Part 7

“His imbecility of character might have been inferred from his proneness to favourites; for all weak princes have this failing.” The preceding would stand very well in a history, and many would pass it over as containing very good inference. Written, however, in the form of a syllogism, it is,

All weak princes are prone to favourites He was prone to favourites ———————————————— ——————————————————————— Therefore He was a weak prince

which is palpably wrong. (Rule 1.) The writer of such a sentence as the preceding might have meant to say, ‘for all who have this failing are weak princes;’ in which case he would have inferred rightly. Every one should be aware that there is much false inference arising out of badness of style, which is just as injurious to the habits of the untrained reader as if the errors were mistakes of logic in the mind of the writer.

‘A is less than B; B is less than C: therefore A is less than C.’ This, at first sight, appears to be a syllogism; but, on reducing it to the usual form, we find it to be,

A is (a magnitude less than B) B is (a magnitude less than C) Therefore A is (a magnitude less than C)

which is not a syllogism, since there is no middle term. Evident as the preceding is, the following additional proposition must be formed before it can be made explicitly logical. ‘If B be a magnitude less than C, then every magnitude less than B is also less than C.’ There is, then, before the preceding can be reduced to a syllogistic form, the necessity of a deduction from the second premiss, and the substitution of the result instead of that premiss. Thus,

A is less than B Less than B is less than C: following from B is less than C. ————————— ————————————————— Therefore A is less than C

But, if the additional argument be examined—namely, if B be less than C, then that which is less than B is less than C—it will be found to require precisely the same considerations repeated; for the original inference was nothing more. In fact, it may easily be seen as follows, that the proposition before us involves more than any simple syllogism can express. When we say that A is less than B, we say that if A were applied to B, every part of A would match a part of B, and there would be parts of B remaining over. But when we say, ‘Every A is B,’ meaning the premiss of a common syllogism, we say that every instance of A is an instance of B, without saying any thing as to whether there are or are not instances of B still left, after those which are also A are taken away. If, then, we wish to write an ordinary syllogism in a manner which shall correspond with ‘A is less than B, B is less than C, therefore A is less than C,’ we must introduce a more definite amount of assertion than was made in the preceding forms. Thus,

Every A is B, and there are Bs which are not As Every B is C, and there are Cs which are not Bs ——————————————————————————————————————————————— Therefore Every A is C, and there are Cs which are not As

Or thus:

The Bs contain all the As, and more The Cs contain all the Bs, and more ——————————————————————————————————— The Cs contain all the As, and more

The most technical form, however, is,

From Every A is B; [Some B is not A] Every B is C; [Some C is not B] Follows Every A is C; [Some C is not A]

This sort of argument is called à fortiori argument, because the premises are more than sufficient to prove the conclusion, and the extent of the conclusion is thereby greater than its mere form would indicate. Thus, ‘A is less than B, B is less than C, therefore, à fortiori, A is less than C,’ means that the extent to which A is less than C must be greater than that to which A is less than B, or B than C. In the syllogism last written, either of the bracketed premises might be struck out without destroying the conclusion; which last would, however, be weakened. As it stands, then, the part of the conclusion, ‘Some C is not A,’ follows it à fortiori.

The argument à fortiori, may then be defined as a universally affirmative syllogism, in which both of the premises are shewn to be less than the whole truth, or greater. Thus, in ‘Every A is X, Every X is B, therefore Every A is B,’ we do not certainly imply that there are more Xs than As, or more Bs than Xs, so that we do not know that there are more Bs than As. But if we are at liberty to state the syllogism as follows,

All the As make up part (and part only) of the Xs Every X is B;

then we are certain that

All the As make up part (and part only) of the Bs.

But if we are at liberty further to say that

All the As make up part (and part only) of the Xs All the Xs make up part (and part only) of the Bs

then we conclude that

All the As make up part of part (only) of the Bs

and the words in Italics mark that quality of the conclusion from which the argument is called à fortiori.

Most syllogisms which give an affirmative conclusion are generally meant to imply à fortiori arguments, except only in mathematics. It is seldom, except in the exact sciences, that we meet with a proposition, ‘Every A is B,’ which we cannot immediately couple with ‘Some Bs are not As.’

When an argument is completely established, with the exception of one assertion only, so that the inference may be drawn as soon as that one assertion is established, the result is stated in a form which bears the name of an hypothetical syllogism. The word hypothesis means nothing but supposition; and the species of syllogism just mentioned first lays down the assertion that a consequence will be true if a certain condition be fulfilled, and then either asserts the fulfilment of the condition, and thence the consequence, or else denies the consequence, and thence denies the fulfilment of the condition. Thus, if we know that

When A is B, it follows that P is Q;

then, as soon as we can ascertain that A is B, we can conclude that P is Q; or, if we can shew that P is not Q, we know that A is not B. But if we find that A is not B, we can infer nothing; for the preceding does not assert that P is Q only when A is B. And if we find out that P is Q, we can infer nothing. This conditional syllogism may be converted into an ordinary syllogism, as follows. Let K be any ‘case in which A is B,’ and Z a ‘case in which P is Q’; then the preceding assertion amounts to ‘Every K is Z.’ Let L be a particular instance, the A of which may or may not be B. If A be B in the instance under discussion, or if A be not B, we have, in the one case and the other,

Every K is Z Every K is Z L is a K L is not a K ———————— ————————————— Therefore L is a Z No conclusion