Authors/Thomas Aquinas/posteriorum/L1/Lect22

Lecture 22 Each science has its own deceptions and areas of ignorance

Latin	English
	Lecture 22 (77b16-78a21) EACH SCIENCE HAS ITS OWN DECEPTIONS AND AREAS OF IGNORANCE
lib. 1 l. 22 n. 1 Postquam ostendit philosophus quod in qualibet scientia sunt propriae interrogationes, responsiones et disputationes; hic ostendit quod in qualibet scientia sunt propriae deceptiones et ignorantiae. Et dividitur in partes duas: in prima, movet quasdam quaestiones; in secunda, solvit; ibi: secundum geometriam vero et cetera.	After showing that each science has its own questions, responses and disputations, the Philosopher shows that each science has its own deceptions and errors. And his treatment is divided into two parts.
lib. 1 l. 22 n. 2 Ponit ergo primo tres quaestiones, quarum prima est. Cum sint quaedam interrogationes geometricae, ut ostensum est, nonne sunt etiam quaedam non geometricae? Et quod quaeritur de geometria, potest de qualibet alia scientia quaeri.	In the first he raises certain questions. In the second he solves them (77b21). Accordingly, he poses three questions, the first of which (77b16) is this: Since there are geometric questions, as we have shown, are there not also non-geometric ones? And what is asked of geometry can be asked of every other science.
Secundam quaestionem ponit; ibi: et secundum unamquamque etc., quae talis est. Utrum interrogationes quae sunt secundum ignorantiam, quae est in unaquaque scientia, possint dici geometricae, et similiter alicui alii scientiae propriae? Dicuntur autem interrogationes secundum ignorantiam alicuius scientiae, quando interrogatur de his, quae sunt contra veritatem scientiae illius.	Then (77b17) he poses the second question, namely: May questions which arise from ignorance bearing on some particular science be called geometric; and likewise for questions proper to any other science? (Questions arising from ignorance bearing on some science are those which ask about matters contrary to the truths of that science).
Tertiam quaestionem ponit; ibi: et utrum secundum ignorantiam etc., quae talis est. In unaquaque quidem scientia accidit decipi per aliquem syllogismum, quem vocat secundum ignorantiam. Contingit autem per aliquem syllogismum deceptionem accidere dupliciter: uno modo, quia peccat in materia, procedens ex falsis; alio modo, quia peccat in forma, non servando debitam figuram et modum. Et est differentia inter hos modos duos: quia ille qui peccat in materia, syllogismus est, cum observentur omnia, quae ad formam syllogismi pertinent. Ille autem qui peccat in forma non est syllogismus, sed paralogismus, idest apparens syllogismus. In dialecticis quidem utroque modo contingit deceptionem fieri. Unde et in I topicorum Aristoteles facit mentionem de litigioso, qui est syllogismus, et de peccante in forma, qui non est syllogismus, sed apparens. Est ergo quaestio, utrum syllogismus ignorantiae, qui fit in scientiis demonstrativis, sit syllogismus ex oppositis scientiae, idest ex falsis procedens, aut paralogismus, scilicet peccans in forma: qui non est syllogismus, sed apparens.	Then (77b19) he poses the third question, namely: In each science it is possible to be deceived by a syllogism which he calls “according to ignorance.” But deception through a syllogism can occur in two ways: in one way when it fails as to form, not observing the correct form and mode of a syllogism. In another way, when it fails in matter, proceeding from the false. Now there is a difference between these two ways, because one that fails in matter is still a syllogism, since everything is observed that pertains to the form of a syllogism. But one that fails in form is not even a syllogism, but a paralogism, i.e., an apparent syllogism. In dialectics, deception can occur in both these ways. Hence in Topics I Aristotle speaks of the contentious, which is a syllogism, and of the one defective in form which is not a syllogism but an apparent one. Hence the question is this: Whether or not a syllogism of ignorance which is used in the demonstrative sciences is a syllogism based on matters opposed to the science, i.e., one that proceeds from false premises, or a paralogism, namely, one that fails in form, which is not a syllogism, but an apparent one?
