Authors/Thomas Aquinas/posteriorum/L1/Lect32

Lecture 32 Solution of some of these doubts hinges upon solution of others of these doubts

Latin	English
	Lecture 32 (82a21-b34) SOLUTION OF SOME OF THESE DOUBTS HINGES UPON SOLUTION OF OTHERS OF THESE DOUBTS
lib. 1 l. 32 n. 1 Postquam philosophus movit quaestiones, hic incipit eas determinare; et dividitur in duas partes. In prima parte, ostendit quod quarumdam dubitationum solutio reducitur ad solutionem aliarum. In secunda, solvit dubitationem quantum ad illa, in quibus per se et principaliter difficultas consistit; ibi: quod autem in illis, si logice et cetera. Circa primum duo facit: primo enim ostendit quod dubitatio, quae potest esse circa media, reducitur ad dubitationem, quae movetur de extremis, et, ea soluta, solvitur; secundo, ostendit quod dubitatio, quae est circa negativas demonstrationes, reducitur ad dubitationem, quae est de affirmativis; ibi: manifestum est autem in privativis et cetera. Circa primum tria facit: primo, proponit quod intendit; secundo, probat propositum; ibi: si enim a praedicante etc.; tertio, excludit quamdam obviationem; ibi: nec si aliquis dicat et cetera.	After raising the questions, the Philosopher here begins to settle them. And his treatment is divided into two parts. In the first he shows that the solution of some of the doubts is reduced to the solution of others. In the second he settles the doubt as to those items in which the difficulty lies per se as in its source (82b34) [L. 33]. Concerning the first he does two things. First, he shows that the doubt bearing on the middles is reduced to the one which is concerned with the extremes and is solved by the solution of the latter. Secondly, he shows that the doubt bearing on negative demonstrations is reduced to the one which is concerned with affirmative demonstrations (82a37). In regard to the first he does three things. First, he states his intended proposition. Secondly, he proves this proposition (82a24). Thirdly, he excludes a subterfuge (82a30).
lib. 1 l. 32 n. 2 Dicit ergo primo quod manifestum est, si quis rationem sequentem consideret, quod non contingit esse media infinita, si praedicationes tam in sursum quam in deorsum stent in aliquibus terminis, scilicet in summo praedicato et in infimo subiecto. Et exponit quid sit procedere praedicationes sursum, et deorsum; et dicit quod sursum ascenditur, quando proceditur ad magis universale, de cuius ratione est quod praedicetur: deorsum autem proceditur, quando itur ad magis particulare, de cuius ratione est quod subiiciatur.	He says therefore first (82a21) that it will be plain to anyone who considers the following reason that “an infinity of middles does not occur,” if the predications both upwards and downwards stop at certain terms, namely, at the highest predicate and the lowest subject. And he explains what upward and downward predication consists in, saying that one proceeds upwards when there is movement to the more universal, one of whose marks it is that it be predicated; but one proceeds downwards when there is movement to the most particular, one of whose marks it is that it functions as a subject.
lib. 1 l. 32 n. 3 Deinde cum dicit: si enim a praedicante etc., ostendit propositum per hunc modum. Sit ita quod a sit summum praedicatum, et c sit infimum subiectum, et sint infinita media, quorum quodlibet vocetur b. Quia igitur a erat primum praedicatum, praedicabitur de aliquo medio sibi propinquiori, et iterum illud medium de alio medio inferiori; et cum media sint infinita, sequitur quod in infinitum procedet praedicatio in descendendo, quod est contra positum. Ponebatur enim quod non descendat praedicatio in infinitum. Similiter etiam si incipiamus a c, quod est infimum subiectum, procedetur ascendendo in infinitum antequam perveniatur ad a, quod etiam est contrarium posito. Si ergo haec sint impossibilia, scilicet quod procedatur praedicando in infinitum sursum vel deorsum, sequetur quod impossibile sit media esse infinita. Et ita patet quod quaestio de infinitate mediorum reducitur ad quaestionem de infinitate extremorum.	Then (82a24) he shows what he has proposed in the following way: Let the case be that A is the highest predicate and F the lowest subject, and that there is an infinitude of middles, each of which we shall call B. Now since A was the first predicate, it will be predicated of some middle near it, and that middle of another middle below it. Since the middles are infinite, it follows that the predication will proceed downwards to infinity—which is contrary to what we are assuming. For it was assumed that the predications do not proceed downwards to infinity. The result is the same if we start at F, which is the lowest subject, and proceed upwards to infinity before A is reached—this too would be against our assumption. Therefore, if these are impossible, namely, that one may proceed to infinity by ascending or by descending, it will be impossible for the middles to be infinite. Thus it is clear that the question of an infinity of middles is reduced to a question of the infinity of the extremes.
