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 [vP][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.
FOOTNOTES:
[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.]