Authors/Thomas Aquinas/posteriorum/L1

Lecture 1 The need for pre-existent knowledge in all learning

Latin	English
	English translation by Fabian R. Larcher, O.P.
lib. 1 l. 1 n. 1 Sicut dicit Aristoteles in principio metaphysicae, hominum genus arte et rationibus vivit: in quo videtur philosophus tangere quoddam hominis proprium quo a caeteris animalibus differt. Alia enim animalia quodam naturali instinctu ad suos actus aguntur; homo autem rationis iudicio in suis actionibus dirigitur. Et inde est quod ad actus humanos faciliter et ordinate perficiendos diversae artes deserviunt. Nihil enim aliud ars esse videtur, quam certa ordinatio rationis quomodo per determinata media ad debitum finem actus humani perveniant. Ratio autem non solum dirigere potest inferiorum partium actus, sed etiam actus sui directiva est. Hoc enim est proprium intellectivae partis, ut in seipsam reflectatur: nam intellectus intelligit seipsum et similiter ratio de suo actu ratiocinari potest. Si igitur ex hoc, quod ratio de actu manus ratiocinatur, adinventa est ars aedificatoria vel fabrilis, per quas homo faciliter et ordinate huiusmodi actus exercere potest; eadem ratione ars quaedam necessaria est, quae sit directiva ipsius actus rationis, per quam scilicet homo in ipso actu rationis ordinate, faciliter et sine errore procedat.	As the Philosopher says in Metaphysics I (980b26), “the human race lives by art and reasonings.” In this statement the Philosopher seems to touch upon that property whereby man differs from the other animals. For the other animals are prompted to their acts by a natural impulse, but man is directed in his actions by a judgment of reason. And this is the reason why there are various arts devoted to the ready and orderly performance of human acts. For an art seems to be nothing more than a definite and fixed procedure established by reason, whereby human acts reach their due end through appropriate means. Now reason is not only able to direct the acts of the lower powers but is also director of its own act: for what is peculiar to the intellective part of man is its ability to reflect upon itself. For the intellect knows itself. In like manner reason is able to reason about its own act. Therefore just as the art of building or carpentering, through which man is enabled to perform manual acts in an easy and orderly manner, arose from the fact that reason reasoned about manual acts, so in like manner an art is needed to direct the act of reasoning, so that by it a man when performing the act of reasoning might proceed in an orderly and easy manner and without error.
lib. 1 l. 1 n. 2 Et haec ars est logica, idest rationalis scientia. Quae non solum rationalis est ex hoc, quod est secundum rationem (quod est omnibus artibus commune); sed etiam ex hoc, quod est circa ipsum actum rationis sicut circa propriam materiam.	And this art is logic, i.e., the science of reason. And it concerns reason not only because it is according to reason, for that is common to all arts, but also because it is concerned with the very act of reasoning as with its proper matter.
lib. 1 l. 1 n. 3 Et ideo videtur esse ars artium, quia in actu rationis nos dirigit, a quo omnes artes procedunt. Oportet igitur logicae partes accipere secundum diversitatem actuum rationis.	Therefore it seems to be the art of the arts, because it directs us in the act of reasoning, from which all arts proceed. Consequently one should view the parts of logic according to the diversity among the acts of reason.
lib. 1 l. 1 n. 4 Sunt autem rationis tres actus: quorum primi duo sunt rationis, secundum quod est intellectus quidam. Una enim actio intellectus est intelligentia indivisibilium sive incomplexorum, secundum quam concipit quid est res. Et haec operatio a quibusdam dicitur informatio intellectus sive imaginatio per intellectum. Et ad hanc operationem rationis ordinatur doctrina, quam tradit Aristoteles in libro praedicamentorum. Secunda vero operatio intellectus est compositio vel divisio intellectus, in qua est iam verum vel falsum. Et huic rationis actui deservit doctrina, quam tradit Aristoteles in libro perihermeneias. Tertius vero actus rationis est secundum id quod est proprium rationis, scilicet discurrere ab uno in aliud, ut per id quod est notum deveniat in cognitionem ignoti. Et huic actui deserviunt reliqui libri logicae.	Now there are three acts of the reason, the first two of which belong to reason regarded as an intellect. One action of the intellect is the understanding of indivisible or uncomplex things, and according to this action it conceives what a thing is. And this operation is called by some the informing of the intellect, or representing by means of the intellect. To this operation of the reason is ordained the doctrine which Aristotle hands down in the book of Predicaments, [i.e., Categories]. The second operation of the intellect is its act of combining or dividing, in which the true or the false are for the first time present. And this act of reason is the subject of the doctrine which Aristotle hands down in the book entitled On Interpretation. But the third act of the reason is concerned with that which is peculiar to reason, namely, to advance from one thing to another in such a way that through that which is known a man comes to a knowledge of the unknown. And this act is considered in the remaining books of logic.
lib. 1 l. 1 n. 5 Attendendum est autem quod actus rationis similes sunt, quantum ad aliquid, actibus naturae. Unde et ars imitatur naturam in quantum potest. In actibus autem naturae invenitur triplex diversitas. In quibusdam enim natura ex necessitate agit, ita quod non potest deficere. In quibusdam vero natura ut frequentius operatur, licet quandoque possit deficere a proprio actu. Unde in his necesse est esse duplicem actum; unum, qui sit ut in pluribus, sicut cum ex semine generatur animal perfectum; alium vero quando natura deficit ab eo quod est sibi conveniens, sicut cum ex semine generatur aliquod monstrum propter corruptionem alicuius principii. Et haec etiam tria inveniuntur in actibus rationis. Est enim aliquis rationis processus necessitatem inducens, in quo non est possibile esse veritatis defectum; et per huiusmodi rationis processum scientiae certitudo acquiritur. Est autem alius rationis processus, in quo ut in pluribus verum concluditur, non tamen necessitatem habens. Tertius vero rationis processus est, in quo ratio a vero deficit propter alicuius principii defectum; quod in ratiocinando erat observandum.	It should be noted that the acts of reason are in a certain sense not unlike the acts of nature: hence so far as it can, art imitates nature. Now in the acts of nature we observe a threefold diversity. For in some of them nature acts from necessity, i.e., in such a way that it cannot fail; in others, nature acts so as to succeed for the most part, although now and then it fails in its act. Hence in this latter case there must be a twofold act: one which succeeds in the majority of cases, as when from seed is generated a perfect animal; the other when nature fails in regard to what is appropriate to it, as when from seed something monstrous is generated owing to a defect in some principle. These three are found also in the acts of the reason. For there is one process of reason which induces necessity, where it is not possible to fall short of the truth; and by such a process of reasoning the certainty of science is acquired. Again, there is a process of reason in which something true in most cases is concluded but without producing necessity. But the third process of reason is that in which reason fails to reach a truth because some principle which should have been observed in reasoning was defective.
lib. 1 l. 1 n. 6 Pars autem logicae, quae primo deservit processui, pars iudicativa dicitur, eo quod iudicium est cum certitudine scientiae. Et quia iudicium certum de effectibus haberi non potest nisi resolvendo in prima principia, ideo pars haec analytica vocatur, idest resolutoria. Certitudo autem iudicii, quae per resolutionem habetur, est, vel ex ipsa forma syllogismi tantum, et ad hoc ordinatur liber priorum analyticorum, qui est de syllogismo simpliciter; vel etiam cum hoc ex materia, quia sumuntur propositiones per se et necessariae, et ad hoc ordinatur liber posteriorum analyticorum, qui est de syllogismo demonstrativo.	Now the part of logic which is devoted to the first process is called the judicative part, because it leads to judgments possessed of the certitude of science. And because a certain and sure judgment touching effects cannot be obtained except by analyzing them into their first principles, this part is called analytical, i.e., resolvent. Furthermore, the certitude obtained by such an analysis of a judgment is derived either from the mere form of the syllogism—and to this is ordained the book of the Prior Ana1ytics which treats of the syllogism as such—or from the matter along with the form, because the propositions employed are per se and necessary [cf. infra, Lectures 10, 13]—and to this is ordained the book of the Posterior Analytics which is concerned with the demonstrative syllogism.
Secundo autem rationis processui deservit alia pars logicae, quae dicitur inventiva. Nam inventio non semper est cum certitudine. Unde de his, quae inventa sunt, iudicium requiritur, ad hoc quod certitudo habeatur. Sicut autem in rebus naturalibus, in his quae ut in pluribus agunt, gradus quidam attenditur (quia quanto virtus naturae est fortior, tanto rarius deficit a suo effectu), ita et in processu rationis, qui non est cum omnimoda certitudine, gradus aliquis invenitur, secundum quod magis et minus ad perfectam certitudinem acceditur. Per huiusmodi enim processum, quandoque quidem, etsi non fiat scientia, fit tamen fides vel opinio propter probabilitatem propositionum, ex quibus proceditur: quia ratio totaliter declinat in unam partem contradictionis, licet cum formidine alterius, et ad hoc ordinatur topica sive dialectica. Nam syllogismus dialecticus ex probabilibus est, de quo agit Aristoteles in libro topicorum. Quandoque vero, non fit complete fides vel opinio, sed suspicio quaedam, quia non totaliter declinatur ad unam partem contradictionis, licet magis inclinetur in hanc quam in illam. Et ad hoc ordinatur rhetorica. Quandoque vero sola existimatio declinat in aliquam partem contradictionis propter aliquam repraesentationem, ad modum quo fit homini abominatio alicuius cibi, si repraesentetur ei sub similitudine alicuius abominabilis. Et ad hoc ordinatur poetica; nam poetae est inducere ad aliquod virtuosum per aliquam decentem repraesentationem. Omnia autem haec ad rationalem philosophiam pertinent: inducere enim ex uno in aliud rationis est. Tertio autem processui rationis deservit pars logicae, quae dicitur sophistica, de qua agit Aristoteles in libro elenchorum.	To the second process of reason another part of logic called investigative is devoted. For investigation is not always accompanied by certitude. Hence in order to have certitude a judgment must be formed, bearing on that which has been investigated. But just as in the works of nature which succeed in the majority of cases certain levels are achieved—because the stronger the power of nature the more rarely does it fail to achieve its effect—so too in that process of reason which is not accompanied by complete certitude certain levels are found accordingly as one approaches more or less to complete certitude. For although science is not obtained by this process of reason, nevertheless belief or opinion is sometimes achieved (on account of the provability of the propositions one starts with), because reason leans completely to one side of a contradiction but with fear concerning the other side. The Topics or dialectics is devoted to this. For the dialectical syllogism which Aristotle treats in the book of Topics proceeds from premises which are provable. At times, however, belief or opinion is not altogether achieved, but suspicion is, because reason does not lean to one side of a contradiction unreservedly, although it is inclined more to one side than to the other. To this the Rhetoric is devoted. At other times a mere fancy inclines one to one side of a contradiction because of some representation, much as a man turns in disgust from certain food if it is described to him in terms of something disgusting. And to this is ordained the Poetics. For the poet’s task is to lead us to something virtuous by some excellent description. And all these pertain to the philosophy of the reason, for it belongs to reason to pass from one thing to another. The third process of reasoning is served by that part of logic which is called sophistry, which Aristotle treats in the book On Sophistical Refutations.
	Lecture I (71al-10) THE NEED FOR PRE-EXISTENT KNOWLEDGE IN ALL LEARNING
lib. 1 l. 1 n. 7 Aliis igitur partibus logicae praetermissis, ad praesens intendendum est circa partem iudicativam, prout traditur in libro posteriorum analyticorum. Qui dividitur in partes duas: in prima, ostendit necessitatem demonstrativi syllogismi, de quo est iste liber; in secunda, de ipso syllogismo demonstrativo determinat; ibi: scire autem opinamur et cetera.	Leaving aside the other parts of logic, we shall fix our attention on the judicative part as it is presented in the book of Posterior Analytics which is divided into two parts. In the first he shows the need for the demonstrative syllogism, with which this book is concerned. In the second part he comes to a decision concerning that syllogism (71b8) [Lect. 4].
lib. 1 l. 1 n. 8 Necessitas autem cuiuslibet rei ordinatae ad finem ex suo fine sumitur; finis autem demonstrativi syllogismi est acquisitio scientiae; unde, si scientia acquiri non posset per syllogismum vel argumentum, nulla esset necessitas demonstrativi syllogismi. Posuit autem Plato quod scientia in nobis non causatur ex syllogismo, sed ex impressione formarum idealium in animas nostras, ex quibus etiam effluere dicebat formas naturales in rebus naturalibus, quas ponebat esse participationes quasdam formarum a materia separatarum. Ex quo sequebatur quod agentia naturalia non causabant formas in rebus inferioribus, sed solum materiam praeparabant ad participandum formas separatas. Et similiter ponebat quod per studium et exercitium non causatur in nobis scientia; sed tantum removentur impedimenta, et reducitur homo quasi in memoriam eorum, quae naturaliter scit ex impressione formarum separatarum. Sententia autem Aristotelis est contraria quantum ad utrumque. Ponit enim quod formae naturales reducuntur in actum a formis quae sunt in materia, scilicet a formis naturalium agentium. Et similiter ponit quod scientia fit in nobis actu per aliquam scientiam in nobis praeexistentem. Et hoc est fieri in nobis scientiam per syllogismum aut argumentum quodcumque. Nam ex uno in aliud argumentando procedimus.	Now the need for anything directed to an end is caused by that end. But the end of the demonstrative syllogism is the attainment of science. Hence if science could not be achieved by syllogizing or arguing, there would be no need for the demonstrative syllogism. Plato, as a matter of fact, held that science in us is not the result of a syllogism but of an impression upon our minds of ideal forms from which, he said, are also derived the natural forms in natural things, which he supposed were participations of forms separated from matter. From this it followed that natural agents were not the causes of forms in natural things but merely prepared the matter for participating in the separated forms. In like fashion he postulated that science in us is not caused by study and exercise, but only that obstacles are removed and man is brought to recall things which he naturally understands in virtue of an imprint of separated forms. But Aristotle’s view is opposed to this on two counts. For he maintains that natural forms are made actual by forms present in matter, i.e., by the forms of natural agents. He further maintains that science is made actual in us by other knowledge already existing in us. This means that it is formed in us through a syllogism or some type of argument. For in arguing we proceed from one thing into another.
lib. 1 l. 1 n. 9 Ad ostendendum igitur necessitatem demonstrativi syllogismi, praemittit Aristoteles quod cognitio in nobis acquiritur ex aliqua cognitione praeexistenti. Duo igitur facit: primo, ostendit propositum; secundo, docet modum praecognitionis; ibi: dupliciter autem et cetera. Circa primum duo facit. Primo, inducit universalem propositionem propositum continentem, scilicet quod acceptio cognitionis in nobis fit ex aliqua praeexistenti cognitione. Et ideo dicit: omnis doctrina et omnis disciplina, non autem omnis cognitio, quia non omnis cognitio ex priori cognitione dependet: esset enim in infinitum abire. Omnis autem disciplinae acceptio ex praeexistenti cognitione fit. Nomen autem doctrinae et disciplinae ad cognitionis acquisitionem pertinet. Nam doctrina est actio eius, qui aliquid cognoscere facit; disciplina autem est receptio cognitionis ab alio. Non autem accipitur hic doctrina et disciplina secundum quod se habent ad acquisitionem scientiae tantum, sed ad acquisitionem cognitionis cuiuscumque. Quod patet, quia manifestat hanc propositionem etiam in disputativis et rhetoricis disputationibus, per quas non acquiritur scientia. Propter quod etiam non dicit ex praeexistenti scientia vel intellectu, sed universaliter cognitione. Addit autem intellectiva ad excludendum acceptionem cognitionis sensitivae vel imaginativae. Nam procedere ex uno in aliud rationis est solum.	Therefore, in order to show the need for demonstrative syllogism Aristotle begins by stating that some of our knowledge is acquired from knowledge already existing. Hence he does two things. First, he states his thesis. Secondly, he explains the character of prior knowledge (71a11) [Lect. 2]. Concerning the first he does two things. First (71a1), he asserts a universal proposition containing his thesis, namely, that the production of knowledge in us is caused from knowledge already existing; hence he says, “Every doctrine and every discipline...” He does not say, “all knowledge,” because not all knowledge depends on previous knowledge, for that would involve an infinite process: but the acquisition of every discipline comes from knowledge already possessed. For the names “doctrine” and “discipline” pertain to the learning process, doctrine being the action exerted by the one who makes us know, and discipline the reception of knowledge from another. Furthermore, “doctrine” and “discipline” are not taken here as pertaining only to the acquisition of scientific knowledge but to the acquiring of any knowledge. That this is so is evidenced by the fact that he explains the proposition even in regard to dialectical and rhetorical disputations, neither of which engenders science. Hence this is another reason why he did not say, “from pre-existent science or intuition,” but “knowledge” universally. However he does add, “intellectual,” in order to preclude knowledge acquired by sense or imagination. For reason alone proceeds from one thing into another.
lib. 1 l. 1 n. 10 Secundo, cum dicit: mathematicae enim etc., manifestat propositionem praemissam per inductionem. Et primo, in demonstrativis in quibus acquiritur scientia. In his autem principaliores sunt mathematicae scientiae, propter certissimum modum demonstrationis. Consequenter autem sunt et omnes aliae artes, quia in omnibus est aliquis modus demonstrationis, alias non essent scientiae.	Then (71a3) he employs induction to prove his thesis; and first of all in regard to those demonstrations in which scientific knowledge is acquired. Of these the best are the mathematical sciences because of their most certain manner of demonstrating. After them come the other arts, because some manner of demonstrating is found in all of them; otherwise they would not be sciences.
lib. 1 l. 1 n. 11 Secundo, cum dicit: similiter autem etc., manifestat idem in orationibus disputativis sive dialecticis, quae utuntur syllogismo et inductione: in quorum utroque proceditur ex aliquo praecognito. Nam in syllogismo accipitur cognitio alicuius universalis conclusi ab aliis universalibus notis. In inductione autem concluditur universale ex singularibus, quae sunt manifesta quantum ad sensum.	Secondly (71a4), he proves the same thing in regard to disputative, i.e., dialectical, arguments, because they employ syllogism and induction, in each of which the process starts from something already known. For in a syllogism the knowledge of some universal conclusion is obtained from other universals already known; in induction, however, a universal is concluded from singulars made known in sense-perception.
lib. 1 l. 1 n. 12 Tertio, cum dicit: similiter autem rhetoricae etc., manifestat idem in rhetoricis, in quibus persuasio fit per enthymema aut per exemplum; non autem per syllogismum vel inductionem completam, propter incertitudinem materiae circa quam versatur, scilicet circa actus singulares hominum, in quibus universales propositiones non possunt assumi vere. Et ideo utitur loco syllogismi, in quo necesse est esse aliquam universalem, aliquo enthymemate; et similiter loco inductionis, in qua concluditur universale, aliquo exemplo, in quo proceditur a singulari, non ad universale, sed ad singulare. Unde patet quod, sicut enthymema est quidam syllogismus detruncatus, ita exemplum est quaedam inductio imperfecta. Si ergo in syllogismo et inductione proceditur ex aliquo praecognito, oportet idem intelligi in enthymemate et exemplo.	Thirdly (71a8), he manifests the same thing in rhetorical arguments, in which persuasion is produced through an enthymeme or example but not through a syllogism or complete induction because of the uncertainty attending the matters discussed, namely, the individual acts of men in which universal propositions cannot be truthfully assumed. Therefore, in place of a syllogism in which there must be something universal, an enthymeme is employed in which it is not necessary to have something universal. Similarly, in place of induction in which a universal is concluded, an example is employed in which one goes from the singular not to the universal but to the singular. Hence it is clear that just as the enthymeme is an abridged syllogism, so an example is an incomplete induction. Therefore, if in the case of the syllogism and induction one proceeds from knowledge already existing, the same must be granted in the case of the enthymeme and example.

Lecture 2 Extent and order of the pre-existent knowledge required for obtaining science

Latin	English
	Lecture 2 (71a11-23) EXTENT AND ORDER OF THE PRE-EXISTENT KNOWLEDGE REQUIRED FOR OBTAINING SCIENCE
lib. 1 l. 2 n. 1 Postquam ostendit philosophus quod omnis disciplina ex praeexistenti fit cognitione nunc ostendit quis sit modus praecognitionis. Et circa hoc duo facit: primo, determinat modum praecognitionis quantum ad illa quae oportet praecognoscere ut habeatur cognitio conclusionis, cuius scientia quaeritur; secundo, determinat modum praecognitionis ipsius conclusionis, cuius scientia per demonstrationem quaeritur; ibi: antequam sit inducere et cetera. In praecognitione autem duo includuntur, scilicet cognitio et cognitionis ordo. Primo ergo, determinat modum praecognitionis quantum ad cognitionem ipsam; secundo, quantum ad cognitionis ordinem; ibi: est autem cognoscere et cetera.	After showing that every discipline is developed from knowledge already existing, the Philosopher shows what is the extent of this preexisting knowledge. Concerning this he does two things. First, he determines the extent of pre-existing knowledge in regard to the things that must be known in order to attain knowledge of the conclusion, of which scientific knowledge is sought. Secondly, he determines the extent of pre-existing knowledge of the conclusion, of which scientific knowledge is sought through demonstration (71a24) [L.3]. Now two things are included in pre-existing knowledge, namely, the knowledge and the order of the knowledge. First, therefore, he determines the extent of pre-existing knowledge so far as the knowledge itself is concerned. Secondly, so far as the order of the knowledge is concerned (71a16).
lib. 1 l. 2 n. 2 Circa primum sciendum est quod id cuius scientia per demonstrationem quaeritur est conclusio aliqua in qua propria passio de subiecto aliquo praedicatur: quae quidem conclusio ex aliquibus principiis infertur. Et quia cognitio simplicium praecedit cognitionem compositorum, necesse est quod, antequam habeatur cognitio conclusionis, cognoscatur aliquo modo subiectum et passio. Et similiter oportet quod praecognoscatur principium, ex quo conclusio infertur, cum ex cognitione principii conclusio innotescat.	In regard to the first it should be noted that the object of which scientific knowledge is sought through demonstration is some conclusion in which a proper attribute is predicated of some subject, which conclusion is inferred from the principles. And because the knowledge of simple things precedes the knowledge of compound things, it is necessary -that the subject and the proper attribute be somehow known before knowledge of the conclusion is obtained. In like manner it is required thatthe principle be known from which the conclusion is inferred, for the conclusion is made known from a knowledge of the principle.
lib. 1 l. 2 n. 3 Horum autem trium, scilicet, principii, subiecti et passionis est duplex modus praecognitionis, scilicet, quia est et quid est. Ostensum est autem in VII metaphysicae quod complexa non definiuntur. Hominis enim albi non est aliqua definitio et multo minus enunciationis alicuius. Unde cum principium sit enunciatio quaedam, non potest de ipso praecognosci quid est, sed solum quia verum est. De passione autem potest quidem sciri quid est, quia, ut in eodem libro ostenditur, accidentia quodammodo definitionem habent. Passionis autem esse et cuiuslibet accidentis est inesse subiecto: quod quidem demonstratione concluditur. Non ergo de passione praecognoscitur quia est, sed quid est solum. Subiectum autem et definitionem habet et eius esse a passione non dependet; sed suum esse proprium praeintelligitur ipsi esse passionis in eo. Et ideo de subiecto oportet praecognoscere et quid est et quia est: praesertim cum ex definitione subiecti et passionis sumatur medium demonstrationis.	Now the extent of pre-existent knowledge of these three items, i.e., of the principle, of the subject, and of the proper attribute, is limited to knowing two things about them, namely, that each is and what each is. But, as stated in Metaphysics VII, complex things are not defined. For there is no definition of “white man,” much less of an enunciation [proposition]. Hence since a principle is an enunciation, there cannot be preexisting knowledge of what it is but only of the fact that it is true. But in regard to the proper attribute, it is possible to know what it is, because, as is pointed out in the same book, accidents do have some sort of definition. Now the being of a proper attribute and of any accident is being in a subject; and this fact is concluded by the demonstration. Consequently, it is not known beforehand that the proper attribute exists, but only what it is. The subject, too, has a definition; moreover, its being does not depend on the proper attribute—rather its own being is known before one knows the proper attribute to be in it. Consequently, it is necessary to know both what the subject is and that it is, especially since the medium of demonstration is taken from the definition of the subject of the proper attribute.
lib. 1 l. 2 n. 4 Propter hoc ergo dicit philosophus quod dupliciter necessarium est praecognoscere: quia duo sunt quae praecognoscuntur de his, quorum praecognitionem habemus, scilicet quia est et quid est. Et quod alia sunt de quibus necesse est primo cognoscere quia sunt, sicut principia de quibus postea exemplificat, ponens in exemplo primum omnium principiorum, scilicet quod de unoquoque est affirmatio vel negatio vera.	This, therefore, is why the Philosopher says (71a11) that it is necessary to know beforehand in two ways; because two items are known beforehand concerning things of which we have pre-existing knowledge, namely, that it is and what it is. [Then he goes on to say] that there are some things concerning which it is necessary first to know that they are, such as principles, concerning which he then gives examples, citing as one example the first of all principles, namely, “There is true affirmation or negation about everything.”
lib. 1 l. 2 n. 5 Alia vero sunt, de quibus oportet praeintelligere quid est quod dicitur, idest quid significatur per nomen, scilicet de passionibus. Et non dicit quid est simpliciter, sed quid est quod dicitur, quia antequam sciatur de aliquo an sit, non potest sciri proprie de eo quid est: non entium enim non sunt definitiones. Unde quaestio, an est, praecedit quaestionem, quid est. Sed non potest ostendi de aliquo an sit, nisi prius intelligatur quid significatur per nomen. Propter quod etiam philosophus in IV metaphysicae, in disputatione contra negantes principia docet incipere a significatione nominum. Exemplificat autem de triangulo, de quo oportet praescire quoniam nomen eius hoc significat, quod scilicet in sua definitione continetur.	Again, there are other things, namely, proper attributes, concerning which it is necessary to know what is said to be predicated, i.e., what is signified by their name. And he does not say unqualifiedly, “what it is,” but “what is said to be predicated,” because one cannot properly know of something what it is before it is known that it is. For there are no definitions of non-beings. Hence the question, whether it is, precedes the question, what it is. But “whether a thing is” cannot be shown unless it is known beforehand what is signified by its name. On this account the Philosopher teaches in Metaphysics IV that in disputing against those who deny principles one must begin with the meanings of names. An example of this is “triangle,” concerning which one must know beforehand that its name signifies such and such, namely what is contained in its definition.
Cum enim accidentia quodam ordine ad substantiam referantur, non est inconveniens id quod est accidens in respectu ad aliquid, esse etiam subiectum in respectu alterius. Sicut superficies est accidens substantiae corporalis: quae tamen superficies est primum subiectum coloris. Id autem quod est ita subiectum, quod nullius est accidens, substantia est. Unde in illis scientiis, quarum subiectum est aliqua substantia, id quod est subiectum nullo modo potest esse passio, sicut est in philosophia prima, et in scientia naturali, quae est de subiecto mobili.	But since accidents are referred to their subjects in a definite order, it is not impossible for something which is an accident in relation to one thing to be a subject in relation to something else: for example, a surface is an accident in relation to a bodily substance, but in relation to color it is the first subject. However, that which is a subject in such a way as never to be an accident of anything else is a substance. Hence in those sciences whose subject is a substance, that which is the subject can never be a proper attribute, as in first philosophy and in natural science, which treats of mobile being.
In illis autem scientiis, quae sunt de aliquibus accidentibus, nihil prohibet id, quod accipitur ut subiectum respectu alicuius passionis, accipi etiam ut passionem respectu anterioris subiecti. Hoc tamen non in infinitum procedit. Est enim devenire ad aliquod primum in scientia illa, quod ita accipitur ut subiectum, quod nullo modo ut passio; sicut patet in mathematicis scientiis, quae sunt de quantitate continua vel discreta. Supponuntur enim in his scientiis ea quae sunt prima in genere quantitatis; sicut unitas, et linea, et superficies et alia huiusmodi. Quibus suppositis, per demonstrationem quaeruntur quaedam alia, sicut triangulus aequilaterus, quadratum in geometricis et alia huiusmodi. Quae quidem demonstrationes quasi operativae dicuntur, ut est illud, super rectam lineam datam triangulum aequilaterum constituere. Quo adinvento, rursus de eo aliquae passiones probantur, sicut quod eius anguli sunt aequales aut aliquid huiusmodi. Patet igitur quod triangulus in primo modo demonstrationis se habet ut passio, in secundo se habet ut subiectum. Unde philosophus hic exemplificat de triangulo ut est passio, non ut est subiectum, cum dicit quod de triangulo oportet praescire quoniam hoc significat.	But in those sciences which bear upon accidents, nothing prevents a same thing from being taken as a subject in reference to one proper attribute, and as an attribute in reference to a more basic subject. Nevertheless, this must not develop into an infinite process, for one must arrive at something which is first in that science and which is taken as a subject in such a way that it is never taken as a proper attribute, as is clear in the mathematical sciences, which treat of continuous or discrete quantity. For in these sciences those things are postulated which are first in the genus of quantity; for example, unit and line and surface and the like. Once these are postulated, certain other things are sought through demonstration, such as the equilateral triangle and the square and so on in geometry. In these cases the demonstrations are said to be, as it were, operational, as when it is required to construct an equilateral triangle on a given straight line. But once it has been constructed, certain proper attributes are proved about it; for example, that its angles are equal, or something of that sort. It is clear, therefore, that in the first type of demonstration “triangle” behaves as a proper attribute, and in the second type as a subject. Hence the Philosopher is using “triangle” as a proper attribute and not as a subject when he says by way of example, “We must assume that triangle means so and so” (71a14).
lib. 1 l. 2 n. 6 Dicit etiam quod quaedam sunt de quibus oportet praescire utrunque, scilicet quid est et quia est. Et exemplificat de unitate quae est principium in omni genere quantitatis. Etsi enim aliquo modo sit accidens respectu substantiae, tamen in scientiis mathematicis, quae sunt de quantitate, non potest accipi ut passio, sed ut subiectum tantum, cum in hoc genere nihil habeat prius.	He says, ‘furthermore, that there are certain things about which we must know beforehand both what each is and whether it is. And he uses the example of “one,” which is the principle in every genus of quantity. For although it is somehow an accident in reference to substance, yet in the mathematical sciences, which treat of quantity, it cannot be taken as a proper attribute but only as a subject, since in this genus [quantity] it has nothing prior to it.
lib. 1 l. 2 n. 7 Rationem autem huiusmodi diversitatis ostendit, quia non est similis modus manifestationis praedictorum, scilicet principii, passionis et subiecti. Non enim est eadem ratio cognitionis in ipsis: nam principia cognoscuntur per actum componentis et dividentis; subiectum autem et passio per actum apprehendentis quod quid est. Quod quidem non similiter competit subiecto et passioni: cum subiectum definiatur absolute, quia in definitione eius non ponitur aliquid, quod sit extra essentiam ipsius; passio autem definitur cum dependentia ad subiectum, quod in eius definitione ponitur. Unde, ex quo non eodem modo cognoscuntur, non est mirum si eorum diversa praecognitio sit.	The reason for this difference is shown by the fact that the manner in which the aforesaid, namely, principle, proper attribute and subject, are manifested is not the same. For the way in which they are known is not the same: for principles are known through the act of composing and dividing, but subject and proper attribute by the act of apprehending the essence. And this, too, does not belong in similar fashion to a subject and to a proper attribute, since a subject is defined absolutely, for nothing outside its essence is mentioned in its definition; but a proper attribute is defined with dependence on the subject which is mentioned in its definition. Therefore, since they are not known in the same way, it is not surprising if they are not foreknown in the same way.
lib. 1 l. 2 n. 8 Deinde cum dicit: est autem cognoscere etc., determinat modum praecognitionis ex parte ipsius ordinis, quem praecognitio importat. Est enim aliquid prius altero et secundum tempus et secundum naturam. Et hic duplex ordo in praecognitione considerandus est. Aliquid enim praecognoscitur sicut prius notum tempore. Et de his dicit quod alia contingit cognoscere aliquem cognoscentem ea prius tempore, quam illa quibus praecognosci dicuntur. Quaedam vero cognoscuntur simul tempore, sed prius natura. Et de his dicit quod quorundam praecognitorum simul tempore est accipere notitiam, et illorum quibus praecognoscuntur.	Then (7a16) he determines the extent of foreknowledge on the part of the order which foreknowing implies. For something is prior to another both in the order of time and in the order of nature. And this twofold order must be considered in regard to pre-existent knowing. For something is known before something else in the sense of being known prior in time. Concerning such things he says that someone could know certain things by knowing them prior to the time when he knows the things to which they are said to be foreknown. But certain others are known at one and the same time, although one is prior by nature to the other. Concerning these he says that one acquires a knowledge of some of these foreknown things at the same time that knowledge of the things to which they are foreknown is acquired.
lib. 1 l. 2 n. 9 Quae autem sint ista manifestat subdens quod huiusmodi sunt quaecumque continentur sub aliquibus universalibus, quorum habent cognitionem, idest de quibus notum est ea sub talibus universalibus contineri.	He indicates what these are when he adds that they are the things contained under certain universals of which we have knowledge, i.e., of which it is known that they are contained under such universals.
Et hoc ulterius manifestat per exemplum. Cum enim ad conclusionem inferendam duae propositiones requirantur, scilicet maior et minor, scita propositione maiori, nondum habetur conclusionis cognitio. Maior ergo propositio praecognoscitur conclusioni non solum natura, sed tempore. Rursus autem si in minori propositione inducatur sive assumatur aliquid contentum sub universali propositione, quae est maior, de quo manifestum non sit quod sub hoc universali contineatur, nondum habetur conclusionis cognitio, quia nondum erit certa veritas minoris propositionis. Si autem in minori propositione assumatur terminus, de quo manifestum sit quod continetur sub universali in maiori propositione, patet veritas minoris propositionis: quia id quod accipitur sub universali habet eius cognitionem, et sic statim habetur conclusionis cognitio. Ut si sic demonstraret aliquis, omnis triangulus habet tres angulos aequales duobus rectis, ista cognita, nondum habetur conclusionis cognitio: sed cum postea assumitur, haec figura descripta in semicirculo, est triangulus, statim scitur quod habet tres angulos aequales duobus rectis. Si autem non esset manifestum quod haec figura in semicirculo descripta est triangulus, nondum statim inducta assumptione sciretur conclusio; sed oporteret ulterius aliud medium quaerere, per quod demonstraretur hanc figuram esse triangulum.	Then he clarifies this with an example. For since two propositions are needed for inferring a conclusion, namely, a major and a minor; when the major proposition is known, the conclusion is not yet known. Therefore, the major proposition is known before the conclusion not only in nature but in time. Further, if in the minor proposition something is introduced or employed which is contained under the universal proposition which is the major, but it is not evident that it is contained under this universal, then a knowledge of the conclusion is not yet possessed, because the truth of the minor proposition will not yet be certain. But if in the minor proposition a term is taken about which it is clear that it is contained under the universal in the major proposition, the truth of the minor proposition is clear, because that which is taken under the universal shares in the same knowledge, and so the knowledge of the conclusion is had at once. Thus, suppose that someone should begin to demonstrate by stating that every triangle has three angles equal to two right angles. When this is known, the knowledge of the conclusion is not yet known. But when it is later assumed that this figure inscribed in a semicircle is a triangle, he knows at once that it has three angles equal to two right angles. However, if it were not clear that this figure inscribed in the semicircle is a triangle, the conclusion would not be known as soon as the minor was stated; rather, it would be necessary to search for a middle through which to demonstrate that this figure is a triangle.
Exemplificans ergo philosophus de his quae cognoscuntur ante conclusionem prius tempore, dicit quod aliquis per demonstrationem conclusionis cognitionem accipiens, hanc propositionem praescivit etiam secundum tempus, scilicet, quod omnis triangulus habet tres angulos duobus rectis aequales. Sed inducens hanc assumptionem, scilicet, quod hoc quod est in semicirculo sit triangulus, simul, scilicet tempore, cognovit conclusionem, quia hoc inductum habet notitiam universalis, sub quo continetur, ut non oporteat ulterius medium quaerere. Et ideo subdit quod, quorundam est hoc modo disciplina, idest eorum accipitur cognitio per se, et non oportet ea cognoscere per aliquod aliud medium, quod sit ultimum in resolutione, qua mediata ad immediata reducuntur. Vel potest legi sic: quod ultimum, idest extremum, quod accipitur sub universali medio, non oportet ut cognoscatur esse sub illo universali per aliquod aliud medium. Et quae sint ista, quae semper habent cognitionem sui universalis, manifestat subdens quod huiusmodi sunt singularia, quae non dicuntur de aliquo subiecto: cum inter singularia et speciem nullum medium possit inveniri.	In giving this example of things which are known at a time prior to the conclusion the Philosopher says that a person obtaining a knowledge of the conclusion through demonstration foreknew this proposition even according to time, namely, that every triangle has three angles equal to two right angles. But inducing this assumption, namely, that this figure in the semicircle is a triangle, he knew the conclusion at the same time, because this induction shares in the evidence of the universal under which it is contained, so that there is no need to search for another middle. He adds, therefore, that “some things are only learnt in this way” (71a23), i.e., learnt in virtue of themselves, so that it is not necessary to learn them through some other middle which is the ultimate reached by analysis in which the mediate is reduced to the immediate. Or it can be read in such a way that the “ultimate,” i.e., the extreme, which is subsumed under the universal middle does not need a further middle to show that it i contained under that universal. And he manifests what those things are which always share the knowledge of their universal, saying that they are the singulars, which are not predicated of any subject, since no middle can be found between singulars and their species.

