Works › Prior Analytics, Books I–II
Volume 1 · pp. 620–623
Chapter V. On the number of propositions in a syllogism both simply one and not simply one, and how the number of propositions is related to the number of terms and conclusions.
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CAPUT V. De numero propositionum in syllogismo et simpliciter uno et non simpliciter uno, et qualiter se habet numerus propositionum ad numerum terminorum et conclusionum.
Hoc autem quod dictum est de numero terminorum sicut dictum est, manifesto sive manifestato, ostendendum est quod syllogismus omnis simpliciter unus ex duabus est propositionibus et non pluribus: nam tres termini secundum substantiam et actum et rationem terminorum materialiter sunt duae propositiones: eo quod medium secundum rationem et actum mediandi necesse in utraque accipi propositione: et nisi sic fiat non erit medium in ratione et actu mediandi: secundum enim unum actum mediandi dicitur medium terminus unus: et hoc modo tres termini sunt completa et sufficiens materia duarum propositionum.
Et si objicitur de enthymemate, ubi sunt duo termini, et una propositio; facile est solvere, dicendo quod enthymema non est syllogismus apertus et perfectus, sed occultus. Quia una propositio ejus, secundum quod occultus syllogismus est, tenetur in mente: et hoc modo habet tres terminos et duas propositiones. Et patet ex ratione jam inducta, quod tres termini ad se invicem secundum rationem et actum terminorum terminati constituunt de necessitate duas propositiones.
Et si objiciatur quod duae propositiones habent quatuor terminos: vel potest dici sicut in praecedenti capitulo hoc solutum est; vel melius potest dici, quod alii sunt termini syllogismi, et alii propositionum. Termini enim syllogismi sunt terminantes sufficienter consequentiam syllogisticam unius ex alio: et tales termini de necessitate sunt tres: quia non concluditur unus de alio nisi per tertium qui est medium. Termini autem propositionis sunt, in quibus stat propositionis resolutio: unde hoc modo duarum propositionum sunt quatuor termini: qui tamen quatuor termini non sunt nisi tres termini syllogismi.
Secundum hunc igitur modum tres termini sunt duae propositiones, nisi aliqui plures termini assumantur ad probandum majorem vel minorem principalis syllogismi sive perfectionem: eo quod aliquando secundum probationem sufficientem syllogismus in suis propositionibus perfectus non est: tunc autem sic coassumptis, pluribus terminis, non est syllogismus simpliciter unus secundum substantiam, quemadmodum in prioribus praecedentis capituli dictum est: quod fit in prosyllogismis qui ad perfectionem syllogismi principalis adducuntur, qui syllogismus finaliter est unus, quamvis secundum substantiam non sit simpliciter unus.
Ex hoc autem manifestum est, quoniam in quacumque oratione syllogistica ad inferendum conclusionem perfectam non sunt nisi pares propositiones, hoc est, duae. Quia duo est primus par numerus, sicut ternarius est primus impar. Propositiones dico, per quas ordinatas in modo fit conclusio syllogismi principalis. Dico autem principalis: quia etiam superiorum conclusionum quae concluduntur in prosyllogismis, necessarium est esse quasdam propositiones, per quas concluduntur: sed sicut conclusio non est principalis, ita nec illae propositiones sunt principalis syllogismi propositiones, et ideo talis oratio non est syllogistica, quia non est simpliciter unus syllogismus: aut plura interrogavit ad conclusionem principalem et pro conclusione: aut non est syllogizata in modo et figura: interrogantur enim aliquando plura ad propositam conclusionem in dialecticis syllogismis, sicut in octavo Topicorum docet Aristoteles.
CHAPTER V. On the number of propositions in a syllogism both simply one and not simply one, and how the number of propositions is related to the number of terms and conclusions.
But when this, which has been said about the number of terms, is manifest or has been made manifest, it must be shown that every syllogism which is simply one is from two propositions and not from more. For three terms according to the substance, act, and reason of terms are materially two propositions, because the middle according to the reason and act of mediating must be accepted in each proposition; and unless this is done, it will not be a middle in the reason and act of mediating. For according to one act of mediating the middle is called one term; and in this way three terms are the complete and sufficient matter of two propositions.
