Syllogisms

Syllogisms.  Besides his categories for representing ontology, Aristotle developed formal logic as a precise method for reasoning with them and about them. His major contribution was the invention of syllogisms as formal patterns for representing rules of inference. The following table lists the names of the four types of propositions used in syllogisms and the corresponding sentence patterns that express them.

With letters such as A and B in the sentence patterns, Aristotle introduced the first known use of variables in history. Each letter represents some category, which the Scholastics called praedicatum in Latin and which became predicate in English. If necessary, the verb form is may be replaced by are, has, or have in order to make grammatical English sentences. Although the patterns may look like English, they are limited to a highly stylized or constrained syntax, which is sometimes called controlled natural language. Such language can be read as if it were natural language, but the people who write it must have some training before they can write it correctly. The advantage of controlled language is that it can be automatically analyzed by computer and be translated to logic.

To make the rules easier to remember, the medieval Scholastics developed a system of mnemonics for naming and classifying them. They started by assigning the vowels A, I, E, and O to the four basic types of propositions. The letters A and I come from the first two vowels of the Latin word affirmo (I affirm), and the letters E and O come from the word nego (I deny). These letters are the vowels used in the names of the valid types of syllogisms. The following table shows examples of the four types of syllogisms named Barbara, Celarent, Darii, and Ferio. The three vowels in each name specify the types of propositions that are used as the two premises and the conclusion.

Barbara, Celarent, Darii, and Ferio are the four types of syllogisms that make up Aristotle's first figure. Another fifteen types are derived from them by rules of conversion, which change the order of the terms or the types of statements. Barbara and Darii are the basis for the modern rule of inheritance in type hierarchies. Celarent and Ferio are used to detect and reason about constraints and constraint violations in a type hierarchy. Those four rules are also the foundation for a subset of first-order logic called description logic, two versions of which are DAML and OIL.