Authors/Thomas Aquinas/posteriorum/L1/Lect39

Lecture 39 Affirmative demonstration is stronger than negative

Latin	English
	Lecture 39 (86a32-b40) AFFIRMATIVE DEMONSTRATION IS STRONGER THAN NEGATIVE
lib. 1 l. 39 n. 1 Postquam philosophus ostendit quod demonstratio universalis est potior quam particularis, hic ostendit quod demonstratio affirmativa sit potior negativa. Et circa hoc ponit quinque rationes.	After showing that universal demonstration is more powerful than particular, the Philosopher here shows that affirmative demonstration is more powerful than negative. In support of this he presents five reasons.
lib. 1 l. 39 n. 2 In quarum prima hoc praesupponit, quod, caeteris paribus, illa demonstratio est dignior, quae procedit ex paucioribus petitionibus aut suppositionibus aut propositionibus. Quae quidem qualiter differant ex supra dictis patet. Nam propositiones possunt dici etiam illae quae sunt per se notae, quae neque suppositiones neque petitiones sunt, ut supra dictum est. Suppositio autem a petitione differt: nam suppositio est propositio non per se nota, sed accipitur sicut a discente opinata; petitio autem est propositio non per se nota, quae non est opinata a discente, sive habeat contrarias opiniones sive non.	In the first reason (86a32) he presupposes that, all else being equal, the better demonstration is the one which proceeds from fewer postulates or suppositions or propositions. How these differ is clear from what has been stated above. For propositions might even be taken to mean those per se known statements which are neither suppositions nor postulates, as stated above. But a supposition is not the same as a postulate: for a supposition is a proposition which is not per se known, but is taken by the learner as opined; a postulate, on the other hand, is a proposition which is not per se known and has not been opined by the learner, nothing being said about whether he has opinions to the contrary or not.
lib. 1 l. 39 n. 3 Quod autem demonstratio sit dignior quae paucioribus utitur, caeteris paribus ostendit dupliciter. Primo quidem, quia si detur quod utraeque propositiones ex quibus proceditur, sint aeque notae, sequitur quod velocius erit cognoscere per pauciores propositiones quam per plures; quia citius terminatur discursus, qui est per pauciores propositiones, quam qui est per plures. Hoc autem est eligibilius seu appetibilius, quod homo citius addiscat. Unde relinquitur quod demonstratio quae ex paucioribus propositionibus procedit, dummodo sint aeque notae, sit melior.	Accordingly, he shows in two ways that, other things being equal, the demonstration which employs fewer of these is better. First, because if it happens that both propositions from which one proceeds are equally known, it follows that one knows more quickly through fewer propositions than through more, because a discourse through fewer propositions comes to an end sooner than one through more. For it is more desirable or more advantageous that a man, learn more quickly. Hence, it remains that a demonstration which proceeds from fewer propositions, provided they are equally known, is better.
lib. 1 l. 39 n. 4 Secundo, probat eamdem propositionem universaliter absque praedicta suppositione, scilicet quod omnes propositiones assumptae sint aequaliter notae. Et ad hoc probandum assumit hanc suppositionem, quod media quae sunt unius ordinis sint aeque nota, sed media quae sunt priora sunt notiora. Hoc enim oportet esse universaliter verum. Hoc igitur supposito, sit una demonstratio, in qua demonstretur quod a sit in e per tria media, quae sunt b c d; quae quidem concludit ex quatuor propositionibus, quae sunt: omne b est a; omne c est b; omne d est c; omne e est d. Alia vero demonstratio sit, quae concludat eamdem conclusionem, scilicet a esse in e, per duo media, quae sunt z h.	Secondly, he proves the same proposition universally without the above supposition, namely, that all the propositions used are equally known. To prove this he assumes this supposition, namely, that the middles which are of one order are equally known, but the middles which are prior are better known. For this must be universally true. Therefore, supposing this, take one demonstration in which it is demonstrated that A is in E in virtue of three middles, B, C, D, so that a conclusion is reached from four propositions, namely, “Every B is A,” “Every C is B,” “Every D is C” and “Every E is D.” Then let the other demonstration be one which concludes the same conclusion, namely, that A is in E, through two middles which are F and G.
