PKOPOSITION Subject (in logic, 2), which is in certain respects different from what follows. (C.S.P.-J.M.B.) I. The Import of Propositions. It fol¬ lows from the definition of the proposition that it must consist of at least three dif¬ ferent members, two terms (between which the relation is said to hold) and another word whose function is to express at once the nature of the connection between them and the asseveration of that connection. (This double force of the copula is adverted to by Bradley, Prino. of Logic, 22.) In Armies conquer countries, we may think of armies and countries as the objects of consciousness, and of conquer as specifying the nature of the relation and at the same time asserting that it holds. But such a proposition as A con¬ quers B can, if there is any occasion for it, be broken up differently, viz. into A is one-of- tlie-conquerors-of-B. Whether B or one-of- the-conquerors-of-I? be regarded as the second of the related objects of consciousness is merely a matter of convenience, and will be deter¬ mined in any actual case by considering whether other propositions, which it is desired to combine with this as data towards conclu¬ sions, have B or one-of-the-conquerors-of-A among their terms. Or, again, we can always decide which is for the moment the way in which we are regarding the proposition by considering whether in its inverted form it is the statement B is conquered by A or One of those who conquer B is A which interests us. To discover the three elements involved in A runs, we have, again, simply to invert it, One who runs is A. And the fact that there is no proposition which cannot be expressed in an exactly equivalent inverted form proves that this analysis of the proposition into two terms and a copulative connecting link is justified. But there is one particular relation that we have by far the most frequently to deal with in reasoning-—the relation of b invariably fol¬ lowing upon a, or of a as the sufficient antece¬ dent of b. This relation is variously expressed in words—a is-followed-by-5, a implies b, a is-indicative-of b, a is-a-sufficient-condition- of b, If a then b, The objects a are-included- among the objects b, or All a is & (where a and b may themselves be propositions, instead of simple terms, without altering the essential character, for logic, of the relation). In order to hold this relation present in consciousness in its purely abstract form, freed from all those variations of language which, rich in meaning though they may be, are entirely inessential to the purposes of logic, it is abso¬ lutely necessary to represent it by some symbol. Formal logic, as ordinarily treated in the books, is only semi-formal. It has been agreed, since the time of the earliest writers upon the subject, to allow terms to enter into propositions shorn of the special implications which follow upon their different meanings, and to represent them by colourless letters of the alphabet ; it is only carrying this admirable device for abstracting from the inessential a little further if we represent the simple copula of All a is b by some symbol. We shall make use of the form <^, a modi¬ fication of that suggested by Peirce, for this purpose, and we shall write a 6 for any one of the copulative relations which have just been variously put into words. De Morgan regard s this relation as sufficiently characterized by the fact that it is transitive, but that is a statement that needs modification. We shall then have for the formal representation of Not all a is b (corresponding to the plan of indicat¬ ing what is not a by a), the same sign with a horizontal mark indicating negation over it, as a <5 A It will also add greatly to facility of expression if we write 00 and o for the Special Teems (q. v.) of logic, everything and nothing (or what exists and what does not exist). Innovations are difficult to make, and there was long and strenuous opposition to the in¬ troduction of the special quantity o into arith¬ metic and algebra ; but it seems that the time has come when these simple aids towards extracting the essential from the accidental in logic should be used. Cf. Symbolic Logic, ad fin., and Teem (negative). This view, that the import of the proposi¬ tion is to affirm some sort of connection be¬ tween two objects of consciousness, dates from Aristotle. A favourite view of recent years is to maintain that in the simple judgment, A is B, there is both an analysis and a syn¬ thesis—that A as being B is given first, as an integral element of consciousness, and that the work of forming the judgment consists in first separating the concepts and then re¬ uniting them by means of the connecting copula (cf. Judgment). This is doubtless a correct account of the manner of forming # O immediate judgments, but it is not correct as a description of propositions. The examples of the proposition which are usually studied by the logicians are so cut and dried that it is difficult to detect its real essence ; it is neces¬ sary to consider it in the process of being 362