DENIAL OF GENERALITY AND GENERALITY OF DENIAL
The conclusion of a certain song[48] about a young man who poisoned his
sweetheart with sheep's-head broth, and was frightened to death by a
voice exclaiming:
"Where's that young maid
What you did poison with my head?"
at his bedside, gives rise to difficulties which are readily solved by a
symbolism that brings into relief the principle that the denial of a
universal and non-existential proposition is a particular and
existential one. The conclusion of the song is:
Now all young men, both high and low,
Take warning by this dismal go!
For if he'd never done nobody no wrong,
He might have been here to have heard this song.
It is an obvious error, say Whitehead and Russell,[49] though one easy
to commit, to assume that the cases: (1) all the propositions of a
certain class are true; and (2) no proposition of the class is true; are
each other's contradictories. However, in the modification[50] of
Frege's symbolism which was used by Russell
(1) is (_x_). _x_,
and (2) is (_x_). not _x_;
while the contradictory of (1) is:
not (_x_). _x_.
The last line but one of the above verse may, then, be written:
(_t_). not (_x_). not not [Greek: ph] (_x_, _t_),
where "[Greek: ph] (_x_, _t_)" denotes the unasserted propositional
function "the doing wrong to the person _x_ at the instant _t_." By
means of the principle of double negation we can at once simplify the
above expression into:
(_t_). not (_x_). [Greek: ph] (_x_, _t_);
which can be thus read: "If at every instant of his life there was at
least one person _x_ to whom he did no wrong (at that instant)." It is
difficult to imagine any one so sunk in iniquity that he would not
satisfy this hypothesis. We are forced, then, unless our imagination for
evil is to be distrusted, to conclude that any one might have been there
to have heard that song. Now this conclusion is plainly false, possibly
on physical grounds, and certainly on aesthetic grounds. It may be added,
by the way, that it is quite possible that De Morgan was mistaken in his
interpretation of the above proposition owing to the fact that he was
unacquainted with Frege's work. In fact, if he had not noticed the fact
that _any_ two of the "not's" cannot be cancelled against one another he
would have concluded that the interpretation was: "If he had never done
any wrong to anybody."
According as the symbol for "not" comes before the (_x_) or between the
(_x_) and the [Greek: ph], we have an expression of what Frege called
respectively the denial of generality, and the generality of denial. The
denial of the generality of a denial is the form of all existential
propositions, while the assertion of or denial of generality is the
general form of all non-existential or universal propositions.
FOOTNOTES:
[48] To which De Morgan drew attention in a letter; see (Mrs.) S. E. De
Morgan, _Memoir of Augustus De Morgan_, London, 1882, p. 324.
[49] _Pa. Ma._, p. 16.
[50] However, here, for the printer's convenience, we depart from Mr.
Russell's usage so far as to write "not" for a curly minus sign.