lib. 1 l. 22 n. 3 Deinde cum dicit: secundum geometriam etc., solvit praedictas quaestiones: et primo, solvit primam; secundo, secundam; ibi: de geometria autem etc.; tertio, tertiam; ibi: in doctrinis autem et cetera. Dicit ergo primo quod interrogatio omnino non geometrica est illa, quae omnino fit ex alia arte, sicut ex musica. Ut si quaeratur in geometria, utrum tonus possit dividi in duo semitonia aequalia; talis interrogatio est omnino non geometrica: quia est ex his, quae nullo modo ad geometriam pertinent.	Then (77b21) he solves these questions. First, he solves the first Secondly, the second (77b23). Thirdly, the third (77b27). He says therefore first (77b21) that a completely non-geometric question is one which is formed entirely from another art, say, music. For example, if one asks in geometry whether a tone could be divided into two equal semi-tones, such a question would be entirely non-geometric, because it concerns matters which do not pertain at all to geometry.
lib. 1 l. 22 n. 4 Deinde cum dicit: de geometria autem etc., solvit secundam quaestionem dicens quod interrogatio de geometria, idest de his quae pertinent ad geometriam, cum interrogatur de aliquo quod est contra veritatem geometriae (sicut si fiat quaestio de hoc quod est parallelas subire, idest lineas aeque distantes concurrere), est quodammodo geometrica et quodammodo non geometrica. Sicut enim arrhythmon, idest quod est sine rhythmo vel sono, dupliciter dicitur, uno modo, quod nullo modo habet sonum, ut lana, alio modo, quod habet pravum sonum, sicut Campana non bene sonans; ita et interrogatio non geometrica dicitur dupliciter. Uno modo, quia est omnino non geometrica, quasi nihil habens geometriae, sicut quaestio de musica proposita. Alio modo, quia prave habet id quod geometriae est; quia videlicet habet contrarium geometricae veritati. Ista ergo interrogatio, quae est de concursu linearum aeque distantium, non est non geometrica primo modo, cum sit de rebus geometricis, sed secundo modo, quia prave habet id quod geometriae est. Et ignorantia haec, scilicet quae est in prave utendo principiis geometriae, contraria est veritati geometriae.	Then (77b23) he solves the second question, saying that a question about geometry, i.e., about matters pertinent to geometry, when a person is asked about something which is against the truth of geometry (for example, if the questions concerned parallels meeting, i.e., equidistant lines coming together), would be geometric in one sense and non-geometric in another. For just as arrythmon, i.e., without rhythm or sound, can be taken in two senses: in one sense for that which has no sound at all, as wool, and in another sense for that which does not give a good sound, as a poorly-sounding bell, so a question is called non-geometric in two ways. In one way, because it is completely alien to geometry, as a question about music. In another way, because it mistakenly holds something in the field of geometry, namely, because it holds something contrary to geometric truth. Therefore, the question about the convergence of parallel lines is non-geometric not in the first way, since it touches on a point of geometry, but in the second way, because it is mistaken about some point of geometry. Such ignorance, namely, which consists in wrongly using the principles of geometry is contrary to the truth of geometry.
lib. 1 l. 22 n. 5 Deinde cum dicit: in doctrinis autem etc., solvit tertiam quaestionem. Et circa hoc duo facit: primo, ostendit quod in demonstrativis scientiis non sit paralogismus in dictione; secundo, quod non sit paralogismus extra dictionem; ibi: non oportet autem et cetera. Cum autem secundum sex locos sophisticos fiat paralogismus in dictione, ex his accipit unum, scilicet paralogismum qui fit secundum aequivocationem, ostendens quod talis paralogismus in scientiis demonstrativis esse non potest: de quo tamen magis videtur. Dicit ergo quod in doctrinis non sit paralogismus, scilicet syllogismus peccans in forma, sicut in dialecticis. In demonstrativis enim oportet medium idem semper esse dupliciter, idest ad duo extrema comparari: quia et de medio maior extremitas universaliter praedicatur, et medium iterum universaliter praedicatur de minori extremitate. Sed quod praedicatur, non dicitur omne, idest signum universale non apponitur ad praedicatum.	Then (77b27) he solves the third question. And he does two things. First, he shows that in demonstrative sciences there is no paralogism in language. Secondly, nor apart from language (77b34). Now although paralogism. in language occurs in any of six sophistical ways, he takes one of them, namely, the paralogism which proceeds by way of equivocation, and shows that such,a paralogism cannot occur in demonstrative sciences, being easier to detect. He says, therefore, that “formal paralogism,” i.e., a syllogism defective in form, as in dialectics, “does not occur in the disciplines.” For in a demonstrative syllogism the middle must always be the same in two ways, i.e., the same middle must be compared to the two extremes: for the major extreme is predicated universally of the middle, and the middle is predicated universally of the minor extreme, even though when it is predicated, we do not say “every,” i.e., the sign of universality is not applied to the predicate.
