Works › Posterior Analytics, Books I–II
Volume 2 · pp. 144–149
Chapter VIII. That the principles of diverse sciences, diverse syllogisms, and diverse demonstrations are diverse.
CAPUT VIII. Quod diversarum scientiarum et syllogismorum diversorum et demonstrationum diversarum sunt principia diversa. Quia autem in ante habitis dictum est, quod una scientia est, in qua principia ex quibus componitur, univocantur per unitatem sui subjecti: et diversa sunt scita, quibus principia non unificantur, sed remanent diversa: et jam ostendimus qualiter unificantur principia scientiae: nunc oportet nos ostendere qualiter diversarum scientiarum principia sunt diversa. Sciamus ergo quod eadem esse principia omnium scientiarum propria est impossibile: et hoc ostendamus primum quidem speculantibus hoc aliquibus logice, hoc est, per media vel omnibus vel pluribus communia. Incipiemus enim hanc speculationem a syllogismo in communi et logica speculatione. Inter syllogismos enim hi veri sunt, ex veris principiis veras concludentes conclusiones. Alii autem sunt falsi, ex principiis falsis falsas concludentes conclusiones: falsorum autem et verorum non sunt eadem principia: ergo non omnium syllogismorum sunt eadem principia. Et si namque aliquis cavillator dicat, quod ex falsis principiis est vel contingit verum syllogizare: et sic eadem videntur esse principia verorum syllogismorum et falsorum. Dicamus quod hoc contingit semel tantum, sicut si gratia exempli A major extremitas de C minori extremitate vera sit: tunc enim erit conclusio vera: medium autem sumitur B, quod falsum sit, et ad A majorem extremitatem, et sit falsum ad C minorem extremitatem: quia nec majori subjicitur, nec de minori praedicatur: nec enim A major extremitas est in B sicut praedicamentum in subjecto, nec B medium est in C minori extremitate sicut praedicatum in subjecto: et ideo utraque praemissarum est falsa, et conclusio vera. Cujus exemplum est: omnis lapis est animal: omnis homo est lapis: ergo omnis homo est animal. Et hoc quidem semel contingit: contingit enim quoad conclusionem propriam et proximam tali conjugationi in qua major extremitas repugnat medio, et idem medium repugnat minori extremitati, et major extremitas non repugnat minori extremitati, sed est de consequentibus ad ipsam:
CHAPTER VIII. That the principles of diverse sciences, diverse syllogisms, and diverse demonstrations are diverse. But because in the things had before it was said that a science is one in which the principles from which it is composed are made univocal through the unity of its subject, and that known things are diverse for which the principles are not unified but remain diverse, and because we have now shown how the principles of science are unified, now we must show how the principles of diverse sciences are diverse. Therefore let us know that it is impossible for the proper principles of all sciences to be the same; and let us show this first, indeed, to those speculating about this logically, that is, through middles common either to all or to many. For we will begin this speculation from syllogism in common and in logical speculation. For among syllogisms these are true: those concluding true conclusions from true principles. But others are false: those concluding false conclusions from false principles. But of false and true syllogisms the principles are not the same; therefore not all syllogisms have the same principles. And if indeed some caviller says that from false principles there is, or there can be, syllogizing a true thing, and thus the principles of true and false syllogisms seem to be the same, let us say that this happens only once, as if, for the sake of example, major extreme A is true of minor extreme C; for then the conclusion will be true. But middle B is taken, which is false both in relation to major extreme A and false in relation to minor extreme C, because it is neither subjected to the major nor predicated of the minor. For major extreme A is not in B as a predicament in a subject, nor is middle B in minor extreme C as a predicate in a subject; and therefore both premises are false and the conclusion true. The example of this is: every stone is an animal; every human being is a stone; therefore every human being is an animal. And this indeed happens once; for it happens with respect to the proper and proximate conclusion for such a pairing, in which the major extreme is repugnant to the middle, and the same middle is repugnant to the minor extreme, and the major extreme is not repugnant to the minor extreme, but is among things consequent upon it:
