Chapter XXVII: Of the Reduction of the Dilemma

§ 796. As the dilemma is only a peculiar variety of the partly conjunctive syllogism, we should naturally expect to find it reducible in the same way to the form of a simple syllogism. And such is in fact the case. The constructive dilemma conforms to the first figure and the destructive to the second.

1) _Simple Constructive Dilemma_.

Barbara.
If A is B or if E is F, C is D. All cases of either A being B or E
being F are cases of C being D.
Either A is B or E is F. All actual cases are cases of either
A being B OP E being F.
.'. C is D. .'. All actual cases are cases of C
being D.

(2) _Simple Destructive_.

Camstres.
If A is B, C is D and E is F. All cases of A being B are cases of
C being D and E being F.
Either C is not D or E is not F. No actual cases are cases of C being
D and E being F.
.'. A is not B. .'. No actual cases are cases of A
being B.

(3) _Complex Constructive_.
Barbara.
If A is B, C is D; and if E is F, All cases of either A being B or
G is H. being F are cases of either C being
D or G being H.
Either A is B or E is F. All actual cases are cases of either A
being B or E being F.
.'. Either C is D or G is H. .'. All actual cases are cases of either C
being D or G being H.

(4) _Complex Destructive_.

If A is B, C is D; and if E is F, All cases of A being B and E being F
G is H. are cases of C being D and G
being H.
Either C is not D Or G is No actual cases are cases of C being
not H D and G being H.
Either A is not B or E is No actual cases are cases of A being
not F. B and E being F.

§ 797. There is nothing to prevent our having Darii, instead of Barbara, in the constructive form, and Baroko, instead of Camestres, in the destructive. As in the case of the partly conjunctive syllogism the remaining moods of the first and second figure are obtained by taking a negative proposition as the consequent of the major premiss, e.g.--

_Simple Constructive_. Celarent or Ferio.
If A is B or if E is F, C is not D No cases of either A being B or E
being F are cases of C being D.
Either A is B or E is F. All (or some) actual cases are cases of
either A being B or E being F
.'. C is not D. .'. All (or some) actual cases are not
cases of C being D.