Contingere autem simul affirmare et negare neque una recipit demonstratio, sed aut si indigeat monstrare et conclusionem sic. Ostenditur autem accipientibus primum de medio, quod verum sit affirmare, negare autem non verum. Medium autem nihil differt esse et non esse accipere, similiter autem et tertium. Si enim assignetur de quo hominem est verum dicere animal, quamvis non hominem verum, sed si solum hominem animal esse omne, non animal autem non; erit enim verum dicere Calliam, et non Calliam, tamen animal, non animal autem non. Causa autem est quod primum non solum de medio dicitur, sed de alio propter id quod de pluribus, quare neque si medium et idem est et non idem, ad conclusionem nihil differt. a10. The law that it is impossible to affirm and deny simultaneously the same predicate of the same subject is not expressly posited by any demonstration except when the conclusion also has to be expressed in that form; in which case the proof lays down as its major premise that the major is truly affirmed of the middle but falsely denied. It makes no difference, however, if we add to the middle, or again to the minor term, the corresponding negative. For grant a minor term of which it is true to predicate man — even if it be also true to predicate not-man of it — still grant simply that man is animal and not not-animal, and the conclusion follows: for it will still be true to say that Callias — even if it be also true to say that not-Callias — is animal and not not-animal. The reason is that the major term is predicable not only of the middle, but of something other than the middle as well, being of wider application; so that the conclusion is not affected even if the middle is extended to cover the original middle term and also what is not the original middle term. Omne autem affirmare aut negare quae est ad impossibile demonstratio accipit, et haec neque semper neque universaliter, sed quantum sufficiens est, sufficiens autem est in genere. Dico autem in genere ut circa quod genus demonstrationes sunt, sicut dictum est prius. a22. The law that every predicate can be either truly affirmed or truly denied of every subject is posited by such demonstration as uses reductio ad impossibile, and then not always universally, but so far as it is requisite; within the limits, that is, of the genus — the genus, I mean (as I have already explained), to which the man of science applies his demonstrations. Communicant autem omnes scientiae secundum communia. Communia autem dico quibus utuntur tanquam ex his demonstrantes, sed non de quibus demonstrant neque quod demonstrant. a26. In virtue of the common elements of demonstration — I mean the common axioms which are used as premises of demonstration, not the subjects nor the attributes demonstrated as belonging to them — all the sciences have communion with one another, Et dialectica de communibus. Et si aliqua universaliter temptet monstrare communia, ut quod est omne affirmare aut negare, aut quod est aequalia ab aequalibus, aut talium quaelibet. —Sed dialectica non est sic definitorum quorumdam, neque generis alicuius unius. Non enim interrogaret; demonstrantem enim non est interrogare propter id quod oppositorum esse non demonstrat idem. Ostensum autem est hoc in his quae de syllogismo. a29. and in communion with them all is dialectic and any science which might attempt a universal proof of axioms such as the law of excluded middle, the law that the subtraction of equals from equals leaves equal remainders, or other axioms of the same kind. Dialectic has no definite sphere of this kind, not being confined to a single genus. Otherwise its method would not be interrogative; for the interrogative method is barred to the demonstrator, who cannot use the opposite facts to prove the same nexus. This was shown in my work on the syllogism. 167. Contingere autem, etc. Postquam determinavit de principiis propriis et communibus, hic ostendit qualiter demonstrativae scientiae ad propria et communia se habeant. The law that it is impossible. After determining about proper and common principles, the Philosopher now shows how the demonstrative sciences behave in regard to the common and to the proper principles. Et dividitur in partes duas: His treatment falls into two parts. in prima ostendit qualiter se habeant demonstrativae scientiae circa communia; In the first he shows how the demonstrative sciences are related to common principles. in secunda qualiter se habeant circa propria, ibi: si autem idem est interrogatio syllogistica, etc. In the second he shows how they are related to proper principles, at: if a syllogistic question (77a36; [173]). Circa primum duo facit: Concerning the first he does two things. primo ostendit quomodo se habeant demonstrativae scientiae circa prima principia inter communia; First, he shows how the demonstrative sciences are related to the first of the common principles. secundo quomodo se habeant communiter circa omnia principia communia, ibi: communicant autem omnes