Authors/Thomas Aquinas/posteriorum/L1/Lect31

Lecture 31 Three questions about proceeding to infinity in confirming demonstrations

Latin	English
	Lecture 31 (81b10-82b20) THREE QUESTIONS ABOUT PROCEEDING TO INFINITY IN CONFIRMING DEMONSTRATIONS
lib. 1 l. 31 n. 1 Postquam philosophus determinavit de syllogismo demonstrativo, ostendens ex quibus et qualibus procedat, et in qua figura demonstrationes fieri possunt; hic inquirit utrum demonstrationes possint in infinitum procedere. Et primo, movet quaestionem; secundo, determinat eam; ibi: quod quidem igitur non contingit media et cetera. Circa primum duo facit: primo, praemittit quaedam, quae sunt necessaria ad intellectum quaestionis; secundo, movet quaestionem; ibi: sit igitur c huiusmodi et cetera. Circa primum duo facit: primo, praemittit de forma syllogistica, quam oportet in demonstrationibus observare; secundo, resumit qualis debeat esse demonstrationis materia; ibi: manifestum igitur est quod principia et cetera.	After the Philosopher has determined concerning the demonstrative syllogism by showing from what and what sort of things it proceeds and in which figure demonstrations can be formed, he now inquires whether demonstrations can proceed to infinity. First, he raises the question. Secondly, he settles it (82a21) [L. 32]. Concerning the first he does two things. First, he sets down certain prefatory remarks needed for understanding the question. Secondly, he raises the question (81b30). Concerning the first he does two things. First, he prefaces something about the syllogistic form one must observe in demonstrations. Secondly, he re views what the matter of demonstration should be (81b14).
lib. 1 l. 31 n. 2 Circa primum tria tangit. Quorum primum est commune omni syllogismo, scilicet quod omnis syllogismus est per tres terminos; ut manifestum est in libro priorum. Secundum autem pertinet ad syllogismum affirmativum; cuius forma est talis quod concludit a esse in c propter id, quod a est in b, et b est in c; et haec est forma syllogistica in prima figura, in qua sola potest concludi affirmativa universalis, quae maxime quaeritur in demonstrationibus. Tertium est quod pertinet ad syllogismum negativum, qui de necessitate unam propositionem habet affirmativam, aliam autem negativam; differenter tamen in prima figura et in secunda, ut patet per ea, quae in libro priorum ostensa sunt.	With respect to the first (81b10) he touches on three things, the first of which is common to every syllogism, namely, that every syllogism is formed in three terms, as is indicated in Prior Analytics I. The second however, pertains to an affirmative syllogism whose form is such that it concludes A to be in C, because A is in B and B in C. And this is the form of a syllogism in the first figure in which alone can be concluded a universal affirmative, the chief quest in demonstration. The third pertains to a negative syllogism which of necessity has one affirmative proposition and one negative, but differently in the first figure and in the second, as is clear from what has been shown in the Prior Analytics.
lib. 1 l. 31 n. 3 Deinde cum dicit: manifestum igitur est etc., resumit quae sit materia demonstrationum. Et circa hoc tria facit: primo enim proponit demonstrationis materiam; secundo, ostendit differentiam huius materiae ad materiam syllogismi dialectici; ibi: secundum quidem igitur opinationem etc.; tertio, differentiam positam manifestat; ibi: habet autem sic se et cetera.	Then (81b14) he reviews what the matter of demonstration should be. In regard to this he does three things. First, he states what this matter is. Secondly, he shows the difference between this matter and the matter of a dialectical syllogism (81b18). Thirdly, he clarifies this difference (81b24).
Dicit ergo primo quod, cum syllogismus habeat tres terminos, ex quibus formantur duae propositiones concludentes tertiam, manifestum est quod hae propositiones, ex quibus proceditur in syllogismo demonstrativo secundum formam praedictam, sunt principia et suppositiones, de quibus in praecedentibus dictum est. Qui enim accipit huiusmodi principia, sic demonstrat per ea, sicut expositum est in forma syllogistica, ut scilicet quia a sit in c probatur per b; et si propositio a.b sit iterum mediata, quod a sit in b demonstratur per aliud medium. Et simile est si propositio minor, scilicet b.c, sit mediata.	He says therefore first (81b14) that since a syllogism has three terms from which are formed the two propositions which conclude the third, it is clear that these propositions, from which one proceeds in a demonstrative syllogism according to the aforesaid form, are the principles and suppositions we discussed earlier. For one who accepts such principles demonstrates through them in the syllogistic form we have mentioned, namely, that A is in C is proved through B; and if the proposition AB is mediate, another middle is used to demonstrate that A is in B. The like is done if the minor proposition, BC, is mediate.
