| (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.