Volume 1 · pp. 527–528

Chapter V. On the mixing of the necessary and inherence in universal syllogisms of the second figure.

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p. 527

CAPUT V. De mixtione necessarii et inesse in secundae figurae universalibus syllogismis.

In secunda autem figura in universalibus syllogismis talis est regula mixtionum, quod scilicet si in primis duobus universalibus modis privativa universalis sit necessaria, ita quod sit de necessario, et affirmativa universalis sit de inesse non ut nunc, sed simpliciter, sequitur conclusio de necessario. Sicut enim diximus, quod in prima figura major appropriat sibi minorem, eo quod sub medio eam accipi oportet, quod quidem medium stat sub majori extremitate: et ideo cum major sit de necessario, oportet minorem esse de inesse non ut nunc, sed simpliciter: ita est etiam in syllogismis secundae figurae, quod negativa appropriat sibi affirmativam, quod cum negativa universalis est de necessario, quod oportet affirmativam esse de inesse, non ut nunc, sed simpliciter. Et hoc est ideo, quia duo primi universales syllogismi secundae figurae oriuntur et formantur a secundo primae figurae: et ideo sicut in prima figura major quae est negativa universalis, appropriat sibi minorem quae est affirmativa universalis, ita fit in secunda figura: et ideo si affirmativa universalis in secunda figura sit de inesse simpliciter, erit conclusio de necessario: si autem erit de inesse per accidens, vel de inesse ut nunc, non potest esse conclusio de necessario. Sic ergo facienda est mixtio, quod universalis privativa sit de necessario, et universalis affirmativa de inesse simpliciter. Si enim e converso fiat, quod universalis negativa sit de inesse, et universalis affirmativa de necessario, non sequitur conclusio de necessario: quia affirmativa in hac figura non potest sibi appropriare negativam ad standum pro inesse simpliciter: sed indifferenter erit de inesse simpliciter, et de inesse ut nunc.

Accipiatur enim primus modus secundae figurae, et sit universalis privativa major de necessario, ita quod A nulli B insit ex necessitate. In minori autem propositione quae est universalis affirmativa, A insit omni C simpliciter, sequitur quod B nulli C inerit de necessitate. Hoc autem ideo est, quia in tali syllogismo, C minor extremitas est sub A, stans substantialiter, et ideo propositio minor est de inesse simpliciter, et non ut nunc.

Similiter autem in secundo modo secundae figurae, si ad C minorem extremitatem ponatur propositio privativa, et ad majorem ponatur affirmativa, ita quod major sit universalis affirmativa de inesse, minor autem sit universalis negativa de necessario: tunc enim iterum sequitur conclusio de necessario. Nam si accipiatur minor negativa, tunc A nulli C continget inesse: et sic est de necessario: quia quod nulli contingit inesse, necessarium est nulli inesse. Si autem sic est quod A nulli C contingit, cum universalis negativa de necessario convertatur simpliciter, tunc sequitur quod etiam e converso nulli A poterit inesse C, et sic est necessarium non inesse. Et tunc per conversionem minoris et transpositionem propositionum constituitur prima figura: quia reducitur primus modus secundae ad secundum primae per simplicem conversionem minoris propositionis. Secundus autem secundae reducitur in secundum primae per conversionem minoris et transpositionem propositionum, et conversionem conclusionis: et sic fit syllogismus, omne B est A, nullum C est A de necessitate, ergo de necessitate nullum C est B. Fit enim prima figura per conversionem minoris et transpositionem propositionum, nullum C B, de necessitate omne A C, ergo nullum A B de necessitate. Si autem nullum A C, convertatur conclusio, et erit conclusio prima quae est, nullum C A de necessitate: et sic probata est syllogismi conjugatio.

CHAPTER V. On the mixing of the necessary and inherence in universal syllogisms of the second figure.