Lecture 3 Pre-existent knowledge of the conclusion

Latin	English
	Lecture 3 (71a24-b9) PRE-EXISTENT KNOWLEDGE OF THE CONCLUSION
lib. 1 l. 3 n. 1 Postquam ostendit philosophus quomodo oportet praecognoscere quaedam alia, antequam de conclusione cognitio sumatur, nunc vult ostendere quomodo ipsam conclusionem contingit praecognoscere, antequam cognitio sumatur de ea per syllogismum vel inductionem. Et circa hoc duo facit.	Having shown the manner in which certain other things must be known before knowledge of the conclusion is obtained, the Philosopher now wishes to show how we know even the conclusion beforehand, i.e., before knowledge of it is obtained through a syllogism or induction. Concerning this he does two things.
Primo namque determinat veritatem, dicens quod, antequam inducatur inductio vel syllogismus ad faciendam cognitionem de aliqua conclusione, illa conclusio quodammodo scitur, et quodammodo non: simpliciter enim nescitur, sed scitur solum secundum quid. Sicut si debeat probari ista conclusio, triangulus habet tres angulos aequales duobus rectis; antequam demonstraretur, ille qui per demonstrationem accipit scientiam eius, nescivit simpliciter, sed scivit secundum quid. Unde quodammodo praescivit, simpliciter autem non. Cuius quidem ratio est, quia, sicut iam ostensum est, oportet principia conclusioni praecognoscere. Principia autem se habent ad conclusiones in demonstrativis, sicut causae activae in naturalibus ad suos effectus (unde in II physicorum propositiones syllogismi ponuntur in genere causae efficientis). Effectus autem, antequam producatur in actu, praeexistit quidem in causis activis virtute, non autem actu, quod est simpliciter esse. Et similiter antequam ex principiis demonstrativis deducatur conclusio, in ipsis quidem principiis praecognitis praecognoscitur conclusio virtute, non autem actu: sic enim in eis praeexistit. Et sic patet quod non praecognoscitur simpliciter, sed secundum quid.	First (71a24), he establishes the truth of the fact, saying that before an induction or syllogism is formed to beget knowledge of a conclusion, that conclusion is somehow known and somehow not known: for, absolutely speaking, it is not known; but in a qualified sense, it is known. Thus, if the conclusion that a triangle has three angles equal to two right angles has to be proved, the one who obtains science of this fact through demonstration already knew it in some way before it was demonstrated; although absolutely speaking, he did not know it. Hence in one sense he already knew it, but in the full sense he did not. And the reason is that, as has been pointed out, the principles of the conclusion must be known beforehand. Now the principles in demonstrative matters are to the conclusion as efficient causes in natural things are to their effects; hence in Physics II the propositions of a syllogism are set in the genus of efficient cause. But an effect, before it is actually produced, pre-exists virtually in its efficient causes but not actually, which is to exist absolutely. In like manner, before it is drawn out of its demonstrative principles, the conclusion is pre-known virtually, although not actually, in its self-evident principles. For that is the way it pre-exists in them. And so it is clear that it is not pre-known in the full sense, but in some sense.
lib. 1 l. 3 n. 2 Secundo; ibi: si vero non etc., excludit ex veritate determinata quandam dubitationem, quam Plato ponit in libro Menonis, sic intitulato ex nomine sui discipuli. Est autem dubitatio talis. Inducit enim quendam, omnino imperitum artis geometricae, interrogatum ordinate de principiis, ex quibus quaedam geometrica conclusio concluditur, incipiendo ex principiis per se notis: ad quae omnia ille, ignarus geometriae, id quod verum est respondit: et sic deducendo quaestiones usque ad conclusiones per singula verum respondit. Ex hoc igitur vult habere, quod etiam illi, qui videntur imperiti aliquarum artium, antequam de eis instruantur, earum notitiam habent. Et sic sequitur quod vel homo nihil addiscat, vel addiscat ea quae prius novit.	Secondly (71a28), in virtue of this established fact, he settles a doubt which Plato maintained in the book, Meno, which gets its title from the name of his disciple. The doubt is presented in the following manner: A person utterly ignorant of the art of geometry is questioned in an orderly Way concerning the per se known principles from which a geometric conclusion is concluded. By starting with principles that are per se known, to each of which this person ignorant of geometry gives a true answer, Aid leading him thus by questions to the conclusion, he gives the true Answer step by step. From this, therefore, he would have it that even those who seem to be ignorant of certain arts really have a knowledge of them before being instructed in them. And so it follows that either a man learns nothing or he learns what he already knew.
lib. 1 l. 3 n. 3 Circa hoc ergo quatuor facit. Primo enim, proponit quod praedicta dubitatio vitari non potest, nisi supposita praedeterminata veritate, scilicet quod conclusio, quam quis addiscit per demonstrationem vel inductionem, erat nota non simpliciter, sed secundum quod est virtute in suis principiis: de quibus aliquis, ignarus scientiae, interrogatus, veritatem respondere potest. Secundum vero Platonis sententiam, conclusio erat praecognita simpliciter; unde non addiscebatur de novo, sed potius per deductionem aliquam rationis in memoriam reducebatur. Sicut etiam de formis naturalibus Anaxagoras ponit, quod ante generationem praeexistebant in materia simpliciter. Aristoteles vero ponit quod praeexistunt in potentia et non simpliciter.	In dealing with this problem he [Aristotle] does four things. First, he suggests that it cannot be settled unless we grant the truth established above, namely, that the conclusion which a person learns through demonstration or induction was already known, not absolutely, but as it was virtually known in its principles concerning which a person ignorant of a science can give true answers. However, according to Plato’s theory the conclusion was pre-known absolutely, so that no one learns afresh but is led to recall by some rational process of deduction. This is similar to Anaxagoras’ position on natural forms, namely, that before they are generated, they already pre-existed in the matter absolutely, whereas Aristotle says that they pre-exist in potency and not absolutely.
lib. 1 l. 3 n. 4 Secundo, cum dicit: non enim sicut etc., ponit falsam quorundam obviationem ad dubitationem Platonis: qui scilicet dicebant quod conclusio antequam demonstraretur vel quocunque modo addisceretur, nullo modo erat cognita. Poterat enim eis obiici secundum dubitationem Platonis hoc modo: si quis interrogaret ab aliquo imperito: nunquid tu scis quod omnis dualitas par est? Et dicente eo, idest concedente se scire, afferret quandam dualitatem, quam ille interrogatus non opinaretur esse, puta illam dualitatem quae est tertia pars senarii; concluderetur quod sciret tertiam partem senarii esse numerum parem: quod erat ei incognitum, sed per demonstrationem inductam addiscit. Et sic videtur sequi quod vel non hoc addisceret vel addisceret quod prius scivit. Ut igitur hanc dubitationem evitarent, solvebant dicentes, quod ille qui interrogatus respondit se scire quod omnis dualitas sit numerus par, non dixit se cognoscere omnem dualitatem simpliciter, sed illam, quam scivit esse dualitatem. Unde cum ista dualitas, quae est allata, fuerit ab eo penitus ignorata, nullo modo scivit quod haec dualitas esset numerus par. Et sic sequitur quod apud cognoscentem principia nullo modo conclusio sit praecognita, nec simpliciter, nec secundum quid.	Secondly, he shows that the way some have answered Plato’s problem is false, namely, by saying that a conclusion is not in any sense known before it is demonstrated or learned by some method or other. For they might face the following objection based on Plato’s problem: If an unlearned person were asked by someone, “Do you know that every duo (pair) is an even number?” and, if upon answering that he does know this, he were presented with a duo which the person interrogated did not know existed, for example, the duo which is one third of six, the conclusion would be that he knew one third of six to be an even number, a fact which had not been known by him but which he learned through the demonstration proposed to him. And so it seems to follow that he either did not freshly learn this or that he learned what he already knew. To avoid this dilemma, they would answer that the person who was questioned and who answered that he knew every duo to be an equal number did not say that he knew every duo absolutely, but those he knew to be duo’s. Hence, since that duo which was proposed was utterly unknown to him, he did not in any sense know that this duo was an even number. And so it follows that when one knows the principles, the conclusion is not in any sense pre-known, either absolutely or in a qualified sense.
lib. 1 l. 3 n. 5 Tertio, ibi: et etiam sciunt etc., improbat hanc solutionem hoc modo. Illud scitur de quo demonstratio habetur, vel de quo de novo accipitur demonstratio. Et hoc dicitur propter addiscentem, qui incipit scire. Addiscentes autem non accipiunt demonstrationem de omni dualitate de qua sciunt, sed de omni simpliciter, et similiter de omni numero aut de omni triangulo. Non ergo verum est quod sciat de omni numero, quem scit esse numerum, aut de omni dualitate, quam scit esse dualitatem, sed de omni simpliciter. Quod autem non sciat de omni numero, quem scit esse numerum, sed de omni simpliciter, probat; ibi: neque enim una propositio etc.: conclusio cum praemissis convenit in terminis: nam subiectum et praedicatum conclusionis sunt maior et minor extremitas in praemissis; sed in praemissis non accipitur aliqua propositio de numero aut de recta linea, cum hac additione, quam tu nosti, sed simpliciter de omni; neque ergo conclusio demonstrationis est cum additione praedicta, sed simpliciter de omni.	Thirdly (71bl), he refutes this solution in the following way: That is known, concerning which a demonstration is had, or concerning which a demonstration is for the first time received. And this is said on account of those learners who begin to know scientifically. But learners do not obtain a demonstration touching every duo they happen to know but every duo absolutely; and the same applies to every number or every triangle. Therefore, it is not true that he knows something about every number which he knows to be a number, or of every duo which he knows to be a duo, but he knows it about every one absolutely. And that he knows it not only of every number he happens to know is a number, but of every number absolutely, is proved at (71b5) on the ground that the conclusion agrees with the premises in its terms. For the subject and predicate of the conclusion are the major and minor extremes in the premises. But in the premises no proposition concerning number or straight line is stated with the addition, “which you know,” but it is stated of all without qualification. Neither, therefore, is the conclusion of the demonstration asserted with the aforesaid qualification, but it is asserted of all without reservation.
lib. 1 l. 3 n. 6 Quarto, ibi: sed nihil etc., ponit veram solutionem dubitationis praedictae secundum praedeterminatam veritatem, dicens quod illud quod quis addiscit, nihil prohibet primo quodammodo scire et quodammodo ignorare. Non enim est inconveniens si aliquis quodammodo praesciat id quod addiscit; sed esset inconveniens si hoc modo praecognosceret, secundum quod addiscit. Addiscere enim proprie est scientiam in aliquo generari. Quod autem generatur, ante generationem neque fuit omnino non ens neque omnino ens, sed quodammodo ens et quodammodo non ens: ens quidem in potentia, non ens vero actu: et hoc est generari, reduci de potentia in actum. Unde nec id quod quis addiscit erat omnino prius notum, ut Plato posuit, nec omnino ignotum, ut secundum solutionem supra improbatam ponebatur; sed erat notum potentia sive virtute in principiis praecognitis universalibus, ignotum autem actu, secundum propriam cognitionem. Et hoc est addiscere, reduci de cognitione potentiali, seu virtuali, aut universali, in cognitionem propriam et actualem.	Fourthly (71b5), he presents the true solution of the problem under discussion in terms of the truth already established, saying that there is nothing to prevent a person from somehow knowing and somehow not knowing a fact before he learns it. For it is not a paradox if one somehow already knows what he learns, but it would be, if he already knew it in the same way that he knows it when he has learned it. For learning is, properly speaking, the generation of science in someone. But that which is generated was not, prior to its generation, a being absolutely, but somehow a being and somehow non-being: for it was a being in potency, although actually non-being. And this is what being generated consists in, namely, in being converted from potency to act. In like fashion, that which a person learns was not previously known absolutely, as Plato preferred; but neither was it absolutely unknown, as they maintained whose answer was refuted above. Rather it was known in potency, i.e., virtually, in the pre-known universal principles; however, it was not actually known in the sense of specific knowledge. And this is what learning consists in, namely, in being brought from potential or virtual or universal knowledge to specific and actual knowledge.

Lecture 4 Nature of the demonstrative syllogism

Latin	English
	Lecture 4 (71b8-72a8) NATURE OF THE DEMONSTRATIVE SYLLOGISM
lib. 1 l. 4 n. 1 Postquam ostendit philosophus necessitatem syllogismi demonstrativi, hic iam incipit de ipso syllogismo demonstrativo determinare. Et dividitur in duas partes: in prima, determinat de syllogismo demonstrativo; in secunda, de medio ex quo syllogismus demonstrativus procedit: et hoc in secundo libro; ibi: quaestiones sunt aequales numero et cetera. Prima dividitur in duas: in prima, determinat de syllogismo demonstrativo absolute; in secunda, comparando demonstrationem demonstrationi; ibi: cum autem demonstratio sit alia quidem universalis et cetera. Prima in duas dividitur: in prima, determinat de syllogismo demonstrativo; in secunda, ostendit quod non sit in demonstrationibus in infinitum procedere; ibi: est autem omnis syllogismus per tres terminos et cetera. Prima dividitur in duas: in prima, determinat de syllogismo demonstrativo per quem acquirimus scientiam: in secunda, ostendit quomodo etiam in nobis per syllogismum acquiritur aliqua ignorantia; ibi: ignorantia autem secundum negationem et cetera. Circa primum tria facit: primo, determinat de syllogismo demonstrativo, ostendendo quid sit; secundo, determinat de materia syllogismi demonstrativi, ostendendo quae et qualia sint, ex quibus est; ibi: quoniam autem impossibile est etc.; tertio, determinat de forma ipsius, ostendendo in qua figura praecipue fiat; ibi: figurarum autem magis faciens scire et cetera. Circa primum tria facit: primo, ostendit de demonstrativo syllogismo, quid est; secundo, notificat quaedam, quae in definitione syllogismi demonstrativi ponuntur; ibi: est autem principium demonstrationis etc.; tertio, excludit quosdam errores, qui ex praemissis circa demonstrationem oriri possunt; ibi: quibusdam quidem igitur et cetera.	After indicating the need for the demonstrative syllogism, the Philosopher now begins to settle questions concerning the demonstrative syllogism itself. And his treatment is divided into two parts. In the first he determines concerning the demonstrative syllogism. In the second he determines concerning the middle from which the demonstrative syllogism proceeds (89b21) [Book II]. The first is divided into two parts. In the first he determines concerning the demonstrative syllogism in itself. In the second he compares demonstration to demonstration (85a12) [L. 37]. The first is divided into two parts. In the first he determines concerning the demonstrative syllogism. In the second he shows that one does not proceed to infinity in demonstrations (81b10) [L. 31] The first is divided into two parts. In the first he determines concerning the demonstrative syllogism through which we acquire science. In the second he shows how we also acquire ignorance through a syllogism (79b23) [L. 27]. Concerning the first he does three things. First, he determines concerning the demonstrative syllogism by showing what it is. Secondly, he determines concerning the matter of the demonstrative syllogism, pointing out the nature and character of the matter out of which it is formed (73a21) [L. 9]. Thirdly, he determines concerning the form of the syllogism, pointing out the figure in which it is chiefly presented (79a17) [L. 26]. Concerning the first he does three things. First, he shows what the demonstrative syllogism is. Secondly, he clarifies certain terms that appear in the definition of the demonstrative syllogism (72a8) [L. 5]. Thirdly, he excludes certain errors that could arise from his doctrine on the nature of demonstration [L. 7].
lib. 1 l. 4 n. 2 Circa primum sciendum est quod in omnibus quae sunt propter finem, definitio quae est per causam finalem, est ratio definitionis, quae est per causam materialem, et medium probans ipsam: propter hoc enim oportet ut domus fiat ex lapidibus et lignis, quia est operimentum protegens nos a frigore et aestu. Sic igitur Aristoteles de demonstratione dat hic duas definitiones: quarum una sumitur a fine demonstrationis, qui est scire; et ex hac concluditur altera, quae sumitur a materia demonstrationis.	In regard to the first it should be noted that in all things which exist for an end, the definition which employs a final cause is both the explanation of the definition which expresses the material cause, and is the middle which proves the latter. For the reason why a house should made of stone and wood is that it is a structure protecting us from the cold and heat. Along these lines, therefore, he gives two definitions demonstration, one of which is expressed in terms of the end of demonstration, which is to know in a scientific manner. And from this one is concluded the other, which is drawn from the matter of a demonstration.
lib. 1 l. 4 n. 3 Unde circa hoc tria facit: primo, definit ipsum scire; secundo, definit demonstrationem per finem eius, qui est ipsum scire; ibi: dicimus autem scire etc.; tertio, ex utraque definitione concludit definitionem demonstrationis, quae sumitur per comparationem materiae demonstrationis; ibi: si igitur est scire ut posuimus et cetera. Circa primum quinque facit. Primo enim, determinat cuiusmodi scire sit, quod definire intendit.	Hence he does three things in regard to this. First, he defines what it is to know in a scientific manner. Secondly, he defines demonstration in terms of its end, which is to know in a scientific manner (71b18). Thirdly, from these two definitions he concludes to that definition of demonstration which is expressed in terms of the matter of demonstration (71b19). Concerning the first he does five things. First (71b8), he determines what the scientific knowing, which he intends to define, bears upon.
lib. 1 l. 4 n. 4 Circa quod sciendum est quod aliquid dicimur scire simpliciter, quando scimus illud in seipso. Dicimur scire aliquid secundum quid, quando scimus illud in alio, in quo est, vel sicut pars in toto, sicut si scientes domum, diceremur scire parietem; vel sicut accidens in subiecto, sicut si scientes Coriscum, diceremur scire venientem; vel sicut effectus in causa, sicut dictum est supra quod conclusionem praescimus in principiis; vel quocunque simili modo. Et hoc est scire per accidens, quia scilicet scito aliquo per se, dicimur scire illud quod accidit ei quocunque modo. Intendit igitur philosophus definire scire simpliciter, non autem scire secundum accidens. Hic enim modus sciendi est sophisticus. Utuntur enim sophistae tali modo arguendi: cognosco Coriscum; Coriscus est veniens; ergo cognosco venientem.	And in regard to this it should be recognized that we are said to know something in a scientific manner absolutely, when we know it in itself. On the other hand, we are said to know something in a scientific manner qualifiedly, when we know it in something else in which it exists either as a part in a whole (as we are said to know a wall through knowing the house), or as an accident in its subject (as in knowing Coriscus we are said to know who is coming toward us), or as an effect in its cause (as in the example given earlier, we know the conclusion in the principles), or indeed in any fashion similar to these. To know in these ways is to know incidentally, because we are said to know that which is somehow accidental to what is known of itself. However, what the Philosopher intends to define here is scientific knowing in the strict sense and not according to an accident. For this form of knowing is sophistical, since Sophists use a form of argument typified by the following: “I know Coriscus; Coriscus is coming toward me: therefore, I know the person coming toward me.”
lib. 1 l. 4 n. 5 Secundo, cum dicit: cum causam arbitramur etc., ponit definitionem ipsius scire simpliciter. Circa quod considerandum est quod scire aliquid est perfecte cognoscere ipsum, hoc autem est perfecte apprehendere veritatem ipsius: eadem enim sunt principia esse rei et veritatis ipsius, ut patet ex II metaphysicae. Oportet igitur scientem, si est perfecte cognoscens, quod cognoscat causam rei scitae. Si autem cognosceret causam tantum, nondum cognosceret effectum in actu, quod est scire simpliciter, sed virtute tantum, quod est scire secundum quid et quasi per accidens. Et ideo oportet scientem simpliciter cognoscere etiam applicationem causae ad effectum. Quia vero scientia est etiam certa cognitio rei; quod autem contingit aliter se habere, non potest aliquis per certitudinem cognoscere; ideo ulterius oportet quod id quod scitur non possit aliter se habere. Quia ergo scientia est perfecta cognitio, ideo dicit: cum causam arbitramur cognoscere; quia vero est actualis cognitio per quam scimus simpliciter, addit: et quoniam illius est causa; quia vero est certa cognitio, subdit: et non est contingere aliter se habere.	Then (71b10) he presents the definition of scientific knowing in the strict sense. Apropos of this it should be noted that to know something scientifically is to know it completely, which means to apprehend its truth perfectly. For the principles of a thing’s being are the same as those of its truth, as is stated in Metaphysics II. Therefore, the scientific knower, if he is to know perfectly, must know the cause of the thing known; hence he says, “when we think that we know the cause” (71b10). But if he were to know the cause by itself, he would not yet know the effect actually—which would be to know it absolutely—but only virtually, which is the same as knowing in a qualified sense and incidentally. Consequently, one who knows scientifically in the full sense must know the application of the cause to the effect; hence he adds, “as the cause of that fact” (71b11). Again, because science is also sure and certain knowledge of a thing, whereas a thing that could be otherwise cannot be known with certainty, it is further required that what is scientifically known could not be otherwise. To repeat: because science is perfect knowledge, lie says, “Men we think that we know the cause”; but because the knowledge through which we know scientifically in the full sense is actual, he adds, “as the cause of that fact.” Finally, because it is certain knowledge, he adds, “and that the fact could not be other than it is (71b11).”
lib. 1 l. 4 n. 6 Tertio, ibi: manifestum igitur etc., manifestat positam definitionem per hoc, quod tam scientes, quam non scientes, existimantes tamen se scire, hoc modo accipiunt scire sicut dictum est: non scientes enim qui existimant se scire, opinantur sic se habere in cognoscendo, sicut dictum est; scientes autem vere sic se habent. Est autem haec recta manifestatio definitionis. Definitio enim est ratio, quam significat nomen, ut dicitur in IV metaphysicae; significatio autem nominis accipienda est ab eo, quod intendunt communiter loquentes per illud nomen significare: unde et in II topicorum dicitur quod nominibus utendum est, ut plures utuntur. Si quis etiam recte consideret, hac notificatione magis ostenditur quid significet nomen, quam directe aliquid significetur: non enim notificat scientiam, de qua proprie posset definitio assignari, cum sit species alicuius generis, sed notificat ipsum scire. Unde et a principio dixit: scire autem opinamur etc., et non dixit: scire est aliquid tale vel tale.	Thirdly (71b12), he explains the definition he laid down, appealing to the fact that both those who know scientifically and those who do not know in that way but believe that they do, take scientific knowing to be as above described. For those who do not know in a scientific manner but believe that they do, are convinced that they know in the manner described, whereas those who know in a scientific manner do know in the manner described. Furthermore, this is the proper way to manifest a definition. For a definition is the notion which a name signifies, as it is stated in Metaphysics IV. But the signification of a name must be based on what is generally meant by those who employ the name. Hence it is stated in Topics II that names must be used as the majority of people use them. Again, careful consideration would indicate that this explanation seems rather to show what the name signifies than to signify something directly. For he does not explain science, concerning which a definition could, properly speaking, be formed, since it is a species of some genus; rather he explains scientific knowing. Hence at the very beginning he said, “We suppose ourselves to possess unqualified scientific knowledge” (71b8) and not that scientific knowledge is such and such.
lib. 1 l. 4 n. 7 Quarto, ibi: quare cuius etc., concludit quoddam corollarium ex definitione posita, scilicet quod illud, de quo simpliciter habetur scientia, oportet esse necessarium, scilicet quod non contingat aliter se habere.	Fourthly (71b14), he draws a corollary from the definition, namely, that that of which there is unqualified scientific knowledge must be something necessary, i.e., which cannot be otherwise.
lib. 1 l. 4 n. 8 Quinto, ibi: si quidem etc., respondet tacitae quaestioni, utrum scilicet sit aliquis alius modus sciendi a praedicto. Quod promittit se in sequentibus dicturum: est enim scire etiam per effectum, ut infra patebit. Dicimur etiam aliquo modo scire ipsa principia indemonstrabilia, quorum non est accipere causam. Sed proprius et perfectus sciendi modus est qui praedictus est.	Fifthly (71b16), he answers a tacit question, namely, whether there is another way of knowing scientifically in addition to the way described here. And he promises to discuss this later. For it is possible to know scientifically through an effect, as will be explained below (cf. L. 23). Furthermore, there is a sense in which we are said to know scientifically the indemonstrable principles to which no cause is ascribed. But the proper and perfect manner of knowing scientifically is the one we have described.
lib. 1 l. 4 n. 9 Deinde, cum dicit: dicimus autem etc., definit syllogismum demonstrativum per comparationem ad finem suum, qui est scire. Circa quod tria facit. Primo, ponit quod scire est finis syllogismi demonstrativi sive effectus eius, cum scire nihil aliud esse videatur, quam intelligere veritatem alicuius conclusionis per demonstrationem.	Then (71b17) he defines the demonstrative syllogism in terms of its end, which is to know in a scientific manner. In regard to this he does three things. First, he asserts that scientific knowing is the end of a demonstrative syllogism or is its effect, since to know scientifically seems to b nothing less than to understand the truth of a conclusion through demonstration.
Secundo, ibi: demonstrationem autem etc., definit syllogismum demonstrativum per huiusmodi finem: dicens quod demonstratio est syllogismus scientialis, idest faciens scire. Tertio, exponit hoc quod dixerat scientialem; ibi: sed scientialem etc., dicens quod scientialis syllogismus dicitur, secundum quem scimus, in quantum ipsum habemus, ne forte aliquis syllogismum scientialem intelligeret, quo aliqua scientia uteretur.	Secondly (71b18), he defines demonstration in terms of the end, saying that a demonstration is a sciential syllogism, i.e., producing scientific knowledge. Thirdly (71b18), he explains, “sciential,” saying that a sciential syllogism is one according to which we know scientifically insofar as we understand it, and not in the sense of a syllogism yielding knowledge to be put to use.
lib. 1 l. 4 n. 10 Deinde, cum dicit: si igitur est scire etc., concludit ex praedictis definitionem syllogismi demonstrativi ex materia sumptam. Et circa hoc duo facit: primo, concludit; secundo, manifestat eam; ibi: verum quidem igitur oportet esse et cetera. Circa primum tria facit. Primo, ponit consequentiam, qua demonstrationis materialis definitio concluditur ex praemissis, dicens quod si scire hoc significat quod diximus, scilicet, causam rei cognoscere etc., necesse est quod demonstrativa scientia, idest quae per demonstrationem acquiritur, procedat ex propositionibus veris, primis et immediatis, idest quae non per aliquod medium demonstrantur, sed per seipsas sunt manifestae (quae quidem immediatae dicuntur, in quantum carent medio demonstrante; primae autem in ordine ad alias propositiones, quae per eas probantur); et iterum ex notioribus, et prioribus, et causis conclusionis.	Then (71b19) he concludes from the foregoing a definition of the demonstrative syllogism that is based on its matter. Concerning this he does two things. First, he concludes it. Secondly, he clarifies it (71b24). Concerning the first he does three things. First (71b20), he sets forth the consequent in which the material definition of demonstration is concluded from the premises laid down above. And he says that if scientific knowing is what we have stated it to be, namely, knowing the cause of a thing, etc., then it is necessary that demonstrative science, i.e., science acquired through demonstration, proceed from propositions which are true, first, and immediate, i.e., not demonstrated by some other mid but clear in virtue of themselves (they are called “immediate,” inasmuch as they do not have a middle demonstrating them, but “first,” in relation to other propositions which are proved through them); and which, furthermore, are better known than, prior to, and causes of, the conclusion.
lib. 1 l. 4 n. 11 Secundo, ibi: sic enim erunt etc., excusat se ab additione alterius particulae, quae videbatur apponenda: quod scilicet demonstratio ex propriis principiis procederet. Sed ipse dicit quod hoc intelligitur per ea, quae dicta sunt. Nam si propositiones demonstrationis sunt causae conclusionis, necesse est quod sint propria principia eius: oportet enim causas esse proportionatas effectibus.	Secondly (71b22), he justifies himself for not adding another element which, it might seem, should be added, namely, that demonstration proceeds from proper principles. But he says that this is understood in virtue of the elements he did state. For since the propositions of a demonstration are causes of the conclusion, they must be its proper principles. For effects require proportionate causes.
lib. 1 l. 4 n. 12 Tertio, ibi: syllogismus quidem etc., manifestat praemissae consequentiae necessitatem, dicens quod licet syllogismus non requirat praemissas conditiones in propositionibus, ex quibus procedit, requirit tamen eas demonstratio: aliter enim non faceret scientiam.	Thirdly (71b23), he manifests the necessity of the aforesaid consequence, saying that although a syllogism does not require these conditions in the premises from which it concludes, a demonstration does require them, for otherwise it would not produce science.
lib. 1 l. 4 n. 13 Deinde, cum dicit: verum quidem etc., manifestat positam definitionem, manifestans etiam quod immediate dixerat, scilicet quod nisi praemissae conditiones demonstrationi adessent, scientiam facere non posset. Primo ergo, ostendit quod semper procedit ex veris ad hoc quod scientiam faciat: quia quod non est, non est scire; sicut diametrum esse symmetrum, idest commensurabilem lateri quadrati (dicuntur enim quantitates incommensurabiles, quarum non potest accipi aliqua mensura communis; et huiusmodi quantitates sunt, quarum non est proportio ad invicem sicut numeri ad numerum; quod de necessitate contingit de diametro quadrati et eius latere, ut patet ex X Euclidis). Quod autem non est verum, non est: nam esse et esse verum convertuntur. Oportet ergo id quod scitur esse verum. Et sic conclusionem demonstrationis, quae facit scire, oportet esse veram, et per consequens eius propositiones: non enim contingit verum sciri ex falsis, etsi concludi possit ex eis, ut infra ostendet.	Then (71b24) he explains this definition as well as the subsequent statement that unless these conditions are fulfilled in a demonstration it cannot beget science. First, therefore, he shows that a demonstration must proceed from true principles in order to beget science, because there cannot be scientific knowledge of that which does not exist, for example, that the diagonal is symmetrical, i.e., commensurable with the side of the square. For those quantities are said to be incommensurable which lack a common measuring unit. These are quantities whose ratio to one another cannot be expressed in terms of one number to another number. That this is the case with the diagonal of a square and its side is plain from Euclid’s sixteenth proposition. Now what is not true does not exist, for to be and to be true are convertible. Therefore, anything scientifically known must be true. Consequently, the conclusion of a demonstration which does beget scientific knowing must be true, and a fortiori its premises. For the true cannot be known in a scientific way from the false, although something true can follow as a conclusion from something false, as he will show later (cf. Lecture 13).
lib. 1 l. 4 n. 14 Secundo, ibi: ex primis autem etc., ostendit quod demonstratio sit ex primis et immediatis, sive indemonstrabilibus. Non enim contingit aliquem habere scientiam, nisi habeat demonstrationem eorum, quorum potest esse demonstratio, et hoc dico per se, et non per accidens. Et hoc ideo dicit, quia possibile esset scire aliquam conclusionem, non habentem demonstrationem praemissorum, etiam si essent demonstrabilia: quia sciret eam per alia principia; et hoc esset secundum accidens.	Secondly (71b27), he shows that the demonstration is composed of first and immediate or indemonstrable principles. For no one can possess scientific knowledge unless he possesses the demonstration of things that can be demonstrated—“and I am speaking per se and not per accidens.” He says this because it would be possible to know some conclusion without having a demonstration of the premises, even were they demonstrable; because one would know it through other principles, and this would be accidental.
Detur ergo quod aliquis demonstrator syllogizet ex demonstrabilibus, sive mediatis: aut ergo habet illorum demonstrationem, aut non habet: si non habet, ergo non scit praemissa, et ita nec conclusionem propter praemissa; si autem habet, cum in demonstrationibus non sit abire in infinitum, ut infra ostendet, tandem erit devenire ad aliqua immediata et indemonstrabilia. Et sic oportet quod demonstratio ex immediatis procedat, vel statim, vel per aliqua media. Unde et in primo libro topicorum dicitur quod demonstratio est ex primis et veris, aut ex his quae per ea fidem sumpserunt.	Suppose, therefore, that a demonstrator syllogizes from demonstrable, i.e., mediate, premises. Now he either possesses a demonstration of those premises or he does not. If he does not, then he does not know the premises in a scientific way; nor consequently, the conclusion because of the premises. But if he does possess their demonstration, then, since one may not proceed to infinity in demonstrations, principles immediate and indemonstrable must be reached. And so it is required that demonstration proceed from principles that are immediate either straightway or through middles. Hence it is stated in Topics I that demonstration is composed of first and true statements or of statements made credible by these.
lib. 1 l. 4 n. 15 Tertio, ibi: causas quoque etc., probat quod demonstrationis propositiones sint causae conclusionis, quia tunc scimus, cum causas cognoscimus. Et ex hoc concludit ulterius quod sint priores et notiores, quia omnis causa est naturaliter prior et notior suo effectu. Oportet autem quod causa conclusionis demonstrativae sit notior, non solum quantum ad cognitionem quid est, sed etiam quantum ad cognitionem quia est. Non enim ad demonstrandum quod eclipsis solis est, sufficit scire quod est lunae interpositio, sed oportet etiam scire quod luna interponitur inter solem et terram. Et quia prius et notius dicitur dupliciter, scilicet quoad nos, et secundum naturam; dicit consequenter quod ea, ex quibus procedit demonstratio, sunt priora et notiora simpliciter et secundum naturam, et non quoad nos.	Thirdly (71b29), he proves that the propositions of a demonstration are the causes of the conclusion, because we know in a scientific manner when we know the causes. And in virtue of this he shows that they are prior and better known, because every cause is by nature prior and better known than its effect. However, the cause of a demonstrated conclusion must be better known not only with respect to the knowledge of what it is, but also with respect to the knowledge that it is. For in order to demonstrate that there is an eclipse of the sun, it is not enough to know, that the moon is interposed; in addition it is necessary to know that the moon is interposed between the sun and the earth. Again, because prior i and better known are taken in two ways, namely, in reference to us and according to nature, he says that the things from which a demonstration, proceeds are prior and better known absolutely and according to nature, and not in reference to us.
Et ad huius expositionem dicit quod priora et notiora simpliciter sunt illa, quae sunt remota a sensu ut universalia. Priora autem et notiora quoad nos sunt proxima sensui, scilicet singularia, quae opponuntur universalibus, sive oppositione prioris et posterioris, sive oppositione propinqui et remoti.	To elucidate this he says that “those things are prior and better known absolutely,” which are farthest from sense, as are universals; but “the prior and better known in reference to us” are nearest to sense, namely, the singulars, which are opposed to universals in the way that the prior and the later are opposite, or in the way that the nearest and the farthest are opposite.
lib. 1 l. 4 n. 16 Videtur autem contrarium huius haberi in I Physic., ubi dicitur quod universalia sunt priora quoad nos, et posteriora secundum naturam. Sed dicendum est quod hic loquitur de ordine singularis ad universale simpliciter, quorum ordinem oportet accipere secundum ordinem cognitionis sensitivae et intellectivae in nobis. Cognitio autem sensitiva est in nobis prior intellectiva, quia intellectualis cognitio ex sensu procedit in nobis. Unde et singulare est prius et notius quoad nos quam universale. In I autem Physic. non ponitur ordo universalis ad singulare simpliciter, sed magis universalis ad minus universale, ut puta, animalis ad hominem, et sic oportet quod quoad nos, universalius sit prius et magis notum.	However, it seems that the contrary of this is found in Physics I where it is stated that universals are prior in reference to us and later according to nature. But it should be said that there [in the Posterior Analytics] he is speaking of the order of singular to universal absolutely; and this order must be taken according to the order of sensitive and intellectual knowledge in us. Now in us sensitive knowledge is prior to intellectual, because intellectual knowledge in us proceeds from sense. For this reason the singular is prior and better known in relation to us than the universal. But in Physics I he is not speaking of the order of the universal to the singular absolutely but of the order of the more universal to the less universal, for example of animal to man. In this case the more universal is prior and better known in reference to us.
In omni enim generatione, quod est in potentia est prius tempore et posterius natura, quod autem est completum in actu est prius natura et posterius tempore. Cognitio autem generis est quasi potentialis, in comparatione ad cognitionem speciei, in qua actu sciuntur omnia essentialia rei. Unde in generatione scientiae nostrae prius est cognoscere magis commune quam minus commune.	For in every instance of generation, that which is in potency is prior in time but is later according to nature; whereas that which is complete in act is prior by nature but later in time. Now one’s knowledge of a genus is, as it were, potential in comparison to one’s knowledge of the species in which all the essentials of a thing are actually known. Hence, too, in the generation of our science, knowledge of the more common precedes knowledge of the less common.
Item, in libro Physic. dicitur quod innata est nobis via ex nobis notioribus; non ergo demonstratio fit ex his quae sunt priora simpliciter, sed quoad nos. Sed dicendum quod hic loquitur secundum quod id quod est in sensu est notius quoad nos, eo quod est in intellectu. Ibi autem secundum quod id quod est notius quoad nos, est etiam in intellectu. Ex singularibus autem quae sunt in sensu, non sunt demonstrationes, sed ex universalibus tantum, quae sunt in intellectu. Vel dicendum quod in omni demonstratione, oportet quod procedatur ex his, quae sunt notiora quoad nos, non tamen singularibus, sed universalibus.	Again, in the Physics it is stated that it is natural for us to proceed from what is better known to us. Therefore, it seems that a demonstration is composed not of things that are prior absolutely but in reference to But it must be said that here he is speaking according to the fact that what is in the sense is better known in reference to us than what is in the intellect; but there he was speaking according to the fact that what is better known in reference to us is also in the intellect. But demonstrations do not proceed from singulars which are in the sense but only from universals, which are in the intellect.
Non enim aliquid potest fieri nobis notum, nisi per id quod est magis notum nobis. Quandoque autem id quod est magis notum quoad nos, est etiam magis notum simpliciter et secundum naturam; sicut accidit in mathematicis, in quibus, propter abstractionem a materia, non fiunt demonstrationes nisi ex principiis formalibus. Et in talibus fiunt demonstrationes ex his quae sunt notiora simpliciter. Item, quandoque id quod est notius quoad nos non est notius simpliciter, sicut accidit in naturalibus, in quibus essentiae et virtutes rerum, propter hoc quod in materia sunt, sunt occultae, sed innotescunt nobis per ea, quae exterius de ipsis apparent. Unde in talibus fiunt demonstrationes ut plurimum per effectus, qui sunt notiores quoad nos, et non simpliciter. Nunc autem non loquitur de hoc modo demonstrationum, sed de primo.	Or it might be said that in every demonstration one must proceed from things better known to us, provided they are not singulars but universals. For something is made known to us only by that which is more known to us. But sometimes that which is more known in reference to us is also more known absolutely and according to nature, as happens in mathematics where on account of abstraction from matter the demonstrations proceed from formal principles alone. In this case the demonstrations proceed from things which are more known absolutely. But sometimes that which is more known in reference to us is not more known absolutely, as happens in natural sciences where the essences and powers of things are hidden, because they are in matter, but are disclosed to us through the things which appear outwardly. Hence in these sciences the demonstrations are for the most part made through effects which are better known in reference to us but not absolutely. But he is not now speaking of this form of demonstration, but of the first.
Quia vero in hac manifestatione hoc etiam omiserat manifestare, quod demonstratio esset ex propriis principiis, consequenter subdit quod hoc habetur etiam ex praemissis. Per hoc enim quod dicitur quod demonstratio est ex primis, habetur quod sit ex propriis principiis, sicut et superius dictum est. Idem enim videtur esse primum et principium: nam primum in unoquoque genere et maximum est causa omnium eorum, quae sunt post, ut dicitur in II methaphysicae.	Finally, because in his explanation he neglected to point out that demonstration should proceed from proper principles, he hastens to add that this fact is easily ascertainable from what he did say. For from the fact that he stated that demonstration is from things which are first, it follows that it is from proper principles, as he stated above. For “first” and “principle” seem to be the same: for that which is first and highest in each genus is the cause of all the things that are after it, as it is stated in Metaphysics II.