et in nulla alia contingit conjugatione, sed in prima figura et in primo modo verum ex falsis syllogizare. Sed hoc patere facile potest si accipiantur horum principiorum quae praemissa sunt media, per quae major falsa et minor falsa concludantur, semper ex falsis sequitur falsum et non verum: et iterum si illa falsa in tertio syllogismo concludi debeat per alia falsa, iterum ex falsis sequitur falsum et non verum: et hoc quantumcumque possit fieri resolutio talis, ut principia syllogismi per alia concludantur, semper syllogizabitur falsum ex falso. Semel ergo contingit verum ex falso syllogizare, et non pluries. Hujus exemplum, quod si debeat syllogizari haec falsa, quae fuit major, omnis lapis est animal, accipiatur medium inter lapidem et animal, et ad eam concludendam, semper ex falso sequitur falsum. Si accipiatur medium ad minorem inter praedicatum et subjectum per quod concludi possit minor, semper ex falsis falsum concludetur quantumcumque fiat reductio ad prosyllogismum majoris et minoris, vel utraque falsa, vel altera. Hoc autem ideo, quia omnis conclusio falsa in prosyllogismis ex falsis falsa est. Verae autem conclusiones sunt ex veris principiis. Alterae autem sunt propositiones verae et alterae falsae propositiones sunt quae sunt syllogismorum principia: ergo verorum syllogismorum et falsorum diversa et altera sunt principia. Scias autem, quod in commento Arabico, quod fuit Alfarabii, ubi haec translatio Boetii habet semel, habet una vice: et glossat Alfarabius, una vice pro raro: dicit enim, quod casu accidit: casum vocans, quod causam ordinatam non habet: quia verum non sequitur ex falso secundum causam qua praemissae sunt causa conclusionis: sed per accidens sequitur ex tali complexione qualis dicta est solum, et non ex habitudine terminorum in syllogismo posito: et haec expositio est plana et facilis. Sic igitur probatum est quod falsorum et verorum syllogismorum non sunt principia eadem: non ergo omnium syllogismorum eadem, sed diversa sunt principia. Postea autem sic probatur, quod nec falsae ratiocinationes syllogismorum falsorum ex eisdem sunt principiis: est enim vel contingit accipere falsas propositiones concludentes quae ad invicem sunt contrariae, et sunt impossibiles, quia ex disparatis: et sunt impossibiles, eo quod praedicatum repugnat subjecto, sicut si dicam justitiam esse injustitiam, falsa est ex compositione contrarii cum contrario: et idem redit si componam propositionem ex genere et specie contrarii, ut si dicam justitiam generalem esse timorem, quae est injustitia specialis. Ex disparatis autem, sicut si dicam hominem esse equum, aut bovem esse hominem, aut equum. Ex impossibilibus autem, sicut si dicam, quod aequale alicui majus est vel minus est eodem: ex terminis enim positis (hoc est, sic dispositis) sic fiunt diversi falsi syllogismi: et falsus syllogismus fit ex sic compositis et concessis talibus propositionibus, aut ex contrariis, aut ex disparatis, aut ex incompossibilibus compositis: cum ista sint diversa ex diversis composita principiis. Adhuc autem nec omnium syllogismorum verorum eadem sunt principia: veri enim syllogismi non in uno tantum genere, sed in multis sunt: eorum enim (quae altera et diversa sunt genere) principia (ex quibus sunt) altera necesse est esse et talia principia: nec etiam conveniunt in genere, sicut unitates punctis non conveniunt in genere: quamvis enim conveniant in uno genere logico, eo quod utrumque reducitur ad quantitatem, tamen non conveniunt in uno genere naturae unius quod possit esse principium passionum ejusdem rationis in genere, vel a qua fluant eadem principia quae in uno subjecto unius communis naturae possint unificari hoc modo, sicut prius dictum est in praehabito capitulo, quod una scientia est unius generis subjecti, in quo unificantur et uniuntur principia, ex quibus ipsa statim componitur. Hoc autem modo unitates ex quibus fit numerus, in genere subjecti non conveniunt punctis, ex quibus fluunt species continuae quantitatis, ut linea, superficies, corpus, et partes illarum, ut angulus, et hujusmodi. Ratio autem propter quam non conveniunt in genere subjecti est, quia hae scilicet unitates non habent positionem: illa autem scilicet puncta habent positionem in continuo. Adhuc autem in his scientiis (quae sunt ex eisdem principiis) necesse est quod scientiae conveniant in mediis, ita quod eadem sint media istius et illius: quia una scientia quae est habitus