removes an objection. For someone could say that even the negative “C is not A” is the one from which one concludes “B is not A” in a demonstration to the impossible. But he excludes this, saying that by the fact that the conclusion is destroyed and from its destruction something in the premises is destroyed, it does not follow that what was first the conclusion is now a principle and vice versa, absolutely and according to nature, but only in a qualified sense. For the relation of conclusion to principle is such that the principle is destroyed by the conclusion’s being destroyed. But that functions as the principle from which a syllogism proceeds, which is related to the conclusion as a whole to a part; while the conclusion is to the principle, as part to whole. For the subject of a negative conclusion is subsumed under the subject of the first proposition. But the propositions AC and AB are not so related to each other that AC is to AB as whole to part. For BA is not subsumed under CA; rather it is just the opposite. Hence it remains that although from the destruction of CA, one concludes to the destruction of BA, nevertheless CA is naturally the conclusion and BA the principle. Consequently, “B is not A” is naturally better known than “C is not A.”
lib. 1 l. 40 n. 7 Et ex hoc sic argumentatur. Illa demonstratio est dignior, quae procedit ex notioribus et prioribus. Sed demonstratio negativa procedit ex notiori et priori quam demonstratio ducens ad impossibile. Utraque enim facit scire per aliquam negativam propositionem; sed demonstratio negativa procedit ad faciendam fidem ex hac propositione negativa, b non est a, quae est naturaliter prior; demonstratio autem ducens ad impossibile procedit ad faciendum fidem ex hac propositione negativa, c non est a, quae est posterior naturaliter. Relinquitur ergo quod demonstratio negativa sit potior ea quae ducit ad impossibile. Sed affirmativa est potior negativa, ut supra ostensum est; ergo demonstratio affirmativa ostensiva est multo potior ea quae ducit ad impossibile.	From this, one argues in the following way: That demonstration is the worthier which proceeds from better known and prior principles. But a negative demonstration proceeds from something better known and prior than does a demonstration leading to the impossible. For each causes one to know something in virtue of a negative proposition: but the negative demonstration proceeds to cause belief from the negative proposition, “B is not A”, which is naturally prior. Demonstration to the impossible, on the other hand, proceeds to cause belief from the negative proposition, “C is not A,” which is naturally posterior. What remains, therefore, is that the negative demonstration is more powerful than one which leads to the impossible. Furthermore, as was shown above, the affirmative is stronger than the negative. Therefore, an affirmative ostensive demonstration is much stronger than one which leads to the impossible.