Authors/Thomas Aquinas/posteriorum/L1/Lect35

Lecture 35 That there is not an infinite process upward or downward in predicates is shown analytically

Latin	English
	Lecture 35 (84a8-b2) ‘THAT THERE IS NOT AN INFINITE PROCESS UPWARD OR DOWNWARD IN PREDICATES IS SHOWN ANALYTICALLY
lib. 1 l. 35 n. 1 Postquam philosophus ostendit logice quod non sit procedere in infinitum in praedicatis in sursum aut deorsum, hic ostendit idem analytice. Et dividitur in duas partes: in prima ostendit principale propositum; in secunda infert quaedam corollaria ex dictis; ibi: monstratis autem his manifestum et cetera. Circa primum duo facit: primo, proponit quod intendit: secundo, probat propositum; ibi: demonstratio quidem enim et cetera.	After showing logically that there is no infinite process upwards or downwards in predicates, the Philosopher now shows the same thing analytically. And his treatment falls into two parts. In the first he shows the principal proposition. In the second he infers certain corrolaries from the aforesaid (84b3) [L. 36]. Concerning the first he does two things. First, he proposes what he intends. Secondly, he proves his proposition (84a10).
lib. 1 l. 35 n. 2 Dicit ergo primo, quod hoc quod non contingit in demonstrativis scientiis, de quibus intendimus, praedicationes in infinitum procedere, neque in sursum neque in deorsum, brevius et citius poterit manifestari analytice quam manifestatum sit logice. Ubi considerandum est quod analytica, idest demonstrativa scientia, quae resolvendo ad principia per se nota iudicativa dicitur, est pars logicae, quae etiam dialecticam sub se continet. Ad logicam autem communiter pertinet considerare praedicationem universaliter, secundum quod continet sub se praedicationem quae est per se, et quae non est per se. Sed demonstrativae scientiae propria est praedicatio per se. Et ideo supra logice probavit propositum, quia ostendit universaliter in omni genere praedicationis non esse processum in infinitum; hic autem intendit ostendere analytice, quia hoc probat solum in his, quae praedicantur per se. Et haec est via expeditior: et ideo sufficit ad propositum, quia in demonstrationibus non utimur nisi tali modo praedicationis.	He says therefore first (84a8) that the fact an infinite process upwards or downwards does not occur in the demonstrative sciences with which we are concerned can be more briefly and quickly manifested analytically than it was logically. Here we might note that analytic, i.e., demonstrative, science which is called judicative, because it resolves to self-evident principles, is a part of logic which even contains dialectics under it. However, it pertains to logic in general to consider predication universally, i.e., as containing under it predication which is per se and predication which is not per se. But predication per se is proper to demonstrative science. Therefore, above he proves his proposition logically, because he showed universally in every genus of predication that there is no infinite process. But here he intends to show it analytically, because he proves it only in things which are predicated per se. And this is a more efficient way; furthermore, it suffices for our purpose, because that is the only mode of predication we use in demonstration.
lib. 1 l. 35 n. 3 Deinde cum dicit: demonstratio quidem etc., ostendit propositum. Et circa hoc tria facit: primo, proponit qua praedicatione analytica, idest demonstrativa scientia, utatur, quia praedicatione per se; secundo, resumit quot sunt modi talis praedicationis; ibi: per seipsa vero etc.; tertio, ostendit quod in nullo modo praedicationis per se possit procedi in infinitum; ibi: horum autem neutra contingunt et cetera.	Then (84a10) he shows his proposition concerning which he does three things. First, he proposes which predication analytic, i.e., demonstrative, science employs, for it uses per se predication. Secondly, he recalls how many modes there are of such predication (84al). Thirdly, he shows that there cannot be an infinite process in any mode of per se predication (8408).
