IMMEDIATE INFERENCE We have now to consider the main types of propositions, their meanings, and their implications. It has already been stated that a proposition is the verbal expression of judgment, and that a judgment is what one thinks, believes, or knows. For purposes of formal logic it is convenient sometimes to use the term propo sition not only for the expressed content of what is actually be lieved, but also for any suggestion that may be believed or dis believed, or merely understood and considered. Now there are three principal types of proposition which must be considered, namely, categorical, hypothetical and disjunctive (or alternative) propositions. A categorical proposition is one in which something is asserted unconditionally. It has the form S is (or is not) P. An hypothetical proposition is one which expresses an antecedent (or condition) and a consequent. Its general form is If A, then C. A disjunctive proposition is one which expresses alternatives. Its general form is either A or B. Categorical propositions usually contain two terms, namely, a subject (S) and a predicate (P), though there are simple forms (the so-called impersonals) which consist of one term only (the predicate) Hypothetical and disjunctive propositions contain three or four terms, or even more. But it is better to think of their component propositions rather than of their component terms. In the above forms, A,B,C, stand for such propositions, not for terms ; e.g., "If a triangle is equilateral (A), then it is equiangular (C)"; "either a line is straight (A), or it is curved (B)." By substituting symbols for actual terms we obtain the form, that is the type, of the propo sition in question. That, in fact, is the simplest way of getting at the various kinds of propositions ; but, of course, it must be borne in mind that in actual reasoning real terms are always employed, for a propositional form is not an actual proposition, just as, in algebra, x, y, z are not actual things or even numbers, but are only devices for dealing with numbers in the most general way possible.
There are important differences between the categorical, hypo thetical and disjunctive types of proposition. The categorical type is that employed when we want to assert a mere brute fact without wishing to suggest that there is any inner connection between the terms of the proposition. The hypothetical type, on the other hand, is just specially designed to express the thought that there is a connection between the antecedent and the con sequent. And the disjunctive type is the most suitable one for expressing alternative possibilities, one or other of which is true, though it is not known which one is true. At the same time, these differences of function need not be exaggerated, and it is usually possible to express what is essentially the same thought in any one of the three types, as the following may serve to illustrate.
Equilateral triangles are equiangular. If a triangle is equilateral, it is equiangular. Either a triangle is equiangular or it is not equi lateral. So, generally, SM is P=if S is Al it is P=either S is P or it is not M. In view of what has just been said and illustrated, it may be remarked that all the essentials of formal inference may be learned by a close study of categorical propositions alone in the first instance. There is no difficulty in applying to hypo thetical and disjunctive arguments what has been found to hold good of categorical arguments. And these are easier to get on with at first. We shall, accordingly, confine ourselves for the present to categorical propositions and inferences.
The Quality and Quantity of Categorical Propositions.— Propositions are either affirmative or negative. These are known as differences of quality. And they are either universal or par titular, that is to say, they assert something either of the whole of the subject term explicitly, or of some indefinite part of it. These are known as differences of quantity. If the two kinds of distinctions are combined in one scheme, we obtain four types of categorical proposition, the forms of which are as follows:— Every S is P (universal affirmative) ; no S is P (universal nega tive) ; some S's are P (particular affirmative) ; some S's are not P (particular negative). The following are examples of the sev eral types: "Every equilateral triangle is equiangular"; "no equi lateral triangle is right-angled"; "some stars are self-luminous"; "some planets are not self-luminous." In this classification of propositions the class universal propositions includes singular propositions (that is, those whose subject is a singular term) like a proper noun, for example, as well as general propositions (that is, those which have a general term, like a common noun, e.g., and assert something about the whole class or kind of object or event which it is the name of). The above four types of cate gorical proposition are commonly called A,E, 1,0, respectively. A and I are the vowels of "affirm," the first being used for the universal, the second for the particular, affirmative ; E and 0, the vowels of "nego" (Latin for I deny), are similarly used for the universal and the particular negative. By using the correspond ing small letters to express the quality and quantity of proposi tions, the above formulae of the four types of categoricals can be expressed respectively as follows: SaP; SeP; SiP; SoP. This is a convenient symbolic method which can be used for all formal inference with great advantage.