Part III, Chapter 9: Of Complex Syllogisms

Antoine Arnauld (1612-1694) and Pierre Nicole · Logic, or the Art of Thinking

Some errors in this text were corrected by AI.

IT must be confessed, that if there are some to whom logic is a help, there are many to whom it is a hindrance; and it must be acknowledged, at the same time, that there are none to whom it is a greater hindrance than to those who pride themselves most upon it, and who affect, with the greatest display, that they are good logicians; for this very affectation, being the mark of a low and shallow mind, it comes to pass that they, attaching themselves more to the exterior of the rules than to good sense, which is the soul of them, are easily led to reject as bad reasoning, some which are very good, since they have not sufficient penetration to adjust them to the rules, which serve no other purpose than to deceive them, because they comprehend them only imperfectly.

In order to avoid this defect, which partakes strongly of that pedantry which is so unworthy in a noble minded man, we ought rather to examine the solidity of a reasoning by the light of nature than by mere forms; and one of the means of satisfying ourselves, when we meet with any difficulty, is to make other reasonings similar to it in different matters, and when it appears clearly to us to afford a good conclusion, by considering only the good sense of it; if we find, at the same time, that it contains something not conformed to the rules, we ought rather to believe that this is owing to some defect in our explication than to its being so in reality.

But the reasonings of which it is more difficult to judge aright, and in which it is more easy to be deceived, are those which, as we have already said, may be called complex, not simply because there were found in them complex propositions, but because the terms of the conclusion being complex, were not taken in all their entirety, in each of the premises, in order to be joined with the middle, but only a part of one of the terms, as in this example—The sun is a thing insensible; The Persians worship the sun; Therefore the Persians worship a thing insensible.

In which we see that the conclusion having its attribute, worship a thing insensible, only a part of this is placed in the major, to wit, a thing insensible, and worshipped in the minor.

Now we shall do two things in relation to these syllogisms. We shall show, in the first place, how they may be reduced to the incomplex syllogisms of which we have hitherto spoken, in order to their being judged by the same rules.

And we shall show, in the second place, that more general rules may be given, for judging at once of the validity or viciousness of these syllogisms, without having recourse to any reduction.

It is a thing strange enough, that although logic has occupied a higher position than it deserved, so that it has been maintained that it was absolutely necessary for acquiring the sciences, it has, nevertheless, been treated of with so little attention, that hardly anything has been said touching aught that is of real use; for logicians commonly content themselves with giving the rules for simple syllogisms, and almost all the examples given of them are composed of incomplex propositions, which are so clear that no one would ever have thought of seriously composing them in any discourse; for who has ever heard of any one making such a syllogism as this: Every man is an animal; Peter is a man; therefore Peter is an animal.

But little pains are taken in applying the rules of syllogism to arguments of which the propositions are complex, though this is often very difficult, and there are many arguments of this nature which appear bad, which are nevertheless very good; and besides, the use of such reasonings is much more frequent than that of syllogisms which are quite simple. This will be shown more easily by examples than by rules.

We have said, for example, that all propositions composed of active verbs are complex in some manner; and of these propositions reasonings are often made, whose form and force are difficult to recognise; as this, which we have already given as an example —

The divine law commands us to honour kings;

Louis XIV. is a king;

Therefore the divine law commands us to honour Louis XIV.

Some persons of small intelligence have accused such reasonings of being defective, because, say they, they are composed of pure affirmatives in the second figure, which is an essential defect. But these persons have shown clearly that they have consulted more the letter and surface of the rules, than the light of reason, by which these rules were discovered; for this reasoning is so true and valid, that if it were opposed to the rule, it would prove that the rule was false, not that the reasoning was bad. I say that, in the first place, this argument is good; for in this proposition, the divine law commands us to honour kings,—this word, kings, is taken generally for all kings in particular, and consequently Louis XIV. is among those whom the divine law commands us to honour.

I say, in the second place, that king, which is the middle term, is not the attribute in this proposition—the divine law commands us to honour kings—though it may be joined with the attribute command, which is a very different thing, for what, which is really the attribute, is affirmed, and agrees. Now, first, king is not affirmed, and does not agree with the law of God; second, the attribute is restricted by the subject, Now the word king is not restricted in this proposition—the divine law commands us to honour kings, since it is taken generally.

But if it is demanded, then, what it really is, it is easy to reply that it is the subject of another proposition involved in this; for when I say that the divine law commands us to honour kings, as I attribute command to the law, I attribute also honour to kings, for it is as if I said, the divine law commands that kings be honoured.

So also in this conclusion—the divine law commands us to honour Louis XIV.—Louis XIV. is not the attribute, although joined to it, and he is, on the contrary, the subject of an involved proposition; for it is the same as if I said, the divine law commands that Louis XIV. be honoured. Thus these propositions are unfolded in the following way: —

The divine law commands that kings be honoured;

Louis XIV. is king;

Therefore the divine law commands that Louis XIV. be honoured.