Dicit ergo primo, quod demonstratio est solum circa illa, quae per se insunt rebus. Tales enim sunt eius conclusiones, et ex talibus demonstrat, ut supra habitum est.	He says therefore first (84a10) that demonstration is concerned exclusively with items that are per se in things. For such are its conclusions and from such does it demonstrate, as was established above.
lib. 1 l. 35 n. 4 Deinde cum dicit: secundum seipsa autem etc., ponit duos modos praedicandi per se. Nam primo quidem praedicantur per se quaecunque insunt subiectis in eo quod quid est, scilicet cum praedicata ponuntur in definitione subiecti. Secundo, quando ipsa subiecta insunt praedicatis in eo quod quid est, idest quando subiecta ponuntur in definitione praedicatorum. Et exemplificat de utroque modo. Nam impar praedicatur de numero per se secundo modo, quia numerus ponitur in definitione ipsius imparis. Est enim impar numerus medio carens. Multitudo autem vel divisibile praedicatur de numero, et ponitur in definitione eius. Unde huiusmodi praedicantur per se de numero primo modo. Alii autem modi, quos supra posuit, reducuntur ad istos.	Then (84a11) he lays down two modes of predicating per se. For in the first place those things are predicated per se which are present in their subjects as constituting their essence, namely, when predicates are placed in the definition of a subject. Secondly, when the subjects themselves are in the essence of the predicate, i.e., when the subjects are placed in the definition of the predicate. And he gives examples of each of these ways: for “odd” is predicated per se of number in the second way, because “number” is placed in the definition of odd. For the odd is a number not divisible by two. Multitude or divisible, however, are predicated of number and are present in its definition; hence these are predicated per se of number in the first way. The other ways, which he mentioned previously, are reduced to these.
lib. 1 l. 35 n. 5 Deinde cum dicit: horum autem neutra etc., ostendit quod in utroque modo praedicationis per se necesse est esse statum. Et circa hoc tria facit: primo, ostendit quod necessarium est esse statum in utroque modo praedicationis per se, tam in sursum quam in deorsum; secundo, concludit quod non possit esse infinitum in mediis; ibi: si autem sic est etc.; tertio, concludit quod non potest procedi in infinitum in demonstrationibus; ibi: si vero hoc et cetera. Circa primum duo facit: primo, ostendit propositum in secundo modo dicendi per se, quando scilicet subiectum ponitur in definitione praedicati; secundo, in primo modo, quando praedicatum ponitur in definitione subiecti; ibi: neque etiam quaecunque et cetera.	Then (84a18) he shows that there must be a stop in each of these modes of per se predication. In regard to this he does three things. First, he shows that there must be a stop both upwards and downwards in both of these modes of per se predication. Secondly, he concludes that there cannot be an infinitude of middles (84a28). Thirdly, he concludes that one cannot proceed to infinity in demonstrations (84a29). Concerning the first he does two things. First, he shows his proposition in regard to the second way of saying per se,” namely, when the subject is placed in the definition of the predicate. Secondly, in the first way, when the predicate is placed in the definition of the subject (84a25). In regard to the first he gives two reasons, in the first of which he proceeds in the following way.
lib. 1 l. 35 n. 6 Circa primum ponit duas rationes. Circa quarum primam sic procedit: primo quidem praemittit propositum, scilicet quod in neutro modo dicendi per se contingit in infinitum procedere; deinde probat hoc in secundo modo, puta cum impar praedicatur de numero. Si enim procedatur ulterius, quod aliquid aliud praedicetur per se de impari secundum istum modum dicendi per se, sequitur quod impar insit in definitione eius. Numerus autem ponitur in definitione imparis: unde sequeretur quod etiam numerus ponatur in definitione illius tertii, quod per se inest impari. Sed hic non contingit abire in infinitum, ut scilicet infinita insint in definitione alicuius, sicut supra probatum est. Relinquitur ergo quod in talibus per se praedicationibus non contingit procedere in infinitum in sursum, idest ex parte praedicati.	First (84a18) he states ‘his proposition, namely, that in neither of these modes of saying per se does an infinite process occur. Then he proves this of the second mode, as when “odd” is predicated of number. For if one goes further and states that something else should be predicated per se of “odd” according to that mode of saying per se, it follows that “odd” is present in its definition. But “number” is placed in the definition of odd; hence it will follow that “number” is also present in the definition of that third thing which is present per se in “odd.” However, this cannot go on to infinity, so that an infinitude of things would be in the definition of something, as was established above. It remains, therefore, that in such per se predications an upward process to infinity does not occur, i.e., on the side of the predicate.