But in the second figure, in universal syllogisms, such is the rule of mixings: namely, if in the first two universal modes the universal privative is necessary, so that it is of the necessary, and the universal affirmative is of inherence not as now but simply, a conclusion of necessity follows. For just as we said that in the first figure the major appropriates the minor to itself, because it is necessary to take it under the middle, which middle stands under the greater extreme, and therefore, when the major is of necessity, the minor must be of inherence not as now but simply, so it is also in syllogisms of the second figure, that the negative appropriates the affirmative to itself, so that when the universal negative is of necessity, the affirmative must be of inherence, not as now, but simply. And this is so because the first two universal syllogisms of the second figure arise and are formed from the second of the first figure; and therefore, just as in the first figure the major, which is universal negative, appropriates to itself the minor, which is universal affirmative, so it happens in the second figure. And therefore, if the universal affirmative in the second figure is of inherence simply, there will be a conclusion of necessity; but if it is of inherence accidentally or of inherence as now, there cannot be a conclusion of necessity. Therefore the mixing must be made in this way, that the universal privative be of necessity and the universal affirmative of inherence simply. For if it is done conversely, so that the universal negative is of inherence and the universal affirmative of necessity, a conclusion of necessity does not follow, because the affirmative in this figure cannot appropriate the negative to itself so that it stands for inherence simply; rather it will indifferently be of inherence simply and of inherence as now.

For let the first mode of the second figure be taken, and let the universal privative major be of necessity, so that A belongs to no B by necessity. But in the minor proposition, which is universal affirmative, let A belong to every C simply; it follows that B will belong to no C by necessity. This is so because in such a syllogism C, the lesser extreme, is under A, standing substantially, and therefore the minor proposition is of inherence simply and not as now.

Similarly in the second mode of the second figure, if a privative proposition is placed at C, the lesser extreme, and an affirmative is placed at the greater, so that the major is universal affirmative of inherence, but the minor is universal negative of necessity, then again a conclusion of necessity follows. For if the negative minor is taken, then A will contingently belong to no C; and thus it is of necessity, because what contingently belongs to none necessarily belongs to none. But if it is so that A contingently belongs to no C, since the universal negative of necessity is converted simply, then it follows that conversely C also will be able to belong to no A, and thus it is necessary not to belong. And then through conversion of the minor and transposition of the propositions the first figure is constituted, because the first mode of the second is reduced to the second of the first by simple conversion of the minor proposition. But the second of the second is reduced into the second of the first through conversion of the minor and transposition of the propositions and conversion of the conclusion; and thus the syllogism is made: every B is A; no C is A of necessity; therefore of necessity no C is B. For the first figure is made through conversion of the minor and transposition of the propositions: no C is B; of necessity every A is C; therefore no A is B of necessity. But if no A is C, let the conclusion be converted, and it will be the first conclusion, which is no C is A of necessity; and thus the conjugation of the syllogism has been proved.

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Si autem in syllogismis universalibus privativa universalis sit de inesse, et universalis affirmativa sit de necessario, erit conjugatio inutilis quantum ad conclusionem de necessario, et peccabit contra proprietatem istius figurae, quae est, quod universalis negativa sit de necessario et universalis affirmativa sit de inesse. Insit enim A omni B ex necessitate, ita quod major sit affirmativa de necessario, C autem insit nulli A, ita quod minor universalis negativa de non inesse tantum: conversa igitur privativa majori ejus fit reductio ad primam figuram in secundo modo primae figurae. Est autem jam in ante habitis ostensum, quod quando in secundo primae universalis negativa quae est ad majorem extremitatem, non est necessaria, non sequitur conclusio de necessario. Et si haec conjugatio est inutilis in prima, sequitur quod etiam inutilis est in secunda: quia secunda figura formatur a prima et reducitur in ipsam.

Amplius autem inutilitas istius conjugationis probatur etiam alio modo sic, scilicet quod aliquid sequitur conclusionem de necessario, sine quo vera potest esse praemissa, quae est de inesse: et sic praemissa potest esse vera sine conclusione: ergo illa conclusio non sequitur ex ipsa. Hoc autem probatur hoc modo: ponatur enim primo haec conjugatio, de necessitate omne B est A, nullum C est A, sequitur secundum hanc conjugationem, si utilis est, ergo de necessitate nullum C est B. Sumatur igitur conversa conclusionis hujus, et conversa majoris quae est affirmativa de necessario, et convertatur in particularem: et arguitur in quarto primae, sic, de necessitate nullum B est C, quae est conversa conclusionis: A est B necessario, quae est conversa majoris: sequitur in quarto primae, quod de necessitate aliquod A non est C: sed conversa universalis negativae de inesse quae est haec, nullum A est C, potest esse vera sine hac particulari, de necessitate aliquod A non est C, ergo haec conclusio non sequebatur ex ipsa: nihil enim in tali conjugatione prohibet A tale accipi cui omni contingit inesse C, ita quod omne A contingit esse C. Et si sic accipiatur, tunc vera erit praemissa sine tali conclusione de necessario.