conclusionis, ex eisdem mediis demonstratur; aut necesse est quod conveniant in extremis: extremum autem majus in prima figura, est in sursum supra medium acceptum: aut est in deorsum, sicut extremum minus quod sub medio accipitur. Vel dicatur in sursum acceptum in secunda figura, quia accipitur supra praedicatum: et in deorsum acceptum medium in tertia figura, quia accipitur sub subjecto. Aut oportet hos quosdam syllogismos media interius habere inter majorem et minorem extremitatem, sicut in prima figura: illos autem alios aliarum figurarum duarum syllogismos oportet media sua exterius extra extremitates habere: sicut secunda medium habet ante extrema primum positione, et tertia habet medium post extrema ultimum positione, ut in libro Priorum est determinatum: sed diversarum scientiarum non eadem sunt media vel extrema, sed potius diversa: ergo eorum quae sunt diversorum generum syllogismorum non sunt eadem principia quae sint media vel extrema. Adhuc de numero principiorum quae communiora sunt, sicut de quolibet affirmatio vel negatio vera, vel totum majus parte, vel si de aequalibus aequalia demas relicta sunt aequalia. Haec autem sunt communiora multis generibus appropriabilia, et talium communium principiorum non possunt esse scientiae diversae, ita quod diversarum eadem sint principia, quia universalia talia communia possunt esse: ex quibus etiam diversa in singulis scientiis demonstrentur, sicut ex principiis. Dico autem communia principia omnibus, ut est illud, quod omne quod est, contingit affirmare vel negare, et alia quae paulo ante posita sunt: genera enim eorum quae sunt de quibus scientiae fiunt, sunt altera et diversa: et quando principia communia appropriantur (sicut dudum in ante habitis dictum est) tunc fiunt et ipsa principia appropriata alia in quantitatibus et scientiis quae de quantitate sunt, et alia in qualitatibus et scientiis quae sunt de qualitate, solum acceptum principium ad illam scientiam appropriant cum quibus propriis demonstrantur conclusiones per communia sic appropriata. Amplius et ad idem, principia ex quibus scientia sive scita conclusio nota est, non sunt multo minora, hoc est, pauciora conclusionibus quae concluduntur ex eis. Et scias quod apud Philosophos usus est in proportionibus numerorum multum plus vel multum minus dicere quod distat in infinitum, et cujus nulla est proportio, ut dicit Alfarabius, sive autem in multo sic dicat in infinitum et non proportionale secundum numerum, sive dicat multo deficientius secundum numerum, semper tenet ratio quam inducemus: quia conclusiones multae et principia cujuslibet conclusionis sunt immediatae propositiones, quae non in multo excedunt conclusiones: propositiones autem sunt aut assumpti termini in duabus figuris, secunda et tertia: aut sunt termini immissi inter majorem et minorem extremitatem, sicut in prima figura: quia assumpto termino supra praedicatum vel sub subjecto, vel immisso inter praedicatum et subjectum, semper apponitur una propositio quae est principium: et sic propositiones non multo sunt pauciores conclusionibus: sed si essent principia omnium eadem, essent propositiones multo pauciores conclusionibus in qualibet scientia, quia omnes conclusiones probarentur per eadem media, quae valde multa sunt. Cum ergo hoc sit inconveniens, patet quod non omnium eadem sint principia immediata nec in una et eadem scientia. Adhuc autem ad idem inconveniens, quia conclusiones infinitae sunt ad minus potentia per medii multiplicationem, ut potest probari de triangulo, quod tres habeat: de quadrato, quod quatuor: de pentagono, quod quinque; et sic in infinitum: quod totum probatur per triangulum. Termini autem immissi vel assumpti (hoc est, medii) sunt finiti secundum numerum determinatum in tribus figuris: et si essent eadem omnium principia, tunc probarentur conclusiones infinitae per principia finita, quod est inconveniens, quia finiti ad infinitum nulla est proportio. Patet ergo quod in una scientia non omnia eadem sunt principia. Hunc puto esse intellectum Aristotelis, et quod nulla alia vis sit rationibus suis. Ad evidentiam autem eorum quae hic dicta sunt (quod termini sic assumpti vel immissi non sunt multo pauciores conclusionibus) intelligendum est quod principia, hoc est, praemissae possunt comparari ad suas conclusiones ultimas, quae ulterius concludi non possunt, eo quod densatae ad immediatas reductae sunt; et sic in veritate non sunt pauciora conclusionibus, quia unaquaeque scientia habet ultimarum hoc modo conclusionum multa media, et