lib. 1 l. 31 n. 4 Deinde cum dicit: secundum quidem igitur etc., ostendit quantum ad praedicta differentiam inter syllogismum demonstrativum et syllogismum dialecticum. Quia enim syllogismus dialecticus ad hoc tendit, ut opinionem faciat, hoc solum est de intentione dialectici, ut procedat ex his, quae sunt maxime opinabilia, et haec sunt ea, quae videntur vel pluribus, vel maxime sapientibus. Et ideo si dialectico in syllogizando occurrat aliqua propositio, quae secundum rei veritatem habeat medium, per quod possit probari, sed tamen non videatur habere medium, sed propter sui probabilitatem videatur esse per se nota; hoc sufficit dialectico, nec inquirit aliud medium, licet propositio sit mediata, et, ex ea syllogizans, sufficienter perficit dialecticum syllogismum.	Then (81b18), apropos of what has been said, he shows the difference between a demonstrative and a dialectical syllogism. For since the latter aims at producing opinion, the sole intent of a dialectician is to proceed from things that are most probable, and these are things that appear to the majority or to the very wise. Hence if a dialectician in syllogizing happens upon a proposition which really has a middle through which it could be proved, but it seems not to have a middle because it appears to be per se known on account of its probability, this is enough for the dialectician: he does not search for a middle, even though the proposition is mediate. Rather he syllogizes from it and completes the dialectical syllogism satisfactorily.
Sed syllogismus demonstrativus ordinatur ad scientiam veritatis; et ideo ad demonstratorem pertinet, ut procedat ex his, quae sunt secundum rei veritatem immediata. Et si occurrat ei mediata propositio, necesse est quod probet eam per medium proprium, quousque deveniat ad immediata, nec est contentus probabilitate propositionis.	The demonstrative syllogism, on the other hand, is ordained to scientific knowledge of the truth; accordingly, it pertains to the demonstrator to proceed from truths which are really immediate. Hence if he happens upon a mediate proposition, he must prove it through its proper middle until he reaches something immediate, because he is not content with the probability of a proposition.
lib. 1 l. 31 n. 5 Deinde cum dicit: habet autem se sic etc., manifestat quod dixerat, dicens quod hoc, quod dictum est, quod demonstrator ad veritatem ex his quae sunt procedit, sic se habet ut dicetur. Invenitur enim aliquid, quod de alio praedicatur, non secundum accidens, et hoc exponit per affirmativam, ostendens quid praedicetur secundum accidens.	Then (81b24) he elucidates what he has said, asserting that his claim that the demonstrator proceeds “to the truth from things that are” is supported by the fact that it is possible to find something “which is not predicated of a thing accidentally.” What this means he explains by showing how in the case of affirmative statements something is predicated accidentally.
Dupliciter enim aliquid praedicatur secundum accidens: uno modo, quando subiectum praedicatur de accidente, puta cum dicimus, album est homo; alio modo dissimiliter, quando accidens praedicatur de subiecto, sicut cum dicitur, homo est albus. Et differt hic modus a primo, quoniam hic, quando accidens praedicatur de subiecto, dicitur, homo est albus, non quia aliquid alterum sit album, sed quia ipse homo est albus: et tamen est propositio per accidens, quia album non convenit homini secundum propriam rationem. Non enim ponitur in definitione eius, neque e converso. Sed quando dicitur, album est homo, hoc non dicitur, quia esse hominem insit albo, sed quia esse hominem inest subiecto albi, cui scilicet accidit esse album. Unde hic modus est magis remotus a praedicatione per se, quam primus.	For something is predicated accidentally in two ways: in one way, when the subject is predicated of an accident, as when we say, “The white thing is a man”; in another way, when the accident is predicated of the subject, as when we say, “The man is white.” Now this way differs from the first, because when the accident is predicated of the subject, it is stated that the man is white not because something else is white but because the man himself is white. Yet it is a per accidens proposition, because “white” does not belong to man according to the specific nature of man. For neither is placed in the definition of the other. But when it is stated that the white thing is a man, it is not so stated because being a man is in the whiteness, but because being a man is in the subject of whiteness, which subject happens to be white. Hence this way is further removed from per se predication than the first.
