_Of the Reduction of the Partly Conjunctive Syllogism._
§ 744. Such syllogisms as those just treated of, if syllogisms they
are to be called, have a major and a middle term visible to the eye,
but appear to be destitute of a minor. The missing minor term is
however supposed to be latent in the transition from the conjunctive
to the simple form of proposition. When we say 'A is B,' we are taken
to mean, 'As a matter of fact, A is B' or 'The actual state of the
case is that A is B.' The insertion therefore of some such expression
as 'The case in hand,' or 'This case,' is, on this view, all that is
wanted to complete the form of the syllogism. When reduced in this
manner to the simple type of argument, it will be found that the
constructive conjunctive conforms to the first figure and the
destructive conjunctive to the second.
_Constructive Mood_. _Barbara_.
If A is B, C is D. \ / All cases of A being B are cases of
\ = / C being D.
A is B. / \ This is a case of A being B.
.'. C is D. / \ .'. This is a case of C being D.
_Destructive Mood._ Camestres.
If A is B, C is D. \ / All cases of A being B are cases of
\ = / C being D.
C is not D. / \ This is not a case of C being D.
.'. A is not B. / \ .'. This is not a case of A being B.
§ 745. It is apparent from the position of the middle term that the
constructive conjunctive must fall into the first figure and the
destructive conjunctive into the second. There is no reason, however,
why they should be confined to the two moods, Barbara and
Carnestres. If the inference is universal, whether as general or
singular, the mood is Barbara or Carnestres; if it is particular, the
mood is Darii or Baroko.
Barbara. Camestres.
If A is B, C is always D. \ If A is B, C is always D. \
A is always B. \ C is never D. \
.'. C is always D. \ .'. A is never B. \
| |
If A is B, C is always D. / If A is B, C is always D. /
A is in this case B. / C is not in this case D. /
.'. C is in this case D. / .'. A is not in this case B. /
Darii. Baroko.
If A is B, C is always D. If A is B, C is never D.
A is sometimes B. C is sometimes not D.
.'. C is sometimes D. .'. A is sometimes not B.
§ 746. The remaining moods of the first and second figure are obtained
by taking a negative proposition as the consequent in the major
premiss.
Celarent. Ferio.
If A is B, C is never D. If A is B, C is never D.
A is always B. A is sometimes B.
.'. C is never D. .'. C is sometimes not D.
_Cesare_. Festino.
If A is B, C is never D. If A is B, C is never D.
C is always D. C is sometimes D.
.'. A is never B. .'. A is sometimes not B.
§ 747. As the partly conjunctive syllogism is thus reducible to the
simple form, it follows that violations of its laws must correspond
with violations of the laws of simple syllogism. By our throwing the
illicit moods into the simple form it will become apparent what
fallacies are involved in them.
_Denial of Anteceded_.
If A is B, C is D. \ / All cases of A being B are cases of C
\ = / being D.
A is not B. / \ This is not a case of A being B.
.'. C is not D. / \ .'. This is not a case of C being D.
Here we see that the denial of the antecedent amounts to illicit
process of the major term.
§ 7481 _Affirmation of Consequent_.
If A is B, C is D. \ / All Cases of A being B are cases of C
| = | being D.
C is D. / \ This is a case of C being D.
Here we see that the affirmation of the consequent amounts to
undistributed middle.
§ 749. If we confine ourselves to the special rules of the four
figures, we see that denial of the antecedent involves a negative
minor in the first figure, and affirmation of the consequent two
affirmative premisses in the second. Or, if the consequent in the
major premiss were itself negative, the affirmation of it would amount
to the fallacy of two negative premisses. Thus--
If A is B, C is not D. \ / No cases of A being B are cases of C
| = | being D.
C is not D. / \ This is not a case of C being D.
§ 750. The positive side of the canon of the conjunctive
syllogism--'To affirm the antecedent is to affirm the consequent,'
corresponds with the Dictum de Omni. For whereas something (viz. C
being D) is affirmed in the major of all conceivable cases of A being
B, the same is affirmed in the conclusion of something which is
included therein, namely, 'this case,' or 'some cases,' or even 'all
actual cases.'
§ 751. The negative side--'to deny the consequent is to deny the
antecedent'--corresponds with the Dictum de Diverse (§ 643). For
whereas in the major all conceivable cases of A being B are included
in C being D, in the minor 'this case,' or 'some cases,' or even 'all
actual cases' of C being D, are excluded from the same notion.
§ 752. The special characteristic of the partly conjunctive syllogism
lies in the transition from hypothesis to fact. We might lay down as
the appropriate axiom of this form of argument, that 'What is true in
the abstract is true--in the concrete,' or 'What is true in theory is
also true in fact,' a proposition which is apt to be neglected or
denied. But this does not vitally distinguish it from the ordinary
syllogism. For though in the latter we think rather of the transition
from a general truth to a particular application of it, yet at bottom
a general truth is nothing but a hypothesis resting upon a slender
basis of observed fact. The proposition 'A is B' may be expressed in
the form 'If A is, B is.' To say that 'All men are mortal' may be
interpreted to mean that 'If we find in any subject the attributes of
humanity, the attributes of mortality are sure to accompany them.'