Lecture 5 First and immediate propositions

Latin	English
	Lecture 5 (72a8-24) FIRST AND IMMEDIATE PROPOSITIONS
lib. 1 l. 5 n. 1 Quia superius philosophus dixerat quod demonstratio est ex primis et immediatis, et haec ab ipso nondum manifestata erant, ideo intendit ista notificare. Et dividitur in partes tres: in prima, ostendit quid sit propositio immediata; in secunda vero ostendit quod oporteat huiusmodi propositiones esse notiores conclusione; ibi: quoniam autem oportet credere et scire etc.; in tertia, excludit quosdam errores, qui ex praedictis occasionem habebant; ibi: quibusdam quidem igitur et cetera. Circa primum duo facit: primo, ostendit quid sit propositio immediata; secundo, dividit ipsam; ibi: immediati autem principii et cetera.	Because the Philosopher had stated above that demonstration is from “first and immediate principles,” but had not yet identified them, he now sets out to identify them. And this is divided into three parts. In the first part he shows what an immediate proposition is. In the second part he shows that such propositions must be better known than the conclusion (72a25) [L. 6]. In the third part he excludes certain errors which arose from the foregoing (72b5) [L. 7]. Concerning the first he does two things. First, he shows what an immediate principle is. Secondly, h divides them (72a15).
Circa primum hoc modo procedit. Primo namque, resumit quod supra dictum erat, scilicet quod principium demonstrationis sit propositio immediata: nam et supra dixerat quod demonstratio est ex primis et immediatis.	With respect to the first he proceeds this way. First (72a8), he recall, what has been said above, namely, that a principle of demonstration I, an immediate proposition, for he had also stated above that a demonstration is composed of things which are first and immediate.
lib. 1 l. 5 n. 2 Secundo, ibi: immediata autem etc., definit immediatam propositionem, et dicit quod immediata propositio est qua non est altera prior. Cuius quidem notificationis ratio ex praedictis apparet. Dictum est enim supra quod demonstratio est ex prioribus. Quandocunque igitur aliqua propositio est mediata, idest habens medium per quod demonstretur praedicatum de subiecto, oportet quod priores ea sint propositiones ex quibus demonstratur: nam praedicatum conclusionis per prius inest medio quam subiecto; cui etiam per prius inest medium quam praedicatum. Relinquitur ergo quod illa propositio, qua non est altera prior, sit immediata.	Secondly (ibid.), he defines the immediate proposition and says that a immediate proposition is one which has no other one prior to it. the reason underlying this description is clear from what has been said. For it has been said above that demonstration is composed of things that, are prior. Accordingly, whenever a proposition is mediate, i.e., has a middle through which the predicate is demonstrated of its subject, it is required that there be prior propositions by which this one is demonstrated. For the predicate of a conclusion is present in the middle previously to being present in the subject; in which, however, the middle is present before the predicate is. Therefore, it follows that that proposition which does not have some other one prior to it is immediate.
lib. 1 l. 5 n. 3 Tertio, ibi: propositio autem est etc., ostendit quid sit propositio, quae ponitur in definitione immediatae propositionis. Et circa hoc tria facit. Primo namque definit propositionem simpliciter, dicens quod propositio est altera pars enunciationis, in qua praedicatur unum de uno. Habet enim enunciatio duas partes, scilicet affirmationem et negationem. Oportet autem quod omnis syllogizans alteram earum proponat, non autem utramque: hoc enim est proprium eius, qui a principio quaestionem movet. Unde per hoc separatur propositio a problemate. Sicut etiam in uno syllogismo non concluditur nisi unum, ita oportet quod propositio, quae est syllogismi principium, sit una. Una autem est in qua est unum de uno. Unde per hoc quod philosophus dicit unum de uno, separatur propositio ab enunciatione, quae dicitur plures, in qua plura de uno vel unum de pluribus praedicatur.	Thirdly (720), he shows what is the nature of the proposition which is mentioned in the definition of an immediate proposition. Concerning this he does three things: First, he defines absolutely what a proposition is, saying that it is one or the other part of an enunciation in which one thing is predicated of one thing. For the enunciation has two parts, namely, affirmation and negation. For anyone who syllogizes must propose one or the other of these parts but not both, for this latter procedure is characteristic of one who first raises a question. (Hence it is on this basis that proposition is distinguished from a problem). For just as one and only one thing is concluded in one syllogism, so the proposition which is a principle of the syllogism should be one—and it is one if one thing is stated of one thing. Hence in asserting that it is “one of one,” he distinguishes the proposition from the enunciation which is said to be “of several,” whether sundry things are said of one thing or of one thing sundry.
lib. 1 l. 5 n. 4 Secundo, ibi: dialectica etc., ponit differentiam inter dialecticam propositionem et demonstrativam, dicens quod cum propositio accipiat alteram partem enunciationis, dialectica indifferenter accipit quancunque earum. Habet enim viam ad utranque partem contradictionis, eo quod ex probabilibus procedit. Unde etiam et in proponendo accipit utramlibet partem contradictionis et quaerendo proponit. Demonstrativa autem propositio accipit alteram partem determinate, quia nunquam habet demonstrator viam, nisi ad verum demonstrandum. Unde etiam semper proponendo accipit veram partem contradictionis. Propter hoc etiam non interrogat, sed sumit, qui demonstrat, quasi notum.	Secondly (72a10), he lays down the difference between the dialectical and the demonstrative proposition, saying that whereas the demonstrative proposition takes one definite side of a question, the dialectical takes either side indifferently. For since dialectic begins with the probable, it can lead to each side of a contradiction. Hence when it lays down its propositions, it employs both parts of a contradiction and presents them in the form of a question [Is an animal that walks on its feet a man, or not?]. But a demonstrative proposition takes one side definitively, because a demonstrator never has any other alternative but to demonstrate the truth. Hence in forming its propositions he always assumes the true side of a contradiction [An animal which walks on two feet is a man, is it not?]. On this account he does not ask but posits something as known in the demonstration.
lib. 1 l. 5 n. 5 Tertio, ibi: enunciatio autem etc., definit enunciationem quae ponitur in definitione propositionis, dicens quod enunciatio complectitur utranque partem contradictionis, ut ex dictis patet. Quid autem sit contradictio consequenter ostendit, dicens quod contradictio est oppositio, cuius non est medium secundum se. Quamvis enim in privatione et habitu, et in contrariis immediatis non sit medium circa determinatum subiectum, tamen est medium simpliciter, nam lapis neque caecus, neque videns est, et albedo neque par, neque impar est. Et hoc etiam quod habent de immediatione circa determinatum subiectum, habent in quantum aliquid participant contradictionis: nam privatio est negatio in subiecto determinato. Et alterum etiam contrariorum habet aliquid privationis. Sed contradictio simpliciter in omnibus caret medio; et hoc non habet ab alio, sed ex seipsa; et propter hoc dicit quod eius non est medium secundum se.	Thirdly (72a11), he defines the term, “enunciation,” which appeared in the definition of a proposition, saying that an enunciation embraces both sides of a contradiction, as is clear from what has been said. Then he shows what contradiction is, saying that contradiction is a form of opposition between whose parts there is of itself no middle. For although between privation and possession and between immediate contraries there is no middle in a given subject, nevertheless, absolutely speaking, there is one; for a stone is neither blind nor seeing, and something white is neither even nor odd. Furthermore, whatever immediacy they have in relation to a definite subject is traced to their participation in contradiction, for privation is negation in a definite subject; and of two things that are immediately contrary, one has some of the marks of privation. But contradiction in the full sense lacks a middle in all cases. And this belongs to it of its very nature and not in virtue of something else. Hence he says that of itself it has no medium.
Exponit etiam consequenter quid sit pars contradictionis: est enim contradictio oppositio affirmationis et negationis. Unde altera pars eius est affirmatio, quae praedicat aliquid de aliquo; altera vero negatio, quae removet aliquid ab aliquo.	He then explains what the parts of a contradiction are. For contradiction is an opposition of affirmation and negation; hence one of its parts is affirmation, which asserts something of something, and the other is negation, which denies something of something.
lib. 1 l. 5 n. 6 Deinde cum dicit: immediati autem etc. dividit immediatum principium. Et circa hoc duo facit: primo dividit; secundo subdividit; ibi: positiones autem quaedam et cetera. Dicit ergo primo quod immediatum principium syllogismi duplex est. Unum est quod dicitur positio, quam non contingit demonstrare et ex hoc immediatum dicitur; neque tamen aliquem docendum, idest qui doceri debet in demonstrativa scientia, necesse est habere, idest mente concipere sive ei assentire. Aliud vero est, quod dicitur dignitas vel maxima propositio, quam necesse est habere in mente et ei assentire quemlibet, qui doceri debet. Et manifestum est quod quaedam principia talia sunt ut probatur in IV metaphysicae de hoc principio, quod affirmatio et negatio non sunt simul vera, cuius contrarium nullus mente credere potest etsi ore proferat. Et in talibus utimur nomine praedicto, scilicet dignitatis vel maximae propositionis, propter huiusmodi principiorum certitudinem ad manifestandum alia.	Then (72a15) he divides immediate principle. Concerning this he does two things. First, he divides. Secondly, he subdivides (72a19). He says therefore first (7205), that there are two types of immediate principles of a syllogism: the first is called a “position” [thesis] and is said to be immediate because one does not demonstrate (neither is it required that the student, i.e., the one being instructed in the demonstrative science, have it, i.e., advert to it or assent to it); the other is called a “dignity” or “maxim,” which anyone who is to be instructed must have in his mind and assent to. That there are such principles is clear from Metaphysics IV, where it is proved that one such is the principle that “affirmation and negation are not simultaneously true,” for no one can believe the contrary of this in his mind ‘ even though he should state it orally. To such principles we give the aforesaid name of “dignity” or “maxim” on account of their certainty in manifesting other things.
lib. 1 l. 5 n. 7 Ad huius autem divisionis intellectum sciendum est quod quaelibet propositio, cuius praedicatum est in ratione subiecti, est immediata et per se nota, quantum est in se. Sed quarundam propositionum termini sunt tales, quod sunt in notitia omnium, sicut ens, et unum, et alia quae sunt entis, in quantum ens: nam ens est prima conceptio intellectus. Unde oportet quod tales propositiones non solum in se, sed etiam quoad omnes, quasi per se notae habeantur. Sicut quod, non contingit idem esse et non esse; et quod, totum sit maius sua parte: et similia. Unde et huiusmodi principia omnes scientiae accipiunt a metaphysica, cuius est considerare ens simpliciter et ea, quae sunt entis.	To clarify this division it should be noted that any proposition whose predicate is included within the notion of its subject is immediate and known in virtue of itself as it stands. However, in the case of some of these propositions the terms are such that they are understood by everyone, as being and one and those other notions that are characteristic of being precisely as being: for being is the first concept in the intellect. Hence it is necessary that propositions of this kind be held as known in virtue of themselves not only as they stand but also in reference to us. Examples of these are the propositions that “It does not occur that the same thing is and is not” and that “The whole is greater than its part,” and others like these. Hence all the sciences take principles of this kind from metaphysics whose task it is to consider being absolutely and the characteristics of being.
Quaedam vero propositiones sunt immediatae, quarum termini non sunt apud omnes noti. Unde, licet praedicatum sit de ratione subiecti, tamen quia definitio subiecti non est omnibus nota, non est necessarium quod tales propositiones ab omnibus concedantur. Sicut haec propositio: omnes recti anguli sunt aequales, quantum est in se, est per se nota sive immediata, quia aequalitas cadit in definitione anguli recti. Angulus enim rectus est, quem facit linea recta super aliam rectam cadens, ita quod ex utraque parte anguli reddantur aequales. Et ideo, cum quadam positione recipiuntur huiusmodi principia.	On the other hand, there are some immediate propositions whose terms are not known by everyone. Hence, although their predicate may be included in the very notion of their subject, yet because the definition of the subject is not known to everyone, it is not necessary that such propositions be conceded by everyone. (Thus the proposition, “All right angles are equal,” is in itself a proposition which is immediate and known in virtue of itself, because equality appears in the definition of a right angle. For a right angle is one which a straight line form when it meets another straight line in such a way that the angles on each side are equal). Therefore, such principles are received as being posited or laid down.
Est et alius modus, quo aliquae propositiones suppositiones dicuntur. Sunt enim quaedam propositiones, quae non possunt probari nisi per principia alterius scientiae; et ideo oportet quod in illa scientia supponantur, licet probentur per principia alterius scientiae. Sicut a puncto ad punctum rectam lineam ducere, supponit geometra et probat naturalis; ostendens quod inter quaelibet duo puncta sit linea media.	There is yet another way, and according to it certain propositions are called “suppositions.” For there are some propositions which can be proved only by the principles of some other science; therefore, they must be supposed in the one science, although they are proved by the principles of the other science. Thus the geometer supposes that he can draw one straight line from one point to another, but the philosopher of nature proves it by showing that there is one straight line between any two points.
lib. 1 l. 5 n. 8 Deinde cum dicit: positionis autem quae est etc., subdividit alterum membrum primae divisionis, scilicet positionem: dicens quod quaedam positio est, quae accipit aliquam partem enunciationis, scilicet affirmationem vel negationem; quod significat cum dicit: ut dico aliquid esse aut non esse. Et haec positio suppositio dicitur, quia tanquam veritatem habens supponitur. Alia autem positio est, quae non significat esse vel non esse, sicut definitio, quae positio dicitur. Ponitur enim ab arithmetico definitio unitatis, tanquam quoddam principium, scilicet quod unitas est indivisibile secundum quantitatem. Sed tamen definitio non dicitur suppositio: illud enim proprie supponitur, quod verum vel falsum significat. Et ideo subdit quod non idem est, quod quid est unitas, quod neque verum, neque falsum significat, et esse unitatem, quod significat verum vel falsum.	Then (72a19) he subdivides a member of the original division, namely, “position,” and says that there is one type of position which takes one side of an enunciation, namely, either affirmation or negation. He refers to this type when he says, “i.e., asserts either the existence or non-existence of a subject.” Such a position is called a “supposition” or “hypothesis,” because it is accepted as having truth. Another type of position is the one which does not signify existence or non-existence: in this way a definition is a position. For the definition of “one” is laid down in arithmetic as a principle, namely, that “one is the quantitatively indivisible.” Nevertheless a definition is not called a supposition, for a supposition, strictly speaking, is a statement which signifies the true or the false. Consequently, he adds that “the definition of ‘one,”’ inasmuch as it signifies neither the true nor the false, “is not the same as ‘to be one,”’ which does signify the true or the false.
lib. 1 l. 5 n. 9 Sed potest quaeri: cum definitio non sit propositio significans esse vel non esse, quomodo ponatur in subdivisione immediatae propositionis. Sed dicendum quod in subdivisione non resumit immediatam propositionem ad subdividendum, sed immediatum principium. Principium autem syllogismi dici potest non solum propositio, sed etiam definitio. Vel potest dici quod licet definitio in se non sit propositio in actu, est tamen in virtute propositio quia cognita definitione, apparet definitionem de subiecto vere praedicari.	Now it might be asked how it is that definition is set down as a member of the subdivision of immediate proposition, if a definition is not a proposition signifying either existence or non-existence. One might answer that in this subdivision he was not subdividing immediate proposition but, immediate principle. Or one might answer that although a definition as such is not an actual proposition, it is one virtually, because once a definition is known, it becomes clear that it is truly predicated of the subject.

Lecture 6 Knowledge of immediate principles

Latin	English
	Lecture 6 (7245-b4) KNOWLEDGE OF IMMEDIATE PRINCIPLES
lib. 1 l. 6 n. 1 Postquam ostendit philosophus quae sunt immediata principia, hic de eorum cognitione determinat. Et circa hoc duo facit: primo, ostendit quod immediata principia sunt magis nota conclusione; secundo, ostendit quod etiam falsitas contrariorum debet esse notissima; ibi: non solum autem et cetera. Circa primum tria facit.	After showing what immediate principles are, the Philosopher now determines concerning our knowledge of them. Apropos of this he does two things. First, he shows that immediate principles are better known than the conclusion. Secondly, that the falsity of their contraries ought to be most evident (72a38). Concerning the first he does three things.
lib. 1 l. 6 n. 2 Primo, proponit intentum dicens quod quia nos credimus alicui rei conclusae et scimus eam per hoc quod habemus syllogismum demonstrativum, et hoc quidem est in quantum scimus syllogismum demonstrativum; necesse est non solum praecognoscere prima principia conclusioni, sed etiam ea magis cognoscere, quam conclusionem.	First (72a25), he states his proposition and says that because we give our assent to a thing which has been concluded and we know it scientifically precisely because we have a demonstrative syllogism (and this insofar as we know the demonstrative syllogism in a scientific way), it is necessary not only to know the first principles of the conclusion beforehand, but also to know them better than we know the conclusion.
Addit autem aut omnia, aut quaedam; quia quaedam principia probatione indigent, ad hoc quod sint nota, et antequam probentur, non sunt magis nota conclusione. Sicut quod, angulus exterior trianguli valeat duos angulos intrinsecos sibi oppositos, antequam probetur, ita ignotum est, sicut quod, triangulus habeat tres angulos aequales duobus rectis. Quaedam vero principia sunt quae, statim proposita, sunt magis nota conclusione. Vel aliter: quaedam conclusiones sunt quae sunt notissimae, utpote per sensum acceptae, sicut quod, sol eclipsetur. Unde principium per quod probatur non est simpliciter magis notum, scilicet quod, luna interponatur inter solem et terram, licet sit magis notum in via rationis procedentis ex causa in effectum. Vel aliter: hoc ideo dicit, quia etiam supra dixerat quod quaedam principia tempore prius cognoscuntur quam conclusio, quaedam vero simul tempore nota fiunt cum conclusione.	He adds, “either all or some,” because some principles require proof in order to be known; so that before they are proved, they are not better known than the conclusion. Thus the fact that an exterior angle of a triangle is equal to its two opposite interior angles is, until proved, as unknown as the fact that a triangle has three angles equal to two right angles. But there are other principles which, once they are posited, are better known than the conclusion. Or, in another way, there are some conclusions which are most evident; for example, those based on sense perception, as that the sun is eclipsed. Hence the principle through which this is proved is not better known absolutely—the principle being that the moon is between the sun and the earth—although it is better known within the reasoning process that goes from cause to effect. Or, in another way, he says this because he had said above that in the order of time certain principles are known before the conclusion, but others are known along with the conclusion at the same moment of time.
lib. 1 l. 6 n. 3 Secundo; ibi: semper enim propter quod etc., probat propositum dupliciter. Primo ratione ostensiva; sic: propter quod unumquodque et illud est magis; sicut si amamus aliquem, propter alterum; ut si magistrum propter discipulum, discipulum magis amamus. Sed conclusiones scimus et eis credimus propter principia. Ergo magis scimus principia et magis eis credimus, quam conclusioni.	Secondly (72a28), he proves his proposition in two ways: first, with an ostensive argument, thus: That in virtue of which something is so, is itself more so; for example, if we love someone because of someone else, as a master because of his disciple, we love the disciple more. But we know conclusions and give our assent to them because of the principles. Therefore, we know the principles with more conviction and give them stronger assent than the conclusion.
lib. 1 l. 6 n. 4 Attendendum est autem circa hanc rationem quod causa semper est potior effectu suo. Quando ergo causa et effectus conveniunt in nomine, tunc illud nomen magis praedicatur de causa quam de effectu; sicut ignis est magis calidus quam ea, quae per ignem calefiunt. Quandoque vero causa et effectus non conveniunt in nomine et tunc licet nomen effectus non conveniat causae, tamen convenit ei aliquid dignius; sicut etsi in sole non sit calor, est tamen in eo virtus quaedam, quae est principium caloris.	Apropos of this reason it should be noted that a cause is always more noble than its effect. When, therefore, cause and effect have the same name, that name is said principally of the cause rather than of the effect; thus fire is primarily called hot rather than things heated by fire. But sometimes the name of the effect is not attributed to the cause. In that case, although the name the effect has does not belong to the cause, nevertheless, something more noble belongs to it. For example, although the sun does not possess heat, nevertheless, there is in it a certain power which is the principle of heat.
lib. 1 l. 6 n. 5 Deinde, cum dicit: non potest autem credere etc., probat idem ratione ducente ad impossibile, quae talis est. Principia praecognoscuntur conclusioni, ut supra habitum est; et sic quando principia cognoscuntur nondum conclusio est cognita. Si igitur principia non essent magis nota quam conclusio, sequeretur quod homo vel plus, vel aequaliter cognosceret ea, quae non novit quam ea quae novit. Hoc autem est impossibile. Ergo impossibile est quod principia non sint magis nota quam conclusio.	Then (72a33) he proves the same thing with a principle which leads to an impossibility. He reasons thus: Principles are known prior to the conclusion, as has been shown above; consequently, when the principles are known, the conclusion is not yet known. If, therefore, the principles were not more known than the conclusion, it would follow that a man would know things he does not know either as well as or better than the things he does know. But this is impossible. Therefore, it is also impossible that the principles not be better known than the conclusion.
Littera sic exponitur: neque sciens, neque alius melius dispositus in cognoscendo quam sciens, si contingeret aliquem esse talem (quod dicit propter intelligentem principia, de quo adhuc non est manifestum), non potest magis credere quae non contingunt, scilicet sciri ab eo, his, quae iam scit. Accidet autem hoc, nisi aliquis de numero credentium conclusionem per demonstrationem, praecognoverit, idest magis cognoverit principia. In Graeco planius habetur sic: non est autem possibile credere magis his, quae novit, quae non existit nec sciens, neque melius dispositus quam si contigerit sciens.	Phrase by phrase this is explained in the following manner: “A man who knows scientifically or even one who knows in a way superior to this, if such there be,” (he says this, having in mind the person who has the intuition of principles, a state he has not yet explained), “cannot give more credence to things he does not know than to things he does know. But this will be the case if one who assents to a conclusion obtained through demonstration did not foreknow,” i.e., did not know the principles better. In Greek it is stated more clearly: “But no one, whether he has scientific knowledge or that form of knowledge which is better than the scientific (if there be such), can believe anything more firmly than the things he knows.”
lib. 1 l. 6 n. 6 Tertio, ibi: magis enim necesse est etc., exponit quod dixerat, dicens quod hoc, quod dictum est, quod magis necesse est credere principiis aut omnibus aut quibusdam quam conclusioni, intelligendum est de illo, qui debet accipere scientiam per demonstrationem. Si enim aliunde conclusio esset nota, sicut per sensum, nihil prohiberet principia non esse magis nota conclusione in via illa.	Thirdly (7206), he clarifies what he had said, saying that his statement to the effect that it is more necessary to believe the principles (either all or some) than the conclusion should be understood as referring to a person who is to acquire a discipline through demonstration. For if the conclusion were more known through some other source, such as sense-perception, nothing would preclude the principles not being better known than the conclusion in that case.
lib. 1 l. 6 n. 7 Deinde, cum dicit: non solum oportet etc., ostendit quod non solum oportet magis cognoscere principia quam conclusionem demonstrandam, sed etiam nihil debet esse certius quam quod opposita principiis sint falsa. Et hoc ideo, quia oportet scientem non esse incredibilem principiis, sed firmissime eis assentire. Quicunque autem dubitat de falsitate unius oppositorum, non potest firmiter inhaerere opposito: quia semper formidat de veritate alterius oppositi.	Then (72a38) he shows that it is not only necessary to know the principles more than the demonstrative conclusion, but nothing should be more certain than the fact that the opposites of the principles are false. And this because the scientific knower must not disbelieve the principles, but assent to them most firmly. But anyone who doubts the falseness of one of two opposites cannot assent firmly to the other, because he will always fear that the opposite one might be true.