Sunt autem quaedam, quae neutro istorum modorum per accidens praedicantur; et ista dicuntur per se. Et talia sunt, ex quibus demonstrator procedit. Sed hoc dialecticus non requirit, et ideo quaestio, quae infra proponitur de huiusmodi quae per se praedicantur, non habet locum in syllogismis dialecticis, sed solum in syllogismo demonstrativo.	But there are certain things which are not predicated per accidens in either of these ways: these are said to be per se. Such are the things from which the demonstrator proceeds. But the dialectician is not so demanding; consequently, the question concerning such things as are predicated per se is not relevant to the dialectical syllogism but only to the demonstrative syllogism.
lib. 1 l. 31 n. 6 Deinde cum dicit: sit igitur c huiusmodi etc., movet quaestiones intentas. Et circa hoc duo facit: primo, movet quaestiones in quibus locum habent; secundo, ostendit in quibus locum non habent; ibi: sed in convertentibus et cetera. Circa primum duo facit: primo, movet quaestiones in demonstrationibus affirmativis; secundo, ostendit quod hae quaestiones similiter locum habent in demonstrationibus negativis; ibi: similiter autem dico et in privativis et cetera. Circa primum duo facit: primo, movet quaestiones; secundo, ostendit ad quid huiusmodi quaestiones pertineant; ibi: est autem hoc intendere et cetera.	Then (81b30) he raises the questions he intended. Concerning this he does two things. First, he raises the questions in regard to things to which they are relevant. Secondly, he shows the cases in which they are not relevant (82a15). Concerning the first he does two things. First, he raises questions in affirmative demonstrations. Secondly, he shows that these questions also have relevance in negative demonstrations (82a9). In regard to the first he does two things. First, he raises the questions. Secondly, he shows where such questions are relevant (82a7).
lib. 1 l. 31 n. 7 Circa primum movet tres quaestiones secundum tres terminos syllogismi. Et primo, movet quaestionem ex parte maioris extremitatis, utrum sit abire in infinitum ascendendo. Et in hac quaestione supponitur ultimum subiectum, quod non praedicatur de alio, et alia praedicantur de ipso. Sit ergo hoc c, et in c primo et immediate sit b, et in b sit e quasi de eo universaliter praedicatum, et iterum f sit in e similiter de eo universaliter praedicatum. Est ergo quaestio: utrum iste ascensus alicubi stet, ita scilicet quod sit devenire ad aliquid quod praedicetur de aliis universaliter, et nihil aliud praedicetur de ipso; aut hoc non sit necesse, sed contingat ascendere in infinitum?	In regard to the first (81b30) he raises three questions corresponding to the three terms of a syllogism. First, he raises a question concerning the major extreme, namely, whether one can go to infinity in ascending order? In this question an ultimate subject is supposed which is not predicated of any other, but other things are predicated of it. Let this subject be C, and let B be in C first and immediately, and let E be in B, as universally predicated of B; furthermore, let F be in E as universally predicated of it. The question is this: Should this ascending process come to a halt somewhere, so that something is reached which is predicated universally of other things but nothing else is predicated of it, or is that not necessary but a process to infinity occurs?
lib. 1 l. 31 n. 8 Secundo, ibi: et iterum si de a quidem etc., movet quaestionem ex parte minoris termini, utrum scilicet sit ire in infinitum descendendo. Et in hac quaestione supponitur esse aliquod primum praedicatum universale, quod de aliis praedicetur, et nihil sit universalius eo, quod praedicetur de ipso. Sit ergo a tale, quod nihil de eo praedicetur sicut totum universale de parte, a vero praedicetur de c primo et immediate, et c de I, et I de b. Est ergo quaestio: utrum necesse sit hic descendendo stare, aut contingat in infinitum ire?	Secondly (81b34) he raises the question on the part of the minor term, namely, whether one can go to infinity in descending. In this question some first universal predicate is supposed which is predicated of other things and nothing is more universal than it so as to be predicated of it. Thus let A be such that nothing is predicated of it as a universal whole of a part, but A is predicated of H both first and immediately, and H of G, and G of B. The question then is this: Is it necessary to come to a halt in this descending process, or may it proceed to infinity?
