“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
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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.
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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
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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
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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.