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 ϕ(x, t),
where “ϕ(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). ϕ(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 æsthetic 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 ϕ, 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.
[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.