lib. 1 l. 35 n. 7 Secundam rationem ponit ibi: at vero necesse est omnia etc., et dicit quod quantumcunque procedatur in huiusmodi per se praedicationibus secundi modi, oportebit quod omnia praedicata per ordinem accepta insint primo subiecto, puta numero, quasi praedicata de eo: quia si impar per se praedicatur de numero, oportet quod quidquid per se praedicatur de impari, etiam per se praedicetur de numero. Et iterum oportet quod numerus omnibus illis insit; quia si numerus ponitur in definitione imparis, oportet quod ponatur in definitione omnium eorum, quae definiuntur per impar. Et ita sequitur quod mutuo sibi invicem insint. Ergo erunt convertibilia et non se invicem excedentia; sic enim propriae passiones se habent ad sua subiecta. Unde si etiam sint infinita per se praedicata secundum hunc modum, non erit ad propositum, quo aliquis intendit ponere infinita in praedicatis esse, vel in sursum vel in deorsum.	He presents the second reason (84a23) and says that no matter how fair one advances in these per se predications of the second mode, it will be required that all the predicates taken in order be in their first subject, say in number, as predicated of it, because if “odd” is predicated per se of number, it will be required that whatever is predicated of odd be also predicated of number. And it is further required that “number” be in all of them, because if “number” is placed in the definition of odd, it has to be placed in the definition of all those things which are defined by “odd.” And thus it follows that they are mutually in one another. Therefore, they will be convertible and none of wider extent than another: for this is the way proper attributes are related to their subjects. Hence, even though there be an infinitude of per se predicates in this way, it offers nothing to the purpose of one who intends to establish that there is an infinitude of predicates upwards or downwards.
lib. 1 l. 35 n. 8 Deinde cum dicit: neque etiam quaecunque sunt etc., probat propositum in primo modo dicendi per se: et dicit quod illa, quae praedicantur in eo quod quid est, idest quasi posita in definitione subiecti, non possunt esse infinita, quia non contingeret definire, ut supra probatum est. Ex hoc ergo concludit quod si omnia, quae praedicantur in demonstrationibus, per se praedicantur, et in praedicatis per se non est procedere in infinitum in sursum, necesse est quod praedicata in demonstrationibus stent in sursum. Et ex hoc etiam sequitur quod stent in deorsum, quia ex quacunque parte ponatur infinitum, tollitur scientia et definitio, ut ex supra dictis patet.	Then (84a25) he proves his own point in regard to the first mode of saying per se, and he states that those things which are predicated in essence, i.e., as pertaining to the definition of the subject, cannot be infinite; otherwise, definition would be impossible, as we showed above. From this, therefore, he concludes that if all the items predicated in demonstrations are predicated per se, and if there is no infinite upward process in per se predicates, it is necessary that the predicates in demonstrations stop in the upward movement. And from this it also follows that they must stop in the downward movement, because no matter on which side infinity is posited, science and definition are destroyed, as is evident from what has been said above.
lib. 1 l. 35 n. 9 Deinde cum dicit: si autem sic est etc., concludit ex praemissis quod si est status in sursum et deorsum, quod media non contingit esse infinita. Supra enim ostendit quod extremis existentibus determinatis, media non possunt esse infinita.	Then (84a28) he concludes from the foregoing that if there is a stop upwards and downwards, the middles cannot be infinite. For it has been established above that if the extremes are determinate, there cannot be an infinitude of middles.
lib. 1 l. 35 n. 10 Deinde cum dicit: si vero hoc est etc., concludit ulterius quod in demonstrationibus non proceditur in infinitum: et dicit quod si praedicta sunt vera, necesse est esse aliqua prima principia demonstrationum, quae non demonstrantur; et sic non omnium erit demonstratio, secundum quod quidam dicunt, ut in principio huius libri dictum est.	Then (84a29) he further concludes that there is no infinite process in demonstrations. And he says that if the above statements are true, it is necessary that there be certain first principles of demonstration that are not demonstrated; consequently, there will not be demonstration of everything, as some claim, as was stated in the beginning of this book.