lib. 1 l. 32 n. 4 Deinde cum dicit: neque enim si aliquis etc., excludit quamdam obviationem. Posset enim aliquis obviare, dicens quod praedicta probatio procedebat, ac si a b c, idest medium et extrema, ita se haberent, quod essent habita ad invicem, ita scilicet, quod inter ea non esset aliquod medium: sic enim definitur habitum in V physicorum, quod scilicet consequenter se habet, cum tangat; et hoc videbatur in praedicta probatione supponi, scilicet quod a praedicaretur de aliquo medio quasi habito, idest immediate sequenti. Sed ille qui ponit media infinita, dicet quod hoc non contingit accipere. Dicit enim quod inter quoscunque terminos acceptos est aliquod medium.	Then (82a30) he excludes an objection. For someone might object saying that the aforesaid proof would hold if ABF, i.e., the middle and the extremes, were so related as to be “had” to one another, i.e., so that there would be no middle between them: for this is the way the Philosopher defines “had” in Physics V, namely, that it is next to something without anything between. And this seemed to be supposed in the above proof, namely, that A is predicated of some middle as though “had” to it, i.e., following it immediately. But one who posits infinite middles will say that this cannot be supposed. For he will say that between any two terms that are taken there is a middle.
lib. 1 l. 32 n. 5 Sed philosophus dicit quod nihil differt, sive sic accipiantur infinita media quod sint habita ad invicem, sicut contingit in discretis; puta, in civitate domus domui est habita, et in numeris unitas unitati: sive non possit inveniri in mediis aliquid habitum, sed semper inter duo media sit aliquod medium accipere; sicut accidit in continuis, in quibus inter quaelibet duo signa, sive inter duo puncta, semper est aliquod medium accipere.	But the philosopher says that it makes no difference whether the infinitude be of middles that are “had” to one another, the way discrete things are (for example, in a city. house is consecutive to house, and in numbers, unity to unity), or whether something “had” cannot be found in the middles although between any two middles it is always possible to find another, as happens in continua, in which, between any two signs, i.e., between two points, another can be found between them.
Et quod hoc nihil differat ad propositum, sive uno modo, sive alio, sic manifestat subdens: quia supposito quod sint infinita media inter a et c, quorum quodlibet vocatur b, quodcunque horum accipio, necesse est quod inter illud et a et c sint infinita media, vel non sint infinita respectu alterius eorum. Verbi gratia: ponamus quod media sint habita ad invicem, sicut accidit in discretis, et accipiamus aliquod medium quod sit habitum ad ipsum a; necesse erit quod inter illud medium et c sint adhuc infinita media. Et similiter si ponantur quaedam finita media inter a et illud medium acceptum. Et eadem ratio est si ponatur medium acceptum immediate coniungi ipsi c, vel per finita media ab eo distare. Ex quo igitur semper a medio accepto oportet accipere infinita media ad alterum extremorum, non differt utrum statim coniungatur alii extremorum, idest sine medio, vel non statim, idest per aliqua media: quia etiam si coniungatur uni extremo sine medio, necesse est quod postea inveniantur infinita media respectu alterius; et ita semper oportebit, si est infinitum in mediis, quod inveniatur infinitum in praedicationibus vel ascendendo vel descendendo, sicut praedicta probatio procedebat.	That this makes no difference one way or the other to the matter at hand he manifests in the following way: Granted that between A and F there is an infinitude of middles, each of which is called B, yet no matter which of these I employ, there is either an infinitude of middles between it and A and F, or there is not an infinitude of them between it and one or the other of the extremes. For example, let us suppose that the middles are mutually “had,” as happens in discrete things, and let us take a middle which is “had” to A; then it will be necessary that between that middle and F there is still an infinitude of middles; and similarly, if we assume a certain finitude of middles between that middle and A. And the same reasoning holds if the middle which is taken be joined immediately to F or is distant from it by a finitude of middles. From the fact, therefore, that from any given middle one must take an infinitude of middles to one or other of the extremes, it makes no difference whether it is joined to either extreme immediately, i.e., without a middle, or not immediately, i.e., through other middles: because even if it be joined to one extreme without a middle, it will still be necessary later to find an infinitude of middles in relation to the other. Consequently, it will always be required, if there is an infinitude of middles, to proceed to infinity in predications either by ascending or by descending, as the above proof showed.
