_Of the Reduction of the Disjunctive Syllogism._
§ 766. We have seen that in the disjunctive syllogism the two
constructive moods alone are formally valid. The first of these,
namely, the denial of the antecedent, will in all cases give a simple
syllogism in the first figure; the second of them, namely, the denial
of the consequent, will in all cases give a simple syllogism in the
second figure.
_Denial of Antecedent_ = Barbara.
Either A is B or C is D.
A is not B.
.'.C is D
is equal to
If A is not B, C is D.
A is not B.
.'. C is D.
is equal to
All cases of A not being B are cases of C being D.
This is a case of A not being B.
.'. This is a case of C being D.
_Denial of Consequent_ = Camestres.
Either A is E or C is D.
C is not D.
.'. A is B.
is equal to
If A is not B, C is D.
C is not D.
.'. A is B.
is equal to
All cases of A not being B are cases of C being D.
This is not a case of C being D.
.'. This is not a case of A being B.
§ 767. The other moods of the first and second figures can be obtained
by varying the quality of the antecedent and consequent in the major
premiss and reducing the quantity of the minor.
§ 768. The invalid destructive moods correspond with the two invalid
types of the partly conjunctive syllogism, and have the same fallacies
of simple syllogism underlying them. Affirmation of the antecedent of
a disjunctive is equivalent to the semi-conjunctive fallacy of denying
the antecedent, and therefore involves the ordinary syllogistic
fallacy of illicit process of the major.
Affirmation of the consequent of a disjunctive is equivalent to the
same fallacy in the semi-conjunctive form, and therefore involves the
ordinary syllogistic fallacy of undistributed middle.
_Affirmation of Antecedent_ = _Illicit Major_.
Either A is B or C is D.
A is B.
.'. C is not D.
is equal to
If A is not B, C is D.
A is B.
.'. C is not D.
is equal to
All cases of A not being B are cases of C being D.
This is not a case of A not being B.
.'. This is not a case of C not being D.
_Affirmation of Consequent_ = _Undistributed Middle_.
Either A is B or C is D.
C is D.
is equal to
If A is not B, C is D.
C is D.
is equal to
All cases of A not being B are cases of C being D.
This is a case of C being D.
§ 769. So far as regards the consequent, the two species of complex
reasoning hitherto discussed are identical both in appearance and
reality. The apparent difference of procedure in the case of the
antecedent, namely, that it is affirmed in the partly conjunctive, but
denied in the disjunctive syllogism, is due merely to the fact that in
the disjunctive proposition the truth of the consequent is involved in
the falsity of the antecedent, so that the antecedent being
necessarily negative, to deny it in appearance is in reality to assert
it.