tot quod habet media, tot habet principia, quia medium principium est conclusionis. Et possunt comparari non ad conclusiones ultimas, sed ad quascumque indefinite: et sic principia sunt non multo pauciora conclusionibus. Hujus exemplum potest esse si ostendatur substantia de isto homine per media quae sunt corpus, animal, rationale, et homo, sex erunt conclusiones et quatuor principia. Concluditur enim substantia de animali per corpus, et de rationali per animal, et de homine per rationale: et sic tres sunt conclusiones. Et iterum concluditur corpus de rationali per animal, et de homine per rationale: et sic sunt conclusiones quinque. Et iterum concluditur animal de homine per rationale: et sic sunt sex conclusiones, et quatuor principia quae sunt propositiones immediatae ad istas conclusiones, sicut patet; sunt enim quatuor penes quatuor media, et sic sex erunt conclusiones, et quatuor principia immediata ad sex conclusiones concludendas: et sic multo sunt pauciora principia conclusionibus: et hoc est quod dicit Aristoteles. <marginal-label>Expositio levis quorumdam.</marginal-label> Sunt tamen qui faciliter hic transeunt, et dicunt quod principia ideo dicuntur non multo pauciora conclusionibus, quia sunt multo plura. Si enim principia sunt mediatae propositiones, ad quamlibet conclusionem duplicantur praemissae: et sic principia sunt in duplo plura, quam conclusiones. Veritas autem secundum intellectum Aristotelis est id quod primo dictum est. Unde translatio Arabica habet, quod principia non sunt elongata a conclusionibus. Adhuc autem cum dicantur propositiones esse termini assumpti vel immissi, non ita intelligitur, quod termini formaliter praedicentur de propositionibus: sed materialiter intelligitur: quia propositiones immediatae sunt materialiter ex talibus terminis. A terminis enim mediis nominantur et numerantur praemissae, quia praemissae super conclusionem nihil addunt nisi tantum medium: quidquid enim est de majori propositione et de minori praeter medium, totum est in conclusione: quia conclusio est medietas utriusque praemissarum. Attendendum etiam quod conclusiones in alia materia et alia vadunt in infinitum. Et si quis dicat quod conclusio est scita et scita non cadunt super infinitum. Dicendum quod conclusiones actu scitae ab homine sunt finitae, et tamen potentia sunt infinitae, ut demonstratur in quarto Euclidis: ex quo probatur quod si poligonia inscribatur circulo angulis multiplicatis in infinitum, ad quantitatem circuli non deveniet: quotiescumque autem accrescit novus angulus, fit nova figura suas habens passiones in infinitum: ergo multiplicantur. Si ergo omnium essent eadem principia, per finita probarentur infinita sicut per propria: quod non esse potest. Amplius ad idem ratio est, quod scilicet non eadem principia omnium sunt: quia constat proxima principia quidem alia sive quaedam esse ex necessitate, sicut necessariorum: alia contingentia contingentium principia, quando sunt propria et essentialia: de talibus enim hic loquimur principiis. Sic ergo tali intentione habitis his rationibus (quae dictae sunt) impossibile est illa principia necessariorum et contingentium esse eadem, et impossibile principia propria esse finita cum infinitae sint conclusiones. Si vero quodammodo ad hominem et non ad rationem instans aliquis cavillator dicat, quod haec quidem determinata principia geometriae, et quod aliae propositiones sint principia syllogismorum (hoc est, scientiae syllogizandi quae est logica) et aliae propositiones sint medicinae: quid utique erit hoc aliud dicere, nisi quod propriarum scientiarum sunt propria principia, et diversarum scientiarum principia diversa? Sed propter hoc quod ista sunt principia et illa principia, dicere omnium principia eadem esse, responsio sophistica est secundum consequens: quia ex hoc sequitur, quod eadem eisdem ipsa sunt eadem. Unde sic omnia fiunt eadem, sicut principiis principia sunt eadem, et principiata ab illis sunt eadem: et sic omnia sunt unum, sicut dixit Melissus: et hoc est etiam speciale medium octavae rationis ad idem, per aliud inconveniens intentionem concludens. At vero etiam non contingit dicere, quod quodlibet, hoc est, omnem conclusionem est demonstrare ex omnibus (hoc est, ex quolibet) ut scilicet omnia demonstrentur ex omnibus indifferenter: horum enim (qui hoc dicunt) est quaerere omnium esse eadem principia: quia sic omnia eisdem demonstrarentur: talia enim dicere multum est insipiens et stultum: hoc enim non possibile in his quae manifestae doctrinae