Works › Prior Analytics, Books I–II
Volume 1 · pp. 510–512
Chapter XIII. On the summary of what has been said and on the formation of two syllogisms concluding indirectly.
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CAPUT XIII. De epilogo dictorum et de formatione duorum syllogismorum indirecte concludentium.
Palam autem ex determinatis quando fit in omnibus figuris, hoc est, in qualibet figurarum, et quando non fit syllogismus, cum praedicativi vel privativi sunt utrique termini, major scilicet et minor. Deinde vero quod in secunda figura non fit syllogismus affirmativus, et in nulla figura syllogismus fit ex omnibus negativis. Similiter etiam quando utrique termini sunt particulares, nihil omnino fit per consequentiam syllogisticam ex tali conjugatione necessarium. Cum autem unus terminorum sit praedicativus, major scilicet, et alter privativus, minor scilicet, ita quod minor qui privativus est sic universaliter sumptus, semper fit syllogismus indirecte concludens, qui est syllogismus minoris extremitatis ad majorem: concludit autem minorem extremitatem de majori: quod est indirecte concludere. Et hujus exemplum est in duobus modis simul inductis, quorum unus habet majorem universalem affirmativam, et minorem universalem negativam: et alter habet majorem particularem affirmativam, et minorem universalem negativam: et hoc in prima figura sic: ut dicatur A quidem omni B vel alicui B inesse, sic, omne B est A, vel aliquod B est A; B autem nulli C dicatur inesse, sic, nullum C B; tunc enim fit syllogismus indirecte concludens minorem extremitatem de majori, conversis propositionibus et transpositis: sequitur enim, quod C alicui A non insit, hoc est, quod aliquod A non est C sic, nullum B C, omne A B, vel aliquod A B: sequitur quod aliquod A non est C. Similiter autem fit et in aliis figuris: quia in talibus syllogismis semper fit reductio ad perfectos syllogismos per conversionem propositionum, et aliquando transpositionem, et per conversionem conclusionis.
Quamvis autem Boetius ponat quinque modos indirecte concludentium in prima figura, de quibus et Aristoteles facit mentionem: quia duos ponit hic, et tres alios in secundo hujus scientiae libro: tamen hic non ponuntur nisi duo qui dicti sunt: quia illi formantur juxta inutiles conjugationes primae figurae, in qua major est semper universalis et minor affirmativa: cujus contrarium est in illis syllogismis indirecte concludentibus, in quibus major affirmativa sive universalis sive particularis, et minor negativa: et ideo similitudinem quamdam habent cum inutilibus conjugationibus: quam similitudinem alii tres modi non habent, qui juxta utiles formantur conjugationes. Isti enim hic ponuntur ad sciendum, quod non omnis inutilis conjugatio est adeo inutilis, quod nihil vel directe vel indirecte possit concludi per eam. Istae enim duae conjugationes quando major universalis affirmativa et minor universalis negativa, et quando major particularis affirmativa et minor universalis negativa inest, inutiles sunt reputatae; et tamen concludunt, sed indirecte.
CHAPTER XIII. On the summary of what has been said and on the formation of two syllogisms concluding indirectly.
From what has been determined it is clear when in all figures, that is, in any figure, a syllogism is made and when it is not made, when both terms, namely the major and the minor, are predicative or privative. Next, that in the second figure no affirmative syllogism is made, and in no figure is a syllogism made from all negatives. Similarly, when both terms are particular, nothing necessary at all is made through syllogistic consequence from such a conjugation. But when one of the terms, namely the major, is predicative, and the other, namely the minor, is privative, so that the minor, which is privative, is taken universally in this way, there is always made a syllogism concluding indirectly, which is a syllogism of the lesser extreme toward the greater; but it concludes the lesser extreme of the greater, and this is to conclude indirectly. An example of this is in the two modes introduced together, of which one has a universal affirmative major and a universal negative minor, and the other has a particular affirmative major and a universal negative minor; and this is so in the first figure: let A indeed be said to belong to every B or to some B, thus: every B is A, or some B is A; but let B be said to belong to no C, thus: no C is B. Then there is made a syllogism concluding indirectly the lesser extreme of the greater, the propositions having been converted and transposed; for it follows that C does not belong to some A, that is, that some A is not C, thus: no B is C; every A is B, or some A is B; it follows that some A is not C. Similarly it occurs also in the other figures, because in such syllogisms there is always a reduction to perfect syllogisms through conversion of propositions, and sometimes through transposition, and through conversion of the conclusion.
But although Boethius posits five modes concluding indirectly in the first figure, of which Aristotle too makes mention, because he posits two here and three others in the second book of this science, nevertheless only the two which have been mentioned are posited here, because they are formed beside the useless conjugations of the first figure, in which the major is always universal and the minor affirmative. The contrary of this is in those syllogisms concluding indirectly, in which the major is affirmative, whether universal or particular, and the minor negative; and therefore they have a certain likeness with useless conjugations, a likeness which the other three modes do not have, which are formed beside useful conjugations. For these are posited here so that it may be known that not every useless conjugation is so useless that nothing can be concluded through it either directly or indirectly. For these two conjugations, when there is a universal affirmative major and universal negative minor, and when there is a particular affirmative major and universal negative minor, are reputed useless; and nevertheless they conclude, but indirectly.