scientiae, etc. Second, how they are related generally to all the common principles, at: in virtue of the common elements (77a29; [170]). Circa primum duo facit: Concerning the first he does two things. primo ostendit quomodo se habeant demonstrativae scientiae circa hoc principium quod non contingit simul affirmare et negare; First, he shows how the demonstrative sciences behave relative to the principle that one should not affirm and deny the same thing. secundo quomodo se habent circa hoc principium, "de quolibet est affirmatio vel negatio vera", ibi: omne autem affirmare, etc. Haec enim duo principia sunt omnium prima, ut probatur in IV Metaphysicae. Second, how they behave relative to the principle that "there is either true affirmation or true negation of each thing", at: the law that every predicate (77a22; [169]). For, as is proved in Metaphysics IV, these two principles are the first of all principles. 168. Dicit ergo primo quod nulla demonstratio accipit hoc principium quod non contingit simul affirmare et negare: si enim aliqua demonstratio eo uteretur ad ostendendum aliquam conclusionem, oporteret quod sic eo uteretur quod acciperet primum, idest maiorem extremitatem, affirmari de medio et non negari, quia, si acciperet affirmationem et negationem ex parte medii, nihil differret utrum sic vel sic esset; et eadem ratio est de tertio, idest de minori extremitate per comparationem ad medium. Verbi gratia: sit ‘animal’ primum, ‘homo’ medium, et ‘Callias’ tertium; si quis vellet uti praedicto principio in demonstratione, oporteret sic arguere: "Omnis homo est animal et non est non animal; Callias est homo; ergo <Callias> est animal et non est non animal"; cum vero dicat, "Omnis homo est animal", nihil differt utrum etiam haec sit vera, "Non homo est animal", vel non sit vera; et similiter in conclusione non differt, ex quo Callias est animal, utrum non Callias sit animal vel non. Et huius causa est quia primum non oportet dici de solo medio, sed potest etiam dici de quodam alio quod est diversum a medio, quod significatur per negationem medii, propter hoc quod primum dicitur de pluribus quandoque quam medium, sicut ‘animal’ de pluribus quam ‘homo,’ unde dicitur de equo, qui est non homo; unde si accipiatur medium idem et non idem, idest si accipiatur affirmativum et negativum, ut cum dico: "Homo et non homo est animal", nihil facit ad conclusionem. He says therefore first (77a10) that no demonstration makes use of the principle that one does not affirm and deny at one and the same time. For if a demonstration were to use it to show some conclusion, it would have to use it in such a way as to assert that the first, that is, the major, extreme is affirmed of the middle and not denied; because if it were to admit affirmation and negation on the part of the middle, it would make no difference whether it was this way or that. And the same reasoning holds for the third, that is, minor, extreme in relation to the middle. For example, take 'animal' as the first, 'man' as the middle’ and 'Callias' as the third. If someone wished to use this principle in a demonstration, he would have to argue in the following way: "Every man is an animal and is not a non-animal; But Callias is a man: Therefore, Callias is an animal and is not a non-animal". For when we say that every man is an animal, it matters not whether it is also true that a non-man is an animal, or whether it is not true. Similarly, in the conclusion it matters not, Callias being an animal, whether non-Callias is an animal or not. Now, the cause of this is that the first is not limited to being said only of the middle, but can also be said of something else diverse from the middle and described by a negation of the middle (since the first is sometimes said of many things other than the middle, as 'animal' is said of things other than 'man'; for it is said of horse, which is not man). Hence if the middle is taken the same and not the same, that is, if an affirmative and a negative middle be taken, as when I say, “man" and "non-man” are animals, it contributes nothing to the conclusion. Cum autem accipitur affirmatio vel negatio ex parte maioris extremitatis, differt quidem quantum ad conclusionem, immo etiam quantum ad veritatem praemissarum: si enim homo esset non animal, non esset verum quod homo est animal neque sequeretur quod Callias esset animal. Tamen nihil plus certificatur cum dicitur: "Homo est animal et non est non animal", quam cum dicitur solum, "Homo est animal": idem enim intelligitur per utrumque. Et sic manifestum est quod demonstrationes non utuntur hoc principio, scilicet quod affirmatio et negatio non sint simul vera, neque ex parte praedicati neque ex parte subiecti. However, if the affirmation and negation are taken on the part of the major extreme, it does make a difference both as to the conclusion and as to the truth of the premises. For if man were not an animal, it would not be true that man is an animal, nor would it follow that Callias is an animal. Yet nothing more is verified by stating that man is an animal and is not a non-animal than by merely stating that man is an animal, for the same thing is conveyed by each. And thus it is clear that demonstrations do not use the principle that affirmation and negation are not simultaneously true, either on the part of the predicate or on the part of the subject. 