syllogisms

A syllogism is a form of deductive reasoning consisting of two premises and a conclusion, in which three terms appear: a major term, a minor term, and a middle term that links the premises. It is a central part of the classical syllogistic developed by Aristotle and is used to derive conclusions that necessarily follow from the premises. The term syllogism derives from Greek syllogismos, meaning “considering together.”

In a syllogism, the major premise asserts something about the major term, the minor premise about the minor term, and the conclusion relates the minor term to the major term. The middle term appears in both premises but not in the conclusion. Example: All humans are mortal (major premise); Socrates is a man (minor premise); Therefore Socrates is mortal (conclusion). This illustrates how a conclusion follows from the premises.

Categorical syllogisms, the best-known type, use propositions of the form All S are P, No S are P, Some S are P, and Some S are not P. They are organized into four figures according to the position of the middle term in the premises. The propositions are labeled A (universal affirmative), E (universal negative), I (particular affirmative), and O (particular negative). Well-known valid moods in Figure 1 include Barbara (AAA-1): All M are P; All S are M; Therefore All S are P; Celarent (EAE-1): No M are P; All S are M; Therefore No S are P; Darii (IAI-1); Ferio (EIO-1).

Validity and modern perspective: a syllogism is valid only if the conclusion necessarily follows from the premises. Core rules include using exactly three terms, ensuring the middle term is distributed in at least one premise, and distributing a term in the conclusion only if it is distributed in its premise. In modern logic, syllogisms are viewed as a special case of predicate logic and can be analyzed with quantifiers and formal inference, often aided by Venn diagrams.