Works › Prior Analytics, Books I–II
Volume 1 · pp. 617–620
Chapter IV. It is proved that no one syllogism has more than three terms.
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CAPUT IV. Probatur quod nullus unus syllogismus habet plus quam tres terminos.
Palam autem hic erit, quoniam omnis demonstratio sive omnis ostensio syllogistica erit per tres terminos et non plures ad eamdem conclusionem contendentes, et unam et eamdem conclusionem inferentes, nisi forte per alia et alia media fiat sive concludatur una et eadem conclusio: sicut fit in dialecticis et probabilibus, in quibus una et eadem conclusio per plura media saepe probatur: sed tunc non est syllogismus unus, sed plures ad unam tendentes conclusionem.
Hoc autem duobus modis fieri potest. Possunt enim plura media ad unam conclusionem ordinata inter se esse non continua sive communicantia, et possunt esse communicantia in aliquo uno quod se habeat ut continuans inter illa et connectens ea. Et exemplum primi est, ut dicatur E esse conclusio principalis intenta, et concludatur per A B, sicut per unum medium, et concludatur etiam per E D, sicut per aliud medium, quae in nullo uno junguntur inter se: quod si connectens ea, et sic ad unam conclusionem ordinata media constituunt duos diversos syllogismos, qui non sunt unus syllogismus, sed sunt unius conclusionis. Exemplum autem secundi est sicut si E conclusio principalis intenta, dicatur concludi per A B sicut per medium unum, et per A C sicut per medium alterum, et per B C sicut per medium tertium: quae tamen media inter se communicant in E quod est commune secundo et tertio, vel in A quod est commune primo et secundo, sicut fit in prosyllogismis, qui inducuntur ad majoris vel minoris declarationem: hos enim necesse est communem habere aliquem terminum, sicut et ipsa major et minor in principali syllogismo in uno termino medio communicant. His enim duobus modis plura esse media ad eamdem conclusionem nihil prohibet.
Haec autem media cum his duobus modis sumpta, non sunt unus secundum substantiam syllogismus, sed sunt plures secundum substantiam syllogismi. Hoc autem fit in prosyllogismis, sicut si A B sint duae propositiones habentes tres terminos ad concludendum quod principalem diximus esse conclusionem, et ad eam probandam sive concludendam sumuntur A propositio, et B propositio, et A propositio sit probanda et probetur per D C, duas propositiones communicantes in uno medio: et rursum B secunda propositio probanda sit per F G, alias duas propositiones in alio medio communicantes: aut unum quidem principalis syllogismi probetur inductione: aliud autem quod est secundum in principali syllogismo, probetur syllogismo, qui prosyllogismus vocatur, inductio enim ad syllogismum habet reduci. Sed omnibus his modis sunt plures syllogismi secundum substantiam, et non unus: cujus signum est, quia sunt plures ab eis conclusae conclusiones, sicut conclusio A quae concluditur in inductione, et conclusio B quae concluditur in prosyllogismo, et conclusio E quae concluditur in syllogismo principali.
Si autem istae conclusiones ab aliquo non dicantur esse plures, sed una, eo quod una est conclusio finaliter intenta, ad quam aliae in prosyllogismis inductae sunt: sic bene conceditur, quod contingit fieri unam ultimo intentam conclusionem per plura media sic ad unum ordinata. Sed hoc modo hoc fieri, quod est principalis conclusio concludatur per A B, sicut per plura media secundum substantiam, est impossibile: quia cum syllogismus sit causa immediata conclusionis, et per suam substantiam sit causa, sequeretur quod plurium causarum secundum unum modum causalitatis esset unus et idem substantialis effectus. Et hoc est impossibile, si causae sunt aeque immediatae et secundum idem genus causalitatis.
CHAPTER IV. It is proved that no one syllogism has more than three terms.
But here it will be plain that every demonstration or every syllogistic showing will be through three terms and not more terms tending to the same conclusion and inferring one and the same conclusion, unless perhaps one and the same conclusion is made or concluded through different middles, as happens in dialectical and probable matters, in which one and the same conclusion is often proved through many middles; but then it is not one syllogism, but many tending to one conclusion.