In fallacia vero aequivocationis est quidem idem medium secundum vocem, non autem secundum rem. Et ideo quando in voce proponitur, latet, sed si ad sensum demonstretur, non potest ibi esse aliqua deceptio. Sicut hoc nomen circulus aequivoce dicitur de figura et de poemate. In rationibus ergo, idest in argumentationibus, latet, idest deceptio potest accidere; ut si dicatur: omnis circulus est figura; poema Homeri est circulus; ergo poema Homeri est figura. Si vero describatur ad sensum circulus, nulla potest esse deceptio: manifestum enim erit quod carmina non sunt circulus.	But in the fallacy of equivocation the middle is the same according to vocal sound but not according to reality. Consequently, it escapes notice when it is proposed orally; but if it is demonstrated to the senses, the deception cannot succeed. For example, the name “circle” is said equivocally of a figure and a poem. Therefore, in reasons, i.e., in argumentations, the ambiguity may go unnoticed, i.e., deception is possible, as for example, if one were to say: “Every circle is a figure; Homer’s poem is a circle: therefore, Homer’s poem is a figure.” But if a circle is drawn for someone to look at, there cannot be deception. For it will be obvious that songs are not circles.
Sicut autem haec deceptio excluditur per hoc quod medium demonstratur ad sensum, ita et in demonstrativis excluditur per hoc quod medium demonstratur ad intellectum. Cum enim aliquid definitur, ita se habet ad intellectum, sicut id quod sensibiliter describitur se habet ad visum. Et ideo dicit quod haec, scilicet definita, in demonstrativis scientiis sunt quae videntur in intellectu. In demonstrationibus autem semper proceditur ex definitionibus. Unde non potest ibi esse deceptio secundum fallaciam aequivocationis: et multo minus secundum alias fallacias in dictione.	Now just as in this case the deception is prevented by presenting the middle to the senses, so in demonstrative syllogisms, deception is prevented by the fact that the middle is shown to the intellect. For when something is defined, it is as plain to the intellect as something sensibly drawn is plain to the sight. Hence he says that in demonstrative syllogisms “these,” i.e., things defined, “are seen by the intellect.” But in demonstrations one always proceeds from definitions. Hence, deception through equivocation has no place there, much less through any of the other fallacies of language.
lib. 1 l. 22 n. 6 Deinde cum dicit: non oportet autem etc., ostendit quod non potest fieri paralogismus in demonstrativis secundum fallaciam extra dictionem. Et quia huiusmodi paralogismis frequenter obviatur ferendo instantiam, per quam ostenditur defectus in forma syllogizandi; ideo primo ostendit qualiter ferenda esset instantia in demonstrativis; secundo, ostendit quod in eis non potest esse paralogismus secundum fallaciam extra dictionem; ibi: contingit autem quosdam et cetera.	Then (77b34) he shows that in demonstrative syllogisms paralogism according to fallacy outside of language cannot occur. And because a paralogism of this kind is frequently challenged by citing an objection, through which the defect in the form of syllogizing is shown: therefore: First, he shows how an objection should be presented in demonstrative matters. Secondly, he shows that in demonstrative matters there cannot be paralogism according to fallacy outside the language (77b40).
Dicit ergo primo, quod non oportet in demonstrativis ferre instantiam in ipsum, idest in aliquem paralogismum, sumendo aliquam propositionem inductivam, idest particularem: nam inductio ex particularibus procedit, sicut syllogismus ex universalibus. Et hoc ideo est, quia in demonstrativis non sumitur propositio, nisi quae est in pluribus: nisi enim sit in pluribus, non erit in omnibus; oportet autem syllogismum demonstrativum ex universalibus procedere. Unde manifestum est quod neque instantia potest esse in demonstrativis, nisi universalis, quia eaedem sunt propositiones et instantiae. Tam enim in dialecticis quam in demonstrativis, illud quod sumitur ut instantia, postea sumitur ut propositio ad syllogizandum contra illum qui proponebat.	He says therefore first (77b34) that “one should not bring an objection against it,” i.e., against a paralogism. by citing an inductive, i.e., particular, proposition. (For an induction proceeds from particulars as a syllogism proceeds from universals). The reason for this is that in demonstrative matters no proposition is admitted unless it is verified in the greater number of cases, for if it is not verified in the greater number it will not be in all. But a demonstrative syllogism must proceed from universals. Therefore, it is obvious that in demonstrative matters an objection must be universal, because the propositions and the objections must be the same. For in dialectical and in demonstrative matters that which is taken as an objection is later used as a proposition to syllogize against the one who proposed.