lib. 1 l. 32 n. 6 Deinde cum dicit: manifestum est autem etc., ostendit quod si in affirmativis demonstrationibus non proceditur in infinitum, neque in privativis in infinitum proceditur; et sic quaestio de demonstrationibus negativis reducitur ad quaestionem de affirmativis. Et circa hoc tria facit: primo, proponit quod intendit; secundo, probat propositum; ibi: tripliciter enim demonstratur etc.; tertio, excludit quamdam obviationem; ibi: manifestum est autem et cetera.	Then (82a37) he shows that if there is no process to infinity in affirmadve demonstrations, then neither in negative demonstrations: and thus the question of negative demonstrations is reduced to the question of affirmative ones. He does three things in regard to this point. First, he proposes what he intends. Secondly, he proves what he proposed (82b4). Thirdly, he excludes an objection (82b28).
Dicit ergo primo, quod manifestum erit ex sequentibus, quod si in praedicativa, idest affirmativa demonstratione statur utrinque, idest in sursum et deorsum, necesse erit quod stetur in negativa demonstratione.	He says therefore first (82a37) that it will be clear from what follows that if in the predicative, i.e., in the affirmative demonstration, a stop is made at both, i.e., upwards and downwards, it will be necessary that a stop be made in the negative demonstration.
Et ad exponendum hoc quod propositum est, dicit: sit ita quod non contingat ab ultimo, idest ab infimo subiecto, ire in sursum in infinitum versus praedicata universalia. Et exponit quid est ultimum, scilicet illud quod non inest alicui alii tanquam minus particulari, sed aliud sit in illo, et sit illud z. Et sit etiam quod incipiendo a primo versus ultimum non procedatur in infinitum. Et exponit quid sit primum illud, scilicet quod praedicatur de aliis, et nihil aliud praedicatur de eo, quasi eo universalius; ut sic primum intelligatur universalissimum, ultimum autem particularissimum. Si igitur ex utraque parte stetur in demonstrationibus affirmativis, dicit consequens esse quod etiam stetur in demonstrationibus negativis.	To elucidate what he is proposing he says: Let the case be such that from the ultimate, i.e., from the lowest subject one cannot go in ascending order to infinity toward universal predicates. And he explains that “ultimate” means that which is not in any other as in a less particular, but something else is in it, and let it be F. And let the case also be that one does not go to infinity when proceeding from the first to the ultimate. And he explains that the “first” means that which is predicated of others but nothing else is predicated of it as more universal than it. Thus the “first” is understood to be the most universal, and the “ultimate” the most particular. If, therefore, on both sides there be a stop in affirmative demonstrations, he says that as a consequence, there is also a stop in negative demonstrations.
lib. 1 l. 32 n. 7 Deinde cum dicit: tripliciter enim etc., probat propositum. Et primo in prima figura; secundo in secunda; ibi: iterum sit b quidem etc.; tertio in tertia; ibi: tertius autem est et cetera. In tribus enim figuris contingit negativam concludi. Dicit ergo primo quod tripliciter potest demonstrari propositio negativa, per quam significatur aliquid non esse. Uno quidem modo in prima figura, secundum hunc modum, quod b insit c universaliter, minori existente universali affirmativa; a vero insit nulli b, maiori existente universali negativa. Quia igitur supponitur quod in affirmativis stetur et in sursum et in deorsum, necesse est quod ista propositio, quae est b-c, affirmativa, si non sit immediata, et quodcunque aliud spatium accipitur, existente aliquo medio inter b et c, necesse erit reducere in immediata; quia ista distantia, quae attenditur secundum habitudinem medii ad minorem extremitatem, est affirmativa, in qua supponitur esse status. Si autem accipiamus alterum spatium, quod est inter b et a, manifestum est quod, si haec propositio, nullum b est a, non est immediata, necesse est quod a removeatur ab aliquo alio per prius quam a b, et illud sit d; quod si accipiatur ut medium inter a et b, necesse est quod praedicetur universaliter de b, quia oportet minorem esse affirmativam. Et iterum si haec non sit immediata, nullum d est a, oportet quod a negetur ab aliquo alio per prius quam a d, puta sit illud e; quod eadem ratione oportebit universaliter praedicari de d. Quia ergo ascendendo statur in affirmativis, ut supponitur, sequitur per consequens quod sit devenire ad aliquid, de quo primo et immediate negetur ipsum a. Alioquin adhuc procederetur amplius in affirmativis, sicut ex praedictis patet.	