And if an objection is made about the enthymeme, where there are two terms and one proposition, it is easy to solve it by saying that an enthymeme is not an open and perfect syllogism, but a hidden one. For one proposition of it, insofar as it is a hidden syllogism, is held in the mind; and in this way it has three terms and two propositions. And it is clear from the reason already introduced that three terms, terminated toward one another according to the reason and act of terms, necessarily constitute two propositions.
And if it is objected that two propositions have four terms, either one can say that this was solved in the preceding chapter, or it can be said better that the terms of a syllogism are one thing and the terms of propositions another. For the terms of a syllogism are those sufficiently terminating the syllogistic consequence of one thing from another; and such terms are necessarily three, because one thing is not concluded of another except through a third, which is the middle. But the terms of a proposition are those in which the resolution of the proposition stands; hence in this way there are four terms of two propositions, which four terms nevertheless are only three terms of the syllogism.
According to this mode, therefore, three terms are two propositions, unless some additional terms are assumed to prove the major or the minor of the principal syllogism or its perfection, because sometimes according to sufficient proof a syllogism is not perfect in its propositions. But then, when several terms are co-assumed in this way, the syllogism is not simply one according to substance, as was said in the preceding parts of the preceding chapter. This happens in prosyllogisms which are brought in for the perfection of the principal syllogism, which syllogism is finally one, although according to substance it is not simply one.
From this it is manifest that in any syllogistic discourse whatever for inferring a perfect conclusion there are only even propositions, that is, two; because two is the first even number, just as three is the first odd number. I mean the propositions by which, when ordered in a mode, the conclusion of the principal syllogism is made. But I say principal, because even of prior conclusions which are concluded in prosyllogisms there must be certain propositions through which they are concluded. But just as the conclusion is not principal, so neither are those propositions the propositions of the principal syllogism; and therefore such discourse is not syllogistic, because it is not simply one syllogism, or it asked several things toward and for the principal conclusion, or it was not syllogized in mode and figure. For sometimes several things are asked toward the proposed conclusion in dialectical syllogisms, as Aristotle teaches in the eighth book of the Topics.
Sed sumptis syllogismis secundum propositiones principales syllogismi, omnis syllogismus est ex propositionibus perfectis, hoc est paribus, quia ex duabus, et est ex terminis abundantibus, hoc est, imparibus, quia ex tribus terminis: terminis enim sunt uno plures, quam propositiones. Quia aliter consequentia extremi ad extremum terminari non potest, nisi per medium. Unde propterea quod unum de uno concluditur, sunt duo termini, et propter consequentiam oportet esse tertium qui est medius terminus, qui bis sumptus duas facit propositiones; et consequentiam tertiae quae est conclusio ex duabus: quam consequentiam facere non posset, nisi referretur ad extremum utriusque propositionis. Propter quod etiam talis proportio est inter propositiones syllogismi et conclusionem, quod conclusio est dimidietas propositionum: quia conclusio principalis est una, et propositiones praemissae duae. Quamvis in prae habitis de mixtionibus ostensum, quod aliquando ex uno syllogismo sequuntur duae conclusiones: tamen illae ambae conclusiones non sunt aeque immediatae, nec aeque principales: sed una est conclusio immediata, et alia secundaria quae concluditur per illam quae est principalis.
Quando autem per antesyllogismos, qui dicuntur prosyllogismi, concluditur altera propositionum praemissarum, super alteram praemissarum vel sub alteram, medium accipiendo ad prosyllogismum: tunc sunt quasi media continua, quasi secundum eamdem lineam ad unam accepta conclusionem: immediato tamen unum accipiendo super vel sub altero extremorum, tunc dicuntur media continua. Aut quando diversae conclusiones concluduntur per plura media non continua, quae secundum unam lineam non accipiuntur ad eamdem conclusionem, sicut in syllogismis qui non sunt unus syllogismus finali conclusione una, sicut si A B sit conclusio quae concludatur per C D duo media continua vel non continua, tunc adhuc multitudo terminorum in propositionibus talium syllogismorum positorum, uno secundum numerum superabit multitudinem propositionum.