His itaque suppositis, manifestum est quod, ex quo ordo cognitionis proportionatur ordini mediorum, quia priora sunt notiora, ut dictum est, necesse est quod aequaliter sit nota haec propositio, omne d est a, in prima demonstratione, et, omne e est a, in secunda, quia utrobique inveniuntur duo media. Sed manifestum est quod in prima demonstratione haec propositio, omne d est a, prior et notior est quam haec propositio, omne e est a; quia haec secunda demonstratur ex priori in prima demonstratione, et ex his quae supra dicta sunt, apparet quod id per quod demonstratur aliquid, est credibilius et notius eo quod per ipsum demonstratur. Ergo relinquitur quod haec propositio, omne e est a, secundum quod concluditur per secundam demonstrationem, sit notior quam eadem propositio, secundum quod concluditur per priorem demonstrationem, quae utebatur pluribus mediis. Relinquitur ergo quod demonstratio quae ex paucioribus procedit, est potior ea quae procedit ex pluribus.	Under these conditions, from the fact that the order of knowing is proportionate to the order of middles, since the prior known are the better known, as has been said, it is obvious that the proposition, “Every D is A” in the first demonstration must be as well known as “Every E is A” in the second, because two middles are used for obtaining each. However, it is also obvious that in the first demonstration this proposition, “Every D is A,” is prior and better known than the proposition, “Every E is A,” because this latter is demonstrated from something prior in the first demonstration-and from what has been established above, it is apparent that that through which something is demonstrated is more credible and more known than that which is demonstrated through it. What remains, therefore, is that this proposition, “Every E is A,” so far as it is concluded by the second demonstration, is better known than the same proposition so far as it is concluded by the first demonstration which used more middles. Consequently, a demonstration which proceeds from fewer is better than one which proceeds from more.
lib. 1 l. 39 n. 5 Maiore igitur propositione probata, Aristoteles assumit quod affirmativa demonstratio ex paucioribus procedit quam negativa: non quidem ex paucioribus terminis, vel ex paucioribus propositionibus secundum materiam, quia utraque demonstratio tam affirmativa quam negativa demonstrat per tres terminos et duas propositiones; sed demonstratio negativa dicitur ex pluribus procedere secundum propositionum qualitatem. Nam demonstratio affirmativa accipit solum ens, idest procedit ex solis propositionibus affirmativis: demonstratio vero negativa accipit esse et non esse, idest assumit affirmativam et negativam simul. Ergo dignior est affirmativa quam negativa.	Therefore, having proved the major proposition, Aristotle assumes that an affirmative proposition proceeds from fewer things than a negative: not indeed from fewer terms or fewer propositions materially, because every demonstration, whether affirmative or negative, demonstrates through three terms and two propositions; but a negative demonstration is said to proceed from more according to the quality of the propositions. For an affirmative demonstration takes only “being,” i.e., proceeds only from affirmative propositions, whereas a negative demonstration takes “being and non-being,” i.e., uses an affirmative and a negative. Therefore, the affirmative is stronger than the negative.