169. Deinde cum dicit: omne autem affirmare vel negare, etc., ostendit quomodo demonstrativae scientiae utantur hoc principio: "de quolibet affirmatio vel negatio vera". Then, at: the law that every predicate (77a22) he shows how demonstrative sciences use the principle that "of anything there is either true affirmation or negation". Et dicit quod hoc principium accipit demonstratio quae est ad impossibile: in hac enim demonstratione probatur aliquid esse verum per hoc quod eius oppositum est falsum, quod nequaquam contingeret si esset possibile utrumque oppositorum esse falsum. Non tamen semper utitur praedicta demonstratio hoc principio, quia quandoque illud oppositum quod ostenditur esse falsum non est negatio, sed contrarium immediatum, sicut si ostenderetur aliquem numerum esse parem per hoc quod falsum est ipsum esse imparem, ducendo ad impossibile; neque etiam utitur hoc principio universaliter, idest in sua universalitate sub his terminis, ‘ens’ et ‘non ens,’ sed quantum sufficiens est in genere aliquo. Et dico de illo genere circa quod sunt demonstrationes; sicut si in geometria accipiatur rectum et non rectum, ut cum ostenditur aliqua linea esse recta per hoc quod est falsum eam esse non rectam, ducendo ad impossibile. And he says that this principle is utilized in a demonstration leading to the impossible. For in this demonstration something is proved to be true by the fact that its opposite is false. (This of course would never happen, if it were possible for the two opposites to be false). Nevertheless such a demonstration does not always employ this principle, for sometimes that opposite which is shown to be false is not a negation but an immediate contrary. For example, if a number were shown to be even on the ground that its opposite, namely, “it is odd,” is false, and this were done by leading to the impossible. Neither does it use this principle universally, that is, in its universality, namely, under the terms 'being' and 'non-being', but only so far as is sufficient for a genus or so far as it is narrowed to a generic subject. And I mean the first genus involved in the demonstration. For example, in leading to the impossible in geometry the terms would be “straight” and “non-straight” in the case where it is shown that some line is straight on the ground that it is false to state that it is not straight. 170. Deinde cum dicit: communicant autem, etc., ostendit qualiter demonstrativae <scientiae> se habeant communiter ad omnia principia communia. Then, at: in virtue of the common elements (77a26) he shows how the sciences as a community function relative to all common principles. Et circa hoc duo facit. In regard to this he does two things. Primo dicit quod omnes scientiae in communibus principiis communicant hoc modo quod omnes utuntur eis sicut ex quibus demonstrant, quod est uti eis ut principiis; sed non utuntur eis ut de quibus aliquid demonstrant, ut de subiectis, neque sicut quod demonstrant, quasi conclusionibus. First, he says that all the sciences share alike in the common principles in the sense that they all use them as items from which they demonstrate-which is to use them as principles. But they do not use them as things about which they demonstrate something, that is, as subjects, or as things which they demonstrate, that is, as conclusions. 171. Secundo ibi: et dialectica, etc., ostendit quod quaedam scientiae utuntur principiis communibus alio modo quam dictum est. Dialectica enim est de communibus, et aliqua <alia> scientia est etiam de communibus, scilicet philosophia prima, cuius subiectum est ens, et considerat ea quae consequuntur ens, ut proprias passiones entis. Second, at: and in communion with them (77a29), he shows that certain sciences employ the common principles in a manner other than has been described. For dialectics is concerned with common things, and a certain other science is concerned with common things, namely, first philosophy, whose subject is being and which considers the things which follow upon being as the proper attributes of being. Sciendum tamen est quod alia ratione dialectica est de communibus et logica et philosophia prima. Philosophia enim prima est de communibus, quia eius consideratio est circa ipsas res communes, scilicet circa ens et partes et passiones entis. —Et, quia circa omnia quae in rebus sunt habet negotiari ratio, logica autem est de operibus rationis, logica etiam erit de his quae communia sunt omnibus, idest de intentionibus rationis, quae ad omnes res se habent; non autem ita quod logica sit de ipsis rebus communibus sicut de subiectis: considerat enim logica sicut subiecta