lib. 1 l. 31 n. 9 Et ostendit consequenter differentiam harum duarum quaestionum, quia in prima quaestione quaerebatur: si aliquis incipiat a particularissimo subiecto, quod nulli inest per modum quo totum universale inest parti, sed alia insunt ei, utrum contingat procedere in infinitum ascendendo? Secunda vero quaestio est: si aliquis incipiat ab universalissimo praedicato, quod praedicatur de aliis sicut totum universale de parte, et nihil hoc modo praedicatur de illo, utrum contingat descendendo procedere in infinitum?	Then he shows the difference between these two questions. For in the first one we asked: If someone begins from a most particular subject which is in nothing else the way a universal whole is in a part but other things are in it, does an infinite ascending process occur? But in the second we are asking: If someone begins with a most universal predicate, which is predicated of other things as a universal whole of its parts but nothing is predicated of it in this way, does an infinite descending process occur?
lib. 1 l. 31 n. 10 Tertio, ibi: amplius media etc., movet tertiam quaestionem ex parte medii termini. Et in hac quaestione supponuntur duo extrema determinata, scilicet universalissimum praedicatum, et particularissimum subiectum; et quaeritur cum hoc, utrum possint esse infinita media: puta, si a sit universalissimum praedicatum, et c sit particularissimum subiectum, et inter a et c sit medium b, et inter a et b iterum sit aliud, et similiter inter b et c, et horum etiam mediorum sint alia media, inter ipsa scilicet et extrema, tam ascendendo quam descendendo. Est ergo quaestio: utrum hoc possit procedere in infinitum, aut hoc sit impossibile?	Thirdly (82a2) he raises the third question on the part of the middle term. In this question two determinate extremes are supposed, namely, a most universal predicate and a most particular subject. The question is whether under these conditions there can be an infinity of middles. Thus, if A is the most universal predicate and C the most particular subject, and if between A and C there is the middle, B, and again between A and B another middle, and likewise between B and C; furthermore, if there are other middles of these middles between them and the extremes both in ascending and in descending order. The question then is this: May these processes go on to infinity or is that impossible?
lib. 1 l. 31 n. 11 Deinde cum dicit: est autem hoc intendere etc., ostendit ad quid tendant huiusmodi quaestiones; in quo declaratur quod huiusmodi quaestiones pertinent ad materiam, de qua nunc agitur, scilicet ad demonstrationes. Dicit ergo quod intendere inquisitioni veritatis in istis quaestionibus idem est ac si quaeratur, utrum demonstrationes procedant in infinitum, vel ascendendo vel descendendo. Ascendendo quidem, ita quod quaelibet propositio, ex qua demonstratio procedit, sit demonstrabilis per aliam priorem demonstrationem; et hoc est quod subiungit, et si est demonstratio omnis, idest cuiuslibet propositionis. Quod quidam existimantes, circa principia erraverunt, ut dicitur in IV metaphysicae. Descendendo autem, si ex qualibet propositione demonstrata contingat iterum ad aliam demonstrationem posteriorem procedere. Et hoc est unum membrum dubitationis, si demonstrationes in infinitum procedunt, vel descendendo vel ascendendo. Aliud autem membrum dubitationis est, si demonstrationes ad invicem terminantur, ita scilicet quod una demonstratio confirmetur per aliam ascendendo, et ex una demonstratione procedat alia descendendo, et hoc usque ad aliquem terminum.	Then (82a7) he shows what is the tenor of these questions. And he says that these questions pertain to the matter now under discussion, namely, to demonstrations. He says, therefore, that the attempt to reach true answers to these questions is the same as trying to settle the question whether demonstrations proceed to infinity by ascending or descending. By ascending, i.e., so that each proposition from which a demonstration proceeds would be demonstrable by another prior proposition. This is what he means when he asks: “Is there a demonstration of everything,” i.e., of every proposition? By so thinking, some have erred in regard to principles, as is stated in Metaphysics IV. And by descending, i.e., whether it is possible from any demonstrated proposition to proceed again to another demonstration subsequent to it. Thus, one element of the doubt is whether demonstrations proceed to infinity either by ascending or by descending. The other element is whether demonstrations are mutually limiting, so that one demonstration may be confirmed by another in an ascending process, and from one demonstration another may proceed by a descending process: and this until a limit is reached.