But this can happen in two ways. For many middles ordered to one conclusion can be non-continuous or non-communicating among themselves, and they can be communicating in some one thing which has itself as continuing and connecting them. An example of the first is that E is said to be the principal intended conclusion, and it is concluded through A B as through one middle, and is also concluded through E D as through another middle, which are not joined to each other in any one thing. If something connects them, and middles thus ordered to one conclusion constitute two diverse syllogisms, they are not one syllogism, but belong to one conclusion. An example of the second is as if E, the principal intended conclusion, is said to be concluded through A B as through one middle, through A C as through another middle, and through B C as through a third middle; yet these middles communicate with one another in E, which is common to the second and third, or in A, which is common to the first and second, as happens in prosyllogisms which are introduced for the declaration of the major or minor. For these must have some common term, just as the major and minor themselves in the principal syllogism communicate in one middle term. In these two ways, then, nothing prevents there being several middles for the same conclusion.
But these middles taken in these two ways are not one syllogism according to substance, but are several according to the substance of the syllogism. This happens in prosyllogisms, as if A B are two propositions having three terms for concluding what we have called the principal conclusion, and for proving or concluding it proposition A and proposition B are taken, and proposition A is to be proved and is proved through D C, two propositions communicating in one middle; and again B, the second proposition, is to be proved through F G, another two propositions communicating in another middle. Or one thing of the principal syllogism is proved by induction, while another thing which is second in the principal syllogism is proved by a syllogism which is called a prosyllogism, for induction has to be reduced to syllogism. But in all these ways there are several syllogisms according to substance and not one, of which the sign is that there are several conclusions concluded by them, such as conclusion A which is concluded in the induction, conclusion B which is concluded in the prosyllogism, and conclusion E which is concluded in the principal syllogism.
But if those conclusions are not said by someone to be many, but one, because one is the finally intended conclusion to which the others have been introduced in prosyllogisms, in this way it is well conceded that one ultimately intended conclusion can happen to be made through several middles thus ordered to one. But for this to be made in such a way that the principal conclusion is concluded through A B as through several middles according to substance is impossible; because since a syllogism is the immediate cause of the conclusion and is a cause through its substance, it would follow that one and the same substantial effect belongs to several causes according to one mode of causality. And this is impossible if the causes are equally immediate and according to the same genus of causality.
Sit enim gratia exempli E principaliter intenta conclusio, conclusum ex A una propositione, et B alia, et ex C et D aliis propositionibus aliud habentibus medium, et sic sit unum conclusum per plura; sequitur ex prius habitis, quod in utroque syllogismo necesse est, quod unum se habeat ad alterum sicut totum se habet ad partem, sicut universalis se habet ad particularem, et unus terminus se habeat ad alterum sicut totum ad partem, sicut major extremitas se habet ad medium et medium ad minorem in prima figura: medium autem se habet ut totum ad utrumque extremum in secunda figura, et extrema se habent ad medium sicut totum ad partem in tertia: quia sine tali habitudine totius ad partem non fit decursus syllogisticus. Diximus enim in prae habitis, quoniam si est syllogismus, necesse est quod aliqui terminorum syllogismi sic se habeant ut totum et pars ad invicem.
Ponamus ergo quod A sic se habeat ad B sicut totum se habet ad partem: sequitur itaque, quod conclusio est ex eis aliquid quod sequitur ex ipsis per consequentiam syllogisticam. Haec ergo conclusio: aut est conclusio E quam diximus esse propriam, aut est alia conclusio quam E, sicut si dicamus quod alterum istorum concludatur quae sunt C D quae alia sunt ab E, sicut si praemissarum altera concludatur per syllogismum. Aut si dicamus quod aliquod aliud concludatur quodcumque quod est aliud et a praemissis et a conclusione principali; et est praeter omnia haec tria.
