A syllogism (properly, a categorical syllogism) is the inference of one proposition from two premises. An example is: all horses have tails; all things with tails are four-legged; so all horses are four-legged. Each premise has one term in common with the conclusion, and one term in common with the other premise. The term that does not occur in the conclusion is called the middle term. The major premise of the syllogism is the premise containing the predicate of the conclusion (the major term), and the minor premise contains its subject (the minor term). So the first premise of the example is the minor premise, the second the major premise, and ‘having a tail’ is the middle term. The four kinds of proposition distinguished in syllogistic reasoning are universal affirmatives (all men are mortal), called A propositions, particular affirmatives (some men are sick), called I propositions, universal negatives (no men are trustworthy), called E propositions, and particular negatives (some men are not lawyers), called O propositions. This enables syllogisms to be classified according to the form of the premises and the conclusions (see also square of opposition). The other classification is by figure, or way in which the middle term is placed in the premises. The conclusion is always of subject-predicate (S-P) form, and the middle term is M. The four figures are illustrated in the diagram:

The example given was a syllogism of the first figure. Mnemonics, in the form of names with the vowels indicating the A, E, I, O, forms, help students to remember the valid forms, called moods of the syllogism. Valid syllogisms of the first figure are Barbara (AAA), Celarent (EAE), Darii (AII), and Ferio (EIO); of the second, Cesare, Camestres, Festino, and Baroco; of the third, Darapti, Disamis, Datisi, Felapton, Bocardo, and Ferison; and of the fourth, Bramantip, Camenes, Dimaris, Fesapo, and Fresison.

Rules exist for converting one form to another, and every valid syllogism may be converted to a syllogism of the first figure. Another set of rules concerns the distribution of terms. Roughly, a term is distributed if it covers all of its class. A test is whether the proposition containing the term must remain true if the term is qualified, or replaced with a term applying to only a subset of the class of things to which the original applies. Thus ‘all men are mortal’ distributes the term ‘man’ because it implies that all blind men are mortal; ‘not every animal is useful’ does not distribute ‘animal’ because it does not imply that not every farmyard animal is useful. In a universal affirmative (A form) the subject is distributed but the predicate is not; in a universal negative (E form) both are distributed; in a particular affirmative (I form) neither are distributed, and in a particular negative (O form) the subject is not distributed but the predicate is. A syllogism cannot be valid unless the middle term is distributed at least once, and any term distributed in the conclusion must be distributed in its premise.

Although the theory of the syllogism dominated logic until the 19th century, it remained a piecemeal affair, able to deal with only a relatively small number of valid forms of argument. There have subsequently been rearguard actions attempting to extend the power of syllogistic reasoning, but in general it has been eclipsed by the modern theory of quantification (see also predicate calculus), which gives greater expressive power for less complexity.