Kyana asked:

Is the syllogism EAO-1 valid and if not what rule does it break?

Answer by Helier Robinson

It is valid. The major premise is MeP, in which both terms are distributed; the minor premise is SaM, in which S is distributed and M is undistributed; and the conclusion is SoP, in which S is undistributed and P is distributed. So no rules are broken: M is distributed at least once, neither illicit major nor illicit minor occur, and the number of negative conclusions, 1, is equal to the number of negative premises.

An example of EAO-1 is:

No mammals are fish
All sheep are mammals
Therefore some sheep are not fish

In modern logic this would be:

(x)(Mx -> ~Fx)
(x)(Sx -> Mx)
Therefore (Ex)(Sx & ~Fx)

and this is invalid.

The reason for the discrepancy between traditional and modern logic in this case is what is called ‘existentional presupposition’: in traditional logic it is assumed that universal propositions (‘All S are P’ and ‘No S are P’) the subject and predicate classes (S and P) have members; in modern logic such existence of members is not assumed but has to be stated explicitly with existential quantifiers. The two logics agree if the existential presupposition is stated explicitly in traditional logic.