sunt sive doctrinales, ut sunt mathematicae. Nec in resolutione alicujus conclusionis in praemissas est hoc posse: sunt enim principia propria propositiones immediatae: accepta autem altera propositione immediata pro praemissa fit, hoc est, sequitur etiam altera conclusio, ut in libro Priorum est probatum: ergo in una et eadem scientia diversarum conclusionum diversa sunt propria principia. Si autem ex his quae dicta sunt (aliquis errorem concipiens secundum ea quae dicta sunt) dicat primas et immediatas propositiones esse principia propria, et hoc principium est unius modi in unoquoque genere uno, et diversum in diversis: si vero (secundum dicta) non ex omnibus (hoc est, ex quolibet) demonstrari quodlibet contingit, ut opus est (hoc est, illo modo quo oportet quodlibet demonstrari) iterum contingit sic ut opus est, ex altera scientia (hoc est, alterius principiis) demonstrare conclusiones scientiae alterius. Sequitur ex hoc uniuscujusque scientiae altera et altera esse principia. Relinquitur ergo (si proxima et non communia sunt principia omnium) quod nulla sunt communia omnium principia, sed ex his propriis principiis demonstratur haec conclusio: ex illis autem aliis propriis demonstratur alia conclusio, et nulla erunt principia communia in omnibus. Si, inquam, hoc dicat aliquis, manifestum est ex superius determinatis, quod non contingit, et est falsum nulla esse communia propter hoc quod in scientiis determinatis alia sunt propria principia, et diversa in diversis. Principia enim sunt duplicia, hoc est, dupliciter dicta vel tripliciter, sicut principia ex quibus est demonstratio, et quae sunt in ipsa demonstratione, et circa quae est demonstratio. Ex quibus autem fit demonstratio, sunt dignitates non ingredientes demonstrationes, sed extra stantes et confirmantes decursum syllogisticum demonstrationis: et haec sunt communia. Quae autem demonstrantur et sunt in ipsa demonstratione, sunt passiones: et circa quae fit demonstratio, sunt subjecta, et propositiones in quibus haec sunt praedicata et subjecta: et haec sunt propria et in diversis diversa. Patet ergo quod principia ex quibus, hoc est, ex quorum virtute est demonstratio, communia sunt: quae autem et circa quae propria, sunt ut numerus in arithmeticis, et magnitudo in continuis, sicut in geometria. Haec est, ut puto, sententia Aristotelis, unde principale intentum est, quod non omnium conclusionum eadem sunt principia. Quod ex hoc patet: quia si omnium conclusionum essent eadem principia: tunc ponamus conclusiones esse E F G, et principia A B C D, concludetur quaelibet conclusio E F G ex quolibet penitus sigillatim A B C D, ita quod quaelibet ex A et quaelibet ex B et sic de aliis. Hoc autem falsum est. <marginal-label>Removet dubium.</marginal-label> Attendendum autem quod licet metaphysicus probet et stabiliat principia omnium: non tamen suum est determinare utrum eadem omnium conclusionum, sed istius est, hoc est, scientiae quae docet componere principia ad conclusionem. <marginal-label>Removet dubium.</marginal-label> Adhuc autem quamvis una sit scientia syllogistica, qua utuntur omnes scientiae, in libro Priorum determinata, non tamen propter hoc eadem omnium sunt principia conclusionum: principia enim eadem sunt quantum ad ea quibus regulatur syllogismi compositio, et habitudo trium terminorum in modo et figura: sed quando dicitur, non sunt eadem principia omnium, intelligitur de principiis quae sunt praemissae relatae ad conclusionem immediate sequentem ex praemissis. Attendendum etiam ut quidam dicunt in explanatione ultimae partis capituli, quod quidam dicunt intendere Aristotelem excludere dictum cavillatoris dicentis, quod licet resolutio conclusionis facta ad proxima et propria principia ostendat diversorum diversa esse principia, tamen resolutio facta ad prima in quolibet genere, ostendit omnium conclusionum eadem esse principia, dicunt, quod non stat resolutio, nisi deducatur usque ad prima ante quae nihil est accipere: prima autem dicunt, quae enuntiantur de subjecto primo: quia est vel praedicatum de subjecto primo per propriam ejus diffinitionem, ut de magnitudine in geometria, et de numero in arithmetica, et sic de aliis: et illas propositiones esse communia principia in scientiis ad diversas conclusiones. Et quando dicit Aristoteles non contingit, glossant, quod hoc non est contingens, sed necessarium. Primum autem quod dictum est, etiam videtur magis esse de intentione Aristotelis. Haec igitur de unitate scientiae ad scientiam et de diversitate dicta sint.
