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
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Therefore L is a Z No conclusion