Amplius hoc idem probatur tertio si aliquis sit terminos ponens, ut si A dicatur esse animal, B autem homo, C vero album, et sint propositiones sumptae similiter quantum ad conjugationem: tunc enim contingit animal nulli albo inesse: et tunc sequitur hac positione facta, quod existente tali positione, quod nec homo inerit alicui albo: quia si nullum animal est album, sequitur quod nullus homo est albus: tamen non sequitur quod nullus homo ex necessitate sit albus, quia homo contingenter est albus. Sed tamen donec manet illa positio, quod nullum animal sit album, sequitur quod nullus homo sit albus de necessitate; sed haec est necessitas positionis, et non simpliciter necessitas secundum ipsam rem: et formetur syllogismus in terminis instantiae sic, omnis homo de necessitate est animal: nullum album est animal, et hoc ponatur: sequitur ergo de necessitate nullum album est homo: hoc non sequitur omni modo necessitatis, sicut dictum est: sequitur ergo in tali conjugatione conclusio de inesse, sed non sequitur conclusio de necessitate inesse simpliciter. Sic ergo probantur conjugationes utiles et inutiles universalium syllogismorum in secunda figura.

But if in universal syllogisms the universal privative is of inherence and the universal affirmative is of necessity, the conjugation will be useless with respect to a conclusion of necessity, and it will offend against the property of this figure, which is that the universal negative be of necessity and the universal affirmative be of inherence. For let A belong to every B of necessity, so that the major is affirmative of necessity, but let C belong to no A, so that the minor is universal negative of non-inherence only. Therefore, when the privative is converted, its reduction to the first figure is made in the second mode of the first figure. But it has already been shown in what was had before that when, in the second of the first, the universal negative which is toward the greater extreme is not necessary, a conclusion of necessity does not follow. And if this conjugation is useless in the first, it follows that it is also useless in the second, because the second figure is formed from the first and is reduced into it.

Furthermore, the uselessness of this conjugation is proved also in another way, namely because something follows the conclusion of necessity without which the premise that is of inherence can be true; and thus the premise can be true without the conclusion; therefore that conclusion does not follow from it. This is proved in this way: first let this conjugation be posited: of necessity every B is A; no C is A; according to this conjugation, if it is useful, it follows: therefore of necessity no C is B. Therefore let the converse of this conclusion be taken, and the converse of the major, which is affirmative of necessity, and let it be converted into a particular; and let an argument be made in the fourth of the first, thus: of necessity no B is C, which is the converse of the conclusion; A is B necessarily, which is the converse of the major; it follows in the fourth of the first that of necessity some A is not C. But the converse of the universal negative of inherence, which is this, no A is C, can be true without this particular, of necessity some A is not C; therefore this conclusion did not follow from it. For nothing in such a conjugation prohibits taking such an A to which C contingently belongs wholly, so that every A contingently is C. And if it is taken thus, then the premise will be true without such a conclusion of necessity.

Furthermore, this same point is proved third if someone is positing terms: for example, let A be said to be animal, B man, and C white, and let the propositions be taken similarly with respect to the conjugation. For then it is contingent that animal belong to no white thing; and then, with this position made, it follows that, while such a position exists, man also will belong to no white thing, because if no animal is white, it follows that no man is white. Nevertheless it does not follow that no man is white of necessity, because man is white contingently. Yet as long as that position remains, that no animal is white, it follows that no man is white of necessity; but this is necessity of the position, and not simply necessity according to the thing itself. And let the syllogism be formed in the terms of the instance thus: every man is animal of necessity; no white thing is animal, and let this be posited; therefore it follows of necessity that no white thing is man. This does not follow with every mode of necessity, as has been said. Therefore in such a conjugation a conclusion of inherence follows, but a conclusion of the necessity of inhering simply does not follow. Thus, therefore, the useful and useless conjugations of universal syllogisms in the second figure are proved.