Et si primum membrum hujus detur divisionis, quod scilicet ex A B propositionibus ut totum et pars se habentibus, concluditur E conclusio principalis: tunc ex A B propositionibus erit syllogismus ad E, et illae duae propositiones non habent nisi tres terminos, in quorum uno communicant et connectuntur. Si autem et aliae propositiones quae sunt C D sic se habeant ad invicem ut totum et pars: tunc etiam aliquid quod est conclusio, sequitur etiam ex illis: aut igitur ex illis sequitur etiam E quae est primi syllogismi conclusio: aut sequitur altera praemissarum in primo syllogismo, sicut aut A aut B major scilicet vel minor: aut sequitur aliud aliquid praeter haec tria, quod nihil horum est. Et si quidem datur hujus divisionis primum membrum, quod scilicet sequitur E, tunc erit syllogizatum per plura media et plures syllogismos. Si autem datur secundum membrum, quod scilicet sequitur alterum, aut A scilicet aut B, erunt plures prosyllogismi: et sic semper aut erunt plures syllogismi sicut prosyllogismi sunt plures, et cadunt in numeris cum principali syllogismo: aut sunt plures syllogismi ad unam conclusionem inducti, sicut diximus paulo ante, quod accidit idem sive eamdem conclusionem per plura media et per plures terminos concludi: hoc enim saepe contingit in dialecticis, et etiam in demonstrativis, ubi idem per plures causas demonstratur demonstrationibus diversis. Si autem datur tertium membrum, scilicet quod concluditur aliquod aliud praeter haec tria quae dicta sunt, scilicet quod nec principalis conclusio, nec major, nec minor syllogismi principalis: tunc plures erunt illi syllogismi et inconjuncti ad invicem et incommunicantes in terminis.
For the sake of example let E be the principally intended conclusion, concluded from A, one proposition, and B, another, and from C and D, other propositions having another middle; and thus let one thing be concluded through many. From what has been had before it follows that in each syllogism one must have itself to the other as the whole has itself to the part, as a universal has itself to a particular; and one term must have itself to another as whole to part, as the greater extreme has itself to the middle and the middle to the lesser in the first figure. But the middle has itself as a whole to each extreme in the second figure, and the extremes have themselves to the middle as whole to part in the third, because without such a relation of whole to part there is no syllogistic course. For we said in the preceding matters that if there is a syllogism, some terms of the syllogism must have themselves to one another as whole and part.
Therefore let us posit that A has itself to B as a whole has itself to a part. And so it follows that the conclusion is something from them which follows from them through syllogistic consequence. Therefore this conclusion is either the conclusion E which we have said is proper, or it is another conclusion than E, as if we say that another of those things which are C D is concluded, which are other than E, as if one of the premises is concluded through a syllogism. Or if we say that some other thing whatever is concluded which is other than both the premises and the principal conclusion, and is beyond all these three.
And if the first member of this division is granted, namely that from the A B propositions having themselves as whole and part the principal conclusion E is concluded, then from propositions A B there will be a syllogism to E, and those two propositions have only three terms, in one of which they communicate and are connected. But if the other propositions which are C D also have themselves to one another as whole and part, then something which is a conclusion also follows from them. Therefore either E, which is the conclusion of the first syllogism, also follows from them, or the other of the premises in the first syllogism follows, namely either A or B, that is, the major or the minor, or something else beyond these three follows, which is none of these. And if the first member of this division is granted, namely that E follows, then it will have been syllogized through several middles and several syllogisms. But if the second member is granted, namely that the other follows, either A or B, there will be several prosyllogisms; and thus always either there will be several syllogisms, just as the prosyllogisms are several and fall into number together with the principal syllogism, or there are several syllogisms introduced for one conclusion, just as we said a little before that it happens that the same thing or the same conclusion is concluded through several middles and through several terms. For this often happens in dialectical matters, and also in demonstrations where the same thing is demonstrated through several causes by diverse demonstrations. But if the third member is granted, namely that something else beyond these three things which have been said is concluded, namely neither the principal conclusion, nor the major, nor the minor of the principal syllogism, then those syllogisms will be many and unjoined to one another and non-communicating in terms.
