Valid or Invalid? - A Brief Introduction to Categorical Syllogisms

To get to grips with the rules for the validity of syllogisms, it is first necessary to undertake a quick whistle stop tour of the various aspects of a categorical syllogism.

A categorical syllogism is a deductive argument comprising three categorical propositions: a major premise, a minor premise and the conclusion. Categorical propositions have four standard forms:

A = All S are P
E = No S are P
I = Some S are P
O = Some S are not P

The mood of a syllogism is defined by which of the forms appear and where. So, for example, All M are P, Some S are M, Therefore, All S are P has the mood: AIA.

A categorical syllogism contains only three categorical terms: a major term, minor term and middle term.

• The major term appears as the predicate in the conclusion, and only once in the major premise (i.e., the first premise).
• The minor term appears as the subject in the conclusion, and only once in the minor premise (i.e,. the second premise).
• The middle term appears once in the major premise, once in the minor premise, and once in the conclusion.

Distribution

A term is said to be distributed when all members of the class denoted by the term are affected by a proposition. This isn't quite as complicated as it sounds. For example, consider the proposition All S are P: in this case, S is distributed because the proposition says something about all members of S - namely, that they are P. But P is not distributed, because the proposition doesn't tell us anything about all members of P. (If it isn't clear why it doesn't, consider that "All cows are mammals" tells us something about all cows, but nothing about all mammals).

Of course, you can always just learn which terms are distributed:

Proposition	Terms Distributed
All S are P	S
No S are P	S and P
Some S are P	None
Some S are not P	P

Okay, you should now have enough information to understand the six rules for the validity of syllogisms.