| 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