Then (82b4) he proves his proposition. First, in the first figure. Secondly, in the second (8513). Thirdly, in the third (82b23). For a negative can be concluded in three figures. He says therefore first (82b4) that there are three ways of demonstrating a negative proposition through which something is signified not to be. In one way in the first figure according to the mode that B is universally in C in the universal affirmative minor, but A is in no B in the universal negative major. Now since we are supposing that there is a stop in affirmatives, both upwards and downwards, it is necessary that the proposition, BC, which is affirmative, if it is not immediate but a space exists with middles between B and C, be reduced to immediates, because that space which exists between the middle term and the extreme is affirmative, in which a stop is supposed. But if we take the other space, which is between B and A, it is clear that if this proposition, “No B is A,” is not immediate, it is necessary that A be removed from something else before being removed from B. Let this be D. But if this D be taken as a middle between A and B, it is necessary that it be universally predicated of B, because the minor must be affirmative. And if this too is not immediate, i.e., “No D is A ‘ “ then A has to be denied of something prior to D, say E, which again will be predicated universally of D for the same reason. Therefore, since there is a stop in affirmatives when we ascend, as supposed, it follows that something is reached of which A should be denied first and immediately; otherwise one would go still further in affirmatives, as is clear from the foregoing.
lib. 1 l. 32 n. 8 Deinde cum dicit: iterum si b quidem etc., probat idem in negativa, quae concluditur in secunda figura. Sit enim ita quod b, quod est medium, praedicetur universaliter de a et negetur universaliter de c, et ex his concludatur quod, nullum c sit a. Si autem negativam iterum demonstrari oporteat, propter hoc quod est mediata, necesse est quod vel demonstretur in prima figura, de quo modo demonstrationis iam ostensum est quod habet statum, si in affirmativis sit status; aut oportet quod demonstretur per hunc modum, idest per secundam figuram; aut per tertium, idest per tertiam figuram. Dictum est autem in prima figura, quod habet statum in negativis, si sit status in affirmativis. Sed hoc quidem demonstrabitur nunc quantum ad secundam figuram.	Then (82b13) he proves the same thing for the negative which is concluded in the second figure. For let the case be that B, which is the middle, is predicated universally of A and denied universally of C, so that the conclusion is “No C is A.” Now if the negative premise needs to be demonstrated because it is mediate, it must be demonstrated either in the first figure in the mode of demonstrating concerning which we have shown that there is a stop, if there is a stop in affirmatives; or it must be demonstrated through this mode, i.e., in the second figure, or through a third mode, i.e., in the third figure. Now it has been established that there is a stop in negatives of the first figure, if there is a stop in affirmatives. Consequently, the same will now be demonstrated as to the second figure.
Demonstretur ergo haec propositio, nullum c est b, sic quod d universaliter praedicetur de b, maiori existente universali affirmativa, et negetur universaliter de c, minori existente universali negativa. Si iterum haec propositio, nullum c est d, est mediata, necesse erit accipere aliquod aliud medium, quod etiam praedicetur de d universaliter, et universaliter removeatur a c. Et ita, sicut proceditur in negativis demonstrationibus, oportebit procedere in affirmativis, scilicet quod b praedicabitur de a, et d de b, et aliquid aliud de d; et sic procedetur in infinitum in affirmativis. Quia ergo supponitur quod in affirmativis stetur in sursum, necesse est etiam quod stetur in negativis, secundum hunc modum, quo negativa demonstratur in secunda figura.	Therefore, let this proposition, “No C is B,” be demonstrated in such a way that D is universally predicated of B in the universal affirmative major and denied universally of C in the universal negative minor. Now if the proposition, “No C is D,” is mediate, it will be required to take some other middle which will be predicated universally of D and universally removed from C. Continuing thus, it will be necessary to proceed in negative demonstrations just as we do in affirmatives, namely, B will be predicated of A and D of B, and something else of D, and so on to infinity in affirmatives. But because we are supposing an upward stop in the affirmatives, it is also necessary to come to a stop in the negatives according to this mode in which a negative is demonstrated in the second figure.