Lecture 7 Discussion of two errors-exclusion of the first one

Latin	English
	Lecture 7 (72b5-24) DISCUSSION OF TWO ERRORS—EXCLUSION OF THE FIRST ONE
lib. 1 l. 7 n. 1 Postquam determinavit philosophus de cognitione principiorum demonstrationis, hic excludit errores ex praedeterminata veritate exortos. Et circa hoc tria facit: primo, ponit errores; secundo, rationes errantium; ibi: supponentes quidem igitur etc., tertio, excludit rationum radices; ibi: nos autem dicimus et cetera.	After determining about the knowledge of the principles of demonstration, the Philosopher now excludes the errors which have arisen from these determinations. Concerning this he does three things. First, he states the errors. Secondly, the reasons they erred (72b8). Thirdly, he removes the roots of these reasons (72b18).
lib. 1 l. 7 n. 2 Dicit ergo primo quod ex una veritate superius determinata duo errores contrarii sunt exorti. Determinatum est enim supra quod oportet principia demonstrationis scire, immo quod etiam ea magis scire oportet. Sed primum solum sufficit ad propositum. Propter hoc autem videtur quibusdam quod nullius rei sit scientia; quibusdam autem videtur quod sit quidem scientia, sed quod omnium possit haberi scientia per demonstrationem. Neutrum autem horum est verum, nec necessario consequitur ex rationibus eorum.	He says therefore first (72b5), that two contrary errors have arisen from one of the truths established above. For it has been established above that the principles of demonstration must be known and must be even better known. But the first of these is sufficient for our purpose. For some, basing themselves on this first statement, have come to believe that there is no science of anything, whereas others believe that there is science, even to the extent of believing that there is science of everything through demonstration. But neither of these positions is true and neither follows necessarily from their reasons.
lib. 1 l. 7 n. 3 Deinde, cum dicit: supponentes etc., ponit rationes quibus in praedictos errores incidunt. Et primo ponit rationem dicentium quod non est scientia. Quae talis est. Principia demonstrationis aut procedunt in infinitum aut est status in eis. Si proceditur in infinitum, non est in eis accipere prima: quia infinita non est transire, ut ad primum veniatur: et ita non est primum cognoscere. Et in hoc recte argumentantur. Nam posteriora non possunt cognosci, ignoratis primis.	Then (72b8) he presents the reasons why they have fallen into these errors. And first of all he presents the reason given by those who say that there is no science, and it is this: The principles of demonstration either proceed to infinity or there is a halt somewhere. But if there is a process to infinity, nothing in that process can be taken as being first, because one cannot exhaust an infinite series and reach what is first. Consequently, it is not possible to know what is first. (They are correct in thus arguing, for the later things cannot be known unless the prior ones are known).
Si autem stetur in principiis, oportet quod prima nesciantur; si scire solum est per demonstrationem: non enim prima habent aliqua priora, per quae demonstrentur. Si autem prima ignorentur, oportet et posteriora iterum non scire simpliciter nec proprie: sed solum sub hac conditione, si principia sunt. Non enim potest per aliquid ignotum aliquid cognosci, nisi sub hac conditione, si illud primum quod ignotum est, sit. Ergo sequitur utroque modo, sive principia stent, sive in eis procedatur in infinitum, quod nullius rei est scientia.	On the other hand, if there is a halt in the principles, then even so, the first things are still not known, if the only way to know scientifically is through demonstration. For first things do not have prior principles through which they are demonstrated. But if the first things are not known, it follows again that the later things are not known in the strict and proper sense, but only on condition that there are principles. For it is not possible for something to be known in virtue of something not known, except on condition that that unknown be a principle. So in either case, whether the principles stop or go on to infinity, it follows that there is no science of anything.
lib. 1 l. 7 n. 4 Secundo, cum dicit: quidam autem etc., ponit rationem dicentium omnium esse scientiam per demonstrationem: quia praemissae radici, scilicet quod non esset scire nisi per demonstrationem, addebant aliam, scilicet quod posset circulariter demonstrari. Sic enim sequebatur quod etiamsi in principiis demonstrationis esset status, prima tamen principia erat scire per demonstrationem: quia illa prima dicebant demonstrari ex posterioribus. Nam circulariter demonstrare est demonstrare ex invicem, idest ut quod primo fuit principium, postmodum fiat conclusio et e converso.	Secondly (72b15), he presents the reasoning of those who say that there is science of everything through demonstration, because to their there is science of everything through demonstration, because to their basic premise-the only way to know scientifically is by demonstration—they added another, namely, that one may demonstrate circularly. From these premises it followed that even if a limit is reached in the series of the principles of demonstration, the first principles are still known through demonstration, because, they said, those principles were demonstrated by previous ones. For a circular demonstration is one which is reciprocal, i.e., something which was first a principle is later a conclusion, and vice versa.
lib. 1 l. 7 n. 5 Deinde, cum dicit: nos autem dicimus etc., excludit falsas radices praedictarum rationum. Et primo, hoc quod supponebant quod non esset scire nisi per demonstrationem. Secundo, hoc quod dicebant quod contingeret circulariter demonstrari; ibi: circulo quoque et cetera. Dicit ergo primo quod non omnis scientia est demonstrativa, idest per demonstrationem accepta; sed immediatorum principiorum est scientia indemonstrabilis, idest non per demonstrationem accepta.	Then (72b18) he cuts away the false bases of these arguments. First, their supposition that the only way to know scientifically is by demonstration. Secondly, their statement that it is legitimate to demonstrate circularly (72b25). He says therefore first (72b18), that not all scientific knowledge is demonstrative, i.e., obtained through demonstration, but the scientific knowledge of immediate principles is indemonstrable, i.e., not obtained by demonstration.
lib. 1 l. 7 n. 6 Sciendum est tamen quod hic Aristoteles large accipit scientiam pro qualibet certitudinali cognitione, et non secundum quod scientia dividitur contra intellectum, prout dicitur quod, scientia est conclusionum et intellectus principiorum.	However, it should be noted that Aristotle is here taking science in a wide sense to include any knowledge that is certain, and not in the sense in which science is set off against understanding, according to the dictum that science deals with conclusions and understanding [intuition] with principles.
lib. 1 l. 7 n. 7 Quod autem hoc necessarium sit, scilicet quod certa cognitio aliquorum habeatur sine demonstratione, sic probat. Necesse est scire priora, ex quibus est demonstratio; sed haec aliquando contingit reducere in aliqua immediata: alias oporteret dicere quod inter duo extrema, scilicet subiectum et praedicatum essent infinita media in actu; immo plus, quod non esset aliqua duo accipere, inter quae non essent infinita media. Qualitercunque autem media assumantur, est accipere aliquid alteri immediatum. Immediata autem, cum sint priora, oportet esse indemonstrabilia. Et ita patet quod necesse est aliquorum scientiam habere sine demonstratione.	But that it is necessary for some things to be held as certain without demonstration he proves in the following way: It is necessary that the prior things from which a demonstration proceeds be known in a scientific way. Furthermore, these must be ultimately reduced to something immediate; otherwise one would be forced to admit that there is an actual infinitude of middles between two extremes—in this case between the subject and predicate. Again, one would have to admit that no two extremes could be found between which there would not be an infinitude of middles. But as it is, the middles are such that it is possible to find two things which are immediate. But immediate principles, being prior, must be indemonstrable. Thus it is clear that it is necessary for some things to be scientifically known without demonstration.
lib. 1 l. 7 n. 8 Si ergo quaeratur quomodo immediatorum habeatur scientia, respondet quod non solum eorum est scientia, immo eorum cognitio est principium quoddam totius scientiae. Nam ex cognitione principiorum derivatur cognitio conclusionum, quarum proprie est scientia. Ipsa autem principia immediata non per aliquod medium extrinsecum cognoscuntur, sed per cognitionem propriorum terminorum. Scito enim quid est totum et quid est pars, cognoscitur quod omne totum est maius sua parte: quia in talibus propositionibus, ut supra dictum est, praedicatum est de ratione subiecti. Et ideo rationabiliter cognitio horum principiorum est causa cognitionis conclusionum: quia semper, quod est per se, est causa eius, quod est per aliud.	Therefore, if someone were to ask how the science of immediate principles is possessed, the answer would be that not only are they known in a scientific manner, but knowledge of them is the source of an science. For one passes from the knowledge of principles to a demonstration of conclusion on which science, properly speaking, bears. But those immediate principles are not made known through an additional middle but through an understanding of their own terms. For as soon as it is known what a whole is and what a part is, it is known that every whole is greater than its part, because in such a proposition, as has been stated above, the predicate is included in the very notion of the subject. And therefore it is reasonable that the knowledge of these principles is the cause of the knowledge of conclusions, because always, that which exists in virtue of itself is the cause of that which exists in virtue of something else.

Lecture 8 The second error is excluded by showing that circular demonstration is not acceptable

Latin	English
	Lecture 8 (72b-73a20) THE SECOND ERROR IS EXCLUDED BY SHOWING THAT CIRCULAR DEMONSTRATION IS NOT ACCEPTABLE
lib. 1 l. 8 n. 1 Postquam philosophus exclusit unam falsam radicem, ostendens quod non omnis scientia est per demonstrationem, hic excludit aliam, ostendens quod non contingit circulariter demonstrare.	After excluding one false basis by showing that not all science depends on demonstration, the Philosopher now excludes another by showing that it is not possible to demonstrate circularly.
lib. 1 l. 8 n. 2 Ad cuius evidentiam sciendum est quod circularis syllogismus dicitur, quando ex conclusione et altera praemissarum conversa concluditur reliqua. Sicut si fiat talis syllogismus: omne animal rationale mortale est risibile; omnis homo est animal rationale mortale; ergo omnis homo est risibilis: assumatur autem conclusio tanquam principium, et adiungatur ei minor conversa, hoc modo: omnis homo est risibilis; omne animal rationale mortale est homo; sequitur quod, omne animal rationale mortale sit risibile: quae erat maior primi syllogismi.	To understand this it should be noted that a demonstration is circular when the conclusion and one of the premises (in converted form) of a syllogism are used to prove the other premise. For example, we might form the following syllogism: Every rational mortal animal is risible; Every man is a rational mortal animal: Therefore, every man is risible. Now if the conclusion were to be used as one principle and the minor in converted form as the other, we would get: Every man is risible; Every rational mortal animal is a man: Therefore, every rational mortal animal is risible—which was the major of the first syllogism.
lib. 1 l. 8 n. 3 Ostendit autem Aristoteles per tres rationes quod non contingit circulariter demonstrare. Quarum prima est. In circulari syllogismo idem fit et principium et conclusio. Principium autem demonstrationis est prius et notius conclusione: ut supra ostensum est. Sequitur ergo quod idem sit prius et posterius respectu unius et eiusdem, et notius et minus notum. Hoc autem est impossibile. Ergo impossibile est circulariter demonstrare.	Accordingly, he. presents three arguments to show that it is not possible to demonstrate circularly. The first of these (72b25) is this: In a circular syllogism the same thing is at once a conclusion and a principle. But a principle of a demonstration is prior to and better known than the conclusion, as has been shown above. Therefore, it follows that a same thing is both prior to and subsequent to one same thing, and also more known and less known. But this is impossible. Therefore, it is impossible to demonstrate circularly.
lib. 1 l. 8 n. 4 Sed posset aliquis dicere quod idem potest esse prius et posterius alio et alio modo, scilicet ut hoc sit prius quoad nos, et illud sit prius simpliciter. Sicut singularia sunt priora quoad nos, et posteriora simpliciter; universalia vero e converso. Hoc autem modo inductio facit notum, scilicet altero modo a demonstratione. Nam demonstratio procedit ex prioribus simpliciter: inductio autem ex prioribus quoad nos.	But someone might say that a same thing can be both prior and subsequent, although not in the same way. For example, this might be prior in reference to us, but that prior absolutely. Thus singulars are prior in reference to us and subsequent absolutely: and conversely for universals. Again, induction makes something known in one way and demonstration in another way. For demonstration proceeds from things that are prior absolutely, but induction from things that are prior in reference to us.
Sed si sic fieret demonstratio circularis, ut scilicet primo concluderetur ex prioribus simpliciter, postea vero ex prioribus quoad nos; sequeretur quod non esset bene determinatum superius quid est scire. Dictum est enim quod scire est causam rei cognoscere. Et ideo ostensum est quod oportet demonstrationem, quae facit scire, ex prioribus simpliciter procedere. Si autem demonstratio, nunc ex prioribus simpliciter, nunc ex prioribus quoad nos procederet; oporteret etiam quod scire non solum esset causam rei cognoscere, sed dupliciter diceretur quia esset etiam quoddam scire per posteriora. Aut ergo oportebit sic dicere, aut oportebit dicere quod altera demonstratio, quae fit ex nobis notioribus, non sit simpliciter demonstratio.	Now if a circular demonstration were so constructed that something is first concluded from things that are absolutely prior, and then from things that are prior in reference to us, it would follow that our doctrine on scientific knowing was not well established. For we stated that to know scientifically is to know the cause of a thing. From this it followed that a demonstration which causes scientific knowledge must proceed from the absolutely prior. But if demonstration were at one time to proceed from the absolutely prior and at another time from things which are prior in reference to us, we would be forced to admit that scientific knowing is not confined to knowing the cause of a thing, but that there is another, namely, that form of knowing which proceeds from what is later. Therefore, one must either admit both or admit that the second form, namely, the demonstration which proceeds from what is better known to us is not a demonstration in the absolute sense.
lib. 1 l. 8 n. 5 Ex his autem apparet quare dialecticus syllogismus potest esse circularis. Procedit enim ex probabilibus. Probabilia autem dicuntur, quae sunt magis nota vel sapientibus vel pluribus. Et sic dialecticus syllogismus procedit ex his quae sunt magis nobis nota. Contingit autem idem esse magis et minus notum quoad diversos; et ideo nihil prohibet syllogismum dialecticum fieri circularem. Sed demonstratio fit ex notioribus simpliciter; et ideo, ut dictum est, non potest fieri demonstratio circularis.	The aforesaid also reveals why a dialectical syllogism can be circular. For it proceeds from things which are probable. But things are said to be probable if they are better known to the wise or to a great number of persons. Consequently, a dialectical syllogism proceeds from things that are better known to us. However, it happens that a same thing is better known to some and less known to others. Consequently, there is nothing to hinder a dialectical syllogism from being circular. But a demonstration is formed from things that are absolutely prior. Therefore, as we have already stated, there cannot be circular demonstration.
lib. 1 l. 8 n. 6 Secundam rationem ponit ibi: accidit autem etc.: quae talis est. Si demonstratio sit circularis, sequitur quod in demonstratione probetur idem per idem; ut si dicamus: si est hoc, est hoc. Sic autem facile est cuilibet demonstrare omnia. Hoc enim poterit facere quilibet, tam sciens quam ignorans. Et sic per demonstrationem non acquiritur scientia: quod est contra definitionem demonstrationis. Non ergo potest esse demonstratio circularis.	Then he sets forth the second argument (72b33) and it is this: If there were circular demonstration, it would follow that a same thing is demonstrated by the same thing, as if I were to say: If it is this, it is this. In this way it is easy for anyone to demonstrate everything, for anyone, wise or ignorant, will be able to do this. Accordingly, science is not acquired through demonstration. But this is against the definition of demonstration. Therefore, there cannot be circular demonstration.
Veritatem autem primae consequentiae hoc modo ostendit. Primo enim, dicit quod manifestum est quod accidit, circulari demonstratione facta, hoc quod prius dictum est, scilicet quod idem probetur per idem, si quis sumat tres terminos. Reflexionem autem fieri per multos, aut per paucos terminos nihil differt (nominat autem hic reflexionem processum, qui fit in demonstratione circulari a principio ad conclusionem, et iterum a conclusionem ad principium.) Huiusmodi ergo reflexio quantum ad vim argumentandi, sive fiat per multa, sive per pauca, non differt. Nec differt de paucis, aut de duobus. Eadem enim virtus arguendi est, si quis sic procedat: si est a, est b, et si est b, est c, et si est c, est d; et iterum reflectat dicens: si est d, est c, et si est c, est b, et si est b, est a; sicut si statim a principio reflecteret dicens: si est a, est b, et si est b, est a. Dicit autem per duos terminos, cum supra dixerit, tribus terminis positis, quia in deductione, quam faciet, utetur tertio termino, qui sit idem cum primo.	He proves the truth of the first consequence in the following way: It is obvious, first of all, that with a circular demonstration the same thing is proved by a same thing, as has been stated above, i.e., if only three terms are employed; although it makes no difference whether the reflexion be made with fewer terms or more. (By reflexion he means the process whereby one goes from principle to conclusion in a demonstration, and then from conclusion to principle). In such a reflexion it makes no difference, so far as the force of the argument is concerned, whether it involves several or fewer terms or even two. For an argument has the same force if one proceeds thus: “If it is A, it is B, and if it is B, it is C, and if it is C, it is D,” and then by reflecting continues, “If it is D, it is C, and if it is C, it is B, and if it is B, it is A”; or if he proceeds by reflecting at the very start, saying: “If it is A, it is B, and if it is B, it is A.” (Although he spoke above of three terms, he restricted himself to two terms in this example, because in the deduction he is about to make he will use a third term, which is the same as the first).
Deinde, cum dicit: cum enim etc., proponit formam argumentandi in tribus terminis, hoc modo: si sit a, est b; et si est b, est c; ergo si est a, ex necessitate est c. Deinde, cum dicit: si igitur cum sit a etc., per formam argumentandi praemissam ostendit quod in circulari demonstratione concluditur idem per idem, sumptis duobus terminis tantum. Erit enim dicere: si est a, est b; et reflectendo (quod est circulariter demonstrare): si est b, est a. Ex quibus duobus sequitur, secundum formam arguendi praemissam, si est a, est a.	Then (72b38) he gives the form of the argument in three terms, namely: “If it is A, it is B, and if it is B, it is C; therefore, if it is A, it is of necessity C.” Then (73a1) he shows by the aforesaid form of arguing that in a circular demonstration a same thing is proved by a same thing, using only two terms. For it consists in saying, “If it is A, it is B,” and then reflecting, “If it is B, it is A”—which is a circular demonstration. Now according to the above given form it follows from these two, that “if it is A, it is A.”
Quod sic patet. Sicut enim in prima deductione, quae fiebat per tres terminos, ad b sequebatur c; ita in deductione reflexa duorum terminorum, ad b sequitur a. Ponatur ergo quod idem significet a in secunda deductione reflexa, quod c significabat in prima directa, quae est per tres terminos. Igitur dicere in secunda deductione, si est b, est a, est hoc ipsum quod erat dicere in prima deductione, si est b, est c. Sed cum dicebatur in prima deductione, si est b, est c, sequebatur, si est a, est c. Ergo in deductione circulari sequitur, si est a, est a: quia c cum a idem ponitur. Et ita leve erit demonstrare omnia, ut dictum est.	That it does follow is obvious: for just as in the first deduction which involved three terms’ C followed from B, so in the reflex deduction of two terms, A followed from B. Let us suppose, then, that the A of the second deduction, i.e., the reflex, signifies the same thing that C signified in the first, i.e., in the direct deduction which was composed of three terms. Therefore, to state in the second deduction that “if it is B it is A” is to state the same thing as was stated in the first deduction, namely, that “if it is B, it is C.” But when it was stated in the first deduction that “if it is B, it is C,” it followed that “if it is A, it is C.” Therefore, in the circular deduction it follows that “if it is A, it is A,” since C is assumed to be the same as A. In this way, it will be easy to demonstrate all things, as has been said.
lib. 1 l. 8 n. 7 Tertiam rationem ponit; ibi: at vero nec hoc possibile est et cetera. Quae talis est. Ponentes omnia posse sciri per demonstrationem, quia demonstratio est circularis, necesse habent dicere quod omnia possunt demonstrari demonstratione circulari; et ita necesse habent dicere quod in demonstratione circulari ex conclusione possit concludi utraque praemissarum. Hoc autem non potest fieri nisi in his, quae ad se invicem convertuntur, idest convertibilia sunt, sicut propria. Sed non omnia sunt huiusmodi. Ergo vanum est dicere quod omnia possunt demonstrari, propter hoc quod demonstratio est circularis.	Then he presents the third argument (73a6) which is this: Those who suppose that everything can be known through demonstration on the ground that demonstration is circular, must grant that anything can be demonstrated by a circular demonstration and, as a consequence, grant that in a circular demonstration each of the premises can be concluded from the conclusion. However, the only cases in which this can be done are those in which mutual conversion is possible, i.e., in things that are convertible, as properties. But not all things are so related. Therefore, it is ridiculous to say that everything can be demonstrated on the ground that there are such things as circular demonstrations.
lib. 1 l. 8 n. 8 Quod autem oporteat in demonstratione circulari omnia esse convertibilia secundum positionem istorum, sic ostendit. Ostensum est in libro priorum quod, uno posito, non sequitur ex necessitate aliud; nec posito uno termino, nec posita una propositione tantum. Nam omnis syllogismus est ex tribus terminis et duabus propositionibus ad minus. Oportet ergo accipere tres terminos convertibiles in demonstratione circulari, scilicet a, b, c; ita quod a insit omni b et omni c, et haec scilicet b et c inhaereant sibi invicem, ita quod omne b sit c, et omne c sit b, et iterum haec insint ipsi a, ita quod omne a sit b, et omne a sit c. Et ita se habentibus terminis, contingit monstrare in prima figura ex alterutris, idest circulariter, omnia quaesita, idest conclusionem ex duabus praemissis et utramlibet praemissarum ex conclusione et altera conversa, sicut ostensum est in his, quae sunt de syllogismo, idest in libro priorum, in quo agitur de syllogismo simpliciter.	Now the reason is obvious why in a circular demonstration all the propositions must be convertible. For it has been shown in the book of Prior Analytics that if one thing is laid down, another does not follow of necessity, whether the thing laid down be one term or one proposition. For every syllogism must start with three terms and two propositions as a minimum. Therefore, in a circular demonstration three terms which are convertible must be taken, namely, A, B, C, such that A is in every B and in every C, and these, namely, B and C, must inhere in each other, so that every B is C and every C is B, and also inhere in A so that every A is B and every A is C. And so, the terms being thus related, it is possible, when using the first figure, to derive any one from any two circularly, i.e., the conclusion from two premises and each premise from the conclusion and the remaining premise, as we pointed out in the Prior Analytics, where we treated the syllogism formally.
Quod sic patet. Sumantur tres termini convertibiles, scilicet risibile, animal rationale mortale, et homo, et syllogizetur sic: omne animal rationale mortale est risibile; omnis homo est animal rationale mortale; ergo omnis homo est risibilis. Et ex hac conclusione potest iterum concludi tam maior quam minor. Maior sic: omnis homo est risibilis; omne animal rationale mortale est homo; ergo omne animal rationale mortale est risibile. Minor sic: omne risibile est animal rationale mortale; omnis homo est risibilis; ergo omnis homo est animal rationale mortale.	The way it is done is this: take the three convertible terms, “risible,” “rational mortal animal” and “man,” and form the syllogism: Every rational mortal animal is risible; Every man is a rational mortal animal: Therefore, every man is risible. Then from the conclusion it is possible to conclude both the major and the minor; the major thus: Every man is risible; But every rational mortal animal is a man: Therefore, every rational mortal animal is risible and the minor thus: Every risible is a rational mortal animal; But every man is risible: Therefore, every man is a rational mortal animal.
lib. 1 l. 8 n. 9 Ostensum est autem in libro priorum quod in aliis figuris, scilicet in secunda et tertia, aut non fit circularis syllogismus, scilicet per quem ex conclusione possit syllogizari utraque praemissarum: aut si fiat, non erit ex acceptis, sed ex aliis propositionibus, quae non sumuntur in primo syllogismo.	However, it has also been proved in the Prior Analytics that in figures other than the first, namely, in the second and third, one cannot form a circular syllogism, i.e., one through which each of the premises can be syllogized from the conclusion; or if one is formed, it is done not by using the premises already used but by using propositions other than those which appear in the first syllogism.
Quod sic patet. In secunda figura non est conclusio nisi negativa: oportet autem alteram praemissarum esse negativam, et alteram affirmativam. Non enim ex duabus negativis potest aliquid concludi, nec ex duabus affirmativis potest concludi negativa. Non est ergo possibile quod ex conclusione et praemissa negativa concludatur affirmativa. Si ergo debet affirmativa probari, oportet quod per alias propositiones probetur, quae non sunt sumptae. Similiter in tertia figura non est conclusio nisi particularis. Oportet autem alteram praemissarum ad minus esse universalem. Si autem in praemissis sit aliqua particularis, non potest concludi universalis. Unde non potest esse quod in tertia figura ex conclusione syllogizetur utralibet praemissarum.	That this is so is obvious. For the second figure always yields a negative conclusion. Consequently, one premise must be affirmative and the other negative. However, it is true that if both are negative, nothing can be concluded; and if both are affirmative, a negative conclusion cannot follow. Therefore, it is not possible to use the negative conclusion and the negative premise to obtain the affirmative premise as a conclusion. Hence, if this affirmative is to be proved, it must be proved through propositions other than the ones originally used. Again, in the third figure the only conclusion ever obtained is particular. However, at least one premise must be universal; furthermore, if either premise is particular, a universal cannot be concluded. Hence it cannot occur that in the third figure each of the premises can be syllogized from the conclusion and the remaining premise.
Et eadem ratione apparet quod nec in prima figura talis circularis syllogismus potest fieri, per quem utraque praemissarum concludatur, nisi in primo modo, in quo solo concluditur universalis affirmativa. Nec etiam in hoc modo potest fieri talis syllogismus circularis, per quem utraque praemissarum concludatur, nisi sumantur tres termini aequales, idest convertibiles. Quod ex hoc patet. Oportet enim ex conclusione, et altera praemissarum conversa, concludere reliquam: sicut dictum est. Non autem potest utraque praemissarum converti, cum utraque sit universalis affirmativa, nisi in terminis convertibilibus.	For the same reasons it is obvious that such a circular syllogism (through which each premise could be concluded) cannot be formed in the first figure except in the first mode, which is the only one that concludes to a universal affirmative. Furthermore, even in this mode the only case in which a circular syllogism could be formed such that each of the premises could be concluded, is when the three terms employed are equal, i.e., convertible. The proof is this: The premise must be concluded from the conclusion and the converse of the other premise, as has been stated. But such a conversion of each premise is impossible (for each is universal), except when the terms happen to be equal.

Lecture 9 How something is said to be predicated of all

Latin	English
	English translation by Fabian R. Larcher, O.P.
	Lecture 9 HOW SOMETHING IS SAID TO BE PREDICATED OF ALL
lib. 1 l. 9 n. 1 Postquam philosophus ostendit quid sit syllogismus demonstrativus, in parte ista incipit ostendere ex quibus et qualibus sit. Et circa hoc tria facit; primo, continuat se ad praecedentia; secundo, interponit quaedam quae sunt necessaria ad praecognoscendum; ibi: primum autem determinabimus etc.; tertio, determinat propositum, scilicet ex quibus sit syllogismus demonstrativus; ibi: si igitur est demonstrativa et cetera.	After showing what a demonstrative syllogism is, the Philosopher in this section begins to show the nature and characteristics of the things that comprise a demonstration. Concerning this he does three things. First, he connects this with what has already been established. Secondly, he explains certain matters that must be understood first (73a25). Thirdly, he establishes what he had in mind, namely, to show what and of what sort are the things that constitute a syllogism (74b5) [L. 12].
lib. 1 l. 9 n. 2 Dicit ergo primo, quod quia dictum est supra, quod impossibile est aliter se habere in definitione eius quod est scire, necessarium erit id quod scitur secundum demonstrationem. Quid autem sit quod est secundum demonstrationem scire exponit, dicens quod demonstrativa scientia est quam habemus in habendo demonstrationem, idest quam ex demonstratione acquirimus. Et sic habetur quod demonstrationis conclusio sit necessaria.	He says therefore first (73a21), that since the definition of scientific knowledge given above spoke of that which cannot be otherwise, that which is scientifically known through demonstration will be necessary. Then he explains what it is to know something in a scientific way through demonstration, saying that demonstrative science is “what we possess in having a demonstration,” i.e., what we acquire through demonstration. Consequently, it follows that the conclusion of a demonstration is necessary.
Quamvis autem necessarium possit syllogizari ex contingentibus, non tamen de necessario potest haberi scientia per medium contingens, ut infra probabitur. Et quia conclusio demonstrationis non solum est necessaria, sed etiam per demonstrationem scita, ut dictum est, sequitur quod demonstrativus syllogismus sit ex necessariis. Et ideo accipiendum est ex quibus necessariis et qualibus sint demonstrationes.	Now although the necessary could be syllogized from the contingent, it is not possible through a contingent middle to obtain scientific knowledge of the necessary, as will be proved later. Furthermore, because the conclusion. of a demonstration is not only necessary, but, as has been said, is known through demonstration, it follows that a demonstrative syllogism proceeds from necessary things. Consequently, we must establish from what and from what sort of necessary things a demonstration proceeds.
lib. 1 l. 9 n. 3 Deinde cum dicit: primum autem etc., interponit ea, quae sunt praeintelligenda ad cognoscendum de his de quibus tractaturus est. Et circa hoc duo facit. Primo, dicit de quo est intentio, dicens quod antequam determinetur in speciali ex quibus et qualibus sit demonstratio, primo determinandum est quid intelligatur cum dicimus de omni, et per se, et universale. Cognoscere enim ista est necessarium ad sciendum ex quibus sit demonstratio. Hoc namque oportet observari in demonstrationibus. Oportet enim in propositionibus demonstrationis aliquid universaliter praedicari, quod significat dici de omni, et per se, et etiam primo, quod significat universale. Haec autem tria se habent ex additione ad invicem. Nam omne quod per se praedicatur, etiam universaliter praedicatur; sed non e converso. Similiter omne quod primo praedicatur, praedicatur per se, sed non convertitur. Unde etiam apparet ratio ordinis istorum.	Then (73a25) he interjects certain things that must be understood as preliminaries to the matters to be discussed. Apropos of these he does two things: First, he states his intention (73a25), saying that before determining specifically the nature and characteristics of the things that form a demonstrative syllogism, we must indicate what is meant when we say, “of all,” and “per se,” i.e., in virtue of itself, and “commensurately universal.” For if we are to understand the nature of the things that form a demonstration, we must know what these terms mean, because they describe things that must be observed in demonstrations. For in the propositions of a demonstration it is required that something be predicated universally~ which he signifies by the term “said of all”—and “per se,” i.e., in virtue of itself, and “first”—which he signifies by the words, “commensurately universal.” But these three things are related by adding something to the previous one. For whatever is predicated per se is predicated universally [i.e., of all], but not vice versa. Again, whatever is predicated first is predicated per se, but not vice versa. This, therefore, shows why they are arranged as they are.
Differentia etiam et numerus istorum trium apparet ex hoc, quod aliquid praedicari dicitur de omni sive universaliter per comparationem ad ea, quae continentur sub subiecto. Tunc enim dicitur aliquid de omni, ut habetur in libro priorum, quando nihil est sumere sub subiecto, de quo praedicatum non dicatur. Per se autem dicitur aliquid praedicari, per comparationem ad ipsum subiectum: quia ponitur in eius definitione, vel e converso, ut infra patebit. Primo vero dicitur aliquid praedicari de altero per comparationem ad ea, quae sunt priora subiecto et continentia ipsum. Nam habere tres angulos etc., non praedicatur primo de isoscele: quia prius praedicatur de priori, scilicet de triangulo.	But why there are three and wherein they differ are explained by the fact that something is said to be predicated “of all” or universally in relation to things contained under the subject. For, as it is stated in Prior Analytics, something is said “of all,” when there is nothing under the extension of ,the subject that does not receive the given predicate. But it is in relation to the subject that something is said to be predicated per se, because the subject is mentioned when this predicate is defined, or vice versa, as will be explained below. Finally, something is said to be predicated of another thing “first” in relation to items that are prior to the, subject and embrace or include it, as the more universal includes the less. Thus to have three angles equal to two right angles is not predicated “first” of isosceles, because it is previously predicated of something prior to isosceles, namely, of triangle.
lib. 1 l. 9 n. 4 Secundo, ibi: de omni quidem etc., determinat propositum. Et dividitur in tres partes. Primo, ostendit quid sit dici de omni; secundo, quid sit dici per se; ibi: per se autem etc.; tertio, quid sit universale; ibi: universale autem dico et cetera. Circa primum duo facit. Primo, ostendit quid sit dici de omni. Ad quod sciendum est quod dici de omni, prout hic sumitur, addit supra dici de omni, prout sumitur in libro priorum. Nam in libro priorum accipitur dici de omni communiter, prout utitur eo et dialecticus et demonstrator. Et ideo non plus ponitur in definitione eius, quam quod praedicatum insit cuilibet eorum quae continentur sub subiecto. Hoc autem contingit vel ut nunc, et sic utitur quandoque dici de omni dialecticus; vel simpliciter et secundum omne tempus, et sic solum utitur eo demonstrator.	Secondly (73a28), he establishes his proposition. And his treatment is divided into three parts. First, he shows what is meant by “said of all.” Secondly, what is meant by “said per se,” i.e., in virtue of itself (73a34) [L. 10]. Thirdly, what is meant by “commensurately universal” (73b27) [L. II]. Concerning the first he does two things. First (73a28), he states what it is to be “said of all.” And it should be noted that the phrase, “said of all,” is taken here in a sense somewhat different from the sense it has in Prior Analytics, where it is taken in a very general sense so as to accommodate both the dialectician and the demonstrator. Therefore, no more is mentioned in its definition than that the predicate be found in each of the things included under its subject. But that might be verified only at a given moment—which is the sense in which the dialectician sometimes uses it; or it might be verified absolutely and at all times—which is the sense to which the demonstrator must always limit himself.
lib. 1 l. 9 n. 5 Et ideo in definitione dici de omni, duo ponit: quorum unum est, ut nihil sit sumere sub subiecto cui praedicatum non insit. Et hoc significat cum dicit: non in quodam quidem sic, in quodam autem non. Aliud est, quod non sit accipere aliquod tempus, in quo praedicatum subiecto non conveniat. Et hoc designat cum dicit: neque aliquando sic, aliquando non; et ponit exemplum. Sicut de omni homine praedicatur animal; et de quocunque verum est dicere quod sit homo, verum est dicere quod sit animal, et quandocunque est homo, est animal. Et similiter se habet de linea et de puncto: nam punctum est in linea qualibet et semper.	Accordingly, two things are mentioned in the definition of “said of all”: one is that there is nothing within the extension of the subject that the predicate does not apply to. And he indicates this when he says, “not of one to the exclusion of others”; the other is that there is no time in which the predicate does not belong. And this he indicates when he says, “not at this or that time only.” And he gives the example of “man” and “animal,” saying that “animal” is predicated of every man; and of anything of which it is true to say that it is a man, it is true to say that it is an animal, and whenever it is a man, it is an animal. The same is true between line and point: for a point is in every line and always in every line.
lib. 1 l. 9 n. 6 Secundo; ibi: signum autem etc., manifestat positam definitionem per signum ab instantiis sumptum. Non enim fertur instantia contra universalem propositionem, nisi quia deficit aliquid eorum, quae per eam significantur. Cum autem interrogamur, an aliquid praedicetur de omni in demonstrativis, dupliciter ferimus instantias; vel quia in quodam eorum quae continentur sub subiecto non est verum; vel quia aliquando non est verum. Unde manifestum est quod dici de omni utrunque praedictorum significat.	Secondly (73a32), he explains this definition, using as evidence the techniques employed in rebuttals. For a universal proposition is not rebutted unless one or other of things it states is not verified. For when we are asked whether something is said “of all” in a demonstration, we can say, “No,” for two reasons, i.e., either because it is not true of each instance of the subject, or because now and then it is not true. Hence it is clear that “being said of all” signifies each of these.