Quaeri tamen potest, utrum isti duo modi indirecte concludentes possint esse tam in secunda et tertia figuris, sicut et in prima. Videtur enim quod non: quia in terminis invenitur instantia, sic, nullus homo est asinus: aliquod animal est asinus: ergo aliquis homo non est animal: patet quod non valet: ergo in secunda figura per talem complexionem non potest indirecte concludi. Similiter videtur instantia in tertia figura, sic, nullus homo est asinus: aliquis homo est animal: ergo aliquis asinus non est animal: patet quod non valet, quamvis indirecta conclusio secundum complexionem eamdem quae dicta est. Ad hoc autem videtur dicendum, quod indirecte concludentes modi possunt esse in omni figura per aliquem modum conversionis conclusionis. Sed quod hic dicitur, non est intelligendum nisi de modis in quibus minor est universalis negativa, et major vel universalis affirmativa, vel particularis affirmativa. In instantiis quae adductae sunt, major est universalis negativa, et minor affirmativa: et ideo non sunt ejusdem conjugationis cum his qui in omni figura possunt indirecte concludere1.
Si autem quaeritur ex quo fiunt isti modi in omni figura, an eodem modo vel diverso perficiantur in omni figura vel non? Sed ad hoc facile patet solutio: quia cum diversae sint figurae, oportet quod diversae sint reductiones ad perfectos modos primae figurae. In prima figura fit perfectio in conversionem utriusque propositionis, et per transpositionem earumdem: hoc enim fieri oportet propter medii dispositionem debitam et extremorum. In secunda autem figura facti isti syllogismi ad primam figuram reducuntur per conversionem minoris, hoc est, universalis negativae, et per propositionum transpositionem. In tertia autem figura facti iidem syllogismi reducuntur per conversionem majoris, quae est affirmativa universalis vel particularis, et per transpositionem propositionum. Et sic in tribus figuris, tribus modis diversis reducuntur: tamen ex omni figura semper reducuntur ad unum modum qui est quartus modus figurae primae.
Scias tamen quod isti modi non in omni figura manent indirecti: quinimo peccant contra directos primae et tertiae figurae, quia in prima figura uterque modus est indirectus, et in tertia similiter; uterque enim modus de duobus dictis modis peccat contra modos directe concludentes in prima et tertia figura, in hoc, quod habet minorem negativam: quod non contingit in modis directe concludentibus in prima et tertia figura. Prima enim et tertia figura semper habent minorem affirmativam in modis directe concludentibus. In secunda autem figura ille tantum peccat contra modos directe concludentes, cujus major est particularis: quia hoc non contingit in modis directe concludentibus in secunda figura. Ille vero qui habet majorem universalem affirmativam et minorem universalem negativam, per talem complexionem non peccat contra modos directos secundae figurae: et ideo ille modus in secunda figura primo et per se directus est, sed potest fieri indirectus per conversionem conclusionis.
Yet it can be asked whether these two modes concluding indirectly can exist in the second and third figures as in the first. For it seems that they cannot, because an instance is found in terms thus: no man is an ass; some animal is an ass; therefore some man is not animal. It is clear that this is not valid; therefore in the second figure one cannot conclude indirectly through such a complexion. Similarly an instance seems to exist in the third figure thus: no man is an ass; some man is animal; therefore some ass is not animal. It is clear that this is not valid, although the indirect conclusion is according to the same complexion which was stated. To this, however, it seems one must say that modes concluding indirectly can exist in every figure through some mode of conversion of the conclusion. But what is said here must be understood only of modes in which the minor is universal negative and the major either universal affirmative or particular affirmative. In the instances that were brought forward, the major is universal negative and the minor affirmative; and therefore they are not of the same conjugation as those which can conclude indirectly in every figure1.
But if it is asked from what these modes are made in every figure, and whether they are perfected in the same way or in a different way in every figure or not, the solution to this is easily clear: because, since the figures are diverse, the reductions to the perfect modes of the first figure must be diverse. In the first figure perfection is made into conversion of each proposition and through transposition of the same; for this must occur on account of the due disposition of the middle and of the extremes. But in the second figure these syllogisms, when made, are reduced to the first figure through conversion of the minor, that is, of the universal negative, and through transposition of the propositions. But in the third figure these same syllogisms, when made, are reduced through conversion of the major, which is affirmative, universal or particular, and through transposition of the propositions. And thus in the three figures they are reduced in three diverse ways; nevertheless from every figure they are always reduced to one mode, which is the fourth mode of the first figure.