and in no other pairing does it happen, but in the first figure and in the first mode, to syllogize the true from false things. But this can easily be clear if the middles of these principles that have been set forth as premises are accepted, through which the false major and the false minor are concluded: from false things a false thing always follows, and not a true one. And again, if that false thing in a third syllogism must be concluded through other false things, again from false things a false thing follows and not a true one; and this is so however much such a resolution can be made, so that the principles of the syllogism are concluded through other things: a false thing will always be syllogized from a false thing. Therefore it happens once to syllogize a true thing from a false one, and not more times. The example of this is that, if this false proposition, which was the major, every stone is an animal, must be syllogized, let a middle be accepted between stone and animal, and for concluding it a false thing always follows from a false thing. If a middle is accepted for the minor between predicate and subject through which the minor can be concluded, a false thing will always be concluded from false things, however much a reduction to a prosyllogism of the major and minor is made, whether both are false or one is false. This is so because every false conclusion in prosyllogisms is false from false things. But true conclusions are from true principles. But true propositions are other, and false propositions, which are the principles of syllogisms, are other; therefore the principles of true syllogisms and false syllogisms are diverse and other. But you should know that in the Arabic commentary, which was Alfarabi's, where this translation of Boethius has once, it has one time, and Alfarabi glosses one time as rarely. For he says that it happens by chance, calling chance what does not have an ordered cause, because the true does not follow from the false according to the cause by which the premises are the cause of the conclusion, but it follows per accidens only from such a complex arrangement as has been stated, and not from the relation of terms in the syllogism set down; and this exposition is plain and easy. Thus, therefore, it has been proved that the principles of false and true syllogisms are not the same; therefore the principles of all syllogisms are not the same but diverse. Afterward it is proved thus that not even the false reasonings of false syllogisms are from the same principles. For it is, or it is possible, to accept false concluding propositions which are contrary to one another and are impossible because they are from disparates; and they are impossible because the predicate is repugnant to the subject, as if I say that justice is injustice, which is false from the composition of contrary with contrary. And the same returns if I compose a proposition from the genus and species of a contrary, as if I say that general justice is fear, which is special injustice. But from disparates, as if I say that a human being is a horse, or that an ox is a human being, or a horse. But from impossibles, as if I say that what is equal to something is greater than or less than the same thing. For from the terms posited, that is, disposed thus, different false syllogisms are made in this way; and a false syllogism is made from propositions thus composed and conceded, either from contraries, or from disparates, or from incompossible things composed, since those diverse things are composed from diverse principles. Moreover, not all true syllogisms have the same principles, for true syllogisms are not in one genus only, but in many. For of those things which are other and diverse in genus, the principles from which they are must necessarily be other and principles of such a kind. Nor do they even agree in genus, just as units do not agree in genus with points; for although they agree in one logical genus, because each is reduced to quantity, nevertheless they do not agree in one genus of one nature which could be the principle of properties of the same account in genus, or from which the same principles could flow that could be unified in one subject of one common nature in this way, just as it was said before in the preceding chapter that one science is of one genus of subject, in which the principles are unified and united, from which it is immediately composed. But in this way the units from which number is made do not agree in the genus of subject with points, from which flow the species of continuous quantity, such as line, surface, body, and their parts, such as angle and things of this kind. But the reason on account of which they do not agree in the genus of subject is that these, namely units, do not have position, but those, namely points, have position in the continuum. Moreover, in these sciences, which are from the same principles, it is necessary that the sciences agree in middles, so that the middles of this and of that are the same, because one science, which is the habit of the conclusion, is demonstrated from the same middles; or it is necessary that they agree in the extremes. But the greater extreme in the first figure is taken upward above the middle; or it is downward, like the lesser extreme which is taken below the middle. Or it is called taken upward in the second figure because it is taken above the predicate; and the middle is taken downward in the third figure because it is taken under the subject. Or these certain syllogisms must have middles inwardly between the major and the minor extreme, as in the first figure; but those other syllogisms of the two other figures must have their middles outwardly outside the extremities, as the second has the middle before the extremes, first by position, and the third has the middle after the extremes, last by position, as is determined in the book of the Prior Analytics. But diverse sciences do not have the same middles or extremes, but rather diverse ones; therefore those things which belong to syllogisms of diverse genera do not have the same principles which are middles or extremes. Moreover, concerning the number of principles which are more common, such as that of anything whatever an affirmation or negation is true, or that the whole is greater than the part, or that if you subtract equal things from equal things the remainders are equal. These, however, are more common things capable of being appropriated to many genera, and there cannot be diverse sciences of such common principles, so that the same principles belong to diverse sciences, because such universal common things can be principles from which diverse things are also demonstrated in individual sciences, as from principles. But I call principles common to all, as is that principle that everything that is can be affirmed or denied, and the others which were placed a little earlier. For the genera of those things concerning which