This is shewn by De Morgan (Formal Logic, p. 13). He takes two universal negative premisses E, E. In whatever figure they may be, they can be reduced by conversion to
No P is M,
No S is M.
Then by obversion they become (without losing any of their force),—
All P is not-M,
All S is not-M ;
and we have undistributed middle. Hence rule 3 is exhibited as a corollary from rule 1. For if any connexion between S and P can be inferred from the first pair of premisses, it must also be inferable from the second pair.
The case in which one of the premisses is particular is dealt with by De Morgan as follows;—“Again, No Y is X, Some Ys are not Zs, may be converted into
Every X is (a thing which is not Y),
Some (things which are not Zs) are Ys,
in which there is no middle term.”
This is not satisfactory, since we may often exhibit a valid syllogism in such a form that there appear to be four terms; e.g., All M is P, All S is M, may be reduced to All M is P, No S is not-M, and there is now no middle term.
The case in question may, however, be disposed of by saying that if we cannot infer anything from two negative premisses both of which are universal, à fortiori we cannot from two negative premisses one of which is particular.[313]