Lecture 10 How something is said to be predicated per se of a thing

Latin	English
	Lecture 10 (73a34-b26) HOW SOMETHING IS SAID TO BE PREDICATED PER SE OF A THING
lib. 1 l. 10 n. 1 Postquam determinavit philosophus de dici de omni, hic determinat de per se. Et circa hoc duo facit: primo, ostendit quot modis dicitur aliquid per se; secundo, ostendit qualiter his modis demonstrator utatur; ibi: quae ergo dicuntur et cetera.	After determining about “said of all,” the Philosopher now determines about “said per se” [i.e., said in virtue of itself] and does three things. First, he shows the number of ways something is said per se. Secondly, how the demonstrator makes use of these ways (73b16). Thirdly, he summarizes (73b25).
lib. 1 l. 10 n. 2 Circa primum sciendum est quod haec praepositio per designat habitudinem causae; designat etiam interdum et situm, sicut cum dicitur aliquis esse per se, quando est solitarius. Causae autem habitudinem designat, aliquando quidem formalis; sicut cum dicitur quod corpus vivit per animam. Quandoque autem habitudinem causae materialis; sicut cum dicitur quod corpus est coloratum per superficiem: quia scilicet proprium subiectum coloris est superficies. Designat etiam habitudinem causae extrinsecae et praecipue efficientis; sicut cum dicitur quod aqua calescit per ignem. Sicut autem haec praepositio per designat habitudinem causae, quando aliquid extrinsecum est causa eius, quod attribuitur subiecto; ita quando subiectum vel aliquid eius est causa eius, quod attribuitur ei, et hoc significat per se.	In regard to the first it should be noted that this preposition per [“in virtue of” or “by”] denotes a causal relationship, although sometimes it also signifies a state, as when someone is said to be per se, i.e., by himself, when he is alone. But when it designates a relationship to a cause, sometimes the cause is formal, as when it is stated that the body lives in virtue of the soul; sometimes the relationship is to a material cause, as when it is stated that a body is colored in virtue of its surface, i.e., because the surface is the subject of color; again, it might even designate a relationship to an extrinsic cause, particularly an efficient cause, as when it is said that water is made hot in virtue of fire. But just as this preposition per designates a relationship to a cause, when something extrinsic is the cause of that which is attributed to the subject, so also when the subject or something pertaining to the subject is the cause of that which is attributed to the subject. This latter is what per se, i.e., in virtue of itself, signifies.
lib. 1 l. 10 n. 3 Primus ergo modus dicendi per se est, quando id, quod attribuitur alicui, pertinet ad formam eius. Et quia definitio significat formam et essentiam rei, primus modus eius quod est per se est, quando praedicatur de aliquo definitio vel aliquid in definitione positum (et hoc est quod dicit quod per se sunt quaecunque insunt in eo, quod quid est, idest in definitione indicante quid est), sive ponatur in recto sive in obliquo. Sicut in definitione trianguli ponitur linea; unde linea per se inest triangulo: et similiter in definitione lineae ponitur punctum; unde punctum per se inest lineae. Rationem autem quare ista ponantur in definitione subiungit dicens: substantia, idest essentia, quam significat definitio ipsorum, idest trianguli et lineae, est ex his, idest ex linea et punctis. Quod non est intelligendum quod linea ex punctis componatur, sed quod punctum sit de ratione lineae, sicut linea de ratione trianguli. Et hoc dicit ad excludendum ea, quae sunt partes materiae et non speciei, quae non ponuntur in definitione, sicut semicirculus non ponitur in definitione circuli, nec digitus in definitione hominis, ut dicitur in VII metaphysicae. Et subiungit quod quaecumque universaliter insunt in ratione dicente quid est, per se attribuuntur alicui.	Therefore, the first way of saying something per se (73a34) is when that which is attributed to a subject pertains to its form. And because the form and essence of a thing are signified by its definition, the first mode of that which is per se is when the definition itself or something expressed in the definition is predicated of the thing defined. This is what he means when he says, “Essential attributes are such as belong to their subject as elements in its essential nature,” i.e., included in the definition which indicates what it is, whether those elements are stated in the nominative case or in one of the oblique cases. Thus, “line” is stated in the definition of triangle. Hence “line” is in triangle per se. Again, in the definition of line, “point” is mentioned; hence “point” is per se in line. And the reason why they are mentioned in the definition is stated when he says, “for the very being or substance” [i.e., the essence, which the definition signifies] “of triangle and line is composed of these elements,” namely, of lines and points. However, this does not mean that a line is formed out of points, but that “point” is involved in the very notion of line, just as “line” is involved in the very notion of triangle. And he asserts this in order to exclude things which are part of a thing’s matter and not of its species: thus, “semicircle” is not mentioned in the definition of circle, or “finger” in the definition of man, as it is stated in Metaphysics VII. He states further that all those items which are found universally in the definition expressing what a thing is are attributed to it per se.
lib. 1 l. 10 n. 4 Secundus modus dicendi per se est, quando haec praepositio per designat habitudinem causae materialis, prout scilicet id, cui aliquid attribuitur, est propria materia et proprium subiectum ipsius. Oportet autem quod proprium subiectum ponatur in definitione accidentis: quandoque quidem in obliquo, sicut cum accidens in abstracto definitur, ut cum dicimus, quod simitas est curvitas nasi; quandoque vero in recto, ut cum accidens definitur in concreto, ut cum dicimus quod simus est nasus curvus. Cuius quidem ratio est, quia cum esse accidentis dependeat a subiecto, oportet etiam quod definitio eius significans esse ipsius contineat in se subiectum. Unde secundus modus dicendi per se est, quando subiectum ponitur in definitione praedicati, quod est proprium accidens eius.	The second mode of saying per se is when this preposition per implies a relationship of material cause, in the sense that that to which something, is attributed is its proper matter and subject. For it is required, when defining an accident, to mention its proper subject in one of the oblique, cases: thus when an accident is defined abstractly, we say that “aquilinity” is a curvature of a nose,” but when it is defined concretely, the subject is put in the nominative case, so that we say that “the aquiline is a curved nose.” Now the reason for this is that since the being of an accident depends on its subject, its definition—which signifies its being—must mention that subject. Hence it is the second mode of saying per se, when the subject is mentioned in the definition of a predicate which is a proper accident of the subject.
Et hoc est quod dicit, et per se dicuntur quibuscunque eorum, idest de numero eorum, quae insunt ipsis, idest subiectis accidentium, ipsa subiecta insunt in ratione demonstrante quid est ipsum accidens, idest in definitione accidentis. Sicut rectum et circulare insunt lineae per se: nam linea ponitur in definitione eorum. Et eadem ratione par et impar per se insunt numero, quia numerus in eorum definitione ponitur: nam par est numerus medium habens. Et similiter primum et compositum per se praedicantur de numero, et numerus in definitione eorum ponitur. Est enim primum in numeris, numerus qui nullo alio numero mensuratur, sed sola unitate, ut septenarius. Compositus autem numerus est, quem etiam alius numerus mensurat, sicut novenarius. Et similiter isopleuros, idest aequilaterum, et scalenon, idest trium inaequalium laterum et altera parte longius, per se insunt triangulo, et triangulus ponitur in definitione eorum. Et ideo subiungit quod, subiecta quae insunt omnibus praemissis accidentibus in ratione dicente quid est, idest in definitione, sicut alicui praedictorum accidentium inest linea, alicui vero numerus, et similiter in aliis, unicuique, inquam, ipsorum subiectorum, per se inesse dico suum accidens.	And this is what he means when he states (73a38), “essential attributes are those such that while they belong to certain subjects,” i.e., to subjects of accidents, “the subjects to which they belong are contained in the attribute’s own defining formula,” i.e., in the expression which describes what the accident is, i.e., in the definition of the accident. “Thus straight and curved belong to line per se.” For “line” is mentioned in their definition. For the same reason “odd” and “even” belong per se to number, because “number” is mentioned in their definition. Again, prime and compound are predicated per se of number, and “number” is mentioned in their definition. (For a prime number, for example, seven, is one which is exactly divisible by no other number but “1”; but a compound number, for example, nine, is one which is exactly divisible by some number greater than “1.” Again, “isoplural,” i.e., equilateral, and scalene, i.e., having three unequal sides, belong per se to triangle, and “triangle” is mentioned in their definition. Accordingly, he adds that their respective subjects belong to each of the aforesaid accidents and are mentioned in the expression which states what each is, i.e., in the definition: thus “line” belongs to some of them, and “number” to others.
lib. 1 l. 10 n. 5 Quae vero praedicata neutraliter insunt, idest neque ita quod ponantur in definitione subiectorum, neque subiecta in definitione eorum, sunt accidentia, idest per accidens praedicantur, sicut musicum et album praedicantur de animali per accidens.	In each of these subjects that have been mentioned, I say that its accident is in it per se. But those predicates which are neutral, i.e., of such a nature as not to be mentioned in the definition of their subjects, nor the subjects in their definition, are accidents, i.e., are predicated per accidens: for example, “musical” and “white” are predicated per accidens of animal.
lib. 1 l. 10 n. 6 Deinde cum dicit: amplius quod non etc., ponit alium modum eius, quod est per se, prout per se significat aliquid solitarium, sicut dicitur quod per se est aliquod particulare, quod est in genere substantiae, quod non praedicatur de aliquo subiecto. Et huius ratio est, quia cum dico, ambulans vel album, non significo ambulans vel album, quasi aliquid per se solitarium existens, cum intelligatur aliquid aliud esse quod sit ambulans vel album. Sed in his, quae significant hoc aliquid, scilicet in primis substantiis, hoc non contingit. Cum enim dicitur Socrates vel Plato, non intelligitur quod sit aliquid alterum, quam id quod vere ipsa sunt, quod scilicet sit subiectum eorum. Sic igitur hoc modo quae non praedicantur de subiecto sunt per se, quae vero dicuntur de subiecto, scilicet sicut in subiecto existentia, accidentia sunt.	Then (73b5) he sets down another mode of that which is per se, i.e., the sense in which it signifies something in isolation. Thus something which is a singular in the genus of substance and which is not predicated of any subject is said to be per se. The reason for this is that when I say, “walking” or “white,” I do not signify either of them as something isolated or apart, since something else which is walking or white is understood. But this is not the case with terms which signify a “this something,” i.e., with terms that signify first substance. For when I say, “Socrates” or “Plato,” it is not to be supposed that there is something else, over and above what they really are, which would be their subject. Therefore, things which are thus not predicated of any subject are per se, but things which are predicated of a subject, as being in the subject, are accidents. However, not all things predicated of a subject, as universals of their inferiors, are accidents.
Nam quae dicuntur de subiecto, sicut universalia de inferioribus, non semper accidentia sunt. Sciendum est autem quod iste modus non est modus praedicandi, sed modus existendi. Unde etiam in principio non dixit, per se dicuntur, sed, per se sunt.	It should be noted, however, that this mode is not a mode of predicating, but a mode of existing; hence at the very start he said that they exist per se and not that they are said per se.
lib. 1 l. 10 n. 7 Deinde cum dicit: item alio modo etc., ponit quartum modum, secundum quod haec praepositio per designat habitudinem causae efficientis vel cuiuscunque alterius. Et ideo dicit quod quidquid inest unicuique propter seipsum, per se dicitur de eo; quod vero non propter seipsum inest alicui, per accidens dicitur, sicut cum dico: hoc ambulante coruscat. Non enim propter id quod ambulat, coruscavit; sed hoc dicitur secundum accidens. Si vero quod praedicatur insit subiecto propter seipsum, per se inest, ut si dicamus quod interfectum interiit: manifestum est enim quod propter id quod illud interfectum est, interiit, et non est accidens quod interfectum interierit.	Then (73b10) he gives the fourth mode, according to which the preposition per designates a relationship of efficient cause or of any other. Consequently, he says that whatever is attributed to a thing because of itself, is said of it per se; but whatever is not so attributed is said per accidens, as when I say, “While he was walking, it lightened.” For it is not the fact that he walks that causes lightning, but this is said by coincidence. Butt if the predicate is in the subject because of itself, it is per se, as when we say, “Slaughtered, it died.” For it is obvious that because something was slaughtered, it died, and it is not a mere coincidence that something slaughtered should die.
lib. 1 l. 10 n. 8 Deinde cum dicit: quae ergo dicuntur etc., ostendit qualiter utatur praedictis modis demonstrator. Ubi notandum est quod cum scientia proprie sit conclusionum, intellectus autem principiorum, proprie scibilia dicuntur conclusiones demonstrationis, in quibus passiones praedicantur de propriis subiectis. Propria autem subiecta non solum ponuntur in definitione accidentium, sed etiam sunt causae eorum. Unde conclusiones demonstrationum includunt duplicem modum dicendi per se, scilicet secundum et quartum.	Then (73b16) he shows how the demonstrator uses the aforesaid modes. But first it should be noted that, since science bears on conclusions, and understanding [intuition] bears on principles, the scientifically knowable are, properly speaking, the conclusions of a demonstration wherein proper attributes are predicated of their appropriate subjects. Now the appropriate subjects are not only placed in the definition of attributes, but they are also their causes. Hence the conclusions of demonstrations involve two modes of predicating per se, namely, the second and the fourth.
Et hoc est quod dicit quod illa quae praedicantur in simpliciter scibilibus, hoc est, in conclusionibus demonstrationum, sic sunt per se, sicut inesse praedicantibus, idest sicut quando subiecta insunt in definitione accidentium, quae de eis praedicantur, aut inesse propter ipsa, idest quando praedicata insunt subiecto propter ipsum subiectum, quod est causa praedicati.	And this is what he means when he says that the predications “in the scientifically knowable in the strict sense,” i.e., in the conclusions of demonstrations are per se in the sense of something contained in the predicates, i.e., in the way that subjects are contained in the definition of accidents which are predicated of the former; or are present on account of them, i.e., in the way that predicates are in a subject by reason of the subject itself, which is the cause of the predicate.
Et consequenter ostendit quod huiusmodi scibilia sunt necessaria: quia non contingit quin proprium accidens praedicetur de subiecto. Sed hoc est duobus modis. Quandoque quidem simpliciter, sicut cum unum accidens convertitur cum subiecto, ut habere tres cum triangulo, et risibile cum homine. Quandoque autem duo opposita sub disiunctione accepta ex necessitate subiecto insunt, ut lineae aut rectum aut obliquum, et numero par aut impar. Cuius rationem ostendit, quia contrarium, privatio et contradictio sunt in eodem genere. Nam privatio nihil aliud est, quam negatio in subiecto determinato. Quandoque etiam contrarium aequiparatur negationi in aliquo genere, sicut in numeris idem est impar quod non par secundum consequentiam. Sicut ergo necesse est affirmare vel negare, ita necesse est alterum eorum, quae per se insunt, proprio inesse subiecto. Deinde epilogat dicens: de omni quidem et cetera. Quod est planum.	Then he shows that such scientifically knowable things are necessary, because it is impossible for a proper accident not to be predicated of its subject. But this can occur in two ways: sometimes it is absolute, as when the accident is convertible with its subject, as “having three angles equal to two right angles” is convertible with triangle, and “risible” with man. At other times, two opposites stated disjunctively are of necessity in the subject, as “straight or oblique” in line, and.”odd or even” in number. He shows that the reason for this is the fact that contrariety, privation and contradiction are in the same genus. For privation is nothing more, than a negation in a determinate subject. Again, a contrary is equivalent to a negation in some genus, as in the genus of numbers, odd is the same as “not even” by way of consequence. Therefore, just as it is necessary, either to affirm or deny, so it is necessary that one of two things that belong per se, be in its proper subject. Then (73b25) he summarizes, and the text is clear.

Lecture 11 How something is said to be predicated as commensurately universal

Latin	English
	Lecture 11 (73b27-74a3) HOW SOMETHING IS SAID TO BE PREDICATED AS COMMENSURATELY UNIVERSAL
lib. 1 l. 11 n. 1 Postquam philosophus determinavit de dici de omni et per se, hic determinat de universali. Et dividitur in duas partes: in prima, ostendit quid sit universale; in secunda, ostendit quomodo in acceptione universalis contingit errare; ibi: oportet autem non latere et cetera. Circa primum duo facit: primo, ostendit quid sit universale; secundo, ostendit quomodo demonstrator universali utatur; ibi: demonstratio autem per se et cetera. Circa primum duo facit: primo, ostendit quod universale continet in se et dici de omni et per se; secundo, ostendit quid supra ea addat; ibi: universale autem et cetera.	After determining about “said of all” and “said per se,” the Philosopher here determines concerning the “universal.” This treatment falls into two parts. In the first he shows what the universal is. Secondly, how error occurs in our understanding of it (74a4) [L. 12]. Concerning the first he does two things. First, he shows what the universal is. Secondly, how the demonstrator uses the universal (74a1). Concerning the first he does two things. First, he shows that the universal contains within itself the attributes of “being said of all” and of “being said per se.” Secondly, he shows what the universal adds to them (73b33).
lib. 1 l. 11 n. 2 Ad evidentiam autem eorum, quae hic dicuntur, sciendum est quod universale non hoc modo hic accipitur, prout omne quod praedicatur de pluribus universale dicitur, secundum quod Porphyrius determinat de quinque universalibus; sed dicitur hic universale secundum quandam adaptationem vel adaequationem praedicati ad subiectum, cum scilicet neque praedicatum invenitur extra subiectum, neque subiectum sine praedicato.	To understand what is being said here it should be noted that “universal” is not to be taken here in the sense that anything predicated of several is a universal, as when Porphyry treats of the five universals; rather “universal” is taken here according to a certain correspondence or commensurateness of the subject with the predicate, so that the predicate is not found outside the subject nor is the subject without the predicate.
lib. 1 l. 11 n. 3 His autem visis, sciendum est quod circa primum tria facit. Primo dicit quod universale, scilicet praedicatum, est quod et de omni est, idest universaliter praedicatur de subiecto, et etiam per se, scilicet inest ei, idest convenit subiecto secundum quod ipsum subiectum est. Multa enim universaliter de aliquibus praedicantur, quae non conveniunt eis per se, et secundum quod ipsa. Sicut omnis lapis coloratus est; non tamen secundum quod lapis, sed secundum quod est superficiem habens.	With this in mind, it should be noted that he does three things with respect to the first point. First (73b27), he says that the universal, namely, the predicate, is both verified of all, i.e., is predicated universally of its subject, and is said per se, i.e., is in and belongs to the subject according to the essential nature of the subject. For many things are said universally of certain things to which they do not belong per se and as such. Thus, every stone is colored, but not precisely as stone, but as it has a surface.
lib. 1 l. 11 n. 4 Secundo; ibi: manifestum igitur etc., infert quoddam corollarium ex dictis, dicens quod, ex quo universale est, quod per se inest; quae autem per se insunt ex necessitate insunt, ut supra ostensum est; manifestum est quod universalia praedicata, prout hic sumuntur, ex necessitate insunt rebus, de quibus praedicantur.	Secondly (73b28), he draws a corollary from this and says that since the universal is something which is per se in a thing, and since it has been shown that whatever things are in something per se are in it of necessity, it is obvious that universal predicates, as they are being taken here, are necessarily present in the things of which they are predicated.
lib. 1 l. 11 n. 5 Tertio; ibi: per se autem etc., ne aliquis crederet aliud esse quod in definitione universalis dixerat per se, et secundum quod ipsum est, ostendit quod per se et secundum quod ipsum est, idem est. Sicut lineae per se inest punctum primo modo, et rectitudo secundo modo: nam utrunque inest ei secundum quod linea est. Et e converso triangulo secundum quod triangulus est insunt duo recti, idest quod valet duos rectos, quia per se triangulo inest.	Thirdly (73b29), lest anyone suppose that “per se” and “precisely as such,” both of which were mentioned in the definition of the universal, are different, he shows that they are the same. Thus, “point” is per se in line in the first way, and “straight” in the second way. For each is in line precisely as it is a line. In like manner, “two right angles” belongs to triangle precisely as triangle, i.e., its angles are equal to two right angles, which is per se in triangle.
lib. 1 l. 11 n. 6 Deinde cum dicit: universale autem etc., ostendit quid addat universale supra dici de omni et per se. Et circa hoc duo facit. Primo, dicit quod tunc est universale praedicatum, cum non solum in quolibet est de quo praedicatur, sed et primo demonstratur inesse ei, de quo praedicatur.	Then (73b33) he shows what “universal” adds to the notions, “being said of all” and “being said per se.” In regard to this he does two things: First, he says that a predicate is “universal,” when it is not only in each thing of which it is asserted, but it is demonstrated to be first or primarily in the thing which receives that predicate.
lib. 1 l. 11 n. 7 Secundo; ibi: ut duos rectos habere etc., manifestat per exemplum, dicens quod habere tres angulos aequales duobus rectis, non inest cuilibet figurae universaliter: licet hoc de figura demonstretur, quia de triangulo demonstratur qui est figura; sed tamen non cuilibet figurae inest, nec demonstrator in sua demonstratione utitur qualibet figura. Quadrangulus enim figura quaedam est, sed non habet tres duobus rectis aequales. Isosceles autem, idest triangulus duorum aequalium laterum, habet quidem universaliter tres angulos aequales duobus rectis, sed non convenit primo isosceli, sed prius triangulo, quia isosceli convenit, in quantum est triangulus. Quod igitur primo demonstratur habere duos rectos, aut quodcunque aliud huiusmodi, huic primo inest praedicatum universale, sicut triangulo.	Secondly (73b34), he clarifies this with an example and says that “having three angles equal to two right angles” is not found in just any figure in general, although this could be demonstrated of some figure, because it is demonstrated of triangle, which is a figure; yet it is not found in any random figure, nor is just any figure used when it is demonstrated. For a rhombus is a figure, but it does not have three angles equal to two right angles. But an isosceles, i.e., a triangle with two equal sides, always has its three angles equal to two right angles. Nevertheless, isosceles is not the primary thing to which this belongs, for it belongs basically to triangle, and belongs to isosceles precisely as it is a triangle. Therefore, whatever is demonstrated basically to have its three angles equal to two right angles (or whatever else be thus demonstrated), the universal predicate is present in it primarily, as in triangle.
lib. 1 l. 11 n. 8 Deinde cum dicit: et demonstratio etc., ostendit qualiter demonstrator universali utatur, et dicit quod demonstratio est per se huius universalis: sed aliorum est quodammodo et non per se. Demonstrator enim demonstrat passionem de proprio subiecto: et si demonstret de aliquo alio, hoc non est nisi in quantum pertinet ad illud subiectum. Sicut passionem trianguli probat de figura et isoscele, in quantum quaedam figura triangulus est, et triangulus quidam isosceles est. Quod autem non primo inest isosceli habere tres, hoc non est quia non universaliter praedicetur de eo, sed quia est frequentius, idest in pluribus quam isosceles, cum hoc commune sit omni triangulo.	Then (74a1) he shows how a demonstrator uses the “universal,” saying that demonstration is concerned per se with such a universal, but with other things qualifiedly and not per se. For a demonstrator demonstrates a proper attribute of its proper subject; and if he demonstrates it of anything else, he does so only insofar as it pertains to that subject. Thus, he proves that some property of triangle belongs to a figure and to an isosceles precisely as some figure is a triangle, and as the isosceles is a triangle. But the reason why “having three” is not in isosceles primarily is not because it is not predicated of it universally, but because it is found more frequently, i.e., in more things than in isosceles, since this is common to every triangle.

Lecture 12 How error occurs in understanding the universal

Latin	English
	Lecture 12 (74a4-b4) HOW ERROR OCCURS IN TAKING THE UNIVERSAL
lib. 1 l. 12 n. 1 Postquam notificavit Aristoteles quid sit universale, hic ostendit quomodo in acceptione universalis errare contingat. Et circa hoc tria facit: primo, dicit quod aliquando circa hoc peccare contingit; secundo, assignat quot modis; ibi: oberramus etc.; tertio, dat documentum quomodo possit cognosci utrum vere acceptum sit universale; ibi: utrum autem secundum quod et cetera.	After specifying what the universal is, the Philosopher here shows how one might err in understanding the universal. In regard to this he does three things. First, he says that sometimes one might err in this matter. Secondly, he tells in how many ways (74a6). Thirdly, he gives the criterion for knowing whether the universal is being employed correctly (74a35).
lib. 1 l. 12 n. 2 Dicit ergo primo quod ad hoc, quod non accidat in demonstratione peccatum, oportet non latere quod multoties videtur demonstrari universale, non autem demonstratur.	He says therefore first (74a4), that in order to avoid mistakes in demonstrating, one should be aware of the fact that quite often something universal seems to be demonstrated, which is not being demonstrated.
lib. 1 l. 12 n. 3 Deinde cum dicit: oberramus autem etc., assignat modos quibus circa hoc errare contingit. Et circa hoc duo facit. Primo, enumerat ipsos modos, dicens quod tripliciter contingit decipi circa acceptionem universalis. Primo quidem, cum nihil aliud sit accipere sub aliquo communi cui primo competit universale, quam hoc singulare, cui inconvenienter assignatur. Sicut si sensibile, quod primo et per se inest animali, assignaretur ut universale primum homini, nullo alio animali existente. Unde notandum quod singulare hic large accipitur pro quolibet inferiori, sicut si species dicatur singulare sub genere contentum. Vel potest dici quod non est possibile invenire aliquod genus, cuius una tantum sit species. Genus enim dividitur in species per oppositas differentias; oportet autem, si unum contrariorum invenitur in natura, et reliquum inveniri, ut patet per philosophum in II de caelo et mundo; et ideo si una species invenitur, invenitur et alia. Una autem species dividitur in diversa individua per divisionem materiae. Contingit autem totam materiam alicui speciei proportionatam, sub uno individuo comprehendi, et tunc non est nisi unum individuum sub una specie. Unde et signanter de singulari mentionem facit.	Secondly (74a6), he indicates the ways in which this mistake can occur. And in regard to this he does two things. First, he enumerates these ways and says that there are three possible errors in understanding a universal. The first is likely to occur when under some common genus there is nothing else to take as the thing to which the universal initially applies than this singular, to which it is incorrectly applied. For example, if man were the only animal existing, and “sensible,” which is initially and per se in animal, were to be assigned as a primary universal to man. (It should be noted that singular is being used here in a wide sense for any inferior, in the way that a species might be called a singular contained under a genus). Or we might say that it is not possible to find a genus with only one species: for a genus is divided into species through opposing differences. But if one contrary is found in nature, so must the other, as the Philosopher explains in On the Heavens II. Therefore, if one species is found, another will be found. However, one species is divided into distinct individuals by the division of matter. But it sometimes happens that all the matter proportionate to a given species is comprehended under one individual, so that in that case there is only one individual under one species. Hence it is significant that he did say, “singular.”
lib. 1 l. 12 n. 4 Secundus modus est, quando est quidem accipere sub aliquo communi multa inferiora, sed tamen illud est commune innominatum, quod invenitur in rebus differentibus specie. Sicut si animali non esset nomen impositum, et sensibile, quod est proprium animalis, assignaretur ut universale primum his quae sub animali continentur, vel divisim vel coniunctim.	The second way is when it is possible to take several inferiors under something common which is verified in things that differ in species, but that common item has no name. For example, if “animality” had no name, and “sensibility,” which is proper to animal, were to be assigned to the inferiors of animal (either collectively or distributively) as their primary universal.
lib. 1 l. 12 n. 5 Tertius modus est, quando illud de quo demonstratur aliquid, ut universale primum, se habet ad id quod demonstratur de eo, sicut totum ad partem. Sicut si posse videre assignaretur animali ut universale primum. Non enim omne animal potest videre. Inest enim his, quae sunt in parte, idest quae particulariter et non universaliter alicui subiecto conveniunt, demonstratio, idest quod demonstrari possint, et erit quidem demonstratio de omni, non tamen respectu huius de quo demonstratur. Posse enim videre demonstratur quidem de aliquo universaliter, non tamen universaliter de animali, sicut de eo cui primo insit. Et exponit quid sit primum, secundum quod demonstratio fertur, quod est universale primum.	The third way is when that of which something is demonstrated to be its primary universal is related to what is demonstrated of it, as a whole is related to a part. For example, if the power to see were assigned as a primary universal to animal: for not every animal can see. In this case the demonstration, i.e., what should have been demonstrated, is “in those things which are in part,” i.e., in some but not all of the things included under the subject; furthermore, it will be a demonstration “of all,” but not of all that the demonstration mentions. For the power to see can indeed be demonstrated universally of something, but not of animal universally, as of that to which it belongs primarily. And he explains why he says, “primarily,” namely, because a demonstration bears on what is both universal and first.
lib. 1 l. 12 n. 6 Secundo; ibi: si igitur etc., subiungit exempla ad praedictos modos, et primo ad tertium, dicens quod, si quis demonstret de lineis rectis quod non intercidant, idest non concurrant, videbitur huiusmodi esse demonstratio, scilicet universalis primi, propter hoc quod non concurrere inest aliquibus lineis rectis. Non autem ita quod hoc fiat, nisi lineae rectae sint aequales, idest aeque distantes. Sed si lineae fuerint aequales, idest aeque distantes, tunc non concurrere convenit eis in quolibet, quia universaliter verum est quod lineae rectae aeque distantes, etiam si in infinitum protrahantur, in neutram partem concurrent.	Secondly (74a13), he gives examples of each of these ways. First, of the third way, saying that if someone were to demonstrate of two straight lines that they do not intersect, i.e., that they do not meet, it might seem that we have a demonstration of this sort, i.e., one that bears on a primary universal, on the ground that “not to meet” is true of certain straight lines, and not that this happens only because the straight lines are equal, i.e., equally distant. But if the lines should be equal, i.e., equidistant, then “not to meet” belongs to any and all of them, because it is universally true that lines which are straight and equally distant, even should they be lengthened ad infinitum, will not meet.
lib. 1 l. 12 n. 7 Secundo; ibi: et si triangulus etc., ponit exemplum ad primum modum, dicens quod si non esset alius triangulus, quam isosceles, qui est triangulus duorum aequalium laterum, quod est trianguli in quantum huiusmodi, videretur esse isoscelis secundum quod est isosceles: nec tamen hoc esset verum.	Secondly (74a17), he gives an example of the first way, saying that if there were no triangle but the isosceles which is a triangle having two equal sides, it might seem that what is true of triangle as triangle should be true of isosceles as isosceles. But this would not be so.
lib. 1 l. 12 n. 8 Tertio; ibi: et proportionale etc., exemplificat de secundo modo. Et videtur hoc ultimo ponere, quia circa hoc diutius immoratur. Et circa hoc tria facit: primo, ponit exemplum; secundo, inducit quoddam corollarium ex dictis; ibi: propter hoc nec si aliquis etc.; tertio, assignat rationem dictorum; ibi: quando igitur non novit et cetera.	Thirdly (74a18), he gives an example of the second way. Now it seems that he saved this for last because he wished to spend more time on it. Concerning it he does three things. First, he gives the example. Secondly, he draws a corollary from what he has said (74a25). Thirdly, he gives a reason for the aforesaid (74a33).
Circa primum sciendum est quod proportio est habitudo unius quantitatis ad alteram, sicut sex ad tria se habent in proportione dupla. Proportionalitas vero est collatio duarum proportionum. Quae, si sit disiuncta, habet quatuor terminos; ut hic: sicut se habent quatuor ad duo, ita sex ad tria: si vero sit coniuncta, habet tres terminos: nam uno utitur ut duobus; ut hic: sicut se habent octo ad quatuor, ita quatuor ad duo.	Concerning the first it should be noted that a ratio is a relation of one quantity to another, as 6 is related to 3 in the ratio of 2 to 1. The com. paring of one ratio to another is a proportion, which, if it is disjoint, has four terms: for example, as 4 is to 2, so 6 is to 3. But if it is joint, it has three terms, one of which is used twice: for example, as 8 is to 4, so 4 is to 2.
Patet autem quod in proportione duo termini se habent ut antecedentia; duo vero ut consequentia; ut hic: sicut se habent quatuor ad duo, ita se habent sex ad tria; sex et quatuor sunt antecedentia: tria vero et duo sunt consequentia. Permutata ergo proportio est quando antecedentia invicem conferuntur, et consequentia similiter. Ut si dicam: sicut se habent quatuor ad duo, ita se habent sex ad tria; ergo sicut se habent quatuor ad sex, ita se habent duo ad tria.	Now it is obvious that in a proportion two terms are antecedents and two are consequents. For example, in the proportion that 6 is to 3 as 4 is to 2, 6 and 4 are the antecedents, 3 and 2 the consequents. Again, a proportion is alternated by bringing the antecedents together and the consequents together. For example, when I say: As 4 is to 2, so 6 is to 3; therefore, as 4 is to 6, so 2 is to 3.
Dicit ergo quod esse proportionale commutabiliter convenit numeris, et lineis, et firmis, idest corporibus, et temporibus. Sicut autem de singulis determinatum est aliquando seorsum, de numeris quidem in arithmetica, de lineis et firmis in geometria, de temporibus in naturali philosophia vel astrologia, ita contingens est, quod de omnibus praedictis commutatim proportionari una demonstratione demonstretur. Sed ideo commutatim proportionari, de singulis horum seorsum demonstratur, quia non est nominatum illud commune, in quo omnia ista sunt unum. Etsi enim quantitas omnibus his communis sit, tamen sub se et alia, praeter haec, comprehendit, sicut orationem et quaedam, quae sunt quantitates per accidens.	What he is saying, therefore, is that alternate proportion is verified in lines, numbers, solids, (i.e., bodies), and times. But just as this is established separately for each of them, namely, for numbers in arithmetic, for lines and solids in geometry, and for times in natural philosophy or astronomy, so it is possible to prove it of all of them in a single demonstration. But the reason why a separate demonstration was employed to prove alternation for each was that the one feature they had in common was unnamed. For although quantity is a feature common to all of them, quantity includes other things besides them; for example, speech and other things that are accidentally quantitative.
Vel melius dicendum quod commutatim proportionari non convenit quantitati in quantum est quantitas, sed in quantum est comparata alteri quantitati secundum proportionalitatem quandam. Et ideo dixerat etiam in principio proportionale esse quod commutabiliter est. Omnibus autem istis, in quantum sunt proportionalia, non est nomen commune positum. Cum autem demonstratur commutatim proportionari de singulis praedictorum divisim, non demonstratur universale. Non enim commutatim proportionari inest numeris et lineis, secundum quod huiusmodi, sed secundum quoddam commune. Demonstrantes autem de lineis seorsum vel de numeris ponunt hoc, quod est commutabiliter proportionari, esse quasi quoddam universale praedicatum lineae secundum quod linea est, aut numeri secundum quod numerus.	Or, better still, it was because alternate proportion does not belong to quantity precisely as quantity, but as compared to another quantity according to a fixed ratio. That is why at the very start he spoke of alternate proportion. But there is no general name for the aforesaid things precisely as they are proportional. Furthermore, when alternate proportion is demonstrated separately for each of them, it is not a universal that is being demonstrated. For “to be alternately proportional” is not in numbers and lines according to what each of them is, but according to something common. Besides, those who demonstrate alternate proportion of lines look upon this attribute as a universal predicate of line precisely as it is a line, or of number precisely as it is number.
lib. 1 l. 12 n. 9 Deinde cum dicit: propter hoc nec si aliquis etc., inducit quoddam corollarium ex dictis, dicens quod eadem ratione, qua non demonstratur universale cum de singulis speciebus aliquid demonstratur, quod est universale praedicatum communis innominati; nec etiam demonstratur universale modo praedicto, si sit commune nomen positum. Sicut si aliquis aut eadem demonstratione aut diversa demonstret de unaquaque specie trianguli, quod habet duos rectos, seorsum scilicet de isoscele et seorsum de gradato, idest de triangulo trium laterum inaequalium, non tamen propter hoc cognovit quod triangulus tres angulos habeat aequales duobus rectis, nisi sophistico modo, idest per accidens: quia non cognovit de triangulo secundum quod est triangulus, sed secundum quod est aequilaterus, aut duorum aequalium laterum, aut trium inaequalium.	Then (74a25), he draws a corollary from the above and says that for the same reason that something universal is not being demonstrated when a common unnamed attribute is proved to be a universal predicate of each species, so too when something is demonstrated this way of a common attribute which does have a name. For example, if someone uses the same demonstration for each species of triangle and proves that each has three angles equal to two right angles, or if he uses different demonstrations, say one for isosceles and another for scalene, even then he does not on that account know that a triangle has three angles equal to two right angles except in a sophistical way, i.e., per accidens: for he does not know it of triangle as triangle, but as equilateral or as having two equal sides or as having three unequal sides.
Neque etiam demonstrans cognovit universale trianguli, idest habet cognitionem de triangulo in universali, etiamsi nullus alius triangulus esset praeter illos, de quibus cognovit. Et hoc ideo, quia non cognovit de triangulo secundum quod est triangulus, sed sub ratione specierum eius. Unde neque cognovit, per se loquendo, omnem triangulum: quia et si secundum numerum cognovit omnem triangulum (si nullus est, quem non novit), tamen secundum speciem non cognovit omnem. Tunc enim cognoscitur aliquid universaliter secundum speciem, quando cognoscitur secundum rationem speciei. Secundum numerum autem et non universaliter, quando cognoscitur secundum multitudinem contentorum sub specie. Nec est differentia quantum ad hoc si comparemus species ad individua vel genera ad species. Nam triangulus est genus aequilateri et isoscelis.	Furthermore, one who demonstrates in this way does not know the universal of triangle, i.e., he does not have a knowledge of triangle in a universal way (even if there happens to be no other triangle besides those of which he knew this: and this because he had not the knowledge of triangle as triangle but under the aspect of its various species). Again, strictly speaking, he did not know every triangle: for although he knew every triangle according to number (if there was none he did not know), yet according to species he did not know each one. For something is known universally according to species, when it is known according to the notion of the species, but according to number and not universally, when it is known according to the multifarious things contained under the species. And in this matter there is no difference whether we compare species to individuals or genera to species. For triangle is the genus of equilateral and isosceles.
lib. 1 l. 12 n. 10 Deinde cum dicit: quando igitur non novit etc., assignat rationem praedictorum, quaerens quando aliquis cognoscat universaliter et simpliciter, ex quo praedicto modo cognoscens non cognoscit universaliter. Et respondet manifestum esse quod, si eadem esset ratio trianguli in communi et uniuscuiusque specierum eius seorsum acceptae aut omnium simul acceptarum, tunc universaliter et simpliciter nosceret de triangulo, quando sciret de aliqua specie eius vel de omnibus simul. Si vero non est eadem ratio, tunc non erit idem cognoscere triangulum in communi et singulas species eius; sed est alterum. Et cognoscendo de speciebus, non cognoscitur de triangulo secundum quod est triangulus.	Then (74a33) he assigns the reason for the aforesaid, first asking when does one know universally and absolutely, if one who knew in the aforesaid way did not know universally. And he answers that obviously if the essence of triangle in general and of each of its species (each taken sepai-ately or all taken together) were the same, then one would know universally and absolutely about triangle, when he knew about any one species of it separately or about all of them together. But if the essence is not the same, then it will not be the same but something different to know triangle and to know its several species. And by knowing something of the species one does not know it of triangle precisely as triangle.
lib. 1 l. 12 n. 11 Deinde cum dicit: utrum autem etc., dat documentum quo proprie possit accipi universale, dicens quod, utrum aliquid sit trianguli secundum quod est triangulus, aut isoscelis, secundum quod est isosceles, et quando id cuius est demonstratio sit primum et universale, secundum hoc, idest secundum aliquod subiectum positum; manifestum est ex hoc quod dicam.	Then (74a35) he gives the rule on how the universal can be properly understood, saying that whether something belongs to triangle precisely as triangle or to isosceles precisely as isosceles, and when it is that what is demonstrated is first and universal in the given subject precisely as this, i.e., precisely according to the subject—all this will be clear from what I shall say.
Quandocumque enim, remoto aliquo, adhuc remanet illud quod assignatur universale, sciendum est quod non est primum universale illius. Sicut, remoto isoscele vel aeneo triangulo, remanet quod habeat tres angulos, scilicet duobus rectis aequales. Unde patet quod habere tres angulos aequales duobus rectis non est universale primum, neque isoscelis, neque aenei trianguli. Remota autem figura non remanet habere tres, nec etiam, remoto termino, qui est superius ad figuram, cum figura sit, quae termino vel terminis clauditur; sed tamen non primo convenit neque figurae, neque termino, quia non convenit eis universaliter.	For whenever some item is removed from the subject and the original universal still applies to what remains, then it is not the first universal of that subject. For example, upon the removal of isosceles or of brazen there still remains the triangle which has three angles equal to two right angles; hence the possession of three angles equal to two right angles is not a first universal of isosceles or of brazen triangle. But if the item “figure” be removed, nothing which has three angles equal to two right angles remains, nor again if you remove “bounded” which is more universal than figure, since a figure is something enclosed by a bound or bounds. Nevertheless, the attribute in question [namely, having three angles equal to two right angles] does not belong first to figure or to bounded, because it does not belong to either of them universally.
Cuius ergo erit primo? Manifestum est quod trianguli, quia secundum triangulum inest aliis, tam superioribus, quam inferioribus: ideo enim competit figurae habere tres, quia triangulus est quaedam figura; et similiter isosceli, quia triangulus est, et de triangulo habere tres universaliter demonstratur. Unde eius est universale primum.	To what then will it be first? Obviously to triangle, because it belongs to the others (both superiors and inferiors) precisely as they are triangles For it belongs to figure to have three angles equal to two right angles only because a triangle is some figure, and similarly to isosceles, only because it is a triangle, and it is of triangle that “having three...” is demonstrated. Hence, it is to triangle that it belongs as a first universal.