syllogismum, enunciationem, praedicamentum aut aliquid huiusmodi. Pars autem logicae quae demonstrativa est, etsi circa communes intentiones versetur docendo, tamen usus demonstrativae scientiae non est in procedendo ex his communibus intentionibus ad aliquid ostendendum de rebus quae sunt subiecta aliarum scientiarum. —Sed hoc dialectica facit, quia ex communibus intentionibus procedit arguendo dialecticus ad ea quae sunt aliarum scientiarum, sive sint propria sive communia, maxime tamen ad communia, sicut argumentatur quod odium est in concupiscibili in qua est amor, ex hoc quod contraria sunt circa idem; est ergo dialectica de communibus non solum quia pertractat intentiones <communes> rationis, quod est commune toti logicae, sed etiam quia circa communia rerum argumentatur. Yet it should be noted that dialectics is concerned with common things under an aspect different from logic and first philosophy. For first philosophy is concerned with common things because its consideration is focused on those common things, namely, on being and on the parts and attributes of being. But because reason occupies itself with all things that are, and logic studies the operations of reason, logic will also be concerned with matters common to all things, that is, with reason’s intentionalities which bear on all things, but not in such a way that logic has these common things as its subject-for logic considers as its subject the syllogism, enunciation, predication, and things of that type. But although the part of logic which is demonstrative is engaged in teaching about common intentionalities, the use of a demonstrative science does not consist in proceeding from common intentionalities to show anything about the things which are the subjects of the other sciences. But dialectics does this, for it goes from common intentions and argues to things that pertain to other sciences, whether they be proper or common things, but mainly common things. Thus, it argues that hatred is in the concupiscible appetite, because love is, on the ground that contraries are concerned with a same thing. Consequently, dialectics is concerned with common things not only because it treats concerning the common intentionalities of reason, which is common to all logic, but also because it argues about common characteristics of things. Quaecunque autem scientia argumentatur circa communia rerum, oportet quod argumentetur circa principia communia, quia veritas principiorum communium est manifesta ex cognitione terminorum communium, ut entis et non entis, totius et partis, et similium. Dicit autem signanter: Et si aliqua scientia temptet monstrare communia, quia philosophia prima non demonstrat principia communia: sunt enim indemonstrabilia simpliciter; sed aliqui errantes attemptaverunt ea demonstrare, ut patet in IV Metaphysicae. Vel etiam quia, etsi non possint demonstrari simpliciter, tamen philosophus primus temptat ea demonstrare eo modo quo possibile est, scilicet contradicendo negantibus ea per ea quae oportet ab eis concedi, non per ea quae sunt magis nota. But any science that argues about the common characteristics of things must argue about the common principles, because the truth of the common principles is made manifest from the knowledge of common terms, as “being” and “non-being,” “whole” and “part” and the like. It is significant that he says, and any science which might attempt, because first philosophy does not demonstrate the common principles, since they are absolutely indemonstrable, although some have unwittingly attempted to demonstrate them, as is stated in Metaphysics IV. Or else, because even though they cannot, strictly speaking, be demonstrated, the first philosopher attempts to uphold them in a way that is possible, namely, by contradicting those who deny them, appealing to things that must be conceded by them, though not to things which are more known.
It should also be noted that the first philosopher demonstrates them not only in this way, but also shows something about them as about subjects: for example, that “it is impossible for the mind to think their opposites,” as is clear from Metaphysics IV. 172. Sciendum est etiam quod primus philosophus non solum hoc modo monstrat ea, sed etiam monstrat aliquid de eis sicut de subiectis, sicut quod impossibile est mente concipere opposita eorum, ut patet in IV Metaphysicae. Cum ergo disputet circa haec principia et philosophus primus et dialecticus, tamen aliter et aliter: dialecticus enim non procedit ex aliquibus principiis determinatis, neque assumit alteram partem contradictionis tantum, sed se habet ad utramque (contingit enim utramque quandoque vel probabilem esse vel ex probabilibus ostendi, quae accipit dialecticus), et propter hoc interrogat; demonstrator autem non interrogat, quia non se habet ad opposita. Et haec differentia utriusque posita est in his quae sunt de syllogismo, idest in libro Priorum. Philosophia vero prima procedit circa communia per modum demonstrationis