Part III, Chapter 4: Of the Figures and Modes of Syllogisms in general
AFTER establishing the general rules which must necessarily be observed in all simple syllogisms, it remains to show how many sorts there are of such syllogisms.
We may say in general that there are as many sorts as there may be different ways of arranging the three propositions of a syllogism, and the three terms of which they are made up, without violating the rules which we have laid down.
The arrangement of the three propositions according to the four differences, A, E, I, O, is called mood,—and the arrangement of the three terms, that is to say, of the middle term with the two terms of the conclusion, is called figure.
Now we may reckon how many moods there are which afford a conclusion, without taking into account the different figures in which the same mood may constitute different syllogisms, for, by the doctrine of combinations, four terms (as A, E, I, O), being taken three by three, can be differently arranged only in sixty-four ways. But of these sixty-four ways, those who will take the trouble to consider each apart, will find that there are of them, —
Twenty-eight excluded by the third and sixth rules,—nothing can be concluded from two negatives, or from two particulars.
Eighteen by the fifth,—that the conclusion follows the weaker part.
Six by the fourth,—that we cannot have a negative conclusion from two affirmatives.
One, I, E, O, to wit, by the third corollary of the general rules.
One, A, E, O, to wit, by the sixth corollary of general rules.
These make in all fifty-four, and, consequently, only ten valid moods remain:
Four affirmative.
But it does not follow from this that there are only ten sorts of syllogisms, since any one of these moods may be made into different syllogisms, according to the other way in which they are diversified, by the different arrangement of the three terms, which we have already said is called figure.
Now, in order to this disposition of the three terms, the two first propositions alone are to be considered, since the conclusion is supposed before we make the syllogism to prove it: and as the middle can be arranged only, with the two terms of the conclusion, in four different ways, there are thus also only four possible figures.
For the middle term is either the subject in the major, and the attribute in the minor, which makes the first figure.
Or it is the attribute in both, which makes the second figure.
Or the subject in both, which makes the third figure.
Or finally, it is the attribute in the major, and subject in the minor, which makes a fourth figure, since it is certain that we may sometimes have a necessary conclusion in this form, which is sufficient to constitute a valid syllogism. Examples of these will be given hereafter.
Nevertheless, since, in this fourth figure, the conclusion is obtained in a way that is by no means natural, and which the mind never takes, Aristotle, and those who have followed him, have not given to this mode of reasoning the name of figure. Galen maintained the contrary; but it is clear that it is only a dispute about words, which ought to be decided by making each party say what they understand by the word figure.
But, without doubt, those are mistaken who apply to the fourth figure (which they blame Aristotle for not recognising) the arguments of the first, of which the major and minor are transposed, as when we say, All body is divisible; all that is divisible is imperfect; therefore all body is imperfect. I am surprised that Gassendi has fallen into this error, for it is ridiculous to take, as a major of a syllogism, the proposition which stands first, and for the minor that which stands second. If this were so, it would be often necessary to take the conclusion itself as the major, or minor of a reasoning, since it is often enough placed first or second of the three propositions which compose it, as in this verse of Horace, the conclusion is the first, the minor second, and the major third.
Qui melior servo, qui liberior sit avarus;
In triviis fixum, cum se dimittit ad assem
Non video: nam qui cupiet, metuet quoque: porro
Qui metuens vivit, liber mihi non erit unquam.
For it is all reducible to this argument: He who is in continual fear is not free, Every miser is in continual fear; Therefore no miser is free.
We are not, therefore, to consider the simple local arrangement of the propositions, which effects no change on the mind; but we are to take, as syllogisms of the first figure, all those in which the middle term is subject, in the proposition where the greater term (that is to say, the attribute of the conclusion) is found, and the attribute in that where the lesser term (that is to say, the subject of the conclusion) is found. And thus it follows, that those syllogisms only are of the fourth figure, where the middle term is attribute in the major, and subject in the minor. And it is in this way that we shall speak of the figures, without any being able to complain of our so doing, since we have stated beforehand that we understand, by this word figure, only a different arrangement of the middle term.