Lecture 13 Demonstration proceeds from necessary things

Latin	English
	Lecture 13 (74b5-75a17) DEMONSTRATION PROCEEDS FROM NECESSARY THINGS
lib. 1 l. 13 n. 1 Postquam determinavit philosophus de dici de omni, et per se, et universali quibus utimur in demonstratione, hic iam incipit ostendere ex quibus demonstratio procedit. Et dividitur in duas partes: in prima, ostendit ex quibus procedat demonstratio propter quid; in secunda, ex quibus procedat demonstratio quia; ibi: sed quia differt et propter quid et cetera. Prima in duas: in prima, ostendit qualia sint ex quibus demonstratio procedit; in secunda, docet quae sint demonstrationis principia; ibi: quid quidem igitur prima significent et cetera. Prima in tres: in prima, ostendit quod demonstratio est ex necessariis; in secunda, quod est ex his, quae sunt per se; ibi: accidentium autem etc.; in tertia, quod procedat ex principiis propriis; ibi: non ergo est ex alio genere et cetera. Circa primum duo facit: primo, ostendit quod demonstratio procedat ex necessariis; secundo, probat quaedam quae supposuerat; ibi: quod autem oporteat ex necessariis et cetera. Circa primum tria facit: primo, continuat se ad praecedentia; secundo, probat propositum; ibi: quae autem sunt per se etc.; tertio, infert ex dictis quandam conclusionem; ibi: manifestum autem ex his et cetera.	Having finished his treatment of that which is said “of all” and “per se” and “universally,” all of which we use in demonstration, the Philosopher now begins to discuss the items from which demonstration proceeds. And there are two parts. In the first he shows what demonstration of the reasoned fact [demonstration propter quid] proceeds from. In the second what demonstration of the fact [demonstration quia] proceeds from (78a22) [L. 23]. The first is divided into two parts. In the first he shows the sort of things from which a demonstration proceeds. In the second, what the principles of demonstration are (76a26) [L. 18]. The first is divided into three parts. In the first he shows that demonstration issues from necessary things. In the second that it issues from things that are per se (76a26) [L. 14]. In the third that it proceeds from proper principles (75a38) [L. 15]. Concerning the first he does two things. First, he shows that demonstration should proceed from necessary things. Secondly, he proves certain things he had presupposed (74b27). In regard to the first he does three things. First, he makes a connection with what went before. Secondly, he proves his proposition (74b6). Thirdly, he draws a conclusion from the aforesaid (74b22).
Dicit ergo primo, ex praedictis inferens, quod si est demonstrativa scientia, idest si scientia per demonstrationem acquiritur, oportet quod sit ex necessariis principiis. Cuius illationis necessitas ex hoc apparet, quia quod scitur impossibile est aliter se habere, ut habitum est in definitione eius quod est scire.	He says therefore first (74b5), as an inference from what has already been established, that if there be demonstrative science, i.e., if science is acquired through demonstration, it must issue from necessary principles. The necessity of this inference is clear, because that which is known scientifically cannot be otherwise than it is, as was pointed out in the definition of scientific knowing.
lib. 1 l. 13 n. 2 Deinde cum dicit: quae autem per se sunt etc., ostendit quod demonstratio sit ex necessariis; et primo, per rationem; secundo, per signum; ibi: signum autem est et cetera. Circa primum ponit duas rationes: quarum prima talis est. Ea, quae per se praedicantur, necessario insunt. Et hoc manifestat in duobus modis per se. In primo quidem, quia ea, quae per se praedicantur, insunt in eo quod quid est, idest in definitione subiecti. Quod autem ponitur in definitione alicuius, necessario praedicatur de eo. In secundo vero, quia quaedam sunt subiecta, quae ponuntur in quod quid est praedicantibus de ipsis, idest in definitione suorum praedicatorum. Quae quidem si sint opposita, necesse est quod alterum eorum subiecto insit; sicut par vel impar numero, ut superius ostensum est. Sed manifestum est quod ex quibusdam principiis huiusmodi, scilicet per se, fit syllogismus demonstrativus: quod probat per hoc, quod omne quod praedicatur, aut praedicatur per se aut per accidens; et ea, quae praedicantur per accidens, non sunt necessaria: ex his autem, quae sunt per accidens, non fit demonstratio, sed magis sophisticus syllogismus. Unde relinquitur quod demonstratio sit ex necessariis.	Then (74b6) he shows that demonstration issues from necessary things. First, with a reason. Secondly, with a sign (74b19). With respect to the first he gives two reasons. The first (74b6) is this: Things predicated per se are in a thing necessarily. (And he manifests this in two of the modes of predicating per se: in the first mode, because the things predicated per se are “in that which is what,” i.e., in the definition of the subject. But whatever is put in the definition of anything is predicated of it necessarily. In the second mode, because certain subjects are put in “that which is the what of things predicated of them,” i.e., in the definition of their predicates. And if these are opposites, one or other of them is necessarily in the subject: thus, either “odd” or “even” is in number, as we showed above). Now it is clear that it is from principles of this kind, namely, per se, that a demonstrative syllogism proceeds. (And this is proved by the fact that whatever is predicated, is predicated either per se or per accidens, and that things predicated per accidens are not necessary). Therefore, since it is not a demonstration but rather a sophistic syllogism that issues from things that are per accidens, it follows that demonstration proceeds from necessary things.
lib. 1 l. 13 n. 3 Sciendum autem est quod cum in demonstratione probetur passio de subiecto per medium, quod est definitio, oportet quod prima propositio, cuius praedicatum est passio et subiectum est definitio, quae continet principia passionis, sit per se in quarto modo; secunda autem, cuius subiectum est ipsum subiectum et praedicatum ipsa definitio, in primo modo. Conclusio vero, in qua praedicatur passio de subiecto, est per se in secundo modo.	Furthermore, it should be noted that, since in a demonstration a proper attribute is proved of a subject through a middle which is the definition, it is required that the first proposition (whose predicate is the proper attribute, and whose subject is the definition which contains the principles of the proper attribute) be per se in the fourth mode, and that the second proposition (whose subject is the subject itself and the predicate its definition) must be in the first mode. But the conclusion, in which the proper attribute is predicated of the subject, must be per se in the second mode.
lib. 1 l. 13 n. 4 Secundam rationem ponit; ibi: aut igitur sic etc.: quae talis est. Demonstratio circa necessarium est et demonstratum, idest demonstrationis conclusio, non potest aliter se habere. Et hoc accipiendum est tanquam principium ad ostendendum propositum, scilicet quod demonstratio ex necessariis procedat; cuius quidem principii veritas ex praemissis apparet, ut iam dictum est. Ex hoc autem principio sic argumentatur. Conclusio necessaria non potest sciri nisi ex principiis necessariis; sed demonstratio facit scire conclusionem necessariam; ergo oportet quod sit ex principiis necessariis.	Then (74b14) he sets forth the second reason and it is this: “Demonstration is concerned with the necessary and the demonstrated,” i.e., the conclusion of the demonstration cannot be other than it is. This statement is to be taken as the principle proving our proposition that demonstration proceeds from necessary things. And from what has been said, the truth of this principle is obvious. From this principle one argues thus: a necessary conclusion cannot be scientifically known save from necessary principles; but a demonstration makes a necessary conclusion scientifically known: therefore, it must proceed from necessary principles.
In quo differt demonstratio ab aliis syllogismis: sufficit enim in aliis syllogismis quod syllogizetur ex veris. Nec est aliquod aliud genus syllogismi, in quo oporteat ex necessariis procedere, sed in demonstratione tantum oportet hoc observare. Et hoc est proprium demonstrationis, scilicet ex necessariis semper procedere.	Herein lies the difference between a demonstration and other syllogisms. For in the latter it is enough if one syllogizes from true principles. Nor is there any other type of syllogism in which it is required to proceed from the necessary: in a demonstration alone must this be observed. And this is proper to a demonstration, i.e., to proceed from the necessary.
lib. 1 l. 13 n. 5 Deinde cum dicit: signum autem etc., probat idem per signum hoc modo. Contra rationem aliquam non infertur instantia, nisi per hoc quod deficit aliquid eorum, quae in ratione illa observanda sunt; sed contra eum, qui opinatur se demonstrare, ferimus instantiam quod non sit necesse ea, ex quibus procedit, esse vera: sive opinemur ea contingere aliter se habere, sive talem instantiam feramus rationis, idest disputationis causa; ergo demonstratio debet procedere ex necessariis.	Then (74b18) he proves the same thing in the following way, using a sign: Suit is brought against an argument solely on the ground that something is wanting that should have been observed in that argument. But against one who believes that he has demonstrated, we bring the charge that the things from which he proceeded are not necessarily true, whether we are convinced that they could be otherwise than stated, or whether we bring a charge of reason, i.e., for the sake of disputing. Therefore, demonstration should proceed from necessary things.
lib. 1 l. 13 n. 6 Deinde cum dicit: manifestum autem etc., infert conclusionem ex dictis: dicens quod manifestum est, ex hoc quod oportet demonstrationem ex necessariis concludere, quod stulti sunt illi, qui opinati sunt bene se principia demonstrationis accipere, si solum propositio accepta sit probabilis vel vera, ut sophistae faciunt, idest illi, qui apparent scientes et non sunt. Nam scire non est nisi per hoc quod scientia habetur, scilicet ex demonstratione; ex hoc autem quod aliquid est probabile vel improbabile non habetur quod sit primum vel non primum: sed tamen oportet illud circa quod fit demonstratio esse primum in genere aliquo et esse verum. Non tamen omne primum accipit demonstrator, sed primum proprium illi generi, circa quod demonstrat; sicut arithmeticus non accipit primum, quod est circa magnitudinem, sed circa numerum.	Then (74b22) he draws a conclusion from the foregoing, saying that “this,” namely, the fact that a demonstration must conclude from necessary things, “shows” that they are obtuse who assumed that they rightly chose the principles of demonstration, if the proposition they chose was probable or true, as the Sophists do, i.e., those who appear wise but are not. For scientific knowing consists in nothing less than having science, namely, from demonstration. But when something is probable or improbable, it is not certain whether it is first or not first, whereas that on which a demonstration bears must be first in some genus and be true. This does not mean that a demonstrator may take anything that is first: it must be first in that genus with respect to which he is demonstrating. Thus, arithmetic does not choose what is first in regard to magnitude, in regard to number.
lib. 1 l. 13 n. 7 Attendendum est autem quod sophistae non sumuntur hic sicut in libro elenchorum, qui procedunt ex his quae videntur probabilia et non sunt, aut videntur syllogizare, non tamen syllogizant. Sicut enim tales sophistae dicuntur, idest apparentes et non existentes, in quantum deficiunt a dialectica argumentatione; ita dialecticae argumentationes si appareant demonstrative probare et non probent, sophisticae sunt, in quantum videntur sua argumentatione scientes, et non sunt.	It should be noted that Sophists are not taken here as they are in the book of Sophistical Refutations, i.e., who proceed from things that seem probable but are not, or seem to syllogize but do not. For just as some are called Sophists, i.e., seeming to be wise but not really so, inasmuch as they appear by their arguments to be scientific knowers but are not, so dialectical arguments, if they seem to prove demonstratively but do not, are sophistic.
lib. 1 l. 13 n. 8 Deinde cum dicit: quod autem ex necessariis etc., ostendit quod supposuerat. Et circa hoc duo facit: primo, ostendit quod conclusio necessaria non potest sciri ex principiis non necessariis; secundo, quod licet non possit sciri necessarium ex non necessariis, tamen syllogizari potest; ibi: cum quidem igitur conclusio et cetera. Primum ostendit duabus rationibus, quarum prima talis est. Si quis non habeat rationem propter quid ostendentem, non efficitur sciens, etiam demonstratione habita: quia scire est causam rei cognoscere, ut supra dictum est. Sed ratio, quae infert conclusionem necessariam ex non necessariis principiis, non ostendit propter quid.	Then (74b27) he proves something he had presupposed. And concerning this he does two things. First, he shows that a necessary conclusion cannot be scientifically known from non-necessary principles. Secondly, that although the necessary cannot be scientifically known from the nonnecessary, it can nevertheless be syllogized (75al). He proves the first with two reasons, one of which (74b27) is this: If one does not have an argument which shows the cause why [i.e., propter quid], he does not know scientifically, even though a syllogism be had; because to know scientifically is to know the cause of a thing, as we stated above. But an argument which infers a necessary conclusion from nonnecessary principles does not show the cause why.
Quod exemplificat in terminis communibus. Ponatur enim quod haec conclusio sit necessaria: omne c est a; et demonstretur per hoc medium b, quod non sit necessarium medium, sed contingens, puta quod haec propositio sit contingens, omne b est a, vel omne c est b, aut utraque; constat quod per hoc medium contingens, quod est b, non potest sciri de conclusione necessaria, quae est, omne c est a, propter quid. Quod sic probatur. Remota causa propter quam est aliquid, oportet quod removeatur effectus; sed hoc medium cum sit contingens, contingit removeri, conclusionem autem removeri non contingit cum sit necessaria; relinquitur ergo quod non potest sciri conclusio necessaria per medium contingens.	To clarify this he gives an example, using general terms. Assume that the conclusion, “Every C is A,” is necessary, and that it is demonstrated by a middle, B, which is not a necessary middle but a contingent one, so that, for example, one of the propositions, “Every B is A,” and “Every C is B,” or both, are contingent. Now it is obvious that through this contingent middle, namely, B, one cannot obtain scientific knowledge propter quid of the necessary conclusion, “Every C is A.” For if the cause on which something depends is removed, the effect must be removed. But this middle, being contingent, can be withdrawn, whereas the conclusion, since it is necessary, cannot be withdrawn. It follows, therefore, that a necessary conclusion cannot be scientifically known through a contingent middle.
lib. 1 l. 13 n. 9 Secundam rationem ponit; ibi: amplius si etc., quae talis est. Si aliquis nunc nescit, cum tamen habeat eandem rationem, quam prius habuit, et salvatus est, idest non desiit esse, salva re, idest etiam re scita non corrupta, et iterum ipse non est oblitus; manifestum est quod etiam neque prius scivit. In hoc autem philosophus innuit quatuor modos, quibus aliquis amittit scientiam, quam prius habuit. Unus modus est quando excidit a mente eius ratio, per quam prius sciebat. Alius modus est per corruptionem ipsius scientis. Tertius per corruptionem ipsius rei scitae, sicut si sciam te sedere, dum sedes, te non sedente, haec scientia perit. Quartus est per oblivionem. Unde nullo istorum modorum existente, si aliquis modo nesciat aliquid, nec prius scivit. Sed ille, qui habet conclusionem necessariam per medium contingens, corrupto medio contingenti, nescit, medio non existente, et tamen eandem rationem habet, et salvus est, et salva est res, et non est oblitus. Ergo neque prius scivit, quando medium non erat corruptum.	The second reason (74b32) is this: “If a man is without scientific knowledge now, then even though he possesses the same argument he had up to now, and he has been kept sound,” i.e., has not ceased to be, “and the thing known has not changed; and if he has not forgotten, then obviously he has not known scientifically.” In this passage the Philosopher mentions four ways in which a person loses the science he once had: one way is when there slips from his mind the argument through which he formerly knew scientifically. Another is through the destruction of the knower. A third is by a change occurring in the thing known: thus if, while you are sitting, I know that you are sitting, this knowledge perishes when you are not sitting. The fourth way is by forgetting. Hence if none of these ways has occurred, then, if a person does not know a thing scientifically now, he never did know it. Now one who holds a necessary (onclusion in virtue of a contingent middle, no longer knows it scientifically when the middle ceases to be, for the middle no longer exists, cven though he retains the same argument and the fact remains the same, and he has not forgotten anything. Therefore, even when the middle had not ceased to be, he did not know scientifically.
Quod autem medium, quod est contingens, corrumpatur, probat, quia id quod non est necessarium, oportet quod aliquando corrumpatur. Si autem dicatur quod medium nondum est corruptum: quia tamen non est necessarium, manifestum est quod contingit ipsum corrumpi.Posito autem contingenti, illud quod accidit non est impossibile, sed possibile et contingens. Quod autem sequebatur erat impossibile, scilicet, quod aliquis scientiam haberet alicuius, quod postea nesciret, manentibus conditionibus supra positis: quod tamen sequitur ex hoc quod est medium esse corruptum; quod et si non sit verum, est tamen contingens, ut dictum est.	But that a contingent middle is liable to perish he proves by the fact that whatever is not necessary must at some time perish. And even though the middle has not perished, nevertheless, because it is not necessary, it is obvious that it is liable to perish. Now when something contingent is put forward, its result is not impossible but possible and contingent. But what followed in our case was impossible, namely, that one should have had science of something and later not have had it, all the conditions described above prevailing. And this follows from the fact that a contingent middle has perished, which, although it may not be true [that it has], is nevertheless liable to, as has been said.
lib. 1 l. 13 n. 10 Deinde cum dicit: cum quidem igitur etc., ostendit quod licet per medium contingens non possit sciri conclusio necessaria, tamen potest syllogizari conclusio necessaria ex medio non necessario. Dicit ergo quod nihil prohibet, cum conclusio necessaria est, medium non necessarium esse per quod ostenditur, syllogismo tamen dialectico, non demonstrativo, qui facit scire. Contingit enim necessarium syllogizari ex non necessariis, sicut contingit syllogizari verum ex non veris: non tamen contingit e converso; quia cum medium est necessarium et conclusio necessaria erit, sicuti ex veris praemissis semper concluditur verum.	Then (75a1) he shows that although a necessary conclusion cannot be scientifically known through a contingent middle, nevertheless a necessary conclusion can be syllogized from a non-necessary middle. Hence he says that nothing hinders a necessary conclusion from being obtained through a non-necessary middle, as witness the case in which something is syllogized with a dialectical syllogism, although not with a demonstrative syllogism which causes science. For the necessary happens to be syllogized from the non-necessary, just as the true happens to be syllogized from the untrue—although the converse does not occur. For when the middle is necessary, the conclusion too will be necessary; similarly, from true premises something true is concluded.
Quod autem ex necessariis semper concludatur necessarium, sic probat: sit enim a de b ex necessitate, idest sit haec propositio necessaria: omne b est a; et hoc de c, idest sit haec etiam necessaria: omne c est b; ex his autem duabus necessariis sequitur tertia necessaria, scilicet conclusio, quod, omne c est a. Ostensum est enim in libro priorum quod ex duabus propositionibus de necessitate sequitur conclusio de necessitate.	But that the necessary is always concluded from necessary premises he proves thus: “Let A be necessary of B,” i.e., let this proposition, “Every B is A,” be necessary, “and the latter of C,” i.e., let “Every C is B” also be necessary. Now from these two necessary things follows a third necessary thing, namely, the conclusion, “Every C is A.” For it has been proved in the Prior Analytics that “from two propositions of necessity follows a’ conclusion of necessity.”
Ostendit etiam consequenter quod si conclusio non esset necessaria, nec medium posset esse necessarium. Ponatur enim quod haec conclusio, omne c est a, sit non necessaria, praemissae autem duae sint necessariae; secundum id quod praeostensum est, sequitur quod conclusio sit necessaria, cum tamen contrarium sit positum, scilicet quod conclusio sit non necessaria.	Then by way of consequence he shows that if the conclusion were not necessary, the middle could not be necessary. For example, suppose that this conclusion, “Every C is A,” is not necessary, but that the two premises are necessary. According to our previous doctrine it follows that the conclusion is necessary—which is contrary to the supposition in our example, namely, that the conclusion not be necessary.
lib. 1 l. 13 n. 11 Deinde cum dicit: quoniam igitur etc., infert conclusionem principaliter intentam ex omnibus praedictis, dicens quod quia oportet necessarium esse aliquid, si demonstratione sciatur, manifestum est ex praemissis quod oportet demonstrationem haberi per medium necessarium: alioquin nesciretur quod conclusio sit necessaria, neque propter quid, neque quia, cum necessarium non possit sciri per non necessarium, ut ostensum est. Sed si aliquis habeat rationem per medium non necessarium, dupliciter potest esse dispositus. Aut enim cum ipse sit non sciens, opinabitur tamen se scire, si accipiat in sua opinione medium non necessarium, tanquam necessarium; aut etiam non opinabitur se scire, si scilicet credat non se habere medium necessarium. Et hoc universaliter intelligendum est, tam de scientia quia, qua scitur aliquid per mediata, quam de scientia propter quid, qua scitur aliquid per immediata. Horum autem differentia posterius ostendetur.	Then (75a13) he infers from the aforesaid the conclusion originally intended, saying that since a thing must be necessary if it is made known by way of demonstration, it is clear from the foregoing that a demonstration must rest on a necessary middle. For otherwise it would not be scientifically known that the conclusion is necessary, neither propter quid nor quia, since the necessary cannot be known through the non-necessary, as we have shown. But if someone rests on an argument based on a non-necessary middle, he will be in one of two states. For since he does not actually know in a scientific way, he will either believe that he does know in a scientific way, if he assumes a non-necessary middle as necessary, or he will not presume that he knows in a scientific way, i.e., if he believes that he does not have a necessary middle. And this is to be universally understood both of scientific knowledge quia, in which something is known through mediate principles, and of science propter quid, in which something is known through immediate principles. The difference between these two will be explained later.

Lecture 14 Demonstration bears upon and proceeds from things which are per se

Latin	English
	Lecture 14 (75al8-37) DEMONSTRATION BEARS UPON AND PROCEEDS FROM THINGS WHICH ARE PER SE
lib. 1 l. 14 n. 1 Postquam ostendit philosophus quod demonstratio est de necessariis et ex necessariis, consequenter ostendit quod est de his, quae sunt per se et ex his, quae sunt per se. Et circa hoc tria facit: primo, ostendit quod demonstratio est de his, quae sunt per se, idest quod conclusiones demonstrationis sunt per se; secundo, movet dubitationem et solvit; ibi: et tamen opponet etc.; tertio, ostendit quod demonstratio est ex his, quae sunt per se, idest quod principia demonstrationis oportet per se esse; ibi: quoniam autem ex necessitate et cetera.	After showing that demonstration proceeds from necessary things, the Philosopher then shows that it is concerned with things that are per se and proceeds from things that are per se. In regard to this he does three things. First, he shows that demonstration is concerned with things that are per se, i.e., that the conclusions of demonstrations are per se. Secondly, he raises a problem (75a21). Thirdly, he shows that demonstration proceeds from things that are per se, i.e., that the principles of demonstrations must be per se (75a28).
lib. 1 l. 14 n. 2 Dicit igitur primo quod demonstrativa scientia non potest esse accidentium, quae non sunt per se, sicut determinatum est per se superius, scilicet quod accidens per se est in cuius definitione ponitur subiectum; sicut par aut impar est per se accidens numeri. Album autem animalis non est per se accidens: quia animal non ponitur in eius definitione.	He says therefore first (75a18) that demonstrative science cannot bear on accidents that are not per se in the way that per se was explained above, namely, that a per se accident is one in whose definition the subject is mentioned, as “even” or “odd” is a per se accident of number. But “white” is not a per se accident of animal, because animal is not mentioned in its definition.
Quod autem de huiusmodi accidentibus, quae non sunt per se, non possit esse demonstratio, sic probat. Accidens, quod non est per se, contingit non inesse (de hoc enim accidente loquimur); si ergo demonstratio fieret de accidente, quod non est per se, sequeretur quod conclusio demonstrationis non esset necessaria: cuius contrarium supra ostensum est.	That there cannot be demonstration bearing on accidents that are not per se he proves in the following way: An accident which is not per se might happen not to be present (and this is the accident under discussion). Therefore, if a demonstration were to bear on an accident which is not per se, it would follow that the conclusion of the demonstration would not be necessary: which is contrary to what has been established.
Quod autem accidens, quod non est per se, non necessario insit, ex hoc potest haberi. Si enim aliquod accidens ex necessitate et semper insit subiecto, oportet quod causam habeat in subiecto, qua posita, non possit accidens non inesse. Quod quidem contingit dupliciter. Uno modo, quando ex principiis speciei accidens causatur; et tale accidens dicitur per se passio vel proprium. Alio modo quando accidens causatur ex principiis individui; et hoc est accidens inseparabile. Omne autem accidens, quod causatur ex principiis subiecti, si debeat definiri, oportet quod subiectum ponatur in sua definitione: nam unumquodque definitur ex propriis principiis; et sic oportet omne accidens, quod ex necessitate inest subiecto, esse accidens per se. Illa ergo quae non sunt per se, non ex necessitate insunt.	That an accident which is not per se does not inhere necessarily can be obtained from the following: If an accident inheres necessarily and always in a subject, it must have its cause in the subject—in which case the accident cannot but inhere. Now this can occur in two ways: in one way, when it is caused from the principles of the species, and such an accident is called a per se attribute or property; in another way, when it is caused from the principles of the individual, and this is called an inseparable accident. In either case every accident which is caused from the principles of the subject must, if it is defined, be defined in such a way as to mention the subject in its definition: for a thing is defined in terms of its proper principles. Thus it is clear that an accident which necessarily inheres in its subject must be a per se accident. Therefore, those that are not per se do not inhere of necessity.
lib. 1 l. 14 n. 3 Videtur autem quod Aristoteles utatur demonstratione circulari, quam supra improbavit. Ostenderat enim supra quod demonstratio necessariorum est ex hoc quod est eorum quae sunt per se; nunc autem e converso ostendit quod demonstratio est eorum quae sunt per se, quia est necessariorum. Sed dicendum quod supra Aristoteles non solum ostendit demonstrationem esse necessariorum propter hoc, quod est eorum quae sunt per se, sed ex definitione eius quod est scire; et hic fuit verus demonstrationis modus. Quod autem ostendit demonstrationem esse necessariorum propter hoc, quod est eorum quae sunt per se, non est vera demonstratio, sed est ostensio ad hominem, apud quem notum est quod demonstratio sit eorum quae sunt per se.	It seems, however, that Aristotle is using the circular demonstration, which he previously repudiated. For he had proved earlier that a demonstration is concerned with necessary things, because it is concerned with, things which are per se; but now he shows that demonstration is con-: 1, cerned with things which are per se, because it is concerned with things which are necessary. The answer is that above Aristotle proved that demonstration is concerned with necessary things not only because it is concerned with things which are per se but also because of the definition of scientific knowing—and this was a true way to demonstrate. But the proof that demonstration is concerned with necessary things because it is concerned with things which are per se, is not a demonstration but an indication directed to a person who already knows that demonstration is concerned with things which are per se.
lib. 1 l. 14 n. 4 Deinde cum dicit: et tamen opponet etc., movet dubitationem quamdam. Et circa hoc duo facit. Primo, ponit dubitationem dicens quod potest aliquis forte opponere: si conclusio, quae sequitur ex contingentibus vel ex his quae sunt per accidens, non est necessaria; quare de contingentibus fit interrogatio sive de his quae sunt per accidens, ut ex iis datis procedatur ad conclusionem, cum tamen in syllogismo requiratur quod conclusio ex necessitate accidat. Et quod interrogatio fiat de contingentibus vel ex his, quae sunt per accidens, manifestat per hoc quod subdit: nihil enim differt, si aliquis interrogatus contingentia, postea dicat conclusionem. Quasi dicat: ita potest inferri conclusio ex contingentibus interrogatis et concessis, sicut ex necessariis: utrisque enim eadem forma syllogizandi est.	Then (75a21) he raises a problem in regard to which he does two things. First, he states the problem, saying that someone may perhaps, object that if the conclusion which follows from contingent things or things which are per accidens is not necessary, why are inquiries made concerning contingent things or things which are per accidens, so that from such data one may reach a conclusion, since it is required of the’ syllogism that the conclusion follow of necessity. That inquiries touching contingent things or things which are per accidens do occur is evident from his next statement, namely, that it makes no difference whether one inquires into contingent things and then gives his conclusion. As if to say: A conclusion can be drawn from contingent things that have been investigated and verified, just as from necessary things. For each employs the same syllogistic form.
lib. 1 l. 14 n. 5 Secundo; ibi: oportet autem etc., solvit dicens quod non ita interrogatur de praemissis contingentibus, quasi conclusio sit necessaria absolute propter interrogata, idest propter praemissa contingentia; sed quia necesse est dicenti praemissa conclusionem dicere, et dicere vera in conclusione, si vera sunt, quae praemissa sunt: quasi dicat quod licet ex praemissis contingentibus non sequatur conclusio necessaria necessitate absoluta, sequitur tamen secundum quod est ibi necessitas consequentiae, secundum quod conclusio sequitur ex praemissis.	Secondly (75a24), he solves the difficulty and says that questions solved by contingent premises are not of such a nature that the conclusion is necessary absolutely, on the basis of the things investigated, i.e., in virtue of the contingent premises, but because it is necessary for the one who admits the premises to admit the conclusion and to admit truth in the conclusion, if the things premised are true. It is as though he were saying that although a conclusion which is necessary with absolute necessity does not follow from contingent premises, yet it follows with the necessity of consequence according to which the conclusion follows from premises.
lib. 1 l. 14 n. 6 Deinde cum dicit: quoniam autem etc., ostendit quod demonstratio sit ex his, quae sunt per se, tali ratione. Demonstratio est ex necessariis et de necessariis. Et hoc ideo, quia est scientifica, idest faciens scire. Ea autem, quae non sunt per se, non sunt necessaria: sunt enim per accidens et huiusmodi non sunt necessaria, ut dictum est. Sed illa sunt ex necessitate circa unumquodque genus, quaecunque sunt per se et conveniunt unicuique secundum quod unumquodque est. Relinquitur ergo quod demonstratio non possit esse nisi ex his, quae sunt per se et de talibus.	Then (75a28) he shows that demonstration proceeds from things which are per se, using the following argument: Demonstration proceeds from necessary things and bears on necessary things—and this because it is scientific, i.e., makes one know scientifically. But things which are not per se are not necessary, for they are per accidens and, being such, are not necessary, as has been stated above. On the other hand, those things bear with necessity on a genus which are per se and belong to it according to what it is. It follows, therefore, that demonstration cannot be from anything or of anything but what is per se.
lib. 1 l. 14 n. 7 Ulterius autem ostendit quod etiam si praemissa essent semper, et necessaria, et vera, et non per se, non tamen sciretur de conclusione propter quid; sicut patet in syllogismis, qui fiunt per signa, in quibus conclusionem, quae est per se, non scit aliquis per se, neque propter quid. Sicut si aliquis probaret quod omne elementum est corruptibile, per hoc quod videtur tempore antiquari, esset quidem probatio per signum, non autem per se, neque propter quid: quia propter quid scire est per causam scire. Oportet ergo medium esse causam eius, quod in demonstratione concluditur. Et hoc manifestum est ex praemissis: quia oportet et medium inesse tertio propter ipsum, idest per se, et similiter primum medio. Primum autem et tertium vocat duas extremitates.	He further shows that even if the premises were always both necessary and true but not per se, one would not know the cause why [propter quid] of the conclusion. This is clear in syllogisms which prove through signs, for although the conclusion be per se, one does not know it per se nor propter quid. For example, if someone were to prove that every element corruptible on the ground that it is seen to grow old. This would be a proof through a sign but neither per se nor propter quid, because to know propter quid one must know through the cause. Therefore, the middle must be the cause of that which is concluded in the demonstration. And this is obvious from the premises: for the middle must inhere in the third causatively, i.e., per se, and likewise the first in the middle. Here he calls the two extremes the first and the third.