lib. 1 l. 39 n. 6 Secundam rationem ponit ibi: amplius quoniam ostensum est etc., et inducitur haec secunda ratio ad confirmationem primae, quae poterat videri deficiens ex hoc quod non assumebatur sub maiori propositione eo modo quo probabatur. Et ideo, ut omnis calumnia excludatur, addit hanc secundam rationem ad confirmationem primae. Ostensum est enim in libro priorum, quod ex duabus propositionibus negativis non potest fieri syllogismus, sed oportet ad minus unam propositionem esse affirmativam et alteram negativam. Ex quo aperte apparet quod propositiones affirmativae habent maiorem efficaciam ad syllogizandum quam negativae. Unde sequitur quod demonstratio affirmativa, quae procedit ex solis affirmativis, sit potior quam demonstratio negativa, quae procedit ex affirmativa et negativa.	Then (86b10) he presents the second reason. Now this second reason is introduced to support the first, which might seem deficient on the ground that it was not subsumed under the major proposition in the way in which it was proved. Therefore, to forestall any subterfuge he adds this second reason to confirm the first. For it has been shown in Prior Analytics I that a syllogism cannot be formed out of two negative propositions, but one proposition at least must be affirmative and the other may be negative. This clearly shows that affirmative propositions have greater efficacy for syllogizing than do negatives. Hence it follows that an affirmative demonstration which proceeds from affirmatives only, is more powerful than the negative demonstration which proceeds from a negative and an affirmative.
lib. 1 l. 39 n. 7 Tertiam rationem ponit ibi: amplius iuxta hoc etc., et dicit quod secundum consequentiam praemissarum rationum possumus hoc accipere, quod quando demonstratio augmentatur, scilicet per resolutionem propositionum in sua principia, necesse est esse plures propositiones affirmativas, sed negativas impossibile est esse plures quam unam. Sit enim talis demonstratio negativa: nullum b est a; omne c est b; ergo nullum c est a. Augeatur ergo demonstratio quantum ad utramque propositionem, si utraque sit mediata, accipiendo medium utriusque; et medium quidem maioris propositionis, scilicet nullum b est a, sit d, medium autem minoris propositionis, scilicet omne c est b, sit e. Et quia haec propositio, omne c est b, est affirmativa, ad quam concludendam non concurrit aliqua negativa, necesse est quod eius medium, quod est e, sit affirmativum ad utramque extremitatem; et sic sumuntur duae propositiones affirmativae, scilicet omne e est b, omne c est e, ex quibus concluditur, omne c est b. Sed maior propositio est negativa, scilicet nullum b est a: negativa autem non concluditur ex duabus negativis; sed in prima figura, per quam maxime fit demonstratio secundum praemissa, maiorem oportet esse negativam et minorem affirmativam. Unde oportebit quod hoc medium d sit affirmativum per comparationem ad b, sit autem negativum per comparationem ad a, tali demonstratione facta: nullum d est a; omne b est d; ergo nullum b est a.	Then (86b12) he presents the third reason, saying that as a consequence of the foregoing reasons, we can assert that when a demonstration is augmented, i.e., by resolving propositions into their principles, it is necessary that there be several affirmative propositions, but no more than one negative. Thus, let us take the following negative demonstration: “No B is A; Every C is B: therefore, No C is A.” Then let this demonstration be augmented as to both propositions, if each is mediate, by taking a middle for each, letting the middle of the major proposition, “No B is A,” be D, and the middle of the minor proposition, “Every C is B,” be E. Now since this proposition, “Every C is B,” is affirmative and was concluded without benefit of a negative, it is necessary that its middle, namely, E, be affirmative to each extreme. Consequently, there are two affirmative propositions, namely, “Every E is B,” and “Every C is E,” from which “Every C is B” is concluded. But the major proposition is negative, namely, “No B is A.” Now a negative is not concluded from two negatives; but in the first figure, in which demonstration is best made, according to what has been said, that major must be negative and the minor affirmative. Consequently, this middle, D, must be affirmative to B and negative to A, as in the following demonstration: “No D is A; Every B is D; therefore No B is A.”