lib. 1 l. 22 n. 7 Deinde cum dicit: contingit autem quosdam etc., ostendit quod in demonstrativis non accidit deceptio per paralogismum extra dictionem. Et sicut supra ostenderat quod non est paralogismus in dictione in demonstrativis, ostendendo de uno, scilicet de paralogismo secundum fallaciam aequivocationis; ita hic ostendit quod in demonstrativis non est paralogismus extra dictionem, ostendendo de uno, qui fit secundum fallaciam consequentis. Patet enim quod secundum alias fallacias extra dictionem non potest esse paralogismus in demonstrativis. Neque enim secundum accidens, cum demonstrationes procedant ex his quae sunt per se; neque secundum quid et simpliciter, cum ea quae in demonstrationibus sumuntur, sint universaliter, et semper, et non secundum quid.	Then (77b40) he shows that in demonstrative matters deception through paralogisms outside of language does not occur. And just as above he showed that there was no paralogism in language in demonstrative matters by showing it for one, namely, for the paralogism which employs the fallacy of equivocation, so now he shows that in demonstrative matters there is not paralogism outside of language by showing it of the one which relies on fallacy of consequent. For it is obvious that paralogism. in demonstrative matters cannot occur according to the other fallacies outside of language: not the fallacy according to accident, because demonstration proceeds from things that are per se; nor according to qualified and absolute, because the statements used in demonstrations are taken universally and always, and without qualification.
Circa hoc ergo duo facit: primo, ostendit qualiter fiat paralogismus secundum fallaciam consequentis; secundo, quod ex hoc modo non accidit deceptio in demonstrativis; ibi: aliquando quidem et cetera.	Therefore, he does two things in regard to his thesis. First, he shows how paralogism according to fallacy of consequent works. Secondly, that deception does not take place in this manner in demonstrative matters (78a5).
lib. 1 l. 22 n. 8 Dicit ergo primo quod quosdam contingit non syllogistice dicere, idest non servare formam syllogismi, propter hoc, quod accipiunt utrisque inhaerentia, idest quia accipiunt medium affirmative praedicatum de utroque extremorum; quod est syllogizare in secunda figura ex duabus propositionibus affirmativis; quod facit fallaciam consequentis. Sicut fecit quidam philosophus nomine Caeneus ad ostendendum quod ignis sit in multiplicata analogia, idest quod in maiori quantitate generatur ignis, quam fuerit corpus ex quo generatur: eo quod ignis, cum sit rarissimum corpus, per rarefactionem ex aliis corporibus generatur. Unde oportet quod materia prioris corporis sub maioribus dimensionibus extendatur, formam ignis assumens. Ad hoc autem probandum utebatur tali syllogismo: quod generatur in multiplicata analogia, cito generatur; sed ignis cito generatur; ergo ignis generatur in multiplicata analogia.	He says therefore first (77b40) that “illogical arguments do sometimes occur,” i.e., do not observe syllogistic form, “because they admit both inherences,” i.e., they accept a middle predicated affirmatively of each extreme, which is the same as syllogizing in the second figure but employing two affirmative propositions, thus committing the fallacy of consequent. This is what the philosopher Caeneus did in order to show that “fire is in multiple proportion,” i.e., that fire is generated in greater quantity than was the body from which it is generated, on the ground that fire, being the most rarified of bodies, is generated from other bodies through rarefaction. Hence it is required that the matter of the previous body be spread out under larger dimensions when assuming the form of fire. To prove this he used the following syllogism: “Whatever is generated in multiplied proportion is generated quickly; but fire is generated quickly: therefore, fire is generated in multiplied proportion.”