et non per modum dialecticae disputationis. Therefore, although both the first philosopher and the dialectician debate about these first principles, yet one does so in one way and the other in another way. For the dialectician neither proceeds from demonstrative principles nor takes only one side of a contradiction but is open to both. For each side might happen to be probable or be upheld by probable statements, which the dialectician utilizes: and that is why he asks [his questions in terms of two alternatives]. But the demonstrator does not ask [in that way], because he is not open to opposites. And this is the difference between the two, as was laid down in the treatment of the syllogism, namely, in the beginning of the Prior Analytics. Therefore, first philosophy in treating common principles proceeds after the manner of a demonstration and not after the manner of a dialectical disputation. Lectio 21 Lecture 21 De interrogationibus, responsionibus et disputationibus in qualibet scientia propriis The questions, responses and disputations peculiar to each science Si autem idem est interrogatio syllogistica et propositio contradictionis, propositiones autem sunt secundum unamquamque scientiam, ex quibus est syllogismus secundum unamquamque, erit utique interrogatio scibilis, ex quibus qui secundum unamquamque proprius fit syllogismus. Manifestum itaque quod non omnis interrogatio geometrica utique erit neque medicinalis, similiter autem est et in aliis. a36. If a syllogistic question is equivalent to a proposition embodying one of the two sides of a contradiction, and if each science has its peculiar propositions from which its peculiar conclusion is developed, then there is such a thing as a distinctively scientific question, and it is the interrogative form of the premises from which the ‘appropriate’ conclusion of each science is developed. Hence it is clear that not every question will be relevant to geometry, nor to medicine, nor to any other science: Sed ex quibus aut demonstratur aliquid de quibus geometria est, aut ex eisdem demonstratur, geometrice, sicut in speculativa. Similiter autem et in aliis. a42. only those questions will be geometrical which form premises for the proof of the theorems of geometry or of any other science, such as optics, which uses the same basic truths as geometry. Of the other sciences the like is true. Et de his quidem et rationem ponendam esse ex geometricis principiis et conclusionibus, sed principiorum rationem non ponendam esse geometrae secundum quod est geometer. Similiter autem et in aliis scientiis. b2. Of these questions the geometer is bound to give his account, using the basic truths of geometry in conjunction with his previous conclusions; of the basic truths the geometer, as such, is not bound to give any account. The like is true of the other sciences. Neque omnem est itaque unumquemque scientem interrogationem interrogare, neque secundum omne interrogatum respondere de unoquoque; sed quae sunt secundum scientiam determinatam. b6. There is a limit, then, to the questions which we may put to each man of science; nor is each man of science bound to answer all inquiries on each several subject, but only such as fall within the defined field of his own science. Si autem disputat cum geometra secundum quod est geometer, sic manifestum est quod et bene, si ex his aliquid demonstret, si vero non, non bene. Manifestum autem quod non arguit geometram, sed aut secundum accidens, quare non utique erit in non geometricis de geometria disputandum: latebit enim prave disputantem. Similiter autem et in aliis se habet scientiis. b8. If, then, in controversy with a geometer qua geometer the disputant confines himself to geometry and proves anything from geometrical premises, he is clearly to be applauded; if he goes outside these he will be at fault, and obviously cannot even refute the geometer except accidentally. One should therefore not discuss geometry among those who are not geometers, for in such a company an unsound argument will pass unnoticed. This is correspondingly true in the other sciences. 173. Si autem idem est interrogatio, etc. Postquam Philosophus ostendit quomodo scientiae demonstrativae se habeant circa communia, hic ostendit quomodo se habeant circa propria. If a syllogistic question. After showing how demonstrative sciences function in relation to common principles, the Philosopher now shows how they employ proper principles. Et dividitur in duas partes: And his treatment falls into two parts. in prima ostendit quod in qualibet scientia sunt propriae interrogationes, responsiones et disputationes; In the first he shows that in each science there are questions, responses and disputations peculiar to each. in secunda ostendit quomodo in qualibet scientia sunt etiam propriae deceptiones, ibi: quoniam autem sunt geometricae, etc. In the second he shows how in each science there are deceptions, peculiar to each, at: since there are “geometrical” questions (77b16; [180]).