It is clear that the whole argument consists in these propositions: —

Kings ought to be honoured; Louis XIV. is king;

Therefore Louis XIV. ought to be honoured. And that this proposition—the divine law commands—which appears the principal, is only an incidental proposition, in this argument, joined to the affirmation, which the divine law helps to prove.

It is clear also that this reasoning is, in Barbara of the first figure, the individual terms, as Louis XIV., standing for universals, as they are taken in all their extension, as we have already remarked.

For the same reason, this argument, which appears to be of the second figure, and conformed to the rules of that figure, is worth nothing: —

We ought to believe the Scripture;

Tradition is not Scripture;

Therefore we ought not to believe tradition. For it ought to be reduced to the first figure in this way —

The Scripture ought to be believed;

Tradition is not Scripture;

Therefore tradition ought not to be believed. Now we are not able to conclude anything from the first figure, from a negative minor.

There are other reasonings, the propositions of which appear to be pure affirmatives in the second figure, which are nevertheless very good, as: —

Every good pastor is ready to give his life for his sheep; Now there are few pastors in the present day who are ready to give their lives for their sheep; Therefore there are in the present day few good pastors. But what makes this reasoning good is, that we conclude affirmatively only in appearance. For the minor is an exclusive proposition, which contains in sense this proposition,—Most of the present pastors are not ready to give their lives for their sheep. And the conclusion, also, may be reduced to this negative,—Many of the present pastors are not good pastors.

Here is another argument, which, being of the first figure, appears to have a negative minor:—

Those who cannot be robbed of what they love are out of the reach of their enemies;

Now, when a man loves God alone, he cannot be robbed of what he loves; Therefore all those who love God alone, are out of the reach of their enemies.

What makes this argument quite valid is, that the minor is negative only in appearance, and is in reality affirmative, for the subject of the major, which ought to be the attribute in the minor, is not those who may be robbed of what they love, but, on the contrary, those who cannot be robbed. Now this is what we affirm of those who love God alone, so that the sense of the minor is, Now all those who love God alone are among the number of those who cannot be robbed of what they love, which is clearly an affirmative proposition.

This is what happens again, where the major is an exclusive proposition, as,

Those only who love God are happy;

Now there are rich men who do not love God;

Therefore there are rich men who are not happy. For the particle only makes the first proposition in this syllogism equal in meaning to these two—the friends of God are happy, and, all others, who are not the friends of God, are not happy.

Now, since it is on this second proposition that the force of the reasoning depends, the minor, which appears to be negative, becomes affirmative, since the subject of the major, which ought to be the attribute in the minor, is not friends of God, but those who are not the friends of God, so that the whole argument ought to stand thus: —

All those who are not the friends of God are not happy;

Now, there are rich men among the number of those who are no friends of God;

Therefore there are rich men who are not happy. But what makes it necessary to express the minor in this way, and to take away from it the appearance of a negative proposition, is, that it is the same thing to say negatively, that a man is not the friend of God, and to say, affirmatively, that he is no friend of God, that is to say, that he is among the number of those who are not the friends of God.

There are many reasonings such as these, of which all the propositions appear negative, and which are, nevertheless very good, because there is in them one which is negative only in appearance, and in reality, affirmative, as we have already shown, and as we may still further see by this example: —

That which has no parts cannot perish by the dissolution of its parts;

The soul has no parts;

Therefore the soul cannot perish by the dissolution of its parts.

There are several who advance such syllogisms to show that we have no right to maintain that this axiom of logic, Nothing can be proved by pure negatives, is true generally, without distinction; but they have not observed that in sense, the minor of this and such other syllogisms, is affirmative, since the middle, which is the subject of the major, is in it the attribute. Now the subject of the major is not that which has parts, but that which has not parts, and thus the sense of the minor is, the soul is a thing without parts, which is a proposition affirmative of a negative attribute.

The same persons sometimes prove, again, that negative reasonings are sometimes conclusive, by these examples: John is not rational, therefore he is not a man. No animal sees, therefore no man sees. But they ought to consider that these examples are only enthymemes, and that no enthymeme is conclusive, save in virtue of a proposition understood, which, consequently, ought to be in the mind, though it be not expressed. Now, in both these examples, the proposition understood is necessarily affirmative. In the first, this—all man is rational, John is not rational, therefore John is not a man; and on the other—every man is an animal, no animal sees, therefore no man sees. Now, we cannot say that these syllogisms are purely negative, and, consequently, the enthymemes, which are conclusive only because they contain these syllogisms complete in the mind of him who uses them, cannot be brought as examples to show that there are some purely negative reasonings which afford valid conclusions.