Si autem dicatur quod C et D propositiones non se habeant ad invicem ut totum et pars, ut facere possint syllogismum sive decursum syllogisticum, tunc varie assumpta sunt ad propositum nihil concludentia: nisi forte dicatur, quod ut praeter necessaria accepta sint: sicut ea quae aut inductionis probantis universalem, vel relationis propositi, vel alterius alicujus talium gratia assumantur. Sed tamen quocumque modo sumantur, semper erunt secundum substantiam plures syllogismi. Si autem dicatur quod ex A B propositionibus non est consequens E conclusio principalis, sed aliquod aliud fiat conclusio ex A B propositionibus quam E, ex C D autem aliis propositionibus assumptis fiat consequens conclusio horum alterum, scilicet aut A aut B sicut in prosyllogismis: aut forte dicatur quod alia quaecumque conclusio sequitur et concluditur ex eo propositionibus praeter haec tria quae dicta sunt: constat quod omnibus his modis plures sunt syllogismi, sed non ad hoc quod est propositum: quia non fiunt ad primam conclusionem. Ponebatur enim a nobis paulo ante, quod A B propositiones sunt ejus conclusionis quam diximus esse E, C D autem propositiones, sicut jam dictum est, non sint ejusdem conclusionis: et sic plures sunt non communicantes sibi syllogismi.
Si autem dicat aliquis, quod ex C D propositionibus nulla fit penitus conclusio omnino, nec eadem quidem primae, nec alia: tunc accidit sive sequitur quod C D propositiones vane sunt assumptae, et non sunt ejus conclusionis quae sequitur ex principio per primum conclusa syllogismum. Propterea manifestum est ex dictis, quod omnis syllogismus et omnis demonstratio erit tantum per tres terminos et non per plures, nisi sint plures syllogismi secundum substantiam.
Si autem aliquis objiciat contra haec et dicat, quod syllogismus ex oppositis non habet nisi duos terminos, sic, nulla disciplina est studiosa: omnis disciplina est studiosa: ergo nulla disciplina est disciplina. Adhuc enthymema quod est conversio propositionis non habet nisi duos terminos: et hoc tamen reducitur in syllogismum, quia omnis argumentatio ad syllogismum habet reduci. Conversio autem propositionis in duabus propositionibus non habet nisi duos terminos: et sic etiam syllogismus in quem tale reducitur enthymema, non nisi duos habebit terminos. Similis objectio est de inductione, quae multos terminos habet, et absque alicujus termini diminutione reducitur in syllogismum. Et similis iterum objectio est de exemplo quod quatuor habet terminos, et absque alicujus termini diminutione reducitur in syllogismum.
But if it is said that propositions C and D do not have themselves to each other as whole and part, so that they could make a syllogism or syllogistic course, then they have been variously assumed for the proposed point, concluding nothing, unless perhaps it is said that they are accepted as things besides the necessary, such as those which are assumed for the sake of induction proving a universal, or for the sake of the relation of the proposed point, or for some other such thing. Yet in whatever way they are taken, they will always be several syllogisms according to substance. But if it is said that from the A B propositions the principal conclusion E is not consequent, but that from the A B propositions some conclusion other than E is made, and from C D, other propositions having been assumed, one of these, namely either A or B, is made the consequent conclusion, as in prosyllogisms; or perhaps it is said that some other conclusion whatever follows and is concluded from those propositions beyond those three things which have been said, it is certain that in all these ways there are several syllogisms, but not to that which was proposed, because they are not made for the first conclusion. For it was posited by us a little before that propositions A B are of that conclusion which we have said is E, but propositions C D, as has now been said, are not of the same conclusion; and thus there are many syllogisms not communicating with one another.
But if someone says that from propositions C D no conclusion at all is made in any way, neither the same as the first nor another, then it happens or follows that propositions C D have been assumed vainly and are not of that conclusion which follows from the beginning through the first concluded syllogism. Therefore it is manifest from what has been said that every syllogism and every demonstration will be only through three terms and not through more, unless there are several syllogisms according to substance.
But if someone objects against these things and says that a syllogism from opposites has only two terms, thus: no discipline is morally good; every discipline is morally good; therefore no discipline is discipline. Again, an enthymeme which is the conversion of a proposition has only two terms, and yet this is reduced to a syllogism, because every argumentation has to be reduced to syllogism. But the conversion of a proposition in two propositions has only two terms; and thus also the syllogism into which such an enthymeme is reduced will have only two terms. There is a similar objection about induction, which has many terms and is reduced into a syllogism without the diminution of any term. And again there is a similar objection about example, which has four terms and is reduced into a syllogism without the diminution of any term.