lib. 1 l. 31 n. 12 Deinde cum dicit: similiter autem dico etc., ostendit quod praedictae dubitationes habent locum etiam in demonstrationibus negativis, quia demonstratio negativa oportet quod utatur propositione affirmativa, in qua subiectum conclusionis contineatur sub medio, a quo praedicatum conclusionis removeatur. Secundum ergo quod est ascensus et descensus in affirmativis, oportet quod sit ascensus et descensus in negativis syllogismis, et propositionibus; ut puta si conclusio demonstrativi syllogismi sit, nullum c est a, et accipiatur sicut medium b, a quo a removeatur. Est ergo primo considerandum utrum a removeatur a b primo, sive immediate, aut sit aliquod medium accipere, a quo primo removeatur a quam a b, puta si prius removeatur ab I, quod oportet universaliter praedicari de b; et iterum erit considerandum utrum a removeatur ab aliquo per prius quam ab I, scilicet a t, quod praedicatur universaliter de I. Ita ergo et in his potest procedi in infinitum in removendo, ut semper sit aliquid accipere, a quo per prius removeatur, vel oportet alicubi stare.	Then (82a9) he shows that these questions are also relevant to negative demonstrations, because a negative demonstration must employ an affirmative proposition in which the subject of the conclusion is contained under the middle and from which the predicate of the conclusion is removed. Therefore, to the extent that there is ascent and descent in affirmative, there must be ascent and descent in negative syllogisms and propositions. For example, if the conclusion of a demonstrative syllogism is, “No C is A,” and the middle taken is B, from which A is removed: the first thing to be considered, therefore, is whether A is removed from B first and immediately, or whether there is another middle G to be taken, from which A would be removed before it would be removed from B. In that case it would be necessary to consider whether A would be removed from something else before G, namely, from H which is predicated universally of G. Therefore in these also the question arises whether one can proceed to infinity in removing (so that something would always remain from which it would have to be removed), or must one stop somewhere.
lib. 1 l. 31 n. 13 Deinde cum dicit: sed in convertibilibus etc., ostendit in quibus praedictae quaestiones locum non habeant. Quia in his, quae aequaliter de se invicem praedicantur et convertuntur ad invicem, non est accipere aliquod prius et posterius secundum illum modum, quo prius est, a quo non convertitur consequentia essendi, prout universalia sunt priora; quia sive praedicata sint infinita, ita scilicet quod procedatur in infinitum in praedicando, sive sint infinita ex utraque parte, idest tam ex parte praedicati quam ex parte subiecti, omnia huiusmodi infinita similiter se habebunt ad omnia; quia quodlibet eorum poterit praedicari de quolibet, et subiici cuilibet convertibilium. Nisi solum quod potest esse talis differentia, quod unum eorum praedicatur ut accidens, et aliud praedicatur sicut praedicamentum, idest sicut substantiale praedicatum. Et haec est differentia proprii et definitionis, quorum utrumque est convertibile; et tamen definitio est praedicatum essentiale, et propter hoc est prius naturaliter proprio, quod est praedicatum accidentale. Et inde est quod in demonstrationibus utimur definitione quasi medio ad demonstrandum propriam passionem de subiecto.	Then (82a15) he shows the cases in which these questions have no relevance: for in cases in which there is mutual predication and mutual conversion there is no prior and subsequent to be taken in the sense in which the prior [notion] is that with which a subsequent [notion] is not convertible, as universals are prior; because no matter whether the predicates be infinite, so that one might proceed to infinity in predicating, or whether there be infinity on both sides, i.e., on the side of the predicate as well as of the subject, all such infinites bear a like relationship to all, because any of them could be predicated of any other and be the subject of any of the convertibles. However, there can be this difference: one of them might be predicated as an accident and another as a predicament, i.e., as a substantial predicate. And this is the difference between a property, and a definition: although the two are convertible with the subject ,nevertheless the definition is an essential predicate and therefore naturally prior to the property, which is an accidental predicate. That is why in demonstrations we use the definition as the middle to demonstrate a proper attribute of the subject.