Hoc autem sic probatur: si enim in continuis terminis secundum unam lineam superioris et inferioris acceptis, accipiatur terminus qui syllogismo principali est extrinsecus, et in prosyllogismo est necessarius, qui etiam intercidens terminus vocatur: aut ille secundum lineam et ordinem terminorum sumetur extrinsecus in linea, aut ponetur intrinsecus ad ordinem medii inter extrema in principali syllogismo posita. Dico autem extrinsecus idem quod ad extrinsecus in ordine terminorum, et sumatur medium aliquod super extremum majus ascendendo, sicut super hominem animal, et super animal vivum, et super vivum corpus, et super corpus substantia: aut sumatur descendendo sub minori extremitate, sicut si medium sit vivum, et sumatur sub vivo animal, et sub animali homo, et sub homine aliquis homo, et sic deinceps: utrumque enim vocatur extrinsecus assumptum. Ad medium autem sive ad intrinsecus dicitur poni intercidens terminus, quando sumitur inter extrema principalis syllogismi, sub majori scilicet, et super minorem extremitatem assumpto termino intercidente: et hoc fit quando non per immediatum medium concluditur extremum distans de extremo, sicut si concludatur substantia de quodam homine per vivum: tunc enim inter vivum et quemdam hominem inferius possunt sumi media animal et homo, et inter substantiam et vivum possunt sumi media corpus et animatum et hujusmodi: quae omnia ad medium, quod est inter extrema, ordinantur: et utroque modo, extrinsecus scilicet et intrinsecus assumendo multiplicantur termini per talium mediorum assumptionem.
But if syllogisms are taken according to the principal propositions of the syllogism, every syllogism is from perfect propositions, that is, even ones, because from two; and it is from abundant terms, that is, odd ones, because from three terms. For there is one more term than propositions. For otherwise the consequence of extreme to extreme cannot be terminated except through a middle. Hence, because one thing is concluded of one, there are two terms; and because of the consequence there must be a third which is the middle term, which, taken twice, makes two propositions and the consequence of the third, which is the conclusion, from the two. It could not make this consequence unless it were referred to the extreme of each proposition. For this reason there is also such a proportion between the propositions of the syllogism and the conclusion, that the conclusion is half the propositions, because the principal conclusion is one and the premise-propositions are two. Although in the preceding matters concerning mixings it was shown that sometimes two conclusions follow from one syllogism, nevertheless those two conclusions are not equally immediate or equally principal, but one is the immediate conclusion and the other secondary, which is concluded through that which is principal.
But when through ante-syllogisms, which are called prosyllogisms, one of the premise-propositions is concluded above or below the other premise, by accepting a middle for the prosyllogism, then they are as continuous middles, as if accepted according to the same line for one conclusion; yet by immediately accepting one above or below the other of the extremes, they are then called continuous middles. Or when diverse conclusions are concluded through several non-continuous middles, which are not accepted according to one line for the same conclusion, as in syllogisms which are not one syllogism with one final conclusion, as if A B is the conclusion which is concluded through C D, two middles continuous or non-continuous, then still the multitude of terms posited in the propositions of such syllogisms will exceed the multitude of propositions by one in number.