Lecture 15 Demonstration does not skip from one genus to an alien genus

Latin	English
	Lecture 15 (75a38-b20) DEMONSTRATION DOES NOT SKIP FROM ONE GENUS TO AN ALIEN GENUS
lib. 1 l. 15 n. 1 Postquam ostendit philosophus quod demonstratio est ex his quae sunt per se, hic concludit quod demonstratio est ex principiis propriis, non extraneis, neque ex communibus. Et dividitur in duas partes: in prima, ostendit quod demonstratio procedit ex propriis principiis; in secunda, determinat quae sint principia propria et quae communia; ibi: difficile autem et cetera. Prima in duas: in prima, ostendit quod demonstratio non procedit ex principiis extraneis; in secunda, ostendit quod non procedit ex principiis communibus; ibi: quoniam autem manifestum est et cetera. Prima in duas: in prima, ex praemissis ostendit quod demonstratio non est ex principiis extraneis; in secunda, ex praemissis etiam ostendit quod demonstrationes non sunt de rebus corruptibilibus, sed de sempiternis; ibi: manifestum autem et si sint propositiones et cetera. Circa primum tria facit: primo, proponit intentum; secundo, probat propositum; ibi: tria enim sunt etc.; tertio, concludit intentum; ibi: propter hoc geometriae et cetera.	After showing that demonstration is from things that are per se, the Philosopher here concludes that demonstration is from principles which are proper and not from extraneous or common principles. His treatment falls into two parts. In the first he shows that demonstration proceeds from proper principles. In the second he establishes which principles are proper and which common (76a26) [L. 18]. The first is divided into two parts. In the first he shows that demonstration does not proceed from extraneous principles. In the second that it does not proceed from common principles (75b37) [L. 17]. The first is divided into two parts. In the first, from the preceding, he shows that demonstration is not from extraneous principles. In the second, also from the preceding, he shows that demonstrations do not bear on corruptible things but on eternal (75b21) [L. 16]. Concerning the first he does three things. First, he states his proposition. Secondly, he proves it (75a39). Thirdly, he concludes to what he intended (75b13).
lib. 1 l. 15 n. 2 Dicit ergo primo quod, ex quo demonstratio est ex his quae sunt per se, manifestum est quod non contingit demonstrare descendentem vel procedentem ex alio genere in aliud genus, sicut non contingit quod geometria ex propriis principiis demonstret aliquid descendens in arithmeticam.	He says therefore first (75a38) that inasmuch as demonstration is from things that are per se, it is plain that demonstration does not consist in descending or skipping from one genus to another, as geometry, demonstrating from its own principles, does not descend to something in arithmetic.
lib. 1 l. 15 n. 3 Deinde cum dicit: tria enim etc., propositum probat. Et circa hoc tria facit. Primo, praemittit quae sint necessaria ad demonstrationem, dicens quod in demonstrationibus tria sunt. Unum est, quod demonstratur, scilicet conclusio, quae quidem continet in se id, quod per se inest alicui generi: per demonstrationem enim concluditur propria passio de proprio subiecto. Aliud autem sunt dignitates, ex quibus demonstratio procedit. Tertium autem est genus subiectum, cuius proprias passiones et per se accidentia demonstratio ostendit.	Then (75a39) he proves this proposition and does three things. First, he lays down the things that are necessary for a demonstration, saying that “three things are necessary in demonstrations. One is that which is demonstrated,” namely, the conclusion which, as a matter of fact, contains within itself that which inheres in its genus per se. For through demonstration a proper attribute is concluded of its proper subject. Another is the dignities [maxims or axioms] from which demonstration proceeds. The third is the generic subject whose proper attributes and per se accidents demonstration reveals.
lib. 1 l. 15 n. 4 Secundo; ibi: ex quibus igitur etc., ostendit quid praedictorum trium possit esse commune diversis scientiis et quid non, dicens quod horum trium unum, scilicet dignitates, ex quibus demonstratio procedit, contingit esse idem in diversis demonstrationibus et etiam in diversis scientiis: sed in illis scientiis, quarum est diversum genus subiectum, sicut in arithmetica, quae est de numeris, et geometria, quae est de magnitudinibus, non contingit quod demonstratio, quae procedit ex principiis unius scientiae, puta arithmeticae, descendat ad subiecta alterius scientiae, sicut ad magnitudines, quae sunt subiecta geometriae; nisi forte subiectum unius scientiae contineatur sub subiecto alterius, sicut si magnitudines contineantur sub numeris (quod quidem qualiter contingat, scilicet subiectum unius scientiae contineri sub subiecto alterius, posterius dicetur). Magnitudines enim sub numeris non continentur, nisi forte secundum quod magnitudines numeratae sunt. Subiecta etiam diversarum demonstrationum sive scientiarum diversa sunt. Arithmetica enim demonstratio semper habet genus proprium circa quod demonstrat. Et aliae scientiae similiter.	Secondly (75b3), he shows which of the three aforesaid can be common to various sciences and which cannot, saying that one of the three, namely, the dignities, from which demonstration proceeds, happens to be the same in diverse demonstrations and even in diverse sciences. But in those sciences whose respective generic subject is diverse, as in arithmetic which is concerned with numbers, and in geometry which is concerned with magnitudes, it does not occur that a demonstration which starts with the principles of one science, say, of arithmetic, descends to the subjects of another science, say to magnitudes, which pertain to geometry; unless perchancp the subject of one science should be contained under the sub., ject of the other, for example, if magnitudes should be contained under, numbers. (How this occurs, namely, how the subject of one science may be contained under the subject of another, will be discussed later). For magnitudes are not contained under numbers except perhaps in the sense that magnitudes are numbered. In any case, the subjects of diverse demonstrations or sciences are diverse. For an arithmetical demonstration always has its proper genus in respect to which it demonstrates, and so do the other sciences.
lib. 1 l. 15 n. 5 Tertio; ibi: quare aut simpliciter etc., probat propositum. Et circa hoc duo facit. Primo, inducit principale propositum per modum conclusionis, eo quod ex praemissis haberi potest, dicens: quare manifestum est quod necesse est, aut esse simpliciter idem genus, circa quod sumuntur principia et conclusiones, et sic non est descensus, neque transitus de genere in genus: aut si debet demonstratio descendere ab uno genere in aliud, oportet esse unum genus sic, idest quodammodo. Aliter enim impossibile est quod demonstretur aliqua conclusio ex aliquibus principiis, cum non sit idem genus vel simpliciter vel secundum quid.	Thirdly (75b8), he proves his proposition and does two things. First he brings in the main intent after the manner of a conclusion on the ground that it can be obtained from the aforesaid, saying: Hence it is clear that it is necessary either that the genus with which the principles and conclusions deal be absolutely the same (in which case there is no descent or skipping from one genus to another); or if the demonstration is to descend from one genus to another, the two must be one thus, i.e., somehow. For otherwise it is impossible for a conclusion to be demonstrated from principles, since there is not the same genus either absolutely or in a qualified sense.
Sciendum est autem quod simpliciter idem genus accipitur, quando ex parte subiecti non sumitur aliqua differentia determinans, quae sit extranea a natura illius generis; sicut si quis per principia verificata de triangulo procedat ad demonstrandum aliquid circa isoscelem vel aliquam aliam speciem trianguli. Secundum quid autem est unum genus, quando assumitur circa subiectum aliqua differentia extranea a natura illius generis; sicut visuale est extraneum a genere lineae, et sonus est extraneus a genere numeri. Numerus ergo simpliciter, qui est genus subiectum arithmeticae, et numerus sonorum, qui est genus subiectum musicae, non sunt unum genus simpliciter. Similiter autem nec linea simpliciter, quam considerat geometra, et linea visualis, quam considerat perspectivus. Unde patet quod quando ea, quae sunt lineae simpliciter, applicantur ad lineam visualem, fit quodammodo descensus in aliud genus: non autem quando ea, quae sunt trianguli, applicantur ad isoscelem.	However, it should be noted that a genus absolutely the same is being taken when on the part of the subject no determinate difference is admitted which is alien to the nature of that genus: for example, if someone using principles verified of triangle should proceed to demonstrate something about isosceles or about any other species of triangle. But a genus is one in a qualified sense when a difference alien to the nature of the genus is admitted of the subject, as “visual” is alien to the genus of line, and “sound” to the genus of number. Therefore, number absolutely, which is the generic subject of arithmetic, and sonant number, which is the generic subject of music, are not one genus absolutely speaking. The same goes for line absolutely, which geometry considers, and the visual line considered in optics. Hence it is clear that when matters pertaining absolutely to the line are applied to the visual line, a descent is being made to another genus—which is not the case when matters pertaining to triangle are applied to isosceles.
lib. 1 l. 15 n. 6 Secundo; ibi: ex eodem enim genere etc., ostendit propositum hoc modo. Oportet in demonstratione eiusdem generis esse media et extrema. Extrema autem in conclusione continentur. Nam maior extremitas in conclusione est praedicatum; minor vero extremitas subiectum; medium autem in praemissis continetur. Oportet igitur principia et conclusiones circa idem genus sumi. Cum autem huic coniunxerimus quod diversae scientiae sint circa diversa genera subiecta; ex necessitate sequitur quod ex principiis unius scientiae non concludatur aliquid in alia scientia, quae non sit sub ea posita.	Secondly (75b10), he manifests the proposition in this way. In demonstration the middles and the extremes must belong to the same genus. Now the extremes are contained in the conclusion: for the major extreme in the conclusion is its predicate, and the minor its subject. But the middle is contained in the premises. It is required, therefore, that the principles and conclusion be taken with respect to the same genus. When we add to this the fact that diverse sciences are of necessity concerned with subjects generically diverse, it follows that from the principles of one science, something in another science not under it may not be concluded.
Quod autem in demonstratione oporteat media et extrema unius generis esse, sic probat. Detur enim quod medium sit alterius generis ab extremis, sicut si extrema sint triangulus et habere tres angulos aequales duobus rectis. Manifestum est quod passio conclusa de triangulo, per se inest ei; non autem per se inest aeneo. Et si e contrario passio per se inesset aeneo, puta sonorum esse, vel aliquid huiusmodi, palam est quod per accidens inesset triangulo.	That the middles and extremes in a demonstration must be of one genus he proves thus: Suppose that a middle belongs to a genus other than that of the extremes which might be, for example, “triangle” and “have three angles equal to two right angles.” Now a proper attribute concluded of triangle is in it per se but is not in “brazen” per se; or, conversely, if the proper attribute is per se in “brazen,” say, “high-sounding” or the like, it is obviously in triangle per accidens.
Unde patet quod oportet omnino, si subiectum conclusionis et medium sint penitus alterius generis, quod passio vel non per se insit medio vel non per se insit subiecto: et ita oportet quod alteri eorum insit per accidens. Et si quidem insit medio per accidens, erit per accidens in praemissis; si autem subiecto, erit in conclusione: et hoc ex parte passionis. Sed utroque modo oportebit per accidens esse in praemissis, quantum ad hoc quod subiectum accipitur sub medio: sicut si triangulus accipiatur sub aeneo aut e converso. Ostensum est autem quod in demonstrationibus tam conclusio, quam praemissae sunt per se et non per accidens. Oportet ergo in demonstrationibus medium et extrema eiusdem generis esse.	This example makes it plain that if the subject of the conclusion and the middle are in entirely different genera, then the proper attribute is either not in the middle per se or else not in the subject per se. Consequently, it is in one of them per accidens. If it is in the middle per accidens, something will be in the premises per accidens. But if it is in the subject per accidens, something will be per accidens in the conclusion: and this on the part of a proper attribute. But either way something will be per accidens in the premises, inasmuch as the subject is subsumed under the middle, for example, triangle under brazen, or vice versa. However, it has been previously established that in a demonstration both the conclusion and the premises are per se and not per accidens. It is required, therefore, in demonstrations that the middle and the extremes be of the same genus.
lib. 1 l. 15 n. 7 Deinde cum dicit: propter hoc geometriae etc., infert duas conclusiones ex praemissis. Quarum prima est quod nulla scientia demonstrat aliquid de subiecto alterius scientiae, sive sit scientiae communioris sive alterius scientiae disparatae; sicut geometria non demonstrat quod contrariorum eadem est scientia: contraria enim pertinent ad scientiam communem, scilicet ad philosophiam primam vel dialecticam. Et similiter geometria non demonstrat quod duo cubi sint unus cubus, idest quod ex ductu unius numeri cubici in alium numerum cubicum surgat numerus cubicus. Dicitur autem numerus cubicus, qui consurgit ex ductu unius numeri in seipsum bis; sicut octonarius est numerus cubicus, surgit enim ex ductu binarii in seipsum bis, quia bis duo bis sunt octo. Et eadem ratione vigintiseptem est numerus cubicus, et radix eius est tria, quia ter tria ter faciunt vigintiseptem. Si ergo ducantur octo in vigintiseptem consurgit numerus cubicus, idest ducenta sexdecim, cuius radix est sex: quia sexies sex sexies sunt ducenta sexdecim. Hoc ergo habet probare arithmeticus, non geometra. Et similiter, quod est unius scientiae non habet probare alia scientia, nisi forte una scientia sit sub altera; sicut se habet perspectiva ad geometriam, et consonantia vel harmonica, idest musica, ad arithmeticam.	Then (75b13) he infers two conclusions from the foregoing. The first conclusion is that no science demonstrates anything about the subject of another science, when this other science is more common than o entirely disparate from the first. Thus, geometry does not demonstrate that one and the same science deals with the both of two contraries: for contraries pertain to the common science, namely, to first philosophy or to dialectics. In like manner, geometry does not demonstrate that “two cubes are one cube,” i.e., that the product of one cube and another cube is a cube. “Cube” here means the number which results from multiply., ing a number by itself twice: for example, 8 is a cube, for it is the result 11of multiplying 2 by itself twice, for 2 times 2 times 2 are 8. Similarly,, 27 is a cube, whose root is 3, because 3 times 3 times 3 make 27. Now if 8 be multiplied by 27 the product is 216, which is a cube whose root is 6, because 6 times 6 times 6 make 216. Therefore, arithmetic, not, geometry has: the power to prove this. Similarly, that which pertains to one science cannot be proved by another science unless the one happens~, to be under the other, as optics is under geometry, and consonance or harmony, i.e., music, is under arithmetic.
lib. 1 l. 15 n. 8 Secunda conclusio ponitur; ibi: neque si aliquid et cetera. Et est quod scientia etiam de proprio subiecto non probat quodlibet accidens, sed accidens quod est sui generis. Sicut si aliquid inest lineis, non secundum quod sunt lineae, neque secundum propria principia linearum, hoc non demonstrat geometra de lineis; sicut quod linea recta sit pulcherrima linearum, aut recta linea si est contraria circulari vel non. Haec enim non sunt secundum proprium genus lineae, sed secundum aliquid communius. Pulchrum enim et contrarium genus lineae transcendunt.	The second conclusion (75b17) states that a science cannot prove just any random accident of its subject, but the accidents proper to its genus Thus, if something belongs to lines not as lines or not according to th proper principles of lines, geometry does not demonstrate it of lines: for example, that a straight line is the most beautiful of lines, or whether straight line is contrary or not to the curved. For these matters are outside the proper genus of line and belong to something more general. For beauty and contrary transcend the genus of line.

Lecture 16 Demonstration is not of perishable but of eternal matters

Latin	English
	Lecture 16 (75b21-36) DEMONSTRATION IS NOT OF PERISHABLE BUT OF ETERNAL MATTERS
lib. 1 l. 16 n. 1 Postquam ex superioribus philosophus concluserat quod demonstratio non concludit ex extraneis principiis, nunc iterum ex superioribus intendit concludere quod demonstratio non est de corruptibilibus. Et circa hoc duo facit: primo, ostendit quod sempiternorum et non corruptibilium est demonstratio; secundo, ostendit qualiter sit eorum quae sunt ut frequenter; ibi: eorum autem et cetera. Circa primum duo facit: primo, ostendit quod demonstratio non sit corruptibilium, sed sempiternorum; secundo, ostendit idem de definitione; ibi: similiter se habet et cetera. Circa primum duo facit: primo, proponit conclusionem intentam; secundo, ponit rationem probantem ipsam; ibi: quod autem universaliter et cetera.	After concluding from the above that demonstration does not conclude from extraneous principles, the Philosopher intends to conclude something else from the above, namely, that demonstration is not of destructible things. Concerning this he does two things. First, he shows that demonstration is of eternal and not of destructible things. Secondly, he shows how it is of things that occur now and then (75b32). Concerning the first he does two things. First, he shows that demonstration is not of perishable but of eternal things. Secondly, he shows that the same is true of definition (75b30). In regard to the first he does two things. First, he proposes the intended conclusion. Secondly, he sets down the reason proving it (75b24).
lib. 1 l. 16 n. 2 Primo ergo ponit duas conclusiones, quarum una sequitur ex altera. Prima est quod necesse est conclusionem demonstrationis huius de qua nunc agitur, et quam possumus dicere simpliciter demonstrationem, esse perpetuam; quod quidem sequitur ex hoc, quod supra habitum est, scilicet quod propositiones, ex quibus fit syllogismus, debent esse universales: quod significavit per dici de omni. Secunda conclusio est quod neque demonstratio, neque scientia est corruptibilium, loquendo simpliciter, sed solum secundum accidens.	Therefore first (75b21), he sets down two conclusions, one of which follows from the other. The first is that the conclusion of this demonstration which we are now discussing and which we can call demonstration in the full sense must be eternal: which, of course, follows from what has been stated so far, namely, that the propositions of a syllogism should be universal: which he signified by “said of all.” The second conclusion is that neither demonstration nor science is of destructible things, i.e., absolutely speaking, but only accidentally.
lib. 1 l. 16 n. 3 Deinde cum dicit: quod autem universaliter etc., inducit rationem ad probandum propositas conclusiones: quae talis est. Conclusionis corruptibilis, et non sempiternae, non est in se continere quod est universaliter, sed aliquando et sic. Dictum est enim supra quod dici de omni duo continet, scilicet quod non in quodam sic et in quodam non, et iterum, quod non aliquando sic et aliquando non. In omnibus autem corruptibilibus invenitur aliquando sic et aliquando non. Unde patet quod in corruptibilibus non invenitur dici de omni, sive quod est universaliter. Sed ubi conclusio est non universalis, oportet aliquam praemissarum esse non universalem. Conclusio ergo corruptibilis oportet quod sequatur ex praemissis, quarum altera non sit universalis. Cum ergo huic coniunxerimus quod demonstratio simpliciter semper debet esse ex universalibus, sequitur quod demonstratio non possit habere conclusionem corruptibilem, sed sempiternam.	Then (75b24) he presents an argument to prove the proposed conclusions. It is this: It is not the character of a destructible non-eternal conclusion to contain what is universally so, but what is so for a time and in certain instances. For it has been established above that “said of all” contains two things, namely, that it is not such as to be so in one case and not in another, or so at one time and not at another. But in destructible things we find that at one time something is so and at another not so. Hence it is clear that “said of all” or “said universally” are not found in destructible things. But where a conclusion is not universal, at least one of the premises is not universal. Therefore, a destructible conclusion must have followed from premises, one of which is not universal. Accordingly, when we add to this the fact that demonstration absolutely must be from what is universal, it follows that a demonstration cannot have a destructible conclusion, but must have an eternal one.
lib. 1 l. 16 n. 4 Deinde cum dicit: similiter se habet etc., ostendit quod etiam definitio est non corruptibilium, sed sempiternorum, tali ratione. Demonstratio quantum ad principia et conclusiones est sempiternorum et non corruptibilium; sed definitio vel est principium, vel conclusio demonstrationis, vel demonstratio positione differens; ergo definitio non est corruptibilium, sed sempiternorum.	Then (75b30) he shows that definition, too, is not of destructible but of eternal things. The reason for this is that demonstration, both as to its principles and conclusions, is of eternal and not of destructible things. But a definition is either a principle of a demonstration, a conclusion of a demonstration, or a demonstration with a different ordering of its terms. Therefore, a definition is not of destructible, but of eternal things.
lib. 1 l. 16 n. 5 Ad intellectum autem huius literae sciendum est quod contingit definitiones diversas dari eiusdem rei, sumptas ex diversis causis. Causae autem ad invicem ordinem habent: nam ex una sumitur ratio alterius. Ex forma enim sumitur ratio materiae: talem enim oportet esse materiam, qualem forma requirit. Efficiens autem est ratio formae: quia enim agens agit sibi simile, oportet quod secundum modum agentis sit etiam modus formae, quae ex actione consequitur. Ex fine autem sumitur ratio efficientis: nam omne agens agit propter finem. Oportet ergo quod definitio, quae sumitur a fine, sit ratio et causa probativa aliarum definitionum, quae sumuntur ex aliis causis.	For a better understanding of this passage it should be noted that it is possible to give different definitions of the same thing, depending on the different causes mentioned. But causes are arranged in a definite order to one another: for the reason of one is derived from another. Thus the reason of matter is derived from the form, for the matter must be such as the form requires. Again, the agent is the reason for the form: for since an agent produces something like unto itself, the mode of the form which results from the action must be according to the mode of the agent. Finally, it is from the end that the reason of the agent is derived, for every agent acts because of an end. Consequently, a definition which is formulated from the end is the reason and cause proving the other definitions which are formulated from the other causes.
Ponamus ergo duas definitiones domus, quarum una sumatur a causa materiali, quae sit talis: domus est cooperimentum constitutum ex lapidibus, cemento et lignis. Alia sumatur ex causa finali, quae sit talis: domus est cooperimentum prohibens nos a pluviis, frigore et calore. Potest ergo prima definitio demonstrari ex secunda, hoc modo: omne cooperimentum prohibens nos a pluviis, frigore et calore oportet quod sit constitutum ex lapidibus, cemento et lignis; domus est huiusmodi; ergo et cetera.	Therefore, let us lay down two definitions, one of which is formulated from the material cause, for example, that a house is a shelter composed of stones, cement and wood; the other being formulated from the final cause, namely, that a house is a shelter protecting us from the rain and heat and cold. Now the first definition can be demonstrated from the second in the following way: Every shelter protecting us from rain, heat and cold should be composed of wood, cement and stones; but a house is such a thing: therefore...
Patet ergo quod definitio, quae sumitur a fine, est principium demonstrationis; illa autem, quae sumitur a materia, est demonstrationis conclusio. Potest tamen utraque coniungi, ut sit una definitio, hoc modo: domus est cooperimentum constitutum ex dictis, defendens a pluvia, frigore et calore. Talis autem definitio continet totum quod est in demonstratione, scilicet medium et conclusionem. Et ideo talis definitio est demonstratio positione differens; quia in hoc solo differt a demonstratione, quia non est ordinata in modo et figura.	Thus it is clear that the definition formulated from the end is the principle of the demonstration, whereas the one formulated from the matter is the conclusion of the demonstration. However, the two can be combined in the following way to form one definition: A house is a shelter composed of the aforesaid, protecting us from rain, cold and heat. But such a definition contains all the elements of a demonstration, namely, a middle and a conclusion. Accordingly, such a definition is a demonstration differing merely in the arrangement of its terms, because the only way it differs from the demonstration is that it is not arranged in a syllogistic mode and figure.
lib. 1 l. 16 n. 6 Sciendum est autem quod quia demonstratio non est corruptibilium, sed sempiternorum, neque definitio, Plato coactus fuit ponere ideas. Cum enim ista sensibilia sint corruptibilia, videbatur quod eorum non posset esse neque demonstratio, neque definitio. Et ideo videbatur quod oporteret ponere quasdam substantias incorruptibiles, de quibus et demonstrationes et definitiones darentur. Et has substantias sempiternas vocabat species vel ideas.	Here we might remark that because demonstration, as well as definition, is not of destructible but of eternal things, Plato was led to posit “Ideas.” For since sensible things are destructible, it appeared that there could be neither demonstration nor definition of them. As a consequence it seemed necessary to postulate certain indestructible substances concerning which demonstrations and definitions could be given. These eternal substances are what he calls “Forms” or “Ideas.”
lib. 1 l. 16 n. 7 Sed huic opinioni occurrit Aristoteles superius dicens quod demonstratio non est corruptibilium nisi per accidens. Etsi enim ista sensibilia corruptibilia sint in particulari, in universali tamen quamdam sempiternitatem habent. Cum ergo demonstratio detur de istis sensibilibus in universali, non autem in particulari, sequitur quod demonstratio non sit corruptibilium, nisi per accidens; sempiternorum autem est per se.	However, Aristotle opposed this opinion above, when he said that demonstration is not of destructible things except per accidens. For although those sensible things are destructible as individuals, nevertheless in the universal they have a certain everlasting status. Therefore, since demonstration beais on those sensible things universally and not individually, it follows that demonstration is not of destructible things except per accidens, but of eternal things per se.
lib. 1 l. 16 n. 8 Deinde cum dicit: eorum autem quae etc., ostendit quomodo eorum, quae sunt ut frequenter, possit esse demonstratio, dicens: quod eorum quae saepe fiunt, sunt etiam demonstrationes et scientiae: sicut de defectu lunae, qui tamen non semper est. Non enim luna semper deficit, sed aliquando. Haec autem quae sunt frequenter, secundum quod huiusmodi sunt, idest secundum quod de eis demonstrationes dantur, sunt semper: sed secundum quod non sunt semper, sunt particularia. De particularibus autem non potest esse demonstratio, ut ostensum est, sed solum de universalibus. Unde patet quod huiusmodi, secundum quod de eis est demonstratio, sunt semper. Et sicut est de defectu lunae, ita est de omnibus aliis similibus.	Then (75b32) he shows how there can be demonstration of things that occur now and then, saying “that there are science and demonstrations of things that occur frequently, as the eclipse of the moon,” which is not always. For the moon is not always being eclipsed, but only now and then. Now things that occur frequently, so far as they are such, i.e., so far as demonstrations are given concerning them, are always; but they are particular, so far as they are not always. But demonstration cannot be of particulars, as we have shown, but only of universals. Hence it is clear that these things, insofar as there is demonstration of them, are always. And as in the case of the eclipse of the moon, so in all kindred matters.
Consideranda tamen est differentia inter ea. Quaedam enim non sunt semper secundum tempus, sunt autem semper per comparationem ad causam: quia nunquam deficit, quin posita tali causa, sequatur effectus; sicut est de defectu lunae. Nunquam enim deficit, quin semper sit lunae eclypsis, quandocunque terra diametraliter interponitur inter solem et lunam. In quibusdam vero contingit quod non semper sunt, etiam per comparationem ad causam: quia videlicet causae impediri possunt. Non enim semper ex semine hominis generatur homo habens duas manus; sed quandoque fit defectus vel propter impedimentum causae agentis vel materiae. In utrisque autem sic ordinandae sunt demonstrationes, ut ex universalibus propositionibus inferatur universalis conclusio, removendo illa, in quibus potest esse defectus vel ex parte temporis tantum vel etiam ex parte causae.	However, there are certain differences to be noted among them. For some are not always with respect to time, but they are always in respect to their cause, because it never fails that under given conditions the effect follows, as in the eclipse of the moon. For the moon never fails to be eclipsed when the earth is diametrically interposed between sun and moon. But others happen not to be always even in respect to their causes, i.e., in those cases where the causes can be impeded. For it is not always that from a human seed a man with two hands is generated, but now and then a failure occurs, owing to a defect in the efficient cause or material cause. However, in both cases the demonstration must be so set up that a universal conclusion may be inferred from universal propositions by ruling out whatever can be an exception either on the part of time alone, or also of some cause.

