Some M is in P.
All M is in S.
Some S is in P.
Let Some S is in P be denied, and No S is in P must be admitted. But if No S is in P and All M is in S, it follows (in Celarent) that No M is in P, which an opponent cannot hold consistently with his admission that Some M is in P.
The beginner sometimes asks: What is the use of reducing the Minor Figures to the First? The reason is that it is only when the relations between the terms are stated in the First Figure that it is at once apparent whether or not the argument is valid under the Axiom or Dictum de Omni. It is then undeniably evident that if the Dictum holds the argument holds. And if the Moods of the First Figure hold, their equivalents in the other Figures must hold too.
Aristotle recognised only two of the Minor Figures, the Second and Third, and thus had in all only fourteen valid moods.
The recognition of the Fourth Figure is attributed by Averroes to Galen. Averroes himself rejects it on the ground that no arguments expressed naturally, that is, in accordance with common usage, fall into that form. This is a sufficient reason for not spending time upon it, if Logic is conceived as a science that has a bearing upon the actual practice of discussion or discursive thought. And this was probably the reason why Aristotle passed it over.
If however the Syllogism of Terms is to be completed as an abstract doctrine, the Fourth Figure must be noticed as one of the forms of premisses that contain the required relation between the extremes. There is a valid syllogism between the extremes when the relations of the three terms are as stated in certain premisses of the Fourth Figure.
III.—The Sorites.
A chain of Syllogisms is called a Sorites. Thus:—