Part III, Chapter 3: General Rules of Simple Incomplex Syllogisms
[ This Chapter, and the following ones until the twelfth, are among the number of those spoken of in the Discourses, containing things which are subtile, and necessary to the speculative part of logic, but which are of little practical utility. ]
WE have seen already, in the preceding chapters, that a simple syllogism ought to have only three terms, two terms for the conclusion, and a single middle term, each of which being repeated twice, constitutes three propositions: the major, into which the middle term and the attribute of the conclusion (which is called the greater term) enter; the minor, into which, also, the middle term and the subject of the conclusion (which is called the smaller term) enter; and the conclusion, of which the lesser term is the subject, and the greater term the attribute.
But because all sorts of conclusions cannot be obtained from all sorts of premises, there are general rules which show that a conclusion cannot be properly obtained in a syllogism in which they are not observed, and these rules are founded on the axioms which were established in the Second Part, touching the nature of propositions affirmative and negative, universal and particular. These, such as they are, we shall only state, having proved them elsewhere.
1. Particular propositions are contained in general ones of the same nature, not the general in the particular,—I in A, and O in E, and not A in I, or E in O.
2. The subject of a proposition, taken universally or particularly, is that which renders it universal or particular.
3. The attribute of an affirmative proposition having never more extension than the subject, is always considered as taken particularly, since it is only by accident that it is sometimes taken generally.
4. The attribute of a negative proposition is always taken generally.
It is mainly on those axioms that the general rules of syllogisms are founded, which rules we cannot violate without falling into false reasonings.
Rule 1
The middle term cannot be taken twice particularly, but it ought to be taken, once at least, universally.
For, before uniting or disuniting the two terms of the conclusion, it is clear that this cannot be done if it is taken for two different parts of the same whole, since it may, perhaps, not be the same part which is united or separated from these terms. Now, if taken twice particularly, it may be taken for two different parts of the same whole, and, consequently, nothing could be concluded, at least necessarily, which is enough to render an argument vicious, since we can only call that a good syllogism, as we have already said, of which the conclusion cannot be false, the premises being true. Thus, in this argument—some man is holy, some man is a thief, therefore some thief is holy, the word man being taken for different parts of mankind, cannot unite them with holy, since it is not the same man who is holy, and who is a thief.
We cannot say the same of the subject and attribute of the conclusion; for, though they be taken twice particularly, they may, nevertheless, unite them together, by uniting one of these terms to the middle, in the whole extension of the middle term; for it follows hence, very clearly, that if this middle is united in some one of its parts to some part of the other term, that first term, which we have already stated to be united to all the middle, will be united also with the term to which some part of the middle is joined. If there are some Frenchmen in every house in Paris, and if there are Germans in some houses in Paris, then there are some houses in which Frenchmen and Germans are together.
If some rich men are fools,
And all rich men are honoured,
Then are some fools honoured;
for the rich men who are fools are also honoured, since all are honoured; and, consequently, in these rich fools which are honoured, the qualities of fool and honour are joined together.