lib. 1 l. 22 n. 9 Deinde cum dicit: aliquando quidem igitur etc., ostendit quod per hunc modum syllogizandi non accidit deceptio in demonstrativis scientiis. Et circa hoc duo facit: primo, manifestat quod ex hoc modo syllogizandi non semper accidit deceptio, dicens quod aliquando, secundum praedictum modum arguendi, non contingit syllogizare ex acceptis, quando scilicet termini non sunt convertibiles. Non enim sequitur, si omnis homo est animal, quod quidquid est animal sit homo. Aliquando vero contingit syllogizare, scilicet in terminis convertibilibus. Sicut enim sequitur: si est homo, est animal rationale mortale; ita etiam sequitur e converso quod, si est animal rationale mortale, est homo. Sed tamen non videtur quod sequatur syllogistice, quia non servatur debita forma syllogismi.	Then (78a5) he shows that deception arising from this form of reasoning does not occur in demonstrative sciences. In regard to this he does two things. First, he makes it clear that this form of syllogizing does not always end in deception, saying that according to this form of arguing there are cases when one cannot syllogize from the premises, namely, when the terms are not convertible-for it does not follow, if every man is an animal, that every animal is a man. But now and then there are cases when one can syllogize, namely, when the terms are convertible. For just as it follows that if a thing is a man, it is a rational mortal animal, so conversely, if a thing is a rational mortal animal, it is a man. However, it does not seem to follow syllogistically, because the due form of a syllogism is not observed.
lib. 1 l. 22 n. 10 Secundo cum dicit: si autem esset impossibile etc., ostendit quod in demonstrativis scientiis contingit praedicto modo syllogizari absque deceptione. Et hoc ostendit tripliciter. Primo sic. Secundum praedictum modum syllogizandi accidit deceptio ex eo, quod non convertitur consequentia, quae putatur converti. In quo non accideret deceptio, si quemadmodum conclusio est vera, ita et praemissae sint verae: tunc enim in convertendo non accidet deceptio. Sicut si dicam de Socrate: Socrates est homo; ergo Socrates est animal; nulla deceptio falsitatis sequitur, sicut si e converso arguatur sic: est animal; ergo est homo.	Secondly (78a7), he shows that it is possible to syllogize in the abovementioned manner without deception. And he shows this in three ways, the first of which is this: According to the above-mentioned manner of syllogizing, deception occurs because the consequence which was assumed convertible is not converted. But in this situation deception would not occur, if to the extent that the conclusion is true, so are the premises true: for there will then be no deception in converting. For if it is stated of Socrates that he is a man, therefore he is an animal; no deception or falsity follows: but it does if it is argued conversely that he is an animal, therefore he is a man.
Sed si praemissa est falsa, conclusione existente vera, tunc in convertendo accidit deceptio. Sicut si dicam: si asinus est homo, est animal; ergo si est animal, est homo. Si ergo impossibile esset ex falsis ostendere verum, et semper oporteret verum ostendi ex veris, tunc facile esset resolvere conclusionem in principia absque deceptione; quia nulla falsitas esset, si ex conclusione inferretur aliqua praemissarum. Tali enim suppositione facta, converterentur de necessitate conclusio et praemissa, quantum ad veritatem. Sicut enim praemissa existente vera, conclusio est vera, ita et e converso. Sit enim, quod a sit; et hoc posito, sequatur ea esse de quibus certum est mihi quod sunt vera, sicut b. Unde cum utrumque sit verum, ex hoc etiam, scilicet ex b, potero iterum inferre a. Sic ergo una ratio est, quare deceptio non accidit in demonstrativis scientiis per fallaciam consequentis, quia in demonstrativis scientiis impossibile est syllogizari verum ex falsis, sicut ostensum est supra.	But if the premise is true and the conclusion false, then deception occurs in converting. For example, if I were to say: “If an ass is a man, it is an animal; therefore, if it is an animal, it is a man.” Consequently, if it were impossible to deduce the true from the false, and it were always necessary to deduce the true from the true, then it would be easy to analyze a conclusion into its principles without deception, because there would be no falsity if either of the premises were to be inferred from the conclusion. On this supposition the conclusion and premises would be converted of necessity as to truth. For just as the conclusion is true if the premise is true, so vice versa. For suppose that A is, and granting this, suppose that the same things follow as are known by me to be true, say B. Hence since both are true, I can then also infer A from B. And so, one reason why deception through fallacy of consequent does not occur in demonstrative sciences is that it is impossible in demonstrative sciences for the true to be syllogized from the false, as we have explained above.
