A syllogism is an argument with two premises and a conclusion, in which the conclusion follows from the premises by their form alone. Aristotle built the first formal system around it in the Prior Analytics.

The structure

The standard example: all men are mortal; Socrates is a man; therefore Socrates is mortal. Three terms appear. The middle term, "man", occurs in both premises and not in the conclusion — it is the link that does the work. The other two terms appear once each in the premises and both in the conclusion.

Aristotle classified syllogisms by figure, determined by where the middle term sits, and by mood, determined by whether each statement is universal or particular and affirmative or negative. He then identified which combinations are valid. Medieval logicians gave the valid moods mnemonic names — Barbara, Celarent, Darii — which encoded the rules for students.

What it established

The achievement was the idea of logical form. Validity was shown to depend on arrangement rather than content, so a schema can be tested once and applied to any subject. That insight founded logic as a discipline and stood without serious rival until the nineteenth century.

The limits

Syllogistic handles only statements in subject-predicate form, so it cannot represent relations. "Every horse is an animal, therefore every horse's head is an animal's head" is valid and cannot be expressed in it — an example Augustus De Morgan used to make the point. Nor can it handle multiple quantifiers.

Frege's predicate logic of 1879 absorbed syllogistic as a small fragment of a far more powerful system. The syllogism survives as the clearest introduction to validity and as the historical origin of formal reasoning.