LOGICAL ADDITION AND THE UTILITY OF SYMBOLISM
Frequently ordinary language contains subtle psychological implications which cannot be translated into symbolic logic except at great length. Thus if a man (say Mr. Jones) wishes to speak collectively of himself and his wife, the order of mentioning the terms in the class considered and the names applied to these terms are, logically speaking, irrelevant. And yet more or less definite information is given about Mr. Jones according as he talks to his friends of:
| (1) Mrs. Jones and I, | |
| (2) I (or me) and my wife (or missus), | |
| (3) My wife and I, | |
| or | (4) I (or me) and Mrs. Jones. |
In case (1) one is probably correct in placing Mr. Jones among the clergy or the small professional men who make up the bulk of the middle-class; in case (2) one would conclude that Mr. Jones belonged to the lower middle-class; the form (3) would be used by Mr. Jones if he were a member of the upper, upper middle, or lower class; while form (4) is only used by retired shopkeepers of the lower middle-class, of which a male member usually combines belief in the supremacy of man with belief in the dignity of his wife as well as himself. A further complication is introduced if a wife is referred to as “the wife.”[35] Cases (2) and (3) then each give rise to one more case. Cases (1) and (4) do not, since nobody has hitherto referred to his wife as “the Mrs. Jones”—at least without a qualifying adjective before the “Mrs.”
On the other hand, certain descriptive phrases and certain propositions can be expressed more shortly and more accurately by means of symbolic logic. Let us consider the proposition “No man marries his deceased wife’s sister.” If we assume, as a first approximation, that all marriages are fertile and that all children are legitimate, then, with only four primitive ideas: the relation of parent to child (P) and the three classes of males, females, and dead people, we can define “wife” (a female who has the relation formed by taking the relative product of P and P̌[36] to a male), “sister,” “deceased wife,” and “deceased wife’s sister” in terms of these ideas and of the fundamental notions of logic. Then the proposition “No man marries his deceased wife’s sister” can be expressed unambiguously by about twenty-nine simple signs on paper, whereas, in words, the unasserted statement consists of no less than thirty-four letters. Although, legally speaking, we should have to adopt somewhat different definitions and possibly increase the complications of our proposition, it must be remembered that, on the other hand, we always reduce the number of symbols in any proposition by increasing the number of definitions in the preliminaries to it.
But the utility of symbolic logic should not be estimated by the brevity with which propositions may sometimes be expressed by its means. Logical simplicity, in fact, can very often only be obtained by apparently complicated statements. For example, the logical interpretation of “The father of Charles II was executed” is, “It is not always false of x that x begat Charles II, and that x was executed and that ‘if y begat Charles II, y is identical with x’ is always true of y.”[37] From the point of view of logic, we may say that the apparently simple is most often very complicated, and, even if it is not so, symbolism will make it seem so,[38] and thus draw attention to what might otherwise easily be overlooked.
[35] Cf. Chapter XXIV below.
[36] C. S. Peirce’s notation for the relation “converse of P.”
[37] Russell, Md., N. S., vol. xiv., October 1905, p. 482.
[38] Russell, International Monthly, vol. iv., 1901, pp. 85-6; cf. M., vol. xxii., 1912, p. 153. [This essay is reprinted in Mysticism and Logic, London and New York, 1918, pp. 74-96.—Ed.]