You should know, however, that these modes do not remain indirect in every figure; rather, they offend against the direct modes of the first and third figures, because in the first figure each mode is indirect, and similarly in the third. For each mode of the two modes mentioned offends against modes concluding directly in the first and third figure in this, that it has the minor negative, which does not occur in modes concluding directly in the first and third figure. For the first and third figure always have the minor affirmative in modes concluding directly. But in the second figure only that one offends against modes concluding directly whose major is particular, because this does not occur in modes concluding directly in the second figure. But the one which has a universal affirmative major and a universal negative minor, through such a complexion, does not offend against the direct modes of the second figure; and therefore that mode in the second figure is first and directly so per se, but can become indirect through conversion of the conclusion.
Attendendum autem quod isti duo modi indirecti in omni figura in qua sunt, non possunt habere nisi conclusiones particulares et negativas. Et hoc contingit ideo quod in tali complexione, quando major est affirmativa universalis vel particularis, et minor negativa universalis, oportet contineri minorem extremitatem sub majori: et cum concluditur indirecte, tunc major extremitas subjicitur in conclusione, et minor praedicatur de majori. Unde quia minor est negativa, oportet conclusionem esse negativam: et quia minor praedicatur in conclusione de majori, oportet quod negetur inferius de suo superiori: et inferius de superiori non potest negari nisi particulariter sic, aliquod animal non est asinus: ideo de necessitate sequitur, quod conclusionem oportet esse et particularem et negativam.
Quod autem in omni figura in tali complexione termini sint sic dispositi, quod minor sit sub majori, patet in terminis in qualibet figura secundum hanc complexionem dispositis. In prima enim talis est secundum hanc complexionem dispositio, omnis vel aliquis homo animal: nullus asinus homo. In secunda vero figura sic disponuntur, aliquod animal homo: nullus asinus homo. In tertia autem sic disponuntur secundum hanc complexionem, omnis vel aliquis homo animal: nullus homo asinus. Et in omnibus his dispositionibus patet quod dictum est, quod scilicet major extremitas est super minori, et minor est inferior majori: et sic manifestum est, quod non potest per tales syllogismos concludi nisi particulariter et negative, quando scilicet in tali sumuntur complexione et indirecte concludentes.
Si autem aliquis quaerat, quare non sunt modi indirecte concludentes, in quibus major sit universalis negativa et minor affirmativa universalis? Dicendum quod talis dispositio terminorum in prima et secunda figura non peccat. In prima enim figura major potest esse universalis negativa et minor affirmativa, et similiter in secunda: et ideo tales syllogismi de necessitate in prima et secunda figura directe concludent: et similiter in tertia major potest esse universalis negativa et minor affirmativa, et directe concludet talis complexionis modus.
Quod autem, in tertia figura non concluditur nisi particularis negativa, quando major est negativa, est propter hoc quod in tali dispositione terminorum major extremitas de minori non nisi particulariter potest negari, sicut aliquod animal non est homo. Hoc autem jam in ante habitis per causam determinatum est: in hac enim tertia figura minor extremitas potest aliquando continere majorem, et esse superius ad ipsam.
But it must be noted that those two indirect modes, in every figure in which they are, can have only particular and negative conclusions. And this happens because in such a complexion, when the major is affirmative, universal or particular, and the minor universal negative, the lesser extreme must be contained under the greater; and when it is concluded indirectly, then the greater extreme is subjected in the conclusion, and the minor is predicated of the major. Hence because the minor is negative, the conclusion must be negative; and because the minor is predicated in the conclusion of the major, the inferior must be denied of its superior; and the inferior cannot be denied of the superior except particularly, thus: some animal is not an ass. Therefore it follows of necessity that the conclusion must be both particular and negative.
But that in every figure in such a complexion the terms are so disposed that the minor is under the major is clear in terms disposed according to this complexion in any figure. For in the first, the disposition according to this complexion is such: every or some man is animal; no ass is man. But in the second figure they are disposed thus: some animal is man; no ass is man. But in the third they are disposed according to this complexion thus: every or some man is animal; no man is ass. And in all these dispositions it is clear what was said, namely that the greater extreme is above the lesser, and the lesser is inferior to the greater; and thus it is manifest that through such syllogisms one cannot conclude except particularly and negatively, namely when they are taken in such a complexion and as concluding indirectly.
But if someone asks why there are no modes concluding indirectly in which the major is universal negative and the minor universal affirmative, one must say that such a disposition of terms does not offend in the first and second figure. For in the first figure the major can be universal negative and the minor affirmative, and similarly in the second; and therefore such syllogisms in the first and second figure will conclude directly of necessity. Similarly in the third, the major can be universal negative and the minor affirmative, and the mode of such a complexion will conclude directly.
But that in the third figure only a particular negative is concluded, when the major is negative, is because in such a disposition of terms the greater extreme cannot be denied of the lesser except particularly, as "some animal is not man." This, however, has already been determined by its cause in what went before; for in this third figure the lesser extreme can sometimes contain the greater and be superior to it.
Notes
- From this it is had that Fapesmo and Fresisomorum can conclude indirectly in every figure. (Ex quo habetur quod fapesmo et fresisomorum possunt in omni figura indirecte concludere.) ↩
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