People who are cynical as to the morality of the English are often unpleasantly surprised to learn that "All trespassers will be prosecuted" does not necessarily imply that "some trespassers will be prosecuted." The view that universal propositions are non-existential is now generally held: Bradley and Venn seem to have been the first to hold this, while older logicians, such as De Morgan,[45] considered universal propositions to be existential, like particular ones. If the Gnat[46] had been content to affirm his proposition about the means of subsistence of Bread-and-Butter flies, in consequence of their lack of which such flies always die, without pointing out such an insect and thereby proving that the class of them is not null, Alice's doubt as to the existence of the class in question, even if it were proved to be well founded, would not have affected the validity of the proposition. This brings us to a great convenience in treating universal propositions as non-existential: we can maintain that all _x_'s are _y_'s at the same time as that no _x_'s are _y_'s, if only _x_ is the null-class. Thus, when Mr. MacColl[47] objected to other symbolic logicians that their premisses imply that all Centaurs are flower-pots, they could reply that their premisses also imply the more usual view that Centaurs are not flower-pots. FOOTNOTES: [45] Cf., e.g., _F. L._, p. 4. [46] See Appendix L. [47] Cf., e.g., _Md._, N. S., vol. xiv., July, 1905, pp. 399-400. CHAPTER XVIII 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: