A syllogism is a classical form of deductive reasoning consisting of two premises and a conclusion, typically organized around a major premise, a minor premise, and a final claim. In a categorical syllogism, three terms interact. The major term is the predicate of the conclusion, the minor term is the subject of the conclusion, and the middle term appears in both premises but not in the conclusion, serving as the bridge that connects the other two. For an example, "all mammals are warm-blooded; dogs are mammals; therefore dogs are warm-blooded" uses "warm-blooded" as the major term, "dogs" as the minor term, and "mammals" as the middle term.
Categorical reasoning also operates with statements that assert relationships between two categories, expressed in four standard forms: "All S are P," "No S are P," "Some S are P," and "Some S are not P." The logical relationships among these forms are mapped by the square of opposition, a diagram that organizes how categorical propositions with the same subject and predicate relate through contradictory, contrary, subcontrary, and subalternation relations. Together, syllogisms and the categorical framework provide the historical foundation of deductive logic.