the sciences are made are other and diverse; and when common principles are appropriated, as was said a while ago in what was previously had, then the appropriated principles themselves become one kind in quantities and in sciences that are about quantity, and another kind in qualities and in sciences that are about quality. They appropriate to that science only the accepted principle together with the proper things by which conclusions are demonstrated through the common principles thus appropriated. Further, for the same point, the principles from which science, or a known conclusion, is known are not much smaller, that is, fewer, than the conclusions that are concluded from them. And you should know that among the Philosophers, in proportions of numbers, the usage is to call that much more or much less which is distant to infinity and of which there is no proportion, as Alfarabi says; but whether in this way he says by much as to infinity and as not proportional according to number, or whether he says much more deficient according to number, the reason that we bring forward always holds, because the conclusions are many and the principles of any conclusion are immediate propositions, which do not exceed the conclusions by much. But the propositions are either terms assumed in the two figures, the second and the third, or they are terms inserted between the major and minor extreme, as in the first figure; because, when a term has been assumed above the predicate or under the subject, or inserted between predicate and subject, one proposition which is a principle is always added. And thus propositions are not much fewer than conclusions; but if the principles of all were the same, the propositions would be much fewer than the conclusions in any science, because all conclusions would be proved through the same middles, which are very many. Therefore, since this is unsuitable, it is clear that the same immediate principles do not belong to all things, not even in one and the same science. Moreover, for the same unsuitable consequence, because conclusions are infinite at least in potency through the multiplication of the middle, as it can be proved of a triangle that it has three, of a quadrangle that it has four, of a pentagon that it has five, and so on to infinity; all of this is proved through triangle. But the terms inserted or assumed, that is, the middles, are finite according to the number determined in the three figures; and if the principles of all things were the same, then infinite conclusions would be proved through finite principles, which is unsuitable, because there is no proportion of the finite to the infinite. Therefore it is clear that in one science not all principles are the same. I think this is the understanding of Aristotle, and that there is no other force in his reasons. But for the evidence of the things that have been said here, namely that terms thus assumed or inserted are not much fewer than conclusions, one must understand that principles, that is, premises, can be compared to their ultimate conclusions, which cannot be concluded further because, having been condensed, they have been reduced to immediate propositions; and thus in truth they are not fewer than conclusions, because each science has many middles for ultimate conclusions in this way, and as many middles as it has, so many principles it has, because the middle is the principle of the conclusion. And they can be compared not to ultimate conclusions but to any conclusions indefinitely; and thus principles are not much fewer than conclusions. The example of this can be if substance is shown of this human being through the middles which are body, animal, rational, and human: there will be six conclusions and four principles. For substance is concluded of animal through body, and of rational through animal, and of human through rational; and thus there are three conclusions. And again body is concluded of rational through animal, and of human through rational; and thus there are five conclusions. And again animal is concluded of human through rational; and thus there are six conclusions, and four principles which are immediate propositions for these conclusions, as is clear. For there are four according to the four middles, and thus there will be six conclusions and four immediate principles for concluding the six conclusions; and thus the principles are much fewer than the conclusions, and this is what Aristotle says. <marginal-label>A slight exposition of certain people.</marginal-label> Yet there are some who pass easily through this place and say that principles are therefore said not to be much fewer than conclusions because they are much more numerous. For if principles are mediate propositions, the premises are doubled for any conclusion, and thus principles are twice as many as conclusions. But the truth according to Aristotle's understanding is what was said first. Hence the Arabic translation has that the principles are not distant from the conclusions. Moreover, when propositions are said to be assumed or inserted terms, it is not understood in such a way that terms are formally predicated of propositions, but it is understood materially, because immediate propositions are materially from such terms. For premises are named and numbered from middle terms, because premises add nothing to the conclusion except only the middle; for whatever belongs to the major proposition and to the minor besides the middle is wholly in the conclusion, because the conclusion is the middle state of each premise. It must also be attended to that conclusions in one matter and another go on to infinity. And if someone says that a conclusion is known and known things do not fall upon the infinite, one must say that conclusions actually known by a human being are finite, and nevertheless in potency they are infinite, as is demonstrated in the fourth book of Euclid, from which it is proved that if a polygon is inscribed in a circle, with angles multiplied to infinity, it will not come to the quantity of the circle; but as often as a new angle increases, a new figure is made having its own properties to infinity; therefore they are multiplied. Therefore, if the principles of all things were the same, infinite things would be proved through finite things as through proper principles; and this cannot be. Moreover, the reason for the same is that, namely, the principles of all things are not the same, because it is established that the proximate principles are indeed other, or certain principles are from necessity, as of necessary things; other contingent principles are principles of contingent things, when they are proper and essential; for here we are speaking of such principles. Therefore with these reasons, which have been stated, held with such an intention, it is impossible that those principles of necessary things and contingent things be the same, and it is impossible for proper principles