Ad haec autem omnia et similia jam olim in hujus scientiae libro responsum est, quod scilicet syllogismus ad plus et minus habet tres terminos, sed aliquando habet tres secundum substantiam et rationem et actum proprium terminorum, sicut in syllogismo perfecto: aliquando habet pauciores, scilicet secundum substantiam, sed tres secundum rationem, sicut in syllogismo ex oppositis, in quo major extremitas secundum substantiam una, duplex est secundum rationem: quia major est etiam minor, quando secundo ponitur in minori propositione. Et simile aliquid isti est in enthymemate, in quo altera propositio retinetur in mente. Aliquando autem habet plures terminos secundum substantiam, sed non secundum rationem, sicut in inductione: in quo omnia singularia sub uno communi inducta rationem habent unius termini, qui est minor extremitas. Aliquando habet plures terminos secundum substantiam et rationem, sicut in exemplo: sed tunc non reducitur argumentatio illa in unum syllogismum, sed in plures sicut fit in exemplo, sicut in consequentibus secundi libri ostendetur.
Ad omnia tamen haec notandum quod quamvis major extremitas aliquando cum seipsa ponat in numerum, sicut in syllogismo ex oppositis, in quo fit ut duo: tamen medium secundum actum mediandi non ponit cum seipso in numerum, ut faciat duos terminos bis positum in duabus propositionibus: nec quatuor erunt termini propter duo subjecta et duo praedicata, quia unus actus unius medii qui est actus uniendi et conjungendi extrema, non potest esse, nisi medium unum referatur ad utrumque extremorum: qui actus mediandi perfici non potest, nisi medium bis ponatur, semel cum uno extremo in propositione majori, et semel cum altero in altera propositione: et sic patet quod non sunt nisi tres termini.
Quamvis etiam in propositione ista medium pluribus litteris significatum sit: tamen non est nisi unum secundum actum et rationem medii, sed pluribus litteris significatur relatio ejus ad diversa: et ideo quando significatur littera una, secundum substantiam significatur: quando autem duabus significatur litteris, significantur duae ejus habitudines: quando autem tribus, significatur ut unum uniens duo: et quando quatuor significatur, significatur ut situm habens in duabus propositionibus, quae duo habent subjecta et duo praedicata.
But to all these and similar things it was already answered long ago in the book of this science, namely that a syllogism, at most and at least, has three terms; but sometimes it has three according to the substance and reason and proper act of the terms, as in a perfect syllogism. Sometimes it has fewer, namely according to substance, but three according to reason, as in a syllogism from opposites, in which the greater extreme, one according to substance, is double according to reason, because the major is also the minor when it is placed a second time in the minor proposition. And something like this occurs in an enthymeme, in which one proposition is retained in the mind. Sometimes it has more terms according to substance, but not according to reason, as in induction, in which all singulars introduced under one common term have the reason of one term, which is the lesser extreme. Sometimes it has more terms according to substance and reason, as in an example; but then that argumentation is not reduced into one syllogism, but into many, as happens in example, as will be shown in the following matters of the second book.
Yet with regard to all these things it must be noted that, although the greater extreme sometimes places itself in the number with itself, as in the syllogism from opposites in which it becomes as two, nevertheless the middle according to the act of mediating does not place itself in number with itself so as to make two terms by being posited twice in two propositions. Nor will there be four terms because of two subjects and two predicates, because one act of one middle, which is the act of uniting and joining the extremes, cannot be unless one middle is referred to both extremes. This act of mediating cannot be perfected unless the middle is posited twice, once with one extreme in the major proposition and once with the other in the other proposition; and thus it is clear that there are only three terms.
Although also in this proposition the middle is signified by several letters, nevertheless it is only one according to the act and reason of the middle; but by several letters its relation to diverse things is signified. And therefore when it is signified by one letter, it is signified according to substance; but when it is signified by two letters, its two relations are signified; and when by three, it is signified as one uniting two; and when by four, it is signified as having position in two propositions which have two subjects and two predicates.
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