Lecture 17 Demonstration does not proceed from common principles, but from principles proper to the thing demonstrated

Latin	English
	Lecture 17 (75b37-76a25) DEMONSTRATION DOES NOT PROCEED FROM COMMON PRINCIPLES, BUT FROM PRINCIPLES PROPER TO THE THING DEMONSTRATED
lib. 1 l. 17 n. 1 Ostenderat supra philosophus quod demonstratio non procedit ex principiis extraneis; hic autem ostendit quod non procedit ex communibus. Et circa hoc duo facit: primo, ostendit propositum; secundo, inducit quandam conclusionem ex dictis; ibi: si autem hoc est et cetera.	Above the Philosopher showed that demonstration does not proceed from extraneous principles; here he shows that it does not proceed from common principles. Concerning this he does two things. First, he states his proposition. Secondly, he draws a conclusion from what he has said (76a17).
Circa primum tria facit. Primo, proponit intentum dicens quod, quia manifestum est quod non contingit unumquodque per unumquodque demonstrare, sed oportet quod demonstratio fiat ex unoquoque principiorum, hoc modo, quod id quod demonstratur sit secundum quod est illud, idest, oportet quod principia demonstrationis insint per se ei, quod demonstratur; si, inquam, ita est, non sufficit, ad hoc quod aliquid sciatur, quod demonstretur ex veris et immediatis, sed oportet ulterius quod demonstretur ex principiis propriis.	Concerning the first he does two things. First (75b37), he states his intention, saying that since it is clear that it is not just through anything at random that something is demonstrated, but it is required that demonstration be from one or another of a thing’s principles in such a way that what is demonstrated be in line with what the thing as such is, i.e., it is required that the principles of the demonstration belong per se to that which is demonstrated; if, I say, this is so, then in order that something be scientifically known, it is not enough that it be demonstrated from true and immediate principles, but it is further required that it be demonstrated from proper principles.
lib. 1 l. 17 n. 2 Secundo; ibi: est enim sic demonstrare etc., probat propositum, scilicet quod non sufficiat ex veris et immediatis aliquid demonstrare, quia sic contingeret aliquid demonstrare, sicut Bryso demonstravit tetragonismum, idest quadraturam circuli, ostendens aliquod quadratum esse circulo aequale per aliqua principia communia, hoc modo: in quocunque genere est invenire aliquid maius et minus alicui, in eodem est invenire et illi aequale; in genere autem quadratorum est invenire aliquod quadratum minus circulo, quod scilicet scribitur intra circulum, et aliquod maius circulo, intra quod circulus describitur; ergo est invenire aliquod quadratum circulo aequale.	Secondly (75b40), he proves his proposition, namely, that it is not enough to demonstrate something from true and immediate principles, because then something could be demonstrated in the way that Bryson proved squaring, i.e., the squaring of a circle. For he showed that some square is equal to a circle, using common principles in the following way: In any matter in which it is possible to have something greater and something less than something else, one can find something equal to it. But in the genus of squares it is possible to find one which is less than a given circle, namely, one inscribed in the circle, and another which is greater, namely, one circumscribed about the circle. Therefore, one can be found which is equal to the circle.
Haec quidem probatio est secundum commune: aequale enim, et maius, et minus, excedunt genus quadranguli et circuli. Unde patet quod huiusmodi rationes demonstrant secundum aliquod commune, quia medium alteri inest, quam ei de quo fit demonstratio; et ideo huiusmodi rationes conveniunt aliis, et non conveniunt istis, de quibus dantur, tanquam proximis. Unde patet quod qui scit per huiusmodi rationes, non scit secundum quod illud est, idest per se, sed per accidens tantum. Si enim esset secundum se, non conveniret demonstratio in aliud genus. Unumquodque enim scimus secundum accidens, cum non cognoscimus illud secundum quod est ex principiis illius, idest secundum quod est ex principiis per se. Sicut habere tres angulos aequales duobus rectis inest per se triangulo, idest secundum quod est ex principiis illius. Quare si per se inesset medium acceptum conclusioni, necesse esset in eadem proximitate esse, idest proximum esse secundum genus conclusioni.	But this proof is according to something common: for equal and greater and less transcend the genus of square and circle. Hence such characteristics demonstrate according to something common, because the middle is present in other things besides the one with which the demonstration is concerned. Therefore, such characteristics, since they belong to other things as well, do not belong to the things of which they are said as to proximate subjects. Consequently, one who knows through such characteristics does not know according to what the thing is as such, i.e., per se, but only per accidens. For if it were based on what the thing is according to itself, the demonstration would not apply to something of another genus. For we know a thing according to what it accidentally is, when we do not know it according to what it is in virtue of its own principles, i.e., according to what comes per se from its principles, as “having three angles equal to two right angles” belongs per se to triangle, i.e., according to what it is in virtue of its principles. Consequently, if the middle which one used belonged per se to the conclusion, it would necessarily have to be in the same proximity, i.e., proximate according to genus, to the conclusion.
lib. 1 l. 17 n. 3 Tertio; ibi: si vero non etc., excludit quandam dubitationem. Contingit enim aliquando medium demonstrationis non esse in eodem genere cum conclusione. Quod qualiter contingat ostendit dicens: si vero non sit medium in eadem proximitate conclusioni, sed hoc modo sicut demonstratur aliquid in harmonica, idest in musica, per arithmeticam; verum quidem est quod huiusmodi etiam similiter demonstratur. Fit enim demonstratio in inferiori scientia per principia superioris scientiae, ut ostensum est; sicut et in scientia superiori per principia superioris.	Thirdly (76a), he excludes a doubt. For it sometimes happens that the middle of a demonstration is not in the same genus as the conclusion. How this can happen he shows when he suggests that if the middle is not in the same proximate genus as the conclusion, but is present in the way that theorems in harmonics, (i.e., in music), are demonstrable by arithmetic, it is nevertheless true that such things are also demonstrated. For in a lower science there is demonstration through the principles of a higher science, as we have established, just as in the higher science there is demonstration through the principles of that higher science.
Sed in hoc differt, quod alterius scientiae, scilicet inferioris, est scire ipsum quia tantum: genus enim subiectum inferioris scientiae est alterum a genere subiecto superioris scientiae, ex qua sumuntur principia. Sed scire propter quid est superioris scientiae, cuius sunt per se illae passiones. Cum enim passio insit subiecto propter medium, illa scientia considerabit propter quid, ad quam pertinet medium, cuius per se est passio, quae demonstratur. Si vero subiectum sit ad aliam scientiam pertinens, illius scientiae non erit propter quid, sed quia tantum; nec tali subiecto per se conveniet passio demonstrata de ipso, sed per medium extraneum. Si vero medium et subiectum pertineant ad eamdem scientiam, tunc illius scientiae erit scire quia et propter quid.	But there is this difference: on the part of the other science, namely, the lower one, there is only knowledge quia [i.e., knowledge of the fact], for the generic subject of the lower science is different from the generic subject of the higher science, from which the principles are borrowed. But knowledge propter quid [i.e., of the cause why] is proper to the higher science to which those proper attributes belong per se. For since it is in virtue of the middle that a proper attribute is in a subject, that science will consider the propter quid to which pertains the middle in which the proper attribute being demonstrated inheres per se. But if the subject belongs to another science, it will not pertain to that science to know propter quid but only quia. Nor will the proper attribute demonstrated of such a subject belong per se to it, but it will be through an alien middle. However, if the middle and the subject pertain to the same science, then it will be characteristic of that science to know quia and propter quid.
Remota autem dubitatione, ulterius conclusionem intentam principaliter inducit, dicens quod ex praedictis patet quod non est demonstrare unumquodque simpliciter, idest quocunque modo, sed secundum hoc quod demonstratur ex propriis principiis uniuscuiusque. Sed et principia propria singularum scientiarum habent aliquod commune prius eis.	Having settled this doubt, he draws the conclusion chiefly intended, saying that in the light of the foregoing it is clear that one should not demonstrate haphazardly, i.e., in any random way, but in such a way that a demonstration is made from the principles proper to each thing. But even the principles proper to the particular sciences have something common prior to them.
lib. 1 l. 17 n. 4 Deinde cum dicit: si autem hoc etc., inducit quandam conclusionem sequentem ex dictis. Et circa hoc tria facit. Primo, inducit conclusionem dicens quod, si hoc verum est, scilicet quod demonstrationes in singulis scientiis non fiunt ex communibus principiis, et iterum quod principia scientiarum habent aliquid prius se, quod est commune; manifestum est quod non est uniuscuiusque scientiae demonstrare principia sua propria. Illa enim priora principia, per quae possent probari singularum scientiarum propria principia, sunt communia principia omnium, et illa scientia, quae considerat huiusmodi principia communia, est propria omnibus, idest ita se habet ad ea, quae sunt communia omnibus, sicut se habent aliae scientiae particulares ad ea, quae sunt propria. Sicut cum subiectum arithmeticae sit numerus, ideo arithmetica considerat ea, quae sunt propria numeri: similiter prima philosophia, quae considerat omnia principia, habet pro subiecto ens, quod est commune ad omnia; et ideo considerat ea, quae sunt propria entis, quae sunt omnibus communia, tanquam propria sibi.	Then (76a17) he draws a conclusion that follows from the aforesaid. And in regard to this he does three things. First, he draws the conclusion that if demonstrations in the particular sciences do not proceed from common principles, and if the principles of those sciences have something prior to them which is common, then it is clear that it is not the business of each science to prove its proper principles. For those prior principles through which the proper principles of the particular sciences can be proved are principles common to all; and the science which considers such common principles is proper to all, i.e., is related to things which are common to all, as those other particular sciences are related to things respectively proper to each. For example, since the subject of arithmetic is number, arithmetic considers things proper to number. In like manner, first philosophy, which considers all principles, has for its subject “being,” which is common to all. Therefore, it considers the things proper to being (which are common to all) as proper to itself.
lib. 1 l. 17 n. 5 Secundo, cum dicit: et namque scivit etc., ostendit praeeminentiam huiusmodi scientiae, quae considerat principia communia, scilicet primae philosophiae, ad alias. Semper enim oportet illud, per quod aliquid probatur, esse magis scitum vel notum. Qui enim scit aliquid ex superioribus causis, oportet quod sit magis intelligens illas causas, quia scivit ex prioribus simpliciter, cum non sciat ex causatis causas: quando enim aliquis scit ex causatis causas, tunc non intelligit ex prioribus et ex magis notis simpliciter, sed ex magis notis et prioribus quoad nos. Cum autem principia inferioris scientiae probantur ex principiis superioris, non proceditur ex causatis in causas, sed e converso. Unde oportet quod talis processus sit ex prioribus et ex magis notis simpliciter. Oportet ergo magis esse scitum quod est superioris scientiae, ex quo probatur id quod est inferioris, et maxime esse scitum id, quo omnia alia probantur, et ipsum non probatur ex alio priori. Et per consequens scientia superior erit magis scientia, quam inferior; et scientia suprema, scilicet philosophia prima, erit maxime scientia.	Secondly (76a19), he shows the pre-eminence of this science which considers common principles, namely, first philosophy, over the others. For that through which something is proved must always be scientifically more known or at least more known. For one whose science of something proceeds from higher causes must understand those causes even better, because his science proceeded from the absolutely prior, since it is not through something caused that he knows these causes. For when one’s knowledge of causes proceeds from caused things, he does not derive his knowledge from the absolutely prior and better known, but from what is prior and better known in reference to us. But when the principles of a lower science are proved by the principles of a higher science, the process is not from caused things to causes, but conversely. Therefore, such a process must go from the absolutely prior and from the absolutely more known. Thus, those items of a higher science that are used in proving matters of a lower science must be more known. Furthermore, that by which all other things are proved and which is itself not proved by anything prior must be the most known. Consequently, a higher science will be science in a fuller sense than a lower; and the highest science, namely, first philosophy, will be science in the fullest sense.
lib. 1 l. 17 n. 6 Tertio, ibi: sed demonstratio etc., redit ad principalem conclusionem: et dicit quod demonstratio non procedit in aliud genus, nisi sicut dictum est quod demonstratio geometriae procedit ad scientias inferiores; sicut sunt artes mechanicae, quae utuntur mensuris; aut speculativae, sicut scientiae quae sunt de visu, ut perspectivae, quae sunt de visuali; et similiter est de arithmetica in comparatione ad harmonicam, idest musicam.	Thirdly (7643), he returns to the principal conclusion and says that demonstration does not cross over into another genus except, as already mentioned, when a demonstration from geometry is applied to certain subordinate sciences, as the mechanical arts, which employ measurements, or the perspective arts, such as the sciences which deal with vision, as optics which deals with the visual. The same applies to arithmetic in relation to harmonics, i.e., music.

Lecture 18 Difference between principles and non-principles, common and proper principles

Latin	English
	Lecture 18 (7646-b22) DIFFERENCE BETWEEN PRINCIPLES AND NON-PRINCIPLES, COMMON AND PROPER PRINCIPLES
lib. 1 l. 18 n. 1 Postquam ostendit quod demonstratio non procedit ex principiis communibus sed ex propriis, hic ad evidentiam praemissorum determinat de principiis propriis et communibus. Et circa hoc duo facit. Primo, ostendit necessitatem huiusmodi determinationis, dicens quod difficile est cognoscere utrum sciamus ex principiis propriis (quod solum est vere scire) aut non ex propriis. Opinantur enim multi se scire, si habeant syllogismum ex aliquibus veris et primis. Sed hoc non est verum: immo oportet, ad hoc quod sciamus, quod principia sint proxima illis quae debent demonstrari (quae hic dicuntur prima, sicut et supra dicebantur extrema); vel oportet proxima esse primis principiis indemonstrabilibus.	After showing that demonstration does not proceed from common but from proper principles, the Philosopher, to elucidate this point, decides questions concerning proper and common principles. With respect to this he does two things. First (76a26), he shows the need for such a determination, saying that it is difficult to discern whether we know from proper principles, which alone is truly scientific knowing, or do not know from proper principles. For many believe that they know scientifically if they possess a syllogism composed of things true and first. But this is not so; indeed, to know in a scientific manner it is required that the principles be proximate to the things to be demonstrated—here they are called “first,” just as above they were called “extremes”—or proximate to the first indemonstrable principles.
lib. 1 l. 18 n. 2 Secundo, ibi: dico autem principia etc., determinat de principiis propriis et communibus. Et circa hoc duo facit: primo enim determinat de principiis propriis et communibus; secundo, ostendit qualiter ad huiusmodi principia se habeant demonstrativae scientiae; ibi: non contingere autem et cetera. Circa primum duo facit: primo, distinguit principia a non principiis; secundo, principia ad invicem; ibi: sunt autem quibus et cetera.	Secondly (76a3l), he determines concerning proper and common principles. And in regard to this he does two things. First he determines concerning common and proper principles. Secondly, he shows how the demonstrative sciences are related to such principles (77a10) [L. 20]. Concerning the first he does two things. First, he distinguishes principles from non-principles. Secondly, he distinguishes the principles from each other (76a37).
lib. 1 l. 18 n. 3 Circa primum duo facit. Primo, ostendit quae sint principia, dicens quod principia in unoquoque genere sunt illa quae, cum sint vera, tamen non contingit ea demonstrare vel simpliciter si sint principia prima, vel ad minus non est demonstrare in illa scientia in qua sumuntur ut principia. Dicit autem, cum sint vera, ad differentiam falsorum, quae non demonstrantur in aliqua scientia.	In regard to the first he does two things. First (76a3l), he shows what the principles are and says that the principles in any genus are those which, since they are true, happen not to be demonstrated, either not at all, if they are first principles, or at least not in that science in which they are accepted as principles. And he says, “the existence of which,” (i.e., whose truth), “cannot be proved,” to distinguish from the false which are not demonstrated in any science.
lib. 1 l. 18 n. 4 Secundo, ibi: quid quidem igitur etc., ostendit convenientiam et differentiam inter principia et non principia. Conveniunt enim principia cum non principiis in hoc, quod de utrisque oportet accipere, quasi supponendo quid significent, et prima, idest principia, et quae sunt ex his, idest quae ex principiis sumuntur: quia quod quid est proprie pertinet ad scientiam quae est de substantia, scilicet ad philosophiam primam, a qua omnes aliae hoc accipiunt.	Secondly (76a32), he shows the points of agreement and of difference between principles and non-principles. For principles agree with non-principles in the fact that with respect to each, i.e., with respect both to the first, namely, the principles, and to the things that proceed from them, i.e., things assumed from the principles, one must accept the supposition signify, because the quod quid est [that which the definition of what they signifies] of a thing pertains properly to the science concerned with substance, i.e., to first philosophy, from which all other sciences accept this.
Sed in hoc differunt principia, et quae sunt ex principiis, quia de principiis oportet accipere supponendo quod sunt; de aliis autem, quae sunt ex principiis, oportet demonstrare quia sunt. Sicut in mathematicis accipitur supponendo et quid est unitas, quae est principium, et quid est rectum, et quid est triangulus, quae non sunt principia, sed passiones: sed quod unitas sit, aut quod magnitudo sit, accipit mathematicus quasi principia; alia vero demonstrat, scilicet quae sunt ex principiis. Demonstrat enim triangulum aequilaterum et angulum rectum, et etiam hanc lineam rectam esse.	But they are unlike in the fact that in regard to the principles one must accept the supposition that they are, whereas in regard to things from principles one is required to demonstrate that they are. Thus in mathematics one accepts by supposing both what unity is (which is a principle) and what straight and triangle are (which are not principles but proper attributes): but the fact that unity is or that magnitude is, the mathematician accepts as principles; but the other things, namely, things that are from principles, he demonstrates. For he demonstrates that a triangle is equilateral, or that an angle is right, or even that this line is straight.
lib. 1 l. 18 n. 5 Deinde cum dicit: sunt autem quibus etc., distinguit principia ad invicem: et primo, principia propria a communibus; secundo, communia ad invicem; ibi: non est autem suppositio et cetera. Prima dividitur in duas; in prima, dividit principia propria et communia; in secunda, manifestat quoddam quod poterat esse dubium; ibi: quasdam tamen scientias et cetera.	Then (76a37) he distinguishes principles from one another. First, the proper from the common. Secondly, the common, one from the other (76b23) [L. 19]. The first is divided into two parts. First, he divides proper and common principles. Secondly, he settles something which might be doubtful (76b16).
lib. 1 l. 18 n. 6 Circa primum tria facit. Primo, ponit divisionem, dicens quod principiorum, quibus utimur in demonstrativis scientiis, alia sunt propria uniuscuiusque scientiae, alia vero communia. Et quia hoc posset videri contrarium ei, quod supra ostensum est, quia scientiae demonstrativae non procedunt ex communibus, ideo subiungit quod communia principia accipiuntur in unaquaque scientia demonstrativa secundum analogiam, idest secundum quod sunt proportionata illi scientiae. Et hoc est quod subdit exponens, quod utile est accipere huiusmodi principia in scientiis, quantum pertinet ad genus subiectum, quod continetur sub illa scientia.	In regard to the first he does three things. First (76a37), he lays down a division, saying that “of the principles which we use in demonstrative sciences, some are proper to each single science but others common.” Then, because this statement might seem contrary to what he established above, namely, that demonstrative sciences do not proceed from common principles, he hastens to add that “the common principles are taken in each demonstrative science according to an analogy,” i.e., as proportionate to that science. And this is what he means when by way of explanation he further states that “it is useful” to accept such principles in the sciences insofar as they pertain to the genus of the subject which is investigated in that science.
lib. 1 l. 18 n. 7 Secundo, ibi: propria principia etc., exemplificat de utrisque, dicens quod propria principia sunt, ut lineam esse huiusmodi, vel rectum. Tam enim subiecti quam passionis definitio in scientiis pro principio habetur. Communia vero principia sunt, ut, si ab aequalibus aequalia demas, quae remanent sunt aequalia, et aliae communes animi conceptiones.	Secondly (76a40), he gives examples of each, saying that proper principles are, for example, that a line or a right angle is such and such. For in the sciences the definitions, both of the subject and of the proper attribute, are held as principles. But common principles are, for example, that if equals be subtracted from equals, the remainders are equal, and other common conceptions in the mind.
lib. 1 l. 18 n. 8 Tertio, ibi: sufficiens autem est etc., ostendit quomodo praemissis principiis scientiae demonstrativae utantur. Et primo quidem de communibus dicit quod sufficiens est accipere unumquodque istorum communium, quantum pertinet ad genus subiectum, de quo est scientia. Idem enim faciet geometria, si non accipiat praemissum principium commune in sua communitate, sed solum in magnitudinibus, et arithmetica in solis numeris. Ita enim poterit concludere geometria, si dicat: si ab aequalibus magnitudinibus aequales auferas magnitudines, quae remanent sunt aequales; sicut si diceret: si ab aequalibus aequalia demas, quae remanent sunt aequalia. Et similiter dicendum est de numeris.	Thirdly (76a42), he shows how the demonstrative sciences use the aforesaid principles. First, in regard to the common principles, he says that “it suffices to accept each of those common ones,” so far as it pertains to the generic subject with which the science is concerned. For geometry does this if it takes the above-mentioned common principle not in its generality but only in regard to magnitudes, and arithmetic in regard to numbers. For the geometer will then be able to reach his conclusion by saying that if equal magnitudes be taken from equal magnitudes, the remaining magnitudes are equal, just as if he were to say that if equals are taken from equals, the remainders are equal. The same must also be said for numbers.
lib. 1 l. 18 n. 9 Secundo, ibi: sunt autem propria etc., ostendit qualiter demonstrativae scientiae utantur propriis principiis, dicens quod propria principia sunt quae supponuntur esse in scientiis, scilicet subiecta, circa quae scientia speculatur ea quae per se insunt eis. Sicut arithmetica considerat unitates, et geometria considerat signa, idest puncta et lineas. Praedictae enim supponunt esse et hoc esse, idest supponunt de eis, et quia sunt et quid sunt. Sed de passionibus supponunt praedictae scientiae quid significet unaquaeque; sicut arithmetica supponit quid est par, et quid est impar, aut quid est numerus quadratus aut cubicus; et geometria supponit quid est rationale in lineis. Dicitur enim linea rationalis, de qua possumus ratiocinari per lineam datam: huiusmodi autem est omnis linea commensurabilis lineae datae; quae vero est ei non commensurabilis, vocatur irrationalis vel surda. Similiter et geometria supponit quid est reflexum aut curvum. Sed praedictae scientiae demonstrant de omnibus praedictis passionibus quod sint per principia communia, et ex illis principiis, quae demonstrantur ex communibus. Et quod dictum est de geometria et arithmetica, intelligendum est etiam de astrologia.	Secondly (76b2), he shows how the demonstrative sciences employ proper principles. And he says that proper principles are things supposed in the sciences as existing, namely, the subjects, whose proper attributes are investigated in the sciences. In this way arithmetic considers unities, and geometry considers “signs,” i.e., points, and lines. For they suppose these things to be and to be this, i.e., they suppose of them that they are and what they are. But in regard to the proper attributes they suppose what each signifies. Thus arithmetic supposes what “even” is and what “odd” is and what a “square or cubic number” is. Similarly, geometry supposes what is rational in lines. (For a rational line is one about which we can reason, the line being given. For example, a rational line is any line commensurable with the given lines; but one which is incommensurable with it is called irrational and surd). In like manner, geometry supposes what a reflex or what a curved line are. However, these sciences demonstrate concerning all the above-mentioned proper attributes that they are, and they do so through common principles and principles demonstrated from the common principles. And what has been said 6f geometry and arithmetic should also be understood of astronomy.
Omnis enim scientia demonstrativa est circa tria: quorum unum est genus subiectum, cuius per se passiones scrutantur; et aliud est communes dignitates, ex quibus sicut ex primis demonstrat; tertium autem passiones, de quibus unaquaeque scientia accipit quid significent.	For every demonstrative science is concerned with three things: one is the generic subject whose per se attributes are investigated; another is the common (axioms) dignities from which, as from basic truths, it demonstrates; the third are the proper attributes concerning which each science supposes what their names signify.
lib. 1 l. 18 n. 10 Deinde cum dicit: quasdam tamen scientias etc., manifestat quoddam, quod poterat esse dubium. Quia enim dixerat quod scientiae supponunt de principiis quia sunt, de passionibus quid sunt, de subiectis autem utrumque, posset aliquis credere quod oporteret specialem fieri mentionem de omnibus istis. Unde hoc removet, dicens quod nihil prohibet quasdam scientias despicere quaedam praedictorum, idest non facere mentionem expressam de praemissis, sicut quandoque non facit mentionem de hoc quod supponat genus subiectum esse, si sit manifestum quod sit, quia non est similiter manifestum de omnibus quod sint, sicut quod sit numerus, et quod sit calidum vel frigidum: quorum unum est propinquum rationi, alterum sensui. Similiter et quaedam scientiae non supponunt de passionibus quid significent, expressam mentionem de eis faciendo. Sicut etiam non oportet quod de communibus principiis semper scientiae faciant mentionem, quia nota sunt. Nihilominus tamen, tria praedicta naturaliter sunt in qualibet scientia supponenda.	Then (76b16) he clarifies something about which there might be doubt. For since he had said that the sciences suppose concerning the principles that they are and concerning the proper attributes what they are, but concerning the subject both that it is and what it is, someone might believe that he should have made special mention of all these. Hence he removes this by saying that nothing hinders certain sciences from neglecting some of the aforesaid, i.e., from making express mention of them, as for example, not mentioning that it takes the existence of its generic subject for granted, if it is already obvious that it does exist. For we not have the same evidence in all cases that they do exist, as we do in the case of number and in the case of hot and cold, the one being close to reason and the other to sense. Again, certain sciences do not suppose what the proper attributes signify in the sense of making express mention of them, just as they do not think it necessary always to make express mention of the common principles, because they are known. Be that as it may, the three above-mentioned items are naturally to be supposed in each science.

Lecture 19 How common principles differ from one another

Latin	English
	Lecture 19 (76b23-77a9) HOW COMMON PRINCIPLES DIFFER FROM ONE ANOTHER
lib. 1 l. 19 n. 1 Postquam divisit Aristoteles principia communia a propriis, hic distinguit communia principia ad invicem. Et dividitur in partes tres: in prima, ponit distinctionem communium principiorum ad invicem; in secunda, ostendit differentiam definitionis a quodam genere principiorum communium, ibi: termini igitur non etc.; in tertia, excludit quemdam errorem, ibi: species quidem igitur et cetera. Circa primum duo facit: primo, distinguit communes animi conceptiones a petitionibus, sive suppositionibus; secundo, petitiones et suppositiones ad invicem, ibi: quaecunque quidem igitur et cetera.	After dividing common principles from proper principles, Aristotle now distinguishes among the common principles. His treatment f Is into three parts. In the first he lays down a distinction among the common principles. In the second he shows the difference between a definition and a certain genus of common principles (76b35). In the third he excludes an error (77a5). Concerning the first he does two things. First, he distinguishes the common conceptions in the mind from postulates or suppositions. Secondly, he distinguishes among the latter (76b27).
lib. 1 l. 19 n. 2 Circa primum considerandum est quod communes animi conceptiones habent aliquid commune cum aliis principiis demonstrationis, et aliquid proprium. Commune quidem habent, quia necesse est tam ista, quam alia principia per se esse vera. Proprium autem est horum principiorum quod non solum necesse est ea per se vera esse, sed etiam necesse est videri quod per se sint vera. Nullus enim potest opinari contraria eorum.	With respect to the first (76b23) it should be noted that the common. conceptions in the mind have something in common with the other principles of demonstration and something proper: something common, because both they and the others must be true in virtue of themselves. But what is proper to the former is that it is necessary not only that they be true of themselves but that they be seen to be such. For no one can think their contraries. He says, therefore, that that principle of which it is not only required that it be in virtue of itself but further required that it be seen, namely, a common conception in the mind or a dignity, is neither a postulate nor a supposition.
lib. 1 l. 19 n. 3 Dicit ergo quod illud principium, quod necesse est non solum per seipsum esse, sed etiam ulterius necesse est, ipsum videri, scilicet communis animi conceptio vel dignitas, non est neque petitio neque suppositio. Quod sic probat. Petitio et suppositio exteriori ratione confirmari possunt, idest argumentatione aliqua. Sed communis animi conceptio non est ad exterius rationem, quia non potest probari per aliquam argumentationem, sed est ad eam, quae est in anima, quia lumine naturalis rationis statim fit nota. Et quod non sit ad exterius rationem patet, quia non fit syllogismus ad probandas huiusmodi communes animi conceptiones. Et quod huiusmodi non sunt notae per exteriorem rationem, sed per interiorem, probat per hoc, quod exteriori rationi potest instari vel vere vel apparenter: interiori autem rationi non est possibile semper instari. Et hoc ideo, quia nihil est adeo verum, quin voce possit negari. Nam et hoc principium notissimum, quod non contingat idem esse et non esse, quidam ore negaverunt. Quaedam autem adeo vera sunt, quod eorum opposita intellectu capi non possunt; et ideo interiori ratione eis obviari non potest, sed solum exteriori, quae est per vocem. Et huiusmodi sunt communes animi conceptiones.	He proves this in the following way: A postulate and a supposition can be confirmed by a reason from without, i.e., by some argumentation; but a common conception in the mind does not bear on a reason from without (because it cannot be proved by any argument), but bears on that reason which is in the soul, because it is made known at once by the natural light of reason. That it does not bear on any reason from without is shown by the fact that a syllogism is not formed to prove such common conceptions of the mind. Furthermore, that these are not made known by an outward reason but by the inward he proves by the fact that it is possible to contest an outward reason, either truly or apparently, but it is not always possible to do so with the inward reason. This is so because nothing is so true that it cannot be denied orally. (For even this most evident principle that the same thing cannot be and not be has been orally denied by some). On the other hand, some things are so true that their opposites cannot be conceived by the intellect. Therefore, they cannot be challenged in the inward reason but only by an outward reason which is by the voice. Such are the common conceptions in the mind.
lib. 1 l. 19 n. 4 Deinde, cum dicit: quaecunque igitur etc., distinguit suppositiones et petitiones ad invicem. Sciendum tamen est, quod aliquid commune habent, et in aliquo differunt. Hoc quidem commune est eis, quod cum sint demonstrabilia, tamen demonstrator accipit ea non demonstrans, et praecipue, quia non sunt demonstrabilia per suam scientiam, sed per aliam, ut supra dictum est. Unde et inter immediata principia computantur, quia demonstrator utitur eis absque medio, eo quod non habeant medium in illa scientia.	Then (76b27) he distinguishes suppositions and postulates from one another. Here, too, it should be noted that they have something in common and something in which they differ. What is common to them is that although they can be demonstrated, the demonstrator assumes them without demonstrating, chiefly because they are not demonstrable by his own science but by another, as explained above. Hence they are reckoned among the immediate principles because the demonstrator uses them without a middle, since they do not have a middle in that science.
Differunt autem ad invicem: quia si quidem talis propositio sit probabilis addiscenti, cui fit demonstratio, dicitur suppositio. Et sic suppositio dicitur non simpliciter, sed ad aliquem. Si vero ille nec sit eiusdem opinionis, neque contrariae, oportet quod demonstrator hoc ab eo petat, et tunc dicitur petitio. Si autem sit contrariae opinionis, tunc erit quaestio, de qua oportet disputari inter eos. Hoc tamen omnibus commune est, quod unoquoque eorum utitur demonstrator non demonstrans, cum sit demonstrabile.	Yet they do differ, because if such a proposition is accepted as reasonable by the pupil to whom a demonstration is being made, it is called a “Supposition.” In that case it is not called a supposition absolutely, but relative to him. But if the pupil has no reason for or against the proposition, the demonstrator must request him to admit it as reasonable, and then it is called a “postulate.” However, if the pupil has a contrary opinion, then it will be a “question,” which must be settled between them. At any rate, what is common to all of them is that the demonstrator uses each of thein without demonstrating, although they are demonstrable.
lib. 1 l. 19 n. 5 Deinde, cum dicit: termini igitur non sunt etc., distinguit definitiones a suppositionibus per duas rationes; quarum secunda incipit ibi: amplius petitio et cetera. Circa primum duo facit: primo, ponit rationem, quae talis est: omnis petitio, vel suppositio dicit aliquid esse vel non esse; termini autem, idest definitiones, non dicunt aliquid esse vel non esse; termini ergo non sunt suppositiones neque petitiones, per se sumpti. Sed in propositionibus assumpti sunt suppositiones; ut cum dicitur, homo est animal rationale mortale. Sed terminos per se sumptos, oportet solum intelligere; intelligere autem non est supponere, sicut nec audire. Sed illa supponuntur quorumcunque existentium, idest ex quibuscumque existentibus fit conclusio, in eo quod illa sunt, idest propter praemissa.	Then (76b35) he distinguishes definitions from suppositions with two reasons, the second of which begins at (77a3). In regard to the first he does two things. First, he presents this reason: Every postulate or supposition declares something to be or not to be; but terms, i.e., definitions, do not declare that something is or is not. Terms, therefore, taken by themselves, are neither postulates nor suppositions. But when they are employed in propositions they are suppositions, as in the statement, “Man is a rational mortal animal.” But terms taken by themselves are only understood; and understanding is not supposing, any more than hearing is. Rather supposition bears on things that exist such that a conclusion is made from them in regard to what they are, i.e., in virtue of the premises.
lib. 1 l. 19 n. 6 Secundo, ibi: neque geometra etc., excludit quamdam dubitationem. Dicebant enim quidam quod geometra falsa suppositione utebatur, cum diceret lineam esse unius pedis, quae non est unius pedis; aut lineam descriptam in pulvere esse rectam, quae non est recta. Sed ipse dicit quod geometra non supponit falsum propter hoc. Cum enim geometra nihil demonstret de particularibus, sed de universalibus, ut supra dictum est; hae autem lineae sunt quaedam particularia; manifestum est quod de his lineis nihil demonstrat, neque etiam ex eis, sed utitur eis ut exemplis universalium, quae per haec exempla intelliguntur, de quibus et ex quibus demonstrat.	Secondly (76b39), he excludes a doubt. For there were some who said that a geometer uses a false supposition when he says that a line not one foot long is one foot long, or that a line traced in the sand is straight when it really is not. But he says that a geometer does not on this account suppose something false. For since the geometer demonstrates nothing of singulars but of universals, as explained above, and these lines are singulars, it is obvious that he is not demonstrating anything about these lines or from these lines; rather he is using them as examples of the universals (which are understood by examples) from which and about which he demonstrates.
lib. 1 l. 19 n. 7 Deinde, cum dicit: amplius petitio etc., ponit secundam rationem, quae talis est: omnis suppositio vel petitio est in toto vel in parte, idest est propositio universalis vel particularis; sed definitiones neutrum horum sunt, quia in eis nihil ponitur sive praedicatur, neque universaliter, neque particulariter; ergo et cetera.	Then (77a3) he presents the second reason which is this: Every supposition or postulate is in whole or in part, i.e., is either a universal proposition or a particular. But definitions are neither of these, because nothing is placed or predicated in them universally or particularly. Therefore...
lib. 1 l. 19 n. 8 Deinde, cum dicit: species quidem esse etc., ostendit ex praemissis quod non est necessarium ponere ideas, ut Plato posuit. Ostensum est enim supra quod demonstrationes de universalibus sunt, et hoc modo sunt de sempiternis. Non igitur necesse est ad hoc quod demonstratio sit, species esse, idest ideas, aut quodcunque unum extra multa, sicut ponebant Platonici mathematica separata cum ideis, ut sic demonstrationes possint esse de sempiternis. Sed necessarium est esse unum in multis et de multis, si demonstratio debet esse, quia non erit universale, nisi sit unum de multis; et si non sit universale, non erit medium demonstrationis; ergo nec demonstratio.	Then (77a5) he shows from the foregoing that it is not necessary to posit “Ideas,” as Plato did. For it was shown above that demonstrations are concerned with universals and in that sense with eternal things. Therefore, it is not necessary for the validity of demonstration that there be “Forms,” i.e., “Ideas,” or a “one outside the many,” as Plato posited separated mathematical beings along with Ideas in order that thereby demonstrations might bear on eternal things. What is required is that there be “one in many and about many,” if there is to be demonstration, because it will not be universal unless it is “one about many.” And if it is not universal, there will not be a middle of demonstration and, consequently, no demonstration.
Et quod oporteat medium demonstrationis esse universale, patet per hoc quod oportet medium demonstrationis esse unum et idem de pluribus praedicatum non aequivoce, sed secundum rationem eamdem: quod est ratio universalis. Si autem aequivocum esset, posset accidere vitium in arguendo.	That the middle of a demonstration must be universal is plain from the fact that the middle of a demonstration must be some one same thing predicated of many not equivocally but according to the same aspect, which is a universal aspect. But if it should happen to be equivocal, a defect in reasoning would occur.