Argument Forms

A familiarity with categorical propositions is essential for the logic. However, what the logic is most concerned with is how these kinds of propositions "hang together"—especially how two or more of them (called premises) might or might not hang together to establish—i.e., to prove, to entail—another one (called the conclusion) by virtue of their forms. The "premises together with their conclusion" is called an argument. In writing an argument it is customary to state the premises first and then to introduce the conclusion with the therefore symbol (∴). If the premises do entail the conclusion by virtue of their forms then the argument form as a whole is valid, while it is invalid if they fail to do so.

The basic kind of categorical argument is the syllogism; the standard form statement of a syllogism consists of two premise sentences and a conclusion sentence in which exactly three terms occur in two of the three sentences each. (Nonstandard form syllogisms may use one term, A, in one of its sentences but employ its complement, nonA, in another which would results in more than three terms.) Perhaps the best known syllogism is the one given earlier, viz.,

but there are 256 syllogistic forms in all. The term that is used in both premise statements—B in this example—is called the middle term, while the terms that occur only in one premise each but also in the conclusion—A and C here— will be called extreme terms. (In traditional language, the subject of the conclusion is called the minor term and the premise in which it also occurs is called the minor premise, while the predicate of the conclusion is called the major term and its premise is called the major premise. But we will not need this nomenclature.)

A categorical argument with more than two premises is a sorites and there can be any number of these for they can have any number of premises. In the simple one given earlier, viz.,

both B and C are middle terms and A and D are the extreme terms.

Although both of these argument forms are valid, most categorical argument forms are not.