lib. 1 l. 22 n. 11 Secundam rationem ponit; ibi: convertuntur autem magis et cetera. In terminis enim convertibilibus non accidit deceptio secundum fallaciam consequentis, eo quod in his consequentia convertitur. Illa vero, quae sunt in mathematicis, idest in demonstrativis scientiis, ut plurimum sunt convertibilia, quia non recipiunt pro medio aliquod praedicatum per accidens, sed solum definitiones, quae sunt demonstrationis principia, ut supra dictum est. Et in hoc differunt ab his, quae sunt in dialogis, idest in dialecticis syllogismis, in quibus frequenter recipiuntur accidentia.	Then (78a10) he gives the second reason. For in cases of convertible terms, deception through fallacy of consequent does not occur, because in those cases the consequent is converted. Now the things which are used in mathematical, i.e., in demonstrative sciences, are for the most part convertible, because these sciences do not admit as a middle anything predicated per accidens, but only definitions which are principles of demonstration, as we have explained. And this is their point of difference from the statements in dialogues, i.e., in dialectical syllogisms, in which accidens are frequently admitted.
lib. 1 l. 22 n. 12 Tertiam rationem ponit ibi: augentur autem etc., quae talis est. In demonstrativis scientiis sunt determinata principia, ex quibus proceditur ad conclusiones. Unde ex conclusionibus potest rediri in principia, sicut ex determinato in determinatum. Quod autem demonstrationes ex determinatis principiis procedant, ex hoc ostendit, quia demonstrationes non augentur per media, idest in demonstrationibus non assumuntur plura media ad unam conclusionem demonstrandam. Quod intelligendum est in demonstrationibus propter quid, de quibus loquitur. Unius enim effectus non potest esse nisi una propria causa, propter quam est.	Then (78a13) he gives the third reason and it is this: In demonstrative sciences there are determinate principles from which one proceeds to the conclusions. Hence it is possible to return from the conclusions to the principles as from something determinate to something determinate. That demonstrations do proceed from determinate principles he shows by the fact that “demonstrations are not increased by middles,” i.e., in demonstrations one does not use a series of different middle terms to demonstrate one conclusion. (This, of course, is to be understood of demonstrations propter quid, of which he is speaking). For of one effect there can be but one sole cause why it is.
Sed licet non multiplicentur per media demonstrationes, multiplicantur tamen duobus modis. Uno modo, in post assumendo, idest in assumendo medium sub medio. Sicut si sub a sumatur b, et sub bc, et sub cd; et sic in infinitum. Sicut cum habere tres angulos probatur de triangulo per hoc, quod est figura habens angulum extrinsecum aequalem duobus intrinsecis sibi oppositis, et de isoscele per hoc quod est triangulus. Alio modo multiplicantur demonstrationes in latus; sicut cum a probatur de c et de e. Verbi gratia: omnis numerus quantus aut est finitus aut infinitus. Et hoc ponatur in quo sit a, scilicet esse finitum vel infinitum. Sed impar numerus est numerus quantus. Et hoc, scilicet numerus quantus, ponatur in quo est b; sed numerus impar ponatur in quo est c. Sequitur ergo quod a praedicetur de c, idest quod numerus impar sit finitus vel infinitus. Et similiter potest idem concludi de numero pari, et per idem medium. Potest autem et haec pars, quae incipit ibi: augentur autem etc., introduci aliter. Ut quia dixerat quod in demonstrativis assumuntur definitiones pro mediis; unius autem rei una est definitio; ex hoc sequitur quod demonstrationes non augeantur per media.	But although demonstrations are not increased by using several middles of demonstration, nevertheless they are increased in two ways: in one way, “by assuming one after another,” i.e., subsuming one middle under another, say B under A, C under B, D under C, and so on. For example, when “having three angles equal to two right angles” is proved of triangle on the ground that it is a figure having an external angle equal to the two opposite interior angles, and is then proved of isosceles on the ground that it is a triangle. Another way they are increased is laterally, as when A is proved of C and of E. For example, “Every quantified number is either finite or infinite.” Let A be this predicate, i.e., “to be finite or infinite.” But an odd number is a quantified number. Let B represent “quantified number,” and C, “odd number.” It follows therefore, that A (to be finite or infinite) is predicated of C (odd number), i.e., that an odd number is either a finite or an infinite number. And he says that it is possible along the same lines to conclude concerning even number, and this through the same middle. The passage (78a13)]], which begins, “A science expands,” can be introduced in another way. Since he had just said that in demonstrative matters, definitions are used for middles, and since there is one sole definition of one thing, it follows that it is not through middles that, demonstrations are increased.