But this is proved thus. For if in continuous terms accepted according to one line of superior and inferior, a term is accepted which is extrinsic to the principal syllogism and necessary in the prosyllogism, which is also called an intervening term, either it will be taken extrinsically in the line according to the line and order of terms, or it will be placed intrinsically with respect to the order of the middle between the extremes posited in the principal syllogism. I call extrinsic the same as toward what is outside in the order of terms, and some middle is taken above the greater extreme by ascending, as animal above man, living above animal, body above living, and substance above body; or it is taken by descending under the lesser extreme, as if the middle is living, and under living animal is taken, and under animal man, and under man some man, and so on. For each is called extrinsically assumed. But an intervening term is said to be placed at the middle or intrinsically when it is taken between the extremes of the principal syllogism, namely under the major and above the lesser extreme, when the intervening term has been assumed. And this happens when a distant extreme is not concluded of an extreme through an immediate middle, as if substance is concluded of some man through living. Then between living and some lower man the middles animal and man can be taken, and between substance and living the middles body and animated and the like can be taken; all of these are ordered to the middle which is between the extremes. And in both ways, namely by assuming extrinsically and intrinsically, terms are multiplied through the assumption of such middles.
Utrumque autem istorum modorum accidit semper uno minus esse intervalla, quae sunt propositiones, quam terminos: semper enim assumptum medium est inter extrema conjungibilia per medium: et talis conjungibilitas non potest esse nisi termini in uno mediante inter duo superent propositiones. Propositiones autem intervalla dicuntur, quia aequales sunt intervallis: et hoc in ante habitis aequaliter sic expositum est.
Et quamvis sic terminorum numerus uno excedat numerum intervallorum, non tamen semper propositiones sic assumptae perfectae sunt, hoc est, secundum numerum, et termini abundantes, hoc est, secundum numerum impares, quia et ad extrinsecus et ad intrinsecus possunt et pariter et impariter assumi. Sed in hoc permutatim se habent termini et propositiones: quia cum propositiones sunt perfectae sive pares secundum numerum, tunc termini assumpti sunt abundantes, sive impares, et e converso cum propositiones sunt impares, sive abundantes, termini sunt pares: uno enim modorum numerus terminorum ad numerum propositionum se habere non potest, cum medium semper sit necessarium ad extrema conjungenda, et cum extrema sint duarum propositionum, necesse quod numerus terminorum uno vincat numerum propositionum in talibus prosyllogismis sic assumptis. Et hujus causa est, quod semper cum termino uno sic assumpto additur una propositio, per illius termini relationem ad medium, undecumque sive extrinsecus, sive intrinsecus sumptum addatur. Propter quod quando termini sunt abundantes, propositiones erunt perfectae, et e converso: et ideo necesse est sic transmutari secundum par et impar terminos et propositiones addita una et eadem forma in terminis.
Conclusiones autem ex talibus terminis et propositionibus conclusae non jam eumdem habebunt ordinem secundum numerum parem vel imparem, neque ad terminos, neque ad propositiones: uno enim termino assumpto addito, conclusiones tunc adjungentur, quod non sunt nisi una unitate pauciores praeexistentibus terminis: quilibet enim superior de inferiori concludi potest, et solus ultimus in tali ordine non potest concludi de aliquo: quia inter ultimum et aliquod inferius sumptum nihil est medium: eo quod nihil sit sub ipso ulterius quod accipi possit sub ipso: ad alios autem terminos superiores omnes fit conclusio superioris de inferiori per medium sumptum. Hujus autem exemplum est, sicut si dicamus, quod eis terminis unius syllogismi qui sunt A B C in quibus A de C per B adjacet terminus assumptus intrinsecus, qui est D, sic A B C D: tunc enim statim duae adjacebunt conclusiones: quia una conclusio est, quae est A majus extremum conclusum de B per C medium, et illa quae est A de B conclusum de C per D medium assumptum. Similiter autem est et in omnibus aliis assumptis terminis, quoad portionem numeri terminorum ad numerum conclusionum in ordine, quando ad extra sumitur terminus, sive ascendendo, sive descendendo.