Sic igitur augmentata demonstratione negativa per resolutionem propositionum in sua principia, erunt quatuor propositiones, quarum una sola est negativa, scilicet nullum d est a; tres autem aliae erunt affirmativae, scilicet omne b est d, omne e est b, omne c est e. Et idem est in omnibus aliis syllogismis, quia semper necesse est medium per quod affirmativae propositiones probantur, esse affirmativum ad ambo extrema. Medium autem per quod probatur negativa propositio, necesse est esse negativum solum ad unam extremitatem; et ita sequitur quod una sola propositio sit negativa, aliae omnes affirmativae.	Hence by augmenting a negative demonstration by resolving the propositions into their principles, there will be four propositions: only one of these is negative, namely, “No D is A,” and the other three will be affirmative, namely, “Every B is D,” “Every E is B,” and “Every C is E.” And the same thing will happen in all other syllogisms, because it is always required that the middle, through which the affirmative propositions are proved, be affirmative to both extremes. But the middle through which the negative proposition is proved must be negative to one of the extremes only. Thus it follows that only one proposition is negative and the others all affirmative.
Ex quo patet quod propositio negativa maxime demonstratur per affirmativas. Si ergo illud per quod aliquid demonstratur, est notius et credibilius eo quod per ipsum demonstratur, cum negativa propositio maxime demonstretur per affirmativam, non autem e converso, sequitur quod affirmativa propositio sit prior et notior et credibilior quam negativa. Unde demonstratio affirmativa erit dignior.	From this it is clear that it is mainly through affirmatives that a negative proposition is demonstrated. Therefore, if that through which something is demonstrated is more known and more credible than that which is demonstrated through it (since a negative proposition is mainly proved by affirmatives and not vice versa), it follows that the affirmative proposition is prior and more known and more credible than the negative. Accordingly, the affirmative demonstration will be more valuable.
lib. 1 l. 39 n. 8 Quartam rationem ponit ibi: amplius si principium syllogismi etc., et dicit quod principium syllogismi demonstrativi est propositio universalis immediata; ita tamen quod affirmativi syllogismi est principium proprium affirmativa propositio, negativi autem syllogismi proprium principium est negativa propositio universalis. Sed nobilioris principii nobilior est effectus. Ergo secundum proportionem propositionis affirmativae ad negativam, est proportio demonstrationis affirmativae ad negativam. Sed affirmativa propositio est potior quam negativa. Quod probat dupliciter: primo quidem, quia affirmativa est prior et notior, cum per affirmativam probetur negativa, et non e converso; secundo, quia affirmatio praecedit naturaliter negationem, sicut esse prius est quam non esse. Quamvis enim in uno et eodem, quod de non esse in esse procedit, non esse sit prius tempore, esse tamen est prius natura, et simpliciter prius etiam tempore; quia non entia non producuntur in esse, nisi ab aliquo ente. Ergo patet quod affirmativa demonstratio est potior quam negativa.	Then (86b30) he presents the fourth reason, saying that the principle of a demonstrative syllogism is an immediate universal proposition, in the sense that the proper principle of an affirmative syllogism is an affirmative proposition and the proper principle of a negative syllogism is a universal negative proposition. But the effect of a nobler principle is itself more noble. Therefore, as an affirmative proposition is to a negative, so is an affirmative demonstration to a negative one. But an affirmative proposition is more powerful than a negative one. (He proves this in two ways: first, because the affirmative is prior and better known, since the negative is proved by the affirmative and not vice versa. Secondly, because affirmation naturally precedes negation, as being is prior to non-being—for although in one and the same thing which passes from non-being to being, the non-being is prior in the order of time, yet in the order of nature, being is prior and, absolutely speaking, is prior even in time, because non-beings are not brought into existence except by something that is). Therefore, it is clear that affirmative demonstration is mo powerful than negative.
lib. 1 l. 39 n. 9 Quintam rationem ponit ibi: adhuc et principalior etc., quae talis est. Illud ex quo aliud dependet, est principalius. Sed demonstratio negativa dependet ex affirmativa; quia non potest esse negativa demonstratio sine affirmativa propositione, quae non probatur nisi per affirmativam demonstrationem. Ergo demonstratio affirmativa est principalior quam negativa.	Then (86b39) he presents the fifth reason and it is this: That upon which something depends is more principal. But negative demonstration depends on affirmative, because there cannot be a negative demonstration without an affirmative proposition, which is not proved except by an affirmative demonstration. Therefore, affirmative demonstration is more principal than negative.