Handbook of the History of Logic: - Fordham University Faculty

330 Henrik Lagerlund
XIV.
Every B is A
Every B is C
Some C is A
XV.
No B is A
Every B is C
Some C is not A
XVI.
Some B is A
Every B is C
Some C is A
XVII.
Every B is A
Some B is C
Some C is A
XVIII.
Some B is not A
Every B is C
Some C is not A
XIX.
No B is A
Some B is C
Some C is not A
(XIV) is reduced to (III) by accidental conversion of the minor premise, and (XV)
is reduced to (IV) in the same way. (XVI) is reduced to (III) by simple conversion
of the major premise and transposing the premises. (XVII) is reduced to (III) by
simple conversion of the minor premise, and (XIX) is reduced to (IV) in the same
way. (XVIII) is proved by impossibility again. The negation of the conclusion is:
‘Every C is A’, and together with the minor premise through (I), ‘Every B is A’
is derived, which contradicts the major premise.
There are two more rules that govern the theory of the syllogisms. They are:
13. In no syllogism with a particular negative indirect conclusion does also a
direct conclusion follow, and vice versa.
14. From a universal conclusion there also follows a particular.
From the second rule five additional moods can be inferred bringing the total
number of valid syllogisms up to 24.