Et quod hoc sequatur ostendit. Posito enim quod sint aliqua principia demonstrationum, necesse est quod illa sint indemonstrabilia; quia cum omnis demonstratio sit ex prioribus, ut supra habitum est, si principia demonstrarentur, sequeretur quod aliquid esset prius principiis; quod est contra rationem principii. Et ita, si non sunt omnia demonstrabilia, sequetur quod non procedant demonstrationes in infinitum.	Then he goes on to show that his consequence follows. For if it is granted that there are certain principles of demonstrations, it is necessary that they be indemonstrable: for since every demonstration proceeds from things that are prior, as has been established above, then if the principles are demonstrated, it will follow that something would be prior to the principles, and this is contrary to the notion of a principle. And so, if not all things are demonstrable, it will follow that demonstrations do not proceed to infinity.
Omnia autem praedicta consequuntur ex hoc quod ostensum est, quod non proceditur in infinitum in mediis: quia nihil est aliud ponere verum esse quodcunque praedictorum, scilicet vel quod demonstrationes procedant in infinitum, vel quod omnia sint demonstrabilia, vel quod nulla sint demonstrationum principia, quam ponere nullum spatium esse immediatum et indivisibile; idest ponere duos terminos sibi invicem non cohaerere in aliqua propositione affirmativa vel negativa, nisi per medium. Si enim aliqua propositio sit immediata, sequitur quod sit indemonstrabilis; quia cum aliquid demonstratur, oportet sumere terminum immittendo, idest, quod sit infra praedicatum et subiectum; de quo scilicet per prius praedicetur praedicatum quam de subiecto, vel a quo prius removeatur. Non autem in demonstrationibus accipitur medium assumendo extrinsecus: hoc enim esset assumere extraneum medium, et non proprium, quod contingit in litigiosis et dialecticis syllogismis. Si ergo demonstrationes contingit in infinitum procedere, sequitur quod sint media infinita inter duos terminos. Sed hoc est impossibile, si praedicationes steterint in sursum et deorsum, ut supra probatum est. Et quod stent praedicationes in sursum et deorsum, prius ostendimus logice, et postea analytice, ut expositum est. Per hanc igitur conclusionem ultimo inductam manifestat intentionem totius capituli, et quare quaelibet propositio sit inducta.	But all these follow from what has been established, namely, from the fact that there is no infinite process in middles, because the position that any of the foregoing statements is true, i.e., that demonstrations proceed to infinity, or that all things are demonstrable, or that there are no principles of demonstrations is tantamount to the position that no distance is immediate and indivisible’ i.e., to the position that the two terms of any affirmative or negative proposition belong together only in virtue of a middle. For if any proposition is immediate, it follows that it is indemonstrable; because when something is demonstrated, it is necessary to take a term by interposing, i.e., by setting it between the subject and predicate, so that the predicate will be predicated of that term before being predicated of the subject-or removed from it. But the middle in demonstrations is not taken by assuming extraneously; for this would be to assume an extraneous middle and not a proper middle-which occurs in contentious and dialectical syllogisms. Therefore, if demonstrations were to proceed to infinity, it would follow that there is an infinitude of middles between two extremes. But this is impossible if, as has been established above, the predications stop in the upward and downward process. But as we have shown, first logically and then analytically, these predications do stop both upwards and downwards, as explained. Therefore, in virtue of this conclusion finally induced, he manifests the intent of the entire chapter and why each proposition was introduced.