to be finite when conclusions are infinite. But if, in some way arguing to the man and not to the reason, some caviller says that these indeed are determined principles of geometry, and that other propositions are principles of syllogisms, that is, of the science of syllogizing, which is logic, and other propositions are principles of medicine, what else certainly will it be to say this, except that there are proper principles of proper sciences and diverse principles of diverse sciences? But because these are principles and those are principles, to say that the principles of all are the same is a sophistical response according to the consequent, because from this it follows that the same things are the same to the same things. Hence in this way all things become the same, just as principles are the same as principles, and the things principled by those are the same; and thus all things are one, as Melissus said. And this is also the special middle of the eighth reason for the same point, concluding the intention through another unsuitable consequence. But indeed it is also not possible to say that anything whatever, that is, every conclusion, is to be demonstrated from all things, that is, from anything whatever, so that all things are demonstrated indifferently from all things. For it belongs to these people who say this to seek that the principles of all things are the same, because thus all things would be demonstrated by the same things. To say such things is very foolish and stupid, for this is not possible in those things which are manifest teachings or doctrinal disciplines, as mathematics are. Nor in the resolution of some conclusion into premises is there this possibility, for proper principles are immediate propositions; but when another immediate proposition has been accepted for a premise, it comes to be, that is, another conclusion also follows, as is proved in the book of the Prior Analytics. Therefore in one and the same science there are diverse proper principles of diverse conclusions. But if from the things that have been said someone, conceiving an error according to the things that have been said, says that first and immediate propositions are proper principles, and that this principle is of one mode in each one genus and diverse in diverse genera; if indeed, according to the things said, it is not possible to demonstrate anything whatever from all things, that is, from anything whatever, as is needed, that is, in that mode in which anything whatever must be demonstrated, then again it happens thus, as is needed, to demonstrate conclusions of one science from another science, that is, from the principles of another. From this it follows that the principles of each science are other and other. Therefore it remains, if the principles of all things are proximate and not common, that there are no common principles of all things, but this conclusion is demonstrated from these proper principles, and another conclusion is demonstrated from those other proper principles, and there will be no principles common in all things. If, I say, someone says this, it is manifest from the things determined above that it does not happen, and it is false that there are no common principles because in determinate sciences some principles are proper and diverse in diverse sciences. For principles are twofold, that is, said in two ways or three ways, as principles from which demonstration is, and things which are in the demonstration itself, and things about which demonstration is. But the things from which demonstration is made are dignities not entering demonstrations, but standing outside and confirming the syllogistic course of demonstration; and these are common. But the things that are demonstrated and are in the demonstration itself are properties; and the things about which demonstration is made are subjects, and propositions in which these are predicates and subjects; and these are proper and diverse in diverse sciences. Therefore it is clear that the principles from which, that is, from whose power there is demonstration, are common, but the principles which and about which are proper, as number in arithmetic and magnitude in continuous things, as in geometry. This is, as I think, Aristotle's meaning, whence the principal intention is that the same principles do not belong to all conclusions. This is clear from this: because if the principles of all conclusions were the same, then let the conclusions be E, F, G, and the principles A, B, C, D; any conclusion E, F, G will be concluded from absolutely any one singly of A, B, C, D, so that any is concluded from A and any from B, and so concerning the others. But this is false. <marginal-label>He removes a doubt.</marginal-label> But it must be attended to that, although the metaphysician proves and establishes the principles of all things, nevertheless it is not his task to determine whether they are the same for all conclusions; rather, this belongs to that science, that is, the science that teaches how to compose principles with a conclusion. <marginal-label>He removes a doubt.</marginal-label> Moreover, although there is one syllogistic science, which all sciences use, determined in the book of the Prior Analytics, nevertheless on account of this the principles of all conclusions are not the same. For the principles are the same as regards those things by which the composition of the syllogism and the relation of the three terms in mode and figure are regulated; but when it is said that the principles of all are not the same, this is understood of principles which are premises referred to the conclusion immediately following from the premises. It must also be attended to, as certain people say in explanation of the last part of the chapter, that certain people say Aristotle intends to exclude the saying of the caviller who says that, although the resolution of the conclusion made to proximate and proper principles shows that diverse things have diverse principles, nevertheless the resolution made to first principles in any genus shows that the principles of all conclusions are the same. They say that the resolution does not stop unless it is led back all the way to the first things before which nothing can be accepted. But they call first those things which are enunciated of the first subject, because it is either a predicate of the first subject through its proper definition, as concerning magnitude in geometry and concerning number in arithmetic, and so concerning the others; and they say that those propositions are common principles in sciences for diverse conclusions. And when Aristotle says it does not happen, they gloss that this is not contingent but necessary. But the first thing that was said also seems more to belong to Aristotle's intention. Therefore let these things have been said concerning the unity of science to science and concerning diversity.
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