lib. 1 l. 32 n. 9 Deinde cum dicit: tertius autem est etc., ostendit idem in tertia figura. Sit ergo medium, ut b, de quo a universaliter praedicetur, c vero ab eo removeatur: sequitur particularis negativa, scilicet quod c negetur a quodam a. Et quod quidem in praemissa affirmativa, quae est, omne b est a, stetur, habetur ex suppositione; quod autem necesse sit stare etiam in hac negativa, nullum b est c, quae est maior, patet, quia si hoc debeat demonstrari, necesse est quod vel demonstretur per superius dicta, idest per primam et secundam figuram, vel demonstrabitur similiter sicut concludebatur conclusio, scilicet per tertiam figuram: ita tamen quod haec maior non assumatur ut universalis, sed ut particularis. Illo autem modo statur, scilicet si procedatur in prima et in secunda figura. Si autem procedatur in tertia figura ad concludendum, quoddam b non esse c, accipiatur medium e, de quo quidem b universaliter affirmetur, c vero ab eo particulariter negetur. Et hoc iterum similiter continget, quod secundum hoc procedetur in demonstratione negativa semper secundum augmentum praedicationis affirmativae in inferius: quia b, quod erat primum medium, praedicabitur de e, et e de quodam alio, et sic in infinitum. Quia igitur supponitur statum esse in affirmativis in deorsum, manifestum est quod stabitur in negativis ex parte ipsius c.	Then (82b23) he shows the same thing in the third figure. Therefore, let B be a middle of which A is universally predicated, but C is universally denied of it: the conclusion will be a particular negative, namely, C is denied of some A. Now that there is a stop in the affirmative premise, “Every B is A,” is granted by our supposition. Furthermore, that there must be a stop in the negative, “No B is C,” which is the major, is evident, because if it had to be demonstrated, it would be done either “through what was said above,” i.e., through the first and second figure or in the way that the conclusion was concluded, namely, through the third figure, in which case this minor is not affirmed as universal but as particular. “But there is a stop in that way,” i.e., if one proceeds in the first and second figure. But if one proceeds in the third figure to conclude that “Some B is not C,” let a middle, E, be taken such that B is universally affirmed of it but C is particularly denied of it. “Then this happens once more in like manner,” i.e., according to this, one will always proceed in the negative demonstration by accumulating affirmative predications in descending order, because B which was the first middle will be predicated of E, and E of something else, and so on to infinity. But since we are supposing that there is a stop in the descending order in affirmatives, it is clear that there will be a stop in the negatives on the part of C.
lib. 1 l. 32 n. 10 Deinde cum dicit: manifestum autem est etc., excludit quamdam obviationem. Posset enim aliquis dicere quod necesse est stare in demonstrationibus negativis, statu existente in affirmativis, si semper syllogizetur secundum eamdem figuram; sed potest in infinitum procedi, si nunc demonstretur per unam figuram, nunc per aliam. Et dicit, manifestum est quod si non procedatur in demonstrationibus una via, sed omnibus, aliquando quidem ex prima figura, aliquando autem ex secunda vel tertia, sic etiam oportebit statum esse in negativis, statu existente in affirmativis. Huiusmodi enim viae diversae demonstrandi sunt finitae, et quaelibet earum multiplicatur non in infinitum, sed finite ascendendo vel descendendo, ut ostensum est. Si autem finita finities accipiantur, necesse est totum esse finitum. Unde relinquitur quod omnibus modis necesse sit in demonstrationibus negativis esse statum, si sit status in affirmativis.	Then (82b28) he excludes an objection. For someone could say that it is necessary to stop in negative propositions when there is a stop in the affirmatives, provided that one always syllogizes according to the same figure; but if one demonstrates now in one figure and now in another, one can go to infinity. And he say’~ that “it is obvious” that if one does not limit himself to one figure\in demonstrating but uses all, proceeding now in the first figure and now in the second and third, there must still be a stop in the negatives if there is one in the affirmatives. For these various ways of demonstrating are finite, and each of them will be enlarged not to infinity but finitely by ascending or descending, as was shown. Now if, finite things be taken a finite number of times, the result is finite. Hence it remains that in all the modes there must be a stop in negative demonstrations, if there is a stop in the affirmatives.