Si autem ad medium intrinsecus sumptus intercidit terminus, eodem modo se habebit quoad numerum conclusionum: quia ad unum solum ultimum in ordine terminorum non faciet syllogismum: quia hoc de nullo concludi potest. Propter quod patet quod in tali assumptione multo plures erunt conclusiones quam termini sint vel propositiones. Est tamen hic attendendum, quod quamvis ultimus terminus in descendendo solus nunquam de aliquo concludi possit in uno ordine terminorum, et primus in ascensu sit de quo nihil concludi possit, eo quod inter ipsum et eum qui concludi deberet de ipso, non potest esse medium: quia aliter sequeretur, quod aliquid esset superius supremo, quod esse non potest: et hoc modo ad neutrum ultimorum potest fieri conclusio: sed ad quodlibet uno mediorum sumpto intrinsecus inter extrema ad medium adduntur duae propositiones: sint enim termini syllogismi principalis A B C, tunc concluditur A de C per B, et sunt istae propositiones, omne A est B, omne C B, et tres termini: sumatur unus terminus inter A et B: et sit ille D, tunc erunt duae novae, novi syllogismi, aliae propositiones, hae scilicet, omne D est A, et omne B est D, et concluditur A de B per D, et tunc videntur esse quatuor termini et quatuor propositiones: et sic tot esse termini quot propositiones. Tamen hoc non est ad unam et eamdem conclusionem, sed ad duas: sed cum propositio in talibus syllogismis non sit intervallum inter duos terminos immediate sumptos, qui sunt subjectum et praedicatum, tunc aliae duae propositiones prioris syllogismi disparent et deficiunt, quae fuerunt omne B A et omne C B, quia tunc B non est immediatum ad A, sed potius conjungitur ei in conclusione nova per D: per D enim concluditur A de B, et sic remanet adhuc, quod duarum propositionum sunt tres termini, et non quatuor: quia priorum propositionum disparet una, et duae novae ex assumptione unius termini generantur.
But in each of those modes it happens that the intervals, which are propositions, are always one less than the terms. For the assumed middle is always between extremes joinable through a middle; and such joinability cannot exist unless the terms, with one mediating between two, exceed the propositions. But propositions are called intervals because they are equal to intervals; and this was explained equally in this way in the preceding matters.
And although in this way the number of terms exceeds the number of intervals by one, nevertheless the propositions thus assumed are not always perfect, that is, even in number, and the terms abundant, that is, odd in number, because they can be assumed both extrinsically and intrinsically both evenly and unevenly. But in this the terms and propositions have themselves by permutation: when the propositions are perfect or even according to number, then the assumed terms are abundant or odd; and conversely, when the propositions are odd or abundant, the terms are even. For the number of terms cannot have itself to the number of propositions in only one of the modes, since the middle is always necessary for joining the extremes, and since the extremes belong to two propositions, it is necessary that the number of terms exceed the number of propositions by one in such prosyllogisms thus assumed. And the cause of this is that, always with one term thus assumed, one proposition is added through the relation of that term to the middle, from wherever, whether extrinsically or intrinsically, it is taken and added. For this reason, when the terms are abundant, the propositions will be perfect, and conversely; and therefore it is necessary that terms and propositions be thus transmuted according to even and odd when one and the same form has been added in the terms.
But conclusions concluded from such terms and propositions will no longer have the same order according to even or odd number, neither with respect to terms nor to propositions. For when one assumed term is added, conclusions are then joined, which are only one unit fewer than the pre-existing terms. For any superior can be concluded of an inferior, and only the last in such an order cannot be concluded of anything, because between the last and something inferior taken there is no middle, since there is nothing further under it which can be accepted under it. But to all other superior terms a conclusion of the superior about the inferior is made through the middle taken. An example of this is as if we say that to the terms of one syllogism, which are A B C, in which A is adjacent to C through B, an intrinsically assumed term, D, is added, thus A B C D. Then two conclusions will immediately be adjacent, because one conclusion is A, the greater extreme, concluded of B through the middle C, and the other is A concluded of B through the assumed middle C through D. It is similar also in all other assumed terms with respect to the portion of the number of terms to the number of conclusions in the order, when a term is taken outwardly, whether by ascending or by descending.
But if a term assumed intrinsically intervenes at the middle, it will have itself in the same way with respect to the number of conclusions, because it will not make a syllogism to only one last term in the order of terms, since this cannot be concluded of anything. Therefore it is clear that in such an assumption there will be many more conclusions than there are terms or propositions. Yet it must be attended here that, although the last term in descending alone can never be concluded of anything in one order of terms, and the first in ascending is that of which nothing can be concluded, because between it and that which ought to be concluded of it there can be no middle, since otherwise it would follow that something is higher than the highest, which cannot be; and in this way no conclusion can be made to either of the last terms. But to any one whatever, when one of the middles has been taken intrinsically between the extremes, two propositions are added to the middle. For let the terms of the principal syllogism be A B C; then A is concluded of C through B, and the propositions are these: every A is B, every C is B, and there are three terms. Let one term be taken between A and B, and let it be D; then there will be two new propositions, of a new syllogism, namely these, every D is A, and every B is D; and A is concluded of B through D. And then there seem to be four terms and four propositions, and thus as many terms as propositions. Nevertheless this is not toward one and the same conclusion, but toward two; but since a proposition in such syllogisms is not an interval between two terms immediately taken, which are subject and predicate, then two other propositions of the prior syllogism disappear and fail, which were every B A and every C B, because then B is not immediate to A, but rather is joined to it in a new conclusion through D. For A is concluded of B through D, and thus it still remains that there are three terms of two propositions and not four, because one of the prior propositions disappears and two new ones are generated from the assumption of one term.
Et omni eodem modo est si inter medium et minorem extremitatem terminus aliquis assumatur, ut cum dicitur, quod termini sint A B C, et inter B et C sumatur E, tunc enim per E concludetur B de C, et hoc modo etiam patet quod uno termino addito duae fiunt conclusiones: quia concluditur A de C per B, et concluditur A de B per D, et similiter est ex alia parte inter medium et minorem extremitatem: tunc enim concludetur medium de minori extremitate, et medium fit extremum per novum assumptum medium. Si autem omnino fiunt diversi syllogismi, sicut si unus sit A B C, et alter D E F, tunc oportet sex esse terminos, et quatuor propositiones: et hoc non est contrarium ad ea quae dicta sunt. Sed ad intrinsecus inter majus extremum et medium, et inter medium et minus extremum sumuntur media in prosyllogismis: omnino autem diversi termini sumuntur ad eos qui positi sunt diversi syllogismos.
Et patet quod in uno ordine terminorum uno addito termino semper additur nova conclusio praeter ultimum, sicut dictum est: quia novum inter extrema medium unam facit conclusionem: sed tunc quae prius fuerunt extrema in syllogismo principali, efficiuntur non extrema nisi aliquod medium alterum sumatur extremorum, cum quo id quod prius erat medium, efficitur extremum; et hoc est quod in ista parte probatur. Et patet quod impossibile est quod ad diversas conclusiones quocumque modo factas, non sint diversae propositiones et diversi termini, si secundum rationem terminorum accipiantur.
And it is in every way the same if some term is assumed between the middle and the lesser extreme, as when it is said that the terms are A B C and E is taken between B and C; for then through E, B will be concluded of C. And in this way also it is clear that when one term is added, two conclusions are made, because A is concluded of C through B, and A is concluded of B through D, and it is similar from the other side between the middle and the lesser extreme. For then the middle will be concluded of the lesser extreme, and the middle becomes an extreme through the newly assumed middle. But if entirely diverse syllogisms are made, as if one is A B C and another D E F, then there must be six terms and four propositions, and this is not contrary to what has been said. But intrinsically, between the greater extreme and the middle, and between the middle and the lesser extreme, middles are taken in prosyllogisms; but altogether diverse terms are taken for those which have been posited as diverse syllogisms.
And it is clear that in one order of terms, when one term has been added, a new conclusion is always added besides the last, as has been said, because a new middle between the extremes makes one conclusion. But then the things which were previously extremes in the principal syllogism are not made non-extremes unless some other middle of the extremes is taken, with which that which was previously the middle becomes an extreme; and this is what is proved in this part. And it is clear that it is impossible that for diverse conclusions made in whatever way there not be diverse